Scipedia: Papers Repository of the International Centre for Numerical Methods in Engineering (CIMNE)

CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter

María Jesús Samper — Mon, 02 Sep 2019 15:43:30 +0200

In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem.

Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection

María Jesús Samper — Mon, 02 Sep 2019 15:32:20 +0200

We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the project of the pressure gradient and adding the difference between the pressure Laplacian and divergence of this new field to the incompressibility equations, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for stationary and the transient of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP.

Numerical modeling of magma withdrawal during explosive caldera-forming eruptions

María Jesús Samper — Mon, 02 Sep 2019 15:26:38 +0200

We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such eruptions as a piston-like system in which the host rock is assumed to subside as a coherent rigid solid. Magma behaves as a Newtonian incompressible fluid below the exsolution level and as a compressible gas-liquid mixture above this level. We consider caldera-forming eruptions within the frame of fluid-structure interaction problems, in which the flow-governing equations are written using an arbitrary Lagrangian-Eulerian (ALE) formulation. We propose a numerical procedure to solve the ALE governing equations in the context of a finite element method. The numerical methodology is based on a staggered algorithm in which the flow and the structural equations are alternatively integrated in time by using separate solvers. The procedure also involves the use of the quasi-Laplacian method to compute the ALE mesh of the fluid and a new conservative remeshing strategy. Despite the fact that we focus the application of the procedure toward modeling caldera-forming eruptions, the numerical procedure is of general applicability. The numerical results have important geological implications in terms of magma chamber dynamics during explosive caldera-forming eruptions. Simulations predict a nearly constant velocity of caldera subsidence that strongly depends on magma viscosity. They also reproduce the characteristic eruption rates of the different phases of an explosive calderaforming eruption. Numerical results indicate that the formation of vortices beneath the ring fault, which may allow mingling and mixing of parcels of magma initially located at different depths in the chamber, is likely to occur for low-viscosity magmas. Numerical results confirm that exsolution of volatiles is an efficient mechanism to sustain explosive caldera-forming eruptions and to explain the formation of large volumes of ignimbrites.

Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations

María Jesús Samper — Mon, 02 Sep 2019 15:17:21 +0200

In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the time discretization we focus on a fully implicit, monolithic scheme. We provide some error estimates for the fully discrete solution which show that the velocity is first order accurate in the time step and attains optimal order accuracy in the mesh size for the given spatial interpolation, both in the spaces L2(Ω) and H10(Ω); the pressure solution is shown to be order 12 accurate in the time step and also optimal in the mesh size. These estimates are proved assuming only a weak compatibility condition on the approximating spaces of velocity and pressure, which is satisfied by equal order interpolations.

Pressure stability in fractional step finite element methods for incompressible flows

María Jesús Samper — Mon, 02 Sep 2019 15:12:17 +0200

The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second order scheme in which part of the extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.

Transmission conditions with constraints in finite element domain decomposition methods for flow problems

María Jesús Samper — Mon, 02 Sep 2019 14:35:32 +0200

This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD algorithm is based on the iterative update of the boundary conditions on the interfaces between the subregions, the so‐called transmission conditions. The transmission conditions involve the essential and natural boundary conditions of the weak form of the problem, and should ensure strong continuity of the velocity and weak continuity of the traction. As a first approach, the transmission conditions are interpolated using the classical Lagrange interpolation functions. Conservation problems might arise when two adjacent subdomains have a sensibly different mesh spacing. In order to conserve any desired quantity of interest, an interface constraining is introduced: continuity of the transmission conditions are constrained under a scalar conservation equation. An example of mass conservation illustrates the algorithm.

A stabilized finite element method for generalized stationary incompressible flows

María Jesús Samper — Mon, 02 Sep 2019 14:25:30 +0200

In this paper we describe a finite element formulation for the numerical solution of the stationary Navier-Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection-diffusion-reaction equations and then we apply it to linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.

On the constitutive modeling of coupled thermomechanical phase-change problems

María Jesús Samper — Wed, 03 Apr 2019 11:24:22 +0200

This paper deals with a thermodynamically consistent numerical formulation for coupled thermoplastic problems including phase-change phenomena and frictional contact. The final goal is to get an accurate, efficient and robust numerical model, able for the numerical simulation of industrial solidification processes. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. All the material properties have been considered to be temperature dependent. The constitutive behavior has been defined by a thermoviscous/elastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. The constitutive model proposed accounts for a continuous transition between the initial liquid state, the intermediate mushy state and the final solid state taking place in a solidification process. In particular, a pure viscous deviatoric model has been used at the initial fluid-like state. A thermomecanical contact model, including a frictional hardening and temperature dependent coupled potential, is derived within a fully consistent thermodinamical theory. The numerical model has been implemented into the computational finite element code COMET developed by the authors. Numerical simulations of solidification processes show the good performance of the computational model developed.

Advances in FE explicit formulation for simulation of metalforming processes

María Jesús Samper — Fri, 18 Jan 2019 12:21:24 +0100

This paper presents some advances of finite element explicit formulation for simulation of metal forming processes. Because of their computational efficiency, finite element programs based on the explicit dynamic formulation proved to be a very attractive tool for the simulation of metal forming processes. The use of explicit programs in the sheet forming simulation is quite common, the possibilities of these codes in bulk forming simulation in our opinion are still not exploited sufficiently. In our paper, we will consider aspects of bulk forming simulation.

We will present new formulations and algorithms developed for bulk metal forming within the explicit formulation. An extension of a finite element code for the thermomechanical coupled analysis is discussed. A new thermomechanical constitutive model developed by the authors and implemented in the program is presented.

A new formulation based on the so-called split algorithm allows us to use linear triangular and tetrahedral elements in the analysis of large plastic deformations characteristic to forming processes. Linear triangles and tetrahedra have many advantages over quadrilateral and hexahedral elements. Linear triangles and tetrahedra based on the standard formulations exhibit volumetric locking and are not suitable for large plastic strain simulation. The new formulation allows to overcome this difficulty.

New formulations and algorithms have been implemented in the finite element code Stampack developed at the International Centre for Numerical Methods in Engineering in Barcelona. Numerical examples illustrate some of the possibilities of the finite element code developed and validate new algorithms.

Combination of the critical displacement method with a damage model for structural instability analysis

María Jesús Samper — Fri, 04 Jan 2019 12:07:16 +0100

The paper describes the extension of the critical
displacement method (CDM), presented by Oñate and Matias in 1996,
to the instability analysis of structures with non-linear material
behaviour using a simple damage model. The extended CDM is useful
to detect instability points using a prediction of the critical
displacement field and a secant load-displacement relationship
accounting for material non-linearities. Examples of application
of CDM to the instability analysis of structures using bar and
solid finite elements are presented.

A scalar damage model with a shear retention factor for the analysis of reinforced concrete structures: theory and validation

María Jesús Samper — Mon, 07 Jan 2019 13:57:07 +0100

A local isotropic single parameter scalar model that can simulate the mechanical behaviour of quasi-brittle materials, such as concrete, is described. The constitutive law needs the mechanical characteristics and the fracture energy of concrete to be completely defined. The damage parameter is obtained directly from the value of an equivalent effective stress in order to reduce the computing effort. Due to the unique damage parameter, this model is suitable for the study of quasi-static problems involving monotonically increasing loads. The problem of localisation and mesh dependency have been partially overcome by using an enhanced local method in which a characteristic internal length related to the mesh dimension is employed instead of the characteristic fracture length. In this work, the model was enriched further with the introduction of a shear retention factor that accounts for the friction between the two surfaces of a crack. These new features assure a real improvement of the damage model, maintaining nevertheless its simplicity and low computing cost and making it suitable for the practical solution of large scale problems. Several numerical simulations of experimental tests, concerning fracture tests on concrete specimens and beams failing in shear, have been performed for the validation of the model. The main results from the numerical analyses are described and compared with the experimental ones.

A finite point method for elasticity problems

María Jesús Samper — Mon, 07 Jan 2019 13:45:56 +0100

The basis of the finite point method (FPM) for the fully meshless solution of elasticity problems in structural mechanics is described. A stabilization technique based on a finite calculus procedure is used to improve the quality of the numerical solution. The efficiency and accuracy of the stabilized FPM in the meshless analysis of simple linear elastic structural problems is shown in some examples of applications.

Rotation‐free triangular plate and shell elements

María Jesús Samper — Wed, 12 Dec 2018 12:50:18 +0100

The paper describes how the finite element method and the finite volume method can be successfully combined to derive two new families of thin plate and shell triangles with translational degrees of freedom as the only nodal variables. The simplest elements of the two families based on combining a linear interpolation of displacements with cell centred and cell vertex finite volume schemes are presented in detail. Examples of the good performance of the new rotation‐free plate and shell triangles are given.

Shape variable definition with C0 , C1 and C2 continuity functions

María Jesús Samper — Fri, 23 Nov 2018 14:27:28 +0100

The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).

An anisotropic elastoplastic constitutive model for large deformation analysis of fibre reinforced composites

María Jesús Samper — Tue, 20 Nov 2018 13:32:06 +0100

In this work a generalized anisotropic elastoplastic constitutive model in large deformation is presented. It is used for the analysis of fiber-reinforced composite materials in the frame of the finite element method. Mixing theory is applied to simulate the behavior of the composite material. The elastic anisotropic behavior is simulated with classical elasticity theory, while that of a non-proportional anisotropic solid is simulated by means of the proposed generalized anisotropic elastoplastic model. The approach assumes the existence of a real anisotropic space and of a fictitious isotropic space where a mapped fictitious problem is solved. Both spaces are related by means of a linear transformation using a fourth order tensor incorporating complete information on the real anisotropic material. Details of the numerical implementations of the model into a non-linear, large deformations finite element solution scheme are provided. Examples of application showing the performance of the model for the analysis of fiber-reinforced composite materials are given.

A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation

María Jesús Samper — Tue, 20 Nov 2018 13:22:17 +0100

A stabilized finite element formulation for incompressible viscous flows is derived. The starting point are the modified Navier-Stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus (FIC) procedure. Application of the standard finite element Galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields. The FIC method also provides a natural explanation for the stabilization terms appearing in all equations for both the Navier-Stokes and the simpler Stokes equations. Transient solution schemes with enhanced stabilization properties are also proposed. Finally a procedure for computing the stabilization parameters is presented.

Desarrollos y aplicaciones de modelos de fractura en la escuela de ingenieros de caminos de Barcelona

María Jesús Samper — Wed, 20 Nov 2019 11:44:53 +0100

El artículo es una panorámica de los aspectos teóricos y algunas aplicaciones prácticas de los modelos de fractura desarrollados por diversos grupos en la Escuela de Ingenieros de Caminos de Barcelona (EICB) durante los últimos quince años para el análisis no lineal de estructuras. La motivación fundamental para el desarrollo de estos modelos se centra en el análisis de la seguridad de estructuras de hormigón en masa y armado. La mayor parte de los modelos se basan en la teoría de daño continuo y utilizan el método de los elementos finitos para la solución numérica. Los modelos de daño se han extendido y aplicado también con éxito al análisis de diversas estructuras de edificios históricos. Los desarrollos más recientes de estos modelos en la EICB incluyen la predicción de fenómenos de localización en estructuras de hormigón y el análisis del comportamiento no lineal de estructuras con materiales compuestos. De todos estos modelos se presentan en el artículo unas breves pinceladas, las aplicaciones más relevantes y las referencias donde pueden encontrarse los detalles sobre cada caso.

Adaptivity based on error estimation for viscoplastic softening materials

María Jesús Samper — Thu, 24 Oct 2019 13:27:02 +0200

This paper focuses on the numerical simulation of strain softening mechanical problems. Two problems arise: (1) the constitutive model has to be regular and (2) the numerical technique must be able to capture the two scales of the problem (the macroscopic geometrical representation and the microscopic behavior in the localization bands). The Perzyna viscoplastic model is used in order to obtain a regularized softening model allowing to simulate strain localization phenomena.
This model is applied to quasistatic examples. The viscous regularization of quasistatic processes is also discussed: in quasistatics, the internal length associated with the obtained band width is no longer only a function of the material parameters but also depends on the boundary value problem (geometry and loads, specially loading velocity). An adaptive computation is applied to softening viscoplastic materials showing strain localization. As the key ingredient of the adaptive strategy, a residual type error estimator is generalized to deal with such highly nonlinear material model. In several numerical examples the adaptive process is able to detect complex collapse modes that are not captured by a first, even if fine, mesh. Consequently, adaptive strategies are found to be essential to detect the collapse mechanism and to assess the optimal location of the elements in the mesh.

Error estimation including pollution assessment for nonlinear finite element analysis

María Jesús Samper — Thu, 24 Oct 2019 13:11:48 +0200

A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure. Pollution errors are taken into account by a feedback strategy, that is, an error estimate based on local computations is used as the input of the pollution analysis. This estimator is used in the frame of an adaptive procedure. Numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions. Moreover, one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms.

Error estimation for adaptive computations on shell structures

María Jesús Samper — Thu, 24 Oct 2019 11:33:01 +0200

The finite element discretization of a shell structure introduces two kinds of
errors: the error in the functional approximation and the error in the geometry
approximation. The first is associated with the finite dimensional interpolation
space and it is present in any finite element computation. The latter is associated
with the piecewise polynomial approximation of a curved surface and is much more
relevant in shell problems than in any other standard 2D or 3D computation. In
the shells framework, formerly the quality control of the finite element solution
has been carried out using flux projection a posteriori error estimators. This
technique exhibits two main drawbacks: 1) the flux smoothing averages stress components
over different elements that may have different physical meaning if the tangent
planes are different and 2) the error estimation process uses only the approximate
solution and hence, the discretized forces and the computational mesh: the data
describing the real geometry and load is therefore not accounted for. In this
work, a residual type error estimator introduced for standard 2D finite element
analysis is generalized to shell problems. This allows to easily account for the
real original geometry of the problem in the error estimation procedure and precludes
the necessity of comparing generalized stress components between non coplanar
elements. This estimator is based on approximating a reference error associated
with a refined reference mesh. In order to build up the residual error equation
the computed solution must be represented (projected) on the reference mesh. The
use of thin shell finite elements requires a proper formulation in order to preclude
shear locking. Following an idea of Donea and Lamain, the interpolation of the
rotations is not unique and requires a particular technique to transfer the information
from the computational mesh to the reference mesh. This technique is also developed
in this work and may be used in any adaptive evolution problem where the solution
must be transferred from one mesh to another.