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

The Papers Repository of the International Centre for Numerical Methods in Engineering (CIMNE) is an online archive aimed at collecting, preserving, and disseminating digital copies of scientific papers published by researchers of the International Center for Numerical Methods in Engineering (CIMNE).

The goal of the repository is to widely disseminate the results of the research performed at CIMNE, by sharing the preprints or postprints -depending on the article sharing policies of the original ... show more

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A finite element reduced order model based on adaptive mesh refinement and artificial neural networks
J. Baiges, R. Codina, I. Castañar, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2019). (prepint) pp. 1-14

Abstract
In this work a reduced order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN-ROM) is presented. The idea is to run a high-fidelity simulation by using an adaptively refined finite element mesh, and compare the results obtained with those of a coarse mesh finite element model. From this comparison, a correction forcing term can be computed for each training configuration. A model for the correction term is built by using an artificial neural network, and the final reduced order model is obtained by putting together the coarse mesh finite element model, plus the artificial neural network model for the correction forcing term. The methodology is applied to non-linear solid mechanics problems, transient quasi-incompressible flows, and a fluid-structure interaction problem. The results of the numerical examples show that the FAN-ROM is capable of improving the simulation results obtained in coarse finite element meshes at a reduced computational cost.
Abstract
In this work a reduced order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN-ROM) is presented. The [...]
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Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows
O. Guasch, R. Codina

Comput. Methods Appl. Mech. Engrg., (2013). Vol. 261-262, pp. 145-166

Abstract
Abstract Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus on the orthogonal subgrid scale (OSS) finite element method and make an analysis of the statistical behavior of its stabilization terms in the quasi static approximation. This is done by resorting to results from classical statistical fluid mechanics concerning two point velocity, pressure and combined correlation functions of various orders. Given a fine enough mesh with characteristic element size h in the inertial subrange of a turbulent flow, it is shown that the rate of transfer of subgrid kinetic energy provided by the OSS stabilization terms does not depend on h and that it equals the molecular physical dissipation rate (up to a dimensionless constant that only depends on the finite element shapes) for a proper redesign of the standard parameters of the formulation. This is a noteworthy fact taking into account that the subgrid stabilization terms do not arise from physical considerations, but from the mathematical necessity to allow equal interpolation for the pressure and velocity fields, as well as to control convection. Therefore, the obtained results contribute somehow to the line of reasoning supporting that pure numerical approaches (i.e., without introducing additional physical models) could probably suffice in the LES simulation of turbulent flows
Abstract
Abstract Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus [...]
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A general algorithm for compressible and incompressible flow—Part II. tests on the explicit form
O. Zienkiewicz, K. Morgan, B. Sai, R. Codina, M. Vasquez

Int. J. Numer. Meth. Fluids (1995). Vol. 20 (8-9), pp. 887-913

Abstract
The algorithm introduced in Part I of this paper is applied in its explicit form to a variety of problems in order to demonstrate its wide range of applicability and excellent performance. Examples range from nearly incompressible, viscous, flows through transonic applications to high speed flows with shocks. In most examples linear triangular elements are used in the finite element approximation, but some use of quadratic approximation, again in triangles, indicates satisfactory performance even in the case of severe shocks.
Abstract
The algorithm introduced in Part I of this paper is applied in its explicit form to a variety of problems in order to demonstrate its wide range of applicability and excellent [...]
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A bonded discrete element method for modeling ship–ice interactions in broken and unbroken sea ice fields
O. Jou Devesa, M. Celigueta, S. Latorre, F. Arrufat, E. Oñate

Computational Particle Mechanics (2019). Vol. 6 (4), pp 739–765

Abstract
This work investigates the failure patterns of ice cakes and floe-icewhen loaded by amoving and sloping structure (ice-breaking ships and cones). In the paper, we introduce the most frequently encountered ice infested scenarios, the main characteristics of ice-breaking ships and the predicted failure modes of floe-ice depending on the loading conditions, the structure type and the ice feature dimensions and thickness. For the simulations, a local bonded discrete element method (DEM) is used to model sea ice and its fractures. The packing of bonded spherical particles which reproduce the ice continuum can break due to ship–ice interactions and the failure modes are studied. A set of validation simulations are first carried out. A level ice sheet breaking against an installed ice-breaking cone with different slope angles is studied, and the results are compared with other DEM simulations. Then, a group of bonded DEM simulations are performed to predict the different failure modes produced when an ice-breaking ship bow contacts with ice cakes and floe-ice of different dimensions and thickness, typical in broken ice fields. Finally, the study of breaking a continuous level ice sheet is carried out by modeling with the bonded DEM an “infinite” large domain of sea ice and loaded by a single-degree-of-freedom model of an ice-breaking ship.
Abstract
This work investigates the failure patterns of ice cakes and floe-icewhen loaded by amoving and sloping structure (ice-breaking ships and cones). In the paper, we introduce [...]
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Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques
Ö. Türk, R. Codina

Applied Numerical Mathematics (2019). (preprint) Vol. 145, pp. 188-200

Abstract
Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method approach can successfully be adapted for the eigenproblem to rectify the associated problems such as the existence of zero diagonal entries in the resulting algebraic system. Two different schemes, namely, the standard penalisation with a small penalty parameter, and the iterative penalisation that enables relatively large parameters, are implemented. The employment of the latter leads to a so-called inhomogeneous generalised eigenvalue problem which requires a special attention. A feasible solution strategy is presented which is adapted from a procedure based on Newton's method proposed for the corresponding standard (inhomogeneous) eigenvalue problems. Concerning the spatial discretisation, among other possible options, the Chebyshev spectral collocation method based on expanding the unknown fields in tensor product of Chebyshev polynomials is employed. It is shown that the method constitutes a novel way of efficiently examining the approximate eigensolutions of the Stokes operator with the use of Chebyshev spectral collocation method directly, without a decoupling of velocity and pressure.
Abstract
Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method [...]
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Logarithmic conformation reformulation in viscoelastic flow problems approximated by a VMS-type stabilized finite element formulation
L. Moreno García, R. Codina, J. Baiges, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 354, pp. 706-731

Abstract
The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina (2014) for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids.
Abstract
The log-conformation reformulation, originally proposed by Fattal and Kupferman (2004), allows computing incompressible viscoelastic problems with high Weissenberg numbers [...]
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Dynamic term-by-term stabilized finite element formulation using orthogonal subgrid-scales for the incompressible Navier–Stokes problem
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 349, pp. 701-721

Abstract
In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for the Navier–Stokes problem that can be viewed as a variational multiscale (VMS) method under some assumptions. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual-based formulations, the introduction of two velocity subscale components. They represent the components of the convective and the pressure gradient terms, respectively, of the momentum equation that cannot be captured by the finite element mesh. A key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field. The method ensures stability in anisotropic space–time discretizations, which is proved using numerical analysis for a linearized problem and demonstrated in classical numerical tests. The work includes a detailed description of the proposed formulation and several numerical examples that serve to justify our claims.
Abstract
In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for [...]
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Time-dependent semidiscrete analysis of the viscoelastic fluid flow problem using a variational multiscale stabilized formulation
G. Barrenechea, E. Castillo, R. Codina

Journal of Numerical Analysis (2019). Vol. 39 (2), pp. 792-819

Abstract
In this article we analyse a stabilized finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and robustness in the different benchmarks tested in the viscoelastic framework, and permitting the use of equal interpolation of the unknown fields. We first present results about a linearized subproblem, for which well-posedness and stability results can be proved. Then, the semidiscrete nonlinear time-dependent case is addressed using a fixed point theorem, which allows us to prove existence of a semidiscrete solution, along with error estimates.
Abstract
In this article we analyse a stabilized finite element formulation recently proposed to approximate viscoelastic fluid flows. The formulation has shown to have accuracy and [...]
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Solution of low Mach number aeroacoustic flows using a Variational Multi-Scale finite element formulation of the compressible Navier–Stokes equations written in primitive variables
C. Bayona, J. Baiges, R. Codina

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 344, pp. 1073-1103

Abstract
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic flows. We develop a Variational Multi-Scale formulation to stabilize the finite element discretization by including the orthogonal, dynamic and non-linear subscales, together with an implicit scheme for advancing in time. Three additional features define the proposed numerical scheme: the splitting of the pressure and temperature variables into a relative and a reference part, the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility, and the approximation of the dynamic stabilization matrix for the time dependent subscales. We also include a weak imposition of implicit non-reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes. The order of accuracy of the method is verified for two- and three-dimensional linear and quadratic elements using steady manufactured solutions. Several benchmark flow problems are studied, including transient examples and aeroacoustic applications.
Abstract
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic [...]
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Computational aeroacoustics to identify sound sources in the generation of sibilant /s/
A. Pont, O. Guasch, J. Baiges, R. Codina, A. Hirtum

Int. J. Numer. Meth. Biomed. Engng (2019). Vol. 35 (1), e3153

Abstract
A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between the upper and lower incisors and then impinges the space between the lower incisors and the lower lip. The flow eddies in that region become a source of direct aerodynamic sound, which is also diffracted by the speech articulators and radiated outwards. The numerical simulation of these phenomena is complex. The spectrum of an /s/ typically peaks between 4 and 10 kHz, which implies that very fine computational meshes are needed to capture the eddies producing such high frequencies. In this work, a large‐scale computation of the aeroacoustics of /s/ has been performed for a realistic vocal tract geometry, resorting to two different acoustic analogies. A stabilized finite element method that acts as a large eddy simulation model has been adopted to solve the flow dynamics. Also, a numerical strategy has been implemented that allows the determination, in a single computational run, of the separate contribution of the sound diffracted by the upper incisors from the overall radiated sound. Results are presented for points located close to the lip opening showing the relative influence of the upper teeth depending on frequency.
Abstract
A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between [...]
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Finite element modeling of nonlinear reaction-diffusion-advection systems of equations
S. Komala, R. Codina

Int. J. Numer. Meth. Heat and Fluid Flows (2018). Vol. 28 (11), pp. 2688-2715

Abstract
Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.
Abstract
Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear [...]
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Stationary and time-dependent numerical approximation of the lid-driven cavity problem for power-law fluid flows at high Reynolds numbers using a stabilized finite element formulation of the VMS type
A. Aguirre, E. Castillo, M. Cruchaga, R. Codina, J. Baiges

Journal of Non-Newtonian Fluid Mechanics (2018). Vol. 257, pp. 22-43

Abstract
In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian fluids, this benchmark case has been extensively studied by many authors for low and moderate Reynolds numbers (up to Re=10,000), giving place to steady flows, using stationary and time-dependent approaches. For more convective flows, the solution becomes unstable, describing an oscillatory behavior. The critical Reynolds number which gives place to this time-dependent fluid dynamics has been defined over a wide range 7, 300 ≲ Re ≲ 35, 000, using different numerical approaches. In the non-Newtonian case, the cavity problem has not been studied deeply for high Reynolds number (Re > 10, 000), specifically, in the oscillatory time-dependent case. A VMS formulation is presented to be validated using existing results, to determine flow conditions at which the instability appears, and lastly, to establish new benchmark solutions for high-Reynolds numbers fluid flows using the power-law model. Obtained results show a good agreement with those reported in the references, and new data related with the oscillatory behavior of the flow has been found for the non-Newtonian case. In this regard, time-dependent flows show dependence on both Reynolds number and power-law index, and the unsteady starting point has been determined for all studied cases. It is determined that the critical Reynolds number (Rec) that defines the first Hopf bifurcation for Newtonian fluid flow is ranged between 8,100 ≲ Rec ≲ 8, 250, whereas for power-law indexes n=0.5 and n=1.5, it is 7,100 ≲ Rec ≲ 7,200 and 18,250 ≲ Rec ≲ 18,500, respectively.
Abstract
In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian [...]
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On hp convergence of stabilized finite element methods for the convection–diffusion equation
R. Codina

Sociedad Española de Matemática Aplicada (2018). Vol. 75 (4), pp. 591-606

Abstract
This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods discussed are classical-residual based stabilization techniques and also projection-based stabilization methods. The theoretical impossibility of obtaining an optimal convergence rate in terms of the polynomial order p for all possible Péclet numbers is explained. The key point turns out to be an inverse estimate that scales as p2. The use of this estimate is not needed in a particular case of (H1-)projection-based methods, and therefore the theoretical lack of convergence described does not exist in this case.
Abstract
This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods [...]
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Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations
C. Bayona, R. Codina, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2018). Vol. 337, pp. 501-526

Abstract
This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled L2-norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.
Abstract
This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. [...]
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Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows
A. Pont, R. Codina, J. Baiges, O. Guasch

Journal of Computational Physics (2018). Vol. 363, pp. 11-29

Abstract
The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. Moreover, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure time derivative. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries without using any tailored damping model will be presented.
Abstract
The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations [...]
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Stabilized mixed finite element method for the M1 radiation model
Q. Schmid, Y. Mesri, E. Hachem, R. Codina

Comput. Methods Appl. Mech. Engrg., (2018). Vol. 335, pp. 69-90

Abstract
In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial differential equations, the so-called M1 model. A Variational Multiscale method is developed for this system, and tested on some illustrative benchmarks. The question of dealing with heterogeneous physical properties is also considered, and treated by means of an immersed method. It combines a level-set approach for representation of interfaces, mixing laws to build effective physical properties and an anisotropic mesh adaptation process, all these ingredients leading to an accurate description of the interface.
Abstract
In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial [...]
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Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling
C. Bayona, J. Baiges, R. Codina

Int. J. Numer. Meth. Fluids (2018). Vol. 86 (5), pp. 313-328

Abstract
A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.
Abstract
A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. [...]
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Variational Multiscale error estimators for solid mechanics adaptive simulations: An Orthogonal Subgrid Scale approach
J. Baiges, R. Codina

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 325, pp. 37-55

Abstract
In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. The main idea is to consider a rich model for the subgrid scales as an error estimator. The subscales are considered to belong to a space orthogonal to the finite element space (Orthogonal Subgrid Scales) and we take into account their contribution both in the element interiors and on the element boundaries (Subscales on the Element Boundaries). A simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the Variational Multiscale Method. Numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non-linear cases.
Abstract
In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. [...]
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Interpolation with restrictions between finite element meshes for flow problems in an ALE setting
A. Pont, R. Codina, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 110 (13), pp. 1203-1226

Abstract
The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission of information between finite element meshes. Because the Lagrangian projection of results from one mesh to another is a dissipative method, a new conservative interpolation method has been developed. A series of constraints, such as the conservation of mass or energy, are applied to the interpolated arrays through Lagrange multipliers in an error minimization problem, so that the resulting array satisfies these physical properties while staying as close as possible to the original interpolated values in the L2 norm. Unlike other conservative interpolation methods that require a considerable effort in mesh generation and modification, the proposed formulation is mesh independent and is only based on the physical properties of the field being interpolated. Moreover, the performed corrections are neither coupled with the main calculation nor with the interpolation itself, for which reason the computational cost is very low.
Abstract
The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission [...]
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Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations
E. Castillo, R. Codina

Mathematical Models and Methods in Applied Sciences (2017). Vol. 51 (4), pp. 1407-1427

Abstract
In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.
Abstract
In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field [...]
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An adaptive Fixed-Mesh ALE method for free surface flows
J. Baiges, R. Codina, A. Pont, E. Castillo

Comput. Methods Appl. Mech. Engrg., (2017). Vol. 313, pp. 159-188

Abstract
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.
Abstract
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background [...]
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Finite element approximation of the viscoelastic flow problem: A non-residual based stabilized formulation
E. Castillo, R. Codina

Computers & Fluids (2017). Vol. 142, pp. 72-78

Abstract
In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns σ−u−p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.
Abstract
In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based [...]
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A pseudo-compressible variational multiscale solver for turbulent incompressible flows
L. Yang, S. Badia, R. Codina

Computational Mechanics (2016). Vol. 58 (6), pp. 1051-1069

Abstract
In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with 5123cells on a 12 cores 64 GB ram machine.
Abstract
In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. [...]
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Residual‐based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics
O. Guasch, P. Sánchez-Martín, A. Pont, J. Baiges, R. Codina

Int. J. Numer. Meth. Fluids (2016). Vol. 82, pp. 839-857

Abstract
The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf–sup compatibility condition can be built in the case of no mean flow, that is, for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual‐based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared‐solenoidal mean flow, and for the aeolian tone generated by flow past a two‐dimensional cylinder.
Abstract
The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics [...]
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A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
Ö. Türk, D. Boffi, R. Codina

Comput. Methods Appl. Mech. Engrg., (2016). Vol. 310, pp. 886-905

Abstract
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.
Abstract
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) [...]
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Variational multi-scale finite element approximation of the compressible Navier-Stokes equations
C. Bayona, J. Baiges, R. Codina

Int. J. Numer. Meth. Heat and Fluid Flows (2016). Vol. 26 (3-4), pp. 1240-1271

Abstract
Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value A complete investigation of the stabilized formulation of the compressible problem is addressed.
Abstract
Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written [...]
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Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics
O. Guasch, A. Pont, J. Baiges, R. Codina

Computers and Fluids (2016). Vol. 133, pp. 129-139

Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.
Abstract
The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) [...]
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A stabilized finite element method for the mixed wave equation in an ALE framework with application to diphthong production
O. Guasch, M. Arnela, R. Codina, H. Espinoza

Acta Acustica united with Acustica (2016). (preprint) Vol. 102, pp. 94-106

Abstract
Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity field. Indeed, this becomes the natural option in many problems, such as those involving waves propagating in moving domains, because the equations can easily be set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Yet, when attempting a standard Galerkin finite element solution (FEM) for them, it turns out that an inf-sup compatibility constraint has to be satisfied, which prevents from using equal interpolations for the approximated acoustic pressure and velocity fields. In this work it is proposed to resort to a subgrid scale stabilization strategy to circumvent this condition and thus facilitate code implementation. As a possible application, we address the generation of diphthongs in voice production.
Abstract
Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity [...]
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Two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched shape functions
E. Castillo, J. Baiges, R. Codina

Journal of Non-Newtonian Fluid Mechanics (2015). Vol. 225, pp. 37-53

Abstract
The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we present a finite element method to solve the two immiscible fluid flow problem using the level set method. When the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The method is tested in two problems: the first example consists in a sloshing case that involves the interaction of a Giesekus and a Newtonian fluid. This example shows that the enriched pressure functions permit the exact resolution of the hydrostatic rest state. The second example is the classical jet buckling problem used to validate our method. To permit the use of equal interpolation between the variables, we use a variational multiscale formulation proposed recently by Castillo and Codina in Comput. Methods Appl. Mech. Engrg. 279 (2014) 579–605, that has shown very good stability properties, permitting also the resolution of the jet buckling flow problem in the the range of Weissenberg number 0 < We < 100, using the Oldroyd-B model without any sign of numerical instability. Additional ingredients of the work are the inclusion of a discontinuity capturing technique for the constitutive equation and some comparisons between a monolithic resolution and a fractional step approach to solve the viscoelastic fluid flow problem from the point of view of computational requirements
Abstract
The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we [...]
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On some time marching schemes for the stabilized finite element approximation of the mixed wave equation
H. Espinoza, R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 296, pp. 295-326

Abstract
In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved.
Abstract
In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, [...]
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A three‐field stabilized finite element method for fluid‐structure interaction: elastic solid and rigid body limit
E. Hachem, S. Feghali, T. Coupez, R. Codina

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 104 (7), pp. 566-584

Abstract
We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing an extra stress in the momentum equation. The obtained three‐field velocity, pressure and stress system is solved using a stabilized finite element method. The key feature of this unified formulation is the ability to describe different kind of interactions between the fluid and the structure, which can be either elastic or a perfect rigid body, without the need of treating this last case via penalization. The level‐set method combined with a dynamic anisotropic mesh adaptation is used to track the fluid‐solid interface.
Abstract
We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing [...]
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Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non‐matching meshes
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2015). Vol. 104 (7), pp. 624-654

Abstract
In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential.
Abstract
In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational [...]
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Numerical modeling of chlorine concentration in water storage tank
R. Codina, J. Principe, C. Muñoz, J. Baiges

Int. J. Numer. Meth. Fluids (2015). Vol. 79 (2), pp. 84-107

Abstract
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed.
Abstract
In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution [...]
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Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics
S. Badia, R. Codina, R. Planas

Archives of Comp. Meths. Engng. (2015). Vol. 22 (4), pp. 621-636

Abstract
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.
Abstract
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]
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Finite element dynamical subgrid-scale model for low Mach number flows with radiative heat transfer
M. Ávila, R. Codina, R. Planas

Int. J. Numer. Meth. Heat and Fluid Flows (2015). Vol. 25, pp. 1361-1384

Abstract
Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields. Design/methodology/approach – The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations. Findings – As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show global conservation of thermal energy. Originality/value – The original contribution of the work is the proposal of keeping the VMS splitting into the nonlinear coupling between the low Mach number and the radiative transport equations, its numerical evaluation and the description of its properties.
Abstract
Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative [...]
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First, second and third order fractional step methods for the three-field viscoelastic flow problem
E. Castillo, R. Codina

Journal of Computational Physics (2015). Vol. 296, pp. 113-137

Abstract
In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier–Stokes problem. These methods can be understood as an inexact LU block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order fractional step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the fractional step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.
Abstract
In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and [...]
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Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows
O. Colomés, S. Badia, R. Codina, J. Principe

Comput. Methods Appl. Mech. Engrg., (2015). Vol. 285, pp. 32-63

Abstract
In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. After a brief review of these models, we discuss some implementation aspects particularly relevant to the simulation of turbulent flows, namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used. We analyze the energy conservation (and numerical dissipation) of the alternative VMS formulations, which is numerically evaluated. In the numerical study, we have considered three well known problems: the decay of homogeneous isotropic turbulence, the Taylor–Green vortex problem and the turbulent flow in a channel. We compare the results obtained using different VMS models, paying special attention to the effect of using orthogonal subscale spaces. The VMS results are also compared against classical LES scheme based on filtering and the dynamic Smagorinsky closure. Altogether, our results show the tremendous potential of VMS for the numerical simulation of turbulence. Further, we study the sensitivity of VMS to the algorithmic constants and analyze the behavior in the small time step limit. We have also carried out a computational cost comparison of the different formulations. Out of these experiments, we can state that the numerical results obtained with the different VMS formulations (as far as they converge) are quite similar. However, some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different. The dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness.
Abstract
In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by [...]
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Variational MultiScale based dissipation models for the estimation of Atmospheric Seeing
J. Baiges, R. Codina

Computers and Fluids (2015). Vol. 107, pp. 141-154

Abstract
In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational Multiscale turbulence model, its main feature being that the numerical mechanisms which are used to deal with stability issues (convection and the inf–sup condition for incompressible flows) do also take care of the modeling of turbulence. Based on this turbulence model, we develop the expressions for the viscous and thermal dissipations, ∊∊num and χnum, which are later used for evaluating the constant of structure of the refraction index Cn2 following the classical model developed by Tatarski. Numerical examples show the behavior of the proposed numerical scheme when applied to turbulent flow practical cases, which include a convective boundary layer, the flow inside a transfer optics room, and a telescope enclosure.
Abstract
In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational [...]
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Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 579-605

Abstract
In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to use equal interpolation for the problem unknowns σ–u–p (elastic deviatoric stress–velocity–pressure) and to stabilize dominant convective terms. Starting from residual-based stabilized formulations, the proposed method introduces a term-by-term stabilization which is shown to have a superior behavior when there are stress singularities. A general discontinuity-capturing technique for the elastic stress component is also proposed, which allows one to eliminate the local oscillations that can appear when the Weissenberg number We is high and the fluid flow finds an abrupt change in the geometry. The formulations are tested in the classical 4:1 planar contraction benchmark up to We=5 in the inertial case, with Reynolds number Re=1, and up to We=6.5 in the quasi non-inertial case, with Re=0.01. The standard Oldroyd-B constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation.
Abstract
In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to [...]
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Stabilized stress–velocity–pressure finite element formulations of the Navier–Stokes problem for fluids with non-linear viscosity
E. Castillo, R. Codina

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 554-578

Abstract
The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability in the calculation of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf–sup conditions, whereas the dominant convection requires a stabilized formulation in any case. This paper proposes two stabilized schemes of Sub-Grid Scale (SGS) type, differing in the definition of the space of the sub-grid scales, and both allowing to use the same interpolation for the variables σ–u–p (deviatoric stress, velocity and pressure), even in problems where the convection component is dominant and the velocity–stress gradients are high. Another aspect considered in this work is the non-linearity of the viscosity, modeled with constitutive models of quasi-Newtonian type. This paper includes a description of the proposed methods, some of their implementation issues and a discussion about benefits and drawbacks of a three-field formulation. Several numerical examples serve to justify our claims.
Abstract
The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. [...]
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Stability, convergence and accuracy of stabilized finite element methods for the wave equation in mixed form
S. Badia, R. Codina, H. Espinoza

SIAM J. Numer. Anal. (2014). Vol. 52 (4), pp. 1729-1752

Abstract
In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns. The variational forms corresponding to different functional settings are treated in a unified manner through the introduction of length scales related to the unknowns. Stability and convergence analysis is performed together with numerical experiments. It is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods.
Abstract
In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element [...]
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Large eddy simulation of low Mach number flows using a dynamical and nonlinear finite element subgrid scale model
M. Ávila, R. Codina, J. Principe

Computers and Fluids (2014). Vol. 99, pp. 44-66

Abstract
Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is to analyze the behavior of our numerical method in the large eddy simulation (LES) of thermally coupled turbulent flows at low Mach number. Methods: Our numerical method is a stabilized finite element approximation based on the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account results in a stable formulation. The quality of the final approximation (accuracy, efficiency as LES model) depends on the particular subscale model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. Another important contribution of this work is the extension of the orthogonal subgrid scale method widely tested for incompressible flows to variable density flows, using a density-weighted L 2 product to define the orthogonality of the subscales and the finite element spaces. Results: Referring to numerical testing, we present numerical results for a laminar testcase validation that shows the dissipative behavior of the different stabilized methods. Then, we present results of the numerical simulation of two turbulent flow problems, the turbulent channel flow with large temperature differences in the wall normal direction at Res ¼ 180, and the turbulent thermally driven cavity with aspect ratio 4. The behavior of the method is evaluated by comparison against results available in the literature obtained using LES and direct numerical simulation (DNS). They are explained based on a careful analysis of the dissipative structure of the method, showing the physical interpretation of the subgrid scale method presented. Conclusion: The material presented here is a clear indication of the potential of the method to model all kinds of turbulent thermally coupled flows. The formulation is the same in laminar and turbulent regimes
Abstract
Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is [...]
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A Sommerfeld non-reflecting boundary condition for the wave equation in mixed form
H. Espinoza, R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2014). Vol. 276, pp. 122-148

Abstract
In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D.
Abstract
In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial [...]
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Error analysis of discontinuous Galerkin methods for the Stokes problem under minimum regularity
S. Badia, R. Codina, T. Gudi, J. Guzmán

IMA Journal of Numerical Analysis (2014). Vol. 34 (2), pp. 800-819

Abstract
In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs to[H10(Ω)]d and the pressurep∈L20(Ω). First, we analyse standard DG methods assuming that the right-hand side f belongs to[H−1(Ω)∩L1(Ω)]d.A DG method that is well defined for f belonging to [H−1(Ω)]d is then investigated. The methods under study include stabilized DG methods using equal-order spaces and inf–sup stable ones where the pressure space is one polynomial degree less than the velocity space.
Abstract
In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs [...]
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On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations
R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2013). Vol. 263, pp. 158-168

Abstract
The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries. Rather than working with this three-field formulation, fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes, although other strategies could be devised. This procedure is applied to four elliptic problems, namely, the convection–diffusion equation (in the diffusion dominated regime), the Stokes problem, the Darcy problem and the Maxwell problem. We justify some well known dG methods with some modifications that in fact allow to improve the performance of the original methods, particularly when the physical properties are discontinuous.
Abstract
The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation [...]
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Immersed stress method for fluid structure interaction using anisotropic mesh adaption
E. Hachem, S. Feghali, R. Codina, T. Coupez

Int. J. Numer. Meth. Engng. (2013). Vol. 94 (9), pp. 805-825

Abstract
This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight.
Abstract
This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with [...]
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Anisotropic adaptive meshing and monolithic variational multiscale method for fluid-structure interaction
E. Hachem, S. Feghali, R. Codina, T. Coupez

Computers and Structures (2013). Vol. 122, pp. 88-100

Abstract
This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. The fluid–solid interfaces are captured using a level-set method. A new a posteriori error estimate, based on the length distribution tensor approach and the associated edge based error analysis, is then used to ensure an accurate capturing of the discontinuities at the fluid–solid interface. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The presence of the structure will be taken into account by means of an extra stress tensor in the Navier–Stokes equations. The system is solved using a stabilized three-field, stress, velocity and pressure finite element (FE) formulation. It consists in the decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales and also in the efficient enrichment of the extra constraint. We assess the accuracy of the proposed formulation by simulating 2D and 3D time-dependent numerical examples such as: falling disk in a channel, turbulent flows behind an airfoil profile and flow behind an immersed vehicle.
Abstract
This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. [...]
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A variational multiscale method with subscales on the element boundaries for the Helmholtz equation
J. Baiges, R. Codina

Int. J. Numer. Meth. Engng. (2013). Vol. 93 (6), pp. 664-684

Abstract
In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method.
Abstract
In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite [...]
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On unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics
S. Badia, R. Codina, R. Planas

Journal of Computational Physics (2013). Vol. 234, pp. 399-416

Abstract
In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation is the fact that it always converges to the physical solution, even for singular ones. A detailed set of numerical experiments have been performed in order to validate our approach.
Abstract
In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]
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A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes
J. Baiges, R. Codina, F. Henke, S. Shahmiri, W. Wall

Int. J. Numer. Meth. Engng. (2012). Vol. 90 (5), pp. 636-658

Abstract
In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element‐discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations.
Abstract
In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic [...]
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A third‐order velocity correction scheme obtained at the discrete level
H. Owen, R. Codina

Int. J. Numer. Meth. Fluids (2012). Vol. 69 (1), pp. 52-72

Abstract
In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed into a discrete pressure Poisson equation and a velocity extrapolation is used. In Badia et al. (IJNMF, 2008, p. 351), where the method was introduced, the discrete Laplacian that appears in the pressure Poisson equation is approximated by a continuous one using an extrapolation for the pressure. In this work we explore the possibility of actually solving the discrete Laplacian. This introduces significant differences because the pressure extrapolation is avoided and only a velocity extrapolation is needed. Our numerical results indicate that it is the second‐order pressure extrapolation which makes third‐order methods unstable. Instead, second‐order velocity extrapolations do not lead to instabilities. Avoiding the pressure extrapolation allows to obtain stable solutions in problems that become unstable when the Laplacian is approximated. A comparison with a pressure correction scheme is also presented to verify the well‐known fact that the use of a second order pressure extrapolation leads to instabilities. Therefore we conclude that it is the combination of a velocity correction scheme with a discrete Laplacian that allows to obtain a stable third‐order scheme by avoiding the pressure extrapolation.
Abstract
In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed [...]
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A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions
S. Badia, R. Codina

SIAM J. Numer. Anal. (2012). Vol. 50 (2), pp. 398-417

Abstract
A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl 0; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal-based interpolations are known to suffer from spurious convergence upon mesh refinement.
Abstract
A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous [...]
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Stokes, Maxwell and Darcy: A single finite element approximation for three model problems
S. Badia, R. Codina

Applied Numerical Mathematics (2012). Vol. 62 (4) pp. 246-263

Abstract
In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantages of our approach. In particular, for Darcyʼs problem we are able to design stabilized methods that yield optimal convergence both for the primal and the dual problems. In the case of Maxwellʼs problem, the formulation we propose allows one to use continuous finite element interpolations that converge optimally to the continuous solution even if it is non-smooth. Once the formulation is presented for the three model problems independently, we also show how it can be used for a problem that combines all the operators of the independent problems. Stability and convergence is achieved regardless of the fact that any of these operators dominates the others, a feature not possible for the methods of which we are aware.
Abstract
In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We [...]
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A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem
S. Badia, R. Codina

Applied Mathematics and Computation (2011). Vol. 218 (8), pp. 4276-4294

Abstract
Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG–dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functins continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.
Abstract
Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are [...]
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The Fixed‐Mesh ALE approach for the numerical simulation of floating solids
J. Baiges, R. Codina, H. Coppola-Owen

Int. J. Numer. Meth. Fluids (2011). Vol. 67 (8), pp. 1004-1023

Abstract
In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. We pay special attention to the tracking of the various interfaces and their intersections, and to the approximate imposition of coupling conditions between the solid and the fluid.
Abstract
In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature [...]
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Finite element dynamical subgrid-scale approximation of low Mach number flow equations
M. Ávila, J. Principe, R. Codina

Journal of Computational Physics (2011). Vol. 230, pp. 7988-8009

Abstract
In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number equations, which are obtained as a limit of the compressible Navier Stokes equations in the small Mach number. In contrast to the commonly used Boussinesq approximation, this model permits to take volumetric deformation into account. Although the former is more general than the later, both systems have similar mathematical structure and their numerical approximation can suffer the same type of instabilities. We propose a stabilized finite element approximation based on the the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account, results in a stable formulation. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. The first ingredient permits to obtain an improved time discretization scheme (higher accuracy, better stability, no restrictions on the time step size). The second ingredient permits to prove global conservation properties. It also allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view. Numerical tests show that nonlinear and dynamic subscales give more accurate solutions than classical stabilized methods.
Abstract
In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number [...]
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Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2011). Vol. 87 (1-5), pp. 386-411

Abstract
Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized finite element approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non‐conforming part, a hybrid formulation for the latter and a simple approximation for the unknowns involved in the hybrid problem.
Abstract
Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are [...]
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Thermal coupling of fluid and structural response of a tunnel induced by fire
B. Schrefler, R. Codina, F. Pesavento, J. Principe

Int. J. Numer. Meth. Engng. (2010). Vol. 87 (1-5), pp. 361-385

Abstract
In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling strategy is based on a Dirichlet/Neumann non‐overlapping domain decomposition of the problem, which is carried out by developing a master code that controls solvers dedicated to the fluid mechanics and to the solid mechanics simulation. The computational fluid dynamics formulation consists of a stabilized finite element approximation of the low Mach number equations based on the subgrid scale concept, that allows us to deal with convection‐dominated problems and to use equal order interpolation of velocity and pressure. The thermo‐structural model of the tunnel vault, that considers a multiphase porous material where pores are partly filled with liquid and partly by gas, is specially devised for the simulation of concrete at high temperatures and consists of balance equations for mass conservation of dry air, mass conservation of water species (both in the liquid and gaseous state), enthalpy conservation and linear momentum conservation taking phase changes into account. The developed algorithm is applied to the problem of the response of a tunnel to a fire. We consider the combustion process as a heat release which can vary usually from 1thinspaceMW for small car fires to 100 MW for catastrophic fires. The released heat is transferred to the concrete walls of the tunnel which could cause extensive and heavy damage of the structure.
Abstract
In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling [...]
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The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems
J. Baiges, R. Codina

Int. J. Numer. Meth. Engng. (2009). Vol. 81 (12), pp. 1529-1557

Abstract
In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed‐mesh (FM)‐ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM‐ALE allows for the use of a single background mesh to solve both the fluid and the structure.
Abstract
In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. [...]
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A free surface finite element model for low Froude number mould filling problems on fixed meshes
H. Coppola-Owen, R. Codina

Int. J. Numer. Meth. Fluids (2011). Vol. 66 (7), pp. 833-851

Abstract
The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position of the interface and the resulting flow field increases. The usual two‐phase flow model provides poor results for such flow. In order to overcome the difficulties, a free surface model that applies boundary conditions at the interface accurately is used. Moreover, the use of wall laws on curved boundaries also fails in the case of low Froude number flows. To solve this second problem, we combine wall laws with ‘do nothing’ boundary conditions. A special feature of our approach is that ‘do nothing’ boundary conditions are only applied in the normal direction. These two key ingredients together with the Level Set method allow us to simulate three‐dimensional mould filling problems borrowed directly from the foundry.
Abstract
The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position [...]
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Approximation of the inductionless MHD problem using a stabilized finite element method
R. Planas, S. Badia, R. Codina

Journal of Computational Physics (2011). Vol. 230 (8), pp. 2977-2996

Abstract
In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers.
Abstract
In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD [...]
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Approximation of the thermally coupled MHD problem using a stabilized finite element method
R. Codina, N. Hernández-Silva

Journal of Computational Physics (2011). Vol. 230 (4), pp. 1281-1303

Abstract
A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization terms, which serve several purposes. First, convective dominated flows in the Navier–Stokes and the heat equation can be dealt with. Second, there is no restriction in the choice of the interpolation spaces of all the variables and, finally, flows highly coupled with the magnetic field can be accounted for. Different aspects related to the design of the final fully discrete and linearized algorithm are also discussed.
Abstract
A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization [...]
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Spatial approximation of the radiation transport equation using a subgrid-scale finite element method
M. Ávila, R. Codina, J. Principe

Comput. Methods Appl. Mech. Engrg., (2011). Vol. 200 (5-8), pp. 425-438

Abstract
In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.
Abstract
In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition [...]
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Long-term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical sub-grid scale modeling
S. Badia, R. Codina, J. Gutiérrez-Santacreu

SIAM J. Numer. Anal. (2010). Vol. 48 (3), pp. 1013-1037

Abstract
Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L 2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation.
Abstract
Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure [...]
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Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling
R. Codina, J. Principe, M. Ávila

Int. J. Numer. Meth. Heat and Fluid Flow (2010). Vol. 20 (5), pp. 492-516

Abstract
The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. This strategy allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view and discuss important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation
Abstract
The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation [...]
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Stabilized continuous and discontinuous Galerkin techniques for Darcy flow
S. Badia, R. Codina

Comput. Methods Appl. Mech. Engrg., (2010). vol. 199 (25-28), pp. 1654-1667

Abstract
We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2-norm.
Abstract
We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. [...]
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On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes
J. Principe, R. Codina

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (21-22), pp. 1386-1402

Abstract
In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated. In order to incorporate the distortion of the mesh into this approximation, we transform the equation for the subgrid scale within each element to the shape-regular reference domain. The expression for the subgrid scale arises from an approximate Fourier analysis and the identification of the wave number direction where instabilities are most likely to occur. The final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation.
Abstract
In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting [...]
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The dissipative structure of variational multiscale methods for incompressible flows
J. Principe, R. Codina, F. Henke

Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (13-16), pp. 791-801

Abstract
In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.
Abstract
In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. [...]
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A numerical strategy to compute optical parameters in turbulent flow. Application to telescopes
R. Codina, J. Baiges, D. Pérez-Sánchez, M. Collados

Computers and Fluids (2010). Vol. 39 (1), pp. 87-98

Abstract
We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier-Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.
Abstract
We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible [...]
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Approximate imposition of boundary conditions in immersed boundary methods
R. Codina, J. Baiges

Int. J. Numer. Meth. Engng. (2009). Vol. 80 (11), pp. 1379-1405

Abstract
We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition
Abstract
We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element [...]
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Computational aeroacoustics of viscous low speed flows using subgrid scale finite element methods
O. Guasch, R. Codina

Journal of Computational Acoustics (2009). Vol. 17 (3), pp. 309-330

Abstract
A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure is followed that makes use of Lighthill's acoustic analogy in the frequency domain. The procedure has been conceptually divided into three steps. In the first one, the incompressible Navier–Stokes equations are solved to obtain the flow velocity field. In the second step, Lighthill's acoustic source term is computed from this velocity field and then Fourier transformed to the frequency domain. Finally, the acoustic pressure field is obtained by solving the corresponding inhomogeneous Helmholtz equation. All equations in the formulation are solved using subgrid scale stabilized finite element methods. The main ideas of the subgrid scale numerical strategy are outlined and its benefits when compared to the Galerkin approach are described. As numerical examples, the aerodynamic noise generated by flow past a two-dimensional cylinder and by flow past two cylinders in parallel arrangement are addressed.
Abstract
A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure [...]
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Mathematical models for thermally coupled low speed flows
J. Principe, R. Codina

Adv. Theor. Appl. Mech. (2009). Vol. 2 (2), pp. 93-112

Abstract
In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat exchange (the isentropic regime) this limit is well understood and rigorous results are available. When heat exchange is considered different simplified models can be obtained, the most famous being the Boussinesq approximation. Here a unified formal justification of these models is presented, paying special attention to the relation between the low Mach number and the Boussinesq approximations. Precise conditions for their validity are given for classical problems in bounded domains.
Abstract
In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat [...]
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Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems
S. Badia, R. Codina

SIAM J. Numer. Anal. (2009). Vol. 47 (3), pp. 1971-2000

Abstract
In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior
Abstract
In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart [...]
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A numerical approximation of the thermal coupling of fluids and solids
J. Principe, R. Codina

Int. J. Numer. Meth. Fluids (2009). Vol. 59 (11), pp. 1181-1201

Abstract
In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context.
Abstract
In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including [...]
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On a multiscale approach to the transient Stokes problem: Dynamic subscales and anisotropic space–time discretization
S. Badia, R. Codina

Applied Mathematics and Computation (2009). Vol. 207 (2), pp. 415-433

Abstract
In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. We define an anisotropic time–space discretization as a family of time–space partitions that does not satisfy the condition h2⩽Cδt with C uniform with respect to h and δt. Standard residual-based stabilization techniques are motivated by a multiscale approach, approximating the effect of the subscales onto the large scales. One of the approximations is to consider the subscales quasi-static (neglecting their time derivative). It is well known that these techniques are unstable for anisotropic time–space discretizations. We show that the use of dynamic subscales (where the subscales time derivatives are not neglected) solves the problem, and prove optimal convergence and stability results that are valid for anisotropic time–space discretizations. Also the improvements related to the use of orthogonal subscales are addressed.
Abstract
In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. [...]
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The fixed-mesh ALE approach for the numerical approximation of flows in moving domains
R. Codina, G. Houzeaux, H. Coppola-Owen, J. Baiges

Journal of Computational Physics (2009). Vol. 228 (5), pp. 1591-1611

Abstract
In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation to be presented falls within the category of fixed-grid methods. Even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed-grid method, our approach can be applied together with any technique to impose approximately boundary conditions, although we also describe the one we actually favor. Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.
Abstract
In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation [...]
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Finite element approximation of the three field formulation of the Stokes problem using arbitrary interpolations
R. Codina

SIAM J. Numer. Anal. (2009). Vol. 47 (1), pp. 699-718

Abstract
The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis.
Abstract
The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of [...]
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Subscales on the element boundaries in the variational two-scale finite element method
R. Codina, J. Principe, J. Baiges

Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (5-8), pp. 838-852

Abstract
In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. In particular, we consider the scalar convection–diffusion–reaction equation, the Stokes problem and Darcy’s problem. For these problems the transmission conditions are the continuity of the unknown and its fluxes through element boundaries. The former is automatically achieved by introducing a single valued subscale on the boundaries (for the conforming approximations we consider), whereas the latter provides the effective condition for approximating these values. The final result is that the subscale on the interelement boundaries must be proportional to the jump of the flux of the finite element component and the average of the subscale calculated in the element interiors.
Abstract
In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. [...]
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A stabilized finite element approximation of low speed thermally coupled flows
J. Principe, R. Codina

Int. J. Num. Meths. for Heat and Fluid Flows. (2008). Vol. 18 (7-8), pp. 835-867

Abstract
Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed. Design/methodology/approach – The formulation, that allows us to consider convection dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity-pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed. Findings – A full Newtown linearization turns out to be very efficient for steady-state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity-pressure and temperature happens to be more appropriate for transient problems. Research limitations/implications – A fractional step scheme, splitting also momentum and continuity equations, could be further analysed. Practical implications – The results presented in the paper are useful to decide the solution strategy for a given problem. Originality/value – The numerical implementation of a stabilized finite element approximation of thermally coupled flows is described. The implementation algorithm is developed considering several possibilities for the solution of the discrete nonlinear problem.
Abstract
Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number [...]
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Algebraic pressure segregation methods for the incompressible Navier-Stokes equations
S. Badia, R. Codina

Archives of Comp. Meths. Engng. (2008). Vol. 15 (3), pp. 1-52

Abstract
This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled.
Abstract
This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block [...]
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Finite element approximation of the modified Boussinesq equations using a stabilized formulation
R. Codina, J. González-Ondina, G. Díaz-Hernández, J. Principe

Int. J. Numer. Meth. Fluids (2008). Vol. 57 (9), pp. 1305-1322

Abstract
In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system of equations, namely, the need to use stable velocity‐depth interpolations, which can be overcome by the use of a stabilization technique. The one described in this paper is based on the splitting of the unknowns into their finite element component and the remainder, which we call the subgrid scale. We also discuss the treatment of high‐order derivatives of the mathematical model and describe the time integration scheme
Abstract
In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system [...]
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Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscale
R. Codina

Applied Numerical Mathematics (2008). Vol. 58 (3), pp. 264-283

Abstract
In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf–sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. It is shown that this formulation is stable and optimally convergent for an adequate choice of the algorithmic parameters on which the method depends.
Abstract
In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility [...]
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Finite element approximation of the hyperbolic wave equation in mixed form
R. Codina

Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (13-16), pp. 1305-1322

Abstract
The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector field to transform the problem into a first-order problem in space and time. We explain why the standard Galerkin method is inappropriate to solve this problem, and propose as alternative a stabilized finite element method that can be cast in the variational multiscale framework. The unknown is split into its finite element component and a remainder, referred to as subscale. As original features of our approach, we consider the possibility of letting the subscales to be time dependent and orthogonal to the finite element space. The formulation depends on algorithmic parameters whose expression is proposed from a heuristic Fourier analysis.
Abstract
The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector [...]
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On some fluid-structure iterative algorithms using pressure segregation methods. Applications to aeroelasticity
S. Badia, R. Codina

Int. J. Numer. Meth. Engng. (2007). Vol. 72 (1), pp. 46-71

Abstract
In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure segregation methods for the solution of the fluid problem and fixed point iterative algorithms for the fluid-structure coupling. These coupling algorithms are applied to the aeroelastic simulation of suspension bridges. We assess flexural and torsional frequencies for a given inflow velocity. Increasing this velocity we reach the value for which the flutter phenomenon appears.
Abstract
In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure [...]
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Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach
S. Badia, R. Codina

Int. J. Numer. Meth. Fluids (2008). Vol. 56 (4), pp. 351-382

Abstract
In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been replaced by a pressure Poisson equation obtained at the discrete level. In these methods it is the velocity instead of the pressure the extrapolated unknown. Moreover, predictor–corrector schemes are suggested, again motivated by the new monolithic system. Key implementation aspects are discussed, and a complete stability analysis is performed. We end with a set of numerical examples in order to compare these methods with classical pressure‐correction schemes.
Abstract
In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been [...]
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Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition
S. Badia, R. Codina

Numerische Mathematik (2007). Vol. 107 (4), pp. 533-557

Abstract
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.
Abstract
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using [...]
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Shockwaves in spillways with the particle finite element method
F. Salazar, J. San-Mauro, M. Celigueta, E. Oñate

Comp. Part. Mech. (2019).

Abstract
Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. These disturbances are propagated downstream and need to be considered in the design of the chute walls. In dam spillways, where flow rates are often high, this phenomenon can have significant implications for the cost and complexity of the solution. It has been traditionally analysed by means of reduced-scale experimental tests, as it has a clear three-dimensional character and therefore cannot be approached with two-dimensional numerical models. In this work, the ability of the particle finite element method (PFEM) to reproduce this phenomenon is analysed. PFEM has been successfully applied in previous works to problems involving high irregularities in free surface. First, simple test cases available in the technical bibliography were selected to be reproduced with PFEM. Subsequently, the method was applied in two spillways of real dams. The results show that PFEM is capable of capturing the shockwave fronts generated both in the contractions and in the expansions that occur behind the spillway piers. This suggests that the method may be useful as a complement to laboratory test campaigns for the design and hydraulic analysis of dam spillways with complex geometries.
Abstract
Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. [...]
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An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics
O. Guasch, R. Codina

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (45-48), pp. 4672-4689

Abstract
An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization.
Abstract
An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element [...]
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A finite element model for free surface flows on fixed meshes
H. Coppola-Owen, R. Codina

Int. J. Numer. Meth. Fluids (2007). Vol. 54 (10), pp. 1151-1171

Abstract
In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow.
Abstract
In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element [...]
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Dynamic subscales in the finite element approximation of thermally coupled incompressible flows
R. Codina, J. Principe

Int. J. Numer. Meth. Fluids (2007). Vol. 54 (6-8), pp. 707-730

Abstract
In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient. They are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. Apart from presenting the main properties of the formulation, we also discuss some computational aspects such as the linearization strategy or the way to integrate in time the equation for the subscales.
Abstract
In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to [...]
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Time dependent subscales in the stabilized finite element approximation of incompressible flow problems
R. Codina, J. Principe, O. Guasch, S. Badia

Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (21-24), pp. 2413-2430

Abstract
In this paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point is that we explore the properties of the discrete formulation that results allowing the subgrid-scales to depend on time. This apparently “natural” idea avoids several inconsistencies of previous formulations and also opens the door to generalizations.
Abstract
In this paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point [...]
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A finite element method for the solution of rotary pumps
G. Houzeaux, R. Codina

Computers and Fluids (2007). Vol. 36 (4), pp. 667-679

Abstract
We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are rotating, the intersection between them changes in time. Therefore, the computational domain should be recomputed in some way at each time step. The strategy used here consists in dividing a cycle into a certain number of time steps and obtaining different computational meshes for each of these time steps. The coupling between two consecutive time steps is achieved by interpolating the flow unknowns in a proper way. This geometrical decomposition enables one to have a plain control over the mesh, particularly in the zones of interest, which are the gap between the gears and the casing, and the engagement and disengagement zones of the gears.
Abstract
We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are [...]
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Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ALE framework
S. Badia, R. Codina

SIAM J. Numer. Anal. (2006). Vol. 44 (5), pp. 2159-2197

Abstract
In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semidiscrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.
Abstract
In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) [...]
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Numerical approximation of the heat transfer between domains separated by thin walls
R. Codina, G. Houzeaux

Int. J. Numer. Meth. Fluids (2006). Vol. 52 (9), pp. 963-986

Abstract
In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin solid wall. First of all, we state the problem in the whole domain with discontinuous physical properties. As an alternative and under certain assumptions on the separating walls, a classical Robin boundary condition between the fluid domains is obtained, thus eliminating the solid wall, and according to which the heat flux is proportional to the temperature difference between the two subdomains. Apart from discussing the relation between both approaches, we consider their numerical approximation, considering different alternatives for the first case, that is, the case in which temperatures are also computed in the solid wall.
Abstract
In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin [...]
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Stabilized finite element approximation of the stationary magneto-hydrodynamics equations
R. Codina, N. Hernández-Silva

Computational Mechanics (2006). Vol. 38 (4-5), pp. 344-355

Abstract
In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational concept. The linearization that yields a well posed linear problem is first identified, and for this linear problem the stabilization method is designed. The key point is the correct behavior of the stabilization parameters on which the formulation depends. It is shown that their expression can be obtained only on the basis of having a correct error estimate. For the stabilization parameters chosen, a stability estimate is proved in detail, as well as the convergence of the numerical solution to the continuous one. The method is then extended to nonlinear problems and its performance checked through numerical experiments.
Abstract
In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational [...]
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Finite element modeling of the lost foam casting process tackling back‐pressure effects
G. Houzeaux, R. Codina

Int. J. Num. Meths. for Heat and Fluid Flow (2006). Vol. 16 (5), pp. 573-589

Abstract
Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – Back-pressure effects are due to the interactions of many physical processes. The strategy proposed herein tries to model all these processes within a simple formula. The main characteristic of the model consists of assuming that the back-pressure is a known function of the external parameters (coating, temperature, gravity, etc.) that affects directly the heat transfer coefficient from the metal to the foam. The general framework of the simulation is a finite element model based on an arbitrary Lagrangian Eulerian (ALE) approach and the use of level set function to capture the metal front advance. Findings – After experimental tunings, the model provides a way to include the back-pressure effects in a simple way. Research limitations/implications – The method is not completely predictive in the sense that a priori tuning is necessary to calibrate the model. Practical implications – Provides more realistic results than classical models. Originality/value – The paper proposes a theoretical framework of a finite element method for the simulation of LFC process. The method uses an ALE method on a fixed mesh and a level-set function to capture metal front advance. It proposes an original formula for the heat transfer coefficient that enables one to include back-pressure effects
Abstract
Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – [...]
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The Characteristic‐Based Split (CBS) scheme—a unified approach to fluid dynamics
P. Nithiarasu, R. Codina, O. Zienkiewicz

Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1514-1546

Abstract
This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the original characteristic schemes are discussed. The important features of the CBS scheme are brought out by studying several problems of compressible and incompressible flows. All special consideration necessary for solving these problems are thoroughly discussed. The CBS scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided. Several invicid and viscous flow examples are also provided to demonstrate the unified CBS approach. For sample two‐dimensional codes, input files and instructions, the readers are referred to ‘www.nithiarasu.co.uk’
Abstract
This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the [...]
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Numerical comparison of CBS and SGS as stabilization techniques for the incompressible Navier–Stokes equations
R. Codina, H. Coppola-Owen, P. Nithiarasu, C. Liu

Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1677-1689

Abstract
In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. 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 other two approaches are particular cases of the subgrid scale (SGS) method. The first, obtained after an algebraic approximation of the subgrid scales, is very similar to the popular Galerkin/least‐squares (GLS) method, whereas in the second, the subscales are assumed to be orthogonal to the finite element space. It is shown that all these formulations display similar stabilization mechanisms, provided the stabilization parameter of the SGS methods is identified with the time step of the CBS approach. This paper provides the numerical experiments for the comparison of formulations made by Codina and Zienkiewicz in a previous article.
Abstract
In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. The first is the characteristic‐based split (CBS). [...]
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On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems
R. Codina, S. Badia

Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (23-24), pp. 2900-2918

Abstract
In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities and the pressure. The first family of schemes consists of classical fractional step methods, of which we discuss several possibilities for the pressure extrapolation and the time integration of first and second order. The second family consists of schemes based on an explicit treatment of the pressure in the momentum equation followed by a Poisson equation for the pressure. It turns out that this “staggered” treatment of the velocity and the pressure is stable. Finally, we present predictor-corrector methods based on the above schemes that aim to converge to the solution of the monolithic time integration method. Apart from presenting these schemes and check its numerical performance, we also present a complete set of stability results for the fractional step methods that are independent of the space stability of the velocity-pressure interpolation, that is, of the classical inf-sup condition.
Abstract
In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities [...]
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Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions
A. Coppola‐Owen, R. Codina

Int. J. Numer. Meth. Fluids (2005). Vol. 49 (12), pp. 1287-1304

Abstract
In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. We propose a solution based on enriching the pressure shape functions on the elements cut by the interface. The enrichment is used to enable the pressure gradient to be discontinuous at the interface, thus improving the ability to simulate the behaviour of fluids with different density under a gravitational force. The additional shape function used is local to each element and the corresponding degree of freedom can therefore be condensed prior to assembly, making the implementation quite simple on any existing finite element code.
Abstract
In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. [...]
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Error estimates for an operator-splitting method for incompressible flows
J. Blasco, R. Codina

Applied Numerical Mathematics (2004). Vol. 51 (1), pp. 1-17

Abstract
In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain first order error estimates in the time step size, both for the intermediate and the end-of-step velocities of the method; we also give some error estimates for the pressure solution.
Abstract
In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity [...]
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A finite element model for the simulation of lost foam casting
G. Houzeaux, R. Codina

Int. J. Numer. Meth. Fluids (2004). Vol. 46 (2), pp. 203-226

Abstract
In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes in play during a casting. We also briefly comment on the possible physical and numerical models to envisage the numerical simulation. Next we present a model which aims at solving “part of” the complexities of the casting, together with a simple energy budget that enables to obtain an equation for the velocity of the metal front advance. Once the physical model is established we develop a finite element method to solve the governing equations. The numerical and physical methodologies are then validated through the solution of a two and a three-dimensional example. Finally, we briefly discuss some possible improvements of the numerical model in order to catch more physical phenomena.
Abstract
In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes [...]
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Approximation of the incompressible Navier–Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes
R. Codina, O. Soto

Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (15-16), pp. 1403-1419

Abstract
This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. The former consists of adding a least-square form of the component orthogonal to the finite element space of the convective and pressure gradient terms; this allows to deal with convection-dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Likewise, we put special emphasis on the use of anisotropic grids. In particular, we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples. The preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described.
Abstract
This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. [...]
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A stabilized finite element predictor–corrector scheme for the incompressible Navier–Stokes equations using a nodal‐based implementation
R. Codina, A. Folch

Int. J. Numer. Meth. Fluids (2004). Vol. 44 (5), pp. 483-503

Abstract
A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to segregate the pressure and a nodal based implementation is presented in this paper. The stabilization consists of adding a least‐squares form of the component orthogonal to the finite element space of the convective and pressure gradient terms, which allows to deal with convection‐dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Finally, the nodal‐based implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements and thus making the calculations much faster. Some issues related to this implementation for fractional step and our predictor–corrector scheme, which is the main contribution of this paper, are discussed
Abstract
A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to [...]
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A Dirichlet/Neumann domain decomposition method for incompressible turbulent flows on overlapping subdomains
G. Houzeaux, R. Codina

Computers and Fluids (2004). Vol. 33 (5-6), pp. 771-782

Abstract
When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility is the use of domain decomposition methods. The domain decomposition method we present in this work aims at coupling overlapping subdomains in relative motion using a Dirichlet/Neumann coupling. The method is applied to the solution of incompressible and turbulent flows.
Abstract
When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility [...]
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An iteration-by-subdomain overlapping Dirichlet/Robin domain decomposition method for advection–diffusion problems
G. Houzeaux, R. Codina

Journal of Computational and Applied Mathematics (2003). Vol. 158 (2), pp. 243-276

Abstract
We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning of the computational domain. We derive interface equations to study the convergence of the method and show its properties through four numerical examples. The mathematical framework is general and can be applied to derive overlapping versions of all the classical nonoverlapping methods.
Abstract
We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning [...]
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A Chimera method based on a Dirichlet/Neumann(Robin) coupling for the Navier–Stokes equations
G. Houzeaux, R. Codina

Comput. Methods Appl. Mech. Engrg., (2003). Vol. 192 (31-32), pp. 3343-3377

Abstract
We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain method based on Dirichlet/Neumann(Robin) coupling. The DD method we propose is not only geometric but also algorithmic, for the solution on each subdomain is obtained on separate processes and the exchange of information between the subdomains is carried out by a master code. This strategy is very flexible as it requires almost no modification to the original numerical code. Therefore, only the master code has to be adapted to the numerical codes and the strategies used on each subdomain. As a basic flow solver, we a use stabilized finite element method.
Abstract
We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain [...]
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Stabilized finite element approximation of transient incompressible flows using orthogonal subscales
R. Codina

Comput. Methods Appl. Mech. Engrg., (2002). Vol. 191 (39-40), pp. 4295-4321

Abstract
In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.
Abstract
In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable [...]
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A numerical model to track two‐fluid interfaces based on a stabilized finite element method and the level set technique
R. Codina, O. Soto

Int. J. Numer. Meth. Fluids (2002). Vol. 40 (1-2), pp. 293-301

Abstract
The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. The stabilization technique employed allows us to use equal velocity–pressure interpolations and to deal with convection–dominated flows. The tracking of the fluid interface is based on the level set technique. A novel smoothing technique of this surface based on ideas of signal processing is presented.
Abstract
The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. [...]
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Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ale framework
S. Badia, R. Codina

Computing and Visualization in Science (2002). Vol. 4 (3), pp. 167-174

Abstract
In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semi-discrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.
Abstract
In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, [...]
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CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter
R. Codina, O. Zienkiewicz

Commun. Numer. Meth. Engng (2001). Vol 18 (2), pp. 99-112

Abstract
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.
Abstract
In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first [...]
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Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection
R. Codina, J. Blasco, G. Buscaglia, A. Huerta

Int. J. Numer. Meth. Fluids (2001). Vol. 37 (4), pp. 419-444

Abstract
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.
Abstract
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 [...]
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Numerical modeling of magma withdrawal during explosive caldera-forming eruptions
A. Folch, J. Martí i Castell, R. Codina

Journal of Geophysical Research (2001). Vol. 106 (B8), pp. 16 163-16 175

Abstract
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.
Abstract
We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such [...]
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Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations
J. Blasco, R. Codina

Applied Numerical Mathematics (2001). Vol. 38 (4), pp. 475-497

Abstract
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.
Abstract
In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the [...]
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Pressure stability in fractional step finite element methods for incompressible flows
R. Codina

Journal of Computational Physics (2001). Vol. 170 (1), pp. 112-140

Abstract
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.
Abstract
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 [...]
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Transmission conditions with constraints in finite element domain decomposition methods for flow problems
G. Houzeaux, R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol 17 (3), pp. 179-190

Abstract
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.
Abstract
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 [...]
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A stabilized finite element method for generalized stationary incompressible flows
R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol. 190 (20-21), pp. 2681-2706

Abstract
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.
Abstract
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 [...]
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Stabilization of incompressibility and convection through orthogonal sub-scale in finite element methods
R. Codina

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 190 (13-14), pp. 1579-1599

Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first of them is the stabilization thought the pressure gradient projection, which consists of adding a certain least-squares form of the difference between the pressure gradient and its L² projection onto the discrete velocity space in the variational equations of the problem. The second is a sub-grid scale method, whose stabilization effect is very similar to that of the Galerkin/least-squares method for the Stokes problem. It is shown here that the first method can also be recast in the framework of sub-grid scale method with a particular choice for the space of sub-scales. This leads to a new stabilization procedure, whose applicability to stabilize convection is also studied in this paper.
Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first [...]
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Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations
R. Codina, J. Blasco

Numerische Mathematik (2000). Vol. 87 (1), pp. 59-81

Abstract
The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the L2 norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this.
Abstract
The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. [...]
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Fourier analysis of an equal‐order incompressible flow solver stabilized by pressure gradient projection
G. Buscaglia, F. Basombrío, R. Codina

Int. J. Numer. Meth. Fluids (2000). Vol. 34 (1), pp. 65-92

Abstract
Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes equations, here denoted by pressure gradient projection (SPGP) method. The analysis is motivated by spurious waves that pollute the computed pressure in start‐up flow simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behaviour of stabilized methods with remarkable accuracy, even though the original Navier–Stokes setting must be significantly simplified to apply them. In the steady state case, good estimates for the stabilization parameters are obtained. In the transient case, spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though this implies additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons with Galerkin–least‐squares (GLS) method and Chorin's projection method are also addressed
Abstract
Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes [...]
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A nodal-based implementation of a stabilized Finite Element Method for Incompressible Flow Problems
R. Codina

Int. J. Numer. Meth. Fluids (2000). Vol. 33 (5), pp. 737-766

Abstract
The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are conducted to check its performance. This method is able to deal with all the instabilities that the standard Galerkin method presents, namely, the pressure instability, the instability arising in convection dominated situations and also the less popular instabilities found when the Navier-Stokes equations have a dominant Coriolis force, or there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal-based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over element and thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this given rise to questions regarding the consistency and the conservation properties of the final scheme that are addressed in this paper.
Abstract
The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are [...]
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On stabilized finite element methods for linear systems of convection-diffusion-reaction equations
R. Codina

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 188 (200), pp. 61-82

Abstract
A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a 1D model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner-Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.
Abstract
A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach [...]
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Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection
R. Codina, J. Blasco

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3-4), pp. 277-300

Abstract
In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the Laplacian of the pressure and the divergence of this new vector field. This leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures. In the case of the transient equations, we consider the possibility of treating the pressure gradient projection either implicitly or explicity. In the first case, the number of unknowns of the problem is substantially increased with respect to the standard Galerkin formulation. Nevertheless, iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns (velocity and pressure). When this vector field is treated explicitly, the increment of computational cost of the stabilized formulation with respect to the Galerkin method is very low. We provide a stability estimate for the case of the simple backward Euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection.
Abstract
In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown [...]
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Finite element simulation of the filling of thin moulds
R. Codina, O. Soto

Int. J. Numer. Meth. Engng. (1999). Vol. 46 (9), pp. 1559-1573

Abstract
In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, such as the Q2/P1 element (tri‐quadratic velocities and piecewise linear, discontinuous pressures) or stabilized finite element formulations. The tracking of the free surface is based on the Volume‐Of‐Fluid (VOF) method. The velocity profile is assumed to be parabolic in the direction normal to the mid‐plane, so that one element along the width of the mould is enough to reproduce this profile if this element is quadratic. The velocity is prescribed to zero on the upper and lower surfaces and the normal to the mid‐plane is also prescribed to zero. In the case of div‐stable elements, the pressure profile is prescribed to be constant along the width of the mould. This can be achieved by using as interpolation degrees of freedom the pressure values at the element centroid and its derivatives in the directions tangent and normal to the mid‐plane, and prescribing the latter to zero. No modifications are needed when stabilized formulations are employed. To advance in time the function used in the VOF technique, we use a constant velocity across the width of the mould, which is taken as the projection on the tangent plane of each element of the nodal velocities. This is needed in order to have mass conservation.
Abstract
In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, [...]
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The characteristic‐based‐split procedure: an efficient and accurate algorithm for fluid problems
O. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vázquez, P. Ortiz

Int. J. Numer. Meth. Fluids (1999). Vol. 31 (1), pp. 359-392

Abstract
In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self‐adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi‐implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic‐based‐split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised.
Abstract
In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them [...]
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Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows
R. Codina, O. Soto

Int. J. Numer. Meth. Fluids (1999). Vol. 30 (3), 309-334

Abstract
The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution.
Abstract
The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε [...]
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A fractional-step finite-element method for the Navier–Stokes equations applied to magma-chamber withdrawal
R. Folch, M. Vázquez, R. Codina, J. Martí i Castell

Computers and Geotechnics (1999). Vol. 25 (3), pp. 263-275

Abstract
We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows the use of equal interpolation spaces for the pressure and velocity fields and can be used to solve both compressible and incompressible flows. The standard Galerkin method is used to space-discretize the equations in which convective terms are dominant because the equations are reformulated in a characteristics co-moving frame providing thus the required artificial diffusion in a consistent way. The algorithm depends on four different parameters. Depending on their values, a fully implicit, a semi-implicit or a fully explicit solution can be obtained. The proposed algorithm may be useful in solving a wide spectrum of problems in geology, engineering and geosciences. Several flows frequently encountered in practical applications such as incompressible, slightly compressible or perfect gas are taken into account. As an example, we use it to model the withdrawal of magma from shallow chambers during explosive volcanic eruptions. Our model constitutes a first attempt to characterize the temporal evolution of the most revelant physical parameters during such a process.
Abstract
We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows [...]
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Numerical solution of the incompressible Navier–Stokes equations with Coriolis forces based on the discretization of the total time derivative.
R. Codina

Journal of Computational Physics (1999). Vol. 148 (2), pp. 467-496

Abstract
In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approximation is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the characteristic Galerkin technique for convection-dominated flows.
Abstract
In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on [...]
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Finite element solution of the Stokes problem with dominating Coriolis force
R. Codina, O. Soto

Comput. Methods Appl. Mech. Engrg., (1997). Vol. 142 (3-4), pp. 215-234

Abstract
The objective of this paper is to present a finite element formulation to solve the Stokes problem with Coriolis force. This force results in a skew-symmetric term in the weak formulation of the problem that deteriorates the stability of the standard Galerkin finite element method when the viscosity is small. We show that the stability is worsened due to the presence of the pressure gradient to enforce the incompressibility of the flow. The relevance of this effect depends on the relative importance of the viscous force and the Coriolis force, which is measured by the Ekman number. When it is small, oscillations occur using the Galerkin approach. To overcome them, we propose two different methods based on a consistent modification of the basic Galerkin formulation. Both methods eliminate the oscillations, keeping the accuracy of the formulation and enhancing its numerical stability.
Abstract
The objective of this paper is to present a finite element formulation to solve the Stokes problem with Coriolis force. This force results in a skew-symmetric term in the [...]
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Shock capturing viscosities for the general fluid mechanics algorithm
P. Nithiarasu, O. Zienkiewicz, B. Sai, K. Morgan, R. Codina, M. Vázquez

Int. J. Numer. Meth. Fluids (1998). Vol. 28 (9), pp. 1325-1353

Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous and inviscid compressible flow problems. Results show that the methods based on the second gradient of pressure give better performance in all situations. For instance, the method constructed from the nodal pressure values and consistent and lumped mass matrices is an excellent choice for inviscid problems. The method based on L2 projection is better than any other method in viscous flow computations. The residual based anisotropic method gives excellent performance in the supersonic range and gives better results in the hypersonic regime if a small amount of residual smoothing is used
Abstract
The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous [...]
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A fractional-step method for the incompressible navier-stokes equations related to a predictor-multicorrector algorithm
J. Blasco, R. Codina, A. Huerta

Int. J. Numer. Meth. Fluids (1998). Vol. 28 (10), pp. 1391-1419

Abstract
An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied in this paper. The method, which is first order accurate in the time-step, is shown to convergence to an exact solution of the equations. By adequately splitting the viscous term, it allows to enforce full Dirichlet boundary conditions on the velocity in all substeps of the scheme, while needing no boundary condition at all for the pressure. It is also shown to be related to an iterative predictor-multicorrector algorithm for evolution equations, when this is applied to the incompressible Naiver-Stokes system. A new derivation of the algorithm in a general setting is provided. Two different finite element interpolations are considered for the implementation of the algorithm; numerical results obtained with them for standard benchmark cases are presented.
Abstract
An implicit fractional-step method for the numerical solution of the time-dependent incompressible Naiver-Stokes equations in primitive variables is developed and studied [...]
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A numerical model for temporal variations during explosive central vent eruptions
R. Folch, J. Marti, R. Codina, M. Vázquez

Journal of Geophysical Research: Solid Earth (1998). Vol. 103, 20883-20899

Abstract
An axisymmetrical numerical model has been developed in order to find the temporal evolution of pressure, the position of the exsolution level, the velocity field, the eruption rate, and the amount of erupted material of a shallow, volatile‐rich, felsic magma chamber during a Plinian central vent eruption. The overpressure necessary to trigger the eruption is assumed to result from crystallization‐driven volatile oversaturation. We solve the resulting set of equations using a finite element method. The results obtained show that the pressure at the conduit entrance decreases exponentially as the eruption proceeds. This produces a shifting of the exsolution level, so that deeper parts of the chamber become progressively volatile oversaturated during the eruption. We assess the influence of chamber geometry and the physical properties of the magma on the computed parameters using several numerical examples. The results are also compared with those predicted by previous models from the literature and are found to be in good agreement with documented eruptions. The model constitutes a first attempt to numerically model the dynamics and the temporal evolution of the most relevant physical parameters during withdrawal from a closed magma chamber.
Abstract
An axisymmetrical numerical model has been developed in order to find the temporal evolution of pressure, the position of the exsolution level, the velocity field, the eruption [...]
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Comparison of some finite element methods for solving the diffusion-convection-reaction equation
R. Codina

Comput. Methods Appl. Mech. Engrg., (1998). Vol. 156, (1-4), pp. 185-210

Abstract
In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard in an effort to emphasize the similarities and differences between them. In particular, it is shown that the classical SUPG method is very similar to an explicit version of the Characteristic-Galerkin method, whereas the Taylor-Galerkin method has a stabilization effect similar to a sub-grid scale model, which is un turn related to the introduction of bubbles functions.
Abstract
In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard [...]
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A general algorithm for compressible and incompressible flows. Part III: The semi‐implicit form
R. Codina, M. Vázquez, O. Zienkiewicz

Int. J. Numer. Meth. Fluids (1998). Vol. 27 (1-4), 13-32

Abstract
In this paper we consider some particular aspects related to the semi‐implicit version of a fractional step finite element method for compressible flows that we have developed recently. The first is the imposition of boundary conditions. We show that no boundary conditions at all need to be imposed in the first step where an intermediate momentum is computed. This allows us to impose the real boundary conditions for the pressure, a point that turns out to be very important for compressible flows. The main difficulty of the semi‐implicit form of the scheme arises in the solution of the continuity equation, since it involves both the density and the pressure. These two variables can be related through the equation of state, which in turn introduces the temperature as a variable in many cases. We discuss here the choice of variables (pressure or density) and some strategies to solve the continuity equation. The final point that we study is the behaviour of the scheme in the incompressible limit. It is shown that the method has an inherent pressure dissipation that allows us to reach this limit without having to satisfy the classical compatibility conditions for the interpolation of the velocity and the pressure
Abstract
In this paper we consider some particular aspects related to the semi‐implicit version of a fractional step finite element method for compressible flows that we have developed [...]
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A finite element formulation for Stokes proble allowing equal velocity-pressure interpolation
R. Codina, J. Blasco

Comput. Methods Appl. Mech. Engrg., (1997). Vol. 143 (3–4), pp. 373-391

Abstract
In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of this method relies on the analysis of a class of fractional-step methods for the Navier-Stokes equations for which it is known that equal interpolations yield good numerical results. The reason for this turns out to be the difference between two discrete Laplacian operators computed in a different manner. The formulation of the Stokes problem considered here aims to reproduce this effect. From the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity-pressure interpolations satisfying of the standard formulation. In particular, this condition is fulfilled by the most common equal order interpolations.
Abstract
In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of [...]
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Split, characteristic based semi‐implicit algorithm for laminar/turbulent incompressible flows
O. Zienkiewicz, B. Sai, K. Morgan, R. Codina

Int. J. Numer. Meth. Fluids (1996). Vol. 23, 787-809

Abstract
In an earlier paper, Zienkiewicz and Codina (Int. j. numer. methods fluids, 20, 869–885 (1995)) presented a general algorithm for the solution of both compressible and incompressible Navier–Stokes equations. The algorithm, based on operator splitting, permits arbitrary interpolation functions to be used while avoiding the Babŭska–Brezzi restriction. In addition, its characteristic based approach introduces a form of rational dissipation. Zienkiewicz et al. (Int. j. numer. methods fluids, 20, 887–913 (1995)) presented the application of this algorithm in its fully explicit form to various inviscid compressible flow problems. They also presented two incompressible flow problems solved by the fully explicit form, employing a pseudo compressibility. The present work deals with the application of the above algorithm it its semi‐implicit form to some incompressible flow benchmark problems. Further, it extends the methodology to turbulent flows by employing both one, and two equation turbulence models. A comparison of results with earlier investigations is presented. Other issues addressed in this study include the effect of additional diffusion terms present in the scheme for both laminar and turbulent flow problems and some practical difficulties associated with local time stepping.
Abstract
In an earlier paper, Zienkiewicz and Codina (Int. j. numer. methods fluids, 20, 869–885 (1995)) presented a general algorithm for the solution of both compressible [...]
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A general algorithm for compressible and incompressible flow—Part I. the split, characteristic‐based scheme
O. Zienkiewicz, R. Codina

Int. J. Numer. Meth. Fluids (1995). Vol. 20, 869-885

Abstract
The paper outlines the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier‐Stokes or Euler equations. Full incompressibility can be dealt with if the algorithm is used in its semi‐explicit form and its structure permits arbitrary interpolation functions to be used avoiding the Babuška‐Brezzi restriction. In a fully explicit version it introduces a rational form of balancing dissipation avoiding the use of arbitrary parameters and forms for this.
Abstract
The paper outlines the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier‐Stokes or Euler equations. Full [...]
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An investigation on thermal performance of wollastonite and bentonite reinforced intumescent fire-retardant coating for steel structures
F. Ahmad, S. Khan, N. Merican, E. Oñate, A. Al-Sehemi, G. Yeoh

Construction and Building Material (2019). Vol. 228, 116734

Abstract
This research emphases on the synthesis of new formulations of intumescent coating with improved thermal performance for steel structures. The coating formulations were based on the expandable graphite reinforced with wollastonite and bentonite. Ten samples of five formulations were synthesized by varying grinding time duration between 1 and 2 min. To analyse the substrate temperature of coated steel, fire test was performed according to ASTM-E119. The char morphology was observed by Field Emission Scanning Electron (FESEM) and Transmission Electron Microscopy (TEM). FTIR and X-ray Diffraction (XRD) test is conducted to analyse the composition of the residual char. The residual char mass was perceived by Thermogravimetric analysis (TGA) of the coating. X-ray Photo Electron Spectroscopy (XPS) was utilized for binding energy and elemental composition of the char. One-hour fire protection test showed 166 _C, the lowest substrate temperature of IFRC5-2 and 40.46% residual mass was obtained by TGA analysis. XRD analysis showed that residual char has aluminum borate and borophosphate and confirmed by functional group analysis using and FTIR. FESEM and TEM illustrated that char relates to hexagonal particles of wollastonite. XPS analysis of IFRC5-2 showed the carbon and oxygen contents were 41.40% and 51.20%. Pyrolysis-Gas Chromatography–Mass Spectrometry (Pyrolysis GC–MS) results showed IFRC-5 produced less concentration of the gaseous products compared to IFRC-C. The formulations developed by grinding solid ingredients for 2 min showed improved thermal performance compared with the formulation produced by grinding solid ingredients for 1 min. Longer grinding time and higher amount of filler improved the thermal properties of the intumescent coating.
Abstract
This research emphases on the synthesis of new formulations of intumescent coating with improved thermal performance for steel structures. The coating formulations were based [...]
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A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation
R. Codina

Comput. Methods Appl. Mech. Engrg., (1993). Vol. 110 (3-4), pp. 325-342

Abstract
To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of a nonlinear crosswind dissipation is proposed. It is shown that the method is less overdiffusive than other discontinuity-capturing techniques and has better numerical behavior. The design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases.
Abstract
To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of [...]
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Stability analysis of the forward Euler scheme for the convection‐diffusion equation using the SUPG formulation in space
R. Codina

Int. J. Numer. Meth. Engng. (1993). Vol. 36, 1445-1464

Abstract
The stability and accuracy of the forward Euler scheme for the semidiscrete problem arising from the space discretization of the convection‐diffusion equation using the SUPG formulation are analysed. Both linear and quadratic finite elements are considered. The stability limits are derived for the one‐dimensional problem and a heuristic criterion is proposed for the multidimensional equation. Some numerical experiments are conducted in order to assess the performance of different finite elements
Abstract
The stability and accuracy of the forward Euler scheme for the semidiscrete problem arising from the space discretization of the convection‐diffusion equation using the [...]
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An iterative penalty method for the finite element solution of the stationary Navier-Stokes equations
R. Codina

Comput. Methods Appl. Mech. Engrg., (1993). Vol. 110 (3-4), pp. 237-262

Abstract
The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, the usual condition on the viscosity and the data that ensures uniqueness is assumed. The method is based on the iterative imposition of the incompressibility condition via penalization. Theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations. Two particular iterative schemes are analysed, namely those based on the Picard and Newton-Raphson algorithms.
Abstract
The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, [...]
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Numerical simulation and visualization of material flow in friction stir welding via particle tracing
N. Dialami, M. Chiumenti, M. Cervera, C. Agelet de Saracibar, J. Ponthot, P. Bussetta

(2014). S. Idelsohn (Ed.), Numerical Simulations of Coupled Problems in Engineering, Part III, Chapter 7 Computational Methods, Vol. 33, Springer, pp. 157-169

Abstract
This work deals with the numerical simulation and material flow visualization of Friction Stir Welding (FSW) processes. The 4-th order Runge-Kutta (RK4) integration method is used for the computation of particle trajectories. The particle tracing method is used to study the effect of input process parameters and pin shapes on the weld quality. The results show that the proposed method is suitable for the optimization of the FSW process.
Abstract
This work deals with the numerical simulation and material flow visualization of Friction Stir Welding (FSW) processes. The 4-th order Runge-Kutta (RK4) integration method [...]
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Challenges to be tackled in the computational modeling and numerical simulation of FSW processes
C. Agelet de Saracibar

Metals (2019). Vol. 9 (5), pp. 573

Abstract
The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes is an extremely challenging task due to the highly nonlinear and coupled nature of the physical problem and the complex computational issues that need to be properly tackled in the numerical model.
Abstract
The computational modeling and numerical simulation of Friction Stir Welding (FSW) processes is an extremely challenging task due to the highly nonlinear and coupled nature [...]
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Conserving algorithms for frictionless and full stick friction contact dynamic problems using the direct elimination method
C. Agelet de Saracibar, D. Di Capua

Int. J. Numer. Meth. Engng. (2017). (preprint) Vol. 113 (6), pp. 910-937

Abstract
In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.
Abstract
In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless [...]
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Software Solutions for ICME
G. Schmitz, A. Engstrom, R. Bernhardt, U. Prahl, L. Adam, J. Seyfarth, M. Apel, C. Agelet de Saracibar, P. Korzhavyi, J. Agren, B. Patzak

Journal of The Minerals, Metals & Materials Society (2016). Vol. 68 (1), pp. 70-76

Abstract
The Integrated Computational Materials Engineering expert group (ICMEg), a coordination activity of the European Commission, aims at developing a global and open standard for information exchange between the heterogeneous varieties of numerous simulation tools. The ICMEg consortium coordinates respective developments by a strategy of networking stakeholders in the first International Workshop on Software Solutions for ICME, compiling identified and relevant software tools into the Handbook of Software Solutions for ICME, discussing strategies for interoperability between different software tools during a second (planned) international workshop, and eventually proposing a scheme for standardized information exchange in a future book or document. The present article summarizes these respective actions to provide the ICME community with some additional insights and resources from which to help move this field forward.
Abstract
The Integrated Computational Materials Engineering expert group (ICMEg), a coordination activity of the European Commission, aims at developing a global and open standard [...]
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A direct elimination algorithm for quasi-static and dynamic contact problems
D. Di Capua, C. Agelet de Saracibar

Finite Elements in Analysis and Design (2015). (preprint) Vol. 93 (1), pp. 107-125

Abstract
This paper deals with the computational modeling and numerical simulation of contact problems at finite deformations using the finite element method. Quasi-static and dynamic problems are considered and two particular frictional conditions, full stick friction and frictionless cases, are addressed. Lagrange multipliers and regularized formulations of the contact problem, such as penalty or augmented Lagrangian methods, are avoided and a new direct elimination method is proposed. Conserving algorithms are also introduced for the proposed formulation for dynamic contact problems. An assessment of the performance of the resulting formulation is shown in a number of selected benchmark tests and numerical examples, including both quasi-static and dynamic contact problems under full stick friction and frictionless contact conditions. Conservation of key discrete properties exhibited by the time stepping algorithm used for dynamic contact problems is also shown in an example
Abstract
This paper deals with the computational modeling and numerical simulation of contact problems at finite deformations using the finite element method. Quasi-static and dynamic [...]
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Discrete element modelling and simulation of sand mould manufacture for the lost foam process
J. Rojek, F. Zárate, C. Agelet de Saracibar, C. Gilbourne, P. Verdot

Int. J. Numer. Meth. Engng. (2004). Vol. 62 (11), pp. 1241-1441

Abstract
This paper presents a numerical model of mould manufacture for the lost foam casting process. The process of mould filling with sand and sand compaction by vibration are modelled using spherical (in 3D) or cylindrical (in 2D) discrete elements. The motion of discrete elements is described by means of equations of rigid body dynamics. Rigid particles interact among one another with contact forces, both in normal and tangential directions. Numerical simulation predicts defects of the mould due to insufficient sand compaction around the pattern. Combining the discrete element model of sand with the finite element model of the pattern allows us to detect possible distortion of the pattern during mould filling and compaction. Results of numerical simulation are validated by comparison with experimental data.
Abstract
This paper presents a numerical model of mould manufacture for the lost foam casting process. The process of mould filling with sand and sand compaction by vibration are modelled [...]
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Numerical analysis of coupled thermomechanical frictional contact problems. Computational model and applications
C. Agelet de Saracibar

Archives of Comp. Meths. Engng. (1998). Vol. 5 (3), pp. 243-301

Abstract
In this paper a numerical model for the analysis of coupled thermomechanical multi-body frictional contact problems at finite deformations is presented. The multi-body frictional contact formulation is fully developed on the continuum setting and then a spatial (Galerkin projection) and temporal (time-stepping algorithm) discretization is applied. A contact pressure and temperature dependent thermal contact model has been used. A fractional step method arising from an operator split of the governing equations has been used to solve the coupled nonlinear system of equations, leading to a staggered solution algorithm. The numerical model has been implemented into an enhanced version of the computational finite element program FEAP. Numerical examples and simulation of industrial metal forming processes show the performance of the numerical model in the analysis of coupled thermomechanical frictional contact problems.
Abstract
In this paper a numerical model for the analysis of coupled thermomechanical multi-body frictional contact problems at finite deformations is presented. The multi-body frictional [...]
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Modeling the effect of storage temperature on the respiration rate and texture of fresh cut pineapple
S. Benítez, M. Chiumenti, F. Sepulcre, I. Achaerandio, M. Pujolá

Journal of Food Engineering (2012). Vol. 113 (4), pp. 527-533

Abstract
The effect of temperature on the respiration rate and texture of fresh cut pineapple was studied over the course of 10 days of storage. The thermal exchange between the pineapple trays and the cooling environment was simulated using the finite element method and tested at 6 °C. The temperatures on pineapple wedges differed between the cold point and points near the surface, indicating that the respiration rate may be affected in pineapple subjected to temperature abuse. The experimental respiration rates obtained were used to develop a model relating respiration to O2 and CO2concentrations at different temperatures using the closed system method. The O2 consumption and CO2 production of pineapple wedges was accurately modeled using Michaelis–Menten kinetics. The texture degradation of pineapple wedges follows a zero-order kinetic reaction at different temperatures and the thermal dependence of the model’s parameters for both respiration rate and texture degradation was described by Arrhenius-type equations.
Abstract
The effect of temperature on the respiration rate and texture of fresh cut pineapple was studied over the course of 10 days of storage. The thermal exchange between the [...]
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Shear band localization via local J2 continuum damage mechanics
M. Cervera, M. Chiumenti, C. Agelet de Saracibar

Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (9-11), pp. 849-880

Abstract
This paper describes a novel formulation for the solution of problems involving shear band localization using a local isotropic J2 continuum damage model and mixed linear simplex (triangles and tetrahedra). Stabilization methods are used to ensure existence and uniqueness of the solution, attaining global and local stability of the corresponding discrete finite element formulation. Consistent residual viscosity is used to enhance robustness and convergence of the formulation. Implementation and computational aspects are also discussed. A simple isotropic local J2 damage constitutive model is considered, either with linear or exponential softening. The softening modulus is regularized according to the material mode II fracture energy and the element size. Numerical examples show that the formulation derived is fully stable and remarkably robust, totally free of volumetric locking and spurious oscillations of the pressure. As a consequence, the results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing very favourably with those obtained with the ill-posed standard approaches.
Abstract
This paper describes a novel formulation for the solution of problems involving shear band localization using a local isotropic J2 continuum damage model and mixed linear [...]
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On the numerical modeling of frictional wear phenomena
C. Agelet de Saracibar, M. Chiumenti

Comput. Methods Appl. Mech. Engrg., (1999). Vol. 177 (3-4), pp. 401-426

Abstract
The evolution of the contact surfaces wear may become particularly important in the definition of the frictional behavior, in particular for frictional contact problems involving large slips, typically in sheet metal forming and bulk forming operations. Despite this fact, most of the current applications reported in the literature are restricted to a standard Coulomb law, using a constant friction coefficient. Such simple models may represent only a limited range of tribological situations and it appears to be necessary to develop a class of models which incorporate the state conditions and their evolution at the contact surfaces, taking into account the influence of complex phenomena such as wear, lubrication and chemical reactions, among others, see Oden and Martins [1]. In this paper a simple numerical model for the simulation of frictional wear behavior, within a fully nonlinear setting, including large slip and finite deformation, is presented. The model relies on the introduction of an internal variable related to the state conditions at the contact surface. Here, two possible definitions of this internal variable have been considered. The fully nonlinear frictional contact formulation, entirely derived first on a continuum setting by Laursen and Simo [2–6], has been extended here to accomodate the characterization of the wear frictional behavior. Within the computational aspects, two families of robust time stepping algorithms, arising from an operator split of the constrained frictional evolution equations, are discussed. Finally, following current approaches, see Lassen [9], Lassen and Bay [10], Owen et al. [11], de Souza et al. [12], Stromberg et al. [13] and Stromberg [14], a long-term tools wear prediction is given by introducing an a priori wear estimate derived from Archard's law, Archard [15]. The numerical model has been implemented into a enhanced version of the computational finite element program FEAP. Numerical examples show the suitability of the proposed model to capture the essential features of the frictional behavior at the contact interfaces and to provide a prediction of tool wear in forming operations.
Abstract
The evolution of the contact surfaces wear may become particularly important in the definition of the frictional behavior, in particular for frictional contact problems involving [...]
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An accurate non-local bonded discrete element method for non linear analysis of solids. Application to concrete fracture tests
M. Celigueta, S. Latorre, F. Arrufat, E. Oñate

Computational Particle Mechanics (2019). (prepint), 10.1007/s40571-019-00278-5

Abstract
We present a numerical procedure for elastic and non linear analysis (including fracture situations) of solid materials and structures using the Discrete Element Method (DEM). It can be applied to strongly cohesive frictional materials such as concrete and rocks. The method consists on defining the non-local constitutive equations at the contact interfaces between discrete particles using the information provided by the stress tensor over the neighbour particles. The method can be used with different yield surfaces and in the paper it is applied to the analysis of fracture of concrete samples. Good comparison with experimental results is obtained.
Abstract
We present a numerical procedure for elastic and non linear analysis (including fracture situations) of solid materials and structures using the Discrete Element Method (DEM). [...]
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Combination of a non-local damage model for quasi-brittle materials with a mesh-adaptive finite element technique
I. de-Pouplana, E. Oñate

Finite Elements in Analysis and Design (2016). Vol. 112, pp. 26-39

Abstract
Progressive fracture in quasi-brittle materials is often treated via strain softening models in continuum damage mechanics. Such constitutive relations favour spurious strain localization and ill-posedness of boundary value problems. The introduction of non-local damage models together with a characteristic length parameter controlling the size of the fracture process zone is known to regularize the problem. In order to account for the non-locality of these models, it is crucial to work with fine spatial discretizations at the damage progress zone. In this paper we present a non-local damage model in combination with a mesh-adaptive finite element technique that can help automatize the analysis of progressive fracture problems in an efficient manner. Classical two-dimensional examples are given to illustrate the presented approach.
Abstract
Progressive fracture in quasi-brittle materials is often treated via strain softening models in continuum damage mechanics. Such constitutive relations favour spurious strain [...]
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El valor del cálculo en los sistemas de ayuda a la toma de decisiones en ingeniería
E. Oñate

Revista de Obras Públicas (2004). Vol. 151 (3449), pp. 41-48

Abstract
El artículo revindica el valor de los métodos de cálculo que se enseñan en las Escuelas de Ingeniería, como herramientas indispensables en los nuevos sistemas de ayuda a la toma de decisiones (SAD). Los SAD integran base de datos, métodos de cálculo y módulos de inteligencia artificial, y su utilización se extiende cada día más para ayuda al diseño y gestión de infraestructuras y servicios de ingeniería. En el texto se describen varios desarrollos y aplicaciones recientes de SAD en ingeniería civil. El artículo acaba con una reflexión sobre los límites de los SAD. The paper stresses the value of the computational methods taught in engineering faculties as indispensable tools within the new decision support systems (DSS). The DSS integrate data bases, computational methods and artificial intelligence modules. They are increasingly used for supporting the design and management of civil constructions and services in engineering. The paper describes recent developments and applications of DSS in civil engineering. The text ends up with a discussion on the limits of the DSS.
Abstract
El artículo revindica el valor de los métodos de cálculo que se enseñan en las Escuelas de Ingeniería, como herramientas indispensables [...]
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An implicit unsteady hydraulic solver for suspended cuttings transport in managed pressure wells
R. Flores, R. Ortega, A. Ilin, D. Simpkins, E. Oñate

Computational Particle Mechanics (2019). Vol. 6 (2), pp 163–175

Abstract
We present a simulation tool for transient events in complex hydraulic networks. The code includes modelling of the transport of suspended cuttings in near-vertical wells. An unstructured finite volume formulation with implicit time integration has been chosen. The unconditional stability of the integrator makes the method suitable for the simulation of transient events over a wide range of characteristic timescales. It handles both very fast transients (e.g. fluid hammer events) and the long-term evolution of the well (e.g. hole cleaning operations). The software has been developed to address the need of the oil industry for a robust and efficient predictive tool allowing effective well control in managed pressure drilling operations. The physical modelling follows the standard practices accepted by the industry (e.g. mud rheology computations). The mathematical foundation of the algorithm is described followed by validation cases that illustrate its capabilities and accuracy. Finally, a practical industrial application example is provided to demonstrate the real-world performance of the software.
Abstract
We present a simulation tool for transient events in complex hydraulic networks. The code includes modelling of the transport of suspended cuttings in near-vertical wells. [...]
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Effect of the integration scheme on the rotation of non-spherical particles with the discrete element method
J. Irazábal González, F. Salazar, M. Santasusana, E. Oñate

Computational Particle Mechanics (2019). Vol. 6 (4), pp 545–559

Abstract
The discrete element method (DEM) is an emerging tool for the calculation of the behaviour of bulk materials. One of the key features of this method is the explicit integration of the motion equations. Explicit methods are rapid, at the cost of a limited time step to achieve numerical stability. First- or second-order integration schemes based on a Taylor series are frequently used in this framework and shown to be accurate for the translational and rotational motion of spherical particles. However, they may lead to relevant inaccuracies when non-spherical particles are used since the orientation implies a modification in the second-order inertia tensor in the inertial reference frame. Specific integration schemes for non-spherical particles have been proposed in the literature, such as the fourth-order Runge–Kutta scheme presented by Munjiza et al. and the predictor–corrector scheme developed by Zhao and van Wachem which applies the direct multiplication algorithm for integrating the orientation. In this work, both methods are adapted to be used together with a velocity Verlet scheme for the translational integration. The performance of the resulting schemes, as well as that of the direct integration method, is assessed, both in benchmark tests with analytical solution and in real-scale problems. The results suggest that the fourth-order Runge–Kutta and the Zhao and van Wachem schemes are clearly more accurate than the direct integration method without increasing the computational time.
Abstract
The discrete element method (DEM) is an emerging tool for the calculation of the behaviour of bulk materials. One of the key features of this method is the explicit integration [...]
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A unified Lagrangian formulation for solid and fluid dynamics and its possibility for modelling submarine landslides and their consequences
X. Zhang, E. Oñate, S. Galindo, J. Bleyer, K. Krabbenhoft

Comput. Methods Appl. Mech. Engrg., (2019). Vol. 343, pp. 314-338

Abstract
Consequences of submarine landslides include both their direct impact on offshore infrastructure, such as subsea electric cables and gas/oil pipelines, and their indirect impact via the generated tsunami. The simulation of submarine landslides and their consequences has been a long-standing challenge majorly due to the strong coupling among sliding sediments, seawater and infrastructure as well as the induced extreme material deformation during the complete process. In this paper, we propose a unified finite element formulation for solid and fluid dynamics based on a generalised Hellinger–Reissner variational principle so that the coupling of fluid and solid can be achieved naturally in a monolithic fashion. In order to tackle extreme deformation problems, the resulting formulation is implemented within the framework of the particle finite element method. The correctness and robustness of the proposed unified formulation for single-phase problems (e.g. fluid dynamics problems involving Newtonian/Non-Newtonian flows and solid dynamics problems) as well as for multi-phase problems (e.g. two-phase flows) are verified against benchmarks. Comparisons are carried out against numerical and analytical solutions or experimental data that are available in the literature. Last but not least, the possibility of the proposed approach for modelling submarine landslides and their consequences is demonstrated via a numerical experiment of an underwater slope stability problem. It is shown that the failure and post-failure processes of the underwater slope can be predicted in a single simulation with its direct threat to a nearby pipeline and indirect threat by generating tsunami being estimated as well.
Abstract
Consequences of submarine landslides include both their direct impact on offshore infrastructure, such as subsea electric cables and gas/oil pipelines, [...]
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Modeling polycrystalline materials with elongated grains
I. Pérez, M. Farias, M. Castro, R. Roselló, C. Morfa, L. Medina, E. Oñate

Int J Numer Methods Eng. (2019). Vol. 118 (3), pp. 121-131

Abstract
A novel algorithm to reproduce the arrangement of grains in polycrystalline materials was recently published by the authors. In this original approach, a dense package of circles (or spheres) with the same distribution as the grains is generated to produce a set of Voronoi cells that are later modified to Laguerre cells representing the original structure. This algorithm was successfully applied to materials with somewhat equidimensional grains; however, it fails for long‐shaped grains. In this paper, modifications are provided in order to overcome these drawbacks. This is accomplished by moving each vertex of the Voronoi cells in such a way that the vertex should be equidistant from the particles with respect to the Euclidean distance. The algorithm is applied to packages of ellipses and spherocylinders in 2D. An example for a package of spheres is also provided to illustrate the application for a simple 3D case. The adherence between the generated packages and the corresponding tessellations is verified by means of the Jaccard coefficient (J). Several packages are generated randomly and the distribution of J coefficients is investigated. The obtained values satisfy the theoretical restraints and the quality of the proposed algorithm is statistically validated.
Abstract
A novel algorithm to reproduce the arrangement of grains in polycrystalline materials was recently published by the authors. In this original approach, a dense package of [...]
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Advances in particle packing algorithms for generating the medium in the Discrete Element Method
C. Recarey, I. Pérez, R. Roselló, M. Muniz, E. Hernández, R. Giraldo, E. Oñate

Computer Methods in Applied Mechanics and Engineering (2019). Vol. 345, pp. 336-362

Abstract
Several theoretical contributions in order to establish a particle packing methodology are presented. In this respect, a generic formulation of a new method for packing particles based on a constructive advancing front method, which uses Monte Carlo techniques for the generation of particle dimensions, is also shown. The method can be used to obtain virtual dense packings of particles with several geometrical shapes. It employs continuous, discrete and empirical statistical distributions in order to generate the dimensions of particles. The packing algorithm is very flexible and allows alternatives for: 1- The direction of the advancing front (inwards or outwards), 2- The selection of the local advancing front, 3- The method for placing a mobile particle in contact with others and 4- The overlap checks. The algorithm also allows obtaining highly porous media when it is slightly modified. Practical applications of the formulations are presented in the end of this paper.
Abstract
Several theoretical contributions in order to establish a particle packing methodology are presented. In this respect, a generic formulation of a new method for packing [...]
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Optimizing rock cutting through computer simulation
J. Rojek, E. Oñate, C. Labra, H. Kargl, J. Akerman

(2010). Technology Innovation in Underground Construction (TUNCONSTRUCT), G. Beer (Ed.)

Abstract
This paper presents an original thermomechanical model of rock cutting with evaluation of tool wear. The model has been developed within the framework of the discrete element method, which is a suitable numerical method to study problems of multiple material fractures likes those of rock cutting. The paper presents brief overview of the theoretical formulation, calibration of the discrete element model, and a number of numerical results obtained in simulation of rock cutting processes typical for underground excavation, using both roadheaders and TBMs.
Abstract
This paper presents an original thermomechanical model of rock cutting with evaluation of tool wear. The model has been developed within the framework of the discrete element [...]
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Advances in the formulation of the rotation-free basic shell triangle
E. Oñate, F. Flores

Computer Methods in Applied Mechanics and Engineering (2005). Vol. 194 (21-24), 2406-2443

Abstract
A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes. This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element. A simple and economic version of the element using a single integration point is presented. The efficiency of the different rotation-free shell triangles is demonstrated in many examples of application including linear and non linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.
Abstract
A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in [...]
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Stabilized solution of the multidimensional advection-diffusion-absorption equation using linear finite elements
E. Oñate, J. Miquel, F. Zárate

Computers and Fluids (2007). Vol. 36 (1), pp. 92-112, 2007

Abstract
A stabilized finite element method (FEM) for the multidimensional steady state advection-diffusion-absorption equation is presented. The stabilized formulation is based on the modified governing differential equations derived via the Finite Calculus (FIC) method. For 1D problems the stabilization terms act as a nonlinear additional diffusion governed by a single stabilization parameter. It is shown that for multidimensional problems an orthotropic stabilizing diffusion must be added along the principal directions of curvature of the solution. A simple iterative algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. Numerical results for 1D and 2D problems with sharp gradients are presented showing the effectiveness and accuracy of the new stabilized formulation.
Abstract
A stabilized finite element method (FEM) for the multidimensional steady state advection-diffusion-absorption equation is presented. The stabilized formulation is based on [...]
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Updated Lagrangian mixed finite element formulation for quasi and fully incompressible fluids
E. Oñate, J. Carbonell

Computational Mechanics (2014). Vol. 54 (6), pp. 1583-1596, 2014

Abstract
We present a mixed velocity-pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. Details of the governing equations for the conservation of momentum and mass are given in both differential and variational form. The finite element interpolation uses an equal order approximation for the velocity and pressure unknowns. The procedure for obtaining stabilized FEM solutions is outlined. The solution in time of the discretized governing conservation equations using an incremental iterative segregated scheme is described. The linearization of these equations and the derivation of the corresponding tangent stiffness matrices is detailed. Other iterative schemes for the direct computation of the nodal velocities and pressures at the updated configuration are presented. The advantages and disadvantages of choosing the current or the updated configuration as the reference configuration in the Lagrangian formulation are discussed.
Abstract
We present a mixed velocity-pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. Details of the governing [...]
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The particle finite element method. An overview
E. Oñate, S. Idelsohn, F. Del Pin, R. Aubry

International Journal of Computational Methods (2004). Vol. 1 (2), pp. 267-307

Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.
Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use [...]
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Residual stress and distortion of rectangular and S-shaped Ti-6Al-4V parts by Directed Energy Deposition: Modelling and experimental calibration
X. Lu, X. Lin, M. Chiumenti, M. Cervera, Y. Hu, X. Ji, L. Ma, H. Yang, W. Huang

Additive Manufacturing (2019). (preprint) Vol. 26, pp. 166-179

Abstract
Residual stresses and distortion in Additive Manufactured (AM) parts are two key obstacles which seriously hinder the wide application of this technology. Nowadays, understanding the thermomechanical behavior induced by the AM process is still a complex task which must take into account the effects of both the process and the material parameters, the microstructure evolution as well as the pre-heating strategy. One of the challenges of this work is to increase the complexity of the geometries used to study the thermomechanical behavior induced by the AM process. The reference geometries are a rectangular and a S-shaped structures made of 44-layers each. The samples have been fabricated by Directed Energy Deposition (DED). In-situ thermal and distortion histories of the substrate are measured in order to calibrate the 3D coupled thermo-mechanical model. Once the numerical results showed a good agreement with the temperature measurements, the validated model has been used to predict the residual stresses and distortions. Different process parameters have been analyzed to study their sensitivity to the process assessment. Different preheating strategies have been also analyzed to check their effectiveness on the mitigation of both distortions and residual stresses. Finally, some simplifications of the actual scanning sequence are proposed to reduce the computational cost without loss of the accuracy of the simulation framework.
Abstract
Residual stresses and distortion in Additive Manufactured (AM) parts are two key obstacles which seriously hinder the wide application of this technology. Nowadays, understanding [...]
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In situ measurements and thermo-mechanical simulation of Ti–6Al–4V laser solid forming processes
X. Lu, X. Lin, M. Chiumenti, M. Cervera, Y. Hu, X. Ji, L. Ma, W. Huang

Int. J. Mechanical Sciences (2019). (preprint) Vol. 153-154, pp. 119-130

Abstract
Residual stresses and distortions are two technical obstacles for popularizing the additive manufacturing (AM) technology. The evolution of the stresses in AM components during the thermal cycles of the metal depositing process is not yet clear, and more accurate in situ measurements are necessary to calibrate and validate the numerical tools developed for its simulation. In this work a fully coupled thermo-mechanical analysis to simulate the laser solid forming (LSF) process is carried out. At the same time, an exhaustive experimental campaign is launched to measure the temperature evolution at different locations, as well as the distortions and both the stress and strain fields. The thermal and mechanical responses of single-wall coupons under different process parameters are recorded and compared with the numerical models. Good agreement between the numerical results and the experimental measurements is obtained. Sensitivity analysis demonstrates that the AM process is significantly affected by the laser power and the feeding rate, while poorly influenced by the scanning speed.
Abstract
Residual stresses and distortions are two technical obstacles for popularizing the additive manufacturing (AM) technology. The evolution of the stresses in AM components during [...]
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Out-of-plane seismic response and failure mechanism of masonry structures using finite elements with enhanced strain accuracy
G. Vlachakis, M. Cervera, G. Barbat, S. Salaustros

Engineering Failure Analysis (2019). (preprint) Vol. 97, pp. 534-555

Abstract
The out-of-plane response is a complex and at the same time key aspect of the seismic vulnerability of masonry structures. It depends on several factors, some of which are the material properties, the quality of the walls, the geometry of the structure, the connections between structural elements and the stiffness of the diaphragms. During the last years, a wide variety of numerical methods has been employed to assess the out-ofplane behaviour of unreinforced masonry structures. Finite element macro-modelling approaches are among the most famous as they allow modelling large structures at a reasonable computational cost. However, macro-modelling approaches may result in a non-realistic representation of localized cracks and a dependency of the numerical solution on the finite element mesh. Mixed strain/displacement finite elements have been recently proposed as a remedy to the above numerical pathologies. Due to the independent interpolation of strains and displacements these finite element formulations are characterized by an enhanced accuracy in strain localization and crack propagation problems, being at the same time practically mesh independent. For these reasons, mixed finite elements are employed in this work for the out-of-plane assessment of unreinforced masonry structures, being at the same time their first real-scale application. A full-scale experimental campaign of two masonry structures, a stone and a brick one, subjected to shaking-table tests is chosen as reference benchmark. Their structural response under seismic actions is numerically assessed through nonlinear static analysis. The proposed approach is validated through the comparison of the numerical results with the experimental ones, as well as with the results obtained using standard irreducible finite elements.
Abstract
The out-of-plane response is a complex and at the same time key aspect of the seismic vulnerability of masonry structures. It depends on several factors, some of which are [...]
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Effect of the tool tilt angle on the material flow and heat generation in Friction Stir Welding
N. Dialami, M. Cervera, M. Chiumenti

Metals (2019). Vol. 9 (1), pp. 28-45

Abstract
This work studies the effect of the tool tilt angle on the generated heat and the material flow in the work pieces joint by Friction Stir Welding (FSW). An apropos kinematic framework together with a two-stage speed-up strategy is adopted to simulate the FSW problem. The effect of tilt angle on the FSWelds is modeled through the contact condition by modifying an enhanced friction model. A rotated friction shear stress is proposed, the angle of rotation depending on the process parameters and the tilt angle. The proposed rotation angle is calibrated by the experimental data provided for a tilt angle 2.5◦ . The differences of generated heat and material flow for the cases of tool with tilt angle of 0◦ and 2.5◦ are discussed. It is concluded that due to the higher temperature, softer material and greater frictional force in the trailing side of the tool, the material flow in the rear side of the FSW tool with the title angle is considerably enhanced, which assists to prevent the generation of defect.
Abstract
This work studies the effect of the tool tilt angle on the generated heat and the material flow in the work pieces joint by Friction Stir Welding (FSW). An apropos kinematic [...]
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Prediction of joint line remnant defect in friction stir welding
N. Dialami, M. Cervera, M. Chiumenti, A. Segatori

Int. J. Mechanical Sciences (2019). Vol. 151, pp. 61-69

Abstract
This work studies computationally and experimentally the joint line remnant defects emerging in friction stir welding (FSW) due to oxide layer propagation into the weld. A finite element-based model is used for the computational simulation. To follow the evolution of the oxide layer originally placed between the butted surfaces, a material tracing technique is incorporated in the numerical model. This approach allows tracking the position of the tracers representing oxide layer particles knowing the nodal velocities. A robust and fast two-stage numerical strategy is adopted for the analysis of FSW process to solve the underlying thermo-mechanical problem. The first stage is a speed-up stage solved on a fixed mesh that allows to quickly obtain the steady state. Oxide layer evolution is traced in the second stage where the rotation of the tool is modelled. Experimentally, to produce a clearly dispersed oxide line in the weld, one of the workpieces is anodized while the other one is taken as extruded. The computationally obtained oxide layer patterns are compared to those obtained experimentally using macrograph analysis of the joint cross section. The effect of the pin features and the process parameters on the final result is studied. The results show that with appropriate modelling of the material tracers in FSW, significant agreement can be attained between the computed and measured post-FSW oxide layer evolved results.
Abstract
This work studies computationally and experimentally the joint line remnant defects emerging in friction stir welding (FSW) due to oxide layer propagation into the weld. A [...]
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Multiscale analysis using a coupled discrete/finite element model
J. Rojek, E. Oñate

Interaction and Multiscale Mechanics: An International Journal (IMMIJ) (2007). Vol. 1 (1), pp. 1-31

Abstract
The present paper presents multiscale modelling via coupling of the discrete and finite element methods. Theoretical formulation of the discrete element method using spherical or cylindrical particles has been briefly reviewed. Basic equations of the finite element method using the explicit time integration have been given. The micr-macro transition for the discrete element method has been discussed. Theoretical formulations for macroscopic stress and strain tensors have been given. Determination of macroscopic constitutive properties using dimensionless micro-macro relationships has been proposed. The formulation of the multiscale DEM/FEM model employing the DEM and FEM in different subdomains of the same body has been presented. The coupling allows the use of partially overlapping DEM and FEM subdomains. The overlap zone in the two coupling algorithms is introduced in order to provide a smooth transition from one discretization method to the other. Coupling between the DEM and FEM subdomains is provided by additional kinematic constraints imposed by means of either the Lagrange multipliers or penalty function method. The coupled DEM/FEM formulation has been implemented in the authors’ own numerical program. Good performance of the numerical algorithms has been demonstrated in a number of examples.
Abstract
The present paper presents multiscale modelling via coupling of the discrete and finite element methods. Theoretical formulation of the discrete element method using spherical [...]
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Challenges, tools and applications of tracking algorithms in the numerical modelling of cracks in concrete and masonry structures
S. Salaustros, M. Cervera, L. Pelà

Archives of Comp. Meths. Engng. (2018). (preprint) Published online, pp. 1-45

Abstract
The importance of crack propagation in the structural behaviour of concrete and masonry structures has led to the development of a wide range of finite element methods for crack simulation. A common standpoint in many of them is the use of tracking algorithms, which identify and designate the location of cracks within the analysed structure. In this way, the crack modelling techniques, smeared or discrete, are applied only to a restricted part of the discretized domain. This paper presents a review of finite element approaches to cracking focusing on the development and use of tracking algorithms. These are presented in four categories according to the information necessary for the definition and storage of the crack-path. In addition to that, the most utilised criteria for the selection of the crack propagation direction are summarized. The various algorithmic issues involved in the development of a tracking algorithm are discussed through the presentation of a local tracking algorithm based on the smeared crack approach. Challenges such as the modelling of arbitrary and multiple cracks propagating towards more than one direction, as well as multi-directional and intersecting cracking, are detailed. The presented numerical model is applied to the analysis of small- and large-scale masonry and concrete structures under monotonic and cyclic loading.
Abstract
The importance of crack propagation in the structural behaviour of concrete and masonry structures has led to the development of a wide range of finite element methods for [...]
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Modeling of microstructure evolution of Ti6Al4V for Additive Manufacturing
E. Salsi, M. Chiumenti, M. Cervera

Metals (2018). Vol. 8 (8), pp. 633-657

Abstract
AM processes are characterized by complex thermal cycles that have a deep influence on the microstructural transformations of the deposited alloy. In this work, a general model for the prediction of microstructure evolution during solid state transformations of Ti6Al4V is presented. Several formulations have been developed and employed for modeling phase transformations in other manufacturing processes and, particularly, in casting. The proposed model is mainly based on the combination and modification of some of these existing formulations, leading to a new overall model specifically dedicated to AM. The accuracy and suitability of the integrated model is enhanced, introducing new dedicated features. In fact the model is designed to deal with fast cooling and re-heating cycles typical of AM processes because: (a) it is able to consider incomplete transformations and varying initial content of phases and (b) it can take into account simultaneous transformations.The model is implemented in COMET, an in-house Finite Element (FE)-based framework for the solution of thermo-mechanical engineering problems. The validation of the microstructural model is performed by comparing the simulation results with the data available in the literature. The sensitivity of the model to the variation of material parameters is also discussed.
Abstract
AM processes are characterized by complex thermal cycles that have a deep influence on the microstructural transformations of the deposited alloy. In this work, a general [...]
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An empirical comparison of machine learning techniques for dam behaviour modelling
F. Salazar, M. Toledo, R. Morán Moya, E. Oñate

Structural Safety (2015). Vol 56, pp. 9-17

Abstract
Predictive models are essential in dam safety assessment. Both deterministic and statistical models applied in the day-to-day practice have demonstrated to be useful, although they show relevant limitations at the same time. On another note, powerful learning algorithms have been developed in the field of machine learning (ML), which have been applied to solve practical problems. The work aims at testing the prediction capability of some state-of-the-art algorithms to model dam behaviour, in terms of displacements and leakage. Models based on random forests (RF), boosted regression trees (BRT), neural networks (NN), support vector machines (SVM) and multivariate adaptive regression splines (MARS) are fitted to predict 14 target variables. Prediction accuracy is compared with the conventional statistical model, which shows poorer performance on average. BRT models stand out as the most accurate overall, followed by NN and RF. It was also verified that the model fit can be improved by removing the records of the first years of dam functioning from the training set.
Abstract
Predictive models are essential in dam safety assessment. Both deterministic and statistical models applied in the day-to-day practice have demonstrated to be useful, although [...]
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Structural shape sensitivity analysis for nonlinear material models with strain softening
G. Bugeda, L. Gil, E. Oñate

Structural Optimization (1999). Vol. 17 (2-3), pp. 162-171

Abstract
This paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the finite element method with a nonlinear constitutive material model. Traditionally, the structural sensitivity analysis is computed using an incremental approach based on the incremental procedures for the solution of the structural equilibrium problem. In this work, a direct (nonincremental) formulation for computing these structural sensitivities, that is valid for some specific nonlinear material models, is proposed. The material models for which the presented approach is valid are characterized by the fact that the stresses at any timet can be expressed in terms of the strains at the timet and, in some cases, the strains at a specific past timet u (t u<t). This is the case of elasticity (linear as well as nonlinear), perfect plasticity and damage models. A special strategy is also proposed for material models with strain softening. For the cases where it is applicable, the sensitivity analysis proposed here allows us to compute the structural sensitivities around any structural equilibrium point after finishing the solution process and it is completely independent of the numerical scheme used to solve the structural equilibrium problem. This possibility is particularized for the case of a damage model considering a strain-softening behaviour. Finally, the quality and reliability of the proposed approach is assessed through its application to some example
Abstract
This paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the finite element method [...]
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Wireless sensor networks for strain monitoring during steel bridges launching
R. Chacón, F. Guzmán, E. Mirabell, E. Real, E. Oñate

Structural Health Monitoring (2009). Vol. 8 (3), pp. 195-205

Abstract
n this paper, an experimental test performed on a hybrid steel plate girder was subjected to concentrated loads at the end of an unstiffened panel is presented. This load was intended to produce a typical reaction of the piers while launching steel girders in bridges. In this test, strain measurements were taken with conventional pre-wired gauges as well as with newly developed wirelessly connected strain-measuring system. Both measurements were carefully compared and the accuracy of the developed system was demonstrated. Results were also compared with numerical simulations deployed with FE models and on such a basis; the reliability of the developed wireless system was proven. Finally, suggestions of potential research trends in this area are provided
Abstract
n this paper, an experimental test performed on a hybrid steel plate girder was subjected to concentrated loads at the end of an unstiffened panel is presented. This load [...]
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Early detection of anomalies in dam performance: A methodology based on boosted regression trees
F. Salazar, M. Toledo, J. González, E. Oñate

Structural Control Health Monitoring (2017). Vol. 24 (11), Article Number: e2012

Abstract
The advances in information and communication technologies led to a general trend towards the availability of more detailed information on dam behaviour. This allows applying advanced data‐based algorithms in its analysis, which has been reflected in an increasing interest in the field. However, most of the related literature is limited to the evaluation of model prediction accuracy, whereas the ulterior objective of data analysis is dam safety assessment. In this work, a machine‐learning algorithm (boosted regression trees) is the core of a methodology for early detection of anomalies. It also includes a criterion to determine whether certain discrepancy between predictions and observations is normal, a procedure to compute a realistic estimate of the model accuracy, and an original approach to identify extraordinary load combinations. The performance of causal and noncausal models is assessed in terms of their ability to detect different types of anomalies, which were artificially introduced on reference time series generated with a numerical model of a 100‐m‐high arch dam. The final approach was implemented in an online application to visualise the results in an intuitive way to support decision making.
Abstract
The advances in information and communication technologies led to a general trend towards the availability of more detailed information on dam behaviour. This allows applying [...]
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Robust multidisciplinary unmanned aerial system design optimisation
D. Lee, J. Periaux, L. González, K. Srinivas, E. Oñate

Structural and Multidisciplinary Optimization (2012). Vol. 45 (3), pp. 433-450

Abstract
There are many applications in aeronautical/aerospace engineering where some values of the design parameters/states cannot be provided or determined accurately. These values can be related to the geometry (wingspan, length, angles) and or to operational flight conditions that vary due to the presence of uncertainty parameters (Mach, angle of attack, air density and temperature, etc.). These uncertainty design parameters cannot be ignored in engineering design and must be taken into the optimisation task to produce more realistic and reliable solutions. In this paper, a robust/uncertainty design method with statistical constraints is introduced to produce a set of reliable solutions which have high performance and low sensitivity. Robust design concept coupled with Multi-Objective Evolutionary Algorithms (MOEAs) is defined by applying two statistical sampling formulas; mean and variance/standard deviation associated with the optimisation fitness/objective functions. The methodology is based on a canonical evolution strategy and incorporates the concepts of hierarchical topology, parallel computing and asynchronous evaluation. It is implemented for two practical Unmanned Aerial System (UAS) design problems; the first case considers robust multi-objective (single-disciplinary: aerodynamics) design optimisation and the second considers a robust multidisciplinary (aero-structures) design optimisation. Numerical results show that the solutions obtained by the robust design method with statistical constraints have a more reliable performance and sensitivity in both aerodynamics and structures when compared to the baseline design. KeywordsRobust/uncertainty design–Multi-objective–Multidisciplinary–UAS–Evolutionary algorithms
Abstract
There are many applications in aeronautical/aerospace engineering where some values of the design parameters/states cannot be provided or determined accurately. These values [...]
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Discrete/finite element modelling of rock cutting with TBM disc cutter
C. Labra, J. Rojek, E. Oñate

Rock Mechanics and Rock Engineering (2017). Vol. 50 (3), pp. 621-638

Abstract
This paper presents advanced computer simulation of rock cutting process typical for excavation works in civil engineering. Theoretical formulation of the hybrid discrete/finite element model has been presented. The discrete and finite element methods have been used in different subdomains of a rock sample according to expected material behaviour, the part which is fractured and damaged during cutting is discretized with the discrete elements while the other part is treated as a continuous body and it is modelled using the finite element method. In this way, an optimum model is created, enabling a proper representation of the physical phenomena during cutting and efficient numerical computation. The model has been applied to simulation of the laboratory test of rock cutting with a single TBM (tunnel boring machine) disc cutter. The micromechanical parameters have been determined using the dimensionless relationships between micro- and macroscopic parameters. A number of numerical simulations of the LCM test in the unrelieved and relieved cutting modes have been performed. Numerical results have been compared with available data from in-situ measurements in a real TBM as well as with the theoretical predictions showing quite a good agreement. The numerical model has provided a new insight into the cutting mechanism enabling us to investigate the stress and pressure distribution at the tool–rock interaction. Sensitivity analysis of rock cutting performed for different parameters including disc geometry, cutting velocity, disc penetration and spacing has shown that the presented numerical model is a suitable tool for the design and optimization of rock cutting process.
Abstract
This paper presents advanced computer simulation of rock cutting process typical for excavation works in civil engineering. Theoretical formulation of the hybrid discrete/finite [...]
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Possibilities of the particle finite element method for fluid-structure interaction problems with free surface waves
E. Oñate, S. Idelsohn, F. Del Pin, R. Aubry

Revue Européenne des Éléments Finis (2004). Vol. 13, pp. 637-666

Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.
Abstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use [...]
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An Implicit Material Point Method Applied to Granular Flows
I. Iaconeta, A. Larese, R. Rossi, E. Oñate

Procedia Engineering (2017). Vol. 175, pp. 226-232

Abstract
The main objective of this work lies in the development of a variational implicit Material Point Method (MPM), implemented in the open source Kratos Multiphysics framework. The ability of the MPM technique to solve large displacement and large deformation problems is widely recognised and its use ranges over many problems in industrial and civil engineering. In the current work the continuum based implicit MPM is applied to engineering applications, where granular material flow is involved. For the resolution of the length and time scale of these particular problems, both continuum and discrete models are typically used. Even if discrete techniques predict more feasible results, nowadays, their use is limited to the investigation of element tests of particles, or to the simulation of reduced systems, not allowing to make important decisions in the analysis and design of granular processes. Some advantages of MPM over discrete methods are tested, such as, the ability to simulate granular flow at the large scale with acceptable computational cost and the capability to get information of stress and strain state in a more straightforward way. The focus of this paper is a comparative study between an irreducible and a mixed formulation, both implemented in the MPM code, to assess the improvement in accuracy and reliability of the numerical results when the latter formulation is adopted.
Abstract
The main objective of this work lies in the development of a variational implicit Material Point Method (MPM), implemented in the open source Kratos Multiphysics framework. [...]
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Modelling and simulation of the sea-landing of aerial vehicles using the Particle Finite Element Method
P. Ryzhakov, R. Rossi, A. Viña, E. Oñate

Ocean Engng. (2013). Vol. 66, pp. 92-100

Abstract
In this paper the Particle Finite Element Method (PFEM) is applied to the simulation of the sea-landing of an unmanned aerial vehicle (UAV). The problem of interest consists in modelling the impact of the vehicle against the water surface, analyzing the main kinematic and dynamic quantities (such as loads exerted upon the capsule at the moment of the impact). The PFEM, a methodology well-suited for free-surface flow simulation is used for modelling the water while a rigid body model is chosen for the vehicle. The vehicle under consideration is characterized by low weight. This leads to difficulties in modelling the fluid–structure interaction using standard Dirichlet–Neumann coupling. We apply a modified partitioned strategy introducing the interface Laplacian into the pressure Poisson's equation for obtaining a convergent FSI solution. The paper concludes with an industrial example of a vehicle sea-landing modelled using PFEM.
Abstract
In this paper the Particle Finite Element Method (PFEM) is applied to the simulation of the sea-landing of an unmanned aerial vehicle (UAV). The problem of interest consists [...]
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An extended class of multilayer perceptron
R. López, E. Oñate

Neurocomputing (2008). Vol. 71 (13-15), pp. 2538-2543

Abstract
In this work an extended class of multilayer perceptron is presented. This includes independent parameters, boundary conditions and lower and upper bounds. In some cases, such extensions contain a priori information of the problem. On some other situations they are necessary in order to define a correct representation for the solution. The use of this augmented class of neural network is illustrated through a case study in the optimal control theory. The numerical results are compared against the analytical solution.
Abstract
In this work an extended class of multilayer perceptron is presented. This includes independent parameters, boundary conditions and lower and upper bounds. In some cases, [...]
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A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells
G. Lomboy, K. Kim, E. Oñate

Mechanics of Advanced Materials and Structures (2007). Vol. 14 (2), pp. 89-105

Abstract
A nonlinear resultant shell element is developed for the solution of problems of composite plates and shells undergoing nonlinear static and nonlinear dynamic behavior with progressive layer failure. The formulation of the tangent stiffness is defined on the mid-surface and is efficient for analyzing thick laminated plates and shells by incorporating bending moments and transverse shear resultant forces in the geometric stiffness. The composite element is free of both membrane and shear locking behavior by using the assumed natural strain method, such that the element also performs very well as thin laminate shells. An equilibrium approach is used to derive the improved transverse shear stiffness, instead of using a shear correction factor. The proposed formulation is computationally efficient and the test results show good agreement with references. The composite shell element is extended to determine ply failures in laminated composite structures undergoing nonlinear static or dynamic behavior. The failure analysis is done by first, computing for the inter-laminar stresses at each gauss point in an element. Having obtained the stresses in each layer, checking for failure is performed based on a chosen failure criterion. Four failure criteria are available to enable the user to adopt the appropriate criterion for the type of problem parameters present.
Abstract
A nonlinear resultant shell element is developed for the solution of problems of composite plates and shells undergoing nonlinear static and nonlinear dynamic behavior with [...]
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Analysis of segmentally constructed prestressed concrete bridges using hexahedral elements with realistic tendon profiles
P. Norachan, K. Kim, E. Oñate

Journal of Structural Engineering (2014). Vol. 140 (6), pp. 04014028/1-17

Abstract
The paper presents a systematically numerical procedure based on the finite-element method for three-dimensional analysis of segmentally constructed prestressed concrete bridges using hexahedral elements including realistic tendon profiles. The enhanced assumed strain (EAS) is used in the formulation of the hexahedral element in order to improve the element performances. Both the geometric nonlinearity and time-dependent effects due to creep, shrinkage, and aging of the concrete and relaxation of the prestress are taken into account, while variations of the structural configuration due to changes of the structural geometry and boundary conditions during the construction process are also incorporated. To include realistic tendon profiles, the idealized prestressing tendon is represented by a system of piecewise linear prestressing segments. Several numerical examples in a wide range of prestressed concrete structures are presented to demonstrate the validity and efficiency of the proposed procedure. Finally, application to the erection of a segmentally erected prestressed concrete bridge is discussed at the end of the paper.
Abstract
The paper presents a systematically numerical procedure based on the finite-element method for three-dimensional analysis of segmentally constructed prestressed concrete bridges [...]
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Numerical analysis of the manufacturing processes of a mock-up of the ITER NHF First Wall Panel
J. González, M. Chiumenti, M. Cervera, C. Agelet de Saracibar, F. Samaniego, I. Cobo

Fusion Engineering and Design (2018). (preprint) Vol. 135, Part A, pp. 65-73

Abstract
The objective of ITER is to build a new Tokamak, with the goal of demonstrating the scientific and technical feasibility of fusion power. The First Wall Panels are the inner component of the reactor, built with different materials that must support high heat flux levels inside the vacuum vessel. The manufacturing processes of the First Wall are a complex procedure including bending, hipping and cutting procedures which, in general, lead to residual stresses and distortions of the fabricated component. In this work, the analysis of the thermo-mechanical response of a simplified prototype of the ITER NHF First Wall Panel is presented from the numerical point of view. The experimental procedure within each phase of the whole manufacturing process is described. Residual stresses and distortions have been measured and analyzed. The numerical simulation of the manufacturing process includes the description of the main hypothesis, the applied loads and the boundary conditions assumed at every stage of the process. Special attention is paid to the simulation of machining and cutting by means of an ad-hoc element deactivation strategy. The numerical results are compared with the experimental evidence to show the prediction capability and the limitations of the proposed numerical model.
Abstract
The objective of ITER is to build a new Tokamak, with the goal of demonstrating the scientific and technical feasibility of fusion power. The First Wall Panels are the inner [...]
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Modeling of ground excavation with the particle finite-element method
J. Carbonell, E. Oñate, B. Suarez

Journal of Engineering Mechanics (2010). Vol. 136 (4), pp. 455-463

Abstract
An excavation process is a nonlinear dynamic problem that includes geometrical, material, and contact nonlinearities. The simulation of ground excavation has to face contact interaction in a changing geometry composed by several solid domains. The particle finite-element method (PFEM) is based on a Lagrangian description for modeling the motion of a continuum medium. The PFEM is particularly suitable for modeling a fluid motion with free surfaces. The application of the PFEM in ground excavation includes the use of the remeshing process, α-shape concepts for detecting the domain boundary, contact mechanics laws, material constitutive models, and surface wear models. Everything is correctly matched to quantify the excavation and the corresponding damage caused to the ground. The erosion and wear parameters of the soil/rock material govern the evolution of the excavation process. The preliminary results presented in this paper show that the PFEM it is a very suitable tool for the simulation of ground excavation processes.
Abstract
An excavation process is a nonlinear dynamic problem that includes geometrical, material, and contact nonlinearities. The simulation of ground excavation has to face contact [...]
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Approximating the Basset force by optimizing the method of van Hinsberg et al.
G. Casas, A. Ferrer, E. Oñate

Journal of Computational Physics (2018). Vol. 352, pp. 142-171

Abstract
In this work we put the method proposed by van Hinsberg et al. to the test, highlighting its accuracy and efficiency in a sequence of benchmarks of increasing complexity. Furthermore, we explore the possibility of systematizing the way in which the method's free parameters are determined by generalizing the optimization problem that was considered originally. Finally, we provide a list of worked-out values, ready for implementation in large-scale particle-laden flow simulations.
Abstract
In this work we put the method proposed by van Hinsberg et al. to the test, highlighting its accuracy and efficiency in a sequence of benchmarks of increasing complexity. [...]
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Volumetric constraint models for anisotropic elastic solids
C. Felippa, E. Oñate

Journal of Applied Mechanics (2004). Vol. 71 (5), pp. 731-734

Abstract
We study three “incompressibility flavors” of linearly-elastic anisotropic solids that exhibit volumetric constraints: isochoric, hydroisochoric and rigidtropic. An isochoric material deforms without volume change under any stress system. An hydroisochoric material does so under hydrostatic stress. A rigidtropic material undergoes zero deformations under a certain stress pattern. Whereas the three models coalesce for isotropic materials, important differences appear for anisotropic behavior. We find that isochoric and hydroisochoric models under certain conditions may be hampered by unstable physical behavior. Rigidtropic models can represent semistable physical materials of arbitrary anisotropy while including isochoric and hydroisochoric behavior as special cases.
Abstract
We study three “incompressibility flavors” of linearly-elastic anisotropic solids that exhibit volumetric constraints: isochoric, hydroisochoric and rigidtropic. [...]
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Active transonic aerofoil design optimization using robust multiobjective evolutionary algorithms
D. Lee, J. Periaux, L. González, E. Oñate, N. Qin

Journal of Aircraft (2011). Vol. 48 (3) pp. 1084-1094

Abstract
The use of adaptive wing/aerofoil designs is being considered, as they are promising techniques in aeronautic/ aerospace since they can reduce aircraft emissions and improve aerodynamic performance of manned or unmanned aircraft. This paper investigates the robust design and optimization for one type of adaptive techniques: active flow control bump at transonic flow conditions on a natural laminar flow aerofoil. The concept of using shock control bump is to control supersonic flow on the suction/pressure side of natural laminar flow aerofoil that leads to delaying shock occurrence (weakening its strength) or boundary-layer separation. Such an active flow control technique reduces total drag at transonic speeds due to reduction of wave drag. The location of boundary-layer transition can influence the position and structure of the supersonic shock on the suction/pressure side of aerofoil. The boundarylayer transition position is considered as an uncertainty design parameter in aerodynamic design due to the many factors, such as surface contamination or surface erosion. This paper studies the shock-control-bump shape design optimization using robust evolutionary algorithms with uncertainty in boundary-layer transition locations. The optimization method is based on a canonical evolution strategy and incorporates the concepts of hierarchical topology, parallel computing, and asynchronous evaluation. Two test cases are conducted: the first test assumes the boundary-layer transition position is at 45% of chord from the leading edge, and the second test considers robust design optimization for the shock control bump at the variability of boundary-layer transition positions. The numerical result shows that the optimization method coupled to uncertainty design techniques produces Pareto optimal shock-control-bump shapes, which have low sensitivity and high aerodynamic performance while having significant total drag reduction.
Abstract
The use of adaptive wing/aerofoil designs is being considered, as they are promising techniques in aeronautic/ aerospace since they can reduce aircraft emissions and improve [...]
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Numerical investigation of wind-tunnel model deformations caused by the twin-sting support system
F. Flores, E. Ortega, E. Oñate

Journal of Aircraft (2010). Vol. 47, pp. 708-714

Abstract
The present work is part of a broader investigation aimed at improving the understanding of the aerodynamics of the rear aircraft section. Complex flow phenomena (e.g., the thick fuselage boundary layer at the rear end) present a challenge to the accurate predictions of aerodynamic performance. Significant efficiency gains are expected from improved-fidelity computations and experimental techniques. As part of a joint effort between several research institutions and industrial partners, a series of wind-tunnel test campaigns focusing on accurate rear-end measurements was planned. To obtain accurate tail-force data, a nonstandard live-rear-end test model was adopted. This technique uses a mechanically separate rear-end section attached to the main body through internal balances, allowing direct measurement of the force acting on the tail. This mechanical setup precludes the use of a conventional tail-coneattached sting. Moreover, to minimize aerodynamic interference on the empennage caused by the support system, a wing-attached twinsting mounting was chosen
Abstract
The present work is part of a broader investigation aimed at improving the understanding of the aerodynamics of the rear aircraft section. Complex flow phenomena (e.g., [...]
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Robust design optimization in aeronautics using stochastic analysis and evolutionary algorithms
J. Pons-Prats, G. Bugeda, F. Zárate, E. Oñate

Journal of Aerospace Engineering (2011). Vol. 225, (10), pp. 1131-1151

Abstract
Uncertainties are a daily issue to deal with in aerospace engineering and applications. Robust optimization methods commonly use a random generation of the inputs and take advantage of multi-point criteria to look for robust solutions accounting with uncertainty definition. From the computational point of view, the application to coupled problems, like computational fluid dynamics (CFD) or fluid–structure interaction (FSI), can be extremely expensive. This study presents a coupling between stochastic analysis techniques and evolutionary optimization algorithms for the definition of a stochastic robust optimization procedure. At first, a stochastic procedure is proposed to be applied into optimization problems. The proposed method has been applied to both CFD and FSI problems for the reduction of drag and flutter, respectively.
Abstