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==Abstract==
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The objective of this work is the derivation and implementation of a unified Finite
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Element formulation for the solution of fluid and solid mechanics, Fluid-Structure Interaction
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(FSI) and coupled thermal problems.
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The unified procedure is based on a stabilized velocity-pressure Lagrangian formulation. Each time
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step increment is solved using a two-step Gauss-Seidel scheme: first the linear momentum equations
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are solved for the velocity increments, next the continuity equation
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is solved for the pressure in the updated configuration.
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The Particle Finite Element Method (PFEM) is used for the fluid domains, while the
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Finite Element Method (FEM) is employed for the solid ones. As a consequence, the domain is
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remeshed only in the parts occupied by the fluid.
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Linear shape functions are used for both the velocity and the pressure fields. In order to deal
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with the incompressibility of the materials, the formulation has been stabilized using an updated
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version of the Finite Calculus (FIC) method. The procedure has been derived for
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quasi-incompressible Newtonian fluids. In this work, the FIC stabilization procedure has been
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extended also to the analysis of quasi-incompressible hypoelastic
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solids.
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Specific attention has been given to the study of free surface flow problems. In particular,
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the mass preservation feature of the PFEM-FIC stabilized procedure has been deeply studied with the
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help of several numerical examples. Furthermore, the conditioning of the problem has been analyzed
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in detail describing the effect of the bulk modulus on the numerical scheme. A strategy based on
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the use of a pseudo bulk modulus for improving the conditioning of the linear system is also
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presented.
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The unified formulation has been validated by comparing its numerical results to experimental
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tests and other numerical solutions for fluid and solid mechanics, and FSI problems. The
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convergence of the scheme has been also analyzed for most of the prob-
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lems presented.
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The unified formulation has been coupled with the heat tranfer problem using a staggered
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scheme. A simple algorithm for simulating phase change problems is also described. The numerical
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solution of several FSI problems involving the temperature is given.
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The thermal coupled scheme has been used successfully for the solution of an industrial problem.
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The objective of study was to analyze the damage of a nuclear power plant pressure vessel induced
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by a high viscous fluid at high temperature, the corium.  The
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numerical study of this industrial problem has been included in this work.
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==Resumen==
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El objectivo de la presente tesis es la derivación e implementación de una formulación unificada con elementos finitos para la solución de problemas de mecánica de fluidos y de sólidos, interacción fluido-estructura (''Fluid-Structure Interaction (FSI)'') y con acoplamiento térmico.  El método unificado está basado en una formulación Lagrangiana estabilizada y las variables incçognitas son las velocidades y la presión. Cada paso de tiempo se soluciona a través de un esquema de dos pasos de tipo Gauss-Seidel. Primero se resuelven  las ecuaciones de momento lineal por los incrementos de velocidad, luego  se calculan las presiones  en la configuración actualizada usando la ecuación de continuidad.  Para los dominios fluidos se utiliza el método de elementos finitos de partículas (''Particle Finite Element Method (PFEM)'') mientras que los sólidos  se solucionan con el método de elementos finitos  (''Finite Element Method (FEM)''). Por lo tanto, se ramalla sólo las partes del dominio ocupadas por el fluido.  Los campos de velocidad y presión se interpolan con funciones de forma lineales. Para poder analizar materiales incompresibles, la formulación ha sido estabilizada con una nueva versión del método ''Finite Calculus (FIC)''. La técnica de estabilización ha sido derivada para fluidos Newtonianos casi-incompresibles. En este trabajo, la estabilización con FIC se usa también para el análisis de sólidos hipoelásticos casi-incompresibles.  En la tesis se dedica particular atención al estudio de flujo con superficie libre. En particular, se analiza en profundidad el tema de las pérdidas de masa y se muestra con varios ejemplos numéricos la capacidad del método de garantizar la conservación de masa en problemas de flujos en superficie libre. Además se estudia con detalle el condicionamiento del  esquema numérico analizando particularmente el efecto del módulo de compresibilidad. Se presenta también una estrategia basada en el uso de un pseudo módulo de compresibilidad para mejorar el condicionamiento del problema.  La formulación unificada ha sido validada comparando sus resultados numéricos con pruebas de laboratorio y resultados numéricos de otras formulaciones. En la mayoría de los ejemplos también se ha estudiado la convergencia del método.  En la tesis también se describe una estrategia segregada para el acoplamiento de la formulación unificada con el problema de transmisión de calor. Además se presenta una simple estrategia para simular el cambio de fase. El esquema acoplado ha sido utilizado para resolver varios problemas de FSI donde se incluye la temperatura y su efecto.  El esquema acoplado con el problema térmico ha sido utilizado con éxito para resolver un problema industrial. El objetivo del estudio era la simulación del daño y la fusión de la vasija de un reactor nuclear provocados por el contacto con un fluido altamente viscoso y a gran temperatura. En la tesis se describe con detalle el estudio numérico realizado para esta aplicación industrial.
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==Acknowledgements==
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I would like to thank all the people that in some way have helped me to complete this doctoral thesis.   I would like to express my gratitude to Prof. Eugenio Oñate for many reasons. Thanks to him, four years ago I could come to Barcelona and start my PhD at CIMNE. He gave me the balanced doses of freedom and pressure and he encouraged me to test innovative technologies. I would like thank Prof. Oñate also for giving me the opportunity and the material support to spend a three-month period at the College of Engineering of Swansea.  Special thanks go to Dr Josep Maria Carbonell.  Without his support I could not realize this work. He devoted me innumerable hours of his time and he gave me technical and not technical suggestions that helped me to make the right choices during my PhD. Moltes gràcies Josep Maria!  I would like to thank Prof. Javier Bonet for giving me the possibility to work with his group during my stay at Swansea. Special thanks go to Prof. Antonio Gil y Dr. Aurelio Arranz for their very kind treatment, for improving my knowledge of the Immersed Boundary Potential Method and for the useful discussions about FSI strategies. I really hope that this is just the first step for a future fruitful collaboration.  I would like to thank also Mr. Kazuto Yamamura and the Nippon Steel & Sumimoto Metal Corporation for giving us the possibility to publish the PFEM simulation  of the damages of a nuclear power plant pressure vessel.  I also acknowledge the Agencia de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) for the financial support.  I would like to thank also Riccardo, Antonia, Jordi, Pablo and Lorenzo for their contributions for this thesis.   This thesis is dedicated to my parents that still represent to me a fundamental and unique support. Grazie per tutti gli sforzi che avete fatto in questi anni e per le numerose visite a Barcellona. Questa tesi è per voi.  Finally, I would like to thank Lucía simply for staying by my side during these years. Her smile has been the gasoline I used for dealing with all this work.
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==Abbreviations==
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'''AL'''  '''A'''ugmented '''L'''agrangian 
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'''ASGS'''  '''A'''lgebraic '''S'''ub'''G'''rid '''S'''cale 
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'''FCM'''  '''F'''uel '''C'''ontaining '''M'''aterial
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'''FEM'''   '''F'''inite '''E'''lement  '''M'''ethod 
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'''FIC'''  '''F'''inite '''I'''ncrement '''C'''alculus 
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'''FSI'''  '''F'''luid-'''S'''tructure '''I'''nteraction 
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'''GLS'''  '''G'''alerkin '''L'''east-'''S'''quares 
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'''IBM'''  '''I'''mmersed  '''B'''oundary '''M'''ethod 
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'''IFEM'''  '''I'''mmersed '''F'''inite '''E'''lement  '''M'''ethod 
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'''ISPM'''  '''I'''mmersed '''S'''tructural  '''P'''otential '''M'''ethod 
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'''LBB'''  '''L'''adyzenskaya-'''B'''abuzka-'''B'''rezzi 
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'''LFCM'''  '''L'''ava-like '''F'''uel '''C'''ontaining '''M'''aterial
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'''NPP'''  '''N'''uclear '''P'''ower '''P'''lant
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'''OSS'''  '''O'''rthogonal '''S'''ub'''S'''cale 
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'''PFEM'''  '''P'''article '''F'''inite '''E'''lement  '''M'''ethod 
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'''SPH'''  '''S'''mooth '''P'''article '''H'''ydrodynamics 
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'''SUPG'''  '''S'''treamline '''U'''pwind '''P'''etrov '''G'''alerkin  
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'''TL'''  '''T'''otal '''L'''agrangian
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'''UL'''  '''U'''pdated '''L'''agrangian
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'''V'''  '''V'''elocity
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'''VMS'''  '''V'''ariational '''M'''ulti-'''S'''cale 
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'''VP'''  '''V'''elocity-'''P'''ressure
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'''VPS'''  '''V'''elocity-'''P'''ressure-'''S'''tabilized
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''Dai diamanti non nasce niente, <br />
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''dal letame nascono i fior.'' 
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"Nothing grows out of precious diamonds,  <br />
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Out of dung the flowers do grow."<br />
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Fabrizio De Andrè
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=Chapter 1. Introduction=
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==1.1 Objectives==
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The objective of this work is to develop a unified formulation for the solution of the fluid and solid mechanics, fluid-structure interaction (FSI) and thermal coupled problems. The aim is to analyze the continuum in a unified manner trying to reduce at minimum the differences between the analysis of fluids and solids. For this purpose the numerical model has been planned in order to meet the specific requirements of  solid and fluid mechanics and  their approximation with the Finite Element Method (FEM) [136], but without limiting excessively the capability of the model. In fact the computational method should be capable to deal with critical problems such as those involving elasto-plastic solids, quasi-incompressible materials, free surface fluids and phase change due to heat transfer solutions.
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According to these considerations, the computational model has been designed for using a stabilized Velocity-Pressure formulation. The formulation has been particularized for hypoelasto-plastic, compressible and quasi-incompressible solids and quasi-incompressible Newtonian fluids. The algorithm for the FSI problems has been inspired by the analogous unified strategy presented in [63].  For the fluid phase, the Particle Finite Element Method (PFEM) [88] has been used, while for the solid the classical FEM is adopted. The scheme has been stabilized with an updated version of the Finite Calculus (FIC) technique [82]. For dealing with thermal coupled problems, this unified formulation has been extended to treat the heat transfer problem via a staggered scheme. For the modeling of the phase change, a simplified procedure has been developed.  The whole formulation has been implemented in a C++ code.
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==1.2 State of the art==
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In this section, an overview of the numerical methods used for simulating a free surface fluid flow interacting with deformable solids is given. For the sake of clarity, the section is divided in three parts representing the main fields involved by this work. First the Eulerian and Lagrangian approaches for free surface fluid dynamics problems are presented. Then an overview of the stabilization for incompressible, or quasi-incompressible, material is given. Finally, the principal algorithms for solving FSI problems are described.
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===1.2.1 Eulerian and Lagrangian approaches for free surface flow analysis===
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Consider the  description of the motion of a general continuum  represented in Figure [[#img-1|1]]. The domain <math display="inline">{\Omega _0}</math> represents the body at the initial state at time <math display="inline">{t=t_0}</math> while the domain <math display="inline">{\Omega }</math> represents the same body at time <math display="inline">{t=t_n}</math> after deformation. The domain <math display="inline">{\Omega _0}</math>  is called ''initial configuration'', whereas <math display="inline">{\Omega }</math> is the ''current,'' or ''deformed'', ''configuration''. In order to describe  the kinematics and the deformation of the body, the ''reference configuration'' has to be defined because the motion is defined with respect to this configuration. <div id='img-1'></div>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
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|-
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|[[Image:draft_Samper_722607179-motionDescription.png|600px|Description of the motion of a general continuum body]]
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|- style="text-align: center; font-size: 75%;"
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| colspan="1" | '''Figure 1:''' Description of the motion of a general continuum body
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|}
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In solid mechanics, the stresses generally depend on the history of deformation and the undeformed configuration must be specified in order to define the strains. Due to the history dependence, Lagrangian descriptions are prevalent in solid mechanics. In fluids however, the stresses do not depend on the history and it is often unnecessary to describe the motion with respect to a reference configuration. For this reason an Eulerian description represents the most reasonable choice. Furthermore, in problems where the fluid contours are fixed, Eulerian meshes are generally preferred to the Lagrangian ones. This is because Eulerian grids are fixed and they do not deform according to the fluid motion, as  shown in Figure [[#img-2|2]].  <div id='img-2'></div>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 100%;"
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|-
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|[[Image:draft_Samper_722607179-eulerianMesh.png|480px|Motion description using an Eulerian mesh]]
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|- style="text-align: center; font-size: 75%;"
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| colspan="1" | '''Figure 2:''' Motion description using an Eulerian mesh
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|}
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Conversely, in the Lagrangian description, the mesh nodes coincide with the fluid particles and the discretization moves and deforms as the fluid flow (Figure [[#img-3|3]]).  <div id='img-3'></div>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 100%;"
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|-
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|[[Image:draft_Samper_722607179-lagrangianMesh.png|480px|Motion description using a Lagrangian mesh]]
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|- style="text-align: center; font-size: 75%;"
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| colspan="1" | '''Figure 3:''' Motion description using a Lagrangian mesh
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|}
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Consequently, on the one hand the non-linear convective term disappears from the problem and on the other hand the mesh undergoes large distortions and it requires to be regenerated.
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In the analysis of free surface flows, the detection of the free surface contours represents a crucial task. Its position is unknown <math display="inline">a~priori</math> and it has to be determined at each time increment in order to solve properly the boundary value problem. For these problems, the Lagrangian description may be preferred to the Eulerian one. In fact, with a Lagrangian approach the free surface is detected automatically by the position of the mesh nodes, while an Eulerian approach requires the implementation of a specific technology for this task.
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Several strategies have been developed in the literature  for tracking the free surface in an Eulerian framework. One of the earliest contributions was given by the so called ''marker and cell'' method [53]. In this approach a set of marker particles that move according the flow are used to detect which regions are occupied by the fluid and which not. An evolution of this technique is the ''volume of fluid'' method [55]. In this case the free surface boundary is detected using a scalar function that assumes the unit value in the fluid cells and the value zero in those ones with no fluid. The cells with an intermedium value are the ones that contain the free surface.  Another possibility is the ''level set'' method [100]. This technique is used in various fields, not only in continuum mechanics, and it allows for detecting shapes or surfaces on a fixed grid without making any parametrization of them. For this reason, this procedure has been also used for matching the free surface contour on an Eulerian mesh [106]. A similar idea was used in [7] where the position of the free surface is detected using a cloud of Lagrangian particles moving over an Eulerian mesh.
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The free surface flows can be solved also using an hybrid Eulerian-Lagrangian technique. This is the so termed ''Arbitrary Lagrangian-Eulerian (ALE)'' approach [137]. The aim of this method is to exploit the best features of the Eulerian and the Lagrangian procedures and to combine them. The mesh nodes can arbitrary be fixed or can move with the fluid [38]. Generally, far from the moving boundaries a fixed Eulerian grid is used, while near to the interface the mesh moves according to the motion of the boundary  [25]. However if the boundary motion is large or unpredictable, also in the ALE methods the grid may suffer large distortions and may require a proper remeshing procedure [116].
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In purely Lagrangian approaches, the  mesh needs to be regenerated whenever a  threshold limit for the distortion is exceeded. This is the basis of a particular class of Lagrangian finite element formulation called the  ''Particle Finite Element Method (PFEM)''. The method was initially developed by the group of professors  Idelsohn and Oñate [66,67]. The PFEM treats the mesh nodes of the domain as particles which can freely move and even separate from the rest of the  fluid domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a classical finite element method. These features make the PFEM the ideal numerical procedure to model and simulate free surface flows. In the last years, many scientific publications have shown the efficacy of the PFEM for solving free surface flow problems, see among others [32,68,117]. The PFEM has also been tested successfully in other kind of problems, such as fluid mechanics including thermal convection-diffusion [5,80,95], multi-fluids  [36,62], granular materials [131], FSI [81,132] and  excavation [19].
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''Mesh free'' methods are other class of Lagrangian techniques. In this strategy the remeshing is not required because the governing equations  are solved over a set of nodes without referring to a mesh. One of the first meshfree techniques is the ''Smooth Particle Hydrodynamics (SPH)'' method. This method was introduced independently by Gingold and Monaghan [50] and Lucy [70] for the simulation of astrophysical problems such as fission of stars. SPH is a particle-based Lagrangian technique where discrete smoothed particles are used to compute approximate values of needed physical quantities and their spatial derivatives. The particles have assigned a characteristic distance, called ‘smoothing length’, over which their properties are ”smoothed” by a kernel function. A typical drawback of the SPH method is that it is hard to reproduce accurately the incompressibility of the materials. The SPH technique has been used successfully for solving fluid-structure interaction problems [3].
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===1.2.2 Stabilization techniques===
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A FEM-based procedure may require to be stabilized when incompressible, or quasi- incompressible, materials are analyzed. For example from the finite element solution of the Navier Stokes equations numerical instabilities may arise from two sources. The first one is due to the presence of the convective term in the linear momentum equations. This term introduces a non-linearity in the equations and it needs a proper stabilization for solving high Reynolds number flows with the FEM [38]. Furthermore, the orders of interpolation of pressure and velocity fields cannot be chosen freely but they have to satisfy the so called <math display="inline">{Ladyzenskaya-Babuzka-Brezzi}</math> (LBB), or <math display="inline">{inf-sup}</math>, condition [15]. If the orders of interpolation of the unknown fields do not satisfy this restriction, a stabilization technique is required in order to avoid numerical instabilities, as the spurious oscillations of the pressure field.
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It is well known that the weak form generated by Galerkin approximation leads to a less diffusive solution than the strong problem. So the main idea of many stabilization techniques consists of adding an artificial diffusion to the problem. The first attempt was made by Von Neumann and Richtmyer [75]. Their solution adds an artificial diffusion to the strong form of the problem. This technique introduces an excessive dissipation to the problem because the diffusivity is added in every direction. An important evolution of this approach was the ''Streamline Upwind Petrov Galerkin (SUPG)'' [58]. In this approach, the artificial diffusion is added by means of the test functions and only on the direction of the streamlines. Furthermore, this is performed in a consistent way: the stabilization terms vanish when the solution is reached. An extension of the SUPG method is the ''Galerking Least-Squares (GLS)'' method [60]. The main difference between the two is that in the GLS method the stabilization terms are applied not only to the convective term but to all the terms of the equation. The ''Variational Multi-Scale (VMS)'' methods [59] split the problem variables in a large-scale and a subscale terms. The large-scale terms represent the part of the solution that can be captured by the finite element mesh, while the subscale part consists of an approximate solution that has to be added to the large-scale term in order to obtain the correct solution. This idea represents the basis of other stabilization methods. Among these, the most largely used are the ''Algebraic Subgrid Scale Formulation (ASGS)'' and the ''Orthogonal Subscales (OSS)'' methods, respectively introduced in [28,29]. Another efficient stabilization technique is the ''Finite Calculus '' (also termed ''Finite Increment Calculus'')  (''FIC'') approach  (see among others [76,77,85,86]). This method  has some analogies with the SUPG technique (for a comparison between these methods see [89]). The FIC approach is based on expressing the equations of balance of mass and momentum in a space-time domain of finite size and retaining higher order terms in the Taylor series expansion typically used for expressing the change in the transported variables within the balance domain. In addition to the standard terms of infinitesimal theory, the FIC form of the momentum and mass balance equations contains derivatives of the classical differential equations in mechanics multiplied by characteristic distances in space and time. In this work an updated version of the FIC method has been derived and tested.
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In solid mechanics a stabilization procedure may be  required for the solution of problems involving  incompressible, or quasi-incompressible solids. Situations of this type are common in forming processes or in the analysis of the rubber-type materials. Many of the stabilization procedures  for fluid dynamics have been also used also for solid dynamics. For example, the VMS method has been applied in quasi-incompressible solid mechanics in [22,24], the OSS in [26]. In  [48,49] a stabilized multi-field Petrov-Galerkin procedure is used. Finally, the  application of the FIC in solid mechanics is reported [94,103].
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===1.2.3 Algorithms for FSI problems===
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Many approaches have been developed for solving FSI problems. Typically the computational techniques for  FSI are distinguished in ''monolithic'' and ''staggered'', or ''partitioned'', approaches. In a monolithic approach the fluid and the solid domains are solved in a single system of equations (see among others [57], or [73]). In this technique the flow of information between the solid and the fluid parts is implicitly performed by the procedure. On the contrary, in  staggered schemes the fluid and the solid dynamics are solved separately and boundary conditions are transferred from a domain to the other at the interface. From the algebraic point of view, the solution is achieved by solving two different linear systems which are coupled by means of the boundary conditions defined along the interface. Within partitioned schemes, depending on the level of coupling between the fluid and the solid dynamics, a further classification can be done. In a ''weakly coupled'' segregated schemes when the transfer of information through the interface is performed only once for each time step,  (for example see [42]), whereas  in ''strongly coupled'' schemes this operation is performed within a convergence loop  (as a  reference see [34]). Clearly, a weakly coupled scheme has a lower computational cost but it can be used only when the interaction between the solid and the fluid domains is not strong or complex. Otherwise the algorithm may not find the correct numerical solution of the problem.
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Partitioned methods allow the reutilization of existing solvers. Furthermore, the solid and the fluid solvers can be updated independently. Staggered schemes lead to smaller and better conditioned linear systems than the ones obtained with monolithic approaches. On the other hand, monolithic strategies are generally more stable and fast than staggered schemes, and they lead to a more accurate solution of FSI problems, because a stronger coupling is ensured.
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Another general classification of the FSI algorithms is based upon the treatment of meshes. In ''conforming mesh'' methods,  in order to allow the transfer of information, the fluid and the solid meshes must have in common the nodes along the interface. Consequently, if the position of the interface nodes changes in a domain, also the other domain must modify its grid in order to guarantee the conformity of the two meshes along the interface.  On the contrary, in ''non-conforming mesh'' methods the interface and the related conditions are treated as constraints imposed on the model equations so that the fluid and solid equations can be solved independently from each other with their respective grids  [21]. This represents an important advantage because, typically, the mesh used for the fluid has an average size lower than the one used for the solid and so it is not necessary to refine the solid finite element grid near to the interface. However, non-conforming mesh algorithms are more complex to implement and it is not trivial to guarantee their robustness.
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In the so-called ''Immersed Boundary Method'' (<math display="inline">{IBM}</math>), the fluid is solved using an Eulerian grid and the solids are immersed on top of this mesh [101,104]. The interaction is ensured by penalizing the Navier-Stokes equations with the momentum forcing sources of the immersed structures. An evolution of the IBM is the <math display="inline">{ Immersed~Structural}</math> <math display="inline">{Potential~Method}</math> (<math display="inline">{ISPM}</math>) where the structure is modeled as a potential energy functional solved over a cloud of integration points that move within the fixed fluid mesh [47,104]. Also in the <math display="inline">{Immersed~Finite}</math> <math display="inline">{Element~Method}</math> (<math display="inline">{IFEM}</math>) [54,129] the structure acts as a momentum forcing source for the fluid governing equations, but in this case a Lagrangian mesh is employed  for the solid domains.
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==1.3 Numerical model==
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The aim of this work is to derive a finite element formulation capable to solve, through a unique set of equations and unknown variables, the mechanics of a general continuum.  The term 'general continuum' refers to a domain that may include compressible and quasi-incompressible solids, fluids or both interacting together. For this reason, the formulation is termed <math display="inline">{Unified}</math>. The Unified formulation is based on a mixed Velocity-Pressure Stabilized procedure and it has been implemented in a sequential <math display="inline">C++</math> code.
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===1.3.1 Reasons===
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There are many reasons for undertaking the above objective.
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The first advantage of the Unified formulation is that it allows for solving fluid and solid dynamics problems by implementing and using a single calculation code.
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Furthermore, if solids and fluids are solved via the same scheme, it  is simpler to implement the solver for FSI problems because it is not required neither changing the variables, neither implementing the transfer of transmission conditions through the interface. With this formulation solids and fluids represent regions of the same continuum and they differ only by the specific values of the material parameters. As a consequence, the FSI solver requires a small computational effort and it can be implemented by introducing just a few  specific functions. This will be explained in detail in the section dedicated to FSI problems.
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Additionally, with the Unified formulation the most natural approach for solving  FSI problem is the monolithic one. This brings in the further advantage that the coupling is ensured strongly and an iteration loop is not required, as for staggered procedures.
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Finally, using the same set of unknowns for the fluid and the solid domains reduces the ill-conditioning of the FSI solver, because the solution system does not include variables of different units of measure.
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===1.3.2 Essential features===
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The Unified formulation is a compromise between a fluid and a solid formulation and it should be capable to satisfy the requirements of each problem mechanics and finite element approximations. In other words, in order to solve adequately fluids and  solids with the same formulation, the solution procedure must take into account the constraints that both models impose.
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Concerning the choice of the unknown variables, these are generally selected depending on the constitutive law of the materials. For Newtonian fluids, the Cauchy stress tensor is related directly to the deformation rate tensor. This implies that the velocities are the most useful unknowns in fluid mechanics. Conversely, in solid mechanics the displacements  are generally preferred as variables, as the stresses are related to the deformations. However, for solids there also exist constitutive laws expressed in rate form, as for hypoelastic models.
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For the analysis of incompressible, or quasi-incompressible, materials a mixed formulation is required in order to overcome the associated numerical instabilities, both in solids and 
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fluids. In fluids, the most popular combination of unknown variables is the velocity-pressure 
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scheme. In solid mechanics, there are several mixed approaches, as the displacement-stress [16], 
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the displacement-strain [23], the displacement-deformation gradient [49], the  
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displacement-deformation gradient-jacobian [48], the  displacement-pressure [26] and the 
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velocity-pressure formulation [110]. The combination of unknowns is selected depending on the 
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desired accuracy for the stress and strain fields and the associated computational cost. Among 
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the mentioned mixed approaches, the displacement- pressure and the velocity-pressure formulations 
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lead to the lowest computational cost, because the number of unknowns is smaller.  However the 
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velocity-pressure formula- tion has the further advantage that is the canonical scheme for fluid 
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mechanics. Hence, according to these considerations, the Unified formulation has been based on a 
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mixed
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velocity-pressure  scheme.
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Another key decision in the design of the Unified formulation concerns the choice of the description framework. Solids are typically solved  using a (total or updated) Lagrangian description, while for fluids, due to the large deformations they undertake, the Eulerian framework is generally preferred. However, for free surface flows, as the ones treated in this work, a Lagrangian description has the important advantage that the free surface boundaries are detected automatically, because the particles of the fluid coincide with the nodes of the mesh. The price for this is that a remeshing procedure is required in order to avoid the excessive distortion of the mesh. According to these considerations, the Unified formulation has been implemented using an updated Lagrangian description.
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In this work, the Particle Finite Element Method (PFEM) [88] has been used for solving the fluid domain only. The PFEM is a Langrangian numerical technique that treats the mesh nodes as moving points which can freely move and even separate  from the domain to which they are initially attached representing, for instance, the effect of water drops. The PFEM is based on a remeshing procedure that efficiently combines the Delaunay tessellation and the  Alpha Shape method [39].  In order to reduce the computational cost associated to the remeshing step, all the unknowns are stored in the nodes and linear shape functions are used for the finite element interpolation. Conversely, the solid domain keeps a fixed mesh during the whole analysis. The reason for this is that in non-linear solid mechanics it is required to preserve not only the nodal unknowns but also elemental information as the stress state of the previous time step. A remeshing implies the elimination of the previous discretization and the creation of new elements. In order to keep the elemental information in the remeshing step, a projection procedure from the elements of the previous mesh to its nodes, and from these to the elements of the new mesh is required. These operations may introduce an interpolation error  in the scheme that affects the solution of solid mechanics problems. In the analysis of Newtonian fluids all the information can be stored in the nodes, so the remeshing does not affect the accuracy of the scheme. For these reasons, in this work the PFEM is used only for the fluid, while solids are solved via the classical FEM.
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In order to improve the conditioning of the solution system, a Gauss-Seidel  segregated scheme is used. This means that the problem is solved iteratively and separating the unkwown variables, the velocities and the pressure. The solution method consists of two steps. In the first one the linear momentum equations are solved for the velocity increments and considering the pressure of the previous non-linear iteration in the residual term. In the second step, the continuity equation is solved for the pressure in the updated configuration  using the velocities computed at the first step. This two-step algorithm leads to a smaller and better conditioned linear system  than using a monolithic approach. A crucial point of this scheme is the derivation of the tangent matrix of the linear momentum equations. The velocity increments are solved by condensing the pressure. According to this procedure, the tangent matrix for the linear momentum equations of the mixed velocity-pressure formulation does not differ from the one of a velocity formulation. Furthermore, the derivation of the tangent matrix for the velocity formulation is easier, as it does not involve the pressure.  For these reasons, the first step towards the unified scheme is the derivation of the Velocity formulation. The mixed Velocity-Pressure formulation will be derived by exploiting the linearization performed for the Velocity formulation. In [79] another procedure for the derivation of the exact tangent matrix for an Updated Lagrangian scheme is described.
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The condensation of the pressure in the tangent matrix of the linear momentum equations may induce the ill-conditioning of the linear system for the analysis of quasi-incompressible materials. This ill-conditioning emanates from the volumetric counterpart of the tangent matrix that  is governed by  the bulk modulus, that typically has high values that may compromise the conditioning of the linear system. In this work, a thorough study of the conditioning drawbacks associated to this scheme has been performed and a useful and easy to implement technique to overcome these inconveniences has been tested and validated.
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The numerical method proposed in this work is an improvement versus other numerical schemes for quasi-incompressible materials where an arbitrarily defined pseudo-bulk modulus was used in both the linear momentum and the mass conservation equations [110,109,108].
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.
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The method  can be also related to the so-called Augmented Lagrangian (AL) procedures for solving the Navier-Stokes equations for incompressible [10,11,41,44,124] and weakly compressible [14,123].
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In order to deal with incompressible (or quasi-incompressible) materials a mixed formulation is required. In this work  both the velocities and the pressure are interpolated using linear shape functions. This combination does not fulfil the <math display="inline">inf-sup</math> condition [15] and the problem needs to be stabilized. The required  stabilization is ensured  using an updated version of the FIC technique [82] applied to the mixed Velocity-Pressure formulation. The FIC stabilization method has a small intrusivity because its terms only affect the continuity equation. In fact for solving the linear momentum equations the same scheme as for the (not stabilized) mixed Velocity-Pressure formulation is used. It will be shown that the PFEM-FIC stabilized procedure guarantees  a good accuracy for the mass conservation of the free surface flows. The stabilization procedure is derived for quasi-incompressible fluids, but it can be easily extended to quasi-incompressible hypoelastic solids.
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The derivation of the Unified stabilized formulation is carried on trying to maintain as much as possible the generality of the scheme. For this reason the constitutive relations are introduced only as the last step. The Velocity (''V'') and the mixed Velocity-Pressure (''VP'') formulations are derived first for a general material and then particularized for specific constitutive laws. For quasi-incompressible materials only the mixed Velocity-Pressure Stabilized (''VPS'') formulation can be used while for compressible materials the Velocity and the standard (not stabilized) mixed Velocity-Pressure formulations are both suitable. In particular, for  compressible solids,  the hypoelastic model has been chosen.
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The FSI problem is solved in a monolithic way, hence solids and fluids are solved by the same linear system. For the solid domain it is possible to choose which formulation to use. Depending on the problem, one may chose the V, the VP or the VPS element.
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The Unified formulation for FSI problems can be easily coupled with the heat transfer problem in order to solve coupled thermal mechanical problem. The coupling is performed via a staggered scheme. The effect of the heat is taken into account by considering the material properties depending on the temperature and including in the strain tensor the deformations induced by the temperature. With the PFEM Unified formulation also phase change problems can be modeled.
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The Unified stabilized VP formulation with thermal coupling has been used for the analysis of an industrial application. The study concerned the analysis of the damages caused by the dropping of a volume of corium at high temperature on the structure of the pressure vessel in a Nuclear Power Plant (NPP).
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===1.3.3 Outline===
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This text is split in the following chapters.
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In the next chapter two velocity-based FEMs are derived for a general compressible material. In the first section the Velocity formulation is derived and the incremental solution scheme is given. Then, the standard mixed Velocity-Pressure formulation is obtained as an extension of the Velocity formulation. The constitutive laws are not introduced for any scheme. In the third section the formulations are adapted for hypoelastic-plastic compressible solids. The chapter ends with several validation examples for non-linear solid mechanics.
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In Chapter [[#3 Unified stabilized formulation for quasi-incompressible materials|3]],  the FIC stabilization strategy is introduced in the mixed stabilized VP scheme. This scheme is adapted for   quasi-incompressible Newtonian fluids and hypoelastic solids, in this order. Then the free surface problem is studied in detail. First the PFEM is described highlighting the advantages and the disadvantages of the method. A useful technique for modeling the slip conditions in Lagrangian flows is also explained. Next the mass conservation feature and the conditioning of the scheme are analyzed in detail. Concerning the former point, it will be shown that the PFEM-FIC formulation guarantees a good preservation of the mass in free surface flows. Regarding  the latter point, a practical strategy for improving the conditioning of the linear system is explained and tested. At the end of the chapter several validation examples for quasi-incompressible Newtoninan fluids and hypoelastic solids are given.
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The FSI solution strategy is described in Chapter [[#4 Unified formulation for FSI problems|4]]. The monolithic scheme for coupling the mechanics of fluids and solids is explained in detail. Finally several validation examples of FSI problems are given.
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Chapter [[#5 Coupled thermal-mechanical formulation|5]]  is dedicated to couple the Unified formulation for FSI with the heat problem. In the first section the heat problem is introduced and discretized using the FEM. Then the coupling strategy is explained and validated with numerical examples. Next, the procedure for modeling  phase change problems is described and an explicative numerical example is presented.
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Chapter [[#6 Industrial application: PFEM Analysis Model of NPP Severe Accident|6]]  is fully devoted to the industrial application of the Unified stabilized  and thermally coupled strategy. The damages of the pressure vessel structure caused by the fall of a volume corium at high temperature are studied using two simplified models where the solid structure melts due to the heat transfer from the corium.
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In Chapter [[#7 Conclusions and future lines of research|7]] the innovative contributions of the work are summarized  and the lines of research opened by this thesis are presented.
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==1.4 Publications==
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The following publications have emanated from the work carried out in this doctoral thesis
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<ol>
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<li> E. Oñate and A. Franci and J.M. Carbonell, Lagrangian formulation for finite ele-ment analysis of quasi-incompressible fluids with reduced mass losses, ''International Journal for Numerical Methods in Fluids'', 74 (10), 699-731, 2014;</li>
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<li> E. Oñate and A. Franci and J.M. Carbonell, A particle finite element method (PFEM) for coupled thermal analysis of quasi and fully incompressible flows and fluid-structure interaction problems, ''Numerical Simulations of Coupled Problems in Engineering''. S.R. Idelsohn (Ed.), 33, 129-164, 2014;</li>
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<li>E. Oñate and A. Franci and J.M. Carbonell, A particle finite element method for analysis of industrial forming processes, ''Computational Mechanics'', DOI: 10.1007/s00466-014-1016-2, 2014; </li>
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<li> A. Franci and E. Oñate and J.M. Carbonell, On the effect of the bulk tangent ma-trix in partitioned solution schemes for nearly incompressible fluids, ''International Journal for Numerical Methods in Engineering'', DOI: 10.1002/nme.4839, 2014;</li>
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<li>A. Franci and E. Oñate and J.M. Carbonell, Unified formulation for solid and fluid mechanics and FSI problems, ''Computer Methods in Applied Mechanics and Engineering'', (in preparation).</li>
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</ol>
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=Chapter 2. Velocity-based formulations for compressible materials=
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In this chapter two velocity-based finite element formulations for compressible materials are presented, namely the Velocity (V) and the mixed Velocity-Pressure (VP) formulations. For both schemes the linear momentum equations are solved iteratively for the velocity increments. The linearization of the governing equations is performed without specifying any constitutive law. The aim of this chapter is  to maintain  as much as possible the generality of the algorithms, leaving the formulations open to different material models. It will be shown that the only requirement demanded to the constitutive laws is that the rate of stress must be linearly related with the rate of deformation.
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There are several reasons that justify the presentation of the Velocity formulation. First of all, the tangent matrix of the linear momentum equations for the Velocity formulation holds also for the mixed Velocity-Pressure formulation but the linearization procedure is easier because the pressure does not appear. Furthermore, the Velocity formulation is useful for making some interesting and didactic comparisons with the mixed formulations.
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After deriving the solution scheme  for the Velocity formulation, the mixed Velocity-Pressure method is presented. The governing equations are the linear momentum and the linear pressure-deformation rate equations. The latter is called continuity, or mass balance, equation for its similarity with the incompressibility constraint of the Navier-Stokes problem. As for the velocity scheme, in the mixed formulation the constitutive law is not specified. In Section [[#2.3 Hypoelasticity|2.3]] the hypoelastic-plastic model is presented and this constitutive law is inserted in both the Velocity and the mixed Velocity-Pressure schemes. The incremental solution scheme is explained in detail for both formulations. At the end of the chapter some validation examples for hypoelastic-plastic solids in statics as in dynamics are given.
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==2.1 Velocity formulation==
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In this section, the velocity formulation for solving transient problems for a general continuum is derived.  The governing equations are the linear momentum equations and they are derived in the updated Lagrangian (UL) framework. This means that the governing equations are integrated over the unknown configuration <math display="inline">{\Omega }</math> (the so-called updated configuration). As a consequence, the space derivatives for the UL description are computed with respect to the spatial coordinates.
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===2.1.1 From the local form to the spatial semi-discretization===
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In this section the spatial semi-discretization of the linear momentum equations is derived.
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For a general continuum, the local form of the linear momentum equations using the UL description reads
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<span id="eq-1"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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|-
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{| style="text-align: left; margin:auto;" 
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|-
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| style="text-align: center;" | <math>\rho (\hbox{X},t) \frac{ \partial v(\hbox{X},t)}{\partial t}-{\partial \sigma (\hbox{X},t) \over \partial \hbox{x}}-b(\hbox{X},t)=0   \quad \quad in  \Omega \times (0,T)   </math>
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|}
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| style="width: 5px;text-align: right;" | (1)
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|}
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where <math display="inline">\rho </math> is the density of the material, <math display="inline">v</math> are the velocity vector, <math display="inline">\sigma </math> is the Cauchy stress tensor and <math display="inline">b</math> is the body force vector. The variables within the brackets are the independent variables:  '''X''' are the Lagrangian  or material coordinates vector, '''x''' the Eulerian or spatial coordinates vector and <math display="inline">t</math> is the time. For simplicity, in what follows the independent variables are not specified. The spatial and material coordinates are related through the motion tensor <math display="inline">{\Phi }</math> as
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\hbox{x}=\Phi (\hbox{X},t),  \hbox{X}= \Phi ^{-1}(\hbox{x},t)  </math>
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|}
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| style="width: 5px;text-align: right;" | (2)
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|}
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The set of governing equations is completed by the following conditions at the Dirichlet (<math display="inline">\Gamma _v</math>) and Neumann (<math display="inline">\Gamma _t</math>) boundaries
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<span id="eq-3"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>v_i -v_i^p =0 \qquad \hbox{on }\Gamma _v </math>
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| style="width: 5px;text-align: right;" | (3)
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|}
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<span id="eq-4"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\sigma _{ij}n_j -t_i^p =0 \qquad \hbox{on }\Gamma _t </math>
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| style="width: 5px;text-align: right;" | (4)
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|}
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where <math display="inline">v_i^p</math> and <math display="inline">t_i^p</math> are the prescribed velocities and the prescribed tractions, respectively, and <math display="inline">n</math> is the unit normal vector.
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In the following summation of terms for repeated indices is assumed, unless otherwise specified.
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The spaces for the trial and test functions are defined, respectively, as
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<span id="eq-5"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>v_ i \in {U},  \quad \quad {U}=\{ v_ i |v_ i  \in C^0,  v_ i=v_i^p   on \Gamma _v \}   </math>
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|}
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| style="width: 5px;text-align: right;" | (5)
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|}
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<span id="eq-6"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>w_ i \in {U_0}, \quad \quad {U_0}=\{ w_ i | w_ i  \in C^0,  w_ i=0   on \Gamma _v \}   </math>
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|}
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| style="width: 5px;text-align: right;" | (6)
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|}
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Multiplying  Eqs.([[#eq-1|1]]) by the test functions and integrating over the updated configuration domain, the following global integral form is obtained
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<span id="eq-7"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\int _\Omega w_i \left(\rho \dot{v}_i- {\partial \sigma _{ij} \over \partial x_j}-b_i\right)d\Omega =0 </math>
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| style="width: 5px;text-align: right;" | (7)
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|}
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where the symbol <math display="inline">\dot{(\cdot )}</math> represents the material time derivative.
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Integrating by parts  the term involving <math display="inline">\sigma _{ij}</math> in Eq.([[#eq-7|7]]) and using the Neumann boundary conditions ([[#eq-4|4]]) yields the weak variational form of the momentum equations as
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<span id="eq-8"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
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{| style="text-align: left; margin:auto;" 
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|-
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| style="text-align: center;" | <math>\int _\Omega w_i \rho \dot{v}_i d\Omega + \int _\Omega \frac{\partial w_i}{\partial x_j} \sigma _{ij} d\Omega - \int _\Omega w_i b_i d\Omega - \int _{\Gamma _t} w_i t_i^p d\Gamma =0  </math>
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| style="width: 5px;text-align: right;" | (8)
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|}
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Eq.([[#eq-8|8]]) is the standard form of the Principle of Virtual Power [9].
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The spatial discretization is introduced using the classical FEM-Galerkin procedure  [136]. Hence both the trial and the test functions are interpolated in space in terms of their nodal values by means of the same shape functions <math display="inline">{N}</math>
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<span id="eq-9"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>v_i = \sum \limits _{I=1}^n N_I(X) \bar v_{iI} \quad ,\quad w_i = \sum \limits _{I=1}^n N_I(X) \bar w_{iI}     </math>
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| style="width: 5px;text-align: right;" | (9)
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|}
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where, assuming the use of  simplicial elements, <math display="inline">n=3/4</math> for 2D/3D problems is the number of the nodes of the element, <math display="inline">\bar{(\cdot )}</math> denotes a nodal value, the capital subscript specifies the node and the lower case subscript represents the cartesian direction. In this work, linear shape functions have been used for <math display="inline">N_I</math>.
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Since Eq.([[#eq-9|9]]) must hold for all the test functions in the interpolation space, introducing the spatial discretization ([[#eq-9|9]]) into  Eq.([[#eq-8|8]]), the spatial semi-discretized form of the momentum equations in the UL framework for the node <math display="inline">{I}</math> reads
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<span id="eq-10"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\underbrace{  \int _\Omega N_I \rho d\Omega \dot{v_i}  }_{\displaystyle{{{f}^{dyn}_{Ii}}}}  +  \underbrace{  \int _\Omega \frac{\partial N_I}{\partial x_j} \sigma _{ij} d\Omega  }_{\displaystyle{{{f}^{int}_{Ii}}}}= \underbrace{  \int _\Omega N_I b_i d\Omega + \int _{\Gamma _t} N_I t_i^p d\Gamma  }_{\displaystyle{{{f}^{ext}_{Ii}}}}  </math>
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| style="width: 5px;text-align: right;" | (10)
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where <math display="inline">{f^{dyn}}</math>, <math display="inline">{f^{int}}</math> and <math display="inline">{f^{ext}}</math> are the dynamic, internal and external forces, respectively, expressed in the UL framework.
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For convenience, the semi-discretized form of the momentum equations in the total Lagrangian (TL) framework is also presented here. This is written as [9]
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<span id="eq-11"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\underbrace{  \int _{\Omega _0} N_I \rho _0 d\Omega \dot{v_i} }_{\displaystyle{{^{TL}{f}^{dyn}_{Ii}}}}+ \underbrace{  \int _{\Omega _0} \frac{\partial N_I}{\partial X_j} P_{ij} d\Omega }_{\displaystyle{{^{TL}{f}^{int}_{Ii}}}} = \underbrace{  \int _{\Omega _0} N_I b_i d\Omega + \int _{\Gamma _0} N_I t^p_{0i} d\Gamma  }_{\displaystyle{{^{TL}{f}^{ext}_{Ii}}}} </math>
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| style="width: 5px;text-align: right;" | (11)
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where <math display="inline">{P}</math> is the first Piola-Kirchhoff stress tensor, or the nominal stress tensor, and <math display="inline">{^{TL}f^{dyn}}</math>, <math display="inline">{^{TL} f^{int}}</math> and <math display="inline">{^{TL} f^{ext}}</math> are the dynamic, internal and external forces, respectively, expressed in the TL framework.  All the variables with vectors subscript <math display="inline">{(\cdot )}_0</math> refer to the last known configuration. Note that Eq.([[#eq-11|11]]) can be obtained from Eq.([[#eq-10|10]]) by pull back transformations on all its terms [9].
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For the sake of clarity in the notation, the terms referred to the TL description are denoted with the left index <math display="inline">{^{TL}(\cdot )}</math>. Unless otherwise specified, the variables belong to the UL description.
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===2.1.2 Time integration===
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In this work, the kinematic variables  have been integrated in time using a second order scheme. In particular, the implicit Newmark's integration rule has been adopted. For the general case, this rule states that accelerations and displacements are computed as
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<span id="eq-12"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>^{n+1}{\dot{v}}</math>
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| style="text-align: right;" | <math>= \frac{1}{\gamma \Delta t} \left({^{n+1}v} -{^{n}v}\right)- \frac{1-\gamma }{\gamma }{^{n}\dot{v}}  </math>
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| style="text-align: center;" | <math> ^{n+1}u</math>
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| style="text-align: right;" | <math>={^{n} u}+ {\Delta t}\frac{\gamma - \beta }{\gamma }{^{n}v} + \Delta t\frac{\beta }{\gamma } {^{n+1}v}+ {\Delta t^2}\frac{\gamma{-} 2\beta }{2\gamma }{^{n} \dot{v}} </math>
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| style="width: 5px;text-align: right;" | (12)
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Where <math display="inline">{\beta }</math> and <math display="inline">{\gamma }</math> are the so-called Newmark's parameters [9]. This time integration scheme is unconditionally stable if the following relation holds
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<span id="eq-13"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\gamma \ge 2 \beta \ge \frac{1}{2} </math>
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| style="width: 5px;text-align: right;" | (13)
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In the present work, the Newmark's parameters chosen are <math display="inline">{\beta = \frac{1}{4}}</math> and <math display="inline">{\gamma = \frac{1}{2}}</math>.
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Replacing the numerical values of the constants in Eq.([[#eq-12|12]]) yields
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<span id="eq-14"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>^{n+1}{\dot{v}}= \frac{2}{ \Delta t} \left({^{n+1}v} - {^{n}v}\right)- {^{n}\dot{v}}  </math>
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| style="width: 5px;text-align: right;" | (14)
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<span id="eq-15"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>^{n+1}u={^{n}u} + \frac{\Delta t}{2}\left({^{n+1}v}+ {^{n}v}\right)  </math>
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| style="width: 5px;text-align: right;" | (15)
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|}
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===2.1.3 Linearization===
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Although the problem is set out in the UL framework, the linearization for the velocities increment of the momentum equations is performed first on the TL  semi-discretized form ([[#eq-11|11]]). The UL linearized form is  obtained by performing a push-forward transformation on the TL form. This is justified by the easier derivation of the tangent matrix in the TL framework. In fact, in Eq.([[#eq-11|11]]) the only variable that depends on time is the nominal stress <math display="inline">{P}</math>, while in the UL form ([[#eq-10|10]]) the time-dependent variables are the updated domain <math display="inline">{\Omega }</math>, the Cauchy stress tensor <math display="inline">{\sigma }</math> and the spatial derivatives <math display="inline">{   \partial N/ \partial x}</math>. For the sake of clarity, the linearization of the internal and dynamic forces will be performed separately.  
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''Internal component of the tangent matrix''
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From  Eq.([[#eq-11|11]]) the internal forces in the TL description are defined as
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<span id="eq-16"></span>
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{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>^{TL}f^{int}_{Ii}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} P_{ij} d\Omega _0  </math>
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| style="width: 5px;text-align: right;" | (16)
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|}
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The constitutive relation is expressed in rate form. Hence it is more convenient to perform the linearization of the material derivative of the internal forces and then integrate for the time step increment <math display="inline">\Delta t</math>. The material time derivative of ([[#eq-16|16]]) is
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<span id="eq-17"></span>
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>^{TL}\dot{f}^{int}_{Ii}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}\dot{P}_{ij} d\Omega _0  </math>
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| style="width: 5px;text-align: right;" | (17)
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|}
530
531
The infinitesimal increment of Eq.([[#eq-17|17]]) is
532
533
<span id="eq-18"></span>
534
{| class="formulaSCP" style="width: 100%; text-align: left;" 
535
|-
536
| 
537
{| style="text-align: left; margin:auto;" 
538
|-
539
| style="text-align: center;" | <math>^{TL}\delta \dot{f}^{int}_{Ii}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} \delta  \dot{P}_{ij} d\Omega _0  </math>
540
|}
541
| style="width: 5px;text-align: right;" | (18)
542
|}
543
544
The first Piola-Kirchhoff stress tensor <math display="inline">{P}</math> is not typically used because it is not symmetric and its rate is a non-objective measure. For these reasons, in the TL framework it is more convenient to work with the second Piola-Kirchhoff stress tensor <math display="inline">{S}</math> and its rate. These stress rate measures are related each other through the following relation
545
546
<span id="eq-19"></span>
547
{| class="formulaSCP" style="width: 100%; text-align: left;" 
548
|-
549
| 
550
{| style="text-align: left; margin:auto;" 
551
|-
552
| style="text-align: center;" | <math>\dot{P}_{ij}=\dot{S}_{ir} F^T_{rj} + S_{ir} \dot{F}^T_{rj}  </math>
553
|}
554
| style="width: 5px;text-align: right;" | (19)
555
|}
556
557
where <math display="inline">{F}</math> is the  deformation gradient tensor defined as
558
559
<span id="eq-20"></span>
560
{| class="formulaSCP" style="width: 100%; text-align: left;" 
561
|-
562
| 
563
{| style="text-align: left; margin:auto;" 
564
|-
565
| style="text-align: center;" | <math>{F}_{ij}= \frac{\partial x_i}{\partial X_j}  </math>
566
|}
567
| style="width: 5px;text-align: right;" | (20)
568
|}
569
570
Substituting Eq.([[#eq-19|19]]) into ([[#eq-18|18]]) yields
571
572
<span id="eq-21"></span>
573
{| class="formulaSCP" style="width: 100%; text-align: left;" 
574
|-
575
| 
576
{| style="text-align: left; margin:auto;" 
577
|-
578
| style="text-align: center;" | <math>^{TL}\delta \dot{f}^{int}_{Ii}= \underbrace{\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} F_{ir} \delta \dot{S}_{jr}d\Omega _0} _{\displaystyle{{^{TL}\delta \dot{f}^{m}_{Ii}}}} +  \underbrace{\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}  S_{ir}  \delta \dot{F}^T_{rj} d\Omega _0} _{\displaystyle{{^{TL}\delta \dot{f}^{g}_{Ii}}}}     </math>
579
|}
580
| style="width: 5px;text-align: right;" | (21)
581
|}
582
583
Eq.([[#eq-21|21]]) shows that the increment of the material time derivative of the  internal forces can be split into material and   geometric parts, <math display="inline">{^{TL} \delta \dot{f}^{m}}</math> and <math display="inline">{^{TL}\delta \dot{f}^{g}}</math>, respectively. The former accounts for the material response through the rate of the second Piola-Kirchhoff stress tensor and the second term is the initial stress term that contains the information of the updated stress field. Note that, up to now, no constitutive relationships have been introduced and the above derivation holds for a general continuum. 
584
585
''Material tangent  matrix''
586
587
From Eq.([[#eq-21|21]]), the material part of the material time derivative of the internal forces reads
588
589
<span id="eq-22"></span>
590
{| class="formulaSCP" style="width: 100%; text-align: left;" 
591
|-
592
| 
593
{| style="text-align: left; margin:auto;" 
594
|-
595
| style="text-align: center;" | <math>{^{TL}\delta \dot{f}^{m}_{Ii}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} F_{ir} \delta \dot{S}_{jr}d\Omega _0}   </math>
596
|}
597
| style="width: 5px;text-align: right;" | (22)
598
|}
599
600
For the derivation of the material tangent matrix, the constitutive relation between the stress and the strain measures is required. In order to maintain the formulation as general as possible, the stress rate measure is related to the deformation rate through a tangent constitutive tensor, as
601
602
<span id="eq-23"></span>
603
{| class="formulaSCP" style="width: 100%; text-align: left;" 
604
|-
605
| 
606
{| style="text-align: left; margin:auto;" 
607
|-
608
| style="text-align: center;" | <math>\dot{S}_{ij}=C_{ijkl} \dot{E}_{kl}  </math>
609
|}
610
| style="width: 5px;text-align: right;" | (23)
611
|}
612
613
where <math display="inline">{C}</math> is a fourth-order  tensor and <math display="inline">{E}</math> is the Green-Lagrange strain tensor. Eq.([[#eq-23|23]]) can also be expressed in Voigt notation as
614
615
<span id="eq-24"></span>
616
{| class="formulaSCP" style="width: 100%; text-align: left;" 
617
|-
618
| 
619
{| style="text-align: left; margin:auto;" 
620
|-
621
| style="text-align: center;" | <math>\{ \dot{S} \}=\left[C\right]\{ \dot{E}\}  </math>
622
|}
623
| style="width: 5px;text-align: right;" | (24)
624
|}
625
626
where <math display="inline">{(\cdot )}</math> denotes a vector with components <math display="inline">\left[(\cdot )_{11}, (\cdot )_{22}, (\cdot )_{33}, (\cdot )_{12}, (\cdot )_{13},  (\cdot )_{23} \right]</math>.  As it will be shown in the following sections, Eq.([[#eq-23|23]]) can represent  both a Kirchhoff solid material and a Newtonian fluid.  If different constitutive laws are used, Eq.([[#eq-23|23]]) should be modified accordingly in order to derive the material part of the tangent matrix.
627
628
The Green-Lagrange strain tensor can be expressed in terms of the nodal velocities as
629
630
<span id="eq-25"></span>
631
{| class="formulaSCP" style="width: 100%; text-align: left;" 
632
|-
633
| 
634
{| style="text-align: left; margin:auto;" 
635
|-
636
| style="text-align: center;" | <math>\dot{E}_{kl}= \frac{\partial N_J}{\partial X_l}  F_{sk}  \bar  v_{Js}  </math>
637
|}
638
| style="width: 5px;text-align: right;" | (25)
639
|}
640
641
In Voigt notation
642
643
<span id="eq-26"></span>
644
{| class="formulaSCP" style="width: 100%; text-align: left;" 
645
|-
646
| 
647
{| style="text-align: left; margin:auto;" 
648
|-
649
| style="text-align: center;" | <math>\{ \dot{E} \}= B_0  \bar{v}  </math>
650
|}
651
| style="width: 5px;text-align: right;" | (26)
652
|}
653
654
where for a plane strain problem
655
656
{| class="formulaSCP" style="width: 100%; text-align: left;" 
657
|-
658
| 
659
{| style="text-align: left; margin:auto;" 
660
|-
661
| style="text-align: right;" | <math>\displaystyle{B_{0}}= \left[                    \begin{matrix}\displaystyle                   \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {X}} & \displaystyle                    \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {X}} \\[.4cm] \displaystyle   \frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {Y}} & \displaystyle                    \frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {Y}} \\[.4cm] \displaystyle    \frac{\partial N_I}{\partial {X}}\frac{\partial x}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial x}{\partial {X}} &   \displaystyle   \frac{\partial N_I}{\partial {X}}\frac{\partial y}{\partial {Y}}+\frac{\partial N_I}{\partial {Y}}\frac{\partial y}{\partial {X}} \\                      \end{matrix} \right] </math>
662
|}
663
| style="width: 5px;text-align: right;" | (30)
664
|}
665
666
Substituting Eq.([[#eq-25|25]]) in  Eq.([[#eq-23|23]]), yields
667
668
<span id="eq-31"></span>
669
{| class="formulaSCP" style="width: 100%; text-align: left;" 
670
|-
671
| 
672
{| style="text-align: left; margin:auto;" 
673
|-
674
| style="text-align: center;" | <math>\dot{S}_{ij}=C_{ijkl}  \frac{\partial N_J}{\partial X_l}  F_{sk} \bar   v_{Js}  </math>
675
|}
676
| style="width: 5px;text-align: right;" | (31)
677
|}
678
679
Similarly, substituting  Eq.([[#eq-26|26]]) in  Eq.([[#eq-24|24]])
680
681
<span id="eq-32"></span>
682
{| class="formulaSCP" style="width: 100%; text-align: left;" 
683
|-
684
| 
685
{| style="text-align: left; margin:auto;" 
686
|-
687
| style="text-align: center;" | <math>\{ \dot{S} \}=\left[C\right] B_0 v  </math>
688
|}
689
| style="width: 5px;text-align: right;" | (32)
690
|}
691
692
Substituting Eq.([[#eq-31|31]]) into Eq.([[#eq-22|22]]) yields
693
694
<span id="eq-33"></span>
695
{| class="formulaSCP" style="width: 100%; text-align: left;" 
696
|-
697
| 
698
{| style="text-align: left; margin:auto;" 
699
|-
700
| style="text-align: center;" | <math>{^{TL}\delta \dot{f}^{m}_{Ii}}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}  F_{ir} C_{jrkl}  \frac{\partial N_J}{\partial X_l}  F_{sk}  d\Omega _0  \delta \bar   v_{Js}  </math>
701
|}
702
| style="width: 5px;text-align: right;" | (33)
703
|}
704
705
where <math display="inline">{ \bar{v}_{Js}}</math> is the <math display="inline">{s}</math>-component of the velocity of node <math display="inline">{J}</math>. In Voigt notation Eq.([[#eq-33|33]]) reads
706
707
<span id="eq-34"></span>
708
{| class="formulaSCP" style="width: 100%; text-align: left;" 
709
|-
710
| 
711
{| style="text-align: left; margin:auto;" 
712
|-
713
| style="text-align: center;" | <math>{^{TL}\delta \dot{f}^{m}}=\int _{\Omega _0} B_0^T \left[C \right]B_0 d\Omega _0 \delta \bar{v}   </math>
714
|}
715
| style="width: 5px;text-align: right;" | (34)
716
|}
717
718
In order to obtain the increment of the internal forces, the material time derivative of  the internal forces increment is integrated over a time step increment <math display="inline">\Delta t</math> as
719
720
<span id="eq-35"></span>
721
{| class="formulaSCP" style="width: 100%; text-align: left;" 
722
|-
723
| 
724
{| style="text-align: left; margin:auto;" 
725
|-
726
| style="text-align: center;" | <math>{^{TL}\delta {f}^{m}}= {^{TL}\delta \dot{f}^{m}} \Delta t  </math>
727
|}
728
| style="width: 5px;text-align: right;" | (35)
729
|}
730
731
From Eq.([[#eq-35|35]]) and Eq.([[#eq-33|33]]), yields
732
733
<span id="eq-36"></span>
734
{| class="formulaSCP" style="width: 100%; text-align: left;" 
735
|-
736
| 
737
{| style="text-align: left; margin:auto;" 
738
|-
739
| style="text-align: center;" | <math>{^{TL}\delta{f}^{m}_{Ii}}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}  F_{ir} \Delta t C_{jrkl}  \frac{\partial N_J}{\partial X_l}  F_{sk}  d\Omega _0  \delta \bar   v_{Js}  </math>
740
|}
741
| style="width: 5px;text-align: right;" | (36)
742
|}
743
744
and, similarly,  from Eq.([[#eq-35|35]]) and Eq.([[#eq-34|34]])
745
746
<span id="eq-37"></span>
747
{| class="formulaSCP" style="width: 100%; text-align: left;" 
748
|-
749
| 
750
{| style="text-align: left; margin:auto;" 
751
|-
752
| style="text-align: center;" | <math>{^{TL}\delta {f}^{m}}=\int _{\Omega _0} B_0^T \Delta t\left[C \right]B_0 d\Omega _0 \delta \bar{v}   </math>
753
|}
754
| style="width: 5px;text-align: right;" | (37)
755
|}
756
757
From Eq.([[#eq-36|36]]) and Eq.([[#eq-37|37]]), the material tangent matrix in the TL description can be computed as
758
759
<span id="eq-38"></span>
760
{| class="formulaSCP" style="width: 100%; text-align: left;" 
761
|-
762
| 
763
{| style="text-align: left; margin:auto;" 
764
|-
765
| style="text-align: center;" | <math>^{TL}{K}^{m}_{IJis}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}  F_{rj} \Delta t C_{ijkl}  \frac{\partial N_J}{\partial X_s}  F_{kl}  d\Omega  </math>
766
|}
767
| style="width: 5px;text-align: right;" | (38)
768
|}
769
770
<span id="eq-39"></span>
771
{| class="formulaSCP" style="width: 100%; text-align: left;" 
772
|-
773
| 
774
{| style="text-align: left; margin:auto;" 
775
|-
776
| style="text-align: center;" | <math>{^{TL}{K}^{m}}=\int _{\Omega _0} B_0^T \Delta t\left[C \right]B_0 d\Omega _0   </math>
777
|}
778
| style="width: 5px;text-align: right;" | (39)
779
|}
780
781
The material tangent matrix for the UL framework is obtained by applying a push-forward transformation on each term of Eq.([[#eq-38|38]]) and integrating over the updated domain <math display="inline">\Omega </math>. The following relations hold
782
783
<span id="eq-40"></span>
784
{| class="formulaSCP" style="width: 100%; text-align: left;" 
785
|-
786
| 
787
{| style="text-align: left; margin:auto;" 
788
|-
789
| style="text-align: center;" | <math>d{\Omega _0}=\frac{ d \Omega }{J}   </math>
790
|}
791
| style="width: 5px;text-align: right;" | (40)
792
|}
793
794
<span id="eq-41"></span>
795
{| class="formulaSCP" style="width: 100%; text-align: left;" 
796
|-
797
| 
798
{| style="text-align: left; margin:auto;" 
799
|-
800
| style="text-align: center;" | <math>\frac{\partial N_I}{\partial X_j}=\frac{\partial N_I}{\partial x_k} F_{kj}  </math>
801
|}
802
| style="width: 5px;text-align: right;" | (41)
803
|}
804
805
<span id="eq-42"></span>
806
{| class="formulaSCP" style="width: 100%; text-align: left;" 
807
|-
808
| 
809
{| style="text-align: left; margin:auto;" 
810
|-
811
| style="text-align: center;" | <math>C_{ijkl}=F^{-1}_{mi} F^{-1}_{nj} F^{-1}_{ok} F^{-1}_{pl} c^{\tau }_{mnop} = F^{-1}_{mi} F^{-1}_{nj} F^{-1}_{ok} F^{-1}_{pl} c^{\sigma }_{mnop} J  </math>
812
|}
813
| style="width: 5px;text-align: right;" | (42)
814
|}
815
816
where <math display="inline">{c^{\tau }}</math> is the tangent moduli corresponding to the material time derivative of the Kirchhoff stress tensor <math display="inline">{\tau ^{\bigtriangledown }}</math> and <math display="inline">{c^{ \sigma }}</math> is the tangent moduli for the rate of the Cauchy stress <math display="inline">{\sigma ^{\bigtriangledown }}</math>. The rate of the Cauchy stress tensor is related to the rate of deformation <math display="inline">{d}</math> through the fourth-order tensor <math display="inline">{c^{\sigma }}</math> by the following expression
817
818
<span id="eq-43"></span>
819
{| class="formulaSCP" style="width: 100%; text-align: left;" 
820
|-
821
| 
822
{| style="text-align: left; margin:auto;" 
823
|-
824
| style="text-align: center;" | <math>\sigma ^{\bigtriangledown }=c^{\sigma } : d  </math>
825
|}
826
| style="width: 5px;text-align: right;" | (43)
827
|}
828
829
Substituting Eqs.([[#eq-40|40]]-[[#eq-42|42]]) into ([[#eq-38|38]]) and using the minor symmetries,  the material tangent matrix  for the UL reads
830
831
<span id="eq-44"></span>
832
{| class="formulaSCP" style="width: 100%; text-align: left;" 
833
|-
834
| 
835
{| style="text-align: left; margin:auto;" 
836
|-
837
| style="text-align: center;" | <math>K^{m}_{IJrs}=\int _{\Omega } \frac{\partial N_I}{\partial x_k} \delta _{ri} \Delta t c^{\sigma }_{kijl}  \frac{\partial N_J}{\partial x_l} \delta _{sj}    d\Omega  </math>
838
|}
839
| style="width: 5px;text-align: right;" | (44)
840
|}
841
842
Using Voigt notation, the same matrix reads
843
844
<span id="eq-45"></span>
845
{| class="formulaSCP" style="width: 100%; text-align: left;" 
846
|-
847
| 
848
{| style="text-align: left; margin:auto;" 
849
|-
850
| style="text-align: center;" | <math>K^{m}_{IJ}=\int _{\Omega ^e} B^{T}_I  \Delta t \left[c^{\sigma } \right] B_J    d\Omega   </math>
851
|}
852
| style="width: 5px;text-align: right;" | (45)
853
|}
854
855
For the node <math display="inline">I</math> of a 3D element,  matrix <math display="inline">B</math> is
856
857
<span id="eq-51"></span>
858
{| class="formulaSCP" style="width: 100%; text-align: left;" 
859
|-
860
| 
861
{| style="text-align: left; margin:auto;" 
862
|-
863
| style="text-align: right;" | <math>\displaystyle{B}_I = \left[\begin{matrix}\displaystyle {\partial N_I \over \partial x} &0&0\\ \displaystyle{0}& \displaystyle {\partial N_I \over \partial y}&0\\ \displaystyle{0}&0&\displaystyle {\partial N_I \over \partial z}\\ \displaystyle {\partial N_I \over \partial y}&\displaystyle {\partial N_I \over \partial x}&0\\[.25cm] \displaystyle {\partial N_I \over \partial z}&0&\displaystyle {\partial N_I \over \partial x}\\[.25cm] \displaystyle{0}&\displaystyle {\partial N_I \over \partial z}&\displaystyle {\partial N_I \over \partial y}          \end{matrix}  \right]  </math>
864
|}
865
| style="width: 5px;text-align: right;" | (51)
866
|}
867
868
869
The geometric tangent matrix for the UL framework is  derived next using the same procedure adopted for the material components. Hence, first the linearization is performed using the TL form and then the UL tangent matrix is obtained by performing the required transformation over the TL terms.
870
871
From Eq.([[#eq-21|21]])
872
873
<span id="eq-52"></span>
874
{| class="formulaSCP" style="width: 100%; text-align: left;" 
875
|-
876
| 
877
{| style="text-align: left; margin:auto;" 
878
|-
879
| style="text-align: center;" | <math>{^{TL}\delta \dot{f}^{g}_{Ii}}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j}  S_{ir}  \delta \dot{F}^T_{rj} d\Omega _0  </math>
880
|}
881
| style="width: 5px;text-align: right;" | (52)
882
|}
883
884
where the rate of the deformation gradient is defined as
885
886
<span id="eq-53"></span>
887
{| class="formulaSCP" style="width: 100%; text-align: left;" 
888
|-
889
| 
890
{| style="text-align: left; margin:auto;" 
891
|-
892
| style="text-align: center;" | <math>\dot{F}_{ij}= \frac{\partial N_J}{\partial X_i}  \bar v_{Jj}  </math>
893
|}
894
| style="width: 5px;text-align: right;" | (53)
895
|}
896
897
Substituting Eq.([[#eq-53|53]]) into Eq.([[#eq-52|52]]), the geometric components of the internal power in the TL description can be written as
898
899
<span id="eq-54"></span>
900
{| class="formulaSCP" style="width: 100%; text-align: left;" 
901
|-
902
| 
903
{| style="text-align: left; margin:auto;" 
904
|-
905
| style="text-align: center;" | <math>{^{TL}\delta \dot{f}^{g}_{Ii}}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} S_{ir} \frac{\partial N_J}{\partial X_r}   d\Omega _0  \delta \bar  v_{Jj}  </math>
906
|}
907
| style="width: 5px;text-align: right;" | (54)
908
|}
909
910
Integrating Eq.([[#eq-54|54]]) on time for a time step increment <math display="inline">\Delta t</math> yields
911
912
<span id="eq-55"></span>
913
{| class="formulaSCP" style="width: 100%; text-align: left;" 
914
|-
915
| 
916
{| style="text-align: left; margin:auto;" 
917
|-
918
| style="text-align: center;" | <math>{^{TL}\delta {f}^{g}_{Ii}}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} \Delta t S_{ir} \frac{\partial N_J}{\partial X_r}   d\Omega  \delta \bar  v_{Jj}  </math>
919
|}
920
| style="width: 5px;text-align: right;" | (55)
921
|}
922
923
From  Eq.([[#eq-55|55]]) the geometric tangent matrix is obtained as
924
925
<span id="eq-56"></span>
926
{| class="formulaSCP" style="width: 100%; text-align: left;" 
927
|-
928
| 
929
{| style="text-align: left; margin:auto;" 
930
|-
931
| style="text-align: center;" | <math>^{TL}K^{g}_{IJij}=\int _{\Omega _0} \frac{\partial N_I}{\partial X_j} \Delta t   S_{ir} \frac{\partial N_J}{\partial X_r} d\Omega _0  </math>
932
|}
933
| style="width: 5px;text-align: right;" | (56)
934
|}
935
936
In order to recover the UL form, the Piola identity has to be recalled, <math display="inline">{}</math>
937
938
<span id="eq-57"></span>
939
{| class="formulaSCP" style="width: 100%; text-align: left;" 
940
|-
941
| 
942
{| style="text-align: left; margin:auto;" 
943
|-
944
| style="text-align: center;" | <math>{S}= {F}^{-1} {\sigma } {F}^{-T} J  </math>
945
|}
946
| style="width: 5px;text-align: right;" | (57)
947
|}
948
949
The geometric tangent matrix in the UL framework is obtained by substituting Eqs.([[#eq-40|40]]), ([[#eq-41|41]]) and ([[#eq-57|57]]) into ([[#eq-56|56]]) and using the symmetries. This leads to
950
951
<span id="eq-58"></span>
952
{| class="formulaSCP" style="width: 100%; text-align: left;" 
953
|-
954
| 
955
{| style="text-align: left; margin:auto;" 
956
|-
957
| style="text-align: center;" | <math>K^{g}_{IJrs}=\int _{\Omega } \frac{\partial N_I}{\partial x_j} \Delta t  {\sigma }_{jk} \frac{\partial N_J}{\partial x_k} d\Omega \delta _{rs}  </math>
958
|}
959
| style="width: 5px;text-align: right;" | (58)
960
|}
961
962
or also
963
964
<span id="eq-59"></span>
965
{| class="formulaSCP" style="width: 100%; text-align: left;" 
966
|-
967
| 
968
{| style="text-align: left; margin:auto;" 
969
|-
970
| style="text-align: center;" | <math>K^{g}_{IJ}=I\int _{\Omega } \beta ^{T}_I \Delta t \sigma   \beta _J    d\Omega   </math>
971
|}
972
| style="width: 5px;text-align: right;" | (59)
973
|}
974
975
where <math display="inline">I</math> is the second order identity tensor  and matrix <math display="inline">\beta </math> for 3D problems is
976
977
<span id="eq-60"></span>
978
{| class="formulaSCP" style="width: 100%; text-align: left;" 
979
|-
980
| 
981
{| style="text-align: left; margin:auto;" 
982
|-
983
| style="text-align: right;" | <math>\displaystyle {\beta }_I = \left[\begin{matrix}\displaystyle {\partial N_I \over \partial x} &  \displaystyle {\partial N_I \over \partial y} &  \displaystyle {\partial N_I \over \partial z} \end{matrix}  \right]^T  </math>
984
|}
985
| style="width: 5px;text-align: right;" | (60)
986
|}
987
988
''Dynamic component of the tangent matrix''
989
990
The dynamic component of the tangent matrix in the UL description can be derived directly from the UL dynamic term <math display="inline">{{f}^{dyn}_{Ii}}</math> of Eq.([[#eq-10|10]]). This reads
991
992
<span id="eq-61"></span>
993
{| class="formulaSCP" style="width: 100%; text-align: left;" 
994
|-
995
| 
996
{| style="text-align: left; margin:auto;" 
997
|-
998
| style="text-align: center;" | <math>f^{dyn}_{Ii}= \int _\Omega N_I \rho d\Omega \dot{v}_i   </math>
999
|}
1000
| style="width: 5px;text-align: right;" | (61)
1001
|}
1002
1003
Eq.([[#eq-61|61]]) has to be discretized in time with the purpose of replacing the accelerations by the velocities using the time integration scheme described in Eq.([[#eq-14|14]]).
1004
1005
Introducing Eq.([[#eq-14|14]]) into ([[#eq-61|61]]) and differentiating for the increment of velocities, the dynamic components of the tangent matrix are obtained as
1006
1007
<span id="eq-62"></span>
1008
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1009
|-
1010
| 
1011
{| style="text-align: left; margin:auto;" 
1012
|-
1013
| style="text-align: center;" | <math>K^{\rho }_{IJij}= \delta _{ij}\int _\Omega N_I \frac{2\rho }{\Delta t}  N_J d\Omega   </math>
1014
|}
1015
| style="width: 5px;text-align: right;" | (62)
1016
|}
1017
1018
Or also
1019
1020
<span id="eq-63"></span>
1021
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1022
|-
1023
| 
1024
{| style="text-align: left; margin:auto;" 
1025
|-
1026
| style="text-align: center;" | <math>K^{\rho }_{IJ}= I\int _{\Omega } N_I \frac{2\rho }{\Delta t}  N_J d\Omega   </math>
1027
|}
1028
| style="width: 5px;text-align: right;" | (63)
1029
|}
1030
1031
===2.1.4 Incremental solution scheme===
1032
1033
The problem is solved through an implicit iterative scheme. At each iteration <math display="inline">{i}</math> the velocity increments are computed as
1034
1035
<span id="eq-64"></span>
1036
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1037
|-
1038
| 
1039
{| style="text-align: left; margin:auto;" 
1040
|-
1041
| style="text-align: center;" | <math>K^{i} \Delta  \bar v=R^i( {^{n+1}{\bar v}^{i}}, {^{n+1}\sigma ^i})   </math>
1042
|}
1043
| style="width: 5px;text-align: right;" | (64)
1044
|}
1045
1046
where <math display="inline">K</math> is the tangent matrix  computed as the sum of the internal, the geometric and the dynamic components, given by Eqs. ([[#eq-45|45]]), ([[#eq-59|59]]) and  ([[#eq-63|63]]) respectively, as
1047
1048
<span id="eq-65"></span>
1049
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1050
|-
1051
| 
1052
{| style="text-align: left; margin:auto;" 
1053
|-
1054
| style="text-align: center;" | <math>K= K^{m}+ K^{g} +K^{\rho }   </math>
1055
|}
1056
| style="width: 5px;text-align: right;" | (65)
1057
|}
1058
1059
<math display="inline">R</math> is the residual and it is computed from Eq.([[#eq-10|10]]) as
1060
1061
<span id="eq-66"></span>
1062
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1063
|-
1064
| 
1065
{| style="text-align: left; margin:auto;" 
1066
|-
1067
| style="text-align: center;" | <math>{^{n+1} R_{Ii}}= \int _\Omega N_I \rho  N_J d\Omega  {^{n+1}\bar{\dot{v}}_{Ji}}+  \int _\Omega \frac{\partial N_I}{\partial x_j} {^{n+1}\sigma _{ij}} d\Omega  -  \int _\Omega N_I {^{n+1}b}_i d\Omega + </math>
1068
|-
1069
| style="text-align: center;" | <math> - \int _{\Gamma _t} N_I {^{n+1}t}_i^{p} d\Gamma   </math>
1070
|}
1071
| style="width: 5px;text-align: right;" | (66)
1072
|}
1073
1074
In Eq.([[#eq-66|66]]) <math display="inline">{^{n+1}R_{Ii}}</math> is the residual of the momentum equations referred to node <math display="inline">{I}</math> and the cartesian direction <math display="inline">{i}</math>. Note that the Cauchy stress tensor still appears in its 'continuum' form because up to now it has not been written as a function of the nodal unknowns. This is done for keeping the generality of the formulation. Only after the introduction of the constitutive laws, it will be possible to compute the Cauchy stress tensor as a function of the nodal unknowns.
1075
1076
In Box 1 the iterative solution incremental scheme of the velocity formulation for a generic time interval <math display="inline">{t[{^nt},{^{n+1}t}]}</math> of duration <math display="inline">{\Delta t}</math> is described.
1077
1078
<div class="center" style="font-size: 85%;">
1079
[[File:Draft_Samper_722607179_7515_Box1.png|600px]]
1080
1081
'''Box 1'''. Iterative incremental solution scheme for the velocity formulation.</div>
1082
1083
==2.2 Mixed velocity-pressure formulation==
1084
1085
In this work, the mixed Velocity-Pressure formulation is derived as an extension of the Velocity formulation presented in the previous section. The governing equations are the linear momentum equations and the linear relation between the time variation of pressure  and the volumetric strain rate. The latter represents a limit for the generality of this formulation because only materials which constitutive law satisfies this relation can be modeled through the mixed Velocity-Pressure formulation.
1086
1087
The problem is solved using a two-step Gauss-Seidel partitioned iterative scheme. First the momentum equations are solved in terms of velocity increments and including the (known) pressures at the previous iteration in the residual expression. Then the continuity equation is solved for the pressure using the updated velocities computed from the momentum equations.  It will be shown that using this not intrusive scheme, it is possible to take advantage of most of the velocity formulation derived in the previous section.  In particular, the incremental velocity scheme for the momentum equations (Box 1) and the structure of the tangent matrix  ([[#eq-65|65]]) hold also for the mixed Velocity-Pressure formulation. The same linear interpolation has been used for the velocity and the pressure fields. It is well known that, for incompressible (or quasi-incompressible) problems, this combination does not fulfill the <math display="inline">{inf-sup}</math> condition [15] and a stabilization method is required.  However, as mentioned in the section devoted to the velocity formulation, the aim of this part is to keep the formulation as general as possible without referring to a specific material.  Hence only in the next chapter, when the mixed Velocity-Pressure formulation is used for solving quasi-incompressible problems, the required stabilization will be introduced in the scheme.
1088
1089
===2.2.1 Quasi-incompressible form of the continuity equation===
1090
1091
Mixed formulations are often used for dealing with incompressible materials. In these problems it is useful to write the stress and the strain measures as the sum of deviatoric and hydrostatic, or volumetric, parts. Hence the Cauchy stress tensor is decomposed as
1092
1093
<span id="eq-67"></span>
1094
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1095
|-
1096
| 
1097
{| style="text-align: left; margin:auto;" 
1098
|-
1099
| style="text-align: center;" | <math>\sigma _{ij}=\sigma '_{ij}+\sigma ^{h}\delta _{ij}  </math>
1100
|}
1101
| style="width: 5px;text-align: right;" | (67)
1102
|}
1103
1104
with
1105
1106
<span id="eq-68"></span>
1107
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1108
|-
1109
| 
1110
{| style="text-align: left; margin:auto;" 
1111
|-
1112
| style="text-align: center;" | <math>\sigma ^{h}=\frac{\sigma _{kk}}{3}  </math>
1113
|}
1114
| style="width: 5px;text-align: right;" | (68)
1115
|}
1116
1117
where <math display="inline">{\sigma '}</math> and <math display="inline">{\sigma }^h I</math> are the deviatoric and the hydrostatic parts of the Cauchy stress tensor, respectively. The pressure <math display="inline">p</math> is defined positive in the tensile state  and equal to the hydrostatic parts of the Cauchy stress tensor <math display="inline">{\sigma ^h}</math>. Hence
1118
1119
<span id="eq-69"></span>
1120
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1121
|-
1122
| 
1123
{| style="text-align: left; margin:auto;" 
1124
|-
1125
| style="text-align: center;" | <math>p:=\sigma ^{h}  </math>
1126
|}
1127
| style="width: 5px;text-align: right;" | (69)
1128
|}
1129
1130
The Cauchy stress tensor can be computed as
1131
1132
<span id="eq-70"></span>
1133
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1134
|-
1135
| 
1136
{| style="text-align: left; margin:auto;" 
1137
|-
1138
| style="text-align: center;" | <math>\sigma _{ij}=\sigma '_{ij}+p\delta _{ij}  </math>
1139
|}
1140
| style="width: 5px;text-align: right;" | (70)
1141
|}
1142
1143
The same decomposition is done for the spatial strain rate tensor <math display="inline">d</math>. So
1144
1145
<span id="eq-71"></span>
1146
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1147
|-
1148
| 
1149
{| style="text-align: left; margin:auto;" 
1150
|-
1151
| style="text-align: center;" | <math>d_{ij}=d'_{ij}+d^{h}\delta _{ij}  </math>
1152
|}
1153
| style="width: 5px;text-align: right;" | (71)
1154
|}
1155
1156
with
1157
1158
<span id="eq-72"></span>
1159
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1160
|-
1161
| 
1162
{| style="text-align: left; margin:auto;" 
1163
|-
1164
| style="text-align: center;" | <math>d^{h}=\frac{d_{kk}}{3}  </math>
1165
|}
1166
| style="width: 5px;text-align: right;" | (72)
1167
|}
1168
1169
where <math display="inline">{d'}</math> and <math display="inline">{d}^h I</math> are the deviatoric and the hydrostatic parts of the strain rate tensor, respectively. The strain rate tensor is computed from the velocities as
1170
1171
<span id="eq-73"></span>
1172
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1173
|-
1174
| 
1175
{| style="text-align: left; margin:auto;" 
1176
|-
1177
| style="text-align: center;" | <math>d_{ij}=\frac{1}{2}\left({\partial v_i \over \partial x_j} +{\partial v_j \over \partial x_i} \right)  </math>
1178
|}
1179
| style="width: 5px;text-align: right;" | (73)
1180
|}
1181
1182
The volumetric strain rate is defined from Eqs.([[#eq-68|68]]) and ([[#eq-73|73]]) as
1183
1184
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1185
|-
1186
| 
1187
{| style="text-align: left; margin:auto;" 
1188
|-
1189
| style="text-align: center;" | <math>{d}^{v}= {d}_{kk}={\partial v_k \over \partial x_k} </math>
1190
|}
1191
| style="width: 5px;text-align: right;" | (74)
1192
|}
1193
1194
The closure equation for the mixed Velocity-Pressure formulation is the linear relation between the change in time of the pressure and the volumetric strain rate. This reads as
1195
1196
<span id="eq-75"></span>
1197
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1198
|-
1199
| 
1200
{| style="text-align: left; margin:auto;" 
1201
|-
1202
| style="text-align: center;" | <math>\frac{1}{\kappa }\frac{\partial p}{\partial t}=  {d}^{v}  </math>
1203
|}
1204
| style="width: 5px;text-align: right;" | (75)
1205
|}
1206
1207
where <math display="inline">{\kappa }</math> is a parameter that depends on the constitutive equation. Typically <math display="inline">{\kappa }</math> is the bulk modulus of the material.
1208
1209
In conclusion, the local form of the whole problem for the mixed Velocity-Pressure formulation is formed by the linear momentum equations (Eq.([[#eq-1|1]])) and the pressure-strain rate relation given by Eq.([[#eq-75|75]]). The linear momentum equations have been already discretized and linearized for the increments of velocities in the previous section devoted to the Velocity formulation. That form holds also for the mixed formulation. So, in the following, only the discretization of  Eq.([[#eq-75|75]]) is given. This equation is a restriction on the generality of the constitutive laws that can be analyzed with the mixed Velocity-Pressure formulation. It will be shown that the constitutive models for hypoelastic solids and quasi-incompressible Newtonian fluids fulfill this relation. In fluid dynamics, Eq.([[#eq-75|75]]) represents the <math display="inline">continuity</math>, or <math display="inline">mass balance</math>, equation for quasi-incompressible fluids. In fact,  Eq.([[#eq-75|75]]) with <math display="inline">\kappa =\infty </math> is the canonical form of the continuity equation of the Navier-Stokes problem. For this reason, from here on Eq.([[#eq-75|75]]) will be called 'continuity equation'.
1210
1211
Multiplying Eq.([[#eq-75|75]]) by arbitrary test functions <math display="inline">q</math> (with dimensions of pressure), integrating over the analysis domain <math display="inline">\Omega </math> and bringing all the terms at the left hand side gives
1212
1213
<span id="eq-76"></span>
1214
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1215
|-
1216
| 
1217
{| style="text-align: left; margin:auto;" 
1218
|-
1219
| style="text-align: center;" | <math>\int _\Omega \frac{q}{\kappa } {\partial p \over \partial t}d\Omega -  \int _\Omega q  d^v d\Omega  =0  </math>
1220
|}
1221
| style="width: 5px;text-align: right;" | (76)
1222
|}
1223
1224
Both the trial and the test functions for the pressure are interpolated in space using the same shape functions <math display="inline">{N}</math>.
1225
1226
<span id="eq-77"></span>
1227
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1228
|-
1229
| 
1230
{| style="text-align: left; margin:auto;" 
1231
|-
1232
| style="text-align: center;" | <math>p = \sum \limits _{I=1}^n N_I \bar p_{I} \quad ,\quad q = \sum \limits _{I=1}^n N_I \bar q_{I}   </math>
1233
|}
1234
| style="width: 5px;text-align: right;" | (77)
1235
|}
1236
1237
where <math display="inline">n=3/4</math> for 2D/3D problems is the number of the nodes of the simplex. In this work,  linear shape functions have been used for <math display="inline">N_I </math>, as for the velocity.
1238
1239
Combining Eq.([[#eq-77|77]]) with Eq.([[#eq-76|76]]) and solving for all the admissible test functions q, yields
1240
1241
<span id="eq-78"></span>
1242
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1243
|-
1244
| 
1245
{| style="text-align: left; margin:auto;" 
1246
|-
1247
| style="text-align: center;" | <math>\int _\Omega {N_I}\frac{1}{\kappa } {N_J}d\Omega  \dot{\bar {p_J}}  - \int _\Omega {N_I}  {\partial N_J \over \partial x_i} d\Omega  \bar {v}_{iJ} =0  </math>
1248
|}
1249
| style="width: 5px;text-align: right;" | (78)
1250
|}
1251
1252
Regarding the time integration a first order scheme has been adopted for the pressure.  Hence, for a time interval <math display="inline">{t[{^nt},{^{n+1}t}]}</math> of duration <math display="inline">{\Delta t}</math> the first and the second variations in time of the pressure are computed as
1253
1254
<span id="eq-79"></span>
1255
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1256
|-
1257
| 
1258
{| style="text-align: left; margin:auto;" 
1259
|-
1260
| style="text-align: center;" | <math>{^{n+1}\dot{p}}= \frac{ {^{n+1}{p}} - {^{n}p} }{\Delta t}  </math>
1261
|}
1262
| style="width: 5px;text-align: right;" | (79)
1263
|}
1264
1265
<span id="eq-80"></span>
1266
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1267
|-
1268
| 
1269
{| style="text-align: left; margin:auto;" 
1270
|-
1271
| style="text-align: center;" | <math>{^{n+1}\ddot{p}}= \frac{^{n+1} {p} - {^{n}p} }{{\Delta t}^2}-\frac{^{n}\dot{p} }{\Delta t}   </math>
1272
|}
1273
| style="width: 5px;text-align: right;" | (80)
1274
|}
1275
1276
Introducing  Eq.([[#eq-79|79]]) in Eq.([[#eq-78|78]]), the discretized form of the continuity equation  is
1277
1278
<span id="eq-81"></span>
1279
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1280
|-
1281
| 
1282
{| style="text-align: left; margin:auto;" 
1283
|-
1284
| style="text-align: right;" | <math>-Q^T {^{n+1}{\bar{v} }} + \frac{1}{\Delta t}M_1 {^{n+1}{\bar{p} }} </math>
1285
| <math>= {^{n}g}   </math>
1286
|}
1287
| style="width: 5px;text-align: right;" | (81)
1288
|}
1289
1290
where
1291
1292
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1293
|-
1294
| 
1295
{| style="text-align: left; margin:auto;" 
1296
|-
1297
| style="text-align: center;" | <math>{M}_{1_{IJ}} =\int _{\Omega ^e} N_I \frac{1}{\kappa } N_J d\Omega    </math>
1298
|}
1299
| style="width: 5px;text-align: right;" | (82)
1300
|}
1301
1302
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1303
|-
1304
| 
1305
{| style="text-align: left; margin:auto;" 
1306
|-
1307
| style="text-align: center;" | <math>Q_{IJ}  =\int _{\Omega ^e} B_I^T m N_J d\Omega </math>
1308
|-
1309
| style="text-align: center;" | 
1310
|}
1311
| style="width: 5px;text-align: right;" | (83)
1312
|}
1313
1314
<span id="eq-84"></span>
1315
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1316
|-
1317
| 
1318
{| style="text-align: left; margin:auto;" 
1319
|-
1320
| style="text-align: center;" | <math>{^{n}g_I}=\int _{\Omega ^n}  N_I \frac{1}{\kappa \Delta t}  N_J d{\Omega } {^{n}\bar p}_J   </math>
1321
|}
1322
| style="width: 5px;text-align: right;" | (84)
1323
|}
1324
1325
where <math display="inline">B</math> has been defined in Eq.([[#eq-51|51]]) and <math display="inline">{m}</math><math display="inline">{=[1,1,1,0,0,0]^T}</math>.
1326
1327
===2.2.2 Solution method===
1328
1329
In the mixed Velocity-Pressure formulation the problem is solved through a partitioned iterative scheme. Specifically, the linear momentum equations are solved for the velocity  increments as in the Velocity formulation. On the other hand, the continuity equation is solved for the pressure in the updated configuration using the velocity field computed with the linear momentum equations. In order to guarantee the coupling between the continuity equation and the linear momentum equations (or equally between the pressure and the velocities) the pressure must appear in the right hand side of the linear momentum equations. For this purpose the Cauchy stress tensor must be computed as the sum of its deviatoric part and the pressure, as Eq.([[#eq-70|70]]). Otherwise, the Velocity-Pressure formulation would be uncoupled and totally equivalent to the Velocity formulation.
1330
1331
In conclusion, for a general time interval <math display="inline">{t[{^nt},{^{n+1}t}]}</math> of duration <math display="inline">{\Delta t}</math> the following linear systems are solved for each iteration <math display="inline">{i}</math>
1332
1333
<span id="eq-85"></span>
1334
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1335
|-
1336
| 
1337
{| style="text-align: left; margin:auto;" 
1338
|-
1339
| style="text-align: center;" | <math>K^{i} \Delta  \bar v=R^i( {^{n+1}{\bar v}^{i}}, {^{n+1}\sigma ' ^i}, {^{n+1}p^i})   </math>
1340
|}
1341
| style="width: 5px;text-align: right;" | (85)
1342
|}
1343
1344
<span id="eq-86"></span>
1345
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1346
|-
1347
| 
1348
{| style="text-align: left; margin:auto;" 
1349
|-
1350
| style="text-align: center;" | <math>\frac{1}{ \Delta t} M_{1}  {^{n+1}\bar p^{i+1}}= {Q}^T{^{n+1}\bar{v}^{i+1}}+ {^ng} {^n\bar p}^{i}   </math>
1351
|}
1352
| style="width: 5px;text-align: right;" | (86)
1353
|}
1354
1355
where <math display="inline">K</math> is the same tangent matrix as for the Velocity formulation (Eq.([[#eq-65|65]])) and the residual <math display="inline">R</math> is computed using the pressure of the previous iteration and the deviatoric part of the Cauchy stress as 
1356
1357
<span id="eq-87"></span>
1358
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1359
|-
1360
| 
1361
{| style="text-align: left; margin:auto;" 
1362
|-
1363
| style="text-align: center;" | <math>{^{n+1}R_{Ii}}= \int _\Omega N_I \rho  N_J d\Omega  {^{n+1}\bar{\dot{v}}_{Ji}}+  \int _\Omega \frac{\partial N_I}{\partial x_j} {^{n+1}\sigma '}_{ij} d\Omega + </math>
1364
|-
1365
| style="text-align: center;" | <math>+ \int _\Omega \frac{\partial N_I}{\partial x_j}\delta _{ij} N_J d\Omega  {^{n+1} \bar  p}_J -  \int _\Omega {N}_I {^{n+1}b}_i d\Omega - \int _{\Gamma _t} N_I {^{n+1}t_i^{p}} d\Gamma  </math>
1366
|}
1367
| style="width: 5px;text-align: right;" | (87)
1368
|}
1369
1370
In Box 2, the iterative incremental solution scheme for a generic continuum via the mixed velocity-pressure formulation  is shown for a time interval <math display="inline">{t[{^nt},{^{n+1}t}]}</math>.
1371
1372
<div class="center" style="font-size: 85%;">
1373
[[File:Draft_Samper_722607179_9439_Box2.png|600px]] 
1374
1375
'''Box 2'''.  Iterative solution scheme for a generic continuum using the mixed velocity-pressure
1376
formulation.</div>
1377
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==2.3 Hypoelasticity==
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1380
Using the definition of Truesdell  [118], a hypoelastic body is a material which may soften or harden in strain but in general has neither preferred state nor preferred stress. The hypoelastic laws were created with the purpose of transferring the linear theory of elasticity from the small to the finite strains regime [118]. In [120] a deep dissertation about the differences between elasticity and hypoelasticity is given.
1381
1382
A hypoelastic body is defined by the constitutive equation [119]
1383
1384
<span id="eq-88"></span>
1385
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1386
|-
1387
| 
1388
{| style="text-align: left; margin:auto;" 
1389
|-
1390
| style="text-align: center;" | <math>\mbox{rate of stress}= f(\hbox{rate of deformation})  </math>
1391
|}
1392
| style="width: 5px;text-align: right;" | (88)
1393
|}
1394
1395
In the  rate theory it is crucial to guarantee the <math display="inline">{objectivity}</math> and the <math display="inline">{frame-invariance}</math>, or <math display="inline">{frame-indifference}</math>, of the scheme. A material is frame invariant if its properties do not depend on the change of observer. An objective constitutive equation is defined to be invariant for all changes of the observer [56]. For guaranteeing the frame indifference, the constitutive law has to be isotropic [121]. This represents a constraint for hypoelastic models. This limitation is even more severe if also plasticity is included in the model. In fact, for hypoelastic-plastic materials, also the yield condition is required to be isotropic [112].
1396
1397
The stress rate cannot be computed simply as a material derivative because it leads to a non-objective measure of stress [9].  In particular, rigid rotations may originate a wrong state of stress if the stress rate is computed as the material time derivative of the Cauchy stress [9]. However, many objective measures of rate of stress are available. The most common ones are the Truesdell's and Jaumann's Cauchy stress rate measures. From here on, an objective rate measure will be denoted by the upper inverse  triangle index <math display="inline">{(\cdot ) ^{\bigtriangledown }}</math>.
1398
1399
Most of hypoelastic laws relate linearly the stress rate to the rate of deformation. Hence, Eq.([[#eq-88|88]]) is now rewritten in the following form
1400
1401
<span id="eq-89"></span>
1402
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1403
|-
1404
| 
1405
{| style="text-align: left; margin:auto;" 
1406
|-
1407
| style="text-align: center;" | <math>\sigma ^{\bigtriangledown }=c : d  </math>
1408
|}
1409
| style="width: 5px;text-align: right;" | (89)
1410
|}
1411
1412
where <math display="inline">{\sigma  ^{\bigtriangledown }}</math> is the Cauchy stress rate tensor, <math display="inline">{c}</math>  is the tangent moduli tensor and <math display="inline">{d}</math> is the deformation rate tensor.
1413
1414
From Eq.([[#eq-89|89]]) it can be deduced that this class of hypoelastic materials has a rate-independent and incrementally linear and reversible behavior. So, as for the elastic materials, in the small deformation regime, the strains and the stresses are totally recovered upon the unloading process. Nevertheless, for large deformations the hypoelastic laws do not guarantee that the work done in a closed deformation path is zero [102]. However this error can be considered negligible if the elastic deformations are small with respect to the total deformations [9]. For this reason the hypoelastic laws are often used for describing the elastic part of elastic-plastic materials: the plastic deformations in fact represent usually the largest part of the overall deformations.
1415
1416
The Jaumann measure for the rate of the Kirchhoff stress tensor <math display="inline">{\tau  ^{\bigtriangledown J}}</math>is
1417
1418
<span id="eq-90"></span>
1419
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1420
|-
1421
| 
1422
{| style="text-align: left; margin:auto;" 
1423
|-
1424
| style="text-align: center;" | <math>\tau  ^{\bigtriangledown J}=c^{\tau J} : d  </math>
1425
|}
1426
| style="width: 5px;text-align: right;" | (90)
1427
|}
1428
1429
where the tangent moduli fourth-order tensor <math display="inline">{c^{\tau J}}</math> for the Jaumann measure of the Kirchhoff stress rate   is
1430
1431
<span id="eq-91"></span>
1432
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1433
|-
1434
| 
1435
{| style="text-align: left; margin:auto;" 
1436
|-
1437
| style="text-align: center;" | <math>c^{\tau J}_{ijkl}= \lambda \delta _{ij} \delta _{kl} + \mu \left(\delta _{ik} \delta _{jl} +  \delta _{il} \delta _{kj} \right) \quad , \quad   c^{\tau J} =\lambda I \otimes I + 2 \mu \mathbf{I}    </math>
1438
|}
1439
| style="width: 5px;text-align: right;" | (91)
1440
|}
1441
1442
where <math display="inline">{\lambda }</math> and <math display="inline">{\mu }</math> are the Lamé constants and they are computed from the Young modulus <math display="inline">E</math> and the Poisson ratio <math display="inline">\nu </math> as
1443
1444
<span id="eq-92"></span>
1445
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1446
|-
1447
| 
1448
{| style="text-align: left; margin:auto;" 
1449
|-
1450
| style="text-align: center;" | <math>\mu =\frac{E}{2(1+\nu )}  </math>
1451
|}
1452
| style="width: 5px;text-align: right;" | (92)
1453
|}
1454
1455
<span id="eq-93"></span>
1456
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1457
|-
1458
| 
1459
{| style="text-align: left; margin:auto;" 
1460
|-
1461
| style="text-align: center;" | <math>\lambda =\frac{\nu E}{(1+\nu )(1-2\nu )}  </math>
1462
|}
1463
| style="width: 5px;text-align: right;" | (93)
1464
|}
1465
1466
and <math display="inline">\mathbf{I}</math> is the fouth-order symmetric identity tensor defined as
1467
1468
<span id="eq-94"></span>
1469
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1470
|-
1471
| 
1472
{| style="text-align: left; margin:auto;" 
1473
|-
1474
| style="text-align: center;" | <math>\mathrm{I}_{ijkl}=\frac{1}{2}\left(\delta _{ik} \delta _{jl} +  \delta _{il} \delta _{kj} \right)  </math>
1475
|}
1476
| style="width: 5px;text-align: right;" | (94)
1477
|}
1478
1479
Separating the volumetric from the deviatoric part, yields
1480
1481
<span id="eq-95"></span>
1482
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1483
|-
1484
| 
1485
{| style="text-align: left; margin:auto;" 
1486
|-
1487
| style="text-align: center;" | <math>c^{\tau J}_{ijkl}= \kappa \delta _{ij} \delta _{kl} + \mu \left(\delta _{ik} \delta _{jl} +  \delta _{il} \delta _{kj} - \frac{2}{3} \delta _{ij} \delta _{kl} \right) \quad , \quad   c^{\tau J} =\kappa I \otimes I + 2 \mu \mathbf{I}'    </math>
1488
|}
1489
| style="width: 5px;text-align: right;" | (95)
1490
|}
1491
1492
where <math display="inline">\kappa </math> is the bulk modulus and it is computed from the Lamé parameters as
1493
1494
<span id="eq-96"></span>
1495
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1496
|-
1497
| 
1498
{| style="text-align: left; margin:auto;" 
1499
|-
1500
| style="text-align: center;" | <math>\kappa =\lambda +\frac{2}{3}\mu  </math>
1501
|}
1502
| style="width: 5px;text-align: right;" | (96)
1503
|}
1504
1505
and <math display="inline">\mathbf{I}'</math> is the fouth-order tensor computed as
1506
1507
<span id="eq-97"></span>
1508
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1509
|-
1510
| 
1511
{| style="text-align: left; margin:auto;" 
1512
|-
1513
| style="text-align: center;" | <math>\mathbf{I}'=\mathbf{I}-\frac{1}{3} I \otimes I  </math>
1514
|}
1515
| style="width: 5px;text-align: right;" | (97)
1516
|}
1517
1518
The Jaumann stress rate measure is defined as
1519
1520
<span id="eq-98"></span>
1521
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1522
|-
1523
| 
1524
{| style="text-align: left; margin:auto;" 
1525
|-
1526
| style="text-align: center;" | <math>\sigma  ^{\bigtriangledown J}=c^{\sigma J} : d  </math>
1527
|}
1528
| style="width: 5px;text-align: right;" | (98)
1529
|}
1530
1531
where <math display="inline">{c^{\sigma J}}</math> is the Jaumann's tangent moduli tensor.
1532
1533
For a anisotropic material the tangent moduli for the Jaumann rate depends on the stress state and it is related to <math display="inline">{c^{\tau J}}</math> as follows
1534
1535
<span id="eq-99"></span>
1536
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1537
|-
1538
| 
1539
{| style="text-align: left; margin:auto;" 
1540
|-
1541
| style="text-align: center;" | <math>c^{\sigma J}_{ijkl}= \frac{c^{\tau J}_{ijkl}}{J} -\sigma _{il} \delta _{kj} \quad , \quad  {c}^{\sigma J} = \frac{c^{\tau J}}{J}- \sigma   \otimes I    </math>
1542
|}
1543
| style="width: 5px;text-align: right;" | (99)
1544
|}
1545
1546
Instead, for isotropic materials, the Jaumann's tangent moduli tensors for the Cauchy stress rate and for the Kirchhoff stress rate are identical [9]. So <math display="inline">{c^{\sigma J}}</math>  can be computed as
1547
1548
<span id="eq-100"></span>
1549
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1550
|-
1551
| 
1552
{| style="text-align: left; margin:auto;" 
1553
|-
1554
| style="text-align: center;" | <math>c^{\sigma J}_{ijkl}= \lambda \delta _{ij} \delta _{kl} + \mu \left(\delta _{ik} \delta _{jl} +  \delta _{il} \delta _{kj} \right) \quad , \quad   c^{\sigma J} =\lambda I \otimes I + 2 \mu \mathbf{I}    </math>
1555
|}
1556
| style="width: 5px;text-align: right;" | (100)
1557
|}
1558
1559
or equally
1560
1561
<span id="eq-101"></span>
1562
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1563
|-
1564
| 
1565
{| style="text-align: left; margin:auto;" 
1566
|-
1567
| style="text-align: center;" | <math>c^{\sigma J}_{ijkl}= \kappa \delta _{ij} \delta _{kl} + \mu \left(\delta _{ik} \delta _{jl} +  \delta _{il} \delta _{kj} - \frac{2}{3} \delta _{ij} \delta _{kl} \right) \quad , \quad   c^{\sigma J} =\kappa I \otimes I + 2 \mu \mathbf{I}'    </math>
1568
|}
1569
| style="width: 5px;text-align: right;" | (101)
1570
|}
1571
1572
For a 3D problem tensor <math display="inline">{c^{\sigma J}}</math> is <math display="inline">\begin{array}{l} \\ {c^{\sigma J}}=\left[                      \begin{array}{cccccc}                         \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0 \\                         \lambda & \lambda + 2\mu & \lambda &0  &0  & 0 \\                         \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\                         0& 0 &0  & \mu & 0 & 0 \\                        0& 0 & 0 & 0 & \mu & 0 \\                        0 & 0 & 0 & 0 & 0 & \mu \\                      \end{array} \right]\\ \\ \end{array}</math>
1573
1574
or, equally, <math display="inline">\begin{array}{l} \\ {c^{\sigma J}}=\left[                      \begin{array}{cccccc}                         \kappa + \frac{4}{3}\mu & \kappa -  \frac{2}{3}\mu & \kappa  -  \frac{2}{3}\mu & 0 & 0 & 0 \\[.1cm]                         \kappa  -  \frac{2}{3}\mu &\kappa{+} \frac{4}{3}\mu & \kappa  -  \frac{2}{3}\mu &0  &0  & 0 \\[.1cm]                         \kappa  -  \frac{2}{3}\mu & \kappa  -  \frac{2}{3}\mu & \kappa + \frac{4}{3}\mu & 0 & 0 & 0 \\[.1cm]                         0& 0 & 0 & \mu & 0 & 0 \\                        0 & 0 & 0 & 0 & \mu & 0 \\                        0 & 0 & 0 & 0 & 0 & \mu \\                      \end{array} \right] \\ \\ \end{array}</math>
1575
1576
A material is said to be isotropic if its behavior is uniform in all directions, so it has no preferred orientations or directions. Many materials, such as metals and ceramics, can be modeled as isotropic for small strains [9]. From the computational point of view, an isotropic constitutive law is much easier to manage than an anisotropic one and it has a lower   computational cost. Isotropic laws are preferred, for example, for their symmetry properties. In fact the anisotropic tangent moduli (Eq.([[#eq-99|99]])) is not symmetric while, the isotropic one (Eq.([[#eq-101|101]])) has both minor and major symmetries, in fact
1577
1578
<span id="eq-102"></span>
1579
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1580
|-
1581
| 
1582
{| style="text-align: left; margin:auto;" 
1583
|-
1584
| style="text-align: center;" | <math>minor ~symmetry \leftrightarrow c^{\sigma J}_{ijkl}= c^{\sigma J}_{jikl}= c^{\sigma J}_{ijlk}   </math>
1585
|}
1586
| style="width: 5px;text-align: right;" | (102)
1587
|}
1588
1589
<span id="eq-103"></span>
1590
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1591
|-
1592
| 
1593
{| style="text-align: left; margin:auto;" 
1594
|-
1595
| style="text-align: center;" | <math>major ~symmetry \leftrightarrow c^{\sigma J}_{ijkl}= c^{\sigma J}_{klij}   </math>
1596
|}
1597
| style="width: 5px;text-align: right;" | (103)
1598
|}
1599
1600
For all these reasons, in this work  the isotropic law has been used for the hypoelastic model.
1601
1602
The tangent moduli <math display="inline">{c^{\sigma J}}</math> will be introduced into the material part <math display="inline">K^m</math> of the global tangent matrix  Eq.([[#eq-45|45]]) for the Velocity and the mixed Velocity-Pressure formulations indifferently.
1603
1604
As it has already pointed out,  the Cauchy stress rate does not coincide with the material derivative of the Cauchy stress tensor. In fact, the following relation holds  between both measures
1605
1606
<span id="eq-104"></span>
1607
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1608
|-
1609
| 
1610
{| style="text-align: left; margin:auto;" 
1611
|-
1612
| style="text-align: center;" | <math>\dot{\sigma }=\sigma ^{\bigtriangledown J}+ \Omega  </math>
1613
|}
1614
| style="width: 5px;text-align: right;" | (104)
1615
|}
1616
1617
where <math display="inline">{\Omega }</math> is a tensor that accounts for the rotations and it is defined as
1618
1619
<span id="eq-105"></span>
1620
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1621
|-
1622
| 
1623
{| style="text-align: left; margin:auto;" 
1624
|-
1625
| style="text-align: center;" | <math>\Omega =W\cdot \sigma  +\sigma \cdot W^T    </math>
1626
|}
1627
| style="width: 5px;text-align: right;" | (105)
1628
|}
1629
1630
where <math display="inline">{ W}</math> is the spin tensor defined as
1631
1632
<span id="eq-106"></span>
1633
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1634
|-
1635
| 
1636
{| style="text-align: left; margin:auto;" 
1637
|-
1638
| style="text-align: center;" | <math>W_{ij}=\frac{1}{2} \left(L_{ij}-L_{ji}\right)= \frac{1}{2} \left(\frac{\partial v_i }{\partial x_j }-\frac{\partial v_j }{\partial x_i }\right) </math>
1639
|}
1640
| style="width: 5px;text-align: right;" | (106)
1641
|}
1642
1643
In this work the tensor <math display="inline">{\Omega }</math> is computed at the end of each time step. Discretizing in time Eq.([[#eq-104|104]])  for the time step interval <math display="inline">{t[{^n}t,{^{n+1}}t]}</math>  and expanding the Cauchy stress rate, yields
1644
1645
<span id="eq-107"></span>
1646
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1647
|-
1648
| 
1649
{| style="text-align: left; margin:auto;" 
1650
|-
1651
| style="text-align: center;" | <math>\frac{{^{n+1} \sigma } - {^{n} \sigma }}{\Delta t}= {c}^{\sigma J} : {^{n+1}d} +  {^n\Omega }  </math>
1652
|}
1653
| style="width: 5px;text-align: right;" | (107)
1654
|}
1655
1656
<math display="inline">{\Omega }</math> can be viewed as a correction of the Cauchy stress tensor. So it can be related to the Cauchy stress tensor of the previous time step as follows
1657
1658
<span id="eq-108"></span>
1659
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1660
|-
1661
| 
1662
{| style="text-align: left; margin:auto;" 
1663
|-
1664
| style="text-align: center;" | <math>^n \hat{\sigma }= { ^n \sigma }+  {^n\Omega }  </math>
1665
|}
1666
| style="width: 5px;text-align: right;" | (108)
1667
|}
1668
1669
Replacing Eq.([[#eq-108|108]]) in Eq.([[#eq-107|107]]), yields
1670
1671
<span id="eq-109"></span>
1672
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1673
|-
1674
| 
1675
{| style="text-align: left; margin:auto;" 
1676
|-
1677
| style="text-align: center;" | <math>\frac{    {^{n+1} \sigma  } - { ^{n}  \hat {\sigma }}}{\Delta t}= {c}^{\sigma J} : {^{n+1}d}   </math>
1678
|}
1679
| style="width: 5px;text-align: right;" | (109)
1680
|}
1681
1682
Substituting in Eq.([[#eq-109|109]]) the relation for <math display="inline">c^{\sigma J}</math> using Eq.([[#eq-101|101]]), yields
1683
1684
<span id="eq-110"></span>
1685
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1686
|-
1687
| 
1688
{| style="text-align: left; margin:auto;" 
1689
|-
1690
| style="text-align: center;" | <math>\frac{    {^{n+1} \sigma  } - { ^{n}  \hat {\sigma }}}{\Delta t} = \left(\kappa I \otimes I + 2 \mu \mathbf{I}' \right): {^{n+1}d}   </math>
1691
|}
1692
| style="width: 5px;text-align: right;" | (110)
1693
|}
1694
1695
Hence,
1696
1697
<span id="eq-111"></span>
1698
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1699
|-
1700
| 
1701
{| style="text-align: left; margin:auto;" 
1702
|-
1703
| style="text-align: center;" | <math>\frac{    {^{n+1} \sigma  } - { ^{n}  \hat {\sigma }}}{\Delta t} =  \underbrace{ \kappa \left(I \otimes I \right): {^{n+1}d}} _{ ^{n+1}\dot{p}} + \underbrace{ 2 \mu \mathbf{I'} :{^{n+1}d} } _{ ^{n+1} \dot{\sigma '}} </math>
1704
|}
1705
| style="width: 5px;text-align: right;" | (111)
1706
|}
1707
1708
The first and the second terms of the right hand side of Eq.([[#eq-111|111]]) represent the increment in the time step of the volumetric and deviatoric parts of the Cauchy stress tensor. From Eq.([[#eq-111|111]]) it can be deduced that for isotropic hypoelastic solids described using the Jaumann measure, the following relation holds
1709
1710
<span id="eq-112"></span>
1711
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1712
|-
1713
| 
1714
{| style="text-align: left; margin:auto;" 
1715
|-
1716
| style="text-align: center;" | <math>\dot{p}= \kappa {d}^v   </math>
1717
|}
1718
| style="width: 5px;text-align: right;" | (112)
1719
|}
1720
1721
Eq.([[#eq-112|112]]) will be used as the closure equation of the mixed Velocity-Pressure formulation for hypoelastic solids. Note that Eq.([[#eq-112|112]]) has the same structure as Eq.([[#eq-75|75]]) analyzed in the previous section.  From Eq.([[#eq-112|112]]) using linear shape functions <math display="inline">N</math> and integrating on time the pressure with a first order scheme, the same matrix form of Eq.([[#eq-81|81]]) obtained for a general material is obtained.
1722
1723
In conclusion the updated stresses can be computed using the velocities only or both the pressure and the velocities, as follows
1724
1725
<span id="eq-113"></span>
1726
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1727
|-
1728
| 
1729
{| style="text-align: left; margin:auto;" 
1730
|-
1731
| style="text-align: center;" | <math>{^{n+1} \sigma } = { ^{n}  \hat {\sigma }}+  \Delta t \left(\kappa I \otimes I + 2 \mu \mathbf{I}' \right): {^{n+1}d}   </math>
1732
|}
1733
| style="width: 5px;text-align: right;" | (113)
1734
|}
1735
1736
<span id="eq-114"></span>
1737
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1738
|-
1739
| 
1740
{| style="text-align: left; margin:auto;" 
1741
|-
1742
| style="text-align: center;" | <math>{^{n+1} \sigma  }= { ^{n}  \hat {\sigma }} + {^{n+1} \Delta p} I +  2  \Delta t \mu \mathbf{I'} :{^{n+1}d}  </math>
1743
|}
1744
| style="width: 5px;text-align: right;" | (114)
1745
|}
1746
1747
Eqs.([[#eq-113|113]])  and  ([[#eq-114|114]]) will be used for computing the Cauchy stress tensor in the Velocity and mixed Velocity-Pressure formulations, respectively.
1748
1749
===2.3.1 Velocity formulation for hypoelastic solids===
1750
1751
The solution scheme for hypoelastic solids is the one derived in Section [[#2.1.4 Incremental solution scheme|2.1.4]] for  a general continuum. The only modifications required are the  definition of the tangent moduli <math display="inline">{c^{\sigma }}</math> in matrix <math display="inline">K^{m}</math> (Eq.([[#eq-45|45]])) and the computation of the Cauchy stress tensor from the nodal velocities according to the hypoelastic model.  This tensor appears in the geometric part of the tangent matrix <math display="inline">K^{g}</math> (Eq.([[#eq-59|59]])) and into the residual <math display="inline">R</math> (Eq.([[#eq-66|66]])). The tangent moduli tensor is taken from the Jaumann isotropic description and it is the tensor <math display="inline">{c^{\sigma J}}</math> of Eq.([[#eq-101|101]]). Concerning the Cauchy stresses, these are computed with  Eq.([[#eq-113|113]]).  The finite element implemented with this hypoelastic velocity formulation is named V-element.
1752
1753
The iterative solution incremental solution scheme for hypoelastic solids using the velocity formulation for a generic time interval <math display="inline">{t[{^n}t,{^{n+1}}t]}</math> is given  in Box 3.
1754
1755
All the matrices and vectors   in Box 3 are collected in Box 4.
1756
1757
<div class="center" style="font-size: 85%;">
1758
[[File:Draft_Samper_722607179_8274_Box3.png|500px]]
1759
1760
'''Box 3'''. Iterative solution scheme for hypoelastic solids using the velocity formulation.
1761
</div>
1762
1763
1764
<div class="center" style="font-size: 85%;">
1765
[[File:Draft_Samper_722607179_1668_Box4.png|500px]]
1766
1767
'''Box 4'''. Element form of the matrices and vectors in Box 3.
1768
</div>
1769
1770
===2.3.2 Mixed Velocity-Pressure formulation for hypoelastic solids===
1771
1772
The solution scheme is like the one presented in Section [[#2.2.2 Solution method|2.2.2]]. As already explained, the tangent matrix of the mixed Velocity-Pressure formulation is the same as for the Velocity formulation. However the Cauchy stress tensor is computed from the nodal velocities and the nodal pressures using Eq.([[#eq-114|114]]). The governing equations are Eqs.([[#eq-85|85]]-[[#eq-86|86]]) particularized with the material parameters of a hypoelastic solid.  The finite element implemented with this hypoelastic mixed velocity-pressure formulation is called VP-element.
1773
1774
In Box 5, the iterative solution incremental scheme for hypoelastic solids using the mixed velocity-pressure formulation  is given for a generic time interval <math display="inline">{t[^nt,{^{n+1}t}]}</math>.
1775
1776
All the matrices and vectors that appear in Box 5 are collected in Box 6.
1777
1778
<div class="center" style="font-size: 85%;">
1779
[[File:Draft_Samper_722607179_9818_Box5.png|550px]]
1780
1781
'''Box 5'''. Iterative solution scheme for hypoelastic solid using mixed Velocity-Pressure formulation.
1782
</div>
1783
1784
<div class="center" style="font-size: 85%;">
1785
[[File:Draft_Samper_722607179_2374_Box6.png|550px]]
1786
1787
'''Box 6'''. Element form of the matrices and vectors in Box 5.
1788
</div>
1789
1790
1791
Note that for the mixed velocity-pressure formulation the material part of the tangent matrix is defined by the same tangent moduli of the velocity scheme (<math display="inline">{c^{\sigma J}}</math>).
1792
1793
In the mixed formulation, the momentum and the continuity equations can be easily decoupled. This is obtained by computing the Cauchy stress tensor using the velocities only (Eq.([[#eq-113|113]])) and not as the sum of its deviatoric part and the pressure (Eq.([[#eq-114|114]])). The uncoupled mixed velocity-pressure formulation is totally equivalent to the velocity formulation. In fact, although the pressures are still computed by solving the continuity equation, they do not appear in the momentum equations and, hence, they do not affect the solution for each the time step.
1794
1795
===2.3.3 Theory of plasticity===
1796
1797
The theory of plasticity is dedicated to those solids that, after being subjected to a loading process, may sustain permanent (<math display="inline">{plastic}</math>) deformations when completely unloaded  [74]. The plasticity is defined <math display="inline">{rate-independent}</math> if the permanent deformations of the material do not depend on the rate of application of the loads. The materials whose behavior can be adequately described by this theory are called <math display="inline">{rate-independent~ plastic}</math> materials.
1798
1799
Elastic-plastic laws are characterized for being path-dependent and dissipative. The stresses cannot be computed as a single-valued function of the strains because they depend on the entire history of the deformation  [9].
1800
1801
The most important properties of the theory of plasticity can be summarized as follows:
1802
1803
<ol>
1804
1805
<li>The increments of strain <math display="inline">{d \varepsilon }</math> are assumed to be additively decomposed into an elastic (reversible) part <math display="inline">{d \varepsilon _{el}}</math> and a plastic (irreversible) part <math display="inline">{d \varepsilon _{pl}}</math>, such that
1806
1807
<span id="eq-115"></span>
1808
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1809
|-
1810
| 
1811
{| style="text-align: left; margin:auto;" 
1812
|-
1813
| style="text-align: center;" | <math>d \varepsilon =d \varepsilon _{el} + d \varepsilon _{pl}   </math>
1814
|}
1815
| style="width: 5px;text-align: right;" | (115)
1816
|}</li>
1817
1818
<li>There exists an ''elastic domain'' where the behavior of the material is purely elastic and permanent deformations are not produced; </li>
1819
1820
<li> The ''yield function'' <math display="inline">{f_Y}</math> delimits the elastic domain. It governs the onset and the continuity of the plastic deformations and it is a functions of the state of stress and of the internal variables <math display="inline">{q}</math>. So
1821
1822
<span id="eq-116"></span>
1823
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1824
|-
1825
| 
1826
{| style="text-align: left; margin:auto;" 
1827
|-
1828
| style="text-align: center;" | <math>f_Y=f_Y (\sigma , q)   </math>
1829
|}
1830
| style="width: 5px;text-align: right;" | (116)
1831
|}</li>
1832
1833
The yield function cannot be positive: it is negative when the stress state is below the yield value and null otherwise (''yield condition'': <math display="inline">{f_Y=0}</math>);
1834
1835
<li> The plastic strain increments are governed by the so called ''flow rule''; </li>
1836
1837
<li> <math display="inline">{\dot \lambda _{pl}}</math> is the ''plastic strain parameter'' and it is positive for a plastic loading and equal to zero for elastic loading or unloading; </li>
1838
1839
<li>The loading-unloading process is described by the Khun-Tucker conditions
1840
1841
<span id="eq-117"></span>
1842
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1843
|-
1844
| 
1845
{| style="text-align: left; margin:auto;" 
1846
|-
1847
| style="text-align: center;" | <math>\dot \lambda _{pl} \geq 0,  f_Y \leq 0,  \dot \lambda _{pl} f_Y =0   </math>
1848
|}
1849
| style="width: 5px;text-align: right;" | (117)
1850
|}</li>
1851
1852
The third condition can also be expressed in the rate form through the so-called <math display="inline">{consistency~condition}</math>, <math display="inline">{\dot f_Y=0}</math>. For plastic loading (<math display="inline">{\dot \lambda _{pl} >0}</math>) the stress state lies on the yield surface (<math display="inline">{f_Y=0}</math>), instead for elastic loading or unloading the yield condition is not reached (<math display="inline">{f_Y<0}</math>) and there is not plastic flow (<math display="inline">{\dot \lambda _{pl} =0}</math>).  
1853
1854
</ol>
1855
1856
====2.3.3.1 Hypoelastic-plastic materials====
1857
1858
Hypoelastic-plastic models are typically used when the elastic strains represent only a small part of the total strains. In other words, when the plastic strains are much larger than the elastic ones. This is because of the inaccuracy of the hypoelastic models in the large strain regime. However, if the elastic strains are small, the energy error introduced by the hypoelastic description of the elastic response is limited and can be considered adequate [9].
1859
1860
Depending on the problem, the yield function can be based on a different constitutive model. For example, for soil plasticity the Drucker-Prager model is the most used, while for porous plastic solids the Gurson model is more adequate. In this work, the <math display="inline">{J_2 ~von ~Mises}</math> flow model is used. This model is particularly indicated for the metal plasticity  [I9].
1861
1862
The hypoelastic-plastic model described in this section has been taken from [9]. According to the von Mises criterion [125], plastic yielding begins when the <math display="inline">{{J_2}}</math> stress deviator invariant reaches a critical value. The <math display="inline">{{J_2}}</math> stress deviator invariant is defined as
1863
1864
<span id="eq-118"></span>
1865
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1866
|-
1867
| 
1868
{| style="text-align: left; margin:auto;" 
1869
|-
1870
| style="text-align: center;" | <math>J_2= \frac{1}{2} {\sigma _{ij}'} {\sigma _{ij}'}   </math>
1871
|}
1872
| style="width: 5px;text-align: right;" | (118)
1873
|}
1874
1875
The key assumption of the von Mises model is that the plastic flow is not affected by the pressure but only by the deviatoric stress. This hypothesis has been experimentally verified for metals [17]. For this reason the von Mises model is called to be <math display="inline">{pressure ~insensitive}</math>.
1876
1877
A yield function for the von Mises criterion can be defined as
1878
1879
<span id="eq-119"></span>
1880
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1881
|-
1882
| 
1883
{| style="text-align: left; margin:auto;" 
1884
|-
1885
| style="text-align: center;" | <math>f(\sigma ,q)= \bar{\sigma } - \sigma _Y  </math>
1886
|}
1887
| style="width: 5px;text-align: right;" | (119)
1888
|}
1889
1890
where <math display="inline">{\sigma _Y}</math> is the uniaxial yield stress and it is related to the shear yield stress <math display="inline">\tau _Y</math> as follows
1891
1892
<span id="eq-120"></span>
1893
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1894
|-
1895
| 
1896
{| style="text-align: left; margin:auto;" 
1897
|-
1898
| style="text-align: center;" | <math>\sigma _Y= \sqrt{3}  \tau _Y  </math>
1899
|}
1900
| style="width: 5px;text-align: right;" | (120)
1901
|}
1902
1903
and in Eq.([[#eq-119|119]])<math display="inline"> {\bar \sigma } </math> is the <math display="inline">{von~ Mises  ~effective}</math> or <math display="inline">{equivalent ~stress}</math>  defined as
1904
1905
<span id="eq-121"></span>
1906
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1907
|-
1908
| 
1909
{| style="text-align: left; margin:auto;" 
1910
|-
1911
| style="text-align: center;" | <math>\bar \sigma =\sqrt{3 J_2}  </math>
1912
|}
1913
| style="width: 5px;text-align: right;" | (121)
1914
|}
1915
1916
Concerning the deformation, the elastic-plastic decomposition described in Eq.([[#eq-115|115]]) is rewritten in terms of rates as
1917
1918
<span id="eq-122"></span>
1919
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1920
|-
1921
| 
1922
{| style="text-align: left; margin:auto;" 
1923
|-
1924
| style="text-align: center;" | <math>{d}={d}_{el} +{d}_{pl}   </math>
1925
|}
1926
| style="width: 5px;text-align: right;" | (122)
1927
|}
1928
1929
where <math display="inline">{{d}_{el}}</math> and <math display="inline">{{d}_{pl}}</math> are the deformation rates associated to the elastic and plastic responses, respectively.
1930
1931
Combining Eq.([[#eq-122|122]]) with Eq.([[#eq-98|98]]), yields
1932
1933
<span id="eq-123"></span>
1934
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1935
|-
1936
| 
1937
{| style="text-align: left; margin:auto;" 
1938
|-
1939
| style="text-align: center;" | <math>\sigma  ^{\bigtriangledown J}=c^{\sigma J}_{el} : \left(d - d_{pl} \right) </math>
1940
|}
1941
| style="width: 5px;text-align: right;" | (123)
1942
|}
1943
1944
The rate of plastic deformations is given by
1945
1946
<span id="eq-124"></span>
1947
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1948
|-
1949
| 
1950
{| style="text-align: left; margin:auto;" 
1951
|-
1952
| style="text-align: center;" | <math>{d}_{pl}=\dot \lambda _{pl} r(\sigma ,q)  </math>
1953
|}
1954
| style="width: 5px;text-align: right;" | (124)
1955
|}
1956
1957
where the plastic flow rate <math display="inline">{\dot \lambda _{pl}}</math> is a scalar and <math display="inline">{r(\sigma ,q)}</math> represents the plastic flow direction.
1958
1959
Substituting Eq.([[#eq-124|124]]) in Eq.([[#eq-123|123]]), yields
1960
1961
<span id="eq-125"></span>
1962
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1963
|-
1964
| 
1965
{| style="text-align: left; margin:auto;" 
1966
|-
1967
| style="text-align: center;" | <math>\sigma  ^{\bigtriangledown J}=c^{\sigma J}_{el} : \left(d - \dot \lambda _{pl} r \right) </math>
1968
|}
1969
| style="width: 5px;text-align: right;" | (125)
1970
|}
1971
1972
During  plastic loading the plastic flow rate is positive and the state of stress remains on the yield surface <math display="inline">f_Y=0</math>. This is consistent with the third Khun-Tucker condition <math display="inline">\dot \lambda f_Y=0</math>. The consistency condition <math display="inline">\dot f_Y=0</math> has the same meaning. Using the chain rule on the consistency condition, yields
1973
1974
<span id="eq-126"></span>
1975
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1976
|-
1977
| 
1978
{| style="text-align: left; margin:auto;" 
1979
|-
1980
| style="text-align: center;" | <math>\dot {f_Y} = {\partial f_Y \over \partial \sigma }:{\dot {\sigma }} +  {\partial f_Y \over \partial  {q}} \cdot \dot{q} =0  </math>
1981
|}
1982
| style="width: 5px;text-align: right;" | (126)
1983
|}
1984
1985
If the yield function depend on the invariant, the following relation holds [9,102]
1986
1987
<span id="eq-127"></span>
1988
{| class="formulaSCP" style="width: 100%; text-align: left;" 
1989
|-
1990
| 
1991
{| style="text-align: left; margin:auto;" 
1992
|-
1993
| style="text-align: center;" | <math>{\partial f_Y \over \partial \sigma }:{\dot {\sigma }} = {\partial f_Y \over \partial \sigma }: {\sigma }^{\bigtriangledown J}     </math>
1994
|}
1995
| style="width: 5px;text-align: right;" | (127)
1996
|}
1997
1998
Combining Eqs.([[#eq-127|127]]) and ([[#eq-125|125]]) and substituting in Eq.([[#eq-126|126]]), yields
1999
2000
<span id="eq-128"></span>
2001
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2002
|-
2003
| 
2004
{| style="text-align: left; margin:auto;" 
2005
|-
2006
| style="text-align: center;" | <math>{\partial f_Y \over \partial \sigma }:c^{\sigma J}_{el} : \left(d - d_{pl} \right)+  {\partial f_Y \over \partial {q}} \cdot \dot{q}  =0  </math>
2007
|}
2008
| style="width: 5px;text-align: right;" | (128)
2009
|}
2010
2011
In the most plastic models the evolution of the function <math display="inline">q</math> of the internal variables <math display="inline">h</math> can be expressed as a function of the plastic strain parameter as follows
2012
2013
<span id="eq-129"></span>
2014
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2015
|-
2016
| 
2017
{| style="text-align: left; margin:auto;" 
2018
|-
2019
| style="text-align: center;" | <math>\dot {q} = \dot \lambda h  </math>
2020
|}
2021
| style="width: 5px;text-align: right;" | (129)
2022
|}
2023
2024
where <math display="inline">h</math> are the internal variables.
2025
2026
Substituting Eqs.([[#eq-124|124]]) and ([[#eq-129|129]]) in [[#eq-128|128]], the following relation for the plastic strain parameter is  obtained
2027
2028
<span id="eq-130"></span>
2029
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2030
|-
2031
| 
2032
{| style="text-align: left; margin:auto;" 
2033
|-
2034
| style="text-align: center;" | <math>\dot \lambda =\frac{{\partial f_Y \over \partial \sigma }:c^{\sigma J}_{el} : d  }{-{\partial f_Y \over \partial {q}} \cdot {h} + {\partial f_y \over \partial \sigma }:c^{\sigma J}_{el} : r}  </math>
2035
|}
2036
| style="width: 5px;text-align: right;" | (130)
2037
|}
2038
2039
The plastic flow vector <math display="inline">{r}</math> is often derived from a plastic flow potential. If the plastic flow potential coincides with the yield function, the plastic flow is termed <math display="inline">{associative}</math>. In this case <math display="inline">{r}</math> is proportional to the normal of the yield surface, that is <math display="inline">{r \propto \frac{\partial f_Y }{\partial \sigma }}</math>. An associative plastic flow has the important advantage that it can lead to a symmetric stiffness matrix [9]. In this work an associative plasticity and a constant plastic modulus <math display="inline">H</math> (for perfect plasticity, <math display="inline">H=0</math>) have been considered. Using these hypotheses  the plastic strain parameter is expressed as
2040
2041
<span id="eq-131"></span>
2042
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2043
|-
2044
| 
2045
{| style="text-align: left; margin:auto;" 
2046
|-
2047
| style="text-align: center;" | <math>\dot \lambda =\frac{r:c^{\sigma J}_{el} : d  }{H + r:c^{\sigma J}_{el} : r}  </math>
2048
|}
2049
| style="width: 5px;text-align: right;" | (131)
2050
|}
2051
2052
Substituing this relation in Eq.([[#eq-125|125]]), yields
2053
2054
<span id="eq-132"></span>
2055
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2056
|-
2057
| 
2058
{| style="text-align: left; margin:auto;" 
2059
|-
2060
| style="text-align: center;" | <math>\sigma  ^{\bigtriangledown J}=c^{\sigma J}_{el} : \left(d - \frac{r:c^{\sigma J}_{el} : d  }{H + r:c^{\sigma J}_{el} : r}   r \right) </math>
2061
|}
2062
| style="width: 5px;text-align: right;" | (132)
2063
|}
2064
2065
The same  can be computed using a tangent moduli over the whole deformation rate as
2066
2067
<span id="eq-133"></span>
2068
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2069
|-
2070
| 
2071
{| style="text-align: left; margin:auto;" 
2072
|-
2073
| style="text-align: center;" | <math>\sigma  ^{\bigtriangledown J}= c^{\sigma J}: d= \left[c^{\sigma J}_{el}-  \frac{\left(c^{\sigma J}_{el} : r\right)\otimes \left(r:  c^{\sigma J}_{el} \right)}{H + r:c^{\sigma J}_{el} : r} \right]: d  </math>
2074
|}
2075
| style="width: 5px;text-align: right;" | (133)
2076
|}
2077
2078
where <math display="inline">{c^{\sigma J}}</math> is the <math display="inline">continuum</math> elasto-plastic tangent modulus.
2079
2080
For associative plasticity, the von Mises plastic flow is computed as
2081
2082
<span id="eq-134"></span>
2083
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2084
|-
2085
| 
2086
{| style="text-align: left; margin:auto;" 
2087
|-
2088
| style="text-align: center;" | <math>r=\frac{\partial f}{\partial \sigma }= \frac{3}{2 \bar \sigma } {\sigma }'  </math>
2089
|}
2090
| style="width: 5px;text-align: right;" | (134)
2091
|}
2092
2093
Because the plastic flow vector <math display="inline">{r}</math> is deviatoric it follows that
2094
2095
<span id="eq-135"></span>
2096
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2097
|-
2098
| 
2099
{| style="text-align: left; margin:auto;" 
2100
|-
2101
| style="text-align: center;" | <math>c^{\sigma J}_{el} : r=2\mu \, r: c^{\tau J}_{el} : r=3\mu  </math>
2102
|}
2103
| style="width: 5px;text-align: right;" | (135)
2104
|}
2105
2106
Form  Eqs.([[#eq-101|101]]), ([[#eq-133|133]]) and ([[#eq-135|135]]), the following expression of the elastoplastic modulus is obtained
2107
2108
<span id="eq-136"></span>
2109
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2110
|-
2111
| 
2112
{| style="text-align: left; margin:auto;" 
2113
|-
2114
| style="text-align: center;" | <math>c^{\sigma J}=\kappa I \otimes I + 2 \mu \mathbf{I}'   - 2 \mu \gamma  n \otimes n  </math>
2115
|}
2116
| style="width: 5px;text-align: right;" | (136)
2117
|}
2118
2119
with
2120
2121
<span id="eq-137"></span>
2122
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2123
|-
2124
| 
2125
{| style="text-align: left; margin:auto;" 
2126
|-
2127
| style="text-align: center;" | <math>\gamma =\frac{1}{1+\left(H/3\mu \right)}  </math>
2128
|}
2129
| style="width: 5px;text-align: right;" | (137)
2130
|}
2131
2132
<span id="eq-138"></span>
2133
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2134
|-
2135
| 
2136
{| style="text-align: left; margin:auto;" 
2137
|-
2138
| style="text-align: center;" | <math>n=\sqrt{\frac{2}{3}}r  </math>
2139
|}
2140
| style="width: 5px;text-align: right;" | (138)
2141
|}
2142
2143
For perfect plasticity <math display="inline">{H=0}</math>, so <math display="inline">{\gamma=1}</math> and Eq.([[#eq-136|136]]) simplifies to
2144
2145
<span id="eq-139"></span>
2146
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2147
|-
2148
| 
2149
{| style="text-align: left; margin:auto;" 
2150
|-
2151
| style="text-align: center;" | <math>c^{\sigma J}=\kappa I \otimes I + 2 \mu \mathbf{I}'   - 2 \mu  n \otimes n  </math>
2152
|}
2153
| style="width: 5px;text-align: right;" | (139)
2154
|}
2155
2156
Note that the continuum elasto-plastic tangent modulus conserves the symmetry properties of its elastic counterpart. For elastic loading or unloading, <math display="inline">{c^{\sigma J}=c^{\sigma J}_{el} }</math>.
2157
2158
For a plane strain state, <math display="inline">{c^{\sigma J}}</math> is 
2159
2160
<math>\begin{array}{l} \\   \left[c^{ \sigma J}\right]= \left[                      \begin{array}{cccccc}                         \kappa +\frac{4}{3}\mu  & \kappa -\frac{2}{3}\mu  & 0 \\                          \kappa -\frac{2}{3}\mu  & \kappa +\frac{4}{3}\mu  & 0 \\                         0  & 0  & \mu \\                      \end{array} \right]- 2 \mu \gamma \left[                      \begin{array}{cccccc}                        n_{xx}n_{xx}  & n_{xx}n_{yy}  & n_{xx}n_{xy}\\                         n_{yy}n_{xx}  & n_{yy}n_{yy}  & n_{yy}n_{xy}\\                         n_{xy}n_{xx}  & n_{xy}n_{yy}  & n_{xy}n_{xy}\\                       \end{array} \right] \\ \\ \end{array}</math>
2161
2162
In order to guarantee the consistency of the elastoplastic incremental scheme, the so-called <math display="inline">{return}</math> <math display="inline">{mapping}</math> algorithm has to be introduced. With this technique, the Khun-Tucker conditions (Eq.([[#eq-117|117]])) are enforced  at the end of a plastic time step  in order to recover exactly the yield condition <math display="inline">{f(\sigma _{n+1})=0}</math>. The return mapping algorithm consists of an initial trial elastic step followed by a plastic corrector one that is activated when the yield function takes a positive value. In Figure [[#img-4|4]] from [13], a graphical representation of the return mapping algorithm is shown. <div id='img-4'></div>
2163
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2164
|-
2165
|[[Image:draft_Samper_722607179-returnMapping.png|390px|Graphical representation of the return mapping algorithm [13].]]
2166
|- style="text-align: center; font-size: 75%;"
2167
| colspan="1" | '''Figure 4:''' Graphical representation of the return mapping algorithm [13].
2168
|}
2169
2170
For associative plasticity, during the plastic corrector step  driven by the increment of the plasticity parameter <math display="inline">{\lambda }</math>, the plastic flow direction <math display="inline">{r}</math> is normal to the yield surface.
2171
2172
For the <math display="inline">{J_2}</math> flow theory and associative plasticity, the return mapping is characterized to be <math display="inline">{radial}</math> [114]. This  because the von Mises yield surface is circular, thus its normal is also radial. In Figure [[#img-5|5]] a graphical representation of the radial return algorithm for <math display="inline">{J_2}</math> plasticity is shown.
2173
2174
<div id='img-5'></div>
2175
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
2176
|-
2177
|[[Image:draft_Samper_722607179-radialReturn.png|450px|Graphical representation of the radial return method for J₂ plasticity.]]
2178
|- style="text-align: center; font-size: 75%;"
2179
| colspan="1" | '''Figure 5:''' Graphical representation of the radial return method for <math>{J_2}</math> plasticity.
2180
|}
2181
2182
The return radial mapping procedure typically starts with the elastic prediction of the stresses. This means that the linear momentum equations are solved using only the elastic part of the continuum tangent modulus (Eq.([[#eq-101|101]])) and the stress tensor <math display="inline">{{\sigma }^{(0)}}</math> is computed with Eq.([[#eq-114|114]]) (the superindex  refers to the iteration index).
2183
2184
Then the effective stress <math display="inline">{\bar  {\sigma }}</math> is computed with Eq.([[#eq-121|121]]) and the yield function Eq.([[#eq-119|119]]) is evaluated. If <math display="inline">{f_Y>0}</math> the return mapping iterative procedure is required. In fact, an elastoplastic step has been computed as purely elastic without fulfilling the consistency condition.
2185
2186
The first step of  the iterative corrector procedure consists of computing the increment of the plastic parameter as
2187
2188
<span id="eq-140"></span>
2189
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2190
|-
2191
| 
2192
{| style="text-align: left; margin:auto;" 
2193
|-
2194
| style="text-align: center;" | <math>\delta \lambda _{pl}^{(k)}= \frac{f^{(k)}}{3\mu + H^{(k)}}  </math>
2195
|}
2196
| style="width: 5px;text-align: right;" | (140)
2197
|}
2198
2199
For perfect plasticity, the plastic modulus <math display="inline">{H}</math> is null, hence the plastic parameter is <math display="inline">{\delta \lambda _{pl}^{(k)}= f^{(k)}/(3\mu )}</math>.
2200
2201
The plastic parameter increment is updated as
2202
2203
<span id="eq-141"></span>
2204
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2205
|-
2206
| 
2207
{| style="text-align: left; margin:auto;" 
2208
|-
2209
| style="text-align: center;" | <math>\Delta \lambda _{pl}^{(k+1)}=\Delta \lambda _{pl}^{(k)}+\delta \lambda _{pl}^{(k)} </math>
2210
|}
2211
| style="width: 5px;text-align: right;" | (141)
2212
|}
2213
2214
If before this step the material has never suffered plastic deformations, then <math display="inline">{\Delta \lambda _{pl}^{(k)}=0}</math>.
2215
2216
Next the plastic strain and the internal variables are updated according to the plastic correction derived from Eq.([[#eq-140|140]]).
2217
2218
The increment of plastic deformation is
2219
2220
<span id="eq-142"></span>
2221
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2222
|-
2223
| 
2224
{| style="text-align: left; margin:auto;" 
2225
|-
2226
| style="text-align: center;" | <math>\Delta \varepsilon _{pl}^{(k)}=- \delta \lambda _{pl}^{(k)} \sqrt{\frac{3}{2}} n </math>
2227
|}
2228
| style="width: 5px;text-align: right;" | (142)
2229
|}
2230
2231
So the total plastic deformations are
2232
2233
<span id="eq-143"></span>
2234
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2235
|-
2236
| 
2237
{| style="text-align: left; margin:auto;" 
2238
|-
2239
| style="text-align: center;" | <math>\varepsilon _{pl}^{(k+1)}=\varepsilon _{pl}^{(k)}+\Delta \varepsilon _{pl}^{(k)} </math>
2240
|}
2241
| style="width: 5px;text-align: right;" | (143)
2242
|}
2243
2244
Once again, for the first plastic step <math display="inline">{\varepsilon _{pl}^{(k)}=0}</math>.
2245
2246
Next the deviatoric stresses are updated as
2247
2248
<span id="eq-144"></span>
2249
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2250
|-
2251
| 
2252
{| style="text-align: left; margin:auto;" 
2253
|-
2254
| style="text-align: center;" | <math>\sigma '^{(k+1)}=\sigma '^{(0)} - 2\mu \Delta \lambda _{pl}^{(k+1)}r^{(0)}  </math>
2255
|}
2256
| style="width: 5px;text-align: right;" | (144)
2257
|}
2258
2259
The plastic flow direction <math display="inline">{r}</math> remains unchanged because also the tensor <math display="inline">{n}</math> remains unchanged (and radial) during the plastic corrector phase (Eq.([[#eq-138|138]])). This is a particular feature of the radial return mapping.
2260
2261
From Eqs.([[#eq-121|121]],[[#eq-144|144]]), the updated effective stress is
2262
2263
<span id="eq-145"></span>
2264
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2265
|-
2266
| 
2267
{| style="text-align: left; margin:auto;" 
2268
|-
2269
| style="text-align: center;" | <math>\bar \sigma ^{(k+1)}=\bar \sigma ^{(0)} - 3\mu \Delta \lambda _{pl}^{(k+1)}  </math>
2270
|}
2271
| style="width: 5px;text-align: right;" | (145)
2272
|}
2273
2274
Then the yield condition (Eq.[[#eq-119|119]]) is verified again with the updated effective stress. If it is not fulfilled, the steps from Eq.([[#eq-140|140]]) to Eq.([[#eq-144|144]]) are repited until <math display="inline">{f_Y^{(k)}=0}</math>.
2275
2276
In Box 5, the return mapping algorithm for  the <math display="inline">{J_2}</math> theory and referred to a general elastoplastic time interval <math display="inline">{t[^nt,{^{n+1}t}]}</math> is summarized.
2277
2278
===2.3.4 Validation examples===
2279
2280
In this section several problems are studied in order to validate and the V and the VP elements and to make interesting comparisons. First an example for small displacements is studied. Then a benchmark problem for non-linear solid mechanics, namely the Cook's membrane, is analyzed. The third example is a uniformly loaded circular plate and it involves also  plasticity. All these examples are analyzed in statics considering a unique unit time step increment for the velocity-based formulations. The last example is solved in dynamics and for both the hypoelastic and hypoelastic-plastic models.
2281
2282
'''''Simply supported beam'''''
2283
2284
The first validated problem is a simply supported beam loaded by its self-weight. The problem has been studied in statics so the inertial forces have not been considered. The geometry of the problem is illustrated in Figure [[#img-6a|6a]] and the problem data are given in Table [[#img-6|2.1]]. 
2285
2286
2287
<div id='img-6a'></div>
2288
<div id='img-6'></div>
2289
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 55%;max-width: 100%;"
2290
|-
2291
|[[Image:draft_Samper_722607179-BeamInput.png|330px|Simply supported beam. Initial geometry.]]
2292
|- style="text-align: center; font-size: 75%;"
2293
| colspan="1" | '''Figure 6:''' Simply supported beam. Initial geometry.
2294
|}
2295
2296
<div class="center" style="font-size: 85%;">
2297
'''Table 2.1.''' Simply supported beam. Problem data.</div>
2298
2299
{|  class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
2300
| style="text-align: center;"|''L''
2301
|style="text-align: center;"| 5 m
2302
|-
2303
| style="text-align: center;"|''H''
2304
|style="text-align: center;"|0.5 m
2305
|-
2306
| style="text-align: center;"|Young modulus
2307
|style="text-align: center;"|196 GPa
2308
|-
2309
| style="text-align: center;"|Density
2310
|style="text-align: center;"|<math>7.85\times 10^3~kg/m^3</math>
2311
|-
2312
| style="text-align: center;"|Poisson ratio
2313
|style="text-align: center;"|0
2314
|}
2315
2316
2317
The material properties of the structure can be assimilated to the ones of a structural steel.
2318
2319
The beam undergoes small displacements under the effect of its self-weight, hence linear elastic theory is suitable for computing a reference solution. The accuracy of the formulation is tested by comparing the computed values for the maximum vertical displacement and the maximum XX-component of the Cauchy stress tensor with the values given by a linear elastic analysis. According to this theory, both maximum values are reached in the central section of the  beam and they are computed as
2320
2321
<span id="eq-146"></span>
2322
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2323
|-
2324
| 
2325
{| style="text-align: left; margin:auto;" 
2326
|-
2327
| style="text-align: center;" | <math>U^{max}_Y = \frac{5gHL^{4}}{384EI}=1.5348\cdot{10}^{-4}m     </math>
2328
|}
2329
| style="width: 5px;text-align: right;" | (146)
2330
|}
2331
2332
<span id="eq-147"></span>
2333
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2334
|-
2335
| 
2336
{| style="text-align: left; margin:auto;" 
2337
|-
2338
| style="text-align: center;" | <math>{\sigma }^{max}_X = \frac{3gHL^{2}}{4H^{2}}=2887818 Pa     </math>
2339
|}
2340
| style="width: 5px;text-align: right;" | (147)
2341
|}
2342
2343
The problem has been solved in 2D using both the Velocity and the mixed Velocity-Pressure formulations using 3-noded triangular elements.  The static problem is solved with only one unit time increment.
2344
2345
In order to verify the convergence of both schemes, the problem has been solved using structured  meshes of 3-noded triangles with the following average sizes: 0.25m, 0.125m, 0.05m, 0.025m, 0.0125m. In Figure [[#img-7|7]] the finest (mesh size=0.00125m, 32000 elements) and the coarsest (mesh size=0.25m, 80 elements) meshes are illustrated.
2346
2347
<div id='img-7'></div>
2348
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2349
|-
2350
|[[Image:draft_Samper_722607179-beam025B.png|400px]]
2351
|-
2352
| style="text-align: center;" | (a) average mesh size= 0.25m
2353
|-
2354
|[[Image:draft_Samper_722607179-beam0125.png|400px]]
2355
|-
2356
| style="text-align: center;" |(b) average mesh size= 0.0125m
2357
|- style="text-align: center; font-size: 75%;"
2358
| colspan="1" | '''Figure 7:''' Simply supported beam. Coarsest and finest meshes used for the analysis.
2359
|}
2360
2361
In Figure [[#img-8|8]] the solutions for the vertical displacement and the XX-component of the Cauchy stress tensor computed   at the Gauss points  obtained with the mesh with average size 0.025m are plotted. 
2362
2363
<div id='img-8'></div>
2364
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2365
|-
2366
|[[Image:draft_Samper_722607179-BeamDispl.png|400px|]]
2367
|- 
2368
| style="text-align: center;" | (a) Displacements in Y-direction
2369
|-
2370
|[[Image:draft_Samper_722607179-BeamCauchy.png|400px]]
2371
|-
2372
| style="text-align: center;" | (b) Cauchy stress, XX-component
2373
|- style="text-align: center; font-size: 75%;"
2374
| colspan="1" | '''Figure 8:''' Simply supported beam. Numerical results.
2375
|}
2376
2377
For the visualization of all the numerical results of this work the pre-postprocessor software GID [128] has been used.
2378
2379
Table [[#table-1|2.2]] collects the values of the maximum vertical displacement (absolute value) and the XX-component of the Cauchy stress tensor computed at the Gauss point using the V-element and the VP-element for different FEM mesh.
2380
2381
<span id='table-1'></span>
2382
<div class="center" style="font-size: 85%;">
2383
'''Table 2.2.''' Simply supported beam.  Cauchy stress XX-component and maximum vertical displacement for different discretizations.</div>
2384
2385
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
2386
|-
2387
| style="border-left: 1px solid;border-right: 1px solid;" |   mesh 
2388
| colspan='2' style="border-left: 1px solid;border-right: 1px solid;border-left: 1px solid;" | '''V-element'''
2389
| colspan='2' style="border-left: 1px solid;border-right: 1px solid;border-left: 1px solid;" | '''VP-element'''
2390
|-
2391
| style="border-left: 1px solid;border-right: 1px solid;" |     size  
2392
| style="border-left: 1px solid;" | <math>\sigma ^{max}_x</math>  
2393
| style="border-right: 1px solid;" | <math>U^{max}_y</math> 
2394
| style="border-left: 1px solid;" | <math>\sigma ^{max}_x</math> 
2395
| style="border-right: 1px solid;" | <math>U^{max}_y</math>
2396
|-
2397
| style="border-left: 1px solid;border-right: 1px solid;" |    2.50E-01
2398
| style="border-left: 1px solid;" | 1.46E+06
2399
| style="border-right: 1px solid;" | 9.92E-05
2400
| style="border-left: 1px solid;" | 1.53E+06
2401
| style="border-right: 1px solid;" | 8.41E-05
2402
|-
2403
| style="border-left: 1px solid;border-right: 1px solid;" |  1.25E-01
2404
| style="border-left: 1px solid;" | 2.29E+06
2405
| style="border-right: 1px solid;" | 1.37E-04
2406
| style="border-left: 1px solid;" | 2.37E+06
2407
| style="border-right: 1px solid;" | 1.29E-04
2408
|-
2409
| style="border-left: 1px solid;border-right: 1px solid;" |  5.00E-02
2410
| style="border-left: 1px solid;" | 2.72E+06
2411
| style="border-right: 1px solid;" | 1.54E-04
2412
| style="border-left: 1px solid;" | 2.80E+06
2413
| style="border-right: 1px solid;" | 1.52E-04
2414
|-
2415
| style="border-left: 1px solid;border-right: 1px solid;" |  2.50E-02
2416
| style="border-left: 1px solid;" | 2.82E+06
2417
| style="border-right: 1px solid;" | 1.56E-04
2418
| style="border-left: 1px solid;" | 2.87E+06
2419
| style="border-right: 1px solid;" | 1.56E-04
2420
|-
2421
| style="border-left: 1px solid;border-right: 1px solid;" |  1.25E-02
2422
| style="border-left: 1px solid;" | 2.86E+06
2423
| style="border-right: 1px solid;" | 1.57E-04
2424
| style="border-left: 1px solid;" | 2.89E+06
2425
| style="border-right: 1px solid;" | 1.57E-04
2426
|}
2427
2428
Simply supported beam.  Cauchy stress XX-component and maximum vertical displacement for different discretizations. 
2429
2430
In the examples presented in this section, for the convergence analysis the percentage error is computed versus the solution obtained with the finest discretization as
2431
2432
<span id="eq-148"></span>
2433
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2434
|-
2435
| 
2436
{| style="text-align: left; margin:auto;" 
2437
|-
2438
| style="text-align: center;" | <math>error = abs\left(\frac{value_{finest ~mesh}-value_{tested ~mesh}}{value_{finest ~mesh}}\right)\cdot 100      </math>
2439
|}
2440
| style="width: 5px;text-align: right;" | (148)
2441
|}
2442
2443
For example, the maximum vertical displacement obtained  with the finest mesh (average size 0.0125m) is the reference solution for both V and VP elements. The convergence curves for both formulations are plotted in Figure [[#img-9|9]]. Both elements show a quadratic convergence rate for this error measure. 
2444
2445
<div id='img-9'></div>
2446
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2447
|-
2448
|[[Image:draft_Samper_722607179-BeamConvergence.png|500px|Simply supported beam. Convergence analysis for the maximum vertical displacement for V and VP elements.]]
2449
|- style="text-align: center; font-size: 75%;"
2450
| colspan="1" | '''Figure 9:''' Simply supported beam. Convergence analysis for the maximum vertical displacement for V and VP elements.
2451
|}
2452
2453
'''''Compressible Cook's membrane'''''
2454
2455
The Cook's membrane is a benchmark problem for solid mechanics. The static problem is solved twice in this thesis. In this section a compressible material is considered; in the next chapter the nearly incompressible case is analyzed.   In both cases the problem has been solved with only one unit time increment.
2456
2457
The initial geometry of the problem, as well the problem data are given in Figure [[#img-10a|10a]]. 
2458
2459
<div id='img-10a'></div>
2460
<div id='img-10'></div>
2461
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2462
|-
2463
|[[Image:draft_Samper_722607179-CMinput.png|400px]]
2464
|[[Image:draft_Samper_722607179-CMcomp5.png|400px]]
2465
|- style="text-align: center; font-size: 75%;"
2466
| (a) Initial geometry
2467
| (b) FEM mesh
2468
|- style="text-align: center; font-size: 75%;"
2469
| colspan="2" | '''Figure 10:''' Cook's membrane. Initial geometry, material data and FEM mesh (5 subdivisions for each edge corresponding to 50 elements).
2470
|}
2471
2472
The self weight of the membrane has not been taken into account in the analysis, so the membrane deforms under only the effect of the external load applied at its free edge.  In this case the structure undergoes large displacements and the solution cannot been computed analytically. The results taken as the  reference ones are those published in [46]. The comparison with the mentioned publication, as well the convergence test are performed for the vertical displacement of point A of Figure [[#img-10a|10a]] with coordinates <math display="inline">(x,y)=(48,52)</math>. According to  [46], under plane stress conditions this displacement is <math display="inline">U^{max}_Y = 23.964</math>.
2473
2474
The domain is discretized with a structured mesh and the edges of the membrane have the same number of partitions.  In Figure [[#img-10|10]] one of the meshes used for this problem is given. The figure refers to the case of 5 elements for each edge of the membrane.
2475
2476
For the 2D simulation a convergence study has been performed using various discretizations. In the finest one the edges have 200 subdivisions, while in the coarsest one only 2.
2477
2478
<div id='img-11'></div>
2479
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 30%;max-width: 100%;"
2480
|-
2481
|[[Image:draft_Samper_722607179-CM3D.png|300px|Cook's membrane. Numerical results for the 3D simulation: the XX-component of the Cauchy stress tensor is plotted over the deformed configuration.]]
2482
|- style="text-align: center; font-size: 75%;"
2483
| colspan="1" | '''Figure 11:''' Cook's membrane. Numerical results for the 3D simulation: the XX-component of the Cauchy stress tensor is plotted over the deformed configuration.
2484
|}
2485
2486
The 3D problem (thickness=1) has been solved for an unstructured mesh with average size 0.5 only. The results for this mesh are given in  Figure [[#img-11|11]], where the XX-component of the Cauchy stress tensor is plotted over the deformed configuration.
2487
2488
For the 3D simulation, the vertical displacements at the central point of the free edge for the V and the VP elements are 23.942 and 23.952, respectively, which correspond to an error versus the reference solution of 0.092% for the V-element and 0.050% for the VP-element.
2489
2490
The  vertical displacement of point A of Figure [[#img-10a|10a]] obtained for all the 2D discretizations and for both the Velocity and the Velocity-Pressure formulations is plotted in the graph of  Figure [[#img-12|12]]. 
2491
2492
<div id='img-12'></div>
2493
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2494
|-
2495
|[[Image:draft_Samper_722607179-CMresults.png|390px|Cook's membrane. Vertical displacement of point A of Figure [[#img-10a|10a]]. Results for V and VP elements compared to the reference solution [46].]]
2496
|- style="text-align: center; font-size: 75%;"
2497
| colspan="1" | '''Figure 12:''' Cook's membrane. Vertical displacement of point A of Figure [[#img-10a|10a]]. Results for V and VP elements compared to the reference solution [46].
2498
|}
2499
2500
In Table [[#table-2|2.3]] the numerical values are given.
2501
2502
<span id='table-2'></span>
2503
<div class="center" style="font-size: 85%;">
2504
'''Table 2.3'''. Cook's membrane.  Maximum vertical displacement  for different discretizations.
2505
</div>
2506
2507
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
2508
|-
2509
| style="border-left: 1px solid;border-right: 1px solid;" |   elements 
2510
| style="border-left: 1px solid;border-right: 1px solid;" |  '''V-element''' 
2511
| style="border-left: 1px solid;border-right: 1px solid;" |  '''VP-element'''
2512
|-
2513
| style="border-left: 1px solid;border-right: 1px solid;" |      per edge  
2514
| style="border-left: 1px solid;border-right: 1px solid;" | <math>U^{max}_y</math>  
2515
| style="border-left: 1px solid;border-right: 1px solid;" | <math>U^{max}_y</math> 
2516
|-
2517
| style="border-left: 1px solid;border-right: 1px solid;" |    2
2518
| style="border-left: 1px solid;border-right: 1px solid;" | 6.707
2519
| style="border-left: 1px solid;border-right: 1px solid;" | 7.8105
2520
|-
2521
| style="border-left: 1px solid;border-right: 1px solid;" |  3
2522
| style="border-left: 1px solid;border-right: 1px solid;" | 9.0274
2523
| style="border-left: 1px solid;border-right: 1px solid;" | 10.901
2524
|-
2525
| style="border-left: 1px solid;border-right: 1px solid;" |  4
2526
| style="border-left: 1px solid;border-right: 1px solid;" | 11.232
2527
| style="border-left: 1px solid;border-right: 1px solid;" | 13.515
2528
|-
2529
| style="border-left: 1px solid;border-right: 1px solid;" |  5
2530
| style="border-left: 1px solid;border-right: 1px solid;" | 13.1755
2531
| style="border-left: 1px solid;border-right: 1px solid;" | 15.5985
2532
|-
2533
| style="border-left: 1px solid;border-right: 1px solid;" |  10
2534
| style="border-left: 1px solid;border-right: 1px solid;" | 19.037
2535
| style="border-left: 1px solid;border-right: 1px solid;" | 20.729
2536
|-
2537
| style="border-left: 1px solid;border-right: 1px solid;" |  15
2538
| style="border-left: 1px solid;border-right: 1px solid;" | 21.272
2539
| style="border-left: 1px solid;border-right: 1px solid;" | 22.332
2540
|-
2541
| style="border-left: 1px solid;border-right: 1px solid;" |  20
2542
| style="border-left: 1px solid;border-right: 1px solid;" | 22.349
2543
| style="border-left: 1px solid;border-right: 1px solid;" | 22.987
2544
|-
2545
| style="border-left: 1px solid;border-right: 1px solid;" |  50
2546
| style="border-left: 1px solid;border-right: 1px solid;" | 23.658
2547
| style="border-left: 1px solid;border-right: 1px solid;" | 23.781
2548
|-
2549
| style="border-left: 1px solid;border-right: 1px solid;" |  100
2550
| style="border-left: 1px solid;border-right: 1px solid;" | 23.878
2551
| style="border-left: 1px solid;border-right: 1px solid;" | 23.913
2552
|-
2553
| style="border-left: 1px solid;border-right: 1px solid;" |  200
2554
| style="border-left: 1px solid;border-right: 1px solid;" | 23.941
2555
| style="border-left: 1px solid;border-right: 1px solid;" | 23.95
2556
|}
2557
2558
2559
Cook's membrane.  Maximum vertical displacement  for different discretizations. CMtable
2560
2561
The convergence curves are given in  Figure [[#img-13|13]]. <div id='img-13'></div>
2562
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2563
|-
2564
|[[Image:draft_Samper_722607179-CMconv.png|390px|Cook's membrane. Convergence analysis for the vertical displacement of point A of Figure [[#img-10a|10a]] for the V and VP elements, V-element and VP-element, respectively.]]
2565
|- style="text-align: center; font-size: 75%;"
2566
| colspan="1" | '''Figure 13:''' Cook's membrane. Convergence analysis for the vertical displacement of point A of Figure [[#img-10a|10a]] for the V and VP elements, V-element and VP-element, respectively.
2567
|}
2568
2569
2570
Also in this case the convergence rate is quadratic for both formulations.
2571
2572
'''''Uniformly loaded circular plate'''''
2573
2574
The problem analyzed in this section is a simply supported circular plate subjected to a uniform pressure <math display="inline">P</math> on its top surface.  The plate constrains are applied on its lower edge. The plate has a radius <math display="inline">R=10</math> and thickness <math display="inline">h=1</math>.   In this work, the axial symmetry of the problem has not been used, and the  plate has been analyzed in 3D using 4-noded tetrahedra. The average size for the tetrahedra is 0.175. This gives 214047 nodes. In Figure [[#img-14|14]] the FEM mesh used is shown. <div id='img-14'></div>
2575
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
2576
|-
2577
|[[Image:draft_Samper_722607179-pericInput.png|540px|Uniformly loaded circular plate. Initial geometry and 3D FEM used.]]
2578
|- style="text-align: center; font-size: 75%;"
2579
| colspan="1" | '''Figure 14:''' Uniformly loaded circular plate. Initial geometry and 3D FEM used.
2580
|}
2581
2582
A hypoelastic-perfectly plastic model has been used,  and  the problem has been solved with the mixed velocity pressure formulation. For the plastic part, a von Mises yield criterion has been considered. The plate has Young modulus <math display="inline">E=10^7</math>, Poisson ratio <math display="inline">\nu=0.24</math> and a uniaxial yield stress <math display="inline">\bar{\sigma }_y=16000</math>. The objective of the study is to determine the limit load for the plate. Using the procedure described in [115], the limit load can be computed analytically by combining the limit analysis and the finite difference method.  According to this theory, the limit load can be approximated as
2583
2584
<span id="eq-149"></span>
2585
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2586
|-
2587
| 
2588
{| style="text-align: left; margin:auto;" 
2589
|-
2590
| style="text-align: center;" | <math>P_{lim} \approx \frac{1.63\bar{\sigma }_y h^2}{R^2}=260.8      </math>
2591
|}
2592
| style="width: 5px;text-align: right;" | (149)
2593
|}
2594
2595
The same problem was solved in [74] using eight-noded axisymmetric quadrilateral elements with four Gauss integration points. The limit load obtained with a relatively coarse mesh (10 finite elements distributed in two layers across the thickness) is <math display="inline">P^{FE}_{lim}=259.8</math> [74].
2596
2597
As in [74], the limit load has been considered as the one for which the non-linear procedure cannot longer converge for a small increment of the load.
2598
2599
In Figure [[#img-15|15]] the maximum vertical displacement of the plate is plotted against the pressure  on the top surface. In Table [[#table-3|2.4]] the numerical values are given. <div id='img-15'></div>
2600
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2601
|-
2602
|[[Image:draft_Samper_722607179-platePericConve.png|390px|Uniformly loaded circular plate. Maximum deflection versus the applied pressure.]]
2603
|- style="text-align: center; font-size: 75%;"
2604
| colspan="1" | '''Figure 15:''' Uniformly loaded circular plate. Maximum deflection versus the applied pressure.
2605
|}
2606
2607
<span id='table-3'></span>
2608
<div class="center" style="font-size: 85%;">
2609
'''Table 2.4.'''. Uniformly loaded circular plate. Numerical values of the maximum vertical deflection for different applied pressures.</div>
2610
2611
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
2612
|-
2613
| style="border-left: 1px solid;border-right: 1px solid;" |   pressure  
2614
| style="border-left: 1px solid;border-right: 1px solid;" |  max. deflection 
2615
| style="border-left: 1px solid;border-right: 1px solid;" |  pressure  
2616
| style="border-left: 1px solid;border-right: 1px solid;" |  max. deflection 
2617
|-
2618
| style="border-left: 1px solid;border-right: 1px solid;" |    101.84
2619
| style="border-left: 1px solid;border-right: 1px solid;" | 0.0758 
2620
| style="border-left: 1px solid;border-right: 1px solid;" |  260.71
2621
| style="border-left: 1px solid;border-right: 1px solid;" | 0.677
2622
|-
2623
| style="border-left: 1px solid;border-right: 1px solid;" |  178.22
2624
| style="border-left: 1px solid;border-right: 1px solid;" | 0.138
2625
| style="border-left: 1px solid;border-right: 1px solid;" |  261.21
2626
| style="border-left: 1px solid;border-right: 1px solid;" | 0.716
2627
|-
2628
| style="border-left: 1px solid;border-right: 1px solid;" |  229.14
2629
| style="border-left: 1px solid;border-right: 1px solid;" | 0.236
2630
| style="border-left: 1px solid;border-right: 1px solid;" |  261.73
2631
| style="border-left: 1px solid;border-right: 1px solid;" | 0.761
2632
|-
2633
| style="border-left: 1px solid;border-right: 1px solid;" |  241.87
2634
| style="border-left: 1px solid;border-right: 1px solid;" | 0.296
2635
| style="border-left: 1px solid;border-right: 1px solid;" |  262.24
2636
| style="border-left: 1px solid;border-right: 1px solid;" | 0.816
2637
|-
2638
| style="border-left: 1px solid;border-right: 1px solid;" |  253.58
2639
| style="border-left: 1px solid;border-right: 1px solid;" | 0.424
2640
| style="border-left: 1px solid;border-right: 1px solid;" |  262.73
2641
| style="border-left: 1px solid;border-right: 1px solid;" | 0.885
2642
|-
2643
| style="border-left: 1px solid;border-right: 1px solid;" |  258.67
2644
| style="border-left: 1px solid;border-right: 1px solid;" | 0.567
2645
| style="border-left: 1px solid;border-right: 1px solid;" |  263.26
2646
| style="border-left: 1px solid;border-right: 1px solid;" | 0.972
2647
|-
2648
| style="border-left: 1px solid;border-right: 1px solid;" |  259.69
2649
| style="border-left: 1px solid;border-right: 1px solid;" | 0.615
2650
| style="border-left: 1px solid;border-right: 1px solid;" |  263.77
2651
| style="border-left: 1px solid;border-right: 1px solid;" | 1.088
2652
|-
2653
| style="border-left: 1px solid;border-right: 1px solid;" |  260.20
2654
| style="border-left: 1px solid;border-right: 1px solid;" | 0.644
2655
| style="border-left: 1px solid;border-right: 1px solid;" |  264.27
2656
| style="border-left: 1px solid;border-right: 1px solid;" | 1.250
2657
|}
2658
2659
Uniformly loaded circular plate. Numerical values of the maximum vertical deflection for different applied pressures. platePericTable
2660
2661
For the present analysis the limit load obtained is <math display="inline">P_{lim}=264.27</math>, the relative percentage errors with respect the solutions given in [115] and [74] are 1.37% and 1.76%, respectively.
2662
2663
In  Figure [[#img-16|16]] the vertical displacements are depicted over the deformed configuration obtained with the limit load. The plate central section is highlighted in the picture.
2664
2665
In Figures [[#img-17|17]] and  [[#img-18|18]] some snapshots of the von Mises effective stress are plotted over the central section of the plate for the different load conditions. The  picture shows clearly the progressive evolution of the plastic zone.   
2666
2667
<div id='img-16'></div>
2668
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
2669
|-
2670
|[[Image:draft_Samper_722607179-pericResults.png|570px|Uniformly loaded circular plate. Vertical displacement contours for the maximum pressure sustained by the plate (P<sub>lim</sub>=264.27).]]
2671
|- style="text-align: center; font-size: 75%;"
2672
| colspan="1" | '''Figure 16:''' Uniformly loaded circular plate. Vertical displacement contours for the maximum pressure sustained by the plate (<math>{P_{lim}=264.27}</math>).
2673
|}
2674
2675
<div id='img-17'></div>
2676
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2677
|-
2678
|[[Image:draft_Samper_722607179-pericVM200.png|400px]]
2679
|-
2680
|(a) Overall load=101.84
2681
|-
2682
|[[Image:draft_Samper_722607179-pericVM350.png|400px]]
2683
|-
2684
| (b) Overall load=178.22
2685
|-
2686
|[[Image:draft_Samper_722607179-pericVM450.png|400px|]]
2687
|-
2688
| (c) Overall load=229.14
2689
|-
2690
|[[Image:draft_Samper_722607179-pericScale3.png|200px]]
2691
|- style="text-align: center; font-size: 75%;"
2692
| colspan="1" | '''Figure 17:''' Uniformly loaded circular plate. Von Mises effective stress over the deformed configurations for different load conditions (only the central section is depicted) (I/II).
2693
|}
2694
2695
<div id='img-18'></div>
2696
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2697
|-
2698
|[[Image:draft_Samper_722607179-pericVM490.png|400px]]
2699
|-
2700
|(a) Overall load=253.58
2701
|-
2702
|[[Image:draft_Samper_722607179-pericVM514.png|400px]]
2703
|-
2704
|(b) Overall load=261.73
2705
|-
2706
|[[Image:draft_Samper_722607179-pericVM519.png|400px|]]
2707
|-
2708
|(c) Overall load=264.27
2709
|-
2710
|[[Image:draft_Samper_722607179-pericScale3.png|200px]]
2711
|- style="text-align: center; font-size: 75%;"
2712
| colspan="1" | '''Figure 18:''' Uniformly loaded circular plate. Von Mises effective stress over the deformed configurations for different load conditions (only the central section is depicted) (II/II).
2713
|}
2714
2715
'''''Plane strain  cantilever in dynamics'''''
2716
2717
The plane strain cantilever illustrated in Figure [[#img-19a|19a]] has been chosen as the reference case for a large displacement dynamics analysis. The problem data are in given in Table 2.5. The problem was introduced and studied in [8].  In the reference publication, the load was applied as a step function at time <math display="inline">t=0s</math> and its magnitude was defined by the equation <math display="inline">f=15(1-y^2/4)</math> where <math display="inline">{y}</math> is the coordinate in the direction of the load. However, in this analysis the load has been considered uniformly distributed over the free edge (the overall value is 40, as in [8]). 
2718
2719
<div id='img-19a'></div>
2720
<div id='img-19'></div>
2721
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;max-width: 100%;"
2722
|-
2723
|[[File:Draft_Samper_722607179_7977_BELYinput.png|300px]]
2724
|- style="text-align: center; font-size: 75%;"
2725
| colspan="1" | '''Figure 19:''' Plane strain cantilever. Initial geometry.
2726
|}
2727
2728
<div class="center" style="font-size: 85%;">
2729
'''Table 2.5.''' Plane strain cantilever. Problema data.</div>
2730
2731
{|  class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
2732
| style="text-align: center;"|''L''
2733
|style="text-align: center;"| 25
2734
|-
2735
| style="text-align: center;"|''D''
2736
|style="text-align: center;"|4
2737
|-
2738
| style="text-align: center;"|Young modulus
2739
|style="text-align: center;"|<math>10^4</math>
2740
|-
2741
| style="text-align: center;"|Poisson ratio
2742
|style="text-align: center;"|0.25
2743
|}
2744
2745
2746
The problem has been solved with both a hypoelastic and  a hypoelastic-plastic models. First the results of the hypoelastic model are given.
2747
2748
''Hypoelastic model''
2749
2750
The problem has been solved in 2D and 3D and using both the V and VP elements. In order to simulate the plane strain state, in the 3D analysis the nodal displacements in the transversal direction to the load have been constrained [8].
2751
2752
The reference solution is the elastic one given in [8].
2753
2754
For the 2D analysis a convergence study has been performed. Structured finite element meshes have been used and the coarsest and the finest ones have a mean size of 1 and 0.125, respectively. Both meshes are given in  Figure [[#img-20|20]]. 
2755
2756
<div id='img-20'></div>
2757
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
2758
|-
2759
|[[Image:draft_Samper_722607179-belyInput1B.png|400px]]
2760
|-
2761
|(a) Average mesh size= 1
2762
|-
2763
|[[Image:draft_Samper_722607179-belyInput0125B.png|400px|Plane strain cantilever. Coarsest (mesh size=1, 200 elements) and finest meshes  (mesh size=0.125, 12800 triangles) used for the 2D analysis.]]
2764
|-
2765
|(b) Average mesh size= 0.125
2766
|- style="text-align: center; font-size: 75%;"
2767
| colspan="1" | '''Figure 20:''' Plane strain cantilever. Coarsest (mesh size=1, 200 elements) and finest meshes  (mesh size=0.125, 12800 triangles) used for the 2D analysis.
2768
|}
2769
2770
For the 3D case, the problem has been solved with the finest mesh only (average size for the 4-noded tetrahedra equal to 0.125). The results for the 3D case obtained with the VP-element are illustrated in Figure [[#img-21|21]] where the pressure contours are plotted over the deformed configuration.  
2771
2772
2773
<div id='img-21'></div>
2774
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2775
|-
2776
|[[Image:draft_Samper_722607179-bely3D.png|390px|Plane strain cantilever. Numerical results for the 3D simulation obtained with the VP-element: pressure contours plotted over the deformed configuration.]]
2777
|- style="text-align: center; font-size: 75%;"
2778
| colspan="1" | '''Figure 21:''' Plane strain cantilever. Numerical results for the 3D simulation obtained with the VP-element: pressure contours plotted over the deformed configuration.
2779
|}
2780
2781
In Figure [[#img-22|22]] the time evolution of the top corner vertical displacement is plotted for each of the FEM meshes. These results have been obtained with the VP-element. <div id='img-22'></div>
2782
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2783
|-
2784
|[[Image:draft_Samper_722607179-bely2Dresults.png|390px|Plane strain cantilever. Time evolution of the top corner vertical displacement for different 2D discretizations. Results obtained with the VP-element.]]
2785
|- style="text-align: center; font-size: 75%;"
2786
| colspan="1" | '''Figure 22:''' Plane strain cantilever. Time evolution of the top corner vertical displacement for different 2D discretizations. Results obtained with the VP-element.
2787
|}
2788
2789
According to [8], the maximum vertical displacement is <math display="inline">U^{max}_Y = 6.88</math>. Table [[#table-4|2.6]] collects the maximum vertical displacement obtained with the V and the VP elements for all the meshes.
2790
2791
<span id='table-4'></span>
2792
<div class="center" style="font-size: 85%;">
2793
'''Table 2.6'''. Plane strain cantilever. Maximum top corner vertical displacement  for different 2D discretizations.</div>
2794
2795
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
2796
|- style="border-top: 2px solid;"
2797
| style="border-left: 2px solid;" |  mesh 
2798
| style="border-left: 1px solid;" | '''V-element''' 
2799
| style="border-left: 1px solid;border-right: 2px solid;" | '''VP-element'''
2800
|-
2801
| style="border-left: 2px solid;border-right: 1px solid;" |   size  
2802
| style="border-left: 1px solid;border-right: 1px solid;" | <math>U^{max}_y</math>
2803
| style="border-left: 1px solid;border-right: 2px solid;" | <math>U^{max}_y</math>
2804
|- style="border-top: 2px solid;"
2805
| style="border-left: 2px solid;" |  1
2806
| style="border-left: 1px solid;" | 5.759
2807
| style="border-left: 1px solid;border-right: 2px solid;" | 6.306
2808
|-
2809
| style="border-left: 2px solid;" | 0.8
2810
| style="border-left: 1px solid;" | 6.144
2811
| style="border-left: 1px solid;border-right: 2px solid;" | 6.534
2812
|-
2813
| style="border-left: 2px solid;" | 0.5
2814
| style="border-left: 1px solid;" | 6.568
2815
| style="border-left: 1px solid;border-right: 2px solid;" | 6.743
2816
|-
2817
| style="border-left: 2px solid;" | 0.25
2818
| style="border-left: 1px solid;" | 6.811
2819
| style="border-left: 1px solid;border-right: 2px solid;" | 6.863
2820
|- style="border-bottom: 2px solid;"
2821
| style="border-left: 2px solid;" | 0.125
2822
| style="border-left: 1px solid;" | 6.875
2823
| style="border-left: 1px solid;border-right: 2px solid;" | 6.895
2824
2825
|}
2826
2827
Plane strain cantilever. Maximum top corner vertical displacement  for different 2D discretizations. BelyTable
2828
2829
The four curves of Figure [[#img-23|23]] are the converged time evolution of the top corner vertical displacement obtained with the V-element in 3D, the VP-element in 2D and 3D and the reference solution [8]. The curves corresponding to the V and VP elements are almost superposed and they match the reference solution. 
2830
2831
<div id='img-23'></div>
2832
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2833
|-
2834
|[[Image:draft_Samper_722607179-belyComparison.png|390px|Plane strain cantilever. Time evolution of the top corner vertical displacement. Solutions for the 2D VP-element and the 3D V and VP elements obtained with the finest mesh (average size 0.125) compared to the reference solution [8].]]
2835
|- style="text-align: center; font-size: 75%;"
2836
| colspan="1" | '''Figure 23:''' Plane strain cantilever. Time evolution of the top corner vertical displacement. Solutions for the 2D VP-element and the 3D V and VP elements obtained with the finest mesh (average size 0.125) compared to the reference solution [8].
2837
|}
2838
2839
''Hypoelastic-plastic model''
2840
2841
The same problem has been solved for an elastic-plastic material with linear hardening.  The yield stress is 300 and the plastic modulus <math display="inline">H</math> is 100. The problem has been solved with the mixed velocity-pressure formulation and by using structured meshes, as the ones of Figure [[#img-20|20]]. The reference solution is taken from [8] where the benchmark was proposed. In [8] the converged value for the maximum top corner vertical displacement is 8.22. The hypoelastic-plastic mixed velocity-pressure formulation converges to 7.97  (error of 2.998% In the graph of  Figure [[#img-24|24]] the time evolution of the top corner vertical displacement is plotted for the different FEM meshes. 
2842
2843
<div id='img-24'></div>
2844
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
2845
|-
2846
|[[Image:draft_Samper_722607179-belyPlasticResults.png|390px|Plane strain elastoplastic cantilever. Time evolution of the top corner vertical displacement for different 2D discretizations.]]
2847
|- style="text-align: center; font-size: 75%;"
2848
| colspan="1" | '''Figure 24:''' Plane strain elastoplastic cantilever. Time evolution of the top corner vertical displacement for different 2D discretizations.
2849
|}
2850
2851
In Table [[#table-5|2.7]] the numerical values for the maximum  and the residual top corner vertical displacements are given for each of the FEM mesh.
2852
2853
<span id='table-5'></span>
2854
<div class="center" style="font-size: 85%;">
2855
'''Table 2.7'''. Plane strain elastoplastic cantilever. Maximum and residual top corner vertical displacements for different discretizations.</div>
2856
2857
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
2858
|-style="border-top: 2px solid;"
2859
| style="border-left: 2px solid;" |   mesh size 
2860
|  <math>U^{max}_y</math>  
2861
| style="border-left: 1px solid;border-right: 2px solid;" | <math>U^{res}_y</math> 
2862
|-
2863
| style="border-left: 2px solid;" |    1
2864
|  6.77
2865
| style="border-right: 2px solid;" | 3.25
2866
|-
2867
| style="border-left: 2px solid;" |  0.8
2868
|  7.17
2869
| style="border-right: 2px solid;" | 3.69
2870
|-
2871
| style="border-left: 2px solid;" |  0.5
2872
|  7.56
2873
| style="border-right: 2px solid;" | 4.14
2874
|-
2875
| style="border-left: 2px solid;" |  0.25
2876
| 7.82
2877
| style="border-right: 2px solid;" | 4.51
2878
|-
2879
| style="border-left: 2px solid;" |  0.125
2880
| 7.92
2881
| style="border-right: 2px solid;" | 4.68
2882
|-
2883
| style="border-left: 2px solid;" |  0.1
2884
|  7.94
2885
| style="border-right: 2px solid;" | 4.72
2886
|-
2887
| style="border-left: 2px solid;" |  0.0625
2888
|  7.97
2889
| style="border-right: 2px solid;" | 4.77
2890
2891
|}
2892
2893
Plane strain elastoplastic cantilever. Maximum and residual top corner vertical displacements for different discretizations. BelyPlasticTable
2894
2895
The problem has been solved also for the 3D problem for a structured mesh of 4-noded tetrahedra with average size 0.125.
2896
2897
In Figures [[#img-25|25]] the von Mises effective stresses are plotted over the deformed configuration at the time instant when the top corner vertical displacement is reached (<math display="inline">t=6.05s</math>). <div id='img-25'></div>
2898
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
2899
|-
2900
|[[Image:draft_Samper_722607179-belyPlastic3DEffectiveStress.png|540px|Plane strain elastoplastic cantilever. Numerical results for the 3D simulation. Von Mises effective stress plotted over the deformed configuration at (t=6.05s).]]
2901
|- style="text-align: center; font-size: 75%;"
2902
| colspan="1" | '''Figure 25:''' Plane strain elastoplastic cantilever. Numerical results for the 3D simulation. Von Mises effective stress plotted over the deformed configuration at (<math>t=6.05s</math>).
2903
|}
2904
2905
In Figure [[#img-26|26]] for the same time instant the XX-component of the Cauchy stress tensor is plotted. 
2906
2907
<div id='img-26'></div>
2908
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
2909
|-
2910
|[[Image:draft_Samper_722607179-belyPlastic3DCauchy.png|540px|Plane strain elastoplastic cantilever. Numerical results for the 3D simulation:  XX-component of the Cauchy stress tensor plotted over the deformed configuration at (t=6.05s).]]
2911
|- style="text-align: center; font-size: 75%;"
2912
| colspan="1" | '''Figure 26:''' Plane strain elastoplastic cantilever. Numerical results for the 3D simulation:  XX-component of the Cauchy stress tensor plotted over the deformed configuration at (<math>t=6.05s</math>).
2913
|}
2914
2915
In  Figure [[#img-27|27]] the 3D solution is compared to the 2D results obtained with a structured mesh with the same average size. The curves coincide almost exactly. 
2916
2917
<div id='img-27'></div>
2918
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 100%;"
2919
|-
2920
|[[Image:draft_Samper_722607179-belyPlastic2Dvs3D.png|480px|Plane strain elastoplastic cantilever. Time evolution of the top corner vertical displacement. Numerical results for the 2D and the 3D simulations for the same average mesh size (0.125).]]
2921
|- style="text-align: center; font-size: 75%;"
2922
| colspan="1" | '''Figure 27:''' Plane strain elastoplastic cantilever. Time evolution of the top corner vertical displacement. Numerical results for the 2D and the 3D simulations for the same average mesh size (0.125).
2923
|}
2924
2925
==2.4 Summary and conclusions==
2926
2927
In this chapter two velocity-based finite element Lagrangian procedures, namely the Velocity and the mixed Velocity-Pressure formulations, have been derived for a general compressible material.
2928
2929
The derivation of both formulations has been carried out with the aim of maintaining the scheme as general as possible. The mixed Velocity-Pressure formulation has been derived exploiting the linearized form of the Velocity formulation. In particular, it has been shown that the tangent matrix of the linear momentum equations is the same for both schemes.
2930
2931
The mixed Velocity-Pressure procedure is based on a two-step Gauss-Seidel solution algorithm. First, the linear momentum equations are solved for the velocity increments, next the continuity equation is solved for the pressure in the updated configuration. At the end of these steps the convergence for the velocities and the pressure is checked. Linear interpolation has been used for both velocity and pressure fields.
2932
2933
Next, both formulations have been particularized for hypoelastic solids. The finite elements generated from the Velocity formulation and the mixed Velocity-Pressure formulations have been called V and VP element, respectively.
2934
2935
The numerical scheme for dealing with <math display="inline">J_2</math> associative plasticity has been also given.
2936
2937
Several numerical examples have been given for validating the V and VP elements for large displacements dynamics problems involving both hypoelastic and hypoelastoplastic compressible solids. It has been shown that both elements are convergent for all the numerical examples analyzed.
2938
2939
=Chapter 3. Unified stabilized formulation for quasi-incompressible materials=
2940
2941
2942
This chapter is devoted to the derivation and validation of the unified stabilized formulation for nearly-incompressible materials. Namely, the cases of quasi-incompressible Newtonian fluids and the quasi-incompressible hypoelastic solids will be analyzed.
2943
2944
Quasi-incompressible materials have a compressibility that is small enough to neglect the variation of density on time but, unlike fully incompressible materials, they are not totally divergence-free and the volumetric strain rate is related to the variation on time of pressure via Eq.([[#eq-75|75]]). This stabilized formulation is based on the mixed velocity-pressure formulation derived in the previous chapter for a general material. In fact a one-field method, as the Velocity formulation presented in Chapter [[#2 Velocity-based formulations for compressible materials|2]], is not sufficient for dealing with the incompressibility constraint. Furthermore the <math display="inline">inf-sup</math> condition [15] imposes the stabilization of the mixed finite element procedure, if an equal order interpolation is used for the velocities and the pressure, as in this work. Consequently, the mixed Velocity-Pressure formulation derived for compressible materials in the previous chapter needs to be stabilized in order to solve quasi-incompressible problems.
2945
2946
In this work a new stabilized Lagrangian method for quasi-incompressible materials is derived. The stabilization procedure is based on the consistent derivation of a residual-based stabilized expression of the mass balance equation using the <math display="inline">{Finite Calculus (FIC)}</math>, also called <math display="inline">{Finite  Increment Calculus}</math>  method [
2947
[43,76,92,94,96–98].  The main ideas of this part of the thesis are taken from [82] where the stabilization technique was derived for  homogeneous viscous  fluids. In this chapter it is shown that the procedure can be extended also for the analysis of quasi-incompressible solids.
2948
2949
The FIC approach in mechanics is based on expressing the equations of balance of mass and momentum in a space-time domain of finite size and retaining higher order terms in the Taylor series expansion typically used for expressing the change in the transported variables within the balance domain. In addition to the standard terms of infinitesimal theory, the FIC form of the balance equations contains derivatives of the classical differential equations in mechanics multiplied by characteristic distances in space and time.
2950
2951
In this work the second order FIC form in space and the first order FIC form in time of the mass balance equation have been used as the basis for the derivation of the stabilized formulation. The discretized variational form of the FIC mass balance equation via the FEM introduces terms in the Neumann boundary of the domain and other terms involving the first and second material time derivatives of the pressure. These terms are relevant for ensuring the consistency of the residual formulation.
2952
2953
The FIC stabilization, although is derived using the linear momentum equations, affects only the continuity equation. This means that the general form of the discretized and linearized momentum equations derived in Chapter  [[#2 Velocity-based formulations for compressible materials|2]] for the mixed Velocity-Pressure formulation still holds for quasi-incompressible materials. Hence, for hypoelastic quasi-incompressible materials the linear momentum equations are solved through the same linear system derived for the VP (compressible) element in Section [[#2.3.2 Mixed Velocity-Pressure formulation for hypoelastic solids|2.3.2]]. To avoid repetitions, the linearized momentum equations for the mixed velocity-pressure formulation will be recalled only for Newtonian fluids by deriving the tangent matrix and the internal forces according to the constitutive law.
2954
2955
For convenience, the stabilized form of the continuity equation is derived for quasi-incompressible Newtonian fluids, as in [82]. Nevertheless it will be shown that the approach can be easily extended to quasi-incompressible hypoelastic solids.
2956
2957
For the fluid analysis a Lagrangian procedure called Particle Finite Element Mehtod (PFEM) [88+ is used. With the PFEM, the  mesh nodes are treated as particles and they move according to the governing equations. The domain is continuously remeshed using a procedure that efficiently combines the Delaunay tessellation and the  Alpha Shape Method [39].
2958
2959
The FIC stabilized formulation here presented  [82] has excellent mass preservation feature in the analysis of free surface fluid problems.  Preservation of mass is a great challenge in the numerical study of flow problems with high values of the bulk modulus that approach the conditions of incompressibility. Mass losses can be induced by the stabilization terms which are typically added to the discretized form of the momentum and mass balance equations in order to account for high convective effects  in the Eulerian description of the flow, and to satisfy the <math display="inline">inf-sup</math> condition imposed by the full incompressibility constraint when equal order interpolation of the velocities and the pressure is used in mixed FEMs [9,38,135,137].
2960
2961
An important source of mass loss  emanates in the numerical solution of free surface flows  due, among other reasons, to the inaccuracies in predicting  the shape of the free surface during large flow motions [65]. Mass losses can also occur in the numerical solution of flows with heterogeneous material properties [64] and in homogeneous viscous flows using the Laplace form of the Navier-Stokes equations [69].
2962
2963
In Lagrangian analysis procedures (such as the PFEM) the motion of the fluid particles is tracked during the transient solution. Hence, the convective terms vanish in the momentum equations and no numerical stabilization is needed for treating those terms. Two other sources of mass loss, however, remain in the numerical solution of Lagrangian flows, i.e. that due to the treatment of the incompressibility constraint by a stabilized numerical method, and that induced by the inaccuracies in tracking the flow particles and, in particular, the free surface.
2964
2965
The discretized variational form of the FIC mass balance equation via the FEM introduces terms in the Neumann boundary of the domain, and other terms involving the first and second material time derivatives of the pressure that are relevant for ensuring the consistency of the residual formulation. These terms are also crucial for preserving the  mass during the transient solution of free surface Lagrangian flows. In addition they enable the computation of the nodal pressures from the stabilized mass balance equation without imposing any  condition on the pressure at the free surface nodes, thus eliminating another source of mass loss which occurs when the pressure is prescribed to a zero value on the free surface in viscous flows.
2966
2967
A section of this chapter is exclusively dedicated to show the excellent mass preservation  features of the PFEM-FIC stabilized formulation for free surface flow problems.
2968
2969
Various approaches have been developed in the recent years for approximating fluid flows by means of a quasi-incompressible material. In practice, they all consider the Navier-Stokes problem with a modified mass conservation equation where a slight compressibility is added to the fluid. In previous works, see for example [27,92,122,137], the fluid compressibility was introduced  by relaxing the incompressibility constraint by means of a penalty parameter. Alternatively, the small compressibility of the fluid can be introduced by considering the actual bulk modulus of the fluid <math display="inline">{\kappa }</math>, which gives this operation a physical meaning, [38,63,137]. The success of quasi-incompressible formulations in fluid mechanics relies on their important advantages from the numerical point of view. The most obvious one is that the quasi-incompressible form of the continuity equation yields a direct relation between the two unknown fields of the Navier-Stokes problem, the velocities and the pressure (Eq.([[#eq-75|75]])). This is useful if the problem is solved using a partitioned scheme because the velocity-pressure relation is crucial for deriving the tangent matrix of the momentum equations. Furthermore, another important drawback of fully incompressible schemes is eluded; the incompressibility constraint leads to a diagonal block of a zero matrix in the global matrix system. Consequently, a pivoting procedure is required to solve numerically this kind of linear  system. It is well known that the computational cost associated to this operation is high and it increases  with the number of degrees of freedom of the problem. The compressibility terms that emanate from the quasi-incompressible form of the continuity equation fill the diagonal of the global matrix, thus overcoming these numerical difficulties.
2970
2971
On the other hand, quasi-incompressible schemes insert in the numerical model parameters that have typically high values and can lead to different numerical instabilities. For example, large values of the penalty parameter or, equally, physical values of the bulk modulus, can compromise the quality of the analyses, or even prevent the convergence of the solution scheme, [137]. For this reason, generally, the value of the actual bulk modulus is reduced arbitrarily to the so-called <math display="inline">{pseudo bulk modulus}</math>. Nevertheless, an excessively small value of the pseudo bulk modulus changes drastically the meaning of the continuity equation of the original Navier-Stokes problem. In other words, the incompressibility constraint would not be satisfied at all. Furthermore, the bulk modulus is proportional to the speed  of sound propagating through the material. Hence, we have to guarantee that the order of magnitude of the velocities of the problem is several times smaller than the velocity of sound in the medium.
2972
2973
In this work, the pseudo-bulk modulus <math display="inline">\kappa _p</math> is used for the tangent matrix of the linear momentum equations, while the actual physical value of the bulk modulus <math display="inline">\kappa </math> is used for the numerical solution of the mass conservation equation. The pseudo bulk modulus is computed as a proportion of the real bulk modulus of the fluid through the parameter <math display="inline">\theta </math> such that <math display="inline">\kappa _p=\theta \kappa </math> with <math display="inline">0<\theta \leq 1</math>. A new numerical strategy for computing a priori the optimum value for the pseudo bulk modulus is also derived.  For free surface flow problems,  it will be shown that the scheme guarantees the good conditioning of the linear system, good convergence and excellent mass preservation features.
2974
2975
The lay-out of this chapter is the following. First the stabilized FIC form of the mass balance equation is derived. The procedure is described for a Newtonian fluid from the local and continuous form until the fully discretized matrix form. Next the linearized and discretized expression of the linear momentum equations derived in Chapter [[#2 Velocity-based formulations for compressible materials|2]] for the mixed Velocity-Pressure formulation is adapted for Newtonian fluids and the complete solution scheme is given. Next the FIC stabilization procedure is extended to  hypoelastic quasi-incompressible materials and the complete solution scheme is given also for this constitutive model.
2976
2977
In Section [[#3.4 Free surface flow analysis|3.4]],  free surface flows are  analyzed in detail. First the essential features of the PFEM are given, then the mass preservation feature of the PFEM-FIC stabilized formulation is shown together with some explicative numerical examples. Finally the conditioning of the linear system is studied and a practical and efficient technique to predict <math display="inline">apriori</math> the optimum value for the pseudo bulk modulus is given.
2978
2979
The chapter ends up with several validation examples for quasi-incompressible solid and fluid mechanics problems.
2980
2981
==3.1 Stabilized FIC form of the mass balance equation==
2982
2983
The FIC stabilized form of the continuity equation is here derived. For convenience, the derivation procedure is carried out for  quasi-incompressible Newtonian fluids.
2984
2985
===3.1.1 Governing equations===
2986
2987
For the sake of clarity, the local forms of the linear momentum <math display="inline">{r_m}_i</math> and the continuity <math display="inline">r_v</math> equations are recalled. From Eqs.([[#eq-1|1]]) and ([[#eq-75|75]]) yields
2988
2989
<span id="eq-150"></span>
2990
{| class="formulaSCP" style="width: 100%; text-align: left;" 
2991
|-
2992
| 
2993
{| style="text-align: left; margin:auto;" 
2994
|-
2995
| style="text-align: center;" | <math>{r_m}_i:=\rho {\partial v_i \over \partial t}-{\partial \sigma _{ij} \over \partial x_j}-b_i=0\quad , \quad i,j=1,n_s \quad  \hbox{in } \Omega  </math>
2996
|}
2997
| style="width: 5px;text-align: right;" | (150)
2998
|}
2999
3000
<span id="eq-151"></span>
3001
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3002
|-
3003
| 
3004
{| style="text-align: left; margin:auto;" 
3005
|-
3006
| style="text-align: center;" | <math>r_v:=-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v=0  </math>
3007
|}
3008
| style="width: 5px;text-align: right;" | (151)
3009
|}
3010
3011
The terms of Eqs.([[#eq-150|150]] and [[#eq-151|151]]) have already been defined in the previous chapters.
3012
3013
The standard form of the constitutive relation for a Newtonian fluid reads
3014
3015
<span id="eq-152"></span>
3016
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3017
|-
3018
| 
3019
{| style="text-align: left; margin:auto;" 
3020
|-
3021
| style="text-align: center;" | <math>\sigma _{ij} =\sigma _{ij}' + p \delta _{ij}= 2\mu d_{ij}' + p \delta _{ij}  </math>
3022
|}
3023
| style="width: 5px;text-align: right;" | (152)
3024
|}
3025
3026
where <math display="inline">\mu </math> is the viscosity and the deviatoric strain rate <math display="inline">d_{ij}'</math> is defined from Eq.([[#eq-71|71]]) as
3027
3028
<span id="eq-153"></span>
3029
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3030
|-
3031
| 
3032
{| style="text-align: left; margin:auto;" 
3033
|-
3034
| style="text-align: center;" | <math>d_{ij}' = d_{ij} - {1 \over 3} d^v \delta _{ij} </math>
3035
|}
3036
| style="width: 5px;text-align: right;" | (153)
3037
|}
3038
3039
where <math display="inline">d^v</math> is the volumetric strain rate.
3040
3041
Substituting Eqs.([[#eq-152|152]]) and ([[#eq-153|153]]) into ([[#eq-150|150]]), gives a useful form of the momentum equations  for the Newtonian fluids as
3042
3043
<span id="eq-154"></span>
3044
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3045
|-
3046
| 
3047
{| style="text-align: left; margin:auto;" 
3048
|-
3049
| style="text-align: center;" | <math>\rho {\partial v_i \over \partial t}-{\partial  \over \partial x_j}(2\mu d_{ij})+{\partial  \over \partial x_i}\left(\frac{2}{3}\mu d^v\right)- {\partial p \over \partial x_i}-b_i=0 \quad , \quad i,j=1,n_s  </math>
3050
|}
3051
| style="width: 5px;text-align: right;" | (154)
3052
|}
3053
3054
===3.1.2 FIC mass balance equation in space and in time===
3055
3056
Previous stabilized FEM formulations for quasi and fully incompressible fluids and solids were based  on the first order form of the Finite Calculus (FIC) balance equation  in space [76-85,92,97,98]. In this work, for the derivation of stabilized formulation both the second order FIC form of the mass balance equation in space [92,96] and the first order FIC form of the mass balance equation in time are used. These forms read respectively I[82]
3057
3058
<span id="eq-155"></span>
3059
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3060
|-
3061
| 
3062
{| style="text-align: left; margin:auto;" 
3063
|-
3064
| style="text-align: center;" | <math>r_v + \frac{h_i^2}{12} \frac{\partial ^2 r_v}{\partial x^2_i}=0\qquad \hbox{in }\Omega \qquad i=1,n_s </math>
3065
|}
3066
| style="width: 5px;text-align: right;" | (155)
3067
|}
3068
3069
and
3070
3071
<span id="eq-156"></span>
3072
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3073
|-
3074
| 
3075
{| style="text-align: left; margin:auto;" 
3076
|-
3077
| style="text-align: center;" | <math>r_v + \frac{\delta }{2} {\partial r_v \over \partial t}=0 \qquad \hbox{in }\Omega  </math>
3078
|}
3079
| style="width: 5px;text-align: right;" | (156)
3080
|}
3081
3082
Eq.([[#eq-155|155]]) is obtained by expressing the balance of mass in a rectangular domain of finite size with dimensions <math display="inline">h_1\times h_2</math> (for 2D problems), where <math display="inline">h_i</math> are arbitrary distances, and retaining up to third order terms in the Taylor series expansions used for expressing the change of mass within the balance domain. The derivation of Eq.([[#eq-155|155]]) for 2D incompressible flows can be found in [96].
3083
3084
Eq.([[#eq-156|156]]), on the other hand, is obtained by expressing the balance of mass in a space-time domain of infinitesimal length in space and finite dimension <math display="inline">\delta </math> in time [76].
3085
3086
The FIC terms in Eqs.([[#eq-155|155]]) and ([[#eq-156|156]]) play the role of space and time stabilization terms respectively. In the discretized problem, the space dimensions <math display="inline">h_i</math> and the time dimension <math display="inline">\delta </math> are related to characteristic element dimensions and the time step increment, respectively as it will be explained later.
3087
3088
Note that for <math display="inline">h_i \to 0</math> and <math display="inline">\delta \to 0</math> the standard form of the mass balance equation ([[#eq-151|151]]), as given by the infinitesimal theory, is recovered.
3089
3090
===3.1.3 FIC stabilized local form of the mass balance equation===
3091
3092
Substituting Eq.([[#eq-151|151]]) into Eqs.([[#eq-155|155]]) and ([[#eq-156|156]]) the second order FIC form in space and the first order FIC form in time of the mass balance equation for a general quasi-incompressible material read
3093
3094
<span id="eq-157"></span>
3095
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3096
|-
3097
| 
3098
{| style="text-align: left; margin:auto;" 
3099
|-
3100
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \frac{h_i^2}{12}\frac{\partial ^2}{\partial  x^2_i}\left(\frac{1}{\kappa }{\partial p \over \partial t} \right)+ \frac{h_i^2}{12}\frac{\partial }{\partial x_i} \left(\frac{\partial d^v}{\partial x_i}\right)=0\qquad \hbox{in }\Omega \qquad i=1,n_s </math>
3101
|}
3102
| style="width: 5px;text-align: right;" | (157)
3103
|}
3104
3105
and
3106
3107
<span id="eq-158"></span>
3108
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3109
|-
3110
| 
3111
{| style="text-align: left; margin:auto;" 
3112
|-
3113
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \frac{\delta }{2 \kappa }{\partial ^2p \over \partial t^2} + \frac{\delta }{2}{\partial d^v \over \partial t}=0  </math>
3114
|}
3115
| style="width: 5px;text-align: right;" | (158)
3116
|}
3117
3118
The FIC form of the mass balance equation is expressed in terms of the momentum equations. Neglecting the space changes of the viscosity <math display="inline">\mu </math>, from Eq.([[#eq-154|154]]) the following expression is obtained
3119
3120
<span id="eq-159"></span>
3121
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3122
|-
3123
| 
3124
{| style="text-align: left; margin:auto;" 
3125
|-
3126
| style="text-align: center;" | <math>\frac{2}{3}\mu \frac{\partial  d^v}{\partial x_i} = - \rho {\partial v_i \over \partial t} + 2\mu {\partial  \over \partial x_j} ( d_{ij}) + {\partial p \over \partial x_i}+b_i = -\rho {\partial v_i \over \partial t} +  \hat r_{m_i} </math>
3127
|}
3128
| style="width: 5px;text-align: right;" | (159)
3129
|}
3130
3131
Hence
3132
3133
<span id="eq-160"></span>
3134
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3135
|-
3136
| 
3137
{| style="text-align: left; margin:auto;" 
3138
|-
3139
| style="text-align: center;" | <math>\frac{\partial  d^v}{\partial x_i} = \frac{3}{2\mu } \left[-\rho {\partial v_i \over \partial t} + \hat r_{m_i}\right] </math>
3140
|}
3141
| style="width: 5px;text-align: right;" | (160)
3142
|}
3143
3144
In the above two equations <math display="inline">\hat r_{m_i}</math> is a ''static momentum term'' defined as
3145
3146
<span id="eq-161"></span>
3147
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3148
|-
3149
| 
3150
{| style="text-align: left; margin:auto;" 
3151
|-
3152
| style="text-align: center;" | <math>\hat r_{m_i}= 2\mu {\partial  \over \partial x_j} ( d_{ij}) + \frac{\partial p}{\partial x_i} + b_i   </math>
3153
|}
3154
| style="width: 5px;text-align: right;" | (161)
3155
|}
3156
3157
Substituting Eq.([[#eq-160|160]]) into Eq.([[#eq-157|157]]) and neglecting the space changes of <math display="inline">c</math> and <math display="inline">\rho </math> in the derivatives, the following form is obtained
3158
3159
<span id="eq-162"></span>
3160
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3161
|-
3162
| 
3163
{| style="text-align: left; margin:auto;" 
3164
|-
3165
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \frac{h_i^2}{12}\frac{\partial ^2}{\partial x^2_i}\left(\frac{1}{\kappa }{\partial p \over \partial t}\right)+ \frac{ h_i^2}{8\mu } \frac{\partial }{\partial x_i} \left(-\rho {\partial v_i \over \partial t}+\hat r_{m_i}\right)=0  </math>
3166
|}
3167
| style="width: 5px;text-align: right;" | (162)
3168
|}
3169
3170
Observation of the term involving the material derivative of <math display="inline">v_i</math> in Eq.([[#eq-162|162]]) gives
3171
3172
<span id="eq-163"></span>
3173
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3174
|-
3175
| 
3176
{| style="text-align: left; margin:auto;" 
3177
|-
3178
| style="text-align: center;" | <math>\frac{\partial }{\partial x_i}  \left(-\rho {\partial v_i \over \partial t}\right)=-\rho {\partial  \over \partial t}  \left({\partial v_i \over \partial x_i}\right)=-\rho {\partial d^v \over \partial t}  </math>
3179
|}
3180
| style="width: 5px;text-align: right;" | (163)
3181
|}
3182
3183
Substituting Eq.([[#eq-163|163]]) into ([[#eq-162|162]]) gives
3184
3185
<span id="eq-164"></span>
3186
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3187
|-
3188
| 
3189
{| style="text-align: left; margin:auto;" 
3190
|-
3191
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \frac{h_i^2}{12 \kappa }\frac{\partial ^2}{\partial x^2_i}\left({\partial p \over \partial t}\right)+ \frac{h_i^2}{8\mu } \left(-\rho {\partial d^v \over \partial t}+ {\partial \hat r_{m_i} \over \partial x_i}   \right)=0  </math>
3192
|}
3193
| style="width: 5px;text-align: right;" | (164)
3194
|}
3195
3196
From Eq.([[#eq-158|158]]),
3197
3198
<span id="eq-165"></span>
3199
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3200
|-
3201
| 
3202
{| style="text-align: left; margin:auto;" 
3203
|-
3204
| style="text-align: center;" | <math>-{\partial d^v \over \partial t}= - \frac{2}{\delta \kappa }{\partial p \over \partial t}+ \frac{2}{\delta }  d^v - \frac{1}{\kappa }{\partial ^2p \over \partial t^2}  </math>
3205
|}
3206
| style="width: 5px;text-align: right;" | (165)
3207
|}
3208
3209
Substituting Eq.([[#eq-165|165]]) into Eq.([[#eq-164|164]]) gives
3210
3211
<span id="eq-166"></span>
3212
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3213
|-
3214
| 
3215
{| style="text-align: left; margin:auto;" 
3216
|-
3217
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v -\frac{h_i^2}{12 \kappa }\frac{\partial ^2}{\partial x_i^2}\left({\partial p \over \partial t}\right)+ \frac{ h_i^2}{8\mu } \left(-\frac{2 \rho }{\delta \kappa }{\partial p \over \partial t} + \frac{2\rho }{\delta }  d^v - \frac{\rho }{\kappa }{\partial ^2p \over \partial t^2}+   {\partial \hat r_{m_i} \over \partial x_i}   \right)=0  </math>
3218
|}
3219
| style="width: 5px;text-align: right;" | (166)
3220
|}
3221
3222
Multiplying Eq.([[#eq-166|166]]) by <math display="inline">\frac{8\mu }{h^2}</math> gives, after grouping some terms,
3223
3224
<span id="eq-167"></span>
3225
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3226
|-
3227
| 
3228
{| style="text-align: left; margin:auto;" 
3229
|-
3230
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}\left( \frac{8\mu }{h_i^2}+ \frac{2 \rho }{\delta } \right) +  d^v \left( \frac{8\mu }{h_i^2}+ \frac{2 \rho }{\delta } \right)  -\frac{2 \mu }{3 \kappa }\frac{\partial ^2}{\partial x_i^2}\left({\partial p \over \partial t}\right) - \frac{\rho }{\kappa }{\partial ^2p \over \partial t^2}+   {\partial \hat r_{m_i} \over \partial x_i} =0  </math>
3231
|}
3232
| style="width: 5px;text-align: right;" | (167)
3233
|}
3234
3235
After some further transformations,
3236
3237
<span id="eq-168"></span>
3238
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3239
|-
3240
| 
3241
{| style="text-align: left; margin:auto;" 
3242
|-
3243
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \frac{2\mu \tau }{3\kappa }  {\partial  \over \partial x_i} \left({\partial  \over \partial x_i} \left({\partial p \over \partial t}\right)\right)- \tau \frac{\rho }{\kappa } {\partial ^2p \over \partial t^2} +\tau {\partial \hat r_{m_i} \over \partial x_i} =0  </math>
3244
|}
3245
| style="width: 5px;text-align: right;" | (168)
3246
|}
3247
3248
Where <math display="inline">\tau </math> is a ''stabilization parameter'' given by
3249
3250
<span id="eq-169"></span>
3251
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3252
|-
3253
| 
3254
{| style="text-align: left; margin:auto;" 
3255
|-
3256
| style="text-align: center;" | <math>\tau = \left(\frac{8\mu }{h^2}+ \frac{2\rho }{\delta } \right)^{-1}  </math>
3257
|}
3258
| style="width: 5px;text-align: right;" | (169)
3259
|}
3260
3261
For ''transient problems'' the stabilization parameter <math display="inline">\tau </math>  is computed as
3262
3263
<span id="eq-170"></span>
3264
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3265
|-
3266
| 
3267
{| style="text-align: left; margin:auto;" 
3268
|-
3269
| style="text-align: center;" | <math>\tau = \left(\frac{8\mu }{(l^e)^2} + \frac{2\rho }{\Delta t}  \right)^{-1} </math>
3270
|}
3271
| style="width: 5px;text-align: right;" | (170)
3272
|}
3273
3274
where <math display="inline">\Delta t</math> is the time step used for the transient solution and <math display="inline">l^e</math> is a characteristic element length.
3275
3276
The coefficient <math display="inline">\frac{2\mu \tau }{3\kappa }</math> multiplying the second space derivatives of <math display="inline">{\partial p \over \partial t}</math> in Eq.([[#eq-168|168]]) is much smaller than the coefficients multiplying the rest of the terms in this equation.  Numerical tests have shown that the results are not affected by this term. Consequently, this second space derivative term will be neglected in the rest of this work.
3277
3278
Hence the FIC stabilized form of the mass balance equation is written as
3279
3280
<span id="eq-171"></span>
3281
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3282
|-
3283
| 
3284
{| style="text-align: left; margin:auto;" 
3285
|-
3286
| style="text-align: center;" | <math>-\frac{1}{\kappa }{\partial p \over \partial t}+ d^v - \tau \frac{\rho }{\kappa } {\partial ^2p \over \partial t^2} +\tau {\partial \hat r_{m_i} \over \partial x_i} =0  </math>
3287
|}
3288
| style="width: 5px;text-align: right;" | (171)
3289
|}
3290
3291
Note that the term <math display="inline">{\partial  \over \partial x_i}  \left(2\mu {\partial  \over \partial x_j} ( d_{ij})\right)</math> within <math display="inline">\hat r_{m_i}</math> in Eq.([[#eq-171|171]]) (see the definition of <math display="inline">\hat r_{m_i}</math> in Eq.([[#eq-161|161]])) vanishes for a linear approximation of the velocity field. This is the case for the simplicial elements used in this work.
3292
3293
===3.1.4 Variational form===
3294
3295
Multiplying Eq.([[#eq-171|171]]) by arbitrary (continuous) test functions <math display="inline">q</math> (with dimensions of pressure) and integrating over the analysis domain <math display="inline">\Omega </math> gives
3296
3297
<span id="eq-172"></span>
3298
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3299
|-
3300
| 
3301
{| style="text-align: left; margin:auto;" 
3302
|-
3303
| style="text-align: center;" | <math>\int _\Omega -\frac{q}{\kappa } {\partial p \over \partial t}d\Omega -\int _\Omega q \frac{\rho }{\kappa }{\partial ^2p \over \partial t^2}d\Omega + \int _\Omega q  d^v d\Omega  + \int _\Omega q \tau {\partial \hat r_{m_i} \over \partial x_i} d\Omega =0  </math>
3304
|}
3305
| style="width: 5px;text-align: right;" | (172)
3306
|}
3307
3308
Integrating by parts the last integral in Eq.([[#eq-172|172]]) (and neglecting the space changes of <math display="inline">\tau </math>) yields
3309
3310
<span id="eq-173"></span>
3311
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3312
|-
3313
| 
3314
{| style="text-align: left; margin:auto;" 
3315
|-
3316
| style="text-align: center;" | <math>\int _\Omega \!-\frac{q}{\kappa } {\partial p \over \partial t}d\Omega -\!\int _\Omega q \tau \frac{\rho }{\kappa } {\partial ^2p \over \partial t^2}d\Omega + \!\int _\Omega q  d^v d\Omega - \!\int _\Omega \tau {\partial q \over \partial x_i} \hat r_{m_i} d\Omega  +\!\underbrace{\int _{\Gamma } q \tau \hat r_{m_i}   n_i d\Gamma }_{BT} =0   </math>
3317
|}
3318
| style="width: 5px;text-align: right;" | (173)
3319
|}
3320
3321
where <math display="inline">n_i</math> are the components of the unit normal vector to the external boundary <math display="inline">\Gamma </math> of <math display="inline"> \Omega </math>.
3322
3323
Using Eq.([[#eq-160|160]]) an equivalent form for the boundary term BT of Eq.([[#eq-173|173]]) is obtained as
3324
3325
<span id="eq-174"></span>
3326
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3327
|-
3328
| 
3329
{| style="text-align: left; margin:auto;" 
3330
|-
3331
| style="text-align: center;" | <math>BT = \int _\Gamma q \tau \hat r_{m_i}   n_i d\Gamma = \int _\Gamma q\tau \left(\rho {\partial v_i \over \partial t}+ \frac{2\mu }{3}{\partial  d^v \over \partial x_i}\right)n_i d\Gamma  = \int _\Gamma q \tau \left(\rho {\partial v_n \over \partial t}+ \frac{2\mu }{3}{\partial  d^v \over \partial n} \right)d\Gamma   </math>
3332
|}
3333
| style="width: 5px;text-align: right;" | (174)
3334
|}
3335
3336
where <math display="inline">{\partial  d^v \over \partial n} </math> is the derivative of the volumetric strain rate in the direction of the normal to the external boundary and <math display="inline">v_n</math> is the velocity normal to the boundary.
3337
3338
The term <math display="inline">\frac{2\mu }{3}{\partial  d^v \over \partial n} </math> can be approximated as follows
3339
3340
<span id="eq-175"></span>
3341
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3342
|-
3343
| 
3344
{| style="text-align: left; margin:auto;" 
3345
|-
3346
| style="text-align: center;" | <math>\frac{2\mu }{3}{\partial  d^v \over \partial n} =\frac{2}{h_n} \left( \frac{2}{3} \mu ^+  d^{v+} -\frac{2}{3} \mu ^- d^{v-}\right)\quad \hbox{at }\Gamma   </math>
3347
|}
3348
| style="width: 5px;text-align: right;" | (175)
3349
|}
3350
3351
where <math display="inline">(\mu ^+, d^{v+})</math> and <math display="inline">(\mu ^-,  d^{v-})</math> are respectively the values of <math display="inline">\mu </math> and <math display="inline"> d^v</math> at exterior and interior points of the boundary <math display="inline">\Gamma </math> and <math display="inline">h_n</math> is a characteristic length in the normal direction to the boundary. Figure [[#img-28|28]] shows an example of the computation of <math display="inline"> \frac{2}{3} \mu {\partial  d^v \over \partial n}</math> at the side of a 3-noded triangle adjacent to the external boundary. The same procedure applies for 4-noded tetrahedra.
3352
3353
<div id='img-28'></div>
3354
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
3355
|-
3356
|[[Image:draft_Samper_722607179-Fig-comput-mu-modificada.png|500px]]
3357
| <math>\begin{array}{l}\displaystyle \frac{2}{3} \mu {\partial  d^v \over \partial n}\vert_{ij} \simeq \frac{2}{h_n} \left( \frac{2}{3} \mu^+ d^{v+}-\frac{2}{3} \mu^- d^{v-} \right)\\
3358
\displaystyle d^{v+} =0 \quad \hbox{outside } \Omega\\
3359
\displaystyle \frac{2}{3} \mu {\partial  d^v \over \partial n}\vert_{ij} \simeq \frac{4}{3h_n} \mu^- d^{v-} = - \frac{4}{3h_n} \mu^e d^{v,e}\end{array}</math>
3360
|- style="text-align: center; font-size: 75%;"
3361
| colspan="2" | '''Figure 28:''' Computation of the term  of <math>\displaystyle \frac{2}{3} \mu {\partial  d^v \over \partial n}</math> at the side <math>ij</math> of a 3-noded triangle <math>ijk</math> adjacent to the external boundary <math>\Gamma </math>.
3362
|}
3363
3364
Clearly, at external boundaries <math display="inline"> d^{v+}=0</math> and <math display="inline"> d^{v-} = d^v</math>. Hence, <math display="inline"> d^{v-}</math> coincides with the volumetric strain in the 3-noded triangular element adjacent to the boundary.
3365
3366
Using above argument Eq.([[#eq-175|175]]) simplifies to
3367
3368
<span id="eq-176"></span>
3369
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3370
|-
3371
| 
3372
{| style="text-align: left; margin:auto;" 
3373
|-
3374
| style="text-align: center;" | <math>\frac{2\mu }{3}{\partial  d^v \over \partial n} =-\frac{4\mu }{3h_n}   d^v\quad \hbox{at }\Gamma   </math>
3375
|}
3376
| style="width: 5px;text-align: right;" | (176)
3377
|}
3378
3379
On the other hand, the stresses at any boundary satisfy the traction equilibrium condition
3380
3381
<span id="eq-177"></span>
3382
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3383
|-
3384
| 
3385
{| style="text-align: left; margin:auto;" 
3386
|-
3387
| style="text-align: center;" | <math>\sigma _{ij} n_j -t_i =0 \quad \hbox{at }\Gamma   </math>
3388
|}
3389
| style="width: 5px;text-align: right;" | (177)
3390
|}
3391
3392
Substituting Eqs.([[#eq-152|152]]) and  ([[#eq-153|153]]) into ([[#eq-177|177]]) and multiplying all terms by <math display="inline">n_i</math>, yields
3393
3394
<span id="eq-178"></span>
3395
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3396
|-
3397
| 
3398
{| style="text-align: left; margin:auto;" 
3399
|-
3400
| style="text-align: center;" | <math>2\mu {\partial v_n \over \partial n} - \frac{2}{3} \mu  d^v + p -t_n =0 \quad \hbox{at }\Gamma   </math>
3401
|}
3402
| style="width: 5px;text-align: right;" | (178)
3403
|}
3404
3405
where <math display="inline">t_n</math> is the normal traction to the boundary (Figure [[#img-28|28]]) and <math display="inline">{\partial v_n \over \partial n}= n_i {\partial v_i \over \partial x_j}n_j</math>.
3406
3407
From Eq.([[#eq-178|178]])
3408
3409
<span id="eq-179"></span>
3410
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3411
|-
3412
| 
3413
{| style="text-align: left; margin:auto;" 
3414
|-
3415
| style="text-align: center;" | <math>\frac{2}{3} \mu  d^v= 2\mu {\partial v_n \over \partial n}+ p -t_n\quad \hbox{at }\Gamma   </math>
3416
|}
3417
| style="width: 5px;text-align: right;" | (179)
3418
|}
3419
3420
Substituting Eq.([[#eq-179|179]]) into Eq.([[#eq-176|176]]) and this one into Eq.([[#eq-174|174]]), gives the  expression of the boundary integral of Eq.([[#eq-173|173]]) as
3421
3422
<span id="eq-180"></span>
3423
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3424
|-
3425
| 
3426
{| style="text-align: left; margin:auto;" 
3427
|-
3428
| style="text-align: center;" | <math>BT =\int _\Gamma q \tau \left(\rho {\partial v_n \over \partial t}-\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} + p -t_n)\right)d\Gamma   </math>
3429
|}
3430
| style="width: 5px;text-align: right;" | (180)
3431
|}
3432
3433
The normal velocity <math display="inline">v_n</math> is fixed at a Dirichlet boundary <math display="inline">\Gamma _v</math> and, hence, <math display="inline">{\partial v_n \over \partial t}=0</math> at <math display="inline">\Gamma _v</math>. Also, accepting that <math display="inline"> d^v=0</math> at <math display="inline">\Gamma _v</math>,   the surface tractions at <math display="inline">\Gamma _v</math> coincide precisely with the reactions computed as <math display="inline">t_n =2\mu {\partial v_n \over \partial n}+p</math>. Hence, the boundary integral can be neglected at a Dirichlet boundary and, therefore, it has a meaning at a Neumann boundary <math display="inline">{\Gamma _t}</math> only. In conclusion,
3434
3435
<span id="eq-181"></span>
3436
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3437
|-
3438
| 
3439
{| style="text-align: left; margin:auto;" 
3440
|-
3441
| style="text-align: center;" | <math>BT = \int _{\Gamma _t} q\tau \left(\rho {\partial v_n \over \partial t}-\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} + p -t_n)\right)d\Gamma      </math>
3442
|}
3443
| style="width: 5px;text-align: right;" | (181)
3444
|}
3445
3446
Substituting Eq.([[#eq-181|181]]) into ([[#eq-173|173]]) and using the expression of <math display="inline">\hat r_{m_i}</math> of Eq.([[#eq-161|161]]) yields the  variational expression of the stabilized mass balance equation, after rearranging the different terms, as
3447
3448
<span id="eq-182"></span>
3449
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3450
|-
3451
| 
3452
{| style="text-align: left; margin:auto;" 
3453
|-
3454
| style="text-align: center;" | <math>\displaystyle  \int _\Omega \frac{q}{\kappa } {\partial p \over \partial t}d\Omega + \int _\Omega q \tau \frac{\rho }{\kappa } {\partial ^2p \over \partial t^2}d\Omega - \int _\Omega q  d^v d\Omega + \int _\Omega \tau {\partial q \over \partial x_i} \left(2\mu {\partial  \over \partial x_i} ( d_{ij})+ {\partial p \over \partial x_i}+b_i\right)d\Omega </math>
3455
|-
3456
| style="text-align: center;" | <math> \displaystyle - \int _{\Gamma _t} q \tau \left[\rho {\partial v_n \over \partial t}-\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} + p -t_n)\right]d\Gamma =0    </math>
3457
|}
3458
| style="width: 5px;text-align: right;" | (182)
3459
|}
3460
3461
3462
Expression ([[#eq-182|182]]) holds for 2D and 3D problems.
3463
3464
The terms involving the first and second material time derivative of the pressure and the boundary term in Eq.([[#eq-182|182]]) are important to preserve the consistency of the residual form of the FIC mass balance equation. This, in turn, is essential for preserving the conservation of mass in the transient solution of free flow problems. The form of Eq.([[#eq-182|182]]) is a key contribution of the new FIC-based stabilized formulation, versus previous works on this topic [77,86,92,96,109].
3465
3466
At an unloaded free surface (Neumann) boundary <math display="inline">t_n=0</math>, and hence
3467
3468
<span id="eq-183"></span>
3469
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3470
|-
3471
| 
3472
{| style="text-align: left; margin:auto;" 
3473
|-
3474
| style="text-align: center;" | <math>BT = \int _{\Gamma _t} q\tau \left(\rho {\partial v_n \over \partial t}-\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} + p)\right)d\Gamma      </math>
3475
|}
3476
| style="width: 5px;text-align: right;" | (183)
3477
|}
3478
3479
For an inviscid fluid <math display="inline">\mu =0</math> and Eq.([[#eq-183|183]]) simplifies to
3480
3481
<span id="eq-184"></span>
3482
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3483
|-
3484
| 
3485
{| style="text-align: left; margin:auto;" 
3486
|-
3487
| style="text-align: center;" | <math>BT = \int _{\Gamma _t} q\tau \left(\rho {\partial v_n \over \partial t}-\frac{2p}{h_n}\right)d\Gamma            </math>
3488
|}
3489
| style="width: 5px;text-align: right;" | (184)
3490
|}
3491
3492
Accounting for the term <math display="inline">{\partial v_n \over \partial t}</math> in the boundary integral of Eqs.([[#eq-182|182]]-[[#eq-184|184]]) has proven to be relevant for the enhanced conservation of mass in free surface flows [82]. On the other hand, the effect of the term involving <math display="inline">{\partial ^2p \over \partial t^2}</math> was negligible in all the problems solved in this work [82].
3493
3494
Eq.([[#eq-182|182]]) is the starting point for deriving a new class of linear triangles with discontinuous pressure field adequate for analysis of incompressible flows with heterogeneous material properties [91].
3495
3496
The discretization of the mass stabilized equation is performed as it has been shown for the continuity equation in the previous chapter. So the analysis domain into finite elements is discretized using 3-noded linear triangles (<math display="inline">n=3</math>) for 2D problems and 4-noded tetrahedra (<math display="inline">n=4</math>) for 3D problems with local linear shape functions <math display="inline">N_I</math> defined for each node <math display="inline">I</math> (<math display="inline">I=1,n</math>) of an element <math display="inline">e</math>.
3497
3498
===3.1.5 FEM discretization and matrix form===
3499
3500
Substituting the approximations ([[#eq-9|9]]) and ([[#eq-77|77]])  into Eq.([[#eq-182|182]]) and choosing a Galerkin form with <math display="inline">q=N_I</math> gives the discretized form of the stabilized mass balance equation, after eliminating the arbitrary test functions as
3501
3502
<span id="eq-185"></span>
3503
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3504
|-
3505
| 
3506
{| style="text-align: left; margin:auto;" 
3507
|-
3508
| style="text-align: center;" | <math>\int _\Omega \frac{1}{\kappa }{N}^T {N} \frac{D\bar {p}}{Dt} d\Omega  + \!\!\int _\Omega \frac{\tau \rho }{\kappa } {N}^T {N} \frac{D^2\bar {p}}{Dt^2} d\Omega{-} \int _\Omega {N}^T  {m}^T {B} \bar {v} d\Omega +  </math>
3509
|-
3510
| style="text-align: center;" | <math> +  \!\!\int _\Omega \tau ({\boldsymbol \nabla }{N})^T {\boldsymbol \nabla }{N} \bar{p} d\Omega  +  \int _{\Gamma _t} \frac{2\tau }{h^n} {N}^T {N} \bar{p} d\Gamma - {f}_p=0 </math>
3511
|}
3512
| style="width: 5px;text-align: right;" | (185)
3513
|}
3514
3515
where
3516
3517
<span id="eq-197"></span>
3518
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3519
|-
3520
| 
3521
{| style="text-align: left; margin:auto;" 
3522
|-
3523
| style="text-align: right;" | <math>\displaystyle{B}_I = \left[\begin{matrix}\displaystyle {\partial N_I \over \partial x} &0&0\\ \displaystyle{0}& \displaystyle {\partial N_I \over \partial y}&0\\ \displaystyle{0}&0&\displaystyle {\partial N_I \over \partial z}\\ \displaystyle {\partial N_I \over \partial y}&\displaystyle {\partial N_I \over \partial x}&0\\[.25cm] \displaystyle {\partial N_I \over \partial z}&0&\displaystyle {\partial N_I \over \partial x}\\[.25cm] \displaystyle{0}&\displaystyle {\partial N_I \over \partial z}&\displaystyle {\partial N_I \over \partial y}          \end{matrix}  \right] \hbox{, } {\boldsymbol \nabla }{N}^T\equiv \left[\begin{matrix}N_1 \\ {\boldsymbol \nabla }{N}_2 \\ \cdots \\  {\boldsymbol \nabla }{N}_N \end{matrix} \right] \hbox{with }  {\boldsymbol \nabla } = \left\{\begin{matrix}\displaystyle {\partial  \over \partial x_1} \\ \displaystyle {\partial  \over \partial x_2}\\ \displaystyle {\partial  \over \partial x_3}   \end{matrix}  \right\}</math>
3524
| style="width: 5px;text-align: right;" | (196)
3525
|-
3526
| style="text-align: right;" | <math> \hbox{and }\displaystyle {N} = [{N}_1, {N}_2,\cdots , {N}_N ]^T   </math>
3527
|}
3528
|}
3529
3530
Eqs.([[#eq-185|185]]) can be written in matrix form as
3531
3532
<span id="eq-197"></span>
3533
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3534
|-
3535
| 
3536
{| style="text-align: left; margin:auto;" 
3537
|-
3538
| style="text-align: center;" | <math>M_1 {\dot{\bar{p}}}+M_2 {\ddot{\bar{p}}}-{Q}^T \bar{v} + ({L}+{M}_b)\bar {p}- {f}_p={0} </math>
3539
|}
3540
| style="width: 5px;text-align: right;" | (197)
3541
|}
3542
3543
The matrices and vectors in Eqs.([[#eq-197|197]]) are assembled from the element contributions as
3544
3545
<span id="eq-198"></span>
3546
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3547
|-
3548
| 
3549
{| style="text-align: left; margin:auto;" 
3550
|-
3551
| style="text-align: center;" | <math>{M}_{1_{IJ}} =\int _{\Omega } \frac{1}{\kappa }{N}_I {N}_J d\Omega    </math>
3552
|}
3553
| style="width: 5px;text-align: right;" | (198)
3554
|}
3555
3556
<span id="eq-199"></span>
3557
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3558
|-
3559
| 
3560
{| style="text-align: left; margin:auto;" 
3561
|-
3562
| style="text-align: center;" | <math>{M}_{2_{IJ}} =\int _{\Omega } \tau \frac{\rho }{\kappa }{N}_I {N}_J d\Omega   </math>
3563
|}
3564
| style="width: 5px;text-align: right;" | (199)
3565
|}
3566
3567
<span id="eq-200"></span>
3568
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3569
|-
3570
| 
3571
{| style="text-align: left; margin:auto;" 
3572
|-
3573
| style="text-align: center;" | <math>{M}_{b_{IJ}} = \int _{\Gamma _t} \frac{2\tau }{h_n} {N}_I {N}_J  d\Gamma    </math>
3574
|}
3575
| style="width: 5px;text-align: right;" | (200)
3576
|}
3577
3578
<span id="eq-201"></span>
3579
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3580
|-
3581
| 
3582
{| style="text-align: left; margin:auto;" 
3583
|-
3584
| style="text-align: center;" | <math>{L}_{IJ}= \int _{\Omega } \tau ({\boldsymbol \nabla }^T  {N}_I) {\boldsymbol \nabla }{N}_J d\Omega   </math>
3585
|}
3586
| style="width: 5px;text-align: right;" | (201)
3587
|}
3588
3589
<span id="eq-202"></span>
3590
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3591
|-
3592
| 
3593
{| style="text-align: left; margin:auto;" 
3594
|-
3595
| style="text-align: center;" | <math>Q_{IJ}  =\int _{\Omega } B_I^{T} m {N}_J d\Omega    </math>
3596
|}
3597
| style="width: 5px;text-align: right;" | (202)
3598
|}
3599
3600
<span id="eq-203"></span>
3601
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3602
|-
3603
| 
3604
{| style="text-align: left; margin:auto;" 
3605
|-
3606
| style="text-align: center;" | <math>{f}_{p_I}=\int _{\Gamma _t}\tau  {N}_I \left[\rho \frac{Dv_n}{Dt}-\frac{2}{h_n} (2\mu  d_n -t_n)\right]d\Gamma - \int _{\Omega ^e} \tau  {\boldsymbol \nabla }^T  {N}_I {b} d\Omega   </math>
3607
|}
3608
| style="width: 5px;text-align: right;" | (203)
3609
|}
3610
3611
The boundary terms in vector <math display="inline">{f}_p</math>  (Eq.([[#eq-203|203]])) can be incorporated with the matrices of Eq.([[#eq-197|197]]). This, however, leads to a not symmetrical set of equations. For this reason  these boundary terms are computed iteratively within the incremental solution scheme.
3612
3613
In a free surface fluid the presence  in Eq.([[#eq-197|197]]) of matrix <math display="inline">{M}_b</math> (Eq.([[#eq-200|200]])) enables the computation of the pressure without the need of prescribing its value at the free surface. This eliminates the error introduced when the pressure is prescribed to zero in free boundaries, which leads to considerable mass losses for viscous flows. Matrix <math display="inline">{M}_b</math> was introduced into the discretized stabilized mass balance equation in [65] using a fractional step method and heuristic arguments.
3614
3615
==3.2 Solution scheme for quasi-incompressible Newtonian fluids==
3616
3617
The solution scheme for  quasi-incompressible  Newtonian fluids has a structure very similar to the two-step procedure presented for a general compressible material in Chapter [[#2 Velocity-based formulations for compressible materials|2]]. The linear momentum equations are solved for the increment of the velocities and the pressures are obtained from the continuity equation in the updated configuration. However, for dealing with the incompressibility, the stabilized form of the continuity equation (Eq.([[#eq-197|197]])) has to be considered.  Furthermore, the linearized form of the momentum equations (Eq.([[#eq-64|64]])) has to be modified according to the constitutive law of Newtonian fluids.
3618
3619
===3.2.1 Governing equations===
3620
3621
'''''Linear momentum equations'''''
3622
3623
For the sake of clarity, the general linearized form of the momentum equations is recalled. For each iteration <math display="inline">{i}</math> the following linear system is solved
3624
3625
<span id="eq-204"></span>
3626
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3627
|-
3628
| 
3629
{| style="text-align: left; margin:auto;" 
3630
|-
3631
| style="text-align: center;" | <math>K^{i} \Delta  \bar v=R^i( {^{n+1}{\bar v}^{i}}, {^{n+1}\sigma ' ^i}, {^{n+1}p^i})   </math>
3632
|}
3633
| style="width: 5px;text-align: right;" | (204)
3634
|}
3635
3636
where:
3637
3638
<span id="eq-205"></span>
3639
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3640
|-
3641
| 
3642
{| style="text-align: left; margin:auto;" 
3643
|-
3644
| style="text-align: center;" | <math>K^{i}= K^{m}({^{n+1}\bar x^{i}},c^{\sigma , i})+              K^{g}({^{n+1}\bar x^{i}},\sigma ^{i}) +             K^{\rho } ({^{n+1}\bar x^{i}})   </math>
3645
|}
3646
| style="width: 5px;text-align: right;" | (205)
3647
|}
3648
3649
with
3650
3651
<span id="eq-206"></span>
3652
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3653
|-
3654
| 
3655
{| style="text-align: left; margin:auto;" 
3656
|-
3657
| style="text-align: center;" | <math>K^{m}_{IJ}=\int _{\Omega } B^{T}_I  \Delta t \left[c^{\sigma } \right] B_J    d\Omega  </math>
3658
|}
3659
| style="width: 5px;text-align: right;" | (206)
3660
|}
3661
3662
<span id="eq-207"></span>
3663
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3664
|-
3665
| 
3666
{| style="text-align: left; margin:auto;" 
3667
|-
3668
| style="text-align: center;" | <math>K^{g}_{IJ}=I\int _{\Omega } \beta ^{T}_I \Delta t \sigma   \beta _J    d\Omega  </math>
3669
|}
3670
| style="width: 5px;text-align: right;" | (207)
3671
|}
3672
3673
<span id="eq-208"></span>
3674
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3675
|-
3676
| 
3677
{| style="text-align: left; margin:auto;" 
3678
|-
3679
| style="text-align: center;" | <math>K^{\rho }_{IJ}= I\int _{\Omega } N_I \frac{2\rho }{\Delta t}  N_J d\Omega    d\Omega   </math>
3680
|}
3681
| style="width: 5px;text-align: right;" | (208)
3682
|}
3683
3684
and
3685
3686
<span id="eq-209"></span>
3687
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3688
|-
3689
| 
3690
{| style="text-align: left; margin:auto;" 
3691
|-
3692
| style="text-align: center;" | <math>{^{n+1}R_{Ii}}= \int _\Omega N_I \rho  N_J d\Omega  {^{n+1}\bar{\dot{v}}_{Ji}}+  \int _\Omega \frac{\partial N_I}{\partial x_j} {^{n+1}\sigma '}_{ij} d\Omega + </math>
3693
|-
3694
| style="text-align: center;" | <math>+ \int _\Omega \frac{\partial N_I}{\partial x_j}\delta _{ij} N_J d\Omega  {^{n+1} \bar  p}_J -  \int _\Omega {N}_I {^{n+1}b}_i d\Omega - \int _{\Gamma _t} N_I {^{n+1}t_i^{p}} d\Gamma  </math>
3695
|}
3696
| style="width: 5px;text-align: right;" | (209)
3697
|}
3698
3699
In order to obtain the linearized form of the momentum equations  for  quasi-incompressible Newtonian fluids, the tangent constitutive tensor in matrix <math display="inline">K^{m}</math> ([[#eq-206|206]]) and the Cauchy stress tensor that appears in the residual vector <math display="inline">R</math> and in <math display="inline">K^{g}</math> have to be computed using the adequate constitutive law.
3700
3701
For a Newtonian fluid, the stresses can be computed as
3702
3703
<span id="eq-210"></span>
3704
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3705
|-
3706
| 
3707
{| style="text-align: left; margin:auto;" 
3708
|-
3709
| style="text-align: center;" | <math>\sigma =\sigma ' + pI =2 \mu d' + pI  </math>
3710
|}
3711
| style="width: 5px;text-align: right;" | (210)
3712
|}
3713
3714
For quasi-incompressible materials Eq.([[#eq-151|151]]) holds within a general time interval <math display="inline">{[n,n+1]}</math>. For clarity purposes, the relation is rewritten as
3715
3716
<span id="eq-211"></span>
3717
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3718
|-
3719
| 
3720
{| style="text-align: left; margin:auto;" 
3721
|-
3722
| style="text-align: center;" | <math>{^{n+1}p}={^{n}p} + \Delta t \kappa {^{n+1}{d}^{v}}  </math>
3723
|}
3724
| style="width: 5px;text-align: right;" | (211)
3725
|}
3726
3727
Considering a time interval <math display="inline">{[n,n+1]}</math> and substituting Eq.([[#eq-211|211]]) into ([[#eq-210|210]]) yields
3728
3729
<span id="eq-212"></span>
3730
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3731
|-
3732
| 
3733
{| style="text-align: left; margin:auto;" 
3734
|-
3735
| style="text-align: center;" | <math>^{n+1}\sigma = \left(2 \mu \mathbf{I}' + \Delta t \kappa I\otimes I  \right): d + {^{n}p}I  </math>
3736
|}
3737
| style="width: 5px;text-align: right;" | (212)
3738
|}
3739
3740
where the fourth-order tensor <math display="inline">\mathbf{I}'</math> has been defined in Eq.([[#eq-97|97]]).
3741
3742
For convenience, Eq.([[#eq-212|212]]) is rewritten in the following form
3743
3744
<span id="eq-213"></span>
3745
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3746
|-
3747
| 
3748
{| style="text-align: left; margin:auto;" 
3749
|-
3750
| style="text-align: center;" | <math>{^{n+1}\Delta \sigma }={^{n+1}\sigma } - {^{n}\sigma }=\left( c^{d} +  c^{\kappa } \right): d </math>
3751
|}
3752
| style="width: 5px;text-align: right;" | (213)
3753
|}
3754
3755
where the following substitutions have been done:
3756
3757
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3758
|-
3759
| 
3760
{| style="text-align: left; margin:auto;" 
3761
|-
3762
| style="text-align: center;" | <math>{^{n}\sigma }= {^{n}p} I </math>
3763
|}
3764
| style="width: 5px;text-align: right;" | (214)
3765
|}
3766
3767
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3768
|-
3769
| 
3770
{| style="text-align: left; margin:auto;" 
3771
|-
3772
| style="text-align: center;" | <math>c^{d}=2 \mu \mathbf{I}' </math>
3773
|}
3774
| style="width: 5px;text-align: right;" | (215)
3775
|}
3776
3777
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3778
|-
3779
| 
3780
{| style="text-align: left; margin:auto;" 
3781
|-
3782
| style="text-align: center;" | <math>c^{\kappa }=  \Delta t \kappa I\otimes I=\Delta t \kappa mm^T </math>
3783
|}
3784
| style="width: 5px;text-align: right;" | (216)
3785
|}
3786
3787
The goal is to obtain a relationship between the measure of rate of stress and the rate of deformation in the form of Eq.([[#eq-43|43]]).  Eq.([[#eq-213|213]]) shows that, according to the constitutive law for Newtonian fluids, the rate of deformation is related to the Cauchy stress and not to rate of the Cauchy stress, as for hypoelastic solids (Eq.([[#eq-98|98]])). For this reason, in fluids preserving the objectivity of the stress rate measures is not a critical issue as for hypoelastic solids. Rigid rotations do not cause any state of stress, because the Cauchy stress is directly obtained from the rate of deformation. For these reasons, the rate of Cauchy stress can be simply defined  from Eq.([[#eq-213|213]]) as
3788
3789
<span id="eq-217"></span>
3790
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3791
|-
3792
| 
3793
{| style="text-align: left; margin:auto;" 
3794
|-
3795
| style="text-align: center;" | <math>\sigma ^{\bigtriangledown }= \dot \sigma =\frac{\Delta  \sigma }{\Delta t}={{c}^{\sigma }}: d= \left(\frac{ c^{d}}{\Delta t} + \frac{c^{\kappa }}{\Delta t}   \right): d  </math>
3796
|}
3797
| style="width: 5px;text-align: right;" | (217)
3798
|}
3799
3800
Note that  Eq.([[#eq-217|217]]) has the same structure as Eq.([[#eq-43|43]]). For convenience, the Newtonian tangent moduli <math display="inline">{{c}^{\sigma }}</math> for the rate of stress is computed as the sum of the deviatoric and volumetric parts. Hence, substituting <math display="inline">{{c}^{\sigma }}</math>  into  Eq.([[#eq-206|206]]) yields that for a quasi-incompressible Newtonian fluid the material part of the tangent matrix can be written as
3801
3802
<span id="eq-218"></span>
3803
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3804
|-
3805
| 
3806
{| style="text-align: left; margin:auto;" 
3807
|-
3808
| style="text-align: center;" | <math>K^{m}_{Nf}=K^{\mu } +K^{\kappa }  </math>
3809
|}
3810
| style="width: 5px;text-align: right;" | (218)
3811
|}
3812
3813
where <math display="inline">{K^{\mu }}</math> and <math display="inline">{K^{\kappa }}</math> are defined for a generic finite element <math display="inline">{e}</math> and the pair of nodes <math display="inline">{I,J}</math> as
3814
3815
<span id="eq-219"></span>
3816
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3817
|-
3818
| 
3819
{| style="text-align: left; margin:auto;" 
3820
|-
3821
| style="text-align: center;" | <math>K^{\mu }_{IJ}=\int _{\Omega ^e} B^{T}_I  \left[ c^{\mu } \right]B_J    d\Omega  </math>
3822
|}
3823
| style="width: 5px;text-align: right;" | (219)
3824
|}
3825
3826
<span id="eq-220"></span>
3827
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3828
|-
3829
| 
3830
{| style="text-align: left; margin:auto;" 
3831
|-
3832
| style="text-align: center;" | <math>K^{\kappa }_{IJ}=\int _{\Omega ^e} B^{T}_I  m \Delta t \kappa  m^T B_J    d\Omega  </math>
3833
|}
3834
| style="width: 5px;text-align: right;" | (220)
3835
|}
3836
3837
<math display="inline">\begin{array}{l} \\ \hbox{with}   {c^{\mu }}=2 \mu \left[                      \begin{array}{cccccc}                        2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\                         &   2/3 & -1/3 &0  &0  & 0 \\                         &  &  2/3 & 0 & 0 & 0 \\                         &  &  & 1/2 & 0 & 0 \\                        {Sym.} &  &  &  & 1/2 & 0 \\                         &  &  &  &  & 1/2 \\                      \end{array} \right] \\ \\ \end{array}</math>
3838
3839
The volumetric part of the tangent matrix <math display="inline">{K^{\kappa }}</math> can compromise the conditioning of the linear system because its terms are orders of magnitude larger than the viscous part. In order to prevent the numerical instabilities originated by the ill-conditioning of the tangent matrix, a reduced pseudo-bulk modulus, computed from the actual bulk modulus <math display="inline">\kappa </math> as <math display="inline">\kappa _p=\theta \kappa </math>, can be used in the expression of <math display="inline">{K^{\kappa }}</math> without altering the numerical results [45].  An adequate selection of the pseudo-bulk modulus also improves the overall accuracy of the numerical solution and the preservation of mass for large time steps [45]. For fully incompressible fluids (<math display="inline">\kappa =\infty </math>), a finite  value of <math display="inline">\kappa </math> is used in <math display="inline">{K^{\kappa }}</math> as this helps to obtaining an accurate solution for velocities and pressure  with reduced mass loss in few iterations per time step [45]. These considerations, however, do not affect the value of <math display="inline">\kappa </math> within matrix <math display="inline">{M}_1</math> in Eq.([[#eq-197|197]]). Clearly, the value of the terms of <math display="inline">{K^{\kappa }}</math> can also be limited by reducing the time step size. This, however, increases the overall computational cost. Another approach for improving mass conservation in incompressible flows was proposed  in [109]. In Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]] the details about this procedure will be given.
3840
3841
''''''FIC stabilized mass balance equation'''''
3842
3843
The pressure is obtained from the stabilized form of the mass balance equation (Eq.([[#eq-197|197]])). Introducing the time integration of the pressure (Eqs.([[#eq-79|79]], [[#eq-80|80]])) into  Eq.([[#eq-197|197]]), yields
3844
3845
<span id="eq-221"></span>
3846
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3847
|-
3848
| 
3849
{| style="text-align: left; margin:auto;" 
3850
|-
3851
| style="text-align: center;" | <math>H \bar p^{i+1}=F_p({\bar v},{{\bar p}}) </math>
3852
|}
3853
| style="width: 5px;text-align: right;" | (221)
3854
|}
3855
3856
where
3857
3858
<span id="eq-222"></span>
3859
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3860
|-
3861
| 
3862
{| style="text-align: left; margin:auto;" 
3863
|-
3864
| style="text-align: center;" | <math>H=\left(\frac{1}{ \Delta t} M_{1} +   \frac{1}{ {\Delta t}^2} M_{2} + {L}+{M}_b \right) </math>
3865
|}
3866
| style="width: 5px;text-align: right;" | (222)
3867
|}
3868
3869
and
3870
3871
<span id="eq-223"></span>
3872
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3873
|-
3874
| 
3875
{| style="text-align: left; margin:auto;" 
3876
|-
3877
| style="text-align: center;" | <math>F_p= \frac{M_{1}}{\Delta t}{^n \bar p} +   \frac{ M_{2}}{{\Delta t}^2} \left({^n \bar p}+ {^n \bar{ \dot p}} \Delta t \right)+ {Q}^T \bar{v} +{f}_p  </math>
3878
|}
3879
| style="width: 5px;text-align: right;" | (223)
3880
|}
3881
3882
where all the matrices and the vectors of Eqs.([[#eq-221|221]]-[[#eq-223|223]]) have been defined in Eqs.([[#eq-198|198]]-[[#eq-203|203]])
3883
3884
===3.2.2 Solution scheme===
3885
3886
The complete solution scheme for a quasi-incompressible Newtonian fluid is described for a generic time interval <math display="inline">{}</math> in Box 8.
3887
3888
In Box 9 all matrices and vectors that appear in Box 8 are given.
3889
3890
<div class="center" style="font-size: 75%;">
3891
[[File:Draft_Samper_722607179_1293_Box8.png|550px]]
3892
3893
'''Box 8'''. Iterative solution scheme for quasi-incompressible Newtonian fluids.
3894
</div>
3895
3896
<div class="center" style="font-size: 75%;">
3897
[[File:Draft_Samper_722607179_5966_Box8.png|550px]]
3898
3899
'''Box 9'''. Element form of the matrices and vectors in Box 8.
3900
</div>
3901
3902
==3.3 Solution scheme for quasi-incompressible hypoelastic solids==
3903
3904
The differences between the  solution scheme for quasi-incompressible and  compressible hypoelastic solids concern only the continuity equation. In fact, the same linearized form of the continuum equations derived in the previous chapter for the compressible hypoelastic model within the mixed Velocity-Pressure formulation holds also for the quasi-incompressible scheme. However, for dealing with incompressibility the scheme needs to be stabilized
3905
3906
In this work the FIC stabilization procedure originally derived for Newtonian fluids in [82] and proposed again in Section [[#3.1 Stabilized FIC form of the mass balance equation|3.1]] of this chapter, has been tested for the first time for quasi-incompressible hypoelastic solids. In order to use the same form of Eqs.([[#eq-221|221]]-[[#eq-223|223]]) for hypoelastic quasi-incompressible solids, the fluid parameters (the fluid    viscosity <math display="inline">\mu _f</math> and the fluid bulk modulus <math display="inline">\kappa _f</math>) should be substituted with the equivalent solid parameters. The similarity between Newtonian fluids and hypoelastic solids is evident comparing the computation of the  Cauchy stress tensor increment for the two mentioned constitutive laws.
3907
3908
For quasi-incompressible Newtonian fluids Eq.([[#eq-213|213]]) holds and, for clarity purposes, is here rewritten as
3909
3910
<span id="eq-224"></span>
3911
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3912
|-
3913
| 
3914
{| style="text-align: left; margin:auto;" 
3915
|-
3916
| style="text-align: center;" | <math>^{n+1} \Delta \sigma _f= 2 {\mu }_f \mathbf{I}' : d + \Delta t {\kappa }_f I\otimes I  : d  </math>
3917
|}
3918
| style="width: 5px;text-align: right;" | (224)
3919
|}
3920
3921
From Eqs.([[#eq-111|111]]), the increment of the Cauchy stress for hypoelastic solids is
3922
3923
<span id="eq-225"></span>
3924
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3925
|-
3926
| 
3927
{| style="text-align: left; margin:auto;" 
3928
|-
3929
| style="text-align: center;" | <math>^{n+1} \Delta \sigma _s= 2 {\Delta t \mu }_s \mathbf{I}' : d + \Delta t {\kappa }_s I\otimes I  : d   </math>
3930
|}
3931
| style="width: 5px;text-align: right;" | (225)
3932
|}
3933
3934
where bulk modulus for the solid <math display="inline">{\kappa }_s</math> is computed from the Lamè constants <math display="inline">\mu _s</math> and <math display="inline">\lambda _s</math> or in terms of the Young modulus and the Poisson ratio of the material (Eqs.([[#eq-92|92]], [[#eq-93|93]],[[#eq-96|96]])).
3935
3936
Eqs.([[#eq-224|224]] and [[#eq-225|225]]) show the duality between hypoelastic and Newtonian quasi-incompressible constitutive laws. In the latter the deviatoric and the volumetric parts of the Cauchy stress tensor are controlled by the dynamic viscosity <math display="inline">\mu _f</math> and the bulk modulus <math display="inline">\kappa _f</math>, respectively. The equivalent roles in hypoelastic solids are taken by the second Lamè constant scaled by the time increment (<math display="inline">\Delta t \mu _s</math>) and the bulk modulus <math display="inline">\kappa _s</math>. Thanks to this equivalence, the stabilized mass continuity equation derived for quasi-incompressible fluids (Eq.([[#eq-197|197]])) holds also for quasi-incompressible hypoelastic solids just by replacing the fluid parameters <math display="inline">\mu _f</math> and <math display="inline">\kappa _f</math> with the equivalent terms for hypoelastic solids <math display="inline">\Delta t \mu _s</math> and <math display="inline">\kappa _s</math>.
3937
3938
Note that using this analogy  the differences between the incremental solution schemes for Newtonian fluids and quasi-incompressible hypoelastic solids are minimal and they essentially differ on the way to compute the stresses and the tangent moduli according to the  specific constitutive law. For hypoelastic quasi-incompressible model, the stress tensor is computed with Eq.([[#eq-114|114]]) and the tangent moduli is obtained from Eq.([[#eq-101|101]]).  The solid finite element implemented with this formulation is called VPS-element.
3939
3940
In Box 10, the iterative solution incremental scheme for quasi-incompressible hypoelastic solids using the stabilized mixed velocity-pressure formulation  is described for a generic time increment <math display="inline">{t[^nt,{^{n+1}t}]}</math>.
3941
3942
In Box 11 all matrices and vectors that appear in Box 10 are given.
3943
3944
<div class="center" style="font-size: 75%;">
3945
[[File:Draft_Samper_722607179_9395_Box10.png|550px]]
3946
3947
'''Box 10'''. Iterative solution scheme for quasi-incompressible hypoelastic solids.
3948
</div>
3949
3950
<div class="center" style="font-size: 75%;">
3951
[[File:Draft_Samper_722607179_7594_Box11.png|550px]]
3952
3953
'''Box 11'''. Element form of the matrices and vectors in Box 10.
3954
</div>
3955
3956
==3.4 Free surface flow analysis==
3957
3958
Free surface flows are  those fluids with at least one side in contact with the air. Their numerical solution is critical because the free surface contours change continuously and they need to be tracked in order to solve accurately the boundary value problem. In this work, this task is carried out using the Lagrangian finite element procedure called <math display="inline">{Particle Finite ElementMethod (PFEM)}</math> [67]. In the PFEM the  mesh nodes are treated as the fluid particles. As a consequence, the free surface contours are automatically detected by the nodes positions. The PFEM will be described and analyzed in detail in  Section [[#3.4.1 The Particle Finite Element Method|3.4.1]].
3959
3960
Another typical drawback associated to free surface flows is the deterioration of the mass preservation of the numerical method. The  indetermination of the free surface position introduces in the scheme an additional  mass loss source to those induced by the inaccuracy of the numerical method. In Section [[#3.4.2 Mass conservation analysis|3.4.2]] the mass conservation feature of  the PFEM-FIC stabilized formulation is studied in detail and many validation examples are given.
3961
3962
In Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]] the key role of the pseudo bulk modulus <math display="inline">\kappa _p</math> for guaranteeing the mass preservation of the quasi-incompressible formulation will be highlighted. In fact, the actual value of the bulk modulus <math display="inline"> \kappa </math> deteriorates the conditioning of the linear system. This may affect the mass preservation feature and even the convergence of the whole scheme. Using a reduced value for the bulk modulus, computed as <math display="inline">\kappa _p=\theta \kappa </math>, all these inconveniences are overcome. In the same section a practical strategy for computing  the pseudo bulk modulus is also given and tested with several numerical examples.
3963
3964
===3.4.1 The Particle Finite Element Method===
3965
3966
The ''Particle Finite ElementMethod (PFEM)'' is a Lagrangian finite element procedure ideated and developed by Idelsohn and Oñate and their work group [66-68,88]. The PFEM addresses those problems where severe changes of topology occur. This may happen in free surface fluid dynamics, in non-linear solid mechanics, in FSI problems or in thermal coupled problems. The essential idea of the PFEM is to follow the topology of the deformed bodies by moving the nodes of the mesh according to the equations of motion in a Lagrangian way. In other words, in the PFEM the  mesh nodes are treated as particles and they transport their momentum together with all their physical properties. All this produces the deformation of the finite elements discretization that needs to be rebuilt whenever a threshold value for the distortion is reached. Remeshing is one of the most characteristic points of the PFEM [61]. This operation is performed via an efficient combination of the  Delaunay Tessellation  [40,111] and the Alpha Shape method [39]. Once the mesh is generated, the differential problem is integrated again over the new mesh in the classical FEM fashion.
3967
3968
In the following parts of this section the essential points of the PFEM will be highlighted. Specifically, first the remeshing procedure will be described, then the basic steps of the PFEM will be summarized and finally the advantages and disadvantages of the technique will be highlighted.
3969
3970
====3.4.1.1 Remeshing====
3971
3972
In order to solve accurately the FEM problem, at each time step it has to be guaranteed a good quality for the mesh. In fact, even a single degenerated element may compromise the entire computation. For this reason the remeshing algorithm must be reliable and robust. In the PFEM this is guaranteed through an efficient algorithm that combines two trustworthy techniques, the Delaunay triangulation and the Alpha Shape method.
3973
3974
Given a cloud of points in the space, the Delaunay tessellation guarantees the creation of the most homogenous discretization for those points. This is obtained by ensuring that the circumball, cirmumcircle in 2D, built on all the nodes of a simplex does not contain any other node of some other simplex [30]. Actually this is guaranteed only in 2D where it is proved that there exists a unique Delaunay triangulation. In 3D there may be the eventuality that the criterion is not fulfilled for certain tetrahedra.  In 2D the Delaunay triangulation has the property of maximizing the minimum inner angle of all the triangles of the mesh (this is not guaranteed in 3D). Note that the maximization of the minimum angle does not imply the minimization of the maximum angle, see Figure [[#img-29|29]]. 
3975
3976
<div id='img-29'></div>
3977
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
3978
|-
3979
|[[Image:draft_Samper_722607179-Delaunay0.png|200px]]
3980
|[[Image:draft_Samper_722607179-Delaunay2.png|200px]]
3981
| [[Image:draft_Samper_722607179-Delaunay3.png|400px]]
3982
|- style="text-align: center; font-size: 85%;"
3983
| (a) Set of points
3984
| (b) Delaunay
3985
| (c) Not  Delaunay
3986
|- style="text-align: center; font-size: 75%;"
3987
| colspan="3" | '''Figure 29:''' 2D Delaunay triangulation of a set of points.
3988
|}
3989
3990
This strategy may be  helpful for avoiding the creation of degenerated elements as the <math display="inline">slivers</math> (simplices with null surface or volume) but it is not sufficient for their complete elimination. It will be explained later that further controls and modifications on the cloud of points are required for this objective.
3991
3992
The Delaunay triangulation alone cannot ensure the detection of the physical boundaries of the bodies. For this purpose, the so-called Alpha Shape method is applied on the tessellation obtained with the Delaunay procedure. The role of the Alpha Shape method is to erase those simplices that  are excessively distorted or big.  In order to do this, the mean mesh size <math display="inline">h</math> and the circumradius of each element <math display="inline">r_e</math> are computed.  Next the following check is done for all the elements
3993
3994
<span id="eq-226"></span>
3995
{| class="formulaSCP" style="width: 100%; text-align: left;" 
3996
|-
3997
| 
3998
{| style="text-align: left; margin:auto;" 
3999
|-
4000
| style="text-align: center;" | <math>if  r_e>\alpha h   \rightarrow \hbox{erase the e-element}   </math>
4001
|}
4002
| style="width: 5px;text-align: right;" | (226)
4003
|}
4004
4005
where <math display="inline">\alpha </math> is a parameter that is chosen arbitraly.
4006
4007
The selection of <math display="inline">\alpha </math> is arbitrary. However, it must be taken into account the following considerations. If <math display="inline">\alpha </math> is excessively large, the Alpha Shape method is worthless because no element will be erased. This means that the Delaunay triangulation remains unchanged. On the other hand,  an excessive small value for <math display="inline">\alpha </math> induces the elimination of all the simplices. Specifically the lowest admissible <math display="inline">\alpha </math> parameter  is the one of the equilateral triangle for which <math display="inline">\alpha=1/ \sqrt 3</math>.  In Figure [[#img-30|30]]  the circumcircle is plotted for simplices with the same mean length for edges. Given a mean size for the edges of a simplex, the equilateral triangle is the one with the minimum circumradius. Equally, for a given circumradius, the equilateral triangle is the simplex which edges have the maximum mean length. 
4008
4009
4010
<div id='img-30a'></div>
4011
<div id='img-30b'></div>
4012
<div id='img-30c'></div>
4013
<div id='img-30'></div>
4014
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4015
|-
4016
|[[Image:draft_Samper_722607179-triangleEquilateral.png|400px|Equilateral triangle]]
4017
|[[Image:draft_Samper_722607179-triangleSharp.png|400px|Flat triangle]]
4018
|- style="text-align: center; font-size: 85%;"
4019
| (a) Equilateral triangle
4020
| (b) Sharp triangle
4021
|- style="text-align: center; font-size: 75%;"
4022
| colspan="2"|[[Image:draft_Samper_722607179-sliver3.png|600px]]
4023
|-style="text-align: center; font-size: 85%;"
4024
| colspan="2"| (c) Flat triangle
4025
|- style="text-align: center; font-size: 75%;"
4026
| colspan="2" | '''Figure 30:''' Triangles with the same mean size and their circumcircles.
4027
|}
4028
4029
The arbitrariness in the choice of the <math display="inline">\alpha </math> parameter may affect the numerical results. In fact, two different values for <math display="inline">\alpha </math> may give two different configurations. See for example the case of Figure [[#img-31|31]]. For the same Delaunay triangulation (Figure [[#img-31b|31b]]), the configuration obtained with <math display="inline">\alpha _1</math> (Figure [[#img-31c|31c]]) is different from the one obtained with <math display="inline">\alpha _2</math> (Figure [[#img-31d|31d]]), where <math display="inline">{\alpha _1 > \alpha _2}</math>.  
4030
4031
4032
<div id='img-31a'></div>
4033
<div id='img-31b'></div>
4034
<div id='img-31c'></div>
4035
<div id='img-31d'></div>
4036
<div id='img-31'></div>
4037
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4038
|-
4039
|[[Image:draft_Samper_722607179-alphaShape1.png|400px]]
4040
|[[Image:draft_Samper_722607179-alphaShape2.png|400px]]
4041
|- style="text-align: center; font-size: 85%;"
4042
| (a) Set of points
4043
| (b) Delaunay triangulation
4044
|-
4045
|[[Image:draft_Samper_722607179-alphaShape3.png|400px]]
4046
|[[Image:draft_Samper_722607179-alphaShape4.png|400px]]
4047
|-style="text-align: center; font-size: 85%;"
4048
| (c) Alpha Shape (<math>{\alpha _1}</math>)
4049
| (d) Alpha Shape (<math>{\alpha _2}</math>)
4050
|- style="text-align: center; font-size: 75%;"
4051
| colspan="2" | '''Figure 31:''' 2D triangulation of a set of points. Alpha shape method applied on the Delaunay triangulation (<math>{\alpha _1 > \alpha _2}</math>).
4052
|}
4053
4054
4055
Specifically, the mesh obtained with <math display="inline">\alpha _1</math> accepts a larger number of elements from the Delaunay discretization than the one given by <math display="inline">\alpha _2</math>, especially in the free surface zone.  However, in the section dedicated to the validation examples  it will be shown that the method is convergent. This means that the error introduced by a certain <math display="inline">\alpha </math>  reduces with the refinement of the mesh.
4056
4057
The remeshing can be performed also in the opposite order, hence using first an Alpha Shape method for detecting the contours of the domains and then performing the Delaunay triangulation with the geometric restrictions of those boundaries [68,87,130]. This can be done by performing the Delaunay triangulation via the ''advancing front technique'' [72].
4058
4059
Both remeshing algorithms may generate particles or elements separated from the rest of the domain. Remember that all the information is stored in the nodes, so the governing equations can be computed also for an isolated particle. This allows, for example, the simulation of detached fluid drops.
4060
4061
However, the first strategy is preferable because it leads to a reduced computational cost and it requires a lower implementation effort. For these reasons, in this work the remeshing is performed applying the Alpha Shape technique after the Delaunay triangulation.
4062
4063
Neither the Alpha Shape method nor the Delaunay triangulation can prevent from the creation of highly distorted elements. Let consider, for example, the triangle depicted in Figure [[#img-30b|30b]]. The circumradius associated to this type of triangle is not excessively large and it may fulfill Eq.([[#eq-226|226]]). The Alpha Shape method and the Delaunay triangulation can only ensure the best physical mesh (so the best discretization that respects the physical boundaries of the domain) for a given cloud of points. However, during the motion, the nodes distribution may be such that it is impossible to avoid the presence of poor quality elements. In order to avoid this eventuality the remeshing with the Delaunay tessellation and  the Alpha Shape method must be combined with some additional controls.
4064
4065
For example, for avoiding the presence of sharp elements in the triangulation, as the one shown in Figure [[#img-32|32]], one should monitor the edge lengths of each triangle.  <div id='img-32'></div>
4066
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4067
|-
4068
|[[Image:draft_Samper_722607179-sliver.png|400px|Distorted element in a 2D triangulation.]]
4069
|- style="text-align: center; font-size: 75%;"
4070
| colspan="1" | '''Figure 32:''' Distorted element in a 2D triangulation.
4071
|}
4072
4073
Whenever the distance between two nodes is smaller than a prearranged critical length, one of those nodes is removed and placed in other zone of the mesh, for example where there are some distorted elements. The reallocation of the nodes is done with the purpose of preserving the number of particles and, consequently, guaranteeing the conservations of the mean mesh size (if the volume does not change).
4074
4075
Another type of critical elements are the slivers, so the simplices with null area or volume. These  may form when there are three points contained in the same line, or  four points laying in the same 3D plane as shown in Figure [[#img-33|33]]. <div id='img-33'></div>
4076
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4077
|-
4078
|[[Image:draft_Samper_722607179-sliver2.png|200px|Sliver element in 3D.]]
4079
|- style="text-align: center; font-size: 75%;"
4080
| colspan="1" | '''Figure 33:''' Sliver element in 3D.
4081
|}
4082
4083
In theory, the Alpha Shape method should not allow the formation of those elements because the associated circumradius is huge, see Figure [[#img-30c|30c]]. However, these elements are critical from the numerical point of view and they can escape to the Alpha Shape control. Furthermore, also in the case that the sliver element is erased from the Delaunay triangulation, this would create a not physical void within the domain. For all these reasons, the formation of sliver elements must be prevented. For these elements the control for the minimum edges lengths is worthless because the sliver may form also when the element nodes are not close.   In this work, the formation of slivers is prevented by checking the element areas. Whenever the element surface or volume is less than a threshold value, computed for example with respect to the mean area of the mesh, one of the nodes of those elements is removed. Note that this control is also beneficious versus the formation of sharp elements.
4084
4085
After these operations over the nodes, the discretization needs to be updated. This can be done modifying locally the mesh or performing again the Delaunay triangulation and the Alpha Shape control. Once the updated mesh is created, the nodal variables of the new particles are assigned, via interpolation, with the values of the neighbor nodes.
4086
4087
All these criteria have been explained for homogeneous meshes. However, they can be easily extended to  heterogeneous meshes. For example, if there is a mesh refinement in a specific zone of the domain, that refined mesh can be preserved if the described checks for the minimum nodes distance and the elements surface are done not for the whole mesh but for the local mesh.
4088
4089
====3.4.1.2 Basic steps====
4090
4091
The basic steps of the PFEM algorithm are here given with the help of a graphic representation (Figure [[#img-34|34]]). <div id='img-34'></div>
4092
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4093
|-
4094
|[[Image:draft_Samper_722607179-PFEMsteps.png|600px|Sequence of steps to update a “cloud” of fluid nodes and a discretized solid domain  from time n   (t=ⁿt)  to   time n+1 (t=ⁿt +∆t).]]
4095
|- style="text-align: center; font-size: 75%;"
4096
| colspan="1" | '''Figure 34:''' Sequence of steps to update a “cloud” of fluid nodes and a discretized solid domain  from time <math>n</math>   (<math>t={^n t}</math>)  to   time <math>n+1</math> (<math>t={^n t} +\Delta t</math>).
4097
|}
4098
Consider a domain <math display="inline">V</math> containing fluid and solid subdomains.
4099
4100
In this work the PFEM is used only for the fluid domains. The solids are computed with the classical FEM and they maintain the same discretization for the whole duration of the analysis. In many other works the PFEM has been also used for modeling  the solid and the fluid domains, indifferently [19,20,95,99].
4101
4102
At the beginning of a time step (first picture of  Figure [[#img-34|34]]) the fluid is represented by a cloud of points, or particles, that store all the information about the physics (for example the density and the viscosity of the fluid), the geometry (information about the local mesh size), the kinematics (the velocity and the acceleration) and the pressure. On the other hand, the solid has the same mesh  of the previous time step and the history for its kinematics and the stress and strain fields is stored in both the nodes and the elements.
4103
4104
The first step represents the distinguishing  step of the PFEM and it consists on the creation of a mesh for the fluid. The algorithm is the one that has been described in Section [[#3.4.1.1 Remeshing|3.4.1.1]]. So first the Delaunay triangulation is performed and then the physical contours of the domains are recognized with the Alpha Shape method. Specifically, the fluid detects automatically all its boundaries, hence the rigid walls, its free surface contours and the interface with the solid. During this step flying subdomains may form or/and some particles may detach from the rest of the domain (second picture of  Figure [[#img-34|34]])
4105
4106
Once all the domains are discretized, it is possible to solve  the equations of continuum mechanics for each of the subdomains using the FEM. The state variables are computed at the next (updated) configuration for <math display="inline">^n t+\Delta t</math>: velocities, pressure and viscous stresses in the fluid and displacements, stresses and strains in the solid. After that, the fluid mesh can be erased and these steps are repeated. For reducing the computational cost associated to the detection of the fluid boundaries it is very useful to memorize those nodes that in the previous time step were located at the contours. In this way, the nodes checked by the Alpha Shape method can be drastically reduced.
4107
4108
Once again, note that in the fluid all the information is stored in the nodes so this operation does not require interpolation procedures for recovering the elemental information. If the PFEM would be used also for the solid mechanics analysis, an algorithm for transferring the elemental information from the previous to the new mesh should be implemented. In [18] an efficient technique  for this purpose is described and validated.
4109
4110
====3.4.1.3 Advantages and disadvantages====
4111
4112
The key differences between the PFEM and the classical FEM are the remeshing technique and the identification of the domain boundary at each time step. The quality of the numerical solution depends on the discretization chosen as in the standard FEM and, as it has been explained, adaptive mesh refinement techniques can be used to improve the solution.  So from this point of view, during the computation step, the PFEM behaves as a classical Lagrangian FEM, with the same advantages and drawbacks.
4113
4114
However with the PFEM the remeshing may introduce a further source of error in the numerical scheme. In fact during this operation some new elements may be created and some other erased. These changes cause local perturbations of the equilibrium reached at the previous time step and they may affect the accuracy and the convergence of the scheme. In fact at the beginning of the time step, each node stores the kinematics and the pressure obtained with a discretization that may be different than the new one. So those values that at the end of the previous step were at equilibrium for the previous configuration and mesh, may not be at equilibrium at the beginning of the new time step for the new configuration created by the remeshing.
4115
4116
Another drawback induced by the remeshing concerns the variation of volume. Although, generally, the volumes gained and lost with the remeshing are compensated, there may be cases where the variation of volume is not negligible. From the personal experience of the author,  remeshing tends to increase the volume of the domain. For avoiding this inconvenience, some additional criteria for limiting the creation of new elements should be added. Generally, the critical zones are the ones near to the free surface, where, typically, the remeshing tends to create new elements that 'close' the surface  waves and increase the overall volume of the fluid, as  shown in Figure [[#img-35|35]]. 
4117
4118
4119
<div id='img-35a'></div>
4120
<div id='img-35b'></div>
4121
<div id='img-35c'></div>
4122
<div id='img-35d'></div>
4123
<div id='img-35'></div>
4124
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4125
|-
4126
|[[Image:draft_Samper_722607179-zoomOUT2.png|400px]]
4127
|[[Image:draft_Samper_722607179-zoomOUT.png|400px|Mesh detail at the end of 
4128
step n]]
4129
|- style="text-align: center; font-size: 85%;"
4130
| (a) Mesh at the end of step <math>n</math>
4131
| (b) Mesh at the beginning of step <math>n+1</math>
4132
|-
4133
|[[Image:draft_Samper_722607179-zoomIN2.png|400px|Mesh detail  at the beginning of step n+1]]
4134
|[[Image:draft_Samper_722607179-zoomIN.png|400px|Example of variation of volume induced by the remeshing.]]
4135
|- style="text-align: center; font-size: 85%;"
4136
| (c) Mesh detail at the end of step <math>n</math>
4137
| (d) Mesh detail  at the beginning of step <math>n+1</math>
4138
|- style="text-align: center; font-size: 75%;"
4139
| colspan="2" | '''Figure 35:''' Example of variation of volume induced by the remeshing.
4140
|}
4141
4142
It is possible to reduce this tendency of the remeshing by locally penalizing the Alpha Shape method. In this work, the free surface nodes have assigned a smaller Alpha Shape parameter <math display="inline">\alpha </math> than the other nodes of the mesh.
4143
4144
This penalization is also good for improving the timing of the contact with the rigid walls or the solid interfaces. In particular, this strategy may  delay the phenomenon illustrated in Figure [[#img-36|36]]. The elements created at the free surface may accelerate the impact between the fluid streams and the containing walls.  
4145
4146
<div id='img-36'></div>
4147
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4148
|-
4149
|[[Image:draft_Samper_722607179-Snapshot3.png|400px|]]
4150
|[[Image:draft_Samper_722607179-Snapshot2.png|400px|Example of detection of contact through the remeshing.]]
4151
|- style="text-align: center; font-size: 85%;"
4152
|(a) End of time step <math>n</math>
4153
|(b) Beginning of time step <math>n+1</math>
4154
|- style="text-align: center; font-size: 75%;"
4155
| colspan="2" | '''Figure 36:''' Example of detection of contact through the remeshing.
4156
|}
4157
The smaller is <math display="inline">\alpha </math> for the interface nodes, the later a contact element between those nodes and the solid or rigid contour is built. In fact, a smaller <math display="inline">\alpha </math> for the interface nodes delays the formation of the contact elements with the rigid (or deformable) contours reproducing better the physical phenomenon of the contact. However, the Alpha Shape cannot be penalized excessively because otherwise the fluid could pass through the wall, or the solid interface.
4158
4159
The remeshing also represents an additional computational cost. In previous works it has been found that the computational time associated to the remeshing grows linearly with the number of nodes [81]. Specifically,  for a single processor Pentium IV PC  the  meshing consumes for 3D problems around 15% of the total CPU time per time step, while the solution of the equations (with typically 3 iterations per time step) and the system assembly consumes approximately 70% and 15% of the CPU time per time step, respectively.  The treatment of the boundary nodes is an issue that deserves a particular attention. In the previous works with the PFEM stick conditions were generally used for the nodes on the rigid walls. However, this may induce pressure concentrations on the boundary elements and deteriorate locally the quality of the mesh. In fact the stick condition affects a zone that has the same order of magnitude of the spatial discretization. This may induce a non-physical behavior, especially when inviscid fluids are analyzed or/and coarse meshes are employed. The problem is even more critical in 3D. Recently,  Cremonesi <math display="inline">et al.</math> [33,35] used slip conditions in the simulation of landslides for better modeling the interaction between a landslide and the substrate interface. In the mentioned works, the wall nodes are treated in an Eulerian way by adding the convective term for the boundary elements.  In this work, the wall particles are still computed in a Lagrangian framework and they are free to move along the direction of the walls. The slip conditions are simulated with a simple algorithm. Essentially it consists on leaving the wall particles move along the direction of the wall until when the separation from the original position is larger than a prearranged critical distance. In that case, during the remeshing procedure, the particle is removed and reallocated at its original position. The reallocation of the particle is done in order to prevent the creation of voids in the walls that may cause fluid leakage. The kinematics and the pressure of the moved particle are obtained via interpolation from the neighbor nodes, in the classical PFEM fashion. In Figure [[#img-37|37]] an example of the application of the algorithm is given. 
4160
4161
4162
<div id='img-37'></div>
4163
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4164
|-
4165
|[[Image:draft_Samper_722607179-movingWallVel2.png|400px|]]
4166
|[[Image:draft_Samper_722607179-movingWallVel3.png|400px|]]
4167
|-
4168
|[[Image:draft_Samper_722607179-movingWallVel4.png|400px|]]
4169
|[[Image:draft_Samper_722607179-movingWallVel6.png|400px|Agorithm for moving the wall particles.]]
4170
|- style="text-align: center; font-size: 75%;"
4171
| colspan="2" | '''Figure 37:''' Agorithm for moving the wall particles.
4172
|}
4173
The pictures refer to the front of a flow.
4174
4175
The void circles follow the position of two nodes of the wall discretization, the full circles denote the original position of those nodes. The pictures show that the wall nodes are free to move along the wall direction and when they reach the maximum separation from the original position (in this case <math display="inline">0.6 h_w</math>, where <math display="inline">h_w</math> is the distance between the node of the wall) they are reallocated at the original position.  In the section dedicated to the validation examples, some interesting comparisons between stick and slip problems are given, both in 2D and 3D. It will be shown that this algorithm can be very helpful for preserving the quality of the mesh and for the overall accuracy of the computation.
4176
4177
All the weak points of the PFEM presented until now are the price to pay in order to gain all the benefits that this Lagrangian technique can give. The PFEM in fact makes simple some tasks that are extremely critical for a complex computational analysis and, in some cases, it allows the solution and the modeling of problems that other methods cannot even face.
4178
4179
The PFEM is designed for those problems where a huge change of topology occurs. The mesh nodes define the evolution of a   domain during its history of deformation. In this way, the detection of the free surface in a fluid flow or the contours of a melting  solid object do not introduce in the scheme any additional complication or implementation work because their contours are known at the beginning of each time step.
4180
4181
With the PFEM the bodies are totally free to deform and move in the space. There is not any limitation in this sense because the domain is continuously updated. Hence the final configuration can be very different then the initial one and occupy a volume of space arbitrary bigger than the initial one. This cannot been done with a classical Eulerian strategy, where a bounding box needs to be defined at the beginning of the analysis.
4182
4183
Furthermore with the PFEM the interface between a fluid stream and a solid is detected automatically via the same Alpha Shape method that generates the mesh. The interface is formed by the nodes of those hybrid elements (elements which nodes belong to both the solid and the fluid) that have been  generated by the Delaunay triangulation and they fulfill the Alpha Shape criteria.
4184
4185
PFEM is also easy to couple with other numerical methods. This thesis is an example of the coupling of the PFEM with the FEM. In other works, it has been shown that the PFEM can also be easily coupled with discrete methods [87]. Furthermore, within the same PFEM, one may chose the preferred numerical method for solving the governing equations after the remeshing. In this work a mixed velocity-pressure stabilized formulation is used for the fluid, but one may implement any other Lagrangian methodology.
4186
4187
All this explains why in the past the PFEM has been employed for facing challenging simulations as tunneling [19,20], forming processes [83,93], melting of polymers [90,95], transport erosion and sedimentation in fluids [81], and fluid-multibody interaction [80].
4188
4189
In conclusion, the PFEM is an extremely powerful technology which shortcomings are legitimized by the complexity of the problems to solve.
4190
4191
===3.4.2 Mass conservation analysis===
4192
4193
This section is focused on the analysis of the mass preservation feature of the PFEM-FIC stabilized formulation for free surface flow problems.
4194
4195
It is useful to distinguish two different sources of variation of volume (in all the examples presented in this work, the density is a constant, so the concepts of mass variation and volume variation are totally equivalent). The first source of volume variation is induced by the numerical solution for each time step increment. The volume may vary during the non-linear iterations due to the inaccuracy of the scheme. So the volume of the fluid domain changes from a volume <math display="inline">{^nV}</math> computed at the beginning of the time step to a slightly different volume <math display="inline">{^{n+1} \bar V}</math> computed after the convergence of the solution scheme. This volume variation caused by the computation is called <math display="inline">\Delta V^{c}</math>.
4196
4197
The second source of mass loss is related to the remeshing. After the achievement of the convergence the fluid domain is remeshed. During this process the volume may vary because some elements are erased or new elements are added to the mesh. The volume at the end of the time step is <math display="inline">{^{n+1} V}</math> and the variation induced by the remeshing with respect to <math display="inline">{^{n+1} \bar V}</math>  is named <math display="inline">\Delta V^{m}</math>. In  Figure [[#img-38|38]]  a graphical representation of the scheme is given. <div id='img-38'></div>
4198
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4199
|-
4200
|
4201
[[File:Draft_Samper_722607179_5417_Fig3-11.png|600px]]
4202
|- style="text-align: center; font-size: 75%;"
4203
| colspan="1" | '''Figure 38:''' Monitoring of volume within a time step interval <math>({}^nt,{}^{n+1}t)</math>.
4204
|}
4205
4206
The sum of both contributions gives the total volume variation for each time step such as
4207
4208
<span id="eq-227"></span>
4209
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4210
|-
4211
| 
4212
{| style="text-align: left; margin:auto;" 
4213
|-
4214
| style="text-align: center;" | <math>\Delta V^{tot}=\Delta V^{c}+\Delta V^{m}  </math>
4215
|}
4216
| style="width: 5px;text-align: right;" | (227)
4217
|}
4218
4219
where
4220
4221
<span id="eq-228"></span>
4222
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4223
|-
4224
| 
4225
{| style="text-align: left; margin:auto;" 
4226
|-
4227
| style="text-align: center;" | <math>\Delta V^{c}= {^{n+1}\bar V} -  {^{n} V}    </math>
4228
|}
4229
| style="width: 5px;text-align: right;" | (228)
4230
|}
4231
4232
<span id="eq-229"></span>
4233
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4234
|-
4235
| 
4236
{| style="text-align: left; margin:auto;" 
4237
|-
4238
| style="text-align: center;" | <math>\Delta V^{m}=  {^{n+1} V} - {^{n+1}\bar V}     </math>
4239
|}
4240
| style="width: 5px;text-align: right;" | (229)
4241
|}
4242
4243
For the proper understanding of the mass preservation feature of a formulation is essential to distinguish these different mass losses sources. Nevertheless, it is improper to define these two sources as uncorrelated. In fact, the variation of mass generated by the remeshing may affect the mass losses of the non-linear scheme. In fact, the remeshing induces a perturbation of the equilibrium reached in the previous time step through the elimination and creation of elements. This may affect the convergence of the numerical scheme and, consequently, the satisfaction of the incompressibility constraint.
4244
4245
The numerical examples presented in this section are taken from [82]. All the examples have been run using the real value of the bulk modulus <math display="inline">\kappa </math>  in matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-220|220]])). This matrix is the component of the tangent matrix <math display="inline">{K}</math> for linear momentum equations  that takes into account the pressure variation. In Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]] it will be shown that  substituting  in matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-220|220]])) the real bulk  modulus <math display="inline">\kappa </math>  with a  pseudo bulk modulus defined as <math display="inline">\kappa _p=\theta \kappa </math>,  is helpful for the conditioning and the overall accuracy of the analysis.
4246
4247
====3.4.2.1 Numerical examples====
4248
4249
In order to verify the mass preservation feature of the formulation, several problems for which guaranteeing the mass conservation is critical, have been solved. In this section,  some of these involving impact and mixing of free surface fluids, are presented.
4250
4251
'''''Sloshing of water in a prismatic tank'''''
4252
4253
The problem has been solved first in 2D. Figure [[#img-39|39]] shows the geometry of the tank, the material properties, the time step size and the initial mesh of 5064 3-noded triangles discretizing the interior fluid. The fluid oscillates due to the hydrostatic forces induced by its original position.   In this and in the next problems, the effect of the surrounding air has not been taken into account.  <div id='img-39'></div>
4254
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
4255
|-
4256
|[[Image:draft_Samper_722607179-Figure2_SL2D_official_input.png|420px|2D analysis of sloshing of water in a prismatic tank. Initial geometry, analysis data and mesh of 5064 3-noded triangles discretizing the water in the tank.]]
4257
|- style="text-align: center; font-size: 75%;"
4258
| colspan="1" | '''Figure 39:''' 2D analysis of sloshing of water in a prismatic tank. Initial geometry, analysis data and mesh of 5064 3-noded triangles discretizing the water in the tank.
4259
|}
4260
4261
This problem, as for  all the ones presented in this section, has been solved using the real bulk modulus in matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-220|220]])), that is equivalent to compute the pseudo bulk modulus <math display="inline">\kappa _p=\theta \kappa </math> with the parameter <math display="inline">\theta=1</math>.
4262
4263
Figures [[#img-40|40]] and  [[#img-41|41]] show some snapshots of the water stream at different times. Pressure contours are plotted over the deformed configuration. <div id='img-40'></div>
4264
4265
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4266
|-
4267
|[[Image:draft_Samper_722607179-Figure4a_SL2D_t=5_7s.png|400px|t=5.7s]]
4268
|[[Image:draft_Samper_722607179-Figure4b_SL2D_t=7_4s.png|400px|2D sloshing of water in a prismatic tank. Snapshots of water geometry at two different times (θ= 1). Colours indicate pressure contours (I/II).]]
4269
|- style="text-align: center; font-size: 75%;"
4270
| <math>t=5.7s</math>
4271
| <math>t=7.4s</math>
4272
|- style="text-align: center; font-size: 75%;"
4273
| colspan="2" | '''Figure 40:''' 2D sloshing of water in a prismatic tank. Snapshots of water geometry at two different times (<math>\theta = 1</math>). Colours indicate pressure contours (I/II).
4274
|}
4275
4276
<div id='img-41'></div>
4277
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4278
|-
4279
|[[Image:draft_Samper_722607179-Figure4c_SL2D_t=13_3s.png|400px|t=13.3st=18.6s]]
4280
|[[Image:draft_Samper_722607179-Figure4d_SL2D_t=18_6s.png|400px|2D sloshing of water in a prismatic tank. Snapshots of water geometry at two different times (θ= 1). Colours indicate pressure contours (II/II).]]
4281
|- style="text-align: center; font-size: 75%;"
4282
| <math>t=13.3s</math>
4283
| <math>t=18.6s</math>
4284
|- style="text-align: center; font-size: 75%;"
4285
| colspan="2" | '''Figure 41:''' 2D sloshing of water in a prismatic tank. Snapshots of water geometry at two different times (<math>\theta = 1</math>). Colours indicate pressure contours (II/II).
4286
|}
4287
4288
The accumulated volume loss (in percentage versus the initial volume) for the method proposed  is approximately  1.33% over 20 seconds of simulation time (curve <math display="inline">A</math> of Figure [[#img-42|42]]). This value has been computed  by summing all the volume variations due to the computation (<math display="inline">\Delta V^c</math>) for all the time steps of the analysis as
4289
4290
<span id="eq-230"></span>
4291
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4292
|-
4293
| 
4294
{| style="text-align: left; margin:auto;" 
4295
|-
4296
| style="text-align: center;" | <math>{^n{\Delta V}_{%}}=\frac{\sum _{i=1}^{n}{^i{\Delta V}^c}}{V_{initial}}\cdot 100  </math>
4297
|}
4298
| style="width: 5px;text-align: right;" | (230)
4299
|}
4300
4301
where <math display="inline">V_{initial}</math> is the initial volume.
4302
4303
In Figure [[#img-42|42]] also the mass losses obtained using a standard first order fractional step method and the PFEM  [82] are plotted. Clearly the method proposed in this  work leads to a reduced overall fluid volume loss. <div id='img-42'></div>
4304
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4305
|-
4306
|[[Image:draft_Samper_722607179-Figure5a_SL2D_allLosses.png|600px|2D sloshing of water in a prismatic tank. Time evolution of the percentage of water volume loss  due to the numerical algorithm. Comparison with the fractional step solution. ]]
4307
|- style="text-align: center; font-size: 75%;"
4308
| colspan="1" | '''Figure 42:''' 2D sloshing of water in a prismatic tank. Time evolution of the percentage of water volume loss  due to the numerical algorithm. Comparison with the fractional step solution. 
4309
|}
4310
4311
In some cases, it may be required to ensure the absolute absence of volume variations. With a small shifting of the free surface it is possible to guarantee the perfect mass conservation in the fluid domain. The method consists on moving the free surface nodes in the normal direction with an offset equal to <math display="inline">\frac{\Delta V}{L_{free~surface}}</math>, where the variation of volume <math display="inline">\Delta V</math> is computed as the sum of the volume variation due to the remeshing at the previous time step <math display="inline">^{n-1}\Delta V^m</math>  and the volume variation due to computation of the current time step <math display="inline">^{n}\Delta V^c</math>. <math display="inline">{L_{free~surface}}</math> is the length of the free surface. In Figure [[#img-43|43]] a graphical representation of the technique in 2D is given.
4312
4313
<div id='img-43'></div>
4314
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 97%;max-width: 100%;"
4315
|-
4316
|[[Image:draft_Samper_722607179-offset.png|582px|Strategy for recovering the initial fluid volume by correcting the free surface at mesh generation level.  (a) Compute total volume variation (∆V) before remeshing. (b) Compute free surface offset = \frac∆VL<sub>free~surface</sub>. (c) Move free surface nodes in the normal direction to the boundary a distance equal to the offset computed in (b).]]
4317
|- style="text-align: center; font-size: 75%;"
4318
| colspan="1" | '''Figure 43:''' Strategy for recovering the initial fluid volume by correcting the free surface at mesh generation level.  (a) Compute total volume variation (<math>\Delta V</math>) before remeshing. (b) Compute free surface offset = <math>\frac{\Delta V}{L_{free~surface}}</math>. (c) Move free surface nodes in the normal direction to the boundary a distance equal to the offset computed in (b).
4319
|}
4320
4321
This procedure induces an arbitrary alteration of the equilibrium configuration, thus it has to be used cautiously. A fundamental requirement for using this technique is that the solution scheme produces very small volume variations and so the correction is minimal.
4322
4323
Figure [[#img-44|44]] shows that the total fluid volume loss can be reduced to almost zero by applying this strategy at each time step. <div id='img-44'></div>
4324
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
4325
|-
4326
|[[Image:draft_Samper_722607179-Fig6a.png|420px|2D sloshing of water in a prismatic tank (θ= 1). Mass losses applying the strategy described in Figure [[#img-43|43]].]]
4327
|- style="text-align: center; font-size: 75%;"
4328
| colspan="1" | '''Figure 44:''' 2D sloshing of water in a prismatic tank (<math>\theta = 1</math>). Mass losses applying the strategy described in Figure [[#img-43|43]].
4329
|}
4330
4331
Figure [[#img-45|45]] shows a comparison of the fluid volume losses between using the real bulk modulus (<math display="inline">\theta =1</math>) and an arbitrarily reduced pseudo bulk modulus (<math display="inline">\theta = 0.08</math>). For both cases, the time step increment is <math display="inline">\Delta t = 10^{-3}s</math>.
4332
4333
<div id='img-45'></div>
4334
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
4335
|-
4336
|[[Image:draft_Samper_722607179-Comparison-value.png|420px|2D sloshing of water in a prismatic tank. Time evolution of percentage of water volume loss obtained using the current method with θ=0.08 (curve A) and θ= 1 (curve B) ∆t = 10⁻³s.]]
4337
|- style="text-align: center; font-size: 75%;"
4338
| colspan="1" | '''Figure 45:''' 2D sloshing of water in a prismatic tank. Time evolution of percentage of water volume loss obtained using the current method with <math>\theta =0.08</math> (curve A) and <math>\theta = 1</math> (curve B) <math>\Delta t = 10^{-3}s</math>.
4339
|}
4340
4341
Figure [[#img-46|46]] shows that a similar improvement in the volume preservation can be obtained using <math display="inline">\theta = 1</math> and reducing the time step to <math display="inline">\Delta t = 10^{-4}s</math>. This, however, increases the cost of the computations.
4342
4343
<div id='img-46'></div>
4344
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
4345
|-
4346
|[[Image:draft_Samper_722607179-volume-preservation.png|420px|2D sloshing of water in a prismatic tank. Time evolution of percentage of mass loss obtained with the current method. Curve A: θ=1 and ∆t = 10⁻⁴s. Curve B: θ= 1 and ∆t = 10⁻³s.]]
4347
|- style="text-align: center; font-size: 75%;"
4348
| colspan="1" | '''Figure 46:''' 2D sloshing of water in a prismatic tank. Time evolution of percentage of mass loss obtained with the current method. Curve A: <math>\theta =1</math> and <math>\Delta t = 10^{-4}s</math>. Curve B: <math>\theta = 1</math> and <math>\Delta t = 10^{-3}s</math>.
4349
|}
4350
4351
These results show that accurate numerical results with reduced volume losses can be obtained by appropriately adjusting the parameter <math display="inline">\theta </math> in the tangent bulk modulus matrix, while keeping the time step size to competitive values in terms of CPU cost. In Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]] this issue will be analyzed in detail.  Specifically, the effects of the bulk modulus <math display="inline">\kappa </math> in terms of volume preservation and overall accuracy will be analyzed and an efficient strategy to predict the best value for the pseudo bulk modulus  (<math display="inline">\kappa _p=\theta \kappa </math>) will be presented.
4352
4353
In Figure [[#img-47|47]] the input data for the 3D case are given. The same sloshing problem of the 2D simulation is solved using a relative coarse initial mesh of 106771 4-noded tetrahedra and <math display="inline">\theta = 1</math>.  <div id='img-47'></div>
4354
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4355
|-
4356
|[[Image:draft_Samper_722607179-Figure7a_SL3D_AS2D_input.png|600px|3D analysis of sloshing of water in prismatic tank (θ= 1). Initial geometry and analysis data.]]
4357
|- style="text-align: center; font-size: 75%;"
4358
| colspan="1" | '''Figure 47:''' 3D analysis of sloshing of water in prismatic tank (<math>\theta = 1</math>). Initial geometry and analysis data.
4359
|}
4360
4361
Figure [[#img-48|48]] shows the results for the 3D analysis at the same time instants of Figure [[#img-40|40]].
4362
4363
<div id='img-48'></div>
4364
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4365
|-
4366
|[[Image:draft_Samper_722607179-Figure7bSL3D_AS2D_5_7.png|400px|t=5.7st=7.4s]]
4367
|[[Image:draft_Samper_722607179-Figure7c_SL3D_AS2D_7_4.png|400px|3D analysis of sloshing of water in prismatic tank (θ= 1). Snapshots of water geometry at two different times (t=5.7s left and t=7.4s right).]]
4368
|- style="text-align: center; font-size: 75%;"
4369
| <math>t=5.7s</math>
4370
| <math>t=7.4s</math>
4371
|- style="text-align: center; font-size: 75%;"
4372
| colspan="2" | '''Figure 48:''' 3D analysis of sloshing of water in prismatic tank (<math>\theta = 1</math>). Snapshots of water geometry at two different times (<math>t=5.7</math>s left and <math>t=7.4</math>s right).
4373
|}
4374
4375
In Figures [[#img-49|49]] and [[#img-50|50]] the mass losses are analyzed. The values of Figure  [[#img-49|49]] have been computed as Eq.([[#eq-230|230]]). It is remarkable that the percentage of total fluid volume loss due to the numerical scheme after 10 seconds of analysis is approximately 1%
4376
4377
The values plotted in the graph of  Figure [[#img-50|50]] are the percentage mass loss due to computation for each time step.
4378
4379
These values have been computed for each time step <math display="inline">n</math> as
4380
4381
<span id="eq-231"></span>
4382
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4383
|-
4384
| 
4385
{| style="text-align: left; margin:auto;" 
4386
|-
4387
| style="text-align: center;" | <math>{^n{\Delta V}_{%}}=\frac{{^n{\Delta V}^c}}{V_{initial}}\cdot 100  </math>
4388
|}
4389
| style="width: 5px;text-align: right;" | (231)
4390
|}
4391
4392
<div id='img-49'></div>
4393
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
4394
|-
4395
|[[Image:draft_Samper_722607179-Fig13b_New.png|390px|3D analysis of sloshing of water in prismatic tank (θ= 1).  Time evolution of accumulated water volume loss (in percentage) due to the numerical algorithm.]]
4396
|- style="text-align: center; font-size: 75%;"
4397
| colspan="1" | '''Figure 49:''' 3D analysis of sloshing of water in prismatic tank (<math>\theta = 1</math>).  Time evolution of accumulated water volume loss (in percentage) due to the numerical algorithm.
4398
|}
4399
4400
<div id='img-50'></div>
4401
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
4402
|-
4403
|[[Image:draft_Samper_722607179-Fig13c_New.png|390px|3D analysis of sloshing of water in prismatic tank (θ= 1). Volume loss per time step over 2 s of analysis. Average volume loss per time step: 1.64×10⁻⁴%]]
4404
|- style="text-align: center; font-size: 75%;"
4405
| colspan="1" | '''Figure 50:''' 3D analysis of sloshing of water in prismatic tank (<math>\theta = 1</math>). Volume loss per time step over <math>2 s</math> of analysis. Average volume loss per time step: 1.64<math>\times 10^{-4}</math>%
4406
|}
4407
4408
'''''Collapse of two water columns in a prismatic tank'''''
4409
4410
This  problem simulates the 2D motion, impact and subsequent mixing of two fluid streams originated by the collapse of two water columns located at the end sides of a prismatic tank. Figure [[#img-51|51]] shows the initial geometry and the problem data. <div id='img-51'></div>
4411
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
4412
|-
4413
|[[Image:draft_Samper_722607179-Figure16_doubleDB_2D_input.png|540px|Collapse and impact of two water columns in a prismatic tank. Analysis data, initial geometry and discretization (3988 3-noded triangles).]]
4414
|- style="text-align: center; font-size: 75%;"
4415
| colspan="1" | '''Figure 51:''' Collapse and impact of two water columns in a prismatic tank. Analysis data, initial geometry and discretization (3988 3-noded triangles).
4416
|}
4417
4418
The two water columns have been discretized  with 3988 3-noded triangles.
4419
4420
The effect of the surrounding air has not been taken into account in the analysis. The problem has been solved for <math display="inline">\theta = 1</math>.
4421
4422
Figure [[#img-52|52]] shows four snapshots of the motion of the water columns after removal of the retaining walls. Alter a few instants the two water streams impact with each other and mix as shown in the figures.
4423
4424
<div id='img-52'></div>
4425
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4426
|-
4427
|[[Image:draft_Samper_722607179-Figure17a_doubleDB056.png|400px|]]
4428
|[[Image:draft_Samper_722607179-Figure17b_doubleDB16.png|400px|]]
4429
|-
4430
|[[Image:draft_Samper_722607179-Figure17c_doubleDB_676.png|400px|]]
4431
|[[Image:draft_Samper_722607179-Figure17d_doubleDB_8.png|400px|Collapse and impact of two water columns. Snapshots of the evolution of the flow at different times.]]
4432
|- style="text-align: center; font-size: 75%;"
4433
| colspan="2" | '''Figure 52:''' Collapse and impact of two water columns. Snapshots of the evolution of the flow at different times.
4434
|}
4435
4436
The evolution of the percentage of the initial fluid volume loss over the simulation time is shown in Figure [[#img-53|53]].  <div id='img-53'></div>
4437
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
4438
|-
4439
|[[Image:draft_Samper_722607179-Fig18a.png|390px|Collapse and impact of two water columns. Accumulated fluid volume (in percentage) over eight seconds of analysis due to the numerical algorithm. Results for θ= 1.]]
4440
|- style="text-align: center; font-size: 75%;"
4441
| colspan="1" | '''Figure 53:''' Collapse and impact of two water columns. Accumulated fluid volume (in percentage) over eight seconds of analysis due to the numerical algorithm. Results for <math>\theta = 1</math>.
4442
|}
4443
4444
The values plotted in Figure  [[#img-53|53]] have been computed using Eq.([[#eq-230|230]]). A maximum of 2.8% of the initial fluid volume is  lost over eight seconds of analysis. This can be considered a low value for a problem of this complexity.
4445
4446
In Figure  [[#img-54|54]] the mass losses for each time step computed with Eq.([[#eq-231|231]]) are plotted. <div id='img-54'></div>
4447
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
4448
|-
4449
|[[Image:draft_Samper_722607179-Fig18b.png|390px|Collapse and impact of two water columns. Volume loss (in %) per time step. Results for θ= 1.]]
4450
|- style="text-align: center; font-size: 75%;"
4451
| colspan="1" | '''Figure 54:''' Collapse and impact of two water columns. Volume loss (in %) per time step. Results for <math>\theta = 1</math>.
4452
|}
4453
4454
'''''Falling of a water sphere in a cylindrical tank containing water'''''
4455
4456
The final example is the 3D analysis of the impact and mixing of a water drop as it falls in a cylindrical tank filled with the same fluid. Figure [[#img-55|55]] shows the material and analysis data and the initial discretization (88892 4-noded tetrahedra). <div id='img-55'></div>
4457
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
4458
|-
4459
|[[Image:draft_Samper_722607179-Figure19_ball_3D_input.png|540px|Falling of a water sphere in a tank filled with water. Analysis data, initial geometry and discretization (88892 4-noded tetrahedra).]]
4460
|- style="text-align: center; font-size: 75%;"
4461
| colspan="1" | '''Figure 55:''' Falling of a water sphere in a tank filled with water. Analysis data, initial geometry and discretization (88892 4-noded tetrahedra).
4462
|}
4463
4464
The problem has been solved using the real bulk modulus also in the tangent matrix (<math display="inline">{i.e.}</math> taking <math display="inline">\theta = 1</math>). Figure [[#img-56|56]] shows four snapshots of the mixing process at different times.
4465
4466
<div id='img-56'></div>
4467
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
4468
|-
4469
|[[Image:draft_Samper_722607179-figure20_ball1.png|300px|t=0.175st=0.275s]]
4470
|[[Image:draft_Samper_722607179-figure20_ball2.png|300px|t=0.500s]]
4471
|- style="text-align: center; font-size: 75%;"
4472
| <math>t=0.175s</math>
4473
| <math>t=0.275s</math>
4474
|-
4475
|[[Image:draft_Samper_722607179-figure20_ball3.png|300px|t=0.900s]]
4476
|[[Image:draft_Samper_722607179-figure20_ball4.png|300px|Falling of a water sphere in tank containing water. Evolution of the impact and mixing of the two liquids at different times. Results for θ= 1.]]
4477
|- style="text-align: center; font-size: 75%;"
4478
| <math>t=0.500s</math>
4479
| <math>t=0.900s</math>
4480
|- style="text-align: center; font-size: 75%;"
4481
| colspan="2" | '''Figure 56:''' Falling of a water sphere in tank containing water. Evolution of the impact and mixing of the two liquids at different times. Results for <math>\theta = 1</math>.
4482
|}
4483
4484
The total water mass lost in the sphere and the tank due to the numerical algorithm was <math display="inline">\simeq </math> 2% after 3 seconds of analysis (Figure [[#img-57|57]]). The average volume loss per time step was <math display="inline">2.54 \times 10^{-4}%</math>.
4485
4486
<div id='img-57'></div>
4487
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
4488
|-
4489
|[[Image:draft_Samper_722607179-Fig26a_NEW.png|390px|Falling of a water sphere in a tank containing water (θ= 1).  Accumulated volume over three seconds of analysis due to the numerical algorithm.]]
4490
|- style="text-align: center; font-size: 75%;"
4491
| colspan="1" | '''Figure 57:''' Falling of a water sphere in a tank containing water (<math>\theta = 1</math>).  Accumulated volume over three seconds of analysis due to the numerical algorithm.
4492
|}
4493
4494
===3.4.3 Analysis of the conditioning of the solution scheme===
4495
4496
This part is devoted to study the effect of the bulk modulus in the iterative matrix (matrix <math display="inline">K</math> (Eq.([[#eq-205|205]])) on the partitioned solution scheme for quasi-incompressible fluids. For clarity purposes, the decomposition of matrix <math display="inline">K</math> is recalled. <math display="inline">K</math> is computed as
4497
4498
<span id="eq-232"></span>
4499
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4500
|-
4501
| 
4502
{| style="text-align: left; margin:auto;" 
4503
|-
4504
| style="text-align: center;" | <math>{K}={K}^{\rho }+{K}^g+{K}^{\mu } +{K}^{\kappa }  </math>
4505
|}
4506
| style="width: 5px;text-align: right;" | (232)
4507
|}
4508
4509
with
4510
4511
<span id="eq-233"></span>
4512
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4513
|-
4514
| 
4515
{| style="text-align: left; margin:auto;" 
4516
|-
4517
| style="text-align: center;" | <math>K^{g}_{IJ}=I\int _{\Omega } \beta ^{T}_I \Delta t \sigma   \beta _J    d\Omega  </math>
4518
|}
4519
| style="width: 5px;text-align: right;" | (233)
4520
|}
4521
4522
<span id="eq-234"></span>
4523
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4524
|-
4525
| 
4526
{| style="text-align: left; margin:auto;" 
4527
|-
4528
| style="text-align: center;" | <math>K^{\rho }_{IJ}= I\int _{\Omega } N_I \frac{2\rho }{\Delta t}  N_J d\Omega   </math>
4529
|}
4530
| style="width: 5px;text-align: right;" | (234)
4531
|}
4532
4533
<span id="eq-235"></span>
4534
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4535
|-
4536
| 
4537
{| style="text-align: left; margin:auto;" 
4538
|-
4539
| style="text-align: center;" | <math>K^{\mu }_{IJ}=\int _{\Omega ^e} B^{T}_I  c^{\mu }  B_J    d\Omega  </math>
4540
|}
4541
| style="width: 5px;text-align: right;" | (235)
4542
|}
4543
4544
<span id="eq-236"></span>
4545
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4546
|-
4547
| 
4548
{| style="text-align: left; margin:auto;" 
4549
|-
4550
| style="text-align: center;" | <math>K^{\kappa }_{IJ}=\int _{\Omega ^e} B^{T}_I  m \Delta t \kappa  m^T B_J    d\Omega  </math>
4551
|}
4552
| style="width: 5px;text-align: right;" | (236)
4553
|}
4554
4555
In this section, it is shown that matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-236|236]]))  can deteriorate the conditioning of the linear system (Eq.([[#eq-204|204]])) and the overall accuracy of the numerical scheme if the real bulk modulus <math display="inline">\kappa </math> is used. In order to avoid these drawbacks, a pseudo bulk modulus <math display="inline">\kappa _p</math> should be used. In this section, a practical rule to set up the value of a pseudo-bulk modulus a priori in matrix <math display="inline">K^{\kappa }</math>  for improving the conditioning of the linear system is presented. The efficiency of the proposed strategy is tested in several problems analyzing the advantage of the modified bulk tangent matrix for the stability of the pressure field, the convergence rate and the computational speed of the analyses. The technique has been tested on the FIC/PFEM Lagrangian formulation presented, but it can be easily extended to other quasi-incompressible stabilized finite element formulations. The method proposed is based on using a pseudo-bulk modulus <math display="inline">\kappa _p</math> in the volumetric component <math display="inline">K^{\kappa }</math> (Eq.([[#eq-236|236]])) of the linear momentum tangent matrix <math display="inline">K</math> (Eq.([[#eq-232|232]])), while the actual physical value of the bulk modulus <math display="inline">\kappa </math> is used for the numerical solution of the mass conservation equation, namely in matrices <math display="inline">M_1</math> (Eq.([[#eq-198|198]])) and <math display="inline">M_2</math> (Eq.([[#eq-199|199]])).  The pseudo-bulk modulus <math display="inline">\kappa _p</math> is defined “a priori” as a proportion of the actual bulk modulus of the fluid (i.e. <math display="inline">\kappa _p =\theta \kappa </math> with <math display="inline">0< \theta \leq 1</math>).   The study here presented recalls the ideas presented in [45].
4556
4557
This section is structured as follows. In the first part, the numerical inconveniences induced by the real bulk modulus are analyzed. Then the strategy for predicting the optimum value for pseudo bulk modulus is explained. Finally several numerical examples are given in order to validate the technique.
4558
4559
====3.4.3.1 Drawbacks associated to the real bulk modulus====
4560
4561
The linearized system  (Eq.([[#eq-204|204]])) suffers from numerical instabilities due to the ill-conditioning of the iteration matrix <math display="inline">{K}</math>  (Eq.([[#eq-232|232]])) caused by the presence of the bulk modulus in matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-236|236]])), so in the part of <math display="inline">{K}</math>  that takes into account the pressure variation. This problem was already pointed out in previous works where similar partitioned schemes were used  [108-110]. The  ill-conditioning  of the iterative matrix of the linear momentum equations originates from the different orders of magnitude of its two main contributions: the mass matrix <math display="inline">{K}^{\rho }</math> (Eq.([[#eq-234|234]])) and the bulk matrix <math display="inline">{K^{\kappa }}</math>(the contributions of the viscous matrix <math display="inline">{K^{\mu }}</math> (Eq.([[#eq-235|235]])) and the geometric part <math display="inline">{K}^{g}</math>  (Eq.[[#eq-233|233]])) are negligible, for low viscous flows). Typically the terms of the bulk matrix are orders of magnitude larger than those of the mass matrix.
4562
4563
A reliable measure of the quality of a matrix is the ''condition number''  [6]. For a general matrix <math display="inline">A</math>, the condition number is defined as
4564
4565
<span id="eq-237"></span>
4566
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4567
|-
4568
| 
4569
{| style="text-align: left; margin:auto;" 
4570
|-
4571
| style="text-align: center;" | <math>C=cond({A}) = \parallel {A} \parallel \cdot  \parallel {A}^{-1} \parallel   </math>
4572
|}
4573
| style="width: 5px;text-align: right;" | (237)
4574
|}
4575
4576
where <math display="inline"> \parallel {A} \parallel </math> denotes here the L2 norm of matrix <math display="inline">{A}</math>.
4577
4578
The condition number <math display="inline">C</math> gives an indication of the accuracy of the results from the matrix inversion and the linear equation solution. Values of <math display="inline">C</math>  close to 1 indicate a well-conditioned matrix.
4579
4580
The deterioration of the quality of  matrix <math display="inline">{K}</math> affects directly the convergence of the iterative linear solver. In this work, the iterative <math display="inline">Bi-Conjugate Gradient</math> (BCG) solver has been used and its tolerance has been fixed to <math display="inline">10^{-6}</math>.
4581
4582
The time step increment affects highly the conditioning of the iterative matrix <math display="inline">{K}</math>, as it has been shown in the first numerical example of Section [[#3.4.2.1 Numerical examples|3.4.2.1]]. In fact, <math display="inline">{K}^{\rho }</math> is inversely proportional to the time increment, while <math display="inline">{K^{\kappa }}</math>  depends linearly on it (see Eqs.([[#eq-234|234]]) and ([[#eq-236|236]])). For this reason, <math display="inline">{K}</math> is well-conditioned only for a tight range of time increments.
4583
4584
For the sake of clarity, a numerical example is used to visualize and quantify the inconveniences caused by the ill-conditioned matrix in the iterative solution scheme. The problem chosen is the 2D water sloshing in a prismatic tank presented in Section [[#3.4.2.1 Numerical examples|3.4.2.1]]. However, with the purpose of reducing the computational cost of the analysis, the problem has been solved using a coarse mesh. So the initial geometry and the material data are the same as in  Figure [[#img-39|39]], but the mesh is the one shown in Figure [[#img-58a|58a]]. In Table [[#img-58|3.1]] all problem data are summarized. 
4585
4586
<div id='img-58a'></div>
4587
<div id='img-58'></div>
4588
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;max-width: 100%;"
4589
|-
4590
|[[Image:draft_Samper_722607179-SL_coarse_input.png|300px]]
4591
|- style="text-align: center; font-size: 75%;"
4592
| colspan="1" | '''Figure 58:'''  2D water sloshing. Initial geometry and finite element mesh.
4593
|}
4594
4595
<div class="center" style="font-size: 75%;width: 80%;">'''Table 3.1'''. 2D water sloshing. Problem data. </div>
4596
4597
{|  class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width: 80%;" 
4598
|- 
4599
| style="text-align:center;"| number of elements
4600
| style="text-align:center;"| 705
4601
|- 
4602
| style="text-align:center;"| number of nodes
4603
| style="text-align:center;"| 428
4604
|- 
4605
| style="text-align:center;"| average mesh size
4606
| style="text-align:center;"| 0.4 m
4607
|- 
4608
| style="text-align:center;"| <math> H_1</math>
4609
| style="text-align:center;"| 7 m
4610
|- 
4611
| style="text-align:center;"| <math> H_2</math>
4612
| style="text-align:center;"| 3 m
4613
|- 
4614
| style="text-align:center;"| ''D''
4615
| style="text-align:center;"| 10 m
4616
|- 
4617
| style="text-align:center;"| viscosity
4618
| style="text-align:center;"| <math> 10^{-3}</math> Pa.s
4619
|- 
4620
| style="text-align:center;"| density
4621
| style="text-align:center;"| <math>10^3</math> kg/m<math>^3</math>
4622
|- 
4623
| style="text-align:center;"| bulk modulus
4624
| style="text-align:center;"| <math>2 \times 15\times 10^9</math> Pa
4625
|}
4626
4627
4628
To highlight the importance of the time step on the conditioning of the iterative matrix, the problem has been solved for two different time increments without reducing the modulus in  matrix <math display="inline">{K}^{\kappa }</math> (<math display="inline">\theta=1</math>). Using a time step of <math display="inline">\Delta t</math>=<math display="inline">10^{-3}s</math>, the iterative matrix has a condition number <math display="inline">C=41</math> while, for <math display="inline">\Delta t</math>= <math display="inline">10^{-2}s</math>, the condition number is <math display="inline">C=3009</math>. If a larger time step is used, the linear system cannot even be solved. This deterioration is reflected by the number of iterations of the linear BCG solver: for the former case the average number of iterations is around <math display="inline">20</math>, while for the latter it is around <math display="inline">177</math>. Clearly, this also leads to a significant increase of computational time for solving a time step.
4629
4630
The ill-conditioning of the linear system also affects the rate of convergence of the iterative loop of the scheme given in Box 8.
4631
4632
In the graph of Figure [[#img-59|59]] the convergence rates <math display="inline">r</math> for the pressure and the velocity fields at <math display="inline">t</math>=1.75<math display="inline">s</math> are displayed. <div id='img-59'></div>
4633
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4634
|-
4635
|[[Image:draft_Samper_722607179-convergencePV.png|360px|2D water sloshing (θ=1). Convergence of the velocities and pressure at t=1.75 s.]]
4636
|- style="text-align: center; font-size: 75%;"
4637
| colspan="1" | '''Figure 59:''' 2D water sloshing (<math>\theta </math>=1). Convergence of the velocities and pressure at t=1.75 s.
4638
|}
4639
4640
For each iteration <math display="inline">i</math> of the non-linear loop, the convergence rates for the nodal velocities and pressures have been computed as
4641
4642
<span id="eq-238"></span>
4643
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4644
|-
4645
| 
4646
{| style="text-align: left; margin:auto;" 
4647
|-
4648
| style="text-align: center;" | <math>\displaystyle r_v=\frac{\Vert \Delta \bar {v}^{i+1}\Vert }{\Vert \bar {v}\Vert }  \le e_v   </math>
4649
|}
4650
| style="width: 5px;text-align: right;" | (238)
4651
|}
4652
4653
<span id="eq-239"></span>
4654
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4655
|-
4656
| 
4657
{| style="text-align: left; margin:auto;" 
4658
|-
4659
| style="text-align: center;" | <math>\displaystyle r_p= \frac{\Vert \bar {p}^{i+1}-\bar {p}^i\Vert }{\Vert \bar{p}\Vert } \le e_p   </math>
4660
|}
4661
| style="width: 5px;text-align: right;" | (239)
4662
|}
4663
4664
where <math display="inline">e_v</math> and <math display="inline">e_p</math> are prescribed error norms for the nodal velocities and the nodal pressures, respectively. In the examples solved in this work, <math display="inline">e_v = e_p=10^{-4}</math> has been chosen.
4665
4666
The values of Figure [[#img-59|59]]  have been obtained considering a time step increment of <math display="inline">\Delta t</math>=<math display="inline">10^{-2}s</math>. The curves show that the convergence of the scheme is slow, especially for the pressure field.  In fact, the pressure error (Eq.([[#eq-239|239]])) after 25 iterations is still larger than the pre-defined tolerance of <math display="inline">e_p=10^{-4}</math>. The lack of a good convergence for the pressure field has two main negative effects: the pressure solution is not accurate and mass conservation is not preserved.
4667
4668
The pressure contours at <math display="inline">t=1.75 s</math> are shown in Figure [[#img-60|60]].
4669
4670
<div id='img-60'></div>
4671
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4672
|-
4673
|[[Image:draft_Samper_722607179-SL_coarse_pressure175.png|360px|2D water sloshing (θ=1). Pressure contours at t=1.75 s.]]
4674
|- style="text-align: center; font-size: 75%;"
4675
| colspan="1" | '''Figure 60:''' 2D water sloshing (<math>\theta </math>=1). Pressure contours at <math>t=1.75 s</math>.
4676
|}
4677
4678
Concerning the mass conservation of the fluid, Figure [[#img-61|61]] shows the accumulated mass variation in absolute value versus time.  This values have been computed at each time step <math display="inline">n</math> as
4679
4680
<span id="eq-240"></span>
4681
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4682
|-
4683
| 
4684
{| style="text-align: left; margin:auto;" 
4685
|-
4686
| style="text-align: center;" | <math>{^n{\Delta V}_{%}}=\frac{\sum _{i=1}^{n}{|^i{\Delta V}^c|}}{V_{initial}}\cdot 100  </math>
4687
|}
4688
| style="width: 5px;text-align: right;" | (240)
4689
|}
4690
4691
where <math display="inline">{\Delta V}^c</math> is the volume loss due to computation and <math display="inline">V_{initial}</math> is the initial volume.
4692
4693
After 20 seconds of simulation, the percentage of mass loss is <math display="inline">9.8%</math>, which corresponds to a mean volume variation of <math display="inline">4.9 \cdot 10^{-3}%</math> per time step.
4694
4695
<div id='img-61'></div>
4696
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4697
|-
4698
|[[Image:draft_Samper_722607179-VolumeLossThetaNo.png|360px|2D water sloshing (θ=1). Accumulated percentage of mass variation in absolute value versus time.]]
4699
|- style="text-align: center; font-size: 75%;"
4700
| colspan="1" | '''Figure 61:''' 2D water sloshing (<math>\theta </math>=1). Accumulated percentage of mass variation in absolute value versus time.
4701
|}
4702
4703
In conclusion, the use of real bulk modulus (<math display="inline">\theta=1</math>) in matrix <math display="inline">{K}^{\kappa }</math> leads to ill-conditioned linear systems and, as consequence, it limits the range of suitable time step increments and deteriorates the convergence of the solution scheme.
4704
4705
These drawbacks were overcome in previous works [109,110] by substituting the physical bulk modulus with a smaller  pseudo bulk modulus <math display="inline">\kappa _p</math>. In this way, it is possible to extend the applicability of the partitioned scheme to a larger range of time increments. However, this strategy is based on heuristic criteria and cannot be used widely because each problem requires a specific value for <math display="inline">\kappa _p</math>. In other words, a pseudo bulk modulus that works well for certain analysis can fail for a different one. In this work,  the optimum pseudo bulk modulus <math display="inline">\kappa _p</math> is computed <math display="inline">apriori</math> and the strategy for predicting its value will be explained in the next section.
4706
4707
====3.4.3.2 Optimum value for the pseudo bulk modulus====
4708
4709
In this work, the pseudo bulk modulus is computed as <math display="inline">\kappa _p=\theta \kappa </math> choosing for the parameter <math display="inline">\theta </math>  a value able to guarantee the well-conditioning of the tangent matrix <math display="inline">{K}</math>  (Eq.([[#eq-236|236]])). The numerical examples presented in this section will show that this reduced value for the volumetric compressibility improves the overall accuracy of the scheme and does not affect the mass conservation. The reason is that the modification only affects the quality of the iterative matrix and the rate of convergence of the linear momentum equations, while the continuity equation, where the mass conservation constraint is imposed, is not modified.  In other words, the physical value of the bulk modulus <math display="inline">\kappa </math> is used in matrices <math display="inline">M_1</math> (Eq.([[#eq-198|198]])) and <math display="inline">M_2</math> (Eq.([[#eq-199|199]])).  This feature represents an innovation versus previous approaches where the pseudo bulk modulus is also used in the mass conservation equation  [108,109].
4710
4711
The parameter <math display="inline">\theta </math> is computed as
4712
4713
<span id="eq-241"></span>
4714
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4715
|-
4716
| 
4717
{| style="text-align: left; margin:auto;" 
4718
|-
4719
| style="text-align: center;" | <math>\theta =\frac{mean(\mid{K}^{\rho }\mid )}{mean(\mid  {K}^{\kappa }\mid )}  </math>
4720
|}
4721
| style="width: 5px;text-align: right;" | (241)
4722
|}
4723
4724
where the operator (<math display="inline">\mid \cdot \mid </math>) denotes the mean of the absolute values of the non-zero matrix components.
4725
4726
For a uniform mesh of elements of characteristic size <math display="inline">h</math>, the here called 'optimum value' of <math display="inline">\theta </math> is estimated as follows
4727
4728
<span id="eq-242"></span>
4729
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4730
|-
4731
| 
4732
{| style="text-align: left; margin:auto;" 
4733
|-
4734
| style="text-align: center;" | <math>\theta \approx \frac{N^2_c \cdot \rho \cdot h^2}{\kappa \cdot {\Delta t}^2}  </math>
4735
|}
4736
| style="width: 5px;text-align: right;" | (242)
4737
|}
4738
4739
where <math display="inline">{N_c}</math> is the value of the shape function <math display="inline">{N_I}</math> at the element center and the following approximation has been used:
4740
4741
<span id="eq-243"></span>
4742
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4743
|-
4744
| 
4745
{| style="text-align: left; margin:auto;" 
4746
|-
4747
| style="text-align: center;" | <math>{\partial N_I \over \partial x} \approx \frac{1}{h}  </math>
4748
|}
4749
| style="width: 5px;text-align: right;" | (243)
4750
|}
4751
4752
If  water is considered (<math display="inline">\rho=10^3 kg/m^{3}</math> and <math display="inline">\kappa=2.15\cdot 10^9 Pa</math>) and linear triangles are used (<math display="inline">N=1/3</math>), <math display="inline">\theta </math> has the following dependency with the mesh size and the time step,
4753
4754
<span id="eq-244"></span>
4755
{| class="formulaSCP" style="width: 100%; text-align: left;" 
4756
|-
4757
| 
4758
{| style="text-align: left; margin:auto;" 
4759
|-
4760
| style="text-align: center;" | <math>\theta \approx 5.17 \cdot 10^{-8}\cdot \left(\frac{ h}{ {\Delta t}} \right)^2  </math>
4761
|}
4762
| style="width: 5px;text-align: right;" | (244)
4763
|}
4764
4765
Typically, the parameter <math display="inline">\theta </math> is calculated at the beginning of the time step at the first iteration of the non-linear loop. It can also be computed  at every time step, at certain instants of the analysis or only once at the beginning of the analysis. This is so because the order of magnitude of <math display="inline">\theta </math> does not vary during the analysis, unless the time step is changed, or a refinement of the mesh is performed. In these cases, <math display="inline">\theta </math> needs to be calculated again because it has a square dependency on both parameters <math display="inline">h</math> and <math display="inline"> \Delta t</math>. The numerical results presented in this work have been obtained computing <math display="inline">{\theta }</math> via Eq.([[#eq-241|241]])  at the beginning of the analyses only.
4766
4767
The parameter <math display="inline">\theta </math> can be computed and assigned locally to each element or globally to the whole mesh. If it is calculated globally, all the elemental bulk contributions to the iteration matrix will have the same value of <math display="inline">\theta </math>. Otherwise  Eq.([[#eq-241|241]]) is used considering separately each element of the mesh; <math display="inline">{i.e.}</math> each element will have a different value of <math display="inline">\theta </math>. The former approach has a reduced computational cost but it works worse than the local approach for not uniform finite element discretizations. In particular, it is recommended to use the local approach when a refinement of the mesh is performed (in the next section, a problem with a refined zone is studied). In this  work, unless otherwise mentioned, the global approach for computing <math display="inline">\theta </math>  is used.
4768
4769
====3.4.3.3 Numerical examples====
4770
4771
In order to show the efficiency of the method described in Section [[#3.4.3.2 Optimum value for the pseudo bulk modulus|3.4.3.2]], two representative free surface problems involving the flow of water have been solved: the sloshing problem introduced in Section [[#3.4.3.1 Drawbacks associated to the real bulk modulus|3.4.3.1]] and the collapse of a water column against a rigid obstacle presented in [57].  The so-called dam break problem has been chosen here to demonstrate that the strategy does not affect the incompressibility constraint at all and the method is able to simulate problems of impact of fluids.  For both sloshing and dam break problems, a comparison of the performances of the scaled bulk matrix and the scheme with <math display="inline">\theta =1</math> is given.
4772
4773
The applicability and generality of the method is studied using very different average mesh sizes and time steps for both problems. Note that the dynamics of the sloshing problem is completely different from the dam break one.
4774
4775
''Water sloshing in a tank''
4776
4777
The problem of Figure [[#img-58a|58a]] is solved using the parameter <math display="inline">\theta </math> computed as in Eq.([[#eq-241|241]]). For <math display="inline">\Delta t=10^{-3}s</math>, <math display="inline">\theta =0.154</math>, while for <math display="inline">\Delta t</math>=<math display="inline">10^{-2}s</math>, <math display="inline">\theta=0.0053</math>. For both time step increments the resulting condition number of the iterative matrix is <math display="inline"> C=23</math> and the average number of iterations of the linear BCG solver  is around 15. The improvement versus using <math display="inline">\theta =1</math>  is evident, if compared to the numbers presented in Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]]. Reducing the number of iterations of the linear solver, also reduces the computational time. For example, using <math display="inline">\Delta t=10^{-2}s</math> for a duration of the sloshing simulation of <math display="inline">20s</math>, the total computational time for <math display="inline">\theta=1</math> is <math display="inline">2746s</math> while for <math display="inline">\theta=0.0053</math> it reduces to <math display="inline">1600s</math>. Also the convergence of the non-linear loop improves with the optimum value of <math display="inline">\theta </math>.
4778
4779
In Figures [[#img-62|62]] and [[#img-63|63]] the convergence of the velocity and pressure fields, respectively, obtained  with <math display="inline">\theta=1</math> and <math display="inline">\theta=0.0053</math>, is compared. The faster convergence of the solution using a smaller value of <math display="inline">\theta </math> is noticeable. 
4780
4781
<div id='img-62'></div>
4782
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4783
|-
4784
|[[Image:draft_Samper_722607179-convergenceV.png|360px|2D water sloshing. Convergence of the velocities at t=1.75s for θ=1 and θ=0.0053 (optimum value).]]
4785
|- style="text-align: center; font-size: 75%;"
4786
| colspan="1" | '''Figure 62:''' 2D water sloshing. Convergence of the velocities at <math>t=1.75s</math> for <math>\theta=1</math> and <math>\theta=0.0053</math> (optimum value).
4787
|}
4788
4789
<div id='img-63'></div>
4790
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4791
|-
4792
|[[Image:draft_Samper_722607179-convergenceP.png|360px|2D water sloshing. Convergence of pressure at t=1.75s for θ=1 and θ=0.0053 (optimum value).]]
4793
|- style="text-align: center; font-size: 75%;"
4794
| colspan="1" | '''Figure 63:''' 2D water sloshing. Convergence of pressure at <math>t=1.75s</math> for <math>\theta=1</math> and <math>\theta=0.0053</math> (optimum value).
4795
|}
4796
4797
In Figure [[#img-64|64]] the pressure solution at time t=1.75<math display="inline">s</math> obtained with <math display="inline">\theta </math>=0.0053 is illustrated. It can be appreciated a remarkable enhancement versus the solution for <math display="inline">\theta </math>=1 (see Figure [[#img-60|60]]). Note that the  elements generated in the free surface region adjacent to the boundaries are due to the coarseness of the mesh and the remeshing criteria and not to the computation. A better result can be obtained using a smaller average mesh size or a refined mesh in the free surface region.
4798
4799
<div id='img-64'></div>
4800
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4801
|-
4802
|[[Image:draft_Samper_722607179-Snapshot1_75_reducedK.png|360px|2D water sloshing (θ=0.0053). Pressure contours at t=1.75s.]]
4803
|- style="text-align: center; font-size: 75%;"
4804
| colspan="1" | '''Figure 64:''' 2D water sloshing (<math>\theta=0.0053</math>). Pressure contours at <math>t=1.75s</math>.
4805
|}
4806
4807
As stated in Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]], the convergence in the continuity equation affects the conservation of mass of the fluid. It has been just shown that the convergence of the pressure improves using a good prediction of <math display="inline">\theta </math>. Consequently, also the mass conservation is better ensured using <math display="inline">\theta=0.0053</math> than using <math display="inline">\theta=1</math>.
4808
4809
<div id='img-65'></div>
4810
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4811
|-
4812
|[[Image:draft_Samper_722607179-VolumeLossComparison.png|360px|2D water sloshing. Accumulated mass variation in absolute value along the duration of the analysis. Solution for θ=1 and θ=0.0053 (optimum value).]]
4813
|- style="text-align: center; font-size: 75%;"
4814
| colspan="1" | '''Figure 65:''' 2D water sloshing. Accumulated mass variation in absolute value along the duration of the analysis. Solution for <math>\theta=1</math> and <math>\theta=0.0053</math> (optimum value).
4815
|}
4816
4817
In the graph of Figure [[#img-65|65]] the percentage of accumulated mass variation (in absolute value) versus time obtained with <math display="inline">\theta=1</math> and <math display="inline">\theta=0.0053</math> are compared. The values plotted in the graph have been computed according to Eq.([[#eq-240|240]]). The better mass preservation of the solution with the smaller value of <math display="inline">\theta </math> is remarkable.
4818
4819
In the graph of Figure [[#img-66|66]] the accumulated mass variation obtained with <math display="inline">\theta=0.0053</math> is illustrated. The values plotted in the graph have been computed using Eq.([[#eq-230|230]]). After 20 seconds of simulation, the scheme with the reduced value of <math display="inline">\theta </math> has an accumulated mass variation of <math display="inline">0.52%</math>, which corresponds to a mean volume variation for each time step of <math display="inline">1.7 \cdot 10^{-3}%</math>. The solution with <math display="inline">\theta=0.0053</math> guarantees a better conservation of mass than with <math display="inline">\theta=1</math>. This represents another evidence that the (quasi)-incompressibility constraint is not affected by reducing the bulk modulus for solving the momentum equations.
4820
4821
<div id='img-66'></div>
4822
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4823
|-
4824
|[[Image:draft_Samper_722607179-VolumeLossOk.png|360px|2D water sloshing (θ=0.0053). Accumulated mass variation versus time.]]
4825
|- style="text-align: center; font-size: 75%;"
4826
| colspan="1" | '''Figure 66:''' 2D water sloshing (<math>\theta </math>=0.0053). Accumulated mass variation versus time.
4827
|}
4828
4829
''Influence of the mesh size''
4830
4831
In order to verify the applicability of the method, the sloshing problem has been solved using <math display="inline">\Delta t</math>=<math display="inline">10^{-3}s</math> and different mesh sizes. In particular, the following average mesh sizes have been used: <math display="inline">h</math> =0.1<math display="inline">m</math>, 0.15<math display="inline">m</math>, 0.2<math display="inline">m</math>, 0.3<math display="inline">m</math> and 0.4<math display="inline">m</math>.
4832
4833
The problem was solved setting <math display="inline">\theta </math>=1 and computing <math display="inline">apriori</math> its reduced value using  Eq.([[#eq-241|241]]).
4834
4835
The curves in Figure [[#img-67|67]] show that the reduced value of <math display="inline">\theta </math> guarantees better results versus using <math display="inline">\theta=1</math>. For all the meshes, the number of iterations required by the linear solver to reach a converged solution is smaller. Furthermore, the results show that the strategy is applicable to coarse and fine meshes.
4836
4837
Table [[#table-6|3.2]] collects all the data and the results. The number of iterations of the linear solver has been considered as a  quality indicator of the analyses. As shown in the previous sections this value is related to the condition number of the iterative matrix.
4838
4839
<div id='img-67'></div>
4840
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4841
|-
4842
|[[Image:draft_Samper_722607179-SLiterationsMesh.png|360px|2D water sloshing. Number of iterations of the linear solver for different numbers of velocity degrees of freedom. Results for θ=1 and the optimum value of θ.]]
4843
|- style="text-align: center; font-size: 75%;"
4844
| colspan="1" | '''Figure 67:''' 2D water sloshing. Number of iterations of the linear solver for different numbers of velocity degrees of freedom. Results for <math>\theta </math>=1 and the optimum value of <math>\theta </math>.
4845
|}
4846
4847
<div class="center" style="font-size: 75%;">
4848
'''Table 3.2.''' 2D water sloshing. Numerical values of the graph of Figure [[#img-67|67]].</div>
4849
4850
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
4851
|- 
4852
|rowspan="2" style="text-align:center;"|  average mesh size
4853
| rowspan="2" style="text-align:center;" |degrees of freedom  (velocities) 
4854
| colspan='2' style="text-align:center;" | number of iterations 
4855
|-
4856
| style="text-align:center;" | <math>\theta </math>=1 
4857
| style="text-align:center;" | optimum <math display="inline">\theta </math> 
4858
|-
4859
| style="text-align:center;"|0.4
4860
| style="text-align:center;"|682
4861
| style="text-align:center;"|''20''
4862
| style="text-align:center;"|<math>15 (\theta =0.535) </math> 
4863
|-
4864
| style="text-align:center;"|0.3
4865
| style="text-align:center;"|1220
4866
| style="text-align:center;"|''27''
4867
| style="text-align:center;"|<math>15 (\theta =0.304) </math> 
4868
|-
4869
| style="text-align:center;"|0.2
4870
| style="text-align:center;"|2782
4871
| style="text-align:center;"|''41''
4872
| style="text-align:center;"|<math>15 (\theta =0.136) </math> 
4873
|-
4874
| style="text-align:center;"|0.15
4875
| style="text-align:center;"|5002
4876
| style="text-align:center;"|''55''
4877
| style="text-align:center;"|<math>16 (\theta =0.0801) </math> 
4878
|-
4879
| style="text-align:center;"|0.1
4880
| style="text-align:center;"|5840
4881
| style="text-align:center;"|''70''
4882
| style="text-align:center;"|<math>15 (\theta =0.0361) </math> 
4883
|}
4884
4885
4886
''Influence of the time step''
4887
4888
The problem of Figure [[#img-58a|58a]] has been solved using different time steps: 0.0001<math display="inline">s</math>, 0.0005<math display="inline">s</math>, 0.001<math display="inline">s</math>, 0.005<math display="inline">s</math>, 0.01<math display="inline">s</math>, 0.02<math display="inline">s</math>. The mesh has a mean size of 0.15<math display="inline">m</math>. The numerical results obtained with the reduced value of <math display="inline">\theta </math> and by setting <math display="inline">\theta </math>=1 are compared.  Once again, the number of  iterations of the linear solver is the parameter chosen to indicate the quality of the analyses: the smaller this value is, the better conditioned the linear system is.
4889
4890
The graph of Figure [[#img-68|68]] shows that the accuracy of the method does not depend on the time step increments, when the suitable reduced value for <math display="inline">\theta </math> is used.  <div id='img-68'></div>
4891
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4892
|-
4893
|[[Image:draft_Samper_722607179-SLiterationsTime.png|360px|2D water sloshing.  Number of iterations of the linear solver for different time step increments. Results for θ=1 and the optimum value of θ.]]
4894
|- style="text-align: center; font-size: 75%;"
4895
| colspan="1" | '''Figure 68:''' 2D water sloshing.  Number of iterations of the linear solver for different time step increments. Results for <math>\theta </math>=1 and the optimum value of <math>\theta </math>.
4896
|}
4897
4898
Table [[#table-7|3.3]] summarizes the problem data and collects the numerical values of the graph of Figure [[#img-68|68]].
4899
4900
<div class="center" style="font-size: 75%;">
4901
'''Table 3.3.''' 2D water sloshing. Numerical values of the graph of Figure [[#img-68|68]].</div>
4902
4903
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
4904
|- 
4905
| rowspan="2" style="text-align:center;"|  <math>\Delta t </math> (s)
4906
| colspan='2' style="text-align:center;" | number of iterations 
4907
|- 
4908
| style="text-align:center;"| <math>\theta =1 </math>
4909
| style="text-align:center;"| optimum <math>\theta</math>
4910
|-
4911
| style="text-align:center;"|0.2
4912
| style="text-align:center;"|''failed''
4913
| style="text-align:center;"|<math>16 (\theta =2.00 \times 10^{-5}) </math> 
4914
|-
4915
| style="text-align:center;"|0.01
4916
| style="text-align:center;"|''523''
4917
| style="text-align:center;"|<math>16 (\theta =8.01 \times 10^{-4}) </math>
4918
|-
4919
| style="text-align:center;"|0.005
4920
| style="text-align:center;"|''244''
4921
| style="text-align:center;"|<math>16 (\theta =3.20 \times 10^{-3}) </math>
4922
|-
4923
| style="text-align:center;"|0.001
4924
| style="text-align:center;"|''55''
4925
| style="text-align:center;"|<math>16 (\theta =8.01 \times 10^{-2}) </math>
4926
|-
4927
| style="text-align:center;"|0.0005
4928
| style="text-align:center;"|''30''
4929
| style="text-align:center;"|<math>16 (\theta =3.89 \times 10^{-1}) </math>
4930
|-
4931
| style="text-align:center;"|0.0001
4932
| style="text-align:center;"|''6''
4933
| style="text-align:center;"|<math>16 (\theta =8.01 ) </math>
4934
|}
4935
4936
4937
For each value of <math display="inline">\Delta t</math>, the number of iterations is 16 and the condition number does not change. Furthermore, using a reduced value of <math display="inline">\theta </math> allows us to solve the problem for each time increment, while if <math display="inline">\theta </math> is fixed to 1, the results are acceptable only until <math display="inline">\Delta t </math>=0.005<math display="inline">s</math>. For larger time steps the results for <math display="inline">\theta </math>=1 are not accurate or the analyses do not even converge.
4938
4939
''Mesh with a refined zone''
4940
4941
As mentioned in the previous sections, the parameter <math display="inline">\theta </math> has a square dependency on the mesh size (see Eq.([[#eq-242|242]])). For this reason, if the discretization is not uniform the global estimation of <math display="inline">\theta </math> might not guarantee the well-conditioning of the linear system. In these cases, a local computation of <math display="inline">\theta </math> is recommended. The sloshing problem is here solved again using the mesh shown in Figure [[#img-69|69]] and <math display="inline">\Delta t= 0.01s</math>. The spatial discretization has two different mean sizes: in the center <math display="inline">h</math>=0.1<math display="inline">m</math> while <math display="inline">h</math>=0.4<math display="inline">m</math> elsewhere. 
4942
4943
<div id='img-69'></div>
4944
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 42%;max-width: 100%;"
4945
|-
4946
|[[Image:draft_Samper_722607179-SL_refined_input.png|300px|2D water sloshing. Finite element mesh with a refined zone.]]
4947
|- style="text-align: center; font-size: 75%;"
4948
| colspan="1" | '''Figure 69:''' 2D water sloshing. Finite element mesh with a refined zone.
4949
|}
4950
4951
4952
The problem is solved both using <math display="inline">\theta </math>=1 and by computing its optimum value globally and locally. The mean number of iterations required by the linear solver to converge for each of these three options is 319, 21 and 17 respectively. Hence, the local approach guarantees the best result for this type of non-uniform meshes.
4953
4954
In Figure [[#img-70|70]] the solution at time <math display="inline"> t= 0.1s</math> obtained with the local approach is illustrated.
4955
4956
<div id='img-70'></div>
4957
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
4958
|-
4959
|[[Image:draft_Samper_722607179-SL_refined_pressure.png|360px|2D sloshing with a refined zone. Solution at time  t= 0.1s obtained with the local approach.]]
4960
|- style="text-align: center; font-size: 75%;"
4961
| colspan="1" | '''Figure 70:''' 2D sloshing with a refined zone. Solution at time <math> t= 0.1s</math> obtained with the local approach.
4962
|}
4963
4964
''Collapse of a water column on a rigid obstacle''
4965
4966
In this section, the  collapse of the water column induced by the instant removal of  a vertical wall is studied.
4967
4968
<div id='img-71a'></div>
4969
<div id='img-71'></div>
4970
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;max-width: 100%;"
4971
|-
4972
|[[Image:draft_Samper_722607179-DB_input.png|300px]]
4973
|- style="text-align: center; font-size: 75%;"
4974
| colspan="1" | '''Figure 71:''' 2D dam break. Initial geometry.
4975
|}
4976
4977
<div class="center" style="font-size: 75%;">
4978
'''Table 3.4.''' 2D dam break. Problem data.</div>
4979
4980
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
4981
|- 
4982
| style="text-align:center;"|L
4983
| style="text-align:center;"|<math>0.146 </math> m
4984
|-
4985
| style="text-align:center;"|H
4986
| style="text-align:center;"|<math>0.048 </math> m
4987
|-
4988
| style="text-align:center;"|D
4989
| style="text-align:center;"|<math>0.024 </math> m
4990
|-
4991
| style="text-align:center;"|viscosity
4992
| style="text-align:center;"|<math>10^{-3} </math> Pa.s
4993
|-
4994
| style="text-align:center;"|density
4995
| style="text-align:center;"|<math>10^3 \hbox{ kg/m}^3 </math> 
4996
|-
4997
| style="text-align:center;"|bulk modulus
4998
| style="text-align:center;"|<math>2.15 \times 10^9 </math> Pa
4999
|}
5000
5001
5002
The initial geometry of the problem is illustrated in Figure [[#img-71a|71a]]. In Table [[#img-71|3.4]] all the problem data are collected. As for the previous example,  the problem is first solved with a very coarse mesh. Then a comparison with the solution obtained for <math display="inline">\theta </math>=1 is given. After that, the same problem is solved varying the mean  mesh size and for different time step increments. The results obtained with the optimum value of <math display="inline">\theta </math> are compared to the experimental results presented in [51]. The objective is to show that the reduction of the bulk modulus in the iteration matrix does not affect the numerical solution of this class of   impact problems which can be solved also with a larger time step. 
5003
5004
The problem is first solved with a coarse discretization (mean element size <math display="inline">h</math>=0.0125<math display="inline">m</math>). The solutions obtained with the optimum value of <math display="inline">\theta </math> and with <math display="inline">\theta </math>=1 are compared in terms of the condition number of the iteration matrix and the number of iterations of the linear solver.  For <math display="inline">\Delta t</math>=<math display="inline">10^{-4}</math>s, matrix <math display="inline">{K}</math> has condition numbers <math display="inline">C</math>=1028 and <math display="inline">C</math>=60,  for <math display="inline">\theta </math>=1 and <math display="inline">\theta </math>=0.0535, respectively, which correspond to 1251 and 14 iterations of the linear solver, respectively. For the same discretization and a time increment of <math display="inline">\Delta t</math>=<math display="inline">10^{-3}s</math>, Eq.([[#eq-241|241]]) yields <math display="inline">\theta </math>=0.000535. The condition number of <math display="inline">{K}</math> and the iterations of the linear solver are the same as using a time step increment ten times smaller. Conversely, for <math display="inline">\theta </math>=1, the condition number grows to <math display="inline">C</math>=102090 and the iterative scheme does not converge.
5005
5006
''Influence of the mesh size''
5007
5008
The problem of Figure [[#img-71a|71a]] has been solved for <math display="inline">\Delta t</math>=<math display="inline">10^{-4}s</math> using unstructured meshes with the following mean element sizes: <math display="inline">{h}</math>= 0.004<math display="inline">m</math>, 0.005<math display="inline">m</math>, 00075<math display="inline">m</math>, 0.01<math display="inline">m</math>, 0.0125<math display="inline">m</math>. The problem was solved both setting <math display="inline">\theta </math>=1 and computing <math display="inline">apriori</math> its optimum value using  Eq.([[#eq-241|241]]). As for the previous section, the number of iterations of the linear solver has been considered as the quality indicator of the analysis. In Figure [[#img-72|72]] and Table [[#table-8|3.5]] all the data and the results are collected.
5009
5010
<div id='img-72'></div>
5011
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
5012
|-
5013
|[[Image:draft_Samper_722607179-DBiterationsMesh.png|360px|2D dam break. Number of iterations of the linear solver for different numbers of velocity degrees of freedom.  Results for θ=1 and the optimum value of θ.]]
5014
|- style="text-align: center; font-size: 75%;"
5015
| colspan="1" | '''Figure 72:''' 2D dam break. Number of iterations of the linear solver for different numbers of velocity degrees of freedom.  Results for <math>\theta </math>=1 and the optimum value of <math>\theta </math>.
5016
|}
5017
5018
5019
<div class="center" style="font-size: 75%;">
5020
'''Table 3.5.''' 2D dam break. Numerical values of the graph of Figure [[#img-72|72]].</div>
5021
5022
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5023
|- 
5024
|rowspan="2" style="text-align:center;"|  average mesh size
5025
| rowspan="2" style="text-align:center;" |degrees of freedom  (velocities) 
5026
| colspan='2' style="text-align:center;" | number of iterations 
5027
|-
5028
| style="text-align:center;" | <math>\theta </math>=1 
5029
| style="text-align:center;" | optimum <math display="inline">\theta </math> 
5030
|-
5031
| style="text-align:center;"|0.0125
5032
| style="text-align:center;"|618
5033
| style="text-align:center;"|''70''
5034
| style="text-align:center;"|<math>17 (\theta =0.0536) </math> 
5035
|-
5036
| style="text-align:center;"|0.01
5037
| style="text-align:center;"|978
5038
| style="text-align:center;"|''82''
5039
| style="text-align:center;"|<math>18 (\theta =0.0345) </math> 
5040
|-
5041
| style="text-align:center;"|0.0075
5042
| style="text-align:center;"|1694
5043
| style="text-align:center;"|''100''
5044
| style="text-align:center;"|<math>16 (\theta =0.0205) </math> 
5045
|-
5046
| style="text-align:center;"|0.005
5047
| style="text-align:center;"|5002
5048
| style="text-align:center;"|''163''
5049
| style="text-align:center;"|<math>16 (\theta =0.00868) </math> 
5050
|-
5051
| style="text-align:center;"|0.004
5052
| style="text-align:center;"|6106
5053
| style="text-align:center;"|''196''
5054
| style="text-align:center;"|<math>17 (\theta =0.00559) </math> 
5055
|}
5056
5057
5058
Figure [[#img-72|72]] and Table  [[#table-8|3.5]] confirm that the efficiency of the method is not affected by the mesh size. In other words, the well-conditioning of the iterative matrix is guaranteed for coarse and fine meshes.
5059
5060
''Influence of the time step''
5061
5062
The same problem was solved for  different time steps: <math display="inline">\Delta t</math>=0.00005<math display="inline">s</math>, 0.0001<math display="inline">s</math>, 0.0005<math display="inline">s</math>, 0.001<math display="inline">s</math>, 0.0025<math display="inline">s</math>. The mesh used has a mean size of <math display="inline">{h}</math>=0.005<math display="inline">m</math>. Table [[#table-9|3.6]] collects the problem data and the resulting number of iterations for all the analyses solved with <math display="inline">\theta </math>=1 and with the optimum value of <math display="inline">\theta </math>.
5063
5064
Figure [[#img-73|73]] and Table [[#table-9|3.6]] confirm the conclusions of the previous section. <div id='img-73'></div>
5065
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
5066
|-
5067
|[[Image:draft_Samper_722607179-DBiterationsTime.png|360px|2D dam break.  Number of iterations of the linear solver for different time steps. Results for θ=1 and the optimum value of θ.]]
5068
|- style="text-align: center; font-size: 75%;"
5069
| colspan="1" | '''Figure 73:''' 2D dam break.  Number of iterations of the linear solver for different time steps. Results for <math>\theta </math>=1 and the optimum value of <math>\theta </math>.
5070
|}
5071
5072
<div class="center" style="font-size: 75%;">
5073
'''Table 3.6.''' 2D dam break.  Numerical values for the graph of Figure [[#img-73|73]].</div>
5074
5075
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5076
|- 
5077
|rowspan="2" style="text-align:center;"|  <math>\Delta t </math> (s)
5078
| colspan='2' style="text-align:center;" | number of iterations 
5079
|- 
5080
| style="text-align:center;"| <math>\theta =1 </math>
5081
| style="text-align:center;"| optimum <math>\theta</math>
5082
|-
5083
| style="text-align:center;"|<math>2.5 \times 10^{-3} </math>
5084
| style="text-align:center;"|''failed''
5085
| style="text-align:center;"|<math>15 (\theta =1.39 \times 10^{-5}) </math> 
5086
|-
5087
| style="text-align:center;"|<math>1.0 \times 10^{-3} </math>
5088
| style="text-align:center;"|''failed''
5089
| style="text-align:center;"|<math>15 (\theta =8.68 \times 10^{-5}) </math>
5090
|-
5091
| style="text-align:center;"|<math>5.0 \times 10^{-4} </math>
5092
| style="text-align:center;"|''1000''
5093
| style="text-align:center;"|<math>16 (\theta =3.47 \times 10^{-4}) </math>
5094
|-
5095
| style="text-align:center;"|<math>1.0 \times 10^{-4} </math>
5096
| style="text-align:center;"|''163''
5097
| style="text-align:center;"|<math>16 (\theta =8.68 \times 10^{-3}) </math>
5098
|-
5099
| style="text-align:center;"|<math>5.0 \times 10^{-5} </math>
5100
| style="text-align:center;"|''79''
5101
| style="text-align:center;"|<math>16 (\theta =3.47 \times 10^{-2}) </math>
5102
|}
5103
5104
5105
The strategy is not affected by the time step and, contrary to the case for <math display="inline">\theta </math>=1, it does not impose limitations to the range of time step increments for solving the problem. For example, for the present problem the time step increment is constrained by the geometry and the dynamics of the problem only. In other words, the maximum  time step increment is the one that guarantees that the fluid particles do not cross through the boundaries. For the present problem and the chosen mesh, the maximum time step increment was <math display="inline">\Delta t</math>=<math display="inline">2.5 \cdot 10^{-3}</math>s.
5106
5107
All the examples presented in this section have shown the positive effect for the accuracy of the numerical scheme given by the use of the predicted value of <math display="inline">\kappa _p</math> in matrix <math display="inline">{K}</math>  (Eq.([[#eq-232|232]])). In all the following problems presented in this work, this strategy has been used. In particular, the technique is still available in  FSI problems for improving the conditioning of fluid counterpart of the global tangent matrix.
5108
5109
==3.5 Validation examples==
5110
5111
This section is devoted to the validation of the unified stabilized formulation for fluids and solids at the incompressible limit.  For each example several discretizations are analyzed in order to verify the convergence of the method. The formulation will be validated by comparing the numerical results to experimental tests and the numerical results of other formulations. In the first part quasi-incompressible Newtonian fluids are analyzed, then a problem involving a hypoelastic quasi-incompressible structure is studied.
5112
5113
===3.5.1 Validation of the Unified formulation for Newtonian fluids===
5114
5115
The purpose of this section is to validate the Unified formulation for the analysis of Newtonian free surface flows.
5116
5117
In all the examples presented in this section, in matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-236|236]])) the pseudo bulk modulus <math display="inline">\kappa _p</math> is considered and its value has been computed according to the strategy presented in Section [[#3.4.3.2 Optimum value for the pseudo bulk modulus|3.4.3.2]].
5118
5119
The effect of the boundary conditions is studied in detail. In particular, the strategy for modeling the slip conditions in the Lagrangian way described in Section [[#3.4.1.3 Advantages and disadvantages|3.4.1.3]] is tested and validated.
5120
5121
All the numerical examples have been solved for several meshes in order to verify the convergence of PFEM stabilized formulation.
5122
5123
''Sloshing of a viscous fluid''
5124
5125
In this section, the sloshing of a viscous fluid in a prismatic tank is analyzed. The initial configuration of the problem is illustrated in Figure [[#img-74a|74a]] and the problem data are given in Table [[#img-74|3.7]]. 
5126
5127
<div id='img-74a'></div>
5128
<div id='img-74'></div>
5129
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 37%;max-width: 100%;"
5130
|-
5131
|[[Image:draft_Samper_722607179-SLinput.png|300px|Sloshing of a viscous fluid. Initial geometry of 2D simulation.]]
5132
|- style="text-align: center; font-size: 75%;"
5133
| colspan="1" | '''Figure 74:''' Sloshing of a viscous fluid. Initial geometry of 2D simulation.
5134
|}
5135
5136
<div class="center" style="font-size: 75%;">
5137
'''Table 3.7.''' Sloshing of a viscous fluid. Problem data.</div>
5138
5139
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5140
|- 
5141
| style="text-align:center;"|D
5142
| style="text-align:center;"|<math>0.5 </math> m
5143
|-
5144
| style="text-align:center;"|<math>H_1</math>
5145
| style="text-align:center;"|<math>0.35 </math> m
5146
|-
5147
| style="text-align:center;"|viscosity
5148
| style="text-align:center;"|<math>5 </math> Pa.s
5149
|-
5150
| style="text-align:center;"|density
5151
| style="text-align:center;"|<math>10^3 \hbox{ kg/m}^3 </math> 
5152
|-
5153
| style="text-align:center;"|bulk modulus
5154
| style="text-align:center;"|<math>2.15 \times 10^9 </math> Pa
5155
|}
5156
5157
5158
5159
Slip conditions have been imposed on the tank walls. The problem has been solved in 2D with different discretizations in order to verify the convergence of the numerical scheme. In particular, the following average mesh sizes of 3-noded triangles  have been used: 0.012m, 0.011m, 0.01m, 0.009m, 0.008m, 0.007m, 0.006m and 0.005m. In Figure [[#img-75|75]] the finest (mesh size=0.005m) and the coarsest (mesh size=0.012m) meshes are illustrated. 
5160
5161
<div id='img-75'></div>
5162
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5163
|-
5164
|[[Image:draft_Samper_722607179-sloshing0012.png|400px]]
5165
|[[Image:draft_Samper_722607179-sloshing0005.png|400px]]
5166
|- style="text-align: center; font-size: 75%;"
5167
| (a) average mesh size= 0.012m
5168
| (b) average mesh size= 0.005m
5169
|- style="text-align: center; font-size: 75%;"
5170
| colspan="2" | '''Figure 75:''' Sloshing of a viscous fluid. Coarsest (1967 triangles) and finest (11440 triangles) meshes used for the analysis.
5171
|}
5172
5173
All the analyses have been run for <math display="inline">{\Delta t=0.001 s}</math> and the total duration of the study is <math display="inline">10s</math>.
5174
5175
In Figure [[#img-76|76]] some representative snapshots of the 2D simulation are given. Each snapshot refers to the maximum height reached by the fluid during its sloshing. The pressure contours have been plotted over the deformed configuration. The pictures  in Figure [[#img-76|76]] assess the smoothness of the pressure field. 
5176
5177
<div id='img-76'></div>
5178
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5179
|-
5180
|[[Image:draft_Samper_722607179-SL000.png|400px|t = 0.00 st = 0.39 s]]
5181
|[[Image:draft_Samper_722607179-SL039.png|400px|t = 0.85 s]]
5182
|[[Image:draft_Samper_722607179-SL085.png|400px|t = 1.34 s]]
5183
|[[Image:draft_Samper_722607179-SL134.png|400px|t = 1.70 s]]
5184
|- style="text-align: center; font-size: 75%;"
5185
| (a) <math>t = 0.00 s</math>
5186
| (b) <math>t = 0.39 s</math>
5187
| (c) <math>t = 0.85 s</math>
5188
| (d) <math>t = 1.34 s</math>
5189
|- style="text-align: center; font-size: 75%;"
5190
|[[Image:draft_Samper_722607179-SL170.png|400px|t = 2.12 s]]
5191
|[[Image:draft_Samper_722607179-SL212.png|400px|t = 3.38 s]]
5192
|[[Image:draft_Samper_722607179-SL338.png|400px|t = 3.81 s]]
5193
|[[Image:draft_Samper_722607179-SL381.png|400px|t = 5.95 s]]
5194
|-style="text-align: center; font-size: 75%;"
5195
| (e) <math>t = 1.70 s</math>
5196
| (f) <math>t = 2.12 s</math>
5197
| (g) <math>t = 3.38 s</math>
5198
| (h) <math>t = 3.81 s</math>
5199
|- style="text-align: center; font-size: 75%;"
5200
|[[Image:draft_Samper_722607179-SL595.png|400px|t = 6.35 s]]
5201
|[[Image:draft_Samper_722607179-SL635.png|400px|t = 9.24 s]]
5202
|[[Image:draft_Samper_722607179-SL924.png|400px|t = 9.67 s]]
5203
|[[Image:draft_Samper_722607179-SL967.png|400px]]
5204
|- style="text-align: center; font-size: 75%;"
5205
| (i) <math>t = 5.95 s</math>
5206
| (j) <math>t = 6.35 s</math>
5207
| (k) <math>t = 9.24 s</math>
5208
| (l) <math>t = 9.67 s</math>
5209
|- style="text-align: center; font-size: 75%;"
5210
| colspan="4" | '''Figure 76:''' Sloshing of a viscous fluid. Pressure contours over the deformed configuration at some time instants.
5211
|}
5212
5213
5214
In Figure [[#img-77|77]]  the time evolution of the fluid level height at the left side of the tank is plotted for the two finest meshes used. 
5215
5216
5217
<div id='img-77'></div>
5218
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
5219
|-
5220
|[[Image:draft_Samper_722607179-SLlast2.png|420px|Sloshing of a viscous fluid. Time evolution of the free surface position at the left side of the tank. Solutions obtained with  the finest meshes: average mesh size 0.006m and 0.005m.]]
5221
|- style="text-align: center; font-size: 75%;"
5222
| colspan="1" | '''Figure 77:''' Sloshing of a viscous fluid. Time evolution of the free surface position at the left side of the tank. Solutions obtained with  the finest meshes: average mesh size 0.006m and 0.005m.
5223
|}
5224
5225
5226
The graph shows that a converged solution has been reached because the two diagrams almost coincide.
5227
5228
The convergence analysis has been performed for the time evolution of the free surface position at the left side of the tank. For the convergence study, the solution obtained with the finest mesh (average size 0.005m) has been considered as the  reference solution. The error is computed as
5229
5230
<span id="eq-245"></span>
5231
{| class="formulaSCP" style="width: 100%; text-align: left;" 
5232
|-
5233
| 
5234
{| style="text-align: left; margin:auto;" 
5235
|-
5236
| style="text-align: center;" | <math>||err||= \frac{\sqrt{\sum _{i=1}^{N} \left(h_{finest~mesh}^i-h_{tested~mesh}^i \right)^2} }{\sqrt{\sum _{i=1}^{N} \left(h_{finest~mesh}^i \right)^2} }      </math>
5237
|}
5238
| style="width: 5px;text-align: right;" | (245)
5239
|}
5240
5241
where <math display="inline">h_{finest~mesh}^i</math> is the value computed for the finest mesh at the <math display="inline">i-th</math> time step  and <math display="inline">N</math> is the total number of time steps.
5242
5243
For a given mesh and for each time step the fluid height at the left side of the tank is compared to the one obtained with the finest mesh at the same time instant. The graph of Figure [[#img-78|78]] shows an overall quadratic slope, despite some oscillations in the curve.  
5244
5245
<div id='img-78'></div>
5246
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5247
|-
5248
|[[Image:draft_Samper_722607179-SLconvergence2.png|600px|Sloshing of a viscous fluid. Convergence analysis for the time evolution of the free surface level at the left side of the tank. Error computed with Eq.([[#eq-245|245]]).]]
5249
|- style="text-align: center; font-size: 75%;"
5250
| colspan="1" | '''Figure 78:''' Sloshing of a viscous fluid. Convergence analysis for the time evolution of the free surface level at the left side of the tank. Error computed with Eq.([[#eq-245|245]]).
5251
|}
5252
5253
5254
A reason for these oscillations is the local character of this convergence study. In order to reduce these drawbacks, a convergence study has been performed also for a global parameter, specifically the potential energy.
5255
5256
The potential energy has been computed as
5257
5258
<span id="eq-246"></span>
5259
{| class="formulaSCP" style="width: 100%; text-align: left;" 
5260
|-
5261
| 
5262
{| style="text-align: left; margin:auto;" 
5263
|-
5264
| style="text-align: center;" | <math>E_{pot}= \sum _{e=1}^{n} m^e g  h^e     </math>
5265
|}
5266
| style="width: 5px;text-align: right;" | (246)
5267
|}
5268
5269
where <math display="inline">m^e</math> is the element mass and <math display="inline">h^e</math> is the mean value for the nodal heights.
5270
5271
The graph of Figure [[#img-79|79]] is the convergence curve for the potential energy error. Once again, the error is computed with respect to the results obtained with the finest mesh in the same form as in Eq.([[#eq-245|245]]).  <div id='img-79'></div>
5272
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5273
|-
5274
|[[Image:draft_Samper_722607179-SLconvergence1.png|600px|Sloshing of a viscous fluid. Convergence graph for the potential energy.  Error computed according to Eq.([[#eq-245|245]]).]]
5275
|- style="text-align: center; font-size: 75%;"
5276
| colspan="1" | '''Figure 79:''' Sloshing of a viscous fluid. Convergence graph for the potential energy.  Error computed according to Eq.([[#eq-245|245]]).
5277
|}
5278
5279
Comparing this graph to the curve of Figure [[#img-78|78]], the oscillations are reduced and the quadratic convergence for the used error measure is confirmed.
5280
5281
''Collapse of a water column on a rigid horizontal plane''
5282
5283
In this section, the collapse of a water column on a rigid horizontal plane is studied. The initial configuration of the problem is illustrated in Figure [[#img-80a|80a]] and the problem data are given in Table [[#img-80|3.8]]. 
5284
5285
<div id='img-80a'></div>
5286
<div id='img-80'></div>
5287
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 45%;max-width: 100%;"
5288
|-
5289
|[[Image:draft_Samper_722607179-DBinput.png|300px]]
5290
5291
|- style="text-align: center; font-size: 75%;"
5292
| colspan="1" | '''Figure 80:''' Collapse of a water column. Initial geometry for the 2D simulation. 
5293
|}
5294
5295
5296
<div class="center" style="font-size: 75%;">
5297
'''Table 3.8.''' Collapse of a water column. Problem data.</div>
5298
5299
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5300
|- 
5301
| style="text-align:center;"|''H''
5302
| style="text-align:center;"|<math>7 </math> m
5303
|-
5304
| style="text-align:center;"|<math>L</math>
5305
| style="text-align:center;"|<math>3 </math> m
5306
|-
5307
| style="text-align:center;"|viscosity
5308
| style="text-align:center;"|<math>10^{-3} </math> Pa.s
5309
|-
5310
| style="text-align:center;"|density
5311
| style="text-align:center;"|<math>10^3 \hbox{ kg/m}^3 </math> 
5312
|-
5313
| style="text-align:center;"|bulk modulus
5314
| style="text-align:center;"|<math>2.15 \times 10^9 </math> Pa
5315
|}
5316
5317
5318
The phenomena involved in this problem are the collapse of a free surface water column and the consequent spread of the water stream over a horizontal plate. The numerical solution is compared to the experimental results given in [71] where many experimental observations of the collapse of free liquid columns are collected. The characteristic variable chosen for the comparison is the residual height <math display="inline">h</math> of the water column (see Figure [[#img-81|81]]).  
5319
5320
<div id='img-81'></div>
5321
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5322
|-
5323
|[[Image:draft_Samper_722607179-DBinputH.png|600px|Graphical definition of parameter h.]]
5324
|- style="text-align: center; font-size: 75%;"
5325
| colspan="1" | '''Figure 81:''' Graphical definition of parameter <math>h</math>.
5326
|}
5327
5328
The problem has been solved for both stick and slip conditions. The 2D simulation has been run for three different discretizations. In particular, the average mesh sizes <math display="inline">0.006m</math>, <math display="inline">0.0045m</math> and <math display="inline">0.00075m</math> have been used. In Figure [[#img-82|82]] the finest (mesh size=<math display="inline">0.00075m</math>) and the coarsest (mesh size=<math display="inline">0.006m</math>) mesh are illustrated.
5329
5330
<div id='img-82'></div>
5331
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5332
|-
5333
|[[Image:draft_Samper_722607179-DB0006.png|400px]]
5334
|[[Image:draft_Samper_722607179-DB000075.png|400px]]
5335
|- style="text-align: center; font-size: 75%;"
5336
| (a) average mesh size= 0.006m
5337
| (b) average mesh size= 0.00075m
5338
|- style="text-align: center; font-size: 75%;"
5339
| colspan="2" | '''Figure 82:''' Collapse of a water column. Coarsest (258 triangles) and finest (13224 triangles) meshes used for the 2D analysis.
5340
|}
5341
5342
In 3D the problem has been solved only for a mesh of 338839 4-noded tetrahedra with average size 0.002m.
5343
5344
All the analyses have been run for <math display="inline">{\Delta t=0.00025 s}</math>.
5345
5346
In Figure [[#img-83|83]] the velocity and the pressure fields obtained with the finest mesh and slip boundary conditions are plotted over the deformed configuration of the fluid.
5347
5348
<div id='img-83'></div>
5349
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5350
|-
5351
|[[Image:draft_Samper_722607179-DBs05_t005_vel.png|400px|Velocity field at t = 0.05 sPressure field at t = 0.05 s]]
5352
|- style="text-align: center; font-size: 75%;"
5353
| (a) Velocity field at <math>t = 0.05 s</math>
5354
|-
5355
|[[Image:draft_Samper_722607179-DBs05_t005.png|400px|Velocity field at t = 0.10 s]]
5356
|- style="text-align: center; font-size: 75%;"
5357
| (b) Pressure field at <math>t = 0.05 s</math>
5358
|-
5359
|[[Image:draft_Samper_722607179-DBs05_t01_vel.png|400px|Pressure field at t = 0.10 s]]
5360
|-style="text-align: center; font-size: 75%;"
5361
| (c) Velocity field at <math>t = 0.10 s</math>
5362
|-
5363
|[[Image:draft_Samper_722607179-DBs05_t01.png|400px|Velocity field at t = 0.20 s]]
5364
|- style="text-align: center; font-size: 75%;"
5365
| (d) Pressure field at <math>t = 0.10 s</math>
5366
|-
5367
|[[Image:draft_Samper_722607179-DBs05_t02_vel.png|400px|Pressure field at t = 0.20 s]]
5368
|-style="text-align: center; font-size: 75%;"
5369
| (e) Velocity field at <math>t = 0.20 s</math>
5370
|-
5371
|[[Image:draft_Samper_722607179-DBs05_t02.png|400px]]
5372
|- style="text-align: center; font-size: 75%;"
5373
| (f) Pressure field at <math>t = 0.20 s</math>
5374
|- style="text-align: center; font-size: 75%;"
5375
| colspan="1" | '''Figure 83:''' Collapse of a water column. Pressure and velocity contours over the deformed configuration at three time instants.
5376
|}
5377
5378
The results for the 3D simulation with slip conditions are shown in  Figure [[#img-84|84]].  
5379
5380
<div id='img-84'></div>
5381
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5382
|-
5383
|[[Image:draft_Samper_722607179-DB3D005velocity.png|400px|Velocity and pressure fields at t = 0.05 s]]
5384
|[[Image:draft_Samper_722607179-DB3D005pressure.png|400px|Velocity and pressure fields at t = 0.10 s]]
5385
|- style="text-align: center; font-size: 75%;"
5386
| colspan="2" | (a) Velocity and pressure fields at <math>t = 0.05 s</math>
5387
|- style="text-align: center; font-size: 75%;"
5388
|[[Image:draft_Samper_722607179-DB3D010velocity.png|400px|]]
5389
|[[Image:draft_Samper_722607179-DB3D010pressure.png|400px|Velocity and pressure fields at t = 0.20 s]]
5390
|-style="text-align: center; font-size: 75%;"
5391
| colspan="2" | (b) Velocity and pressure fields at <math>t = 0.10 s</math>
5392
|-
5393
|[[Image:draft_Samper_722607179-DB3D020velocity.png|400px|]]
5394
|[[Image:draft_Samper_722607179-DB3D020pressure.png|400px]]
5395
|-style="text-align: center; font-size: 75%;"
5396
| colspan="2" | (c) Velocity and pressure fields at <math>t = 0.20 s</math>
5397
|- style="text-align: center; font-size: 75%;"
5398
| colspan="2" | '''Figure 84:''' Collapse of a water column. 3D simulation: velocity contours plotted over the 3D geometry and pressure contours drawn over the central section.
5399
|}
5400
5401
In Figure [[#img-85|85]] the time evolution of the height at the left wall obtained with the finest mesh is compared to the values measured in the experimental tests [71].
5402
5403
<div id='img-85'></div>
5404
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
5405
|-
5406
|[[Image:draft_Samper_722607179-DBbestSolution.png|390px|Collapse of a water column: residual height h vs time. Comparison between the best numerical solution and the experimental results.]]
5407
|- style="text-align: center; font-size: 75%;"
5408
| colspan="1" | '''Figure 85:''' Collapse of a water column: residual height <math>h</math> vs time. Comparison between the best numerical solution and the experimental results.
5409
|}
5410
5411
The graph shows a very good agreement between the numerical and the experimental results.
5412
5413
For this problem, the solution obtained for slip and stick boundary conditions have been compared. In Figure [[#img-86|86]] the velocity contours obtained for both types of boundary conditions are plotted over the deformed configuration for some time instants.  The mesh used is the coarsest one (average size 0.006 m).
5414
5415
<div id='img-86'></div>
5416
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5417
|-
5418
|[[Image:draft_Samper_722607179-DBs60_t005_vel.png|400px|slip c., velocity field at t = 0.05 sstick c., velocity field at t = 0.05 s]]
5419
|[[Image:draft_Samper_722607179-DBs60stick_t005_vel.png|400px|slip c., velocity field at t = 0.10 s]]
5420
|- style="text-align: center; font-size: 75%;"
5421
| (a) slip c., velocity field at <math>t = 0.05 s</math>
5422
| (b) slip c., velocity field at <math>t = 0.05 s</math>
5423
|-
5424
|[[Image:draft_Samper_722607179-DBs60_t01_vel.png|400px|stick c., velocity field at t = 0.10 s]]
5425
|[[Image:draft_Samper_722607179-DBs60stick_t01_vel.png|400px|slip c., velocity field at t = 0.20 s]]
5426
|- style="text-align: center; font-size: 75%;"
5427
| (c) stick c., velocity field at <math>t = 0.10 s</math>
5428
| (d) slip c., velocity field at <math>t = 0.10 s</math>
5429
|-
5430
|[[Image:draft_Samper_722607179-DBs60_t02_vel.png|400px|stick c., velocity field at t = 0.20 s]]
5431
|[[Image:draft_Samper_722607179-DBs60stick_t02_vel.png|400px]]
5432
|- style="text-align: center; font-size: 75%;"
5433
| (e) stick c., velocity field at <math>t = 0.20 s</math>
5434
| (f) slip c., velocity field at <math>t = 0.20 s</math>
5435
|- style="text-align: center; font-size: 75%;"
5436
| colspan="2" | '''Figure 86:''' Collapse of a water column. 2D results for slip and stick conditions and a coarse mesh (average size 0.006m). Velocity  contours over the deformed configuration at three time instants.
5437
|}
5438
5439
In Figure [[#img-87|87]] the pressure field obtained for both boundary conditions is given.  
5440
5441
<div id='img-87'></div>
5442
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5443
|-
5444
|[[Image:draft_Samper_722607179-DBs60_t005.png|400px|slip c., pressure field at t = 0.05 sstick c., pressure field at t = 0.05 s]]
5445
|[[Image:draft_Samper_722607179-DBs60stick_t005.png|400px|slip c., pressure field at t = 0.10 s]]
5446
|- style="text-align: center; font-size: 75%;"
5447
| (a) slip c., velocity field at <math>t = 0.05 s</math>
5448
| (b) slip c., velocity field at <math>t = 0.05 s</math>
5449
|-
5450
|[[Image:draft_Samper_722607179-DBs60_t01.png|400px|stick c., pressure field at t = 0.10 s]]
5451
|[[Image:draft_Samper_722607179-DBs60stick_t01.png|400px|slip c., pressure field at t = 0.20 s]]
5452
|- style="text-align: center; font-size: 75%;"
5453
| (c) stick c., velocity field at <math>t = 0.10 s</math>
5454
| (d) slip c., velocity field at <math>t = 0.10 s</math>
5455
|-
5456
|[[Image:draft_Samper_722607179-DBs60_t02.png|400px|stick c., pressure field at t = 0.20 s]]
5457
|[[Image:draft_Samper_722607179-DBs60stick_t02.png|400px]]
5458
|- style="text-align: center; font-size: 75%;"
5459
| (e) stick c., velocity field at <math>t = 0.20 s</math>
5460
| (f) slip c., velocity field at <math>t = 0.20 s</math>
5461
|- style="text-align: center; font-size: 75%;"
5462
| colspan="2" | '''Figure 87:''' Collapse of a water column. 2D results for slip and stick conditions and a coarse mesh (average size 0.006m). Pressure  contours over the deformed configuration at three time instants.
5463
|}
5464
5465
The stick condition affects a layer that has a size of the same order of magnitude than the discretization. Hence, the coarser the mesh is, the bigger is the zone affected by the stick condition. When a coarse mesh is used, as in this case, imposing the stick condition penalizes excessively the motion and it can provoke the instability illustrated in the right column of  Figure [[#img-87|87]]. The stream obtained with the stick conditions is less uniform than the one obtained in the slip case and there are pressure concentrations. On the contrary, with a slip condition  a fine solution is obtained also for the pressure field despite the coarseness of the mesh used in this example. This is because the motion of the wall particles helps in the remeshing step and in the global motion of the fluid stream.
5466
5467
In Figure [[#img-88|88]] the results for the velocity and the pressure fields obtained  with the mesh with average size <math display="inline">0.00075 m</math> and stick conditions are given. The pictures show that also for the stick case, a good solution for the pressure field can be obtained if the mesh is sufficiently fine.
5468
5469
<div id='img-88'></div>
5470
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5471
|-
5472
|[[Image:draft_Samper_722607179-DBstickVel05.png|400px|velocity field at t = 0.05 spressure field at t = 0.05 s]]
5473
|[[Image:draft_Samper_722607179-DBstickPres05.png|400px|velocity field at t = 0.10 s]]
5474
|- style="text-align: center; font-size: 75%;"
5475
| (a) velocity field at <math>t = 0.05 s</math>
5476
| (b) pressure field at <math>t = 0.05 s</math>
5477
|-
5478
|[[Image:draft_Samper_722607179-DBstickVel1.png|400px|pressure field at t = 0.10 s]]
5479
|[[Image:draft_Samper_722607179-DBstickPres1.png|400px|velocity field at t = 0.20 s]]
5480
|- style="text-align: center; font-size: 75%;"
5481
| (c) velocity field at <math>t = 0.10 s</math>
5482
| (d) pressure field at <math>t = 0.10 s</math>
5483
|-
5484
|[[Image:draft_Samper_722607179-DBstickVel2.png|400px|pressure field at t = 0.20 s]]
5485
|[[Image:draft_Samper_722607179-DBstickPres2.png|400px]]
5486
|- style="text-align: center; font-size: 75%;"
5487
| (e) velocity field at <math>t = 0.20 s</math>
5488
| (f) pressure field at <math>t = 0.20 s</math>
5489
|- style="text-align: center; font-size: 75%;"
5490
| colspan="2" | '''Figure 88:''' Collapse of a water column. Results obtained with stick condition and a mesh with average size of 0.00075m. Velocity and pressure contours over the deformed configuration at three time instants.
5491
|}
5492
5493
The graph plotted in Figure [[#img-89|89]] shows that, also with the coarsest mesh tested in this example  (average mesh size <math display="inline">0.006m</math>), the solution obtained with the slip condition is very close to the experimental results, while the numerical solution obtained for the same mesh but imposing stick conditions on the walls is quite far from the expected one.  
5494
5495
<div id='img-89'></div>
5496
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5497
|-
5498
|[[Image:draft_Samper_722607179-DBslipVSstick06B.png|600px|Collapse of a water column. Results for slip and stick conditions for the coarset tested mesh (average size 0.006m). Time evolution of the residual height h of the water column.]]
5499
|- style="text-align: center; font-size: 75%;"
5500
| colspan="1" | '''Figure 89:''' Collapse of a water column. Results for slip and stick conditions for the coarset tested mesh (average size <math>0.006m</math>). Time evolution of the residual height <math>h</math> of the water column.
5501
|}
5502
5503
However, as it is shown in the  graphs of Figure [[#img-90|90]], the solution obtained with the stick condition tends to the slip solution with mesh refinement. In particular, using a discretization with an average mesh size of <math display="inline">0.00075 m</math> the two solutions almost coincide. 
5504
5505
<div id='img-90'></div>
5506
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5507
|-
5508
|[[Image:draft_Samper_722607179-DBslipVSstick045B.png|400px|h vs t, mesh size= 0.00045mh vs t, mesh size= 0.00075m]]
5509
|-style="text-align: center; font-size: 75%;"
5510
| (a) <math>h</math> vs <math>t</math>, mesh size= 0.00045m
5511
|-
5512
|[[Image:draft_Samper_722607179-DBslipVSstick075B.png|400px]]
5513
|- style="text-align: center; font-size: 75%;"
5514
| (b)  <math>h</math> vs <math>t</math>, mesh size= 0.00075m
5515
|- style="text-align: center; font-size: 75%;"
5516
| colspan="1" | '''Figure 90:''' Collapse of a water column. Results for slip and stick conditions for different average mesh sizes. Time evolution of the residual height <math>h</math> of the water column.
5517
|}
5518
5519
''Collapse of  a water column over a rigid step''
5520
5521
The collapse of a water column over a rigid step is here studied in 2D and 3D. The initial configuration of the problem is illustrated in Figure [[#img-91a|91a]] and the problem data are given in Table [[#img-91|3.9]]. 
5522
5523
<div id='img-91a'></div>
5524
<div id='img-91'></div>
5525
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;max-width: 100%;"
5526
|-
5527
|[[Image:draft_Samper_722607179-DBobstacleInput.png|300px|Collapse of a water column over a rigid step. Initial geometry of 2D simulation.]]
5528
|- style="text-align: center; font-size: 75%;"
5529
| colspan="1" | '''Figure 91:''' Collapse of a water column over a rigid step. Initial geometry of 2D simulation.
5530
|}
5531
5532
<div class="center" style="font-size: 75%;">
5533
'''Table 3.9.''' Collapse of a water column over a rigid step. Problem data.</div>
5534
5535
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5536
|- 
5537
| style="text-align:center;"|''L''
5538
| style="text-align:center;"|<math>0.146 </math> m
5539
|-
5540
| style="text-align:center;"|<math>H</math>
5541
| style="text-align:center;"|<math>0.048 </math> m
5542
|-
5543
| style="text-align:center;"|<math>D</math>
5544
| style="text-align:center;"|<math>0.024 </math> m
5545
|-
5546
| style="text-align:center;"|viscosity
5547
| style="text-align:center;"|<math>10^{-3} </math> Pa.s
5548
|-
5549
| style="text-align:center;"|density
5550
| style="text-align:center;"|<math>10^3 \hbox{ kg/m}^3 </math> 
5551
|-
5552
| style="text-align:center;"|bulk modulus
5553
| style="text-align:center;"|<math>2.15 \times 10^9 </math> Pa
5554
|}
5555
5556
5557
The phenomena involved in this problem are the collapse of the water column, the impact against the rigid step, the subsequent  creation of the wave, the impact against the vertical wall and the final mixing of the fluid. The numerical solution is compared to the experimental results given in [51], where many experimental observations of the collapse of free liquid columns are collected.
5558
5559
The problem has been solved for both stick and slip conditions. A convergence study for different mesh sizes has been performed for the 2D simulation. In particular, for the convergence test the following average mesh sizes of 3-noded triangles have been used: 0.011m, 0.0095m, 0.009m, 0.0085m, 0.008m, 0.0075m, 0.007m, 0.0065m, 0.006m, 0.0055m, 0.005m, 0.0045m, 0.004m, 0.003m, 0.0025m and 0.002m. In Figure [[#img-92|92]] the finest (mesh size=0.0002m) and the coarsest (mesh size=0.0011m) mesh are illustrated. 
5560
5561
5562
<div id='img-92'></div>
5563
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5564
|-
5565
|[[Image:draft_Samper_722607179-DBOBS0011.png|400px]]
5566
|[[Image:draft_Samper_722607179-DBOBS0002.png|400px]]
5567
|- style="text-align: center; font-size: 75%;"
5568
| (a) average mesh size= 0.011m
5569
| (b) average mesh size= 0.002m
5570
|- style="text-align: center; font-size: 75%;"
5571
| colspan="2" | '''Figure 92:''' Collapse of a water column over a rigid step. Coarsest (1014 triangles) and finest (24668 triangles) meshes used for the 2D  analysis.
5572
|}
5573
5574
All the analyses have been run for <math display="inline">{\Delta t=0.0002 s}</math>. <div id='img-93'></div>
5575
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5576
|-
5577
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-DBs40vel_t01.png|300px|]]
5578
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-01comp.png|300px|]]
5579
|-
5580
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-DBs40vel_t02.png|300px|]]
5581
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-02comp.png|300px|]]
5582
|-
5583
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-DBs40vel_t03.png|300px|]]
5584
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-03comp.png|300px|]]
5585
|-
5586
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-DBs40vel_t04.png|300px|]]
5587
|style="vertical-align:bottom;"|[[Image:draft_Samper_722607179-04comp.png|300px|Collapse of a water column over a rigid step. Velocity contours over the deformed configuration at time: t=0.1s, 0.2s, t=0.3s and 0.4s. Results obtained with the finest mesh and slip conditions on the walls.]]
5588
|- style="text-align: center; font-size: 75%;"
5589
| colspan="2" | '''Figure 93:''' Collapse of a water column over a rigid step. Velocity contours over the deformed configuration at time: <math>t=0.1s</math>, <math>0.2s</math>, <math>t=0.3s</math> and <math>0.4s</math>. Results obtained with the finest mesh and slip conditions on the walls.
5590
|}
5591
5592
In Figure [[#img-93|93]] the numerical results obtained for the 2D simulation with the finest mesh are compared to the experimental observations [51]  for the same instants. The velocity contours are plotted over the deformed configurations.
5593
5594
The 3D problem has been solved considering a width of 0.146m and an average size of 0.008m for the tetrahedra of the FEM mesh. This gives an initial mesh of 170711 4-noded tetrahedra. In Figure [[#img-94|94]] the 2D results (green) are superposed to the cut performed at the center of the 3D domain (grey) for the same time instants. An average size of 0.008m has been used for the edges of  both the triangles and the tetrahedra for the 2D and 3D discretizations.
5595
5596
<div id='img-94'></div>
5597
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5598
|-
5599
|[[Image:draft_Samper_722607179-DBobstacleSup3D01.png|300px|t = 0.1 st = 0.2 s]]
5600
|-style="text-align: center; font-size: 75%;"
5601
| (a) <math>t = 0.1 s</math>
5602
|-
5603
|[[Image:draft_Samper_722607179-DBobstacleSup3D02.png|300px|t = 0.3 s]]
5604
|- style="text-align: center; font-size: 75%;"
5605
| (b) <math>t = 0.2 s</math>
5606
|-style="text-align: center; font-size: 75%;"
5607
| colspan="2"|[[Image:draft_Samper_722607179-DBobstacleSup3D03.png|300px]]
5608
|- style="text-align: center; font-size: 75%;"
5609
| (c) <math>t = 0.3 s</math>
5610
|- style="text-align: center; font-size: 75%;"
5611
| colspan="1" | '''Figure 94:''' Collapse of a water column over a rigid step. Superposition of the results obtained with an average mesh size of 0.008 m for the 2D and 3D simulations (green and grey colours, respectively) and with slip conditions on the walls.
5612
|}
5613
5614
The Figure [[#img-95|95]] refers to the slip problem only and it shows that the solutions obtained with the three finest meshes are very similar. This is a further evidence of the convergence of the PFEM strategy here presented.
5615
5616
<div id='img-95'></div>
5617
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5618
|-
5619
|[[Image:draft_Samper_722607179-DBobstaclesSlipLast3sup.png|300px|Collapse of a water column over a rigid step. Superposition of the 2D results obtained at t=0.3s with the meshes with average size 0.002m (orange), 0.0025m (pink) and 0.003m (green) (slip conditon on the walls).]]
5620
|- style="text-align: center; font-size: 75%;"
5621
| colspan="1" | '''Figure 95:''' Collapse of a water column over a rigid step. Superposition of the 2D results obtained at <math>t=0.3s</math> with the meshes with average size 0.002m (orange), 0.0025m (pink) and 0.003m (green) (slip conditon on the walls).
5622
|}
5623
5624
In Figure [[#img-96|96]] the solution obtained at <math display="inline">t=0.3s</math> with a stick condition  and the finest mesh is given. The differences between the streams are very small.
5625
5626
<div id='img-96'></div>
5627
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5628
|-
5629
|[[Image:draft_Samper_722607179-DBobsSup03slipVSstick.png|300px|Collapse of a water column over a rigid step. Superposition of the 2D results obtained at t=0.3s with the mesh with average size 0.002m for the slip (orange colour) and the stick (pink) conditions.]]
5630
|- style="text-align: center; font-size: 75%;"
5631
| colspan="1" | '''Figure 96:''' Collapse of a water column over a rigid step. Superposition of the 2D results obtained at <math>t=0.3s</math> with the mesh with average size 0.002m for the slip (orange colour) and the stick (pink) conditions.
5632
|}
5633
5634
As already mentioned, the convergence analysis has been performed for both cases of slip and stick conditions imposed on the walls. Specifically, the study of convergence has been performed first for the maximum velocity and then the kinetic energy of the whole domain. Both values have been computed at <math display="inline">t=0.1s</math>, so just before the impact with the rigid step.
5635
5636
In the graph of Figure [[#img-97|97]] the Y-axis is the maximum velocity obtained for the stick and slip conditions and the X-axis refers to the degrees of freedom (<math display="inline">dof</math>) for the velocity. The solutions obtained with the stick condition suffer from oscillations for the coarsest meshes, but for finer meshes tend to those of the slip condition.  <div id='img-97'></div>
5637
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
5638
|-
5639
|[[Image:draft_Samper_722607179-DBobstacleVelocity.png|390px|Collapse of a water column over a rigid step. Maximum velocity at t=0.1s for slip and stick conditions (2D simulation).]]
5640
|- style="text-align: center; font-size: 75%;"
5641
| colspan="1" | '''Figure 97:''' Collapse of a water column over a rigid step. Maximum velocity at <math>t=0.1s</math> for slip and stick conditions (2D simulation).
5642
|}
5643
5644
A convergence study for the kinetic energy of the whole domain at <math display="inline">t=0.1s</math> has been performed. The kinetic energy has been computed as
5645
5646
<span id="eq-247"></span>
5647
{| class="formulaSCP" style="width: 100%; text-align: left;" 
5648
|-
5649
| 
5650
{| style="text-align: left; margin:auto;" 
5651
|-
5652
| style="text-align: center;" | <math>E_{kin}= \sum _{e=1}^{n} 0.5  m^e  (v^e)^2      </math>
5653
|}
5654
| style="width: 5px;text-align: right;" | (247)
5655
|}
5656
5657
where <math display="inline">m^e</math> is the element mass and <math display="inline">v^e</math> is the mean value for the nodal velocities of the element.
5658
5659
The graph of Figure [[#img-98|98]] is the evolution of the kinetic energy for the first 0.1 seconds of analysis, for both cases of slip and stick conditions and for three different meshes: the coarsest one (mesh size=0.011m), the finest one (mesh size=0.002m) and an intermediate discretization (mesh size=0.006m). <div id='img-98'></div>
5660
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
5661
|-
5662
|[[Image:draft_Samper_722607179-DBobsKinEne.png|390px|Collapse of a water column over a rigid step. Time evolution of the kinetic energy for slip and stick conditions and for three different FEM discretizations (2D simulation).]]
5663
|- style="text-align: center; font-size: 75%;"
5664
| colspan="1" | '''Figure 98:''' Collapse of a water column over a rigid step. Time evolution of the kinetic energy for slip and stick conditions and for three different FEM discretizations (2D simulation).
5665
|}
5666
5667
5668
The curves for the slip case are almost superposed, contrary to the ones for the stick problem. However, for the finest mesh, the curves for the slip and stick conditions are very close.
5669
5670
In the graph of Figure [[#img-99|99]] the Y-axis refers to the kinetic energy at <math display="inline">t=0.1s</math> for the stick and slip conditions and  the X-axis refers to the degrees of freedom for the velocity. With slip conditions, the kinetic energy has almost the same value for all the discretizations (the difference between the values given by the coarsest and the finest discretization is lower than 0.5% However for stick conditions the kinetic energy grows with the refinement of the mesh and it seems to converge to the value of the slip case.  
5671
5672
<div id='img-99'></div>
5673
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
5674
|-
5675
|[[Image:draft_Samper_722607179-DBobsKinResults.png|390px|Collapse of a water column over a rigid step. Kinetic energy at t=0.1s for slip and stick conditions.]]
5676
|- style="text-align: center; font-size: 75%;"
5677
| colspan="1" | '''Figure 99:''' Collapse of a water column over a rigid step. Kinetic energy at <math>t=0.1s</math> for slip and stick conditions.
5678
|}
5679
5680
The convergence curve for the kinetic energy evolution on time for the first <math display="inline">0.1 s</math> of analysis is given in Figure [[#img-100|100]]. For both slip and stick cases, the error has been computed using to Eq.([[#eq-245|245]]). The slope is almost 2 for both curves. 
5681
5682
<div id='img-100'></div>
5683
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
5684
|-
5685
|[[Image:draft_Samper_722607179-DBconvergence2.png|360px|Collapse of a water column over a rigid step. Convergence analysis for the  kinetic energy evolution on time for the first 0.1 s of analysis  for slip and stick conditions (2D case).]]
5686
|- style="text-align: center; font-size: 75%;"
5687
| colspan="1" | '''Figure 100:''' Collapse of a water column over a rigid step. Convergence analysis for the  kinetic energy evolution on time for the first <math>0.1 s</math> of analysis  for slip and stick conditions (2D case).
5688
|}
5689
5690
In Figure [[#img-101|101]] the time evolution of the pressure until <math display="inline">t=0.1s</math> is studied. The graph on the left refers to a coarse mesh and the one in the right to the finest mesh. The plots show that the boundary conditions affect the pressure field. In particular the stick condition induces oscillations in the pressure solution which amplitude reduces  by refining the mesh.
5691
5692
<div id='img-101'></div>
5693
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5694
|-
5695
|[[Image:draft_Samper_722607179-DBobsPres06.png|400px|]]
5696
|[[Image:draft_Samper_722607179-DBobsPres02.png|400px]]
5697
|- style="text-align: center; font-size: 75%;"
5698
| colspan="2" | '''Figure 101:''' Collapse of a water column over a rigid step. Time evolution of the pressure for a mesh with average size 0.006 m (top) and 0.002 m (bottom). Solution for stick and slip conditions (2D simulation).
5699
|}
5700
5701
This analysis has shown that for coarse meshes there are significant differences between the solution given by imposing stick or slip conditions. However, for fine meshes the problems solved with either slip or stick conditions tend to the same solution. In fact, for coarse meshes, the  stick conditions affect highly the motion of the fluid and a refinement is required in order to obtain a solution close to the expected one. Instead, with slip conditions on the walls it is possible to obtain fine results, for both the velocity and pressure fields, also for coarse meshes.
5702
5703
===3.5.2 Validation of the Unified formulation for quasi-incompressible hypoelastic solids===
5704
5705
The Unified formulation is here validated for quasi-incompressible solids by solving  a benchmark problem for non-linear solid mechanics, the Cook's membrane. In Section [[#2.3.4 Validation examples|2.3.4]] the same structure was analyzed considering a compressible material. In this section, the nearly-incompressible problem is analyzed. In particular, first the problem is solved for <math display="inline">\nu=0.4999</math> with all the solid elements derived in this thesis, the V, the VP and the VPS elements comparing the numerical results for the displacements, the pressure and the stresses. Then the maximum displacement given by the three elements solving the same problem for the range of Poisson ratio from 0.3 to 0.499999 is compared.
5706
5707
''Nearly incompressible Cook's membrane''
5708
5709
In Figure [[#img-102a|102a]] the initial geometry and the problem data are given. <div id='img-102a'></div>
5710
<div id='img-102'></div>
5711
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5712
|-
5713
|[[Image:draft_Samper_722607179-CMincInput.png|250px|Initial geometry]]
5714
|[[Image:draft_Samper_722607179-CMinc.png|200px|Nearly incompressible Cook's membrane. Initial geometry, material data and FEM mesh (127 elements).]]
5715
|- style="text-align: center; font-size: 75%;"
5716
| (a) Initial geometry
5717
| (b) FEM mesh
5718
|- style="text-align: center; font-size: 75%;"
5719
| colspan="2" | '''Figure 102:''' Nearly incompressible Cook's membrane. Initial geometry, material data and FEM mesh (127 elements).
5720
|}
5721
5722
The problem has been solved in 2D for different discretizations in order to verify the convergence of the scheme. In this case, unstructured meshes have been used. In  Figure [[#img-102|102]] the coarsest finite element discretization (average size 5) used in this example is depicted. The finest mesh tested in this problem has mean size 0.25 that gives a mesh of 52372 triangles.
5723
5724
As for the previous case, the problem is studied in statics and the self-weight of the membrane has not been taken into account.  The reference solution for this problem is taken from [105], where the problem was solved with an Incompatible Bubble element. In this publication the tip vertical displacement obtained <math display="inline">U^{max}_Y</math> by other formulations is also given. Specifically the results are provided for a FEM formulation with linear displacements and constant pressure and the Enhanced Assumed Strain formulation [12,13]. For all these formulations and the finest mesh tested in [105], this value is around <math display="inline">U^{max}_Y = 7.71</math>.
5725
5726
The Poisson ratio 0.4999 represents a material that is almost incompressible. It is well known that values of the Poisson ratio close to 0.5 generate numerical problems to non-stabilized FEM schemes, such as the locking of the solution.  Apart from this, the proximity to the incompressible limit also produces ill-conditioned matrices and  deteriorates the convergence of the solution. In order to overcome these drawbacks a stabilized formulation is required. In this work, the stabilization of the quasi incompressible mixed formulation for the  solid element is achieved via the FIC strategy presented in Section [[#3.1 Stabilized FIC form of the mass balance equation|3.1]] using the VPS-element. The quasi-incompressible membrane is here also solved using the non-stabilized V and VP elements derived in Chapter [[#2 Velocity-based formulations for compressible materials|2]]. Both these elements, although in different ways, suffer from instability near the incompressible limit of the material. Despite that, this example is presented with the purpose of showing how each formulation is affected by the mentioned drawbacks and to underline the superiority of a mixed formulation for dealing with nearly incompressible materials.
5727
5728
<div id='img-103'></div>
5729
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
5730
|-
5731
|[[Image:draft_Samper_722607179-CMresults2.png|390px|Nearly incompressible Cook's membrane. Top corner vertical displacement for the V, VP and the VPS elements.]]
5732
|- style="text-align: center; font-size: 75%;"
5733
| colspan="1" | '''Figure 103:''' Nearly incompressible Cook's membrane. Top corner vertical displacement for the V, VP and the VPS elements.
5734
|}
5735
5736
In Figure [[#img-103|103]] the top corner vertical displacements obtained with the V, the VP and the VPS elements for different FEM meshes are plotted.
5737
5738
In Table [[#table-10|3.10]] the numerical values are given.
5739
5740
<span id='table-10'></span>
5741
5742
<div class="center" style="font-size: 75%;">'''Table 3.10.''' Nearly incompressible Cook's membrane. Top corner vertical displacement  for different formulations and discretizations.</div>
5743
5744
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5745
|-
5746
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |    mesh size 
5747
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |  number of elements
5748
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |  '''V-element''' 
5749
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |  '''VP-element''' 
5750
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |   '''VPS-element'''
5751
|-
5752
| style="border-left: 2px solid;border-right: 2px solid;" | <math>U_y</math>  
5753
| style="border-left: 2px solid;border-right: 2px solid;" | <math>U_y</math> 
5754
| style="border-left: 2px solid;border-right: 2px solid;" | <math>U_y</math> 
5755
|-
5756
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |    5
5757
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 127
5758
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 4.411
5759
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 7.031
5760
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 7.268
5761
|-
5762
| style="border-left: 2px solid;border-right: 2px solid;" |  4
5763
| style="border-left: 2px solid;border-right: 2px solid;" | 194
5764
| style="border-left: 2px solid;border-right: 2px solid;" | 4.365
5765
| style="border-left: 2px solid;border-right: 2px solid;" | 7.178
5766
| style="border-left: 2px solid;border-right: 2px solid;" | 7.338
5767
|-
5768
| style="border-left: 2px solid;border-right: 2px solid;" |  3
5769
| style="border-left: 2px solid;border-right: 2px solid;" | 361
5770
| style="border-left: 2px solid;border-right: 2px solid;" | 4.648
5771
| style="border-left: 2px solid;border-right: 2px solid;" | 7.401
5772
| style="border-left: 2px solid;border-right: 2px solid;" | 7.508
5773
|-
5774
| style="border-left: 2px solid;border-right: 2px solid;" |  2
5775
| style="border-left: 2px solid;border-right: 2px solid;" | 802
5776
| style="border-left: 2px solid;border-right: 2px solid;" | 5.643
5777
| style="border-left: 2px solid;border-right: 2px solid;" | 7.551
5778
| style="border-left: 2px solid;border-right: 2px solid;" | 7.603
5779
|-
5780
| style="border-left: 2px solid;border-right: 2px solid;" |  1.5
5781
| style="border-left: 2px solid;border-right: 2px solid;" | 1441
5782
| style="border-left: 2px solid;border-right: 2px solid;" | 4.937
5783
| style="border-left: 2px solid;border-right: 2px solid;" | 7.632
5784
| style="border-left: 2px solid;border-right: 2px solid;" | 6.655
5785
|-
5786
| style="border-left: 2px solid;border-right: 2px solid;" |  1
5787
| style="border-left: 2px solid;border-right: 2px solid;" | 3288
5788
| style="border-left: 2px solid;border-right: 2px solid;" | 5.690
5789
| style="border-left: 2px solid;border-right: 2px solid;" | 7.695
5790
| style="border-left: 2px solid;border-right: 2px solid;" | 7.714
5791
|-
5792
| style="border-left: 2px solid;border-right: 2px solid;" |  0.75
5793
| style="border-left: 2px solid;border-right: 2px solid;" | 5806
5794
| style="border-left: 2px solid;border-right: 2px solid;" | 6.199
5795
| style="border-left: 2px solid;border-right: 2px solid;" | 7.707
5796
| style="border-left: 2px solid;border-right: 2px solid;" | 7.729
5797
|-
5798
| style="border-left: 2px solid;border-right: 2px solid;" |  0.5
5799
| style="border-left: 2px solid;border-right: 2px solid;" | 13015
5800
| style="border-left: 2px solid;border-right: 2px solid;" | 6.731
5801
| style="border-left: 2px solid;border-right: 2px solid;" | 7.745
5802
| style="border-left: 2px solid;border-right: 2px solid;" | 7.755
5803
|-
5804
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" |  0.25
5805
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 52372
5806
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 7.272
5807
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 7.765
5808
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 7.771
5809
5810
|}
5811
5812
Nearly incompressible Cook's membrane. Top corner vertical displacement  for different formulations and discretizations. CMincTable
5813
5814
For the V, VP and VPS the percentage errors for the tip displacement with respect to the reference solution are 5.68%, 0.707 % and 0.791%, respectively.
5815
5816
For the analysis of stress results, the case of the mesh with average size 1.5 is studied. In Figure [[#img-104|104]] the X-component of the Cauchy stress tensor obtained with V, VP, and VPS elements  is plotted over the deformed configurations. From the pictures it is clear that both the V and VP solutions deteriorate for values of <math display="inline">\nu </math> close to  the incompressible limit of the material, while the stress field given by the VPS-element is good and smooth.  <div id='img-104'></div>
5817
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5818
|-
5819
|[[Image:draft_Samper_722607179-CMuncCauchy.png|400px]]
5820
|[[Image:draft_Samper_722607179-CMcouCauchy.png|400px]]
5821
|[[File:Draft_Samper_722607179_1036_CMcouScauchy.png|400px]]
5822
|- style="text-align: center; font-size: 75%;"
5823
| (a) V-element
5824
| (b) VP-element
5825
| (c) VPS-element
5826
|- style="text-align: center; font-size: 75%;"
5827
| colspan="2" | '''Figure 104:''' Nearly incompressible Cook's membrane. Results of the XX component of the Cauchy stress tensor for V, VP and VPS elements.
5828
|}
5829
5830
In Figure [[#img-105|105]] the pressure solution obtained with the VP-element and the VPS-element are given.  <div id='img-105'></div>
5831
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5832
|-
5833
|[[Image:draft_Samper_722607179-CMcouPressure.png|200px]]
5834
|[[Image:draft_Samper_722607179-CMcouSpressure.png|200px]]
5835
|- style="text-align: center; font-size: 75%;"
5836
| (a) VP-element
5837
| (b) VPS-element
5838
|- style="text-align: center; font-size: 75%;"
5839
| colspan="2" | '''Figure 105:''' Nearly incompressible Cook's membrane. Pressure solution for the VP and VPS elements.
5840
|}
5841
5842
The non-stabilized VP-element yields a pressure field that is completely untrustworthy exhibiting the classical checkerboard modes. Instead, the solution of the VPS element is smooth and accurate.
5843
5844
From the kinematic point of view, the displacements obtained by the mixed formulation (VP and VPS elements) are close to the expected solution, while the solution given by the V-element is totally locked. For the mesh displayed in  Figures [[#img-104|104]]-[[#img-105|105]] (average mesh size 1.5) the errors for the  top corner displacement with respect the reference solution are  36.5%, 1.9% and 1.61% for the V, the VP and the VPS elements, respectively.
5845
5846
The same problem is solved for different Poisson ratios, from 0.3 to 0.499999, using the same FEM discretization (average mesh size equal to 1). For the stabilized formulation the Poisson ratio that appears in the tangent matrix of the linear momentum equations has been limited to 0.4999. Instead, in the equation of the pressure the actual value of the Poisson ratio is used. This technique is similar to the one used for the stabilization of the fluid in Section [[#3.4.3 Analysis of the conditioning of the solution scheme|3.4.3]] where the bulk modulus is reduced only in the tangent matrix of the linear momentum equations but not in the continuity equation. The graphs of Figure [[#img-106|106]] are the plots of  the top corner vertical displacement obtained for the different Poisson ratios using the V, the VP and the VPS elements (the graph at the right is a zoom of the last part of the graph at the left).  <div id='img-106'></div>
5847
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5848
|-
5849
|[[Image:draft_Samper_722607179-CMSincPoisResults.png|400px|]]
5850
|[[Image:draft_Samper_722607179-CMSincPoisResultsZoom.png|400px|Nearly incompressible Cook's membrane. Maximum vertical displacement for different Poisson ratios. Results for V, VP and VPS elements. The graph at the right is a zoom of the last part of the graph at the left.]]
5851
|- style="text-align: center; font-size: 75%;"
5852
| colspan="2" | '''Figure 106:''' Nearly incompressible Cook's membrane. Maximum vertical displacement for different Poisson ratios. Results for V, VP and VPS elements. The graph at the right is a zoom of the last part of the graph at the left.
5853
|}
5854
5855
In Table [[#table-11|3.11]] the numerical values are given.
5856
5857
<span id='table-11'></span>
5858
5859
<div class="center" style="font-size: 75%;">'''Table 3.11'''. Nearly incompressible Cook's membrane. Numerical values for the maximum vertical displacement for different Poisson ratios  obtained with V, VP and VPS elements.
5860
</div>
5861
5862
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
5863
|-
5864
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |   Poisson ratio 
5865
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |  '''V-element''' 
5866
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |  '''VP-element'''
5867
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |   '''VPS-element'''
5868
|-
5869
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" |   0.3
5870
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 9.176
5871
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 9.212
5872
| style="border-left: 2px solid;border-right: 2px solid;border-top: 2px solid;" | 9.245
5873
|-
5874
| style="border-left: 2px solid;border-right: 2px solid;" |  0.4
5875
| style="border-left: 2px solid;border-right: 2px solid;" | 8.557
5876
| style="border-left: 2px solid;border-right: 2px solid;" | 8.601
5877
| style="border-left: 2px solid;border-right: 2px solid;" | 8.628
5878
|-
5879
| style="border-left: 2px solid;border-right: 2px solid;" |  0.45
5880
| style="border-left: 2px solid;border-right: 2px solid;" | 8.143
5881
| style="border-left: 2px solid;border-right: 2px solid;" | 8.203
5882
| style="border-left: 2px solid;border-right: 2px solid;" | 8.230
5883
|-
5884
| style="border-left: 2px solid;border-right: 2px solid;" |  0.49
5885
| style="border-left: 2px solid;border-right: 2px solid;" | 7.658
5886
| style="border-left: 2px solid;border-right: 2px solid;" | 7.833
5887
| style="border-left: 2px solid;border-right: 2px solid;" | 7.866
5888
|-
5889
| style="border-left: 2px solid;border-right: 2px solid;" |  0.495
5890
| style="border-left: 2px solid;border-right: 2px solid;" | 7.503
5891
| style="border-left: 2px solid;border-right: 2px solid;" | 7.781
5892
| style="border-left: 2px solid;border-right: 2px solid;" | 7.817
5893
|-
5894
| style="border-left: 2px solid;border-right: 2px solid;" |  0.499
5895
| style="border-left: 2px solid;border-right: 2px solid;" | 6.988
5896
| style="border-left: 2px solid;border-right: 2px solid;" | 7.735
5897
| style="border-left: 2px solid;border-right: 2px solid;" | 7.766
5898
|-
5899
| style="border-left: 2px solid;border-right: 2px solid;" |  0.4999
5900
| style="border-left: 2px solid;border-right: 2px solid;" | 5.690
5901
| style="border-left: 2px solid;border-right: 2px solid;" | 7.695
5902
| style="border-left: 2px solid;border-right: 2px solid;" | 7.714
5903
|-
5904
| style="border-left: 2px solid;border-right: 2px solid;" |  0.49999
5905
| style="border-left: 2px solid;border-right: 2px solid;" | 4.872
5906
| style="border-left: 2px solid;border-right: 2px solid;" | 7.532
5907
| style="border-left: 2px solid;border-right: 2px solid;" | 7.714
5908
|-
5909
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" |  0.499999
5910
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 4.706
5911
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 6.696
5912
| style="border-left: 2px solid;border-right: 2px solid;border-bottom: 2px solid;" | 7.713
5913
5914
|}
5915
5916
Nearly incompressible Cook's membrane. Numerical values for the maximum vertical displacement for different Poisson ratios  obtained with V, VP and VPS elements. CMpoiTable
5917
5918
The graphs show that for Poisson ratios bigger than 0.45, the velocity formulation exhibits locking. Instead, the mixed  formulations despite the increasing of ill-conditioning of the linear system, yieldsa good solution until a Poisson ratio of 0.49999. Beyond this value,  the non-linear solver does not even converge for the non-stabilized VP-element, while the solution given by the VPS-element is still a good one.
5919
5920
==3.6 Summary and conclusions==
5921
5922
In this chapter the Unified Stabilized formulation for quasi-incompressible materials has been derived. This numerical procedure is based on the mixed Velocity-Pressure formulation derived in Chapter [[#2 Velocity-based formulations for compressible materials|2]] for compressible materials.
5923
5924
In order to deal with material incompressibility using a linear interpolation for both the pressure and the velocity fields, the scheme required to be stabilized. The necessary stabilization has been introduced using an enhanced version of the Finite Calculus (FIC) method.  In Section [[#3.1 Stabilized FIC form of the mass balance equation|3.1]] the complete derivation of the stabilized form of the mass balance equation has been presented. The FIC stabilization has been derived for quasi-incompressible fluids and has been extended also to hypoelastic quasi-incompressible solids. The solid finite element based on the mixed Velocity-Pressure stabilized formulation  has been called VPS element.
5925
5926
The solution schemes for solving quasi-incompressible Newtonian fluids and hypoelastic solids with the stabilized mixed Velocity-Pressure formulation  have been given and explained in detail.
5927
5928
Section [[#3.4 Free surface flow analysis|3.4]] has been fully devoted to the analysis of free surface fluids with the Unified stabilized formulation. First the Particle Finite Element Method (PFEM) has been explained and its advantages and disadvantages have been highlighted. A simple technique for modeling the slip conditi
5929
5930
ons moving the wall particles have been also given. Then mass preservation properties of the PFEM-FIC stabilized formulation have been tested with several numerical examples. It has been shown that the method  yields excellent results  for a variety of 2D and 3D free surface flow problems  involving surface waves, water splashing, violent impact of flows with containment walls  and mixing of fluids. Next the conditioning of the scheme has been studied and the numerical inconveniences associated to the high value of the bulk modulus have been highlighted. An efficient and easy to implement technique for improving the conditioning and the global convergence of the problem using a scaled value for the pseudo bulk modulus has been given. The strategy has been validated for two benchmark problems for free surface flows.
5931
5932
In the last section of this chapter several numerical examples have been presented for validating the unified stabilized formulation for solving fluids and solids close to the incompressible limit. The proposed method has been validated  versus both experimental tests and numerical results from other formulations and it has been shown that the method is convergent to the expected solution. Particular attention has been given to the analysis of the boundary conditions. In particular, it has been shown that for inviscid fluids and coarse meshes it is preferable to use slip conditions in order to avoid  the pressure concentrations induced by the stick conditions.
5933
5934
=Chapter 4. Unified formulation for FSI problems=
5935
5936
5937
==4.1 Introduction==
5938
5939
In the previous chapters the Unified formulation was used for solving the mechanics of fluids and solids separately. In this chapter it will be shown that the method is also suitable for solving fluid-structure interaction (FSI) problems.  The algorithm for coupling the three solid formulations (the V, the VP and the VPS elements) with the mixed Velocity-Pressure FIC-stabilized formulation for Newtonian fluids is described. It will be shown that  the proposed method gives the possibility to select the formulation for the solid depending on the problem to solve. The numerical results will be validated by comparing the solution of benchmark FSI problems with the results of the literature.
5940
5941
In this work, the FSI problems are solved in a monolithic way. This means that  fluids and  solids are solved within the same linear system and the iterations between the fluid and the solid solver are not required, as for staggered schemes. The unified formulation  for both fluid and solid mechanics solution uses the same framework (Lagrangian description), the same spatial discretization and the same  temporal integration schemes (linear shape functions and implicit integration of time), the same unknown variables (velocities and pressures),  and the same solution method (two-step partitioned scheme). In conclusion, solids and fluids just represent different regions of the same global domain and they differ on the characteristic material parameters only.  All this simplifies the implementation of the FSI solver. In fact, it is not required any variable transformation  and the implementation effort for coupling the mechanics of  fluids and solids is reduced to a proper assembly of the global linear system.
5942
5943
Concerning the mesh, there must be a correspondence at the interface between the solid and the fluid nodes, in the spirit of conforming mesh methods. It will be shown that the PFEM guarantees the conformity.
5944
5945
In conclusion, for explaining the extension of the Unified formulation to FSI problems only the assembly of the global linear system and the way to detect the fluid-solid interface have to be described. This will be  done in the next section. Then the solution schemes for solving  FSI problems coupling the FIC-stabilized mixed VP  formulation for quasi-incompressible Newtonian fluids with the V, the VP and the VPS elements for hypoelastic solids are given. Finally, several numerical examples of benchmark FSI problems are solved and all the schemes are validated and compared.
5946
5947
==4.2 FSI algorithm==
5948
5949
The assembly of the global linear system is performed by making a loop over all the nodes of the mesh. Each node provides the contributions of the elements that share the node, and each element is computed according to the specific constitutive law and solution scheme. So, when an interface node is analyzed, it is necessary to sum the contributions of both materials in the global linear system (see Figure [[#img-107|107]] for a graphic representation of the global assembly). <div id='img-107'></div>
5950
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5951
|-
5952
|[[Image:draft_Samper_722607179-FSImatrix.png|600px|Graphic representation of domain contributions to the global linear system.]]
5953
|- style="text-align: center; font-size: 75%;"
5954
| colspan="1" | '''Figure 107:''' Graphic representation of domain contributions to the global linear system.
5955
|}
5956
5957
Those nodes that belong to a single domain assemble to the global matrix with  the elemental contributions of one material only.
5958
5959
Concerning the degrees of freedom, each node of the mesh is characterized by a single set of kinematic variables. So they move according to a unique velocity. This means that the degrees of freedom for the solid and fluid velocities coincide also at the interface nodes. On the other hand, in order to guarantee the correct boundary conditions for the stresses,  each interface node  has a   degree of freedom for the pressure of the fluid and, for the VP and the VPS elements, for the pressure of the solid. This means that, for each interface node the  fluid elements assemble only the contributions for  the degree of freedom of the fluid pressure, while the   solid elements do that for the solid pressure. This requires solving twice the continuity equation: once for the fluid domain and once for the VP-element or the VPS-element for the solid domain.
5960
5961
In order to ensure the coupling, the fluid and the solid meshes must have in common the nodes along the interface. In other words, there must be a node to node conformity. In this work, the solid is solved using the FEM while the fluid is computed by the PFEM. In terms of meshing, this means that the solid domain maintains the same grid during all the analysis, whereas  the fluid is remeshed whenever its discretization becomes excessively distorted. The conformity of the fluid and solid meshes on the interface is guaranteed by exploiting the capability of the PFEM for detecting the boundaries. In practice, the fluid detects the solid interface nodes (nodes that  are located on the external boundaries of the solid fixed mesh) in the same way it recognizes its rigid contours. As it has been explained in Section [[#3.4.1.1 Remeshing|3.4.1.1]], this is done by an efficient combination between the Alpha Shape method and the Delaunay triangulation. According to this strategy, if the separation of the fluid contour from the solid domain is small enough so that that the Alpha Shape criteria are fulfilled, a fluid element connecting the fluid domain to the solid domain is generated. Otherwise the two domains keep apart from each other.  In Figure [[#img-108|108]] a graphic representation of the method for detecting the interface is given [31].  <div id='img-108a'></div>
5962
<div id='img-108b'></div>
5963
<div id='img-108'></div>
5964
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
5965
|-
5966
|[[Image:draft_Samper_722607179-contactDetection1.png|400px|]]
5967
|- style="text-align: center; font-size: 75%;"
5968
|(a)
5969
|-
5970
|[[Image:draft_Samper_722607179-contactDetection2.png|400px|Detection of an interface with the PFEM [31].]]
5971
|- style="text-align: center; font-size: 75%;"
5972
| (b) 
5973
|- style="text-align: center; font-size: 75%;"
5974
| colspan="2" | '''Figure 108:''' Detection of an interface with the PFEM [31].
5975
|}
5976
5977
In the situation described in the pictures of Figure [[#img-108a|108a]] none of the contact elements generated by the Delaunay triangulation fulfill the Alpha Shape criteria. Hence for the forthcoming time step interval there is not interaction between the solid and the  fluid domains. Instead in Figure [[#img-108b|108b]] some hybrid elements that share solid and fluid nodes have been generated. In this case, the coupling is active and the interface nodes assemble as described before.
5978
5979
For the good use of this procedure, it is essential to select a similar mesh size for the fluid and the solid domain in the interface zone. Otherwise, some typical drawbacks may arise. For example,  if the solid mesh is much finer than the fluid one some distorted elements may form in the interface zone. The opposite case is even worse. If the solid mesh at the boundaries  is much coarser than the fluid one, some fluid particles may cross the interface and compromise the whole computation. Using the same mesh size (and a reasonable time step increment) neither of these problems arise. Note that this behavior is characteristic of PFEM. In fact all this can occur also in the contact zone of a fluid domain with a rigid boundary in a fluid dynamic analysis, as explained in the Section [[#3.4.1 The Particle Finite Element Method|3.4.1]]. This confirms again that, from the remeshing point of view, there is not difference between the contour nodes of a deformable body and the nodes forming a rigid wall.
5980
5981
==4.3 Coupling with the Velocity formulation for the solid==
5982
5983
The coupling of the PFEM-FIC stabilized VP formulation with the V-element for the solid consists essentially on assembling adequately the linear momentum equations. For each degree of freedom for the interface nodal velocity, the solid and the fluid contributions sum. On the other hand, the stabilized continuity equation is assembled and solved only for the fluid elements because the pressure is not an unknown variable of the Velocity formulation.
5984
5985
In order to avoid ambiguities in the notation, the variables and the matrices referred to the fluid domain will be marked by subscript '''f'' ', and those ones belonging to the solid by '''s'' '. When a shared variable is considered (for example, the unknowns referred to the nodes of the interface) the subscript will be '''s,f'' '. Instead, the subscript that marks all the nodes of the domain is  '''s+f'' '.
5986
5987
In Box 12, the solution scheme for a generic time interval <math display="inline">{}</math> is given. In the scheme only the contributions of the interface nodes are analyzed. For the rest of nodes the problem is locally uncoupled, hence the schemes described in Box 3 for hypoelastic solids and in Box 8 for Newtonian fluids still hold.
5988
5989
All the matrices and the vectors that appear in Box 12 were already defined in Boxes 4 and 9.
5990
5991
<div class="center" style="font-size: 75%;">
5992
[[File:Draft_Samper_722607179_1135_Box12.png|550px]]
5993
5994
'''Box 12'''. Iterative solution scheme for FSI problem solved with the V-element for the solid.
5995
</div>
5996
5997
==4.4 Coupling with the mixed Velocity-Pressure formulation for the solid==
5998
5999
The coupling of the fluid stabilized VP formulation with the VP-element is performed similarly to the V-element. In particular, the solution scheme of the linear momentum equations does not change, because also in this case the degrees of freedom of the  velocity are not duplicated. Hence the fluid velocity coincides with the solid velocity.
6000
6001
The main differences concern the pressure solution. The solid and the fluid pressures are two different degrees of freedom. As a consequence, the continuity equations are solved separately for the fluid and the solid. For the solid, both the stabilized and the not stabilized equations,  Eq.([[#eq-86|86]]) and  Eq.([[#eq-112|112]]) respectively, can be solved. Depending on the compressibility of the solid bodies one may choose one of the two schemes.
6002
6003
For the fluid counterpart, matrix <math display="inline">K^{\kappa }</math> (Eq.([[#eq-220|220]])) is computed using the pseudo bulk modulus <math display="inline">\kappa _p</math> and its value is predicted using the strategy described in Section [[#3.4.3.2 Optimum value for the pseudo bulk modulus|3.4.3.2]].
6004
6005
In Box 13, the solution scheme for a FSI problem solved using the VP-element or the VPS-element for the solid is given for a time interval <math display="inline">{}</math>. Once again, all the matrices and vectors that appear Box 13 refer only to the contributions of the interface nodes and they have already been defined in Box 6 for the VP-element, Box 11 for the VPS-element and Box 9 for the stabilized VP formulation for quasi-incompressible Newtonian fluids.
6006
6007
For the nodes that do not belong to the interface, the schemes described in Box 5 and 8 still hold.
6008
6009
<div class="center" style="font-size: 75%;">
6010
[[File:Draft_Samper_722607179_6066_Box13.png|550px]]
6011
6012
'''Box 13'''. Iterative solution scheme for FSI problem solved with the VP-element or/and the VPS-element for the solid.
6013
</div>
6014
6015
==4.5 Numerical examples==
6016
6017
''Falling of a cylinder in a viscous fluid''
6018
6019
This example is the two-dimensional abstraction of the moving of a circular cylinder between two parallel walls. The cylinder moves  perpendicularly  to its axis due to the gravity force increasing the falling velocity until an asymptotic value.
6020
6021
The distance from the rigid walls and the axis of the cylinder is <math display="inline">l=0.02m</math>. The radius of the circle is <math display="inline">a=0.0025m</math>.   The geometry of the problem and the material data are given in Figure [[#img-109a|109a]] and in Table 4.1. 
6022
6023
<div id='img-109a'></div>
6024
<div id='img-109'></div>
6025
6026
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 36%;max-width: 100%;"
6027
|-
6028
|[[Image:draft_Samper_722607179-cylinderInputB.png|150px]]
6029
|- style="text-align: center; font-size: 75%;"
6030
| colspan="1" | '''Figure 109:''' Falling of a cylinder in a viscous fluid. Initial geometry.
6031
|}
6032
6033
6034
<div class="center" style="font-size: 75%;">
6035
'''Table 4.1.''' Falling of a cylinder in a viscous fluid. Problem data.</div>
6036
6037
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
6038
|- 
6039
| colspan="2" | '''Geometric data'''
6040
|-
6041
| style="text-align:center;"|''l''
6042
| style="text-align:center;"|<math>0.02 </math> m
6043
|-
6044
| style="text-align:center;"|<math>a </math>
6045
| style="text-align:center;"|<math>0.0025 </math> m
6046
|-
6047
| style="text-align:center;"|<math>g</math>
6048
| style="text-align:center;"|<math>9.81~m/s^2 </math> 
6049
|- 
6050
| colspan="2" | '''Fluid data'''
6051
|-
6052
| style="text-align:center;"|Density
6053
| style="text-align:center;"|<math>1.0 \times 10^3 \hbox{ kg/m}^3 </math> 
6054
|-
6055
| style="text-align:center;"|Viscosity
6056
| style="text-align:center;"|<math>0.1 </math> Pa.s
6057
|- 
6058
| colspan="2" | '''Solid data'''
6059
|-
6060
| style="text-align:center;"|Density
6061
| style="text-align:center;"|<math>1.2 \times 10^3 \hbox{ kg/m}^3 </math> 
6062
|-
6063
| style="text-align:center;"|Viscosity
6064
| style="text-align:center;"|<math>10^7 </math> GPa
6065
|-
6066
| style="text-align:center;"|Poisson ratio
6067
| style="text-align:center;"|<math>0.35 </math> 
6068
|}
6069
6070
6071
The same problem has been studied in many other works  [47,127]. The numerical solution can be also compared to the analytical study of the motion of a rigid cylinder with constant velocity <math display="inline">U</math> between two parallel plane walls. For this problem, the analytical solution is obtained  studying the creeping motion equations using stream functions [52]. Considering stick conditions on the walls, for a unit of length of the cylinder the drag force in stationary conditions is
6072
6073
<span id="eq-248"></span>
6074
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6075
|-
6076
| 
6077
{| style="text-align: left; margin:auto;" 
6078
|-
6079
| style="text-align: center;" | <math>F = \frac{4 \pi \mu U}{ln(l/a)-0.9157+1.7244(a/l)^2-1.7302(a/l)^4}     </math>
6080
|}
6081
| style="width: 5px;text-align: right;" | (248)
6082
|}
6083
6084
where <math display="inline">U</math> is the velocity of the rigid cylinder.
6085
6086
This relation for the drag force holds also for the stationary conditions of the problem studied in this section. From the equilibrium of the drag force with the Archimedes' force yields
6087
6088
<span id="eq-249"></span>
6089
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6090
|-
6091
| 
6092
{| style="text-align: left; margin:auto;" 
6093
|-
6094
| style="text-align: center;" | <math>F = \Delta \rho g \pi a^2     </math>
6095
|}
6096
| style="width: 5px;text-align: right;" | (249)
6097
|}
6098
6099
where <math display="inline">\Delta \rho </math> is the difference between the density of the two materials involved.  Combining Eq.([[#eq-248|248]]) and Eq.([[#eq-249|249]]), the terminal velocity of a rigid circular cylinder with radius <math display="inline">a</math> is
6100
6101
<span id="eq-250"></span>
6102
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6103
|-
6104
| 
6105
{| style="text-align: left; margin:auto;" 
6106
|-
6107
| style="text-align: center;" | <math>U_{max} = \frac{ln(l/a)-0.9157+1.7244(a/l)^2-1.7302(a/l)^4}{4 \mu }\Delta \rho g  a^2     </math>
6108
|}
6109
| style="width: 5px;text-align: right;" | (250)
6110
|}
6111
6112
For this problem this relation gives
6113
6114
<span id="eq-251"></span>
6115
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6116
|-
6117
| 
6118
{| style="text-align: left; margin:auto;" 
6119
|-
6120
| style="text-align: center;" | <math>U_{max} = 0.0365 m/s      </math>
6121
|}
6122
| style="width: 5px;text-align: right;" | (251)
6123
|}
6124
6125
This value for the terminal velocity is near to the asymptotic ones obtained by the mentioned works [47,127].
6126
6127
Numerical studies show that this expression holds only for a tight range of values of the viscosity and, specifically, is not suitable for a fluid with small viscosity.
6128
6129
In this work, the solid has been modeled not as a rigid body but with a hypoelastic model, and using a high value for the Young modulus. The VP formulation has been used for the solid. This means that the example belongs properly to the class of the FSI problems. The problem has been solved for both  stick and slip conditions on the vertical walls of the container. Due to the dominant role of the viscosity, the solution of the stick problem does not tend to the solution of the slip problem.
6130
6131
<div id='img-110'></div>
6132
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6133
|-
6134
|[[Image:draft_Samper_722607179-falling70slip0375.png|400px]]
6135
|[[Image:draft_Samper_722607179-falling70slip0675.png|400px]]
6136
|[[Image:draft_Samper_722607179-falling70slip1000.png|400px]]
6137
|- style="text-align: center; font-size: 75%;"
6138
| (a) t = 0.375 s
6139
| (b) t = 0.675 s
6140
| (c) t = 1.000 s
6141
|- style="text-align: center; font-size: 75%;"
6142
| colspan="3" | '''Figure 110:''' Falling of a cylinder in a viscous fluid. Snapshots of the cylinder motion at different instants of the 2D simulation (slip conditions on the boundaries). Velocity contours are depicted over the solid and fluid domains.
6143
|}
6144
6145
In Figures [[#img-110|110]] and  [[#img-111|111]] some snapshots of the simulation with the velocity contours are given.  Figure [[#img-110|110]]  refers to the slip case,  Figure [[#img-111|111]] to the stick one.  <div id='img-111'></div>
6146
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6147
|-
6148
|[[Image:draft_Samper_722607179-falling70stick0375.png|400px]]
6149
|[[Image:draft_Samper_722607179-falling70stick0675.png|400px]]
6150
| colspan="2"|[[Image:draft_Samper_722607179-falling70stick1000.png|400px]]
6151
|- style="text-align: center; font-size: 75%;"
6152
| (a) t = 0.375 s
6153
| (b) t = 0.675 s
6154
| (c) t = 1.000 s
6155
|- style="text-align: center; font-size: 75%;"
6156
| colspan="3" | '''Figure 111:''' Falling of a cylinder in a viscous fluid. Snapshots of the cylinder motion at different instants of the 2D simulation (stick conditions on the boundaries). Velocity contours are depicted over the solid and fluid domains.
6157
|}
6158
6159
The numerical results plotted in Figures  [[#img-110|110]] and  [[#img-111|111]] have been obtained with a mesh with average size <math display="inline">0.0007m</math>. The pictures show the importance of the boundary conditions for this case. It is evident that the velocity fields obtained for the slip and the stick cases, as well the terminal velocity of the cylinder, are significantly different.
6160
6161
For the same mesh, the resulting pressure field for the slip and stick cases is illustrated in Figure [[#img-112|112]]. The pictures show that the perturbation caused by the motion of the cylinder is almost imperceptible and there are not significant differences between the slip and stick cases. 
6162
6163
<div id='img-112'></div>
6164
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6165
|-
6166
|[[Image:draft_Samper_722607179-falling70slip1000pressure.png|400px|slip conditions, t=1 s]]
6167
|[[Image:draft_Samper_722607179-falling70stickpressure.png|400px]]
6168
|- style="text-align: center; font-size: 75%;"
6169
| (a) slip conditions, t=1 s
6170
| (b) stick conditions, t=1 s
6171
|- style="text-align: center; font-size: 75%;"
6172
| colspan="2" | '''Figure 112:''' Falling of a cylinder in a viscous fluid. Pressure field obtained for the slip and stick cases.
6173
|}
6174
6175
In the graph of Figure [[#img-113|113]] the time evolution of the vertical velocity of the cylinder obtained with the finest meshes  (average size=<math display="inline">0.0004m</math>) are given for both the slip and  stick cases. <div id='img-113'></div>
6176
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
6177
|-
6178
|[[Image:draft_Samper_722607179-CylinderComparison.png|420px|Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder. Results for the slip and the stick cases.]]
6179
|- style="text-align: center; font-size: 75%;"
6180
| colspan="1" | '''Figure 113:''' Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder. Results for the slip and the stick cases.
6181
|}
6182
6183
The terminal velocities of the cylinder obtained with slip and  stick conditions are <math display="inline">0.0377 m/s</math> and <math display="inline">0.0336 m/s</math>, respectively.
6184
6185
For this example the transmission conditions between the solid and the fluid domain have been  monitored. The curves of Figure [[#img-114|114]] represent the time evolution of the Neumann condition in the X-direction (horizontal) at the points <math display="inline">A, B, C</math>   located at the boundary of the cylinder and depicted in  Figure [[#img-109a|109a]]. Specifically, the value plotted in the curves is the mean value of the X-component of  vector <math display="inline">\sigma n</math> (<math display="inline">\sigma _{xx} n_x+\tau _{xy} n_y</math>) computed for the fluid and the solid elements at the points <math display="inline">A, B, C</math> of   Figure [[#img-109a|109a]]. <div id='img-114'></div>
6186
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
6187
|-
6188
|[[Image:draft_Samper_722607179-NeumannBound.png|450px|Falling of a cylinder in a viscous fluid. Time evolution of the X-component of σₓₓ nₓ+σ<sub>xy</sub> n<sub>y</sub> computed at the point A, B, C of   Figure [[#img-109a|109a]].]]
6189
|- style="text-align: center; font-size: 75%;"
6190
| colspan="1" | '''Figure 114:''' Falling of a cylinder in a viscous fluid. Time evolution of the X-component of <math>\sigma _{xx} n_x+\sigma _{xy} n_y</math> computed at the point <math>A, B, C</math> of   Figure [[#img-109a|109a]].
6191
|}
6192
The graph shows that the transmission condition is guaranteed during all the analysis.
6193
6194
<div id='img-115'></div>
6195
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6196
|-
6197
|[[Image:draft_Samper_722607179-fallingCoarseMesh.png|400px|Mesh size= 0.0015m]]
6198
|[[Image:draft_Samper_722607179-fallingFineMesh.png|400px]]
6199
|- style="text-align: center; font-size: 75%;"
6200
| (a) Mesh size= 0.0015m
6201
| (b) Mesh size= 0.0004m
6202
|- style="text-align: center; font-size: 75%;"
6203
| colspan="2" | '''Figure 115:''' Falling of a cylinder in a viscous fluid. Coarsest (average mesh size=0.0015) and finest (average mesh size=0.0004) meshes used for the example depicted on the solid domain.
6204
|}
6205
6206
A convergence study was also performed. In Figure [[#img-115|115]] the coarsest (average mesh size=0.0015m) and the finest (average mesh size=0.0004m) meshes used for the example are depicted on the solid body. The whole domain has been discretized using 2983 and 42873 3-noded triangular elements, respectively.
6207
6208
The graph of Figure [[#img-116|116]] shows the time evolution of the vertical velocity obtained with five of the meshes tested  for the slip case. <div id='img-116'></div>
6209
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
6210
|-
6211
|[[Image:draft_Samper_722607179-CylinderSlipResultsSmall.png|390px|Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder for five different meshes and slip conditions on the walls.]]
6212
|- style="text-align: center; font-size: 75%;"
6213
| colspan="1" | '''Figure 116:''' Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder for five different meshes and slip conditions on the walls.
6214
|}
6215
6216
In Figure [[#img-117|117]] the results obtained with the same meshes assuming stick conditions on the walls are given. <div id='img-117'></div>
6217
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
6218
|-
6219
|[[Image:draft_Samper_722607179-CylinderStickResultSmall.png|390px|Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder for five different meshes and stick conditions on the walls.]]
6220
|- style="text-align: center; font-size: 75%;"
6221
| colspan="1" | '''Figure 117:''' Falling of a cylinder in a viscous fluid. Time evolution of the vertical velocity of the cylinder for five different meshes and stick conditions on the walls.
6222
|}
6223
6224
The convergence graphs for both the stick and slip conditions are plotted in Figure [[#img-118|118]]. Both cases show the same convergence rate. <div id='img-118'></div>
6225
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 62%;max-width: 100%;"
6226
|-
6227
|[[Image:draft_Samper_722607179-CylinderComparisonConv.png|372px|Falling of a cylinder in a viscous fluid. Convergence curves for the slip and stick cases.]]
6228
|- style="text-align: center; font-size: 75%;"
6229
| colspan="1" | '''Figure 118:''' Falling of a cylinder in a viscous fluid. Convergence curves for the slip and stick cases.
6230
|}
6231
6232
The problem has been solved also for a quasi-incompressible solid using the VPS-element. For this case, a Poisson ratio of 0.4999  and the same Young modulus used for the problem solved with the VP-element, have been considered. In Figure [[#img-119|119]] the velocity and the pressure fields for the solid and the fluid computed at <math display="inline">t=1s</math> for stick conditions on the walls and using a mean mesh size of <math display="inline">0.007m</math> are given.
6233
6234
In the graph of Figure [[#img-120|120]] the time evolution of the vertical velocity obtained with the VPS-element for <math display="inline">\nu=0.4999</math> is compared with the solution obtained with the VP-element for <math display="inline">\nu=0.35</math> and  the same average mesh size and boundary conditions.
6235
6236
As expected, the solutions are almost the same and the vicinity to the incompressible limit does not compromise the quality of the results.
6237
6238
<div id='img-119'></div>
6239
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6240
|-
6241
|[[Image:draft_Samper_722607179-stick70incVelocity.png|400px|Velocity field]]
6242
|[[Image:draft_Samper_722607179-stick70incFluidPres.png|400px|Solid Cauchy stress]]
6243
| colspan="2"|[[Image:draft_Samper_722607179-stick70incCauchy.png|400px]]
6244
|- style="text-align: center; font-size: 75%;"
6245
| (a) Velocity field
6246
| (b) Fluid pressure
6247
| (c) Solid Cauchy stress
6248
|- style="text-align: center; font-size: 75%;"
6249
| colspan="3" | '''Figure 119:''' Falling of a quasi incompressible cylinder in a viscous fluid.  Velocity field, fluid pressure and solid Cauchy stress (YY component) at <math>t=1s</math> for stick conditions on the walls and a mean mesh size of <math>0.007m</math>.
6250
|}
6251
6252
<div id='img-120'></div>
6253
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
6254
|-
6255
|[[Image:draft_Samper_722607179-CylinderInc.png|420px|Falling of a cylinder in a viscous fluid. Solutions obtained using for the solid the VP-element (ν=0.35) and the VPS-element (ν=0.4999).]]
6256
|- style="text-align: center; font-size: 75%;"
6257
| colspan="1" | '''Figure 120:''' Falling of a cylinder in a viscous fluid. Solutions obtained using for the solid the VP-element (<math>\nu=0.35</math>) and the VPS-element (<math>\nu=0.4999</math>).
6258
|}
6259
6260
''Water entry of a nylon sphere''
6261
6262
The problem was presented by Aristoff <math display="inline">{et al.}</math> in [4]. In the mentioned work the experimental results of the water entry of spheres of different materials are studied.
6263
6264
In this section, the case of a nylon sphere is analyzed. The numerical results given by the Unified formulation (with the VP-element for the solid) are compared to the results of the laboratory test. The sphere impacts the water in the tank with a vertical velocity of 2.17 <math display="inline">{m/s}</math> and its diameter is 2.54 cm.  The density of nylon is 1140 <math display="inline">{kg/m^3}</math> and the Young modulus and Poisson ratio are 3 <math display="inline">{GPa}</math> and 0.2, respectively. The water was modeled considering a density of 1000 <math display="inline">{kg/m^3}</math>, a dynamic viscosity 0.00089 <math display="inline">{Pa \cdot s}</math> and a bulk modulus 2.15 <math display="inline">{GPa}</math>. In order to simulate  this problem correctly, a very fine mesh is necessary. For this reason the whole domain was discretized with 1059924 tetrahedra. The time step used for the analysis is <math display="inline">{\Delta t =10^{-4} s}</math>.
6265
6266
In Figure [[#img-121|121]] the numerical results are compared to the experimental ones [4]. The comparison shows the good agreement of the results obtained with the PFEM Unified formulation  with the laboratory values and its capability to simulate the complex phenomena such as the creation of the void (the air has not been taken into account) within the fluid domain.
6267
6268
<div id='img-121'></div>
6269
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6270
|-
6271
|[[Image:draft_Samper_722607179-aristoffReal1.png|400px|]]
6272
|-
6273
|[[Image:draft_Samper_722607179-aristoffNumerical1.png|400px|]]
6274
|-
6275
|[[Image:draft_Samper_722607179-aristoffReal2.png|400px|]]
6276
|-
6277
|[[Image:draft_Samper_722607179-aristoffNumerical2.png|400px|Water entry of a nylon sphere: comparison between experimental and numerical results.]]
6278
|- style="text-align: center; font-size: 75%;"
6279
| colspan="1" | '''Figure 121:''' Water entry of a nylon sphere: comparison between experimental and numerical results.
6280
|}
6281
6282
''Filling of an elastic container with a viscous fluid''
6283
6284
This example is inspired from a similar problem presented in [31]. A volume of a viscous fluid drops from a rigid container over a thin and highly deformable membrane. The impact of the fluid mass causes an initial huge stretching of the  structure and its subsequent oscillations. A hypoelastic law is used for the material of the structure.  The problem was solved in 2D for two different values of the fluid viscosity, namely 50 and 100 <math display="inline">Pa \cdot s</math>, and with both the V and VP elements. The purpose was to compare the formulations and to show that both solid elements can be used for the modeling of  standard elastic solids in FSI problems. The initial geometry of the problem is given in Figure [[#img-122a|122a]] and the material data are given in Table [[#img-122a|4.2]]. 
6285
6286
<div id='img-122a'></div>
6287
<div id='img-122b'></div>
6288
<div id='img-122'></div>
6289
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 32%;max-width: 100%;"
6290
|-
6291
|[[Image:draft_Samper_722607179-elasticContNoMesh.png|200px|Filling of an elastic container with a viscous fluid. Initial geometry.]]
6292
|- style="text-align: center; font-size: 75%;"
6293
| colspan="1" | '''Figure 122:''' Filling of an elastic container with a viscous fluid. Initial geometry.
6294
|}
6295
6296
<div class="center" style="font-size: 75%;">
6297
'''Table 4.2.''' Filling of an elastic container with a viscous fluid. Problem data.</div>
6298
6299
{| class="wikitable" style="text-align: center; margin: 1em auto;font-size:85%;"
6300
|- 
6301
| colspan="2" | '''Geometry data'''
6302
|-
6303
| style="text-align:center;"|''h''
6304
| style="text-align:center;"|<math>2.5 </math> 
6305
|-
6306
| style="text-align:center;"|<math>H </math>
6307
| style="text-align:center;"|<math>3.75 </math> 
6308
|-
6309
| style="text-align:center;"|<math>R</math>
6310
| style="text-align:center;"|<math>2.25 </math> 
6311
|-
6312
| style="text-align:center;"|<math>b</math>
6313
| style="text-align:center;"|<math>1.3 </math> 
6314
|-
6315
| style="text-align:center;"|<math>B</math>
6316
| style="text-align:center;"|<math>4.8714 </math> 
6317
|-
6318
| style="text-align:center;"|<math>s</math>
6319
| style="text-align:center;"|<math>0.2 </math> 
6320
|- 
6321
| colspan="2" | '''Fluid data'''
6322
|-
6323
| style="text-align:center;"|Density
6324
| style="text-align:center;"|<math>1000 \hbox{ kg/m}^3 </math> 
6325
|-
6326
| style="text-align:center;"|Viscosity
6327
| style="text-align:center;"|<math>50,100 </math> Pa.s
6328
|- 
6329
| colspan="2" | '''Solid data'''
6330
|-
6331
| style="text-align:center;"|Density
6332
| style="text-align:center;"|<math>20  \hbox{ kg/m}^3 </math> 
6333
|-
6334
| style="text-align:center;"|Young modulus
6335
| style="text-align:center;"|<math>2.1 \times 10^7 </math> GPa
6336
|-
6337
| style="text-align:center;"|Poisson ratio
6338
| style="text-align:center;"|<math>0.3 </math> 
6339
|}
6340
6341
6342
In the graph of Figure [[#img-123|123]] the results for the less viscous case (<math display="inline">\mu </math>=50 Pa<math display="inline">\cdot </math>s) obtained using the V and the VP elements for the solid are given.  The comparison is performed for the vertical displacement of the lowest point of the elastic structure. The curves are almost coincident and only after <math display="inline">4.5 s</math> of simulation some slight differences appear.
6343
6344
For the same problem, some representative snapshots are collected in Figure [[#img-124|124]]. The pressure contours are depicted over the solid domain, while over the fluid one the mesh is plotted. The numerical results  refer to the simulation  using the VP-element for the solid.
6345
6346
<div id='img-123'></div>
6347
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 59%;max-width: 100%;"
6348
|-
6349
|[[Image:draft_Samper_722607179-elasticContainerComparison.png|354px|Filling of an elastic container with a viscous fluid (μ=50 Pa ⋅s). Vertical displacement of the bottom of the elastic container obtained using the V and the VP elements for the solid domain.]]
6350
|- style="text-align: center; font-size: 75%;"
6351
| colspan="1" | '''Figure 123:''' Filling of an elastic container with a viscous fluid (<math>\mu=50 Pa \cdot s</math>). Vertical displacement of the bottom of the elastic container obtained using the V and the VP elements for the solid domain.
6352
|}
6353
6354
6355
<div id='img-124'></div>
6356
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6357
|-
6358
|[[Image:draft_Samper_722607179-elasticContainer0920.png|400px|t = 0.920 s]]
6359
|[[Image:draft_Samper_722607179-elasticContainer1045.png|400px|t = 1.545 s]]
6360
|[[Image:draft_Samper_722607179-elasticContainer1545.png|400px|t = 2.670 s]]
6361
|- style="text-align: center; font-size: 75%;"
6362
| (a) <math>t = 0.920 s</math>
6363
| (b) <math>t = 1.045 s</math>
6364
| (c) <math>t = 1.545 s</math>
6365
|-
6366
|[[Image:draft_Samper_722607179-elasticContainer2670.png|400px|t = 3.170 s]]
6367
|[[Image:draft_Samper_722607179-elasticContainer3170.png|400px|t = 7.320 s]]
6368
|[[Image:draft_Samper_722607179-elasticContainer7320.png|400px]]
6369
|- style="text-align: center; font-size: 75%;"
6370
| (d) <math>t = 2.670 s</math>
6371
| (e) <math>t = 3.170 s</math>
6372
| (f) <math>t = 7.320 s</math>
6373
|- style="text-align: center; font-size: 75%;"
6374
| colspan="2" | '''Figure 124:''' Filling of an elastic container with  a viscous fluid (<math>\mu=50 Pa \cdot s</math>). Snapshots at different instants of the 2D simulation. Pressure contours  depicted  over the solid domain.
6375
|}
6376
6377
In Figure [[#img-125|125]]   the snapshots of the most viscous case (<math display="inline">\mu </math>=100 Pa<math display="inline">\cdot </math>s)  are given  for the same time instants of Figure  [[#img-124|124]]. The numerical results refer to the solution obtained using the VP-element for the solid domain.
6378
6379
<div id='img-125'></div>
6380
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6381
|-
6382
|[[Image:draft_Samper_722607179-elasticContainer0920V.png|400px|t = 0.920 st = 1.045 s]]
6383
|[[Image:draft_Samper_722607179-elasticContainer1045V.png|400px|t = 1.545 s]]
6384
|[[Image:draft_Samper_722607179-elasticContainer1545V.png|400px|t = 2.670 s]]
6385
|- style="text-align: center; font-size: 75%;"
6386
| (a) <math>t = 0.920 s</math>
6387
| (b) <math>t = 1.045 s</math>
6388
| (c) <math>t = 1.545 s</math>
6389
|-
6390
|[[Image:draft_Samper_722607179-elasticContainer2670V.png|400px|t = 3.170 s]]
6391
|[[Image:draft_Samper_722607179-elasticContainer3170V.png|400px|t = 7.320 s]]
6392
|[[Image:draft_Samper_722607179-elasticContainer7320V.png|400px]]
6393
|- style="text-align: center; font-size: 75%;"
6394
| (d) <math>t = 2.670 s</math>
6395
| (e) <math>t = 3.170 s</math>
6396
| (f) <math>t = 7.320 s</math>
6397
|- style="text-align: center; font-size: 75%;"
6398
| <math>t = 7.320 s</math>
6399
|- style="text-align: center; font-size: 75%;"
6400
| colspan="2" | '''Figure 125:''' Filling of an elastic container with  a viscous fluid (<math>\mu=100 Pa \cdot s</math>). Snapshots at different instants of the 2D simulation. Pressure contours depicted  over the solid domain.
6401
|}
6402
6403
Also for the most viscous case, the results obtained using  the velocity and the mixed velocity-pressure formulations for the solid are compared analyzing the time evolution of the vertical displacement at the bottom of the elastic structure.  The two curves  of  Figure [[#img-126|126]] represent the solutions obtained with the velocity and the mixed  velocity-pressure formulations.
6404
6405
Once again, the differences between the results of the two formulations are very small and this is a further evidence of the validity and flexibility of the proposed Unified formulation.  This example evidenced the possibility to choose for the solid a velocity or a mixed formulation also for solving FSI problems.
6406
6407
<div id='img-126'></div>
6408
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 59%;max-width: 100%;"
6409
|-
6410
|[[Image:draft_Samper_722607179-elasticContainerComparisonVisc.png|354px|Filling of an elastic container with a viscous fluid (μ=100 Pa ⋅s). Vertical displacement of the bottom of the elastic container obtained using the V and the VP elements for the solid domain.]]
6411
|- style="text-align: center; font-size: 75%;"
6412
| colspan="1" | '''Figure 126:''' Filling of an elastic container with a viscous fluid (<math>\mu=100 Pa \cdot s</math>). Vertical displacement of the bottom of the elastic container obtained using the V and the VP elements for the solid domain.
6413
|}
6414
6415
''Collapse of a water column on a deformable membrane''
6416
6417
The problem illustrated in Figure [[#img-127|127]] was introduced by Walhorn <math display="inline">{et al.}</math> [126].
6418
6419
<div id='img-127'></div>
6420
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
6421
|-
6422
|[[Image:draft_Samper_722607179-FSIinput.png|420px|Collapse of a water column on a deformable membrane. Initial geometry and problem data.]]
6423
|- style="text-align: center; font-size: 75%;"
6424
| colspan="1" | '''Figure 127:''' Collapse of a water column on a deformable membrane. Initial geometry and problem data.
6425
|}
6426
6427
The water column collapses by instantaneously removing the vertical wall. This originates the flow of water within the tank, the formation of a jet after the water stream hits the rigid ground, and the subsequent sloshing of the fluid as it impacts a highly deformable elastic membrane. The membrane bends and starts oscillating under the effect of its inertial forces and the impact with the water stream.
6428
6429
The problem has been solved in 2D as in 3D, considering stick conditions for the rigid walls.  In Figure  [[#img-128|128]]  some representative snapshots of the 2D simulation are given. 
6430
6431
<div id='img-128'></div>
6432
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6433
|-
6434
|[[Image:draft_Samper_722607179-FSIDB2D0238B.png|400px|t = 0.238 st = 0.350 s]]
6435
|[[Image:draft_Samper_722607179-FSIDB2D0350B.png|400px|t = 0.526 s]]
6436
|[[Image:draft_Samper_722607179-FSIDB2D0526B.png|400px|t = 0.670 s]]
6437
|- style="text-align: center; font-size: 75%;"
6438
| (a) <math>t = 0.238 s</math>
6439
| (b) t = 0.350 s
6440
| (c) <math>t = 0.526 s</math>
6441
|-
6442
|[[Image:draft_Samper_722607179-FSIDB2D0670B.png|400px|t = 1.500 s]]
6443
|[[Image:draft_Samper_722607179-FSIDB2D1500B.png|400px|t = 2.000s]]
6444
|[[Image:draft_Samper_722607179-FSIDB2D2000B.png|400px]]
6445
|- style="text-align: center; font-size: 75%;"
6446
| (d) <math>t = 0.670 s</math>
6447
| (e) <math>t = 1.500 s</math>
6448
| (f) <math>t = 2.000s</math>
6449
|- style="text-align: center; font-size: 75%;"
6450
| colspan="3" | '''Figure 128:''' Collapse of a water column on a deformable membrane. Snapshots  of the 2D simulation at different instants. The VP element is used for the solid.
6451
|}
6452
6453
The results obtained with the present formulation have been compared to the ones computed in [31,57,63].  In the graph of Figure [[#img-129|129]] the time evolution of the horizontal deflection of the left top corner is illustrated. <div id='img-129'></div>
6454
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
6455
|-
6456
|[[Image:draft_Samper_722607179-fsiDBcomparison.png|450px|Collapse of a water column on a deformable membrane. Horizontal deflection of the left top corner on time. Numerical results obtained with V and VP elements for the solid. Comparison with numerical results obtained in [31,63,126].]]
6457
|- style="text-align: center; font-size: 75%;"
6458
| colspan="1" | '''Figure 129:''' Collapse of a water column on a deformable membrane. Horizontal deflection of the left top corner on time. Numerical results obtained with V and VP elements for the solid. Comparison with numerical results obtained in [31,63,126].
6459
|}
6460
6461
The diagram shows that, for the first part of the analysis, the proposed formulation agrees well with the results reported in the literature. After around <math display="inline">0.5s</math> of simulation, the numerical results of  each formulation starts to diverge, although for all the formulations the membrane oscillates two times around its vertical position before the time instant <math display="inline">t=1s</math>. The first part of the simulation is easier to analyze than the second one because the phenomena to model are less aleatory and the fluid splashes do not affect the results, as it occurs in the second part of the simulation. Furthermore,  the initial deformation of the elastic structure affects highly the rest of the simulation. In fact, a smaller bending of the membrane induces an impact of the water stream at a higher height of the containing wall and with a bigger tangential component of the impact velocity. Consequently, the fluid stream impacts   against the right side of the elastic membrane later and with reduced inertial forces. These considerations have an even greater effect for the 3D simulation.
6462
6463
In the graph of Figure [[#img-130|130]]  the results of the 2D and 3D simulations obtained using the V-element are compared. <div id='img-130'></div>
6464
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
6465
|-
6466
|[[Image:draft_Samper_722607179-fsiDB3DVel.png|420px|Collapse of a water column on a deformable membrane. Horizontal deflection of the left top corner on time. Comparison between 2D and 3D analyses (V-element for the solid part).]]
6467
|- style="text-align: center; font-size: 75%;"
6468
| colspan="1" | '''Figure 130:''' Collapse of a water column on a deformable membrane. Horizontal deflection of the left top corner on time. Comparison between 2D and 3D analyses (V-element for the solid part).
6469
|}
6470
6471
The graph shows that there is a good agreement between the 2D and 3D analyses in the first part of the graph. Then the graphs are  quite different. In particular, in the 3D problem the fluid stream impacts later against the right side of the membrane producing a reduced displacement of the elastic structure. This occurs because, in the 3D simulation, the fluid stream after the impact against the vertical rigid wall (this occurs after around <math display="inline">0.35s</math> of analysis), can move in the direction transversal to the main direction of its motion and this reduces the impact force of the water stream. This behavior is intensified if stick boundary conditions are considered and the mesh is not sufficiently fine.
6472
6473
In Figure [[#img-131|131]] the numerical results of the 3D simulation are given.
6474
6475
<div id='img-131'></div>
6476
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6477
|-
6478
|[[Image:draft_Samper_722607179-FSIDB3B023.png|400px|t = 0.23 st = 0.37 s]]
6479
|[[Image:draft_Samper_722607179-FSIDB3B037.png|400px|t = 0.49 s]]
6480
|- style="text-align: center; font-size: 75%;"
6481
| <math>t = 0.23 s</math>
6482
| <math>t = 0.49 s</math>
6483
|-
6484
|[[Image:draft_Samper_722607179-FSIDB3B049.png|400px|t = 0.89 s]]
6485
|[[Image:draft_Samper_722607179-FSIDB3B089.png|400px|Collapse of a water column on a deformable membrane. Snapshots of the 3D simulation of different instants. The V-element is used for the solid.]]
6486
|- style="text-align: center; font-size: 75%;"
6487
| <math>t = 0.89 s</math>
6488
|- style="text-align: center; font-size: 75%;"
6489
| colspan="2" | '''Figure 131:''' Collapse of a water column on a deformable membrane. Snapshots of the 3D simulation of different instants. The V-element is used for the solid.
6490
|}
6491
6492
==4.6 Summary and conclusions==
6493
6494
In this chapter the Unified Stabilized formulation has been used for the solution of FSI problems. Specifically, the numerical schemes for coupling the stabilized velocity-pressure formulation for quasi-incompressible Newtonian fluids with the V, the VP and the VPS elements for the solid have been presented. It has been shown that, depending on the specific need, one may choose any of these solid elements.
6495
6496
It has been shown that the implementation effort for extending the Unified formulation to the coupled FSI problems is small. Essentially, it consists of assembling adequately the global linear system and exploiting the capability of the PFEM for the detection of the interface.
6497
6498
Several validating examples have been given. The numerical solution obtained with the proposed scheme has been compared to analytical solutions and numerical results of other formulations for free surface FSI problems. Good agreement has been found in all cases. It has also been shown that the method is convergent.
6499
6500
=Chapter 5. Coupled thermal-mechanical formulation=
6501
6502
6503
==5.1 Introduction==
6504
6505
This chapter is devoted to the coupling of the unified formulation with the heat transfer problem. The objective is to solve general problems involving fluids, solids or both of them which behavior is temperature-dependent and where the phase change of the materials is allowed.
6506
6507
There is a growing interest in industry  in having a technology capable to solve coupled thermo-mechanical problems. In fact, industrial problems generally lack of an analytical solution and often the unique way to predict their solution is based on  experience. An alternative is represented by experimental tests. However, these cannot be carried out for all the required problems; for example due to the complexity of the geometry,  and, in many cases, the high cost associated to these. For these reasons, computer-based methods represent often the most reliable alternative for the analysis of coupled thermal-mechanical problems.
6508
6509
Thermo-mechanical problems involving fluid and solids however, represent a challenge for the numerical analysis due to its multi-field nature and high non-linearity. From the computational point of view, there are many complications associated to these problems.
6510
6511
First of all, the coupling with the heat problem increases the non-linearity of the mechanical problem. The solution of the mechanical problem depends on the heat transfer via temperature dependence of the material properties, at the same time, the heat problem depends on the solution of the mechanical problem due to the change of the configuration.
6512
6513
The complexity of the problem increases if  phase change is also considered. This phenomenon not only increases the non-linearity of the problem due to the huge changes of topology that it may induce, but it also requires the implementation of a specific technology for its simulation. Furthermore accounting for phase change may induce problems related to the quality of the mesh. Consider, for example, the melting of a solid body. If a Lagrangian mesh is used, as it generally occurs with solids, the body that  melts undergoes large deformations that may compromise the quality of the mesh and the results of the overall simulation.
6514
6515
Another critical point is the imposition of the boundary conditions for the temperature. For this it is required to track the contours of the domains involved in the simulation for all the duration of the analysis. This task may not  be trivial if the problem involves a complex geometry, severe changes of topology or  phase change.
6516
6517
With the PFEM many of these complexities are overcome thanks to its Lagrangian nature. In fact, as it has already explained in the previous chapters, the boundaries of the bodies are automatically detected by the position of the boundary nodes. Furthermore, if PFEM is also used for modeling the solid dynamics, the drawbacks associated to the large deformations of the solid caused by  heat are overcome through the remeshing which guarantees a fine discretization for all the duration of the analysis. Finally, in the Lagrangian formulation the convective term disappears from the heat transfer equation. For all these reasons, other authors faced this problem  in the past using the PFEM [90,95,107].
6518
6519
In this work, the PFEM is only used for the fluid domain. So in order to overcome the decay of quality of the solid mesh in a melting process, a further technology is required. The algorithm that has been used in this work consists on transforming the solid elements into fluid elements when they reach a threshold value. This means that the part of the solid body that is melting is considered as a fluid. Consequently it benefits from the PFEM technology and it can be remeshed if required. The details of the algorithm will be given in the following sections.
6520
6521
Due to the complexity and the vastness of the issue, in this treatise various assumptions and simplifications have been accepted. This has been in some cases at the expense of the accuracy of the analyses. Nevertheless, the main objective of this part of this work is to show that the PFEM unified formulation for FSI problems can be easily coupled to the heat problem in order to solve simulations where also the temperature affects. The principal assumptions of this work are described in  the following.
6522
6523
In all the simulations, the air has not been taken into account, so heat transmission only occurs via the contact between solid and liquid bodies, or through the contours where the boundary conditions for the temperature have been imposed. Furthermore, in the phase change modeling, there is not a transition where a multi-fluid analysis is employed. The material that changes its state takes instantaneously the  properties of the other material involved in the analysis, without considering  a layer formed by a transition material. Note that these simplifications have been assumed just for convenience and they are not required by the formulation itself. With an additional implementation work both assumptions can be avoided. A proof of this are the successful attempts made in previous works where the PFEM was successfully used for modeling multi-fluids [64].
6524
6525
In order to guarantee a strong thermal-mechanical coupling, the heat problem is solved within the same iteration loop used for solving the mechanical problem. This strategy increases the non-linearity of the mechanical problem because of its dependence on the temperature.  A simpler way to solve the global problem is to consider a constant temperature during the non-linear iterations of the mechanical problem and to solve the heat problem only once the mechanical problem has converged. This approach leads to a lower computational cost and it reduces the non-linearity of the mechanical problem. However, the resulting coupling is weaker. Both strong and weak coupling strategies belong to the staggered solution schemes because the mechanical and the heat problem are solved in different linear systems. Conversely, in monolithic approaches both problems are solved simultaneously. On the one hand, this scheme enables the stability and the convergence of the whole coupled problem, on the other, it leads to longer and worse conditioned algebraic systems. However,  in this work this scheme has not been chosen essentially for another reason. Monolithic schemes require modifications also in the formulation for the mechanical solution. This occurs because in the assembly of the linear system also the degrees of freedom of the temperature have to be taken into account. Instead, in staggered schemes,  the FSI formulation remains the same because the mechanical and the heat transfer problems are two different blocks and they may be implemented separately. This, apart from reducing the implementation effort, gives us the possibility to test the unified formulation in its original version.
6526
6527
This chapter is split into the following sections. First, the governing equations of the heat are introduced and discretized using FEM. Also the scheme for the solution of a general time step is given. Then the thermal coupling algorithm is described and validated with three numerical examples. After that, the strategy for modeling  phase change is explained. The chapter ends with an example where the thermal-mechanical coupling technique is applied.
6528
6529
==5.2 Heat problem==
6530
6531
The heat  transfer problem in a Lagrangian setting is governed by the following differential equation
6532
6533
<span id="eq-252"></span>
6534
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6535
|-
6536
| 
6537
{| style="text-align: left; margin:auto;" 
6538
|-
6539
| style="text-align: center;" | <math>\rho c {\partial T \over \partial t} - {\partial  \over \partial x_i} \left(k {\partial T \over \partial x_i}\right)+ Q =0  \quad i=1,n_s  \quad \hbox{in }\Omega   </math>
6540
|}
6541
| style="width: 5px;text-align: right;" | (252)
6542
|}
6543
6544
where <math display="inline">T</math> is the temperature, <math display="inline">c</math> is the thermal capacity, <math display="inline">k</math> is the heat conductivity and <math display="inline">Q</math> is the heat source.
6545
6546
Notice that the convective term does not appear in Eq.([[#eq-252|252]]).
6547
6548
The temperature and the heat flux at the boundaries are prescribed with the following boundary conditions
6549
6550
<span id="eq-253"></span>
6551
<span id="eq-254"></span>
6552
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6553
|-
6554
| 
6555
{| style="text-align: left; margin:auto;" 
6556
|-
6557
| style="text-align: center;" | <math>\phi - \phi ^p =0 \quad \hbox{on }\Gamma _\phi </math>
6558
| style="width: 5px;text-align: right;" | (253)
6559
|-
6560
| style="text-align: center;" | <math> k {\partial \phi  \over \partial n} + q_n^p =0 \quad \hbox{on }\Gamma _q  </math>
6561
| style="width: 5px;text-align: right;" | (254)
6562
|}
6563
|}
6564
6565
where <math display="inline">\phi ^p</math> and <math display="inline">q_n^p</math> are the prescribed temperature and the prescribed normal heat flux at the boundaries <math display="inline">\Gamma _\phi </math> and <math display="inline">\Gamma _q</math>, respectively, and <math display="inline">n</math> is the direction normal to the boundary.
6566
6567
The problem is completed by the initial condition
6568
6569
<span id="eq-255"></span>
6570
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6571
|-
6572
| 
6573
{| style="text-align: left; margin:auto;" 
6574
|-
6575
| style="text-align: center;" | <math>T(t=0)= \bar T \quad \hbox{in }\Omega   </math>
6576
|}
6577
| style="width: 5px;text-align: right;" | (255)
6578
|}
6579
6580
The space for the test functions for the temperature is defined as
6581
6582
<span id="eq-256"></span>
6583
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6584
|-
6585
| 
6586
{| style="text-align: left; margin:auto;" 
6587
|-
6588
| style="text-align: center;" | <math>\hat w_ i \in {U_0}, \quad \quad {U_0}=\{ \hat w_ i | \hat w_ i  \in C^0,  \hat w_ i=0   on \Gamma _\phi \}   </math>
6589
|}
6590
| style="width: 5px;text-align: right;" | (256)
6591
|}
6592
6593
Multiplying  Eq.([[#eq-252|252]]) by the test functions and integrating over the updated configuration domain, the following global integral form is obtained
6594
6595
<span id="eq-257"></span>
6596
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6597
|-
6598
| 
6599
{| style="text-align: left; margin:auto;" 
6600
|-
6601
| style="text-align: center;" | <math>\int _\Omega  \hat w_ i   \left[ \rho c {\partial T \over \partial t} - {\partial  \over \partial x_i} \left(k {\partial T \over \partial x_i}\right)+ Q \right]d\Omega =0 </math>
6602
|}
6603
| style="width: 5px;text-align: right;" | (257)
6604
|}
6605
6606
Integrating Eq.([[#eq-257|257]]) by parts, using Eq.([[#eq-254|254]]) and neglecting the space changes of the conductivity, the following weak variational form of the heat problem is obtained
6607
6608
<span id="eq-258"></span>
6609
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6610
|-
6611
| 
6612
{| style="text-align: left; margin:auto;" 
6613
|-
6614
| style="text-align: center;" | <math>\int _\Omega \hat w \rho c {\partial T \over \partial t} d\Omega + \int _\Omega {\partial \hat w  \over \partial x_i}  k {\partial T \over \partial x_i}d\Omega +\int _\Omega \hat w Q d\Omega + \int _{\Gamma _q} \hat w q_n^p d\Gamma =0 </math>
6615
|}
6616
| style="width: 5px;text-align: right;" | (258)
6617
|}
6618
6619
===5.2.1 FEM discretization and solution for a time step===
6620
6621
For the heat problem, the same procedure for the spatial discretization of the mechanical one is used. Hence the analysis domain is discretized into finite elements with <math display="inline">n</math> nodes in the standard manner, leading to a mesh with <math display="inline">N_e</math> elements and <math display="inline">N</math> nodes. For 2D problems 3-noded linear triangles are used, while 3D domains are discretized using 4-noded tetrahedra. The temperature is interpolated over the mesh in terms of its nodal values, in the same manner as for the velocities and the pressure, using the  global linear shape functions <math display="inline">N_j</math> spanning over the elements sharing node <math display="inline">j</math> (<math display="inline">j=1,N</math>). In matrix form,
6622
6623
<span id="eq-259"></span>
6624
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6625
|-
6626
| 
6627
{| style="text-align: left; margin:auto;" 
6628
|-
6629
| style="text-align: center;" | <math>\quad T = {N}_T \bar {T}  </math>
6630
|}
6631
| style="width: 5px;text-align: right;" | (259)
6632
|}
6633
6634
where <math display="inline">{N}_T = [{N}_1, { N}_2,\cdots , {N}_N ]</math> and
6635
6636
<span id="eq-260"></span>
6637
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6638
|-
6639
| 
6640
{| style="text-align: left; margin:auto;" 
6641
|-
6642
| style="text-align: center;" | <math>\begin{array}{c}{\bar T}= \left\{\begin{matrix}\bar{T}_1\\\bar{T}_2\\\vdots \\ \bar{T}_N\end{matrix}  \right\} \end{array} </math>
6643
|}
6644
| style="width: 5px;text-align: right;" | (260)
6645
|}
6646
6647
In Eq.([[#eq-259|259]]) vector <math display="inline">\bar {T}</math>  contains the nodal  temperatures for the whole mesh. Substituting Eq.([[#eq-259|259]]) into Eq.([[#eq-258|258]]) and choosing a Galerkin formulation with <math display="inline"> \hat w_i =N_i</math> leads to the following algebraic equation
6648
6649
<span id="eq-261"></span>
6650
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6651
|-
6652
| 
6653
{| style="text-align: left; margin:auto;" 
6654
|-
6655
| style="text-align: center;" | <math>{C} {\dot{\bar{T}}}+\hat {L} \bar{T} + {f}_T={0}  </math>
6656
|}
6657
| style="width: 5px;text-align: right;" | (261)
6658
|}
6659
6660
The matrices and vectors in Eq.([[#eq-261|261]]) are assembled from the element contributions given in Box 5.1.
6661
6662
<div class="center" style="font-size: 75%;">
6663
[[File:Draft_Samper_722607179_3738_Box 5-1.png]]
6664
6665
'''Box 5.1'''. Element form of the matrices and vectors in Eq.(261)
6666
</div>
6667
6668
Eq.([[#eq-261|261]]) is solved using  a standard forward Euler scheme. For the <math display="inline">i-th</math> iteration within a time interval <math display="inline">[n,n+1]</math> the following linear system is solved
6669
6670
<span id="eq-262"></span>
6671
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6672
|-
6673
| 
6674
{| style="text-align: left; margin:auto;" 
6675
|-
6676
| style="text-align: center;" | <math>\left[\frac{1}{\Delta t}{C} + \hat{L}   \right]\Delta \bar{T}=- {}^{n+1} \bar{r}^i_T \quad ,\quad  {r}_T= {C} {\dot{\bar{T}}} + \hat{L} \bar{T} + {f}_T </math>
6677
|}
6678
| style="width: 5px;text-align: right;" | (262)
6679
|}
6680
6681
with
6682
6683
<span id="eq-263"></span>
6684
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6685
|-
6686
| 
6687
{| style="text-align: left; margin:auto;" 
6688
|-
6689
| style="text-align: center;" | <math>{}^{n+1}\bar{T}^{i+1}= {}^{n}\bar{T}^i + \Delta \bar{T} </math>
6690
|}
6691
| style="width: 5px;text-align: right;" | (263)
6692
|}
6693
6694
If the following conditions are verified, the next time step is considered.
6695
6696
<span id="eq-264"></span>
6697
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6698
|-
6699
| 
6700
{| style="text-align: left; margin:auto;" 
6701
|-
6702
| style="text-align: center;" | <math>\begin{array}{c}\displaystyle \Vert {}^{n+1}\bar {T}^{i+1} - {}^{n+1}\bar {T}^i\Vert \le e_T  \Vert {}^{n}\bar{T}\Vert  \end{array}   </math>
6703
|}
6704
| style="width: 5px;text-align: right;" | (264)
6705
|}
6706
6707
where <math display="inline">e_T</math> is a prescribed error norm.
6708
6709
==5.3 Thermal coupling==
6710
6711
In a thermo-coupled mechanical problem, the deformation induced by the heat has to be taken into account. The total deformation rate is the sum of the elastic, the plastic and thermal parts. Hence Eq.([[#eq-122|122]]) for a thermal-elastoplastic problem is:
6712
6713
<span id="eq-265"></span>
6714
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6715
|-
6716
| 
6717
{| style="text-align: left; margin:auto;" 
6718
|-
6719
| style="text-align: center;" | <math>{D}={D}_{el} +{D}_{pl} +{D}_{th}   </math>
6720
|}
6721
| style="width: 5px;text-align: right;" | (265)
6722
|}
6723
6724
where the thermal part of the deformation rate tensor <math display="inline">{D_{th}}</math>  is defined as
6725
6726
<span id="eq-266"></span>
6727
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6728
|-
6729
| 
6730
{| style="text-align: left; margin:auto;" 
6731
|-
6732
| style="text-align: center;" | <math>D_{th}= \alpha \dot{T} mm^T  </math>
6733
|}
6734
| style="width: 5px;text-align: right;" | (266)
6735
|}
6736
6737
where <math display="inline">\alpha </math> is the thermal expansion coefficient and <math display="inline">{m}</math><math display="inline">{=[1,1,1,0,0,0]^T}</math>.
6738
6739
In this work, the heat problem is solved after the mechanical problem in the same iteration loop (Figure [[#img-132|132]]). 
6740
6741
<div id='img-132'></div>
6742
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6743
|-
6744
|
6745
[[File:Draft_Samper_722607179_5638_Fig5.1.png|600px]]
6746
|- style="text-align: center; font-size: 75%;"
6747
| colspan="1" | '''Figure 132:''' Inner staggered thermal coupling scheme for a general time step <math>({}^nt,{}^{n+1}t)</math>.
6748
|}
6749
6750
This strategy belongs to the class of staggered schemes because the heat and the mechanical problems are solved in two different linear systems. This method is called '<math display="inline">internal</math> scheme' in order to distinguish it from another staggered scheme that will be presented later.  This strategy increases the non-linearity of the problem because the properties of the material may depend on the temperature. The convergence is tested at the end of each iteration for the velocities, the pressure and the temperature.
6751
6752
The thermal coupling can be also performed through a different staggered scheme, that will be called '<math display="inline">external</math> scheme'. In this case the heat problem is solved after the convergence of the mechanical problem (Figure [[#img-133|133]]).
6753
6754
<div id='img-133'></div>
6755
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6756
|-
6757
|
6758
[[File:Draft_Samper_722607179_3378_Fig5.2.png|600px]]
6759
|- style="text-align: center; font-size: 75%;"
6760
| colspan="1" | '''Figure 133:''' External staggered thermal coupling  scheme for a general time step <math>({}^nt,{}^{n+1}t)</math>.
6761
|}
6762
6763
The external staggered scheme reduces the non-linearity of the mechanical problem because during the non-linear iterations the temperature is considered as a constant. It also gives the possibility to choose different time step increments for the mechanical and the heat problems. On the other hand, the resulting coupling is weaker than for the internal scheme. For guaranteeing a stronger coupling the mechanical and the thermal problems should be solved iteratively within the time step. However, this strategy increases  the computational cost of the analyses. In practice, their duration increases by a factor equal to the number of  the global iterations of the thermal-mechanical problem.
6764
6765
===5.3.1 Numerical examples===
6766
6767
In this section, three numerical examples are presented. First, the heat transfer problem is validated comparing the numerical solution of a heated plate to the analytical one. Next, the sloshing of a fluid in a heated tank is presented. Finally, the heating of three solid objects in a fluid contained in a rectangular tank is studied.
6768
6769
''Heating  of a plate''
6770
6771
The objective of this first example is to verify the implementation of the thermal problem. The problem consists of a square plate at initial temperature T=100 cooled by three of its edges at constant temperature T=0 keeping the other edge insulated. Figure [[#img-134|134]] shows a graphical representation of the problem.
6772
6773
<div id='img-134'></div>
6774
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6775
|-
6776
|[[Image:draft_Samper_722607179-heatedPlateInput.png|300px|Heating of a plate. Initial geometry, thermal boundary and initial conditions.]]
6777
|- style="text-align: center; font-size: 75%;"
6778
| colspan="1" | '''Figure 134:''' Heating of a plate. Initial geometry, thermal boundary and initial conditions.
6779
|}
6780
6781
The thermal conductivity of the plate is <math display="inline">k=1</math>. The length of the plate edges   is <math display="inline">L=1</math>.
6782
6783
The problem can be computed analytically by solving the partial differential equations of the heat problem with the proper initial and boundary conditions. The analytical solution for the temperature field is:
6784
6785
<span id="eq-267"></span>
6786
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6787
|-
6788
| 
6789
{| style="text-align: left; margin:auto;" 
6790
|-
6791
| style="text-align: center;" | <math>T(x,y,t)= \sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{1600}{(2m-1)(2n-1){\pi }^2}\sin \left(\frac{(2m-1)\pi y}{2L} \right) \cdot </math>
6792
|-
6793
| style="text-align: center;" | <math> \cdot \sin \left(\frac{(2n-1)\pi x}{L} \right)exp\left(-  \frac{(2m-1)^2+4(2n-1)^2}{4L^2} {\pi }^2 t \right)   </math>
6794
|}
6795
| style="width: 5px;text-align: right;" | (267)
6796
|}
6797
6798
In the graph of Figure [[#img-135|135]] the temperature evolution with time at the central point of the top edge (with coordinates <math display="inline">(x,y)=(0.5,1)</math>) obtained with the proposed approach is compared to the analytical solution. Apart from the initial instants, the curves are almost identical. <div id='img-135'></div>
6799
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6800
|-
6801
|[[Image:draft_Samper_722607179-heatedPlateComp.png|600px|Heating of a plate. Temperature evolution on time at the point (x,y)=(0.5,1). Analytical and numerical solutions.]]
6802
|- style="text-align: center; font-size: 75%;"
6803
| colspan="1" | '''Figure 135:''' Heating of a plate. Temperature evolution on time at the point <math>(x,y)=(0.5,1)</math>. Analytical and numerical solutions.
6804
|}
6805
6806
''Sloshing  of a fluid in heated tank''
6807
6808
This fluid dynamics problem was presented in [84]. A fluid at initial temperature <math display="inline">T=20\,^{\circ } C</math> oscillates due to the hydrostatic forces induced by its initial position in a rectangular tank heated to a uniform and constant temperature of <math display="inline">T=75\,^{\circ } C</math>. The geometry and the problem data of the 2D simulation are shown in Figure [[#img-136|136]].  <div id='img-136'></div>
6809
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6810
|-
6811
|[[Image:draft_Samper_722607179-ThermalSloshingInput.png|300px|]]
6812
|[[Image:draft_Samper_722607179-ThermalSLData.png|200px|2D sloshing of a fluid in a heated tank. Initial geometry, problem data, thermal boundary and initial conditions.]]
6813
|- style="text-align: center; font-size: 75%;"
6814
| colspan="2" | '''Figure 136:''' 2D sloshing of a fluid in a heated tank. Initial geometry, problem data, thermal boundary and initial conditions.
6815
|}
6816
6817
The fluid domain has been initially discretized with 2828 3-noded triangles. The coupled thermal-fluid dynamics simulation has been run for <math display="inline">100 s</math> using a time step increment of <math display="inline">\Delta t</math>=<math display="inline">0.005 s</math>.
6818
6819
The main purpose of this example is to show the capability of the proposed Lagrangian technique for dealing with thermal coupled problems involving severe changes of topology. For this reason, some simplifications have been accepted. For example, the properties of the fluids do not depend on temperature. Furthermore, with the purpose of reducing the computational time and visualizing better the temperature contours, a very high (and not realistic) thermal conductivity has been used. These assumptions reduce clearly the truthfulness of the problem but, as it has already pointed out, this is not the principal objective of this example.
6820
6821
In Figure [[#img-137|137]] the evolution of the  temperature with time at the points <math display="inline">A, B</math> and <math display="inline">C</math> of  Figure [[#img-136|136]] is plotted. The coordinates of these sampling points are (<math display="inline">m,0.1m</math>), (<math display="inline">m,0.4m</math>) and (<math display="inline">m,0.1m</math>), respectively.  
6822
6823
<div id='img-137'></div>
6824
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
6825
|-
6826
|[[Image:draft_Samper_722607179-ThermalSloshingGraph.png|360px|2D sloshing of a fluid in a heated tank. Time evolution of the temperature at the points A, B and C of  Figure [[#img-136|136]].]]
6827
|- style="text-align: center; font-size: 75%;"
6828
| colspan="1" | '''Figure 137:''' 2D sloshing of a fluid in a heated tank. Time evolution of the temperature at the points <math>A, B</math> and <math>C</math> of  Figure [[#img-136|136]].
6829
|}
6830
6831
The fluid, because of its high thermal conductivity, changes its temperature quickly due to the contact with the hotter tank walls. The heat flux along the free surface has been considered to be null.
6832
6833
Figure [[#img-138|138]] shows some snapshots of the numerical simulation. The temperature contours have been superposed on the fluid domain at different time instants. <div id='img-138'></div>
6834
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 99%;max-width: 100%;"
6835
|-
6836
|[[Image:draft_Samper_722607179-ThermalSloshing.png|594px|2D sloshing of a fluid in a heated tank. Snapshots of fluid geometry at six different time instants. Colours indicate temperature contours.]]
6837
|- style="text-align: center; font-size: 75%;"
6838
| colspan="1" | '''Figure 138:''' 2D sloshing of a fluid in a heated tank. Snapshots of fluid geometry at six different time instants. Colours indicate temperature contours.
6839
|}
6840
6841
Figures [[#img-138|138]]  and [[#img-137|137]]  show that the fluid does not heat uniformly because of the convection effect that is automatically captured by the Lagrangian technique here presented.
6842
6843
''Falling of three objects in a heated tank filled with fluid''
6844
6845
This 2D example is taken from [83]. Three solid objects with the same shape fall from the same height into a tank containing a fluid at rest. For the solid objects a hypoelastic model has been used. The geometry and the problem data, as well the initial thermal conditions, are shown in Figure [[#img-139|139]].  <div id='img-139'></div>
6846
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6847
|-
6848
|[[Image:draft_Samper_722607179-ballsInput2.png|400px|]]
6849
|[[Image:draft_Samper_722607179-ballDataGeo.png|150px|]]
6850
|-
6851
| colspan="2"|[[Image:draft_Samper_722607179-ballDataMat.png|500px|Falling of three objects in a heated tank filled with fluid. Initial geometry, thermal conditions and material properties.]]
6852
|- style="text-align: center; font-size: 75%;"
6853
| colspan="2" | '''Figure 139:''' Falling of three objects in a heated tank filled with fluid. Initial geometry, thermal conditions and material properties.
6854
|}
6855
6856
Once again, the material properties  are assumed to be temperature independent. The fluid in the tank has  initial temperature T=340<math display="inline">K</math>, while the solid bodies from left to right, have initial temperatures T=180<math display="inline">K</math>, T=200<math display="inline">K</math>, T=220<math display="inline">K</math>, respectively. The solid and the fluid domains have been discretized with a mesh composed of 9394 3-noded triangular elements. The simulation has been run for a total duration of <math display="inline">8 s</math> using a time step increment <math display="inline">\Delta t</math>=<math display="inline">0.0001 s</math>. The heat flux in the normal direction is assumed to be null for the boundaries in contact with the air or the walls.
6857
6858
In the graph of Figure [[#img-140|140]] the evolution of the temperature at the central point of the three objects is plotted.  <div id='img-140'></div>
6859
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
6860
|-
6861
|[[Image:draft_Samper_722607179-REPBalls6hGraph.png|420px|Falling of three objects in a heated tank filled with fluid. Evolution of the temperature at the center of the three objects.]]
6862
|- style="text-align: center; font-size: 75%;"
6863
| colspan="1" | '''Figure 140:''' Falling of three objects in a heated tank filled with fluid. Evolution of the temperature at the center of the three objects.
6864
|}
6865
6866
Figure [[#img-141|141]] collects six representative snapshots of the numerical simulation with the temperature results plotted over the fluid and the solid domains.
6867
6868
<div id='img-141'></div>
6869
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6870
|-
6871
|[[File:Draft_Samper_722607179_8286_balls664.png|400px|t=0.67st=1.82s ]]
6872
|[[File:Draft_Samper_722607179_4424_balls1816.png|400px|t=2.66s]]
6873
|- style="text-align: center; font-size: 75%;"
6874
| (a) t=0.67s
6875
| (b) t=1.82s 
6876
|-
6877
|[[File:Draft_Samper_722607179_9821_balls2656.png|400px|t=4.50s ]]
6878
|[[File:Draft_Samper_722607179_5100_balls4500.png|400px|t=7.00s]]
6879
|- style="text-align: center; font-size: 75%;"
6880
| (c) t=2.66s 
6881
| (d) t=4.50s 
6882
|-
6883
|[[File:Draft_Samper_722607179_4584_balls7000.png|400px|t=8.00s ]]
6884
|[[File:Draft_Samper_722607179_1033_balls8000.png|400px|Falling of three objects in a heated tank filled with fluid. Snapshots with temperature contours at different time steps.]]
6885
|- style="text-align: center; font-size: 75%;"
6886
| (e) t=7.00s
6887
| (f) t=8.00s 
6888
|- style="text-align: center; font-size: 75%;"
6889
| colspan="2" | '''Figure 141:''' Falling of three objects in a heated tank filled with fluid. Snapshots with temperature contours at different time steps.
6890
|}
6891
6892
==5.4 Phase change==
6893
6894
For dealing with the phase change transformation, a term that takes into account the latent heat released or absorbed during the melting process need to be considered [37]. Hence, the heat equation for phase change problems reads
6895
6896
<span id="eq-268"></span>
6897
{| class="formulaSCP" style="width: 100%; text-align: left;" 
6898
|-
6899
| 
6900
{| style="text-align: left; margin:auto;" 
6901
|-
6902
| style="text-align: center;" | <math>\rho c \frac{DT}{Dt} - {\partial  \over \partial x_i} \left(k {\partial T \over \partial x_i}\right)+ Q +\rho \mathcal{L}_{PC} \frac{\partial f_L}{\partial t} =0  </math>
6903
|}
6904
| style="width: 5px;text-align: right;" | (268)
6905
|}
6906
6907
where <math display="inline">{\mathcal{L}_{PC}}</math> is the latent heat of the phase transformation and <math display="inline">{f_L}</math> is the liquid fraction function which is equal to one in the liquid phase and null in the solid one, as shown in Figure [[#img-142|142]]. <div id='img-142'></div>
6908
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
6909
|-
6910
|[[Image:draft_Samper_722607179-fluidFunction.png|350px|General liquid fraction function for phase change modeling.]]
6911
|- style="text-align: center; font-size: 75%;"
6912
| colspan="1" | '''Figure 142:''' General liquid fraction function for phase change modeling.
6913
|}
6914
6915
The phase change is modeled in the following way.
6916
6917
The solid elements transform into fluid elements if at least one of the two following conditions is verified:
6918
6919
<ol>
6920
6921
<li>''Due to temperature'': if the mean temperature of the solid elements that share the same node is greater than the melting temperature of the material. In this case, all these solid elements shift to the fluid domain, as illustrated in Figure [[#img-143|143]] for a 2D problem. <div id='img-143'></div>
6922
</li>
6923
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6924
|-
6925
|[[Image:draft_Samper_722607179-fusion1.png|600px|Graphic representation of the change of phase algorithm.]]
6926
|- style="text-align: center; font-size: 75%;"
6927
| colspan="1" | '''Figure 143:''' Graphic representation of the change of phase algorithm.
6928
|}
6929
6930
<li>''Due to plastic deformation'': if a solid element has accumulated a plastic deformation greater than a pre-defined limit value. </li>
6931
6932
</ol>
6933
6934
The transformation from fluid to solid or viceversa entails just a few of changes and  small computational work. The main ones concern the change of the  material properties and the remeshing procedure.
6935
6936
With respect to the former point, in this work the properties of the melted or solidified material changes instantaneously taking the values of the other material analyzed in the problem. This represents a strong simplification and it may affect the accuracy of the numerical results. For a better modeling of the problem, a multi-fluid analysis should be performed considering different properties for the melted or solidified materials. Despite this more accurate strategy has not been implemented or tested, from the author point of view, it can be easily linked to the unified formulation. This  opinion is sustained by the fact that the proposed strategy is open to analyze different materials at the same time and the PFEM has already been used successfully in the past for solving multi-fluid problems [64].
6937
6938
Regarding  remeshing, as already explained in the previous chapters, the mesh is regenerated just on the fluid domain. So when a solid element passes to the fluid domain, its nodes, except the ones that still belong to the contour of the remaining part of solid domain, become involved in the remeshing procedure, as the rest of the  elements in the fluid domain. The remaining part of the solid domain is not remeshed but it just loses the part of its domain involved in the melting.
6939
6940
The above two changes are essentially the only ones required by the thermally coupled unified formulation for dealing with phase change problems.
6941
6942
===5.4.1 Numerical example: melting of an ice block===
6943
6944
This 2D example is presented to show an application of the phase change strategy implemented. The problem was presented in  [83]. An ice block at initial temperature T=270<math display="inline">K</math> is dropped into a tank containing water at rest at temperature T=340<math display="inline">K</math>. The initial geometry, the problem data and the initial thermal conditions are shown in Figure [[#img-144|144]].
6945
6946
<div id='img-144'></div>
6947
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6948
|-
6949
|[[Image:draft_Samper_722607179-iceInput2.png|400px|]]
6950
|[[Image:draft_Samper_722607179-iceDataGeo.png|100px|]]
6951
|-
6952
| colspan="2"|[[Image:draft_Samper_722607179-iceDataMat.png|600px|Melting of an ice block. Geometry, material data and initial thermal conditions.]]
6953
|- style="text-align: center; font-size: 75%;"
6954
| colspan="2" | '''Figure 144:''' Melting of an ice block. Geometry, material data and initial thermal conditions.
6955
|}
6956
6957
Ice is treated as a hypoelastic solid until some of its elements reach the fusion temperature (T=273.15<math display="inline">K</math>). These elements pass to the fluid domain taking all its physical properties. For this analysis the following assumptions have been made: the mechanical and thermal properties of the water and the ice do not change with the temperature and the heat normal flux along the boundaries in contact with the air or the walls have been considered to be null.
6958
6959
In Figure [[#img-145|145]]  snapshots of some representative instants of the analysis are shown. The temperature contours are plotted over the water and the ice.
6960
6961
<div id='img-145'></div>
6962
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6963
|-
6964
|[[Image:draft_Samper_722607179-Ice44.png|400px|t=0.44s]]
6965
|[[Image:draft_Samper_722607179-Ice110.png|400px|t=2.63s]]
6966
|- style="text-align: center; font-size: 75%;"
6967
| (a) <math>t=0.44s</math>
6968
| (b) <math>t=1.10s </math>
6969
|-
6970
|[[Image:draft_Samper_722607179-Ice263.png|400px|t=4.55s]]
6971
|[[Image:draft_Samper_722607179-Ice455.png|400px|t=10.00s]]
6972
|- style="text-align: center; font-size: 75%;"
6973
| (c) <math>t=2.63s</math>
6974
| (d) <math>t=4.55s</math>
6975
|-
6976
|[[Image:draft_Samper_722607179-Ice1001.png|400px|t=16.00s]]
6977
|[[Image:draft_Samper_722607179-Ice1600.png|400px]]
6978
|- style="text-align: center; font-size: 75%;"
6979
| (e) <math>t=10.00s</math>
6980
| (f) <math>t=16.00s</math>
6981
|- style="text-align: center; font-size: 75%;"
6982
| colspan="2" | '''Figure 145:''' Melting of an ice block. Snapshots with temperature contours at different time steps.
6983
|}
6984
6985
In Figure [[#img-146|146]] the detail of the melting of the ice piece is illustrated. The finite element mesh is drawn over the solid and the fluid domains. 
6986
6987
<div id='img-146'></div>
6988
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
6989
|-
6990
|style="padding-right:10px"|[[File:Draft_Samper_722607179_6940_iceZoom281.png|400px|t=2.81s]]
6991
|style="padding-right:10px"|[[File:Draft_Samper_722607179_7428_iceZoom470.png|400px|t=9.77s]]
6992
|[[File:Draft_Samper_722607179_4036_iceZoom977.png|400px|t=12.74s]]
6993
|- style="text-align: center; font-size: 75%;"
6994
| (a) <math>t=2.81s</math>
6995
| (b) <math>t=4.70 s</math>
6996
| (c) <math>t=9.77s</math>
6997
|-
6998
|style="padding-right:10px"|[[File:Draft_Samper_722607179_7550_iceZoom1274.png|400px|t=14.69s]]
6999
|style="padding-right:10px"|[[File:Draft_Samper_722607179_4837_iceZoom1469.png|400px|t=15.47s]]
7000
|[[File:Draft_Samper_722607179_1247_iceZoom1547.png|400px]]
7001
|- style="text-align: center; font-size: 75%;"
7002
| (d) <math>t=12.74s</math>
7003
| (e) <math>t=14.69s</math>
7004
| (f) <math>t=15.47s</math>
7005
|- style="text-align: center; font-size: 75%;"
7006
| colspan="2" | '''Figure 146:''' Melting of an ice block. Zoom on the piece of ice at different time steps.
7007
|}
7008
7009
==5.5 Summary and conclusions==
7010
7011
The purpose of this chapter has been to show that the Unified PFEM formulation can be easily coupled with the heat transfer problem in order to solve coupled thermal mechanical problems.
7012
7013
The coupling has been ensured via a staggered scheme for which the mechanical and the thermal problems are solved within the same iteration loop. For the temperature field the same linear shape functions of the velocity and pressure fields have been used.
7014
7015
Several numerical examples have been presented with the objective of showing the applicability of the Unified coupled thermal-mechanical formulation for solid and fluid dynamics  problems involving the temperature.
7016
7017
The algorithm for the phase change modeling has been explained and an explicative numerical example has been also given.
7018
7019
The numerical examples presented have shown the possibility of the Unified PFEM formulation for dealing with thermal-mechanical problems.
7020
7021
=Chapter 6. Industrial application: PFEM Analysis Model of NPP Severe Accident=
7022
7023
7024
==6.1 Introduction==
7025
7026
In this chapter the analysis of an industrial problem solved using the Unified formulation is presented. The project required the modeling of the damages in a pressure vessel structure caused by the dropping of a volume of corium at high temperature.
7027
7028
In a nuclear power plant, the reactor pressure vessel (Figure [[#img-147|147]]) is the structure that contains the reactor core, the nuclear reactor coolant and the core shroud.
7029
7030
<div id='img-147'></div>
7031
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7032
|-
7033
|[[File:Draft_Samper_722607179_3245_Fig6-1a.png|400px]]
7034
|[[File:Draft_Samper_722607179_2063_Fig6-1b.png|400px]]
7035
|- style="text-align: center; font-size: 75%;"
7036
| colspan="2" | '''Figure 147:''' Pressure vessels for nuclear power plants. From [2].
7037
|}
7038
7039
The core of a nuclear reactor is the place where the nuclear reactions occur. The coolant is used to remove the heat form the nuclear reactor and the core shroud is a strainless structure used to direct the cooling water flow.
7040
7041
Corium is a heterogeneous material, also called Fuel Containing Material (FCM) or Lava-like Fuel Containing Material (LFCM). Its formation is the result of the combustion and melting of the reactor's components  and the subsequent  chemical and radioactive reactions. Consequently, the corium's composition  depends on the type of the reactor and specifically on the materials of its components. Generally it is composed by nuclear fuel, fission products, control rods, structural materials from the affected parts of the reactor, products of their chemical reaction with air, water and steam, and, in case the reactor vessel is breached, molten concrete from the floor of the reactor room.
7042
7043
The temperature of corium depends on its internal heat generation dynamics, specifically on the chemical and radioactive reactions that may occur during the accident. Potentially, the corium can reach  temperatures over 2800 C.
7044
7045
Corium accumulates at the bottom of the reactor vessel and, in case of adequate cooling, solidifies. Otherwise, the corium may melt through the reactor vessel and flow out (Figure [[#img-148|148]]).  
7046
7047
<div id='img-148'></div>
7048
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7049
|-
7050
|[[Image:draft_Samper_722607179-coriumReal.png|400px]]
7051
|- style="text-align: center; font-size: 75%;"
7052
| colspan="1" | '''Figure 148:''' A blob of corium in the Chernobyl Nuclear Reactor. From  [1].
7053
|}
7054
7055
The seriousness of this kind of accident explains the high interest of this study.
7056
7057
The purpose of the analysis was to model with the PFEM the interaction between the corium and the pressure vessel structure. The scope was to evaluate the capability of the method for simulating such a complex problem. For the project, the PFEM unified formulation with thermal coupling  presented in this work has been used.
7058
7059
The study is a multi-physics and highly non-linear problem and it involves many critical topics for a computational analysis, such as free-surface flow, fluid-structure interaction, plasticity with thermal softening and phase change.
7060
7061
Specifically, the analysis consisted of  two models:
7062
7063
* Basic model. A sphere of corium falls from a certain height over a prismatic plate;
7064
*  Detailed model. A corium volume is placed all around a control rod housing located in the center of a steel shell.
7065
7066
The first model represents a general case and its geometry is not necessary related to a specific component of the pressure vessel. In Figure [[#img-149|149]] a graphic and simplified representation of the study is given.
7067
7068
<div id='img-149'></div>
7069
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7070
|-
7071
|[[Image:draft_Samper_722607179-coriumSlump.png|300px|Basic model. Graphic representation of the phenomena required to model.]]
7072
|- style="text-align: center; font-size: 75%;"
7073
| colspan="1" | '''Figure 149:''' Basic model. Graphic representation of the phenomena required to model.
7074
|}
7075
7076
Instead, in the second model the geometry is more complex and it reproduces a part of the bottom head of a reactor pressure vessel. In Figure [[#img-150|150]] a graphic representation of the area of study is shown.
7077
7078
<div id='img-150'></div>
7079
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7080
|-
7081
|[[Image:draft_Samper_722607179-pressureVessel.png|300px|]]
7082
|[[Image:draft_Samper_722607179-pressureVesselDet.png|300px|Graphic representation of the accumulation of corium at the bottom of the pressure vessel (images provided by NSSMC).]]
7083
|- style="text-align: center; font-size: 75%;"
7084
| colspan="2" | '''Figure 150:''' Graphic representation of the accumulation of corium at the bottom of the pressure vessel (images provided by NSSMC).
7085
|}
7086
7087
For both models, the phenomena to model are:
7088
7089
<ol>
7090
7091
<li>Dropping and slumping of a high temperature volume of corium on the pressure vessel; </li>
7092
<li>Melting, deforming and distruction of the structure due to the heating caused by the corium; </li>
7093
<li>Slumping of corium through the vessel. </li>
7094
7095
</ol>
7096
7097
===6.1.1 Assumptions allowed by the specification===
7098
7099
Due to the complexity of the analysis the following assumptions were considered to be admissible in the specifications provided by the contractor for the analysis of the two models:
7100
7101
* The data analysis given in the specifications (geometric data, material properties, boundary conditions...) were open to changes after consultation with the contractor;
7102
* Simplified constitutive models could be considered whenever necessary;
7103
* The dimensions of the models could be scaled down in consideration to the speed of the analyses;
7104
* The cooling water and its effects (boiling, flow etc.) are not considered in the study;
7105
* The corium is assumed to be a highly viscous fluid at constant temperature;
7106
* The corium could not mix or react with the surrounding media;
7107
*  The melting and solidification of steel are treated with a simple model.
7108
7109
==6.2 Numerical method==
7110
7111
For simulating both models described in the previous section, the unified formulation with thermal-mechanical coupling has been used. The  multi-physics of the  project   represented a great opportunity to verify the efficiency of the formulation in its completeness. In fact, the analysis involves most of the technical aspects presented in this work. The unified formulation allows the simulation of the interaction of free-surface fluids with solids with plasticity, as  shown in Chapters [[#2 Velocity-based formulations for compressible materials|2]], [[#3 Unified stabilized formulation for quasi-incompressible materials|3]] and [[#4 Unified formulation for FSI problems|4]]. Finally Chapter [[#5 Coupled thermal-mechanical formulation|5]] showed that with just a small implementation work, also coupled thermal-mechanical and phase change problems can also be simulated.
7112
7113
From the practical point of view, in industrial projects the computational cost of the analysis has to be taken in serious consideration. Due to the complexity of the (3D) geometry, the duration of the physical phenomena and the required accuracy of the results, some industrial problems may lead to prohibitive computational times. Hence, in some cases it is necessary to accept a compromise between the accuracy and computational cost of the analysis. In order to calibrate at best the model, some preliminary studies of the simulations should be done. In this work, for example, this phase has been extremely useful for defining the adequate time step increment and the best FEM discretizations, deciding where a finer mesh was necessary and where not, and also for setting up some geometric and material values. For both models considered a preliminary study has been done and in the following some details of this phase are given.
7114
7115
Concerning the numerical method used for these simulations, the mechanical problem was solved with the unified formulation using the VP-element for the solid with an hypoelastic-plastic model, a von Mises yield criterion and thermal softening. The phase change was modeled following the criteria presented in Section [[#5.4 Phase change|5.4]]. In both models there is only melting of the solids and not solidification of the fluids. For simplicity, it was assumed that the solid elements that become fluid take instantaneously the same material properties of the fluid. As already explained, for a more realistic simulation, a multi-fluid analysis should be performed taking into account different properties for the melted material.
7116
7117
==6.3 Basic Model==
7118
7119
===6.3.1 Problem data===
7120
7121
The basic model consists of a corium sphere at <math display="inline">{T=2000 K}</math> that falls from a height <math display="inline">{H}</math> over a prismatic plate at <math display="inline">{T=293 K}</math>.
7122
7123
The geometry of the problem is illustrated in Figure [[#img-151|151]].
7124
7125
<div id='img-151'></div>
7126
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7127
|-
7128
|[[Image:draft_Samper_722607179-coriumBasicInputXZ.png|400px|XZ viewXY view]]
7129
|[[Image:draft_Samper_722607179-coriumBasicInputXY.png|400px|Basic model: XZ and XY views of the initial geometry.]]
7130
|- style="text-align: center; font-size: 75%;"
7131
| (a) XZ view
7132
| (b) XY view
7133
|- style="text-align: center; font-size: 75%;"
7134
| colspan="2" | '''Figure 151:''' Basic model: XZ and XY views of the initial geometry.
7135
|}
7136
7137
The boundary conditions of the plate are illustrated in Figure [[#img-152|152]].
7138
7139
<div id='img-152'></div>
7140
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7141
|-
7142
|[[Image:draft_Samper_722607179-BasicModelConstr.png|600px|Basic model: constraints of the plate.]]
7143
|- style="text-align: center; font-size: 75%;"
7144
| colspan="1" | '''Figure 152:''' Basic model: constraints of the plate.
7145
|}
7146
7147
The properties of the corium are the ones given in the specification for <math display="inline">{T=2000 K}</math> and they are collected in Table [[#table-12|6.1]]. It has been assumed that the corium keeps the same temperature during the analysis.
7148
7149
<span id='table-12'></span>
7150
7151
<div class="center" style="font-size: 75%;">'''Table 6.1'''. Material data for the corium at T=2000K.</div>
7152
7153
{| class="wikitable" style="text-align: center; margin: 1em auto;"
7154
|- style="border-top: 2px solid;"
7155
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{\rho }</math>  
7156
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{k}</math>
7157
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{c_v}</math>
7158
|- style="border-top: 2px solid;"
7159
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{kg/m^3}</math>  
7160
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{W/(m \cdot K)}</math>
7161
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{J/(kg \cdot K)}</math>
7162
|- style="border-top: 2px solid;border-bottom: 2px solid;"
7163
| style="border-left: 2px solid;border-right: 2px solid;" |  7865  
7164
| style="border-left: 2px solid;border-right: 2px solid;" | 22 
7165
| style="border-left: 2px solid;border-right: 2px solid;" | 500  
7166
7167
|}
7168
7169
Material data for the corium at T=2000K. MatDataCorium
7170
7171
Concerning the steel plate, the material data at <math display="inline">{T=293 K}</math> are collected in  Table [[#table-13|6.2]]. These material properties are comparable to those of the stainless steels and they can represent a Stainless Steel 309.
7172
7173
<span id='table-13'></span>
7174
<div class="center" style="font-size: 75%;">'''Table 6.2'''. Material data for the plate at T=293 K.</div>
7175
7176
{| class="wikitable" style="text-align: center; margin: 1em auto;"
7177
|- style="border-top: 2px solid;"
7178
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{\rho }</math> 
7179
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{E}</math>
7180
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\nu }</math>
7181
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{k}</math>
7182
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{c_v}</math>
7183
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\alpha }</math>
7184
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\sigma _Y}</math>
7185
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{T_{melting}}</math>
7186
|- style="border-top: 2px solid;"
7187
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{kg/m^3}</math> 
7188
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{GPa}</math>
7189
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{-}</math>
7190
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{W/(m \cdot K)}</math>
7191
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{J/(kg \cdot K)}</math>
7192
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{1/ K}</math>
7193
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{GPa}</math>
7194
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{K}</math>
7195
|- style="border-top: 2px solid;border-bottom: 2px solid;"
7196
| style="border-left: 2px solid;border-right: 2px solid;" |  7850 
7197
| style="border-left: 2px solid;border-right: 2px solid;" | 196  
7198
| style="border-left: 2px solid;border-right: 2px solid;" | 0.33 
7199
| style="border-left: 2px solid;border-right: 2px solid;" | 16  
7200
| style="border-left: 2px solid;border-right: 2px solid;" | 500 
7201
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0.000012</math>
7202
| style="border-left: 2px solid;border-right: 2px solid;" | 0.282 
7203
| style="border-left: 2px solid;border-right: 2px solid;" | 1693
7204
7205
|}
7206
7207
Material data for the plate at T=293 K. MatDataPlate
7208
7209
The thermal influence on the material properties has been taken into account by multiplying the Young modulus <math display="inline">{E}</math>, the heat conductivity <math display="inline">{ k}</math>, the specific heat <math display="inline">{c_v}</math> and the yield stress <math display="inline">{\sigma _Y}</math> by the coefficients <math display="inline">{C_E}</math>, <math display="inline">{ C_k}</math>, <math display="inline">{C_c}</math> and <math display="inline">{C_Y}</math>, respectively. Hence:
7210
7211
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7212
|-
7213
| 
7214
{| style="text-align: left; margin:auto;" 
7215
|-
7216
| style="text-align: center;" | <math>E(T)=  C_E(T) \cdot E(T=293 K) </math>
7217
|}
7218
| style="width: 5px;text-align: right;" | (269)
7219
|}
7220
7221
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7222
|-
7223
| 
7224
{| style="text-align: left; margin:auto;" 
7225
|-
7226
| style="text-align: center;" | <math>k(T)=  C_k(T) \cdot k(T=293 K) </math>
7227
|}
7228
| style="width: 5px;text-align: right;" | (270)
7229
|}
7230
7231
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7232
|-
7233
| 
7234
{| style="text-align: left; margin:auto;" 
7235
|-
7236
| style="text-align: center;" | <math>c_v(T)=  C_c(T) \cdot c_v(T=293 K) </math>
7237
|}
7238
| style="width: 5px;text-align: right;" | (271)
7239
|}
7240
7241
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7242
|-
7243
| 
7244
{| style="text-align: left; margin:auto;" 
7245
|-
7246
| style="text-align: center;" | <math>\sigma _Y(T)=  C_Y(T) \cdot \sigma _Y(T=293 K) </math>
7247
|}
7248
| style="width: 5px;text-align: right;" | (272)
7249
|}
7250
7251
In Table [[#table-14|6.3]] the values of the thermal coefficients are given for certain temperatures. For other temperatures, the coefficients are obtained via a linear interpolation.
7252
7253
<span id='table-14'></span>
7254
<div class="center" style="font-size: 75%;">'''Table 6.3'''. Thermal coefficients for the plate.</div>
7255
7256
{| class="wikitable" style="text-align: center; margin: 1em auto;"
7257
|- style="border-top: 2px solid;"
7258
| colspan='2' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | Temperature 
7259
| colspan='4' style="border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | Thermal coefficients
7260
|- style="border-top: 2px solid;"
7261
| style="border-left: 2px solid;border-right: 2px solid;" |    [C] 
7262
| style="border-left: 2px solid;border-right: 2px solid;" | [K] 
7263
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{C_E}</math>
7264
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{C_k}</math>
7265
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{C_c}</math>
7266
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{C_Y}</math>
7267
|- style="border-top: 2px solid;"
7268
| style="border-left: 2px solid;border-right: 2px solid;" |  20 
7269
| style="border-left: 2px solid;border-right: 2px solid;" | 293  
7270
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7271
| style="border-left: 2px solid;border-right: 2px solid;" | 1  
7272
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7273
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7274
|- style="border-top: 2px solid;"
7275
| style="border-left: 2px solid;border-right: 2px solid;" |  93 
7276
| style="border-left: 2px solid;border-right: 2px solid;" | 366 
7277
| style="border-left: 2px solid;border-right: 2px solid;" | 0.992 
7278
| style="border-left: 2px solid;border-right: 2px solid;" | 1.082 
7279
| style="border-left: 2px solid;border-right: 2px solid;" | 1.036 
7280
| style="border-left: 2px solid;border-right: 2px solid;" | 1
7281
|- style="border-top: 2px solid;"
7282
| style="border-left: 2px solid;border-right: 2px solid;" |  204 
7283
| style="border-left: 2px solid;border-right: 2px solid;" | 477 
7284
| style="border-left: 2px solid;border-right: 2px solid;" | 0.969  
7285
| style="border-left: 2px solid;border-right: 2px solid;" | 1.208 
7286
| style="border-left: 2px solid;border-right: 2px solid;" | 1.091 
7287
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7288
|- style="border-top: 2px solid;"
7289
| style="border-left: 2px solid;border-right: 2px solid;" |  315 
7290
| style="border-left: 2px solid;border-right: 2px solid;" | 588 
7291
| style="border-left: 2px solid;border-right: 2px solid;" | 0.914 
7292
| style="border-left: 2px solid;border-right: 2px solid;" | 1.333 
7293
| style="border-left: 2px solid;border-right: 2px solid;" | 1.146 
7294
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7295
|- style="border-top: 2px solid;"
7296
| style="border-left: 2px solid;border-right: 2px solid;" |  427 
7297
| style="border-left: 2px solid;border-right: 2px solid;" | 700 
7298
| style="border-left: 2px solid;border-right: 2px solid;" | 0.910 
7299
| style="border-left: 2px solid;border-right: 2px solid;" | 1.460 
7300
| style="border-left: 2px solid;border-right: 2px solid;" | 1.202 
7301
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
7302
|- style="border-top: 2px solid;"
7303
| style="border-left: 2px solid;border-right: 2px solid;" |  538 
7304
| style="border-left: 2px solid;border-right: 2px solid;" | 811  
7305
| style="border-left: 2px solid;border-right: 2px solid;" | 0.864 
7306
| style="border-left: 2px solid;border-right: 2px solid;" | 1.586 
7307
| style="border-left: 2px solid;border-right: 2px solid;" | 1.257 
7308
| style="border-left: 2px solid;border-right: 2px solid;" | 0.781 
7309
|- style="border-top: 2px solid;"
7310
| style="border-left: 2px solid;border-right: 2px solid;" |  649 
7311
| style="border-left: 2px solid;border-right: 2px solid;" | 922 
7312
| style="border-left: 2px solid;border-right: 2px solid;" | 0.828 
7313
| style="border-left: 2px solid;border-right: 2px solid;" | 1.711 
7314
| style="border-left: 2px solid;border-right: 2px solid;" | 1.312 
7315
| style="border-left: 2px solid;border-right: 2px solid;" | 0.767
7316
|- style="border-top: 2px solid;"
7317
| style="border-left: 2px solid;border-right: 2px solid;" |  760 
7318
| style="border-left: 2px solid;border-right: 2px solid;" | 1033 
7319
| style="border-left: 2px solid;border-right: 2px solid;" | 0.900 
7320
| style="border-left: 2px solid;border-right: 2px solid;" | 1.837 
7321
| style="border-left: 2px solid;border-right: 2px solid;" | 1.367 
7322
| style="border-left: 2px solid;border-right: 2px solid;" | 0.636 
7323
|- style="border-top: 2px solid;"
7324
| style="border-left: 2px solid;border-right: 2px solid;" |  871 
7325
| style="border-left: 2px solid;border-right: 2px solid;" | 1144  
7326
| style="border-left: 2px solid;border-right: 2px solid;" | 0.778 
7327
| style="border-left: 2px solid;border-right: 2px solid;" | 1.962 
7328
| style="border-left: 2px solid;border-right: 2px solid;" | 1.422 
7329
| style="border-left: 2px solid;border-right: 2px solid;" | 0.267 
7330
|- style="border-top: 2px solid;"
7331
| style="border-left: 2px solid;border-right: 2px solid;" |   982 
7332
| style="border-left: 2px solid;border-right: 2px solid;" | 1255 
7333
| style="border-left: 2px solid;border-right: 2px solid;" | 0.733 
7334
| style="border-left: 2px solid;border-right: 2px solid;" | 2.087  
7335
| style="border-left: 2px solid;border-right: 2px solid;" | 1.477  
7336
| style="border-left: 2px solid;border-right: 2px solid;" | 0.146 
7337
|- style="border-top: 2px solid;"
7338
| style="border-left: 2px solid;border-right: 2px solid;" |   1093 
7339
| style="border-left: 2px solid;border-right: 2px solid;" | 1366 
7340
| style="border-left: 2px solid;border-right: 2px solid;" | 0.733 
7341
| style="border-left: 2px solid;border-right: 2px solid;" | 2.087  
7342
| style="border-left: 2px solid;border-right: 2px solid;" | 1.477  
7343
| style="border-left: 2px solid;border-right: 2px solid;" | 0.010 
7344
|- style="border-top: 2px solid;border-bottom: 2px solid;"
7345
| style="border-left: 2px solid;border-right: 2px solid;" |   1377 
7346
| style="border-left: 2px solid;border-right: 2px solid;" | 1650 
7347
| style="border-left: 2px solid;border-right: 2px solid;" | 0.733 
7348
| style="border-left: 2px solid;border-right: 2px solid;" | 2.087  
7349
| style="border-left: 2px solid;border-right: 2px solid;" | 1.477  
7350
| style="border-left: 2px solid;border-right: 2px solid;" | 0 
7351
7352
|}
7353
7354
Thermal coefficients for the plate. TherDataPlate
7355
7356
===6.3.2 Preliminary study===
7357
7358
In order to calibrate the model, the problem was studied in 2D first. The original purpose of this preliminary study was just to understand better the difficulties of the problem and to determine a range of suitable values for both the mesh size and the time step increment. However,  this preliminary study allowed to identify that some  parameters of the proposed model needed to be modified in order to simulate the desired phenomenon of heating and melting of the steel plate.
7359
7360
In fact, in the initial configuration proposed in the specifications, the sphere of corium was located at the height <math display="inline">{H=200 mm}</math>. However, the preliminary studies showed that it would be not possible to simulate the desired problem if the corium would fall from that height. In fact, the inertial forces accumulated by the corium during its fall generate a huge splashing of the fluid over the plate. In order to allow the laying of the corium on the plate and the consequent heating of the structure, the initial height of the corium was reduced in the PFEM analyses. After some tests the height <math display="inline">{H= 80 mm}</math> was chosen for the simulation.
7361
7362
The preparatory tests also showed that with the value of corium viscosity proposed in the specifications (<math display="inline">{\mu =0.01 Pa \cdot s}</math>), the corium flows progressively towards the edges of the plate and finally a part of this falls down by the plate contour. In order to avoid this undesired phenomenon,  a non-linear viscosity was used for the basic model. In particular, corium was modeled using the constitutive law proposed and successfully applied for the simulation of melted metals in  [133,134]. According to the mentioned publications, the viscosity of the corium was computed as:
7363
7364
<span id="eq-273"></span>
7365
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7366
|-
7367
| 
7368
{| style="text-align: left; margin:auto;" 
7369
|-
7370
| style="text-align: center;" | <math>\mu = \frac{\sigma _Y(T)}{\sqrt{3} \bar{\epsilon }_d}  </math>
7371
|}
7372
| style="width: 5px;text-align: right;" | (273)
7373
|}
7374
7375
where <math display="inline">{\bar{\epsilon }_d}</math> is the deviatoric strain invariant and <math display="inline">{\sigma _Y}</math> is the yield stress.
7376
7377
===6.3.3 Numerical results===
7378
7379
The simulation was run using a finite element mesh composed by 327451 four-noded tetrahedral elements with 61452 nodes; 27936 nodes for the fluid  and 33516 for the solid. The fluid was discretized using an uniform mesh with an average size of <math display="inline">{5 mm}</math>. On the other hand, the solid plate was not meshed uniformly. In its central part, for guaranteeing a good contact between the solid and the fluid, an unstructured mesh with the same average size of the fluid was used, while for the surrounding zone the average mesh size is <math display="inline">{8 mm}</math>.
7380
7381
For the first phase, when the splashing occurs  the time step increment <math display="inline">{\Delta t= 0.002 s}</math> was used. In Figure [[#img-153|153]] some snapshots referred to this phase are given. <div id='img-153'></div>
7382
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7383
|-
7384
|[[Image:draft_Samper_722607179-initial1.png|400px|]]
7385
|-
7386
|[[Image:draft_Samper_722607179-initial2.png|400px|]]
7387
|-
7388
| colspan="2"|[[Image:draft_Samper_722607179-initial3.png|400px|Basic model. Snapshots of the initial splashing of the corium on the plate.]]
7389
|- style="text-align: center; font-size: 75%;"
7390
| colspan="2" | '''Figure 153:''' Basic model. Snapshots of the initial splashing of the corium on the plate.
7391
|}
7392
7393
In order to reduce the computational time of the analysis, for the other phases of the simulation the time step increment was increased to <math display="inline">{\Delta t= 0.02 s}</math>.
7394
7395
After the splashing and spreading of the corium on the steel plate, the plate starts heating due to the contact with the hotter corium and progressively melts in its central part. In Figure [[#img-154|154]] a few snapshots of this phase are given.  
7396
7397
<div id='img-154'></div>
7398
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7399
|-
7400
|[[Image:draft_Samper_722607179-melted560.png|400px|t = 56.0 st = 83.0 s]]
7401
|-style="text-align: center; font-size: 75%;"
7402
| (a) <math>t = 56.0 s</math>
7403
|-
7404
|[[Image:draft_Samper_722607179-melted830.png|400px|t =92.7 s]]
7405
|- style="text-align: center; font-size: 75%;"
7406
| (b) <math>t = 83.0 s</math>
7407
|-
7408
| [[Image:draft_Samper_722607179-melted927.png|400px]]
7409
|- style="text-align: center; font-size: 75%;"
7410
| (c) <math>t =92.7 s</math>
7411
|- style="text-align: center; font-size: 75%;"
7412
| colspan="2" | '''Figure 154:''' Basic model. Snapshots of the melting of the plate and plot of temperature contours (blue and red colors correspond to 293 K and 2000 K, respectively). 
7413
|}
7414
7415
The melting of the plate starts at <math display="inline">{t=30.3 s}</math>. The time evolution of the accumulated melted volume of the solid material  is illustrated in the graph of Figure [[#img-155|155]].
7416
7417
<div id='img-155'></div>
7418
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 52%;max-width: 100%;"
7419
|-
7420
|[[Image:draft_Samper_722607179-meltedvOLUME.png|312px|Basic model. Time evolution of the melted volume of the steel plate.]]
7421
|- style="text-align: center; font-size: 75%;"
7422
| colspan="1" | '''Figure 155:''' Basic model. Time evolution of the melted volume of the steel plate.
7423
|}
7424
7425
From the graph one can deduce that the law is non-linear. This behavior is in part a consequence of the hypothesis of constant temperature for the corium. For this reason the melted particles of the solid elements take immediately the temperature of the corium that is greater than the melting temperature of the solid. A more reliable law would be obtained by considering variable the temperature of the corium and by using a higher heat capacity for the solid elements.
7426
7427
Another interesting point of the graph is represented by the increasing of the velocity of melting during the final instants of the simulation. The reason is that during this phase there is an increasing number of solid elements that become part of the fluid domain not due to the temperature criterion (see Figure [[#img-143|143]]) but due to the plasticity effect. In fact, in that moment of the analysis a huge part of the width of the plate has already melted and the thin remaining layer of the plate has to sustain all the weight of the corium and the melted steel. In this situation, that zone of the plate plastifies and it undergoes huge plastic deformations. A proof of this can be deduced from the graph of Figure [[#img-156|156]].
7428
7429
<div id='img-156'></div>
7430
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 62%;max-width: 100%;"
7431
|-
7432
|[[Image:draft_Samper_722607179-temp.png|372px|Basic model. Time evolution of the temperature at the center of the lower side of the plate.]]
7433
|- style="text-align: center; font-size: 75%;"
7434
| colspan="1" | '''Figure 156:''' Basic model. Time evolution of the temperature at the center of the lower side of the plate.
7435
|}
7436
7437
The evolution of the temperature in the plate bottom does not present that pronounced peak in the final part. This explains that the final increasing in the number of melted elements is not related directly with the effect of the temperature.
7438
7439
The collapsing of the plate occurs at <math display="inline">{t=93.8 s}</math>. The hole created by the corium enlarges quickly  due to the large deformations accumulated in the center of the plate.
7440
7441
This effect can be clearly seen in Figures [[#img-157|157]] and [[#img-158|158]], where several snapshots of this phase of the simulation are given.
7442
7443
<div id='img-157'></div>
7444
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7445
|-
7446
|[[Image:draft_Samper_722607179-938.png|400px|t = 93.8 s]]
7447
|[[Image:draft_Samper_722607179-938S.png|400px|t = 94.1 s]]
7448
|- style="text-align: center; font-size: 75%;"
7449
| colspan="2"| (a) <math>t = 93.8 s</math>
7450
|-
7451
|[[Image:draft_Samper_722607179-941.png|400px|]]
7452
|[[Image:draft_Samper_722607179-941S.png|400px|t = 94.4 s]]
7453
|- style="text-align: center; font-size: 75%;"
7454
| colspan="2"| (b) <math>t = 94.1 s</math>
7455
|-style="text-align: center; font-size: 75%;"
7456
|[[Image:draft_Samper_722607179-944.png|400px|]]
7457
|[[Image:draft_Samper_722607179-944S.png|400px]]
7458
|-style="text-align: center; font-size: 75%;"
7459
| colspan="2"| (c) <math>t = 94.4 s</math>
7460
|- style="text-align: center; font-size: 75%;"
7461
| colspan="2" | '''Figure 157:''' Basic model. Snapshots of the piercing of the melted plate by the corium (I/II). 
7462
|}
7463
7464
<div id='img-158'></div>
7465
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7466
|-
7467
|[[Image:draft_Samper_722607179-945.png|400px|t = 94.5 s]]
7468
|[[Image:draft_Samper_722607179-945S.png|400px|t = 94.6 s]]
7469
|- style="text-align: center; font-size: 75%;"
7470
| colspan="2"| (a) <math>t = 94.5 s</math>
7471
|-
7472
|[[Image:draft_Samper_722607179-946.png|400px|]]
7473
|[[Image:draft_Samper_722607179-946S.png|400px|t = 94.8 s]]
7474
|- style="text-align: center; font-size: 75%;"
7475
| colspan="2"| (b) <math>t = 94.6 s</math>
7476
|-
7477
|[[Image:draft_Samper_722607179-948.png|400px|]]
7478
|[[Image:draft_Samper_722607179-948S.png|400px|Basic model. Snapshots of the piercing of the melted plate by the corium (II/II).]]
7479
|- style="text-align: center; font-size: 75%;"
7480
| colspan="2"|  (c) <math>t = 94.8 s</math>
7481
|- style="text-align: center; font-size: 75%;"
7482
| colspan="2" | '''Figure 158:''' Basic model. Snapshots of the piercing of the melted plate by the corium (II/II).
7483
|}
7484
7485
In Figure [[#img-159|159]]  the failure of the plate is represented in its deformed configuration enlarged by a factor of <math display="inline">{10^5}</math>. From this representation it can be appreciated better the localization of the strains in the central part of the plate.
7486
7487
<div id='img-159'></div>
7488
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7489
|-
7490
|[[Image:draft_Samper_722607179-defMeshE5t898.png|400px|t = 89.8 st = 93.0 s]]
7491
|[[Image:draft_Samper_722607179-defMeshE5t930.png|400px|t = 93.3 s]]
7492
|- style="text-align: center; font-size: 75%;"
7493
| (a) <math>t = 89.8 s</math>
7494
| (b) <math>t = 93.0 s</math>
7495
|-
7496
|[[Image:draft_Samper_722607179-defMeshE5t933.png|400px|t = 93.5 s]]
7497
|[[Image:draft_Samper_722607179-defMeshE5t935.png|400px|t = 93.8 s]]
7498
|- style="text-align: center; font-size: 75%;"
7499
| (c) <math>t = 93.3 s</math>
7500
| (d) <math>t = 93.5 s</math>
7501
|-
7502
|[[Image:draft_Samper_722607179-defMeshE5t938.png|400px|t =94.2 s]]
7503
|[[Image:draft_Samper_722607179-defMeshE5t942.png|400px ]]
7504
|- style="text-align: center; font-size: 75%;"
7505
| (e) <math>t = 93.8 s</math>
7506
| (f) <math>t =94.2 s</math>
7507
|- style="text-align: center; font-size: 75%;"
7508
| colspan="2" | '''Figure 159:''' Basic model. Deformed mesh of the plate during the last phase of the simulation (enlargement factor=<math>{10^5}</math>). 
7509
|}
7510
7511
In Figure [[#img-160|160]] the plate deformation at the time <math display="inline">{t=93 s}</math> is illustrated in a 3D-view with the same enlargement factor (<math display="inline">{10^5}</math>).  <div id='img-160'></div>
7512
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
7513
|-
7514
|[[Image:draft_Samper_722607179-defMeshXYZ.png|350px|Basic model. 3D-view of the deformed mesh of the plate at t=93.0 s (enlargement factor=10⁵).]]
7515
|- style="text-align: center; font-size: 75%;"
7516
| colspan="1" | '''Figure 160:''' Basic model. 3D-view of the deformed mesh of the plate at t=93.0 s (enlargement factor=<math>{10^5}</math>).
7517
|}
7518
7519
In Figure [[#img-161|161]] the XZ-view of the steel plate at the end of the analysis is given.
7520
7521
<div id='img-161'></div>
7522
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7523
|-
7524
|[[Image:draft_Samper_722607179-lastViewSolid.png|400px|Basic model. Final configuration of the steel plate.]]
7525
|- style="text-align: center; font-size: 75%;"
7526
| colspan="1" | '''Figure 161:''' Basic model. Final configuration of the steel plate.
7527
|}
7528
7529
==6.4 Detailed model==
7530
7531
===6.4.1 Problem data===
7532
7533
The problem consists of a volume of corium at <math display="inline">{T = 2000K}</math> that leans against a steel shell and surrounds a rod that is located in the middle of the steel structure. In Figures  [[#img-162|162]]  and  [[#img-163|163]] the initial geometry of the problem is illustrated. All the solid components of the problem have an initial temperature of <math display="inline">{T = 293K}</math>. The objective of the analysis is to simulate the melting of the solid structure due to the heating caused by the corium.
7534
7535
<div id='img-162'></div>
7536
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7537
|-
7538
|[[Image:draft_Samper_722607179-DetailedInputXZ.png|400px|XZ view of the steel shell and corium]]
7539
|[[Image:draft_Samper_722607179-DetailedModelRod.png|400px|Detailed model. Initial geometry (dimensions in mm). ]]
7540
|- style="text-align: center; font-size: 75%;"
7541
| (a) XZ view of the steel shell and corium
7542
| (b) XY view of the rod
7543
|- style="text-align: center; font-size: 75%;"
7544
| colspan="2" | '''Figure 162:''' Detailed model. Initial geometry (dimensions in <math>{mm}</math>). 
7545
|}
7546
7547
<div id='img-163'></div>
7548
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7549
|-
7550
|[[Image:draft_Samper_722607179-DetailedModelXY.png|400px|XY view3D view]]
7551
|[[Image:draft_Samper_722607179-DetailedModelXYZ.png|400px]]
7552
|- style="text-align: center; font-size: 75%;"
7553
| (a) XY view
7554
| (b) 3D view
7555
|- style="text-align: center; font-size: 75%;"
7556
| colspan="2" | '''Figure 163:''' Detailed model. 3D  views of the initial geometry. 
7557
|}
7558
7559
The shell is constrained in the same manner as the plate of the basic model Figure [[#img-152|152]] while the rod is constrained at its top.
7560
7561
The material properties of the rod components, as well their dependency on the temperature, are the ones given in the specification and they are collected in  Tables  [[#table-15|6.4]] and  [[#table-16|6.5]], respectively.
7562
7563
<span id='table-15'></span>
7564
<div class="center" style="font-size: 75%;">'''Table 6.4'''. Material data for the rod components at T=293 K.
7565
</div>
7566
7567
{| class="wikitable" style="text-align: center; margin: 1em auto;"
7568
|- style="border-top: 2px solid;font-size:85%;"
7569
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" | material   
7570
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\rho }</math>
7571
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{E}</math>
7572
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\nu }</math>
7573
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{k}</math>
7574
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{c_v}</math>
7575
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{\sigma _Y}</math>
7576
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{T_{melting}}</math>
7577
|-
7578
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{kg/m^3}</math>
7579
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{GPa}</math>
7580
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{-}</math>
7581
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{W/(m \cdot K)}</math>
7582
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{J/(kg \cdot K)}</math>
7583
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{GPa}</math>
7584
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{K}</math>
7585
|- style="border-top: 2px solid;"
7586
| style="border-left: 2px solid;border-right: 2px solid;" |  Inco. 600S 
7587
| style="border-left: 2px solid;border-right: 2px solid;" | 8430 
7588
| style="border-left: 2px solid;border-right: 2px solid;" | 216  
7589
| style="border-left: 2px solid;border-right: 2px solid;" | 0.33 
7590
| style="border-left: 2px solid;border-right: 2px solid;" | 14.86  
7591
| style="border-left: 2px solid;border-right: 2px solid;" | 444 
7592
| style="border-left: 2px solid;border-right: 2px solid;" | 0.282 
7593
| style="border-left: 2px solid;border-right: 2px solid;" | 1693
7594
|- style="border-top: 2px solid;border-bottom: 2px solid;"
7595
| style="border-left: 2px solid;border-right: 2px solid;" |  SUS310S 
7596
| style="border-left: 2px solid;border-right: 2px solid;" | 8030 
7597
| style="border-left: 2px solid;border-right: 2px solid;" | 200  
7598
| style="border-left: 2px solid;border-right: 2px solid;" | 0.33 
7599
| style="border-left: 2px solid;border-right: 2px solid;" | 13.31  
7600
| style="border-left: 2px solid;border-right: 2px solid;" | 502 
7601
| style="border-left: 2px solid;border-right: 2px solid;" | 0.242 
7602
| style="border-left: 2px solid;border-right: 2px solid;" | 1693
7603
7604
|}
7605
7606
Material data for the rod components at T=293 K. MatDataRod
7607
7608
7609
<span id='table-16'></span>
7610
<div class="center" style="font-size: 75%;">'''Table 6.5'''. Material data for the rod components at different temperatures.
7611
</div>
7612
7613
{| class="wikitable" style="text-align: center; margin: 1em auto;"
7614
|- style="border-top: 2px solid;font-size:85%;"
7615
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" | ''property''   
7616
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  ''material''
7617
| colspan='6' style="border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | ''temperature [K]''
7618
|-
7619
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{293 }</math>
7620
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{366 }</math>
7621
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{477}</math>
7622
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{588 }</math>
7623
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{700}</math>
7624
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{ 811}</math>
7625
|- style="border-top: 2px solid;font-size:85%;"
7626
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  E [GPa]
7627
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7628
| style="border-left: 2px solid;border-right: 2px solid;" | 216.7 
7629
| style="border-left: 2px solid;border-right: 2px solid;" | 215.7 
7630
| style="border-left: 2px solid;border-right: 2px solid;" | 209.9 
7631
| style="border-left: 2px solid;border-right: 2px solid;" | 198.1 
7632
| style="border-left: 2px solid;border-right: 2px solid;" | 197.1  
7633
| style="border-left: 2px solid;border-right: 2px solid;" | 187.3 
7634
|-                                         
7635
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S   
7636
| style="border-left: 2px solid;border-right: 2px solid;" | 200.1  
7637
| style="border-left: 2px solid;border-right: 2px solid;" | -         
7638
| style="border-left: 2px solid;border-right: 2px solid;" | -         
7639
| style="border-left: 2px solid;border-right: 2px solid;" | -         
7640
| style="border-left: 2px solid;border-right: 2px solid;" | 170.6  
7641
| style="border-left: 2px solid;border-right: 2px solid;" | 157.9
7642
|- style="border-top: 2px solid;font-size:85%;"
7643
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline"> \sigma_Y </math> MPa
7644
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7645
| style="border-left: 2px solid;border-right: 2px solid;" | 282 
7646
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7647
| style="border-left: 2px solid;border-right: 2px solid;" | -
7648
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7649
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7650
| style="border-left: 2px solid;border-right: 2px solid;" | 220 
7651
|-                                                              
7652
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S   
7653
| style="border-left: 2px solid;border-right: 2px solid;" | 242  
7654
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7655
| style="border-left: 2px solid;border-right: 2px solid;" | -
7656
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7657
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7658
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7659
|- style="border-top: 2px solid;font-size:85%;"
7660
| rowspan="2"  style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline">k [W=(m \cdot K)] </math>
7661
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7662
| style="border-left: 2px solid;border-right: 2px solid;" | 14.86 
7663
| style="border-left: 2px solid;border-right: 2px solid;" | 15.74  
7664
| style="border-left: 2px solid;border-right: 2px solid;" | 17.46 
7665
| style="border-left: 2px solid;border-right: 2px solid;" | 19.18  
7666
| style="border-left: 2px solid;border-right: 2px solid;" | 20.93 
7667
| style="border-left: 2px solid;border-right: 2px solid;" | 22.78 
7668
|-                                                                  
7669
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S    
7670
| style="border-left: 2px solid;border-right: 2px solid;" | 13.31 
7671
| style="border-left: 2px solid;border-right: 2px solid;" | 14.24  
7672
| style="border-left: 2px solid;border-right: 2px solid;" | 16.24 
7673
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7674
| style="border-left: 2px solid;border-right: 2px solid;" | 21.02 
7675
| style="border-left: 2px solid;border-right: 2px solid;" | 18.84 
7676
|- style="border-top: 2px solid;font-size:85%;"
7677
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline"> c_v [J/kg\cdot K)]   </math>
7678
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7679
| style="border-left: 2px solid;border-right: 2px solid;" | 444 
7680
| style="border-left: 2px solid;border-right: 2px solid;" | 465  
7681
| style="border-left: 2px solid;border-right: 2px solid;" | 486 
7682
| style="border-left: 2px solid;border-right: 2px solid;" | 507 
7683
| style="border-left: 2px solid;border-right: 2px solid;" | 528 
7684
| style="border-left: 2px solid;border-right: 2px solid;" | 553 
7685
|-                                                                          
7686
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S    
7687
| style="border-left: 2px solid;border-right: 2px solid;" | 502 
7688
| style="border-left: 2px solid;border-right: 2px solid;" | -     
7689
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7690
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7691
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7692
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7693
|- style="border-top: 2px solid;font-size:85%;"
7694
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> \alpha [1/K]   </math>
7695
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7696
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7697
| style="border-left: 2px solid;border-right: 2px solid;" | 1.33E-5 
7698
| style="border-left: 2px solid;border-right: 2px solid;" | 1.39E-5 
7699
| style="border-left: 2px solid;border-right: 2px solid;" | 1.42E-5 
7700
| style="border-left: 2px solid;border-right: 2px solid;" | 1.46E-5 
7701
| style="border-left: 2px solid;border-right: 2px solid;" | 1.51E-5 
7702
|-                                                                          
7703
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S    
7704
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7705
| style="border-left: 2px solid;border-right: 2px solid;" | 1.59E-5    
7706
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7707
| style="border-left: 2px solid;border-right: 2px solid;" | 1.62E-5 
7708
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7709
| style="border-left: 2px solid;border-right: 2px solid;" | 1.69E-5 
7710
|- style="border-top: 2px solid;font-size:85%;"
7711
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  ''property''
7712
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  ''material''
7713
| colspan='5' style="border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | ''temperature [K]''
7714
|-style="font-size:85%;"
7715
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{922 }</math>
7716
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{1033}</math>
7717
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{1144 }</math>
7718
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{1255}</math>
7719
| style="border-left: 2px solid;border-right: 2px solid;" | <math>{1366}</math>
7720
|-style="font-size:85%;"
7721
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  E [GPa]
7722
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600  
7723
| style="border-left: 2px solid;border-right: 2px solid;" | 179.5 
7724
| style="border-left: 2px solid;border-right: 2px solid;" | 168.7 
7725
| style="border-left: 2px solid;border-right: 2px solid;" | 158.9 
7726
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7727
| style="border-left: 2px solid;border-right: 2px solid;" | -
7728
|-     style="font-size:85%;"                                   
7729
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S    
7730
| style="border-left: 2px solid;border-right: 2px solid;" | 145.1  
7731
| style="border-left: 2px solid;border-right: 2px solid;" | 131.4 
7732
| style="border-left: 2px solid;border-right: 2px solid;" | 114.7 
7733
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7734
| style="border-left: 2px solid;border-right: 2px solid;" | -
7735
|-style="font-size:85%;"
7736
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline"> \sigma_Y </math> MPa
7737
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600  
7738
| style="border-left: 2px solid;border-right: 2px solid;" | 216.7 
7739
| style="border-left: 2px solid;border-right: 2px solid;" | 179.5 
7740
| style="border-left: 2px solid;border-right: 2px solid;" | 75.5 
7741
| style="border-left: 2px solid;border-right: 2px solid;" | 41.2 
7742
| style="border-left: 2px solid;border-right: 2px solid;" | -
7743
|-  style="font-size:85%;"                                                 
7744
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S      
7745
| style="border-left: 2px solid;border-right: 2px solid;" | 146.1  
7746
| style="border-left: 2px solid;border-right: 2px solid;" | 133.4 
7747
| style="border-left: 2px solid;border-right: 2px solid;" | 84.3 
7748
| style="border-left: 2px solid;border-right: 2px solid;" | 44.1 
7749
| style="border-left: 2px solid;border-right: 2px solid;" | 21.6
7750
|--style="font-size:85%;"
7751
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">k [W=(m \cdot  K)] </math>
7752
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7753
| style="border-left: 2px solid;border-right: 2px solid;" | 24.83 
7754
| style="border-left: 2px solid;border-right: 2px solid;" | 26.84 
7755
| style="border-left: 2px solid;border-right: 2px solid;" | 28.85 
7756
| style="border-left: 2px solid;border-right: 2px solid;" | 31.02 
7757
| style="border-left: 2px solid;border-right: 2px solid;" | -
7758
|-   style="font-size:85%;"                                                            
7759
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S     
7760
| style="border-left: 2px solid;border-right: 2px solid;" | 26.38 
7761
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7762
| style="border-left: 2px solid;border-right: 2px solid;" | 30.86  
7763
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7764
| style="border-left: 2px solid;border-right: 2px solid;" | -
7765
|-style="font-size:85%;"
7766
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> c_v [J/kg\cdot K)]   </math> 
7767
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7768
| style="border-left: 2px solid;border-right: 2px solid;" | 586 
7769
| style="border-left: 2px solid;border-right: 2px solid;" | 607  
7770
| style="border-left: 2px solid;border-right: 2px solid;" | 624 
7771
| style="border-left: 2px solid;border-right: 2px solid;" | 
7772
| style="border-left: 2px solid;border-right: 2px solid;" | 
7773
|-   style="font-size:85%;"                                                                      
7774
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S   
7775
| style="border-left: 2px solid;border-right: 2px solid;" | - 
7776
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7777
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7778
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7779
| style="border-left: 2px solid;border-right: 2px solid;" | -
7780
|-style="font-size:85%;"
7781
| rowspan="2" style="border-left: 2px solid;border-right: 2px solid;" |  <math display="inline"> \alpha [1/K]   </math>
7782
| style="border-left: 2px solid;border-right: 2px solid;" | Inco. 600 
7783
| style="border-left: 2px solid;border-right: 2px solid;" | 1.55E-5 
7784
| style="border-left: 2px solid;border-right: 2px solid;" | 1.60E-5 
7785
| style="border-left: 2px solid;border-right: 2px solid;" | 1.64E-5 
7786
| style="border-left: 2px solid;border-right: 2px solid;" | 1.67 E-5 
7787
| style="border-left: 2px solid;border-right: 2px solid;" | 
7788
|-    style="font-size:85%;"                                                                    
7789
| style="border-left: 2px solid;border-right: 2px solid;" | SUS310S   
7790
| style="border-left: 2px solid;border-right: 2px solid;" | 1.73E-5 
7791
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7792
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7793
| style="border-left: 2px solid;border-right: 2px solid;" | -  
7794
| style="border-left: 2px solid;border-right: 2px solid;" | 1.92E-5
7795
7796
|}
7797
7798
Material data for the rod components at different temperatures. MatDataRodT
7799
7800
Concerning the plate and the corium, the material properties are the ones already introduced for the basic model (Figures [[#table-12|12]], [[#table-13|13]] and [[#table-14|14]]). As for the basic model, it has been assumed that the corium keeps the same temperature (<math display="inline">{T = 2000K}</math>)  for all the duration of the analysis.
7801
7802
===6.4.2 Preliminary study===
7803
7804
In the detailed model the steel structure is not plane as for the basic model, so the corium can not spread over the structure and fall by its edges. For this reason, in this case, it was not necessary to use the non-linear viscosity of Eq.([[#eq-273|273]]).
7805
7806
However, with the proposed value of the viscosity <math display="inline">{\mu=0.01 Pa \cdot s}</math>, the corium should have a water-type behavior. In fact, the ratio <math display="inline">{\rho /\mu }</math> of the corium is very similar to that of  water:
7807
7808
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7809
|-
7810
| 
7811
{| style="text-align: left; margin:auto;" 
7812
|-
7813
| style="text-align: center;" | <math>\left[\rho /\mu \right]_{corium}=7865/0.01=0.7865 \cdot 10^6 </math>
7814
|}
7815
| style="width: 5px;text-align: right;" | (274)
7816
|}
7817
7818
{| class="formulaSCP" style="width: 100%; text-align: left;" 
7819
|-
7820
| 
7821
{| style="text-align: left; margin:auto;" 
7822
|-
7823
| style="text-align: center;" | <math>\left[\rho /\mu \right]_{water}=1000/0.001=1.0 \cdot 10^6 </math>
7824
|}
7825
| style="width: 5px;text-align: right;" | (275)
7826
|}
7827
7828
It is well known that the Reynolds number depends on this ratio. Hence, the dynamics of the problem involving corium should be similar to that of the same problem involving water. With the proposed value of viscosity, the corium initially splashes over the shell, then starts oscillating until a hydrostatic state is reached. This is in contradiction with the description of corium as  a 'highly viscous fluid' given in the specifications.   Furthermore, from the computational point of view, for modeling this kind of problem a small time step increment should be used and the discretization of the upper surface of the plate should be fine. The union of these two requirements would increase highly the computational cost of the analysis.
7829
7830
For all these reasons, a higher value of viscosity has been considered. More specifically, <math display="inline">{\mu=1 Pa \cdot s}</math> has been used.
7831
7832
===6.4.3 Numerical results===
7833
7834
The finite element mesh used for the simulation consists of 396984 four-noded tetrahedra (71992 nodes). Both the fluid and the solid have been discretized using a non-structured mesh. In the central parts of the shell and the rod a finer mesh has been adopted. In particular, in those zones of the solid and in all the fluid domain the mesh has an average size of <math display="inline">{3.8 mm}</math>.
7835
7836
As for the basic model, the time step increment has been set depending on the phase of the simulation. In the first phase, when the corium is spreading on the shell and around the rod, the time step increment <math display="inline">{\Delta t= 0.002 s}</math> was used. In Figure [[#img-164|164]], some snapshots of this phase are given. After <math display="inline">{0.6 s}</math> the viscous fluid is almost at rest.
7837
7838
<div id='img-164'></div>
7839
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7840
|-
7841
|[[Image:draft_Samper_722607179-Snapshot01.png|400px|t = 0.1 s]]
7842
|[[Image:draft_Samper_722607179-Snapshot02.png|400px|t = 0.3 s]]
7843
|[[Image:draft_Samper_722607179-Snapshot03.png|400px|t = 0.4 s]]
7844
|- style="text-align: center; font-size: 75%;"
7845
| (a) <math>t = 0.1 s</math>
7846
| (b) <math> t = 0.2 s</math>
7847
| (c) <math>t = 0.3 s</math>
7848
|-
7849
|[[Image:draft_Samper_722607179-Snapshot04.png|400px|t = 0.5 s]]
7850
|[[Image:draft_Samper_722607179-Snapshot05.png|400px|t = 0.6 s]]
7851
|[[Image:draft_Samper_722607179-Snapshot06.png|400px]]
7852
|- style="text-align: center; font-size: 75%;"
7853
| (d) <math>t = 0.4 s</math>
7854
| (e) <math>t = 0.5 s</math>
7855
| (f) <math>t = 0.6 s</math>
7856
|- style="text-align: center; font-size: 75%;"
7857
| colspan="3" | '''Figure 164:''' Detailed model. Snapshots of the initial spreading of corium.
7858
|}
7859
7860
For the rest of the simulation the time step increment was increased to <math display="inline">{\Delta t= 0.005 s}</math>.
7861
7862
In Figure [[#img-165|165]] some representative snapshots of the rest of the analysis are given.
7863
7864
<div id='img-165'></div>
7865
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7866
|-
7867
|[[Image:draft_Samper_722607179-Snapshot07.png|400px|t = 15.0 st = 15.5 s]]
7868
|[[Image:draft_Samper_722607179-Snapshot155.png|400px|t = 15.9 s]]
7869
|[[Image:draft_Samper_722607179-Snapshot159.png|400px|t = 16.3 s]]
7870
|- style="text-align: center; font-size: 75%;"
7871
| (a) <math>t = 15.0 s</math>
7872
| (b) <math>t = 15.5 s</math>
7873
| (c) <math>t = 15.9 s</math>
7874
|-
7875
|[[Image:draft_Samper_722607179-Snapshot163.png|400px|t = 16.7 s]]
7876
|[[Image:draft_Samper_722607179-Snapshot167.png|400px|t = 17.1  s]]
7877
|[[Image:draft_Samper_722607179-Snapshot171.png|400px]]
7878
|- style="text-align: center; font-size: 75%;"
7879
| (d) <math>t = 16.3 s</math>
7880
| (e) <math>t = 16.7 s</math>
7881
| (f) <math>t = 17.1  s</math>
7882
|- style="text-align: center; font-size: 75%;"
7883
| colspan="3" | '''Figure 165:''' Detailed model. Snapshots of the final phase of the simulation. A hole is created in the rod and the corium passes through it. 
7884
|}
7885
7886
The results show that the failure of the structure occurs in the rod. This can be clearly visualized in Figure [[#img-166|166]] where only the structure is represented: the central part of the rod is completely melted, while no parts of the shell have reached the melting temperature of the material.
7887
7888
<div id='img-166'></div>
7889
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
7890
|-
7891
|[[Image:draft_Samper_722607179-Solidmelting123.png|400px|t = 12.3 s]]
7892
|[[Image:draft_Samper_722607179-Solidmelting131.png|400px|t = 14.7 s]]
7893
|[[Image:draft_Samper_722607179-Solidmelting147.png|400px|t = 15.1 s]]
7894
|- style="text-align: center; font-size: 75%;"
7895
| (a) <math>t = 12.3 s</math>
7896
| (b) <math>t = 13.1 s s</math>
7897
| (c) <math>t = 14.7 s</math>
7898
|-
7899
|[[Image:draft_Samper_722607179-Solidmelting151.png|400px|t = 15.5 s]]
7900
|[[Image:draft_Samper_722607179-Solidmelting155.png|400px|t = 15.9  s]]
7901
|[[Image:draft_Samper_722607179-Solidmelting159.png|400px|t =16.3 s]]
7902
|- style="text-align: center; font-size: 75%;"
7903
| (d) <math>t = 15.1 s</math>
7904
| (e) <math>t = 15.5 s</math>
7905
| (f) <math>t = 15.9  s</math>
7906
|-
7907
|[[Image:draft_Samper_722607179-Solidmelting163.png|400px|t = 16.7 s]]
7908
|[[Image:draft_Samper_722607179-Solidmelting167.png|400px|t = 17.1 s]]
7909
| colspan="2"|[[Image:draft_Samper_722607179-Solidmelting171.png|400px]]
7910
|- style="text-align: center; font-size: 75%;"
7911
| (g) <math>t =16.3 s</math>
7912
| (h) <math>t = 16.7 s</math>
7913
| (i) <math>t = 17.1 s</math>
7914
|- style="text-align: center; font-size: 75%;"
7915
| colspan="3" | '''Figure 166:''' Detailed model. Snapshots of rod melting.
7916
|}
7917
7918
In the graph of Figure [[#img-167|167]] the time evolution of the temperature on the external surfaces of the rod and the shell is plotted. This graph confirms that the rod is heating faster that the shell.
7919
7920
<div id='img-167'></div>
7921
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 62%;max-width: 100%;"
7922
|-
7923
|[[Image:draft_Samper_722607179-detailedTemperature.png|372px|Detailed model. Time evolution of the temperature on the rod and shell external surfaces.]]
7924
|- style="text-align: center; font-size: 75%;"
7925
| colspan="1" | '''Figure 167:''' Detailed model. Time evolution of the temperature on the rod and shell external surfaces.
7926
|}
7927
7928
The melting of the rod starts at <math display="inline">{t=9.53 s}</math>. The time evolution of the accumulated melted volume of the solid part of the domain is illustrated in the graph of Figure [[#img-155|155]].
7929
7930
<div id='img-168'></div>
7931
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;max-width: 100%;"
7932
|-
7933
|[[Image:draft_Samper_722607179-detailedMeltedVolume.png|360px|Detailed model. Time evolution of the melted volume.]]
7934
|- style="text-align: center; font-size: 75%;"
7935
| colspan="1" | '''Figure 168:''' Detailed model. Time evolution of the melted volume.
7936
|}
7937
7938
Note that in this case the peak of melting velocity that occurs in the basic model (see Figure [[#img-155|155]]) does not happen. That is because in this problem the phase change of the solid is essentially governed by the temperature. The plastic zones are local and they do not create any plastic mechanisms, as it occurs for the basic problem (see Figure [[#img-159|159]]).
7939
7940
In Figure [[#img-169|169]] the deformation of the solid structure at <math display="inline">{t=15.0 s}</math> is illustrated in a 3D-view with an  enlargement factor of <math display="inline">{10^5}</math>.  <div id='img-169'></div>
7941
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 100%;"
7942
|-
7943
|[[Image:draft_Samper_722607179-detailedDeformed.png|480px|Detailed model. 3D-view of the deformed configuration of the shell and the rod at t=15.0 s (enlargement factor=10⁵).]]
7944
|- style="text-align: center; font-size: 75%;"
7945
| colspan="1" | '''Figure 169:''' Detailed model. 3D-view of the deformed configuration of the shell and the rod at t=15.0 s (enlargement factor=<math>{10^5}</math>).
7946
|}
7947
7948
==6.5 Summary and conclusions==
7949
7950
In this chapter the potentialities of the PFEM have been tested for the solution of two very complex problems related to the melting of two structures in two specific situations of NPP severe accident. The simulations involved many critical phenomena for a finite element analysis, such as free-surface flows, thermal fluid-structure interaction, and phase change.
7951
7952
The objectives of the study have been reached for both the basic and detailed models proposed. In particular, the phenomena required by the contractor have been modeled successfully.
7953
7954
The analyses were run by using a slightly different model than the proposed one. Preliminary tests indicated that with the  geometry and viscosity of the corium proposed initially by the contractor, the fluid would splash and fall down without melting the adjacent structure. For this reason, with the agreement of the contractor, some input data were modified. In particular, the corium was modeled using a higher value of the viscosity and, for the basic model, a smaller initial height was considered.
7955
7956
Due to the complexity of the analysis, some approximations and simplifications were accepted. The purpose of most of these was to reduce the computational cost of the analyses.
7957
7958
Among these assumptions, the most invasive ones were the use of the constant temperature for the corium and the simplified modeling of the different material properties of the solid melted volume. In particular, the constant temperature for the corium may have accelerated the melting of the structure. A more accurate simulation would be obtained by considering the corium to have a variable temperature and a higher heat capacity.
7959
7960
=Chapter 7. Conclusions and future lines of research=
7961
7962
7963
The aim of this chapter is to summarize all the work done for this thesis highlighting its innovative points and main contributions. The thesis opens new perspectives to several developments. The future lines of work are given in the last section of this chapter.
7964
7965
==7.1 Contributions==
7966
7967
The objective of this thesis was the derivation and the implementation in a <math display="inline">C++</math> code of a unified formulation for fluid and solid mechanics, fluid-structure interaction and thermal coupled problems using the PFEM.
7968
7969
The finite element procedure was derived starting from the Velocity formulation presented in Chapter [[#2 Velocity-based formulations for compressible materials|2]]. In the same chapter, the mixed Velocity formulation was obtained exploiting the linearized form of the Velocity formulation. The mixed Velocity-Pressure scheme is based on a two-step Gauss-Seidel procedure where first the linear momentum equations are solved for the velocity increments and then the continuity equation is solved for the pressure in the updated configuration. Linear interpolation was used for both the velocity and the pressure fields.  Both velocity-based finite element Lagrangian procedures were derived first for a general compressible material and then particularized for the hypoelastic model. The hypoelastic solid elements generated from the Velocity formulation and the mixed Velocity-Pressure formulations were called V and VP elements, respectively. The V and the VP elements were validated for several benchmark problems for elastoplastic compressible structures. It has been shown that both elements are convergent for all the numerical examples analyzed.
7970
7971
In Chapter [[#3 Unified stabilized formulation for quasi-incompressible materials|3]] the Unified Stabilized formulation for quasi-incompressible materials was derived. This numerical procedure essentially consists of the mixed Velocity-Pressure formulation derived in Chapter [[#3 Unified stabilized formulation for quasi-incompressible materials|3]] where for the continuity equation the FIC stabilized form is used. The FIC stabilization was derived for the case of quasi-incompressible fluids and was extended also to hypoelastic quasi-incompressible solids. The stabilized element for quasi-incompressible solids was called VPS element. The solution schemes for solving quasi-incompressible Newtonian fluids and hypoelastic solids with the stabilized mixed Velocity-Pressure formulation  were given and explained in detail. The entire Section [[#3.4 Free surface flow analysis|3.4]] was devoted to the analysis of free surface fluids with the Unified stabilized formulation. First the Particle Finite Element Method (PFEM) was explained analyzing its advantages and disadvantages. An innovative strategy for modeling slip conditions in a Lagrangian way has also described. The excellent mass preservation properties of the PFEM-FIC stabilized formulation by solving a variety of 2D and 3D free surface flow problems involving surface waves, water splashing, violent impact of flows with containment walls  and mixing of fluids. Also the conditioning of the scheme was studied and the effect of the bulk modulus on the numerical scheme highlighted. It has been shown that using a scaled value for the bulk modulus, pseudo bulk modulus, in the linear momentum tangent matrix improves the conditioning and the global convergence of the linear system. A simple and efficient strategy for calibrating <math display="inline">a priori</math> the optimum value for the pseudo bulk modulus was also given. The strategy was successfully validated for two benchmark problems for free surface flow. In the last section of Chapter  [[#3 Unified stabilized formulation for quasi-incompressible materials|3]] the unified stabilized formulation for fluids and solids at the incompressible limit was validated comparing the numerical results to both experimental tests and numerical results from other formulations. It was shown that the method is convergent to the expected solution. Specific attention was given to the study of the boundary conditions highlighting the beneficious effect of using slip conditions for inviscid fluids and coarse meshes.
7972
7973
Chapter [[#4 Unified formulation for FSI problems|4]] was devoted to the application of the Unified Stabilized to FSI problems. It was shown that this operation requires a small implementation effort. It is only required to assemble properly the global linear system and to detect the interface exploiting the capability of the PFEM for detecting the contours. It was shown that, depending on the specific need, one may choose any of  the V, the VP and the VPS elements  for modeling hypoelastic solids. The numerical solution given by the proposed scheme was validated with analytical solutions and numerical results of other formulations for free surface FSI problems.
7974
7975
In Chapter [[#5 Coupled thermal-mechanical formulation|5]] the Unified formulation was used for solving coupled thermal-mechanical problem. The coupling was ensured via a staggered scheme and for the temperature field the same linear shape functions of the velocity and pressure fields were used.  The numerical method was tested with several  solid and fluid dynamics  problems involving the temperature. The algorithm for the phase change modeling was explained and an explicative numerical example was given.
7976
7977
In Chapter [[#6 Industrial application: PFEM Analysis Model of NPP Severe Accident|6]] the Unified stabilized thermal coupled formulation has been tested with an industrial problem. The analysis concerned the melting of two structures in two specific situations of Nuclear Power Plant (NPP) accident. The simulations involved many critical issues for a finite element analysis, such as free-surface flows, thermal fluid-structure interaction and phase change. Because of the complexity of the analysis, some approximations and simplifications were accepted by the contractor. The project ended successfully and all the phenomena required by the contractor in the specifications were modeled.
7978
7979
The innovative points and the contributions of this thesis can be summarized by the following list:
7980
7981
<ol>
7982
7983
<li>Derivation and implementation of the updated version of the FIC stabilization technique for quasi-incompressible fluids; </li>
7984
<li>Study of the mass conservation in free surface fluid flows; </li>
7985
<li> Extension of the FIC procedure to quasi-incompressible solids;</li>
7986
<li> Analysis of the conditioning of the partitioned solution method for quasi-incompressible fluids;</li>
7987
<li>Development of a procedure for improving the conditioning of the linear system; </li>
7988
<li>Development of an innovative technique for treating the boundary nodes in the PFEM; </li>
7989
<li> Slight modifications to the unified scheme for FSI problems proposed in [63];</li>
7990
<li>Development of a simplified technique for phase change modeling. </li>
7991
7992
</ol>
7993
7994
==7.2 Lines for future work==
7995
7996
This section outlines the possible lines of research opened by this thesis. The following enumeration summarizes most of these
7997
7998
<ol>
7999
8000
<li> '''''More solid consitutive laws'''''. Although it has been shown that the hypoelastic model can be used for modeling many critical non-linear solid mechanics problems, it could very useful to extend the Unified formulation to other type of constitutive models (such as a hyperelastic model). In particular, it would be interesting to introduce a displacements-based model. This would be important also for showing the generality of the formulation;</li>
8001
<li> '''''Improvement of the PFEM thechnology'''''. The PFEM technology implemented in this work allows the solving of one fluid only. Extension of the PFEM to multi-fluid would be of interest for many practical problems. For example, it has been remarked that in the viscous fluid problems the effect of the air has not been taken into account. This for some problems, i.e. casting process, can affect the truthfulness of the analyses. Previous works showed that the PFEM can deal with multi-fluid problems and this formulation does not constrain the implementation of a technology for the study of these problems. Another interesting improvement would be the modeling of the contact between solids and between a deformable solid and a rigid wall. This would extend the applicability of the proposed formulation;</li>
8002
<li> '''''Generalization of the Lagrangian modeling of slip conditions'''''. In this work it has been presented a simple but efficient strategy for modeling slip boundary conditions in a Lagrangian way. The method has been applied to horizontal and vertical walls only. The technology has to be generalized for every type of walls. This could be done using Lagrangian multipliers or rotation matrices;</li>
8003
<li> '''''Convergence analysis'''''. The convergence studies performed in this work were deveoted to show qualitatively the convergence of the formulation more than evaluating and computing the convergence rate of the method. In order to do this, the L2 norm error should be studied. Its computation for free surface flow problems may be complex due to the high deformation of the domain and its discretization;</li>
8004
<li> '''''Improvement  and  validation of the  thermal coupled  solver'''''. The coupling of the unified formulation with the thermal problem was not the main objective of this thesis. The thermal coupling has been analyzed essentially with the purpose of showing the versatility and potential of the unified PFEM formulation. For this reason, for the analysis of thermal coupled problems some simplifications have been assumed, above all for phase change modeling. The thermal and the mechanical solution were validated separately, however the work misses  validation examples for  coupled thermal-mechanical problem;</li>
8005
<li> '''''Code optimization'''''. The computational time of the analysis is a not secondary issue. The C++ code implemented and used for this thesis may have not optimized routines that reduce the speed of the analyses. More work should be inverted in the future for reducing the computational cost of each function of the code;</li>
8006
<li> '''''Parallelization'''''. The C++ code used in this work is fully sequential. In order to have a computational technology able to compete with commercial software, a parallel version of the code should be implemented.</li>
8007
8008
</ol>
8009
8010
==BIBLIOGRAPHY==
8011
8012
8013
[1]   http://islandbreath.blogspot.it/2013/07/fukushima-burns-on-and-on.htm).
8014
8015
[2]   www.thehindu.com/news/national/tamil-nadu/kudankulam-plant-not-to-
8016
draw-water-from-pechipaarai-dam-tamirabharani/article3443184.ece                  and 
8017
http://www.world-nuclear-news.org/nn  nuclear  plans  forge  ahead  160609.htm.
8018
8019
[3] C. Antoci, M. Gallati, and S. Sibilla. A numerical approach to the testing of the fission hypothesis. Computers and Structures, 85:879–890, 2007.
8020
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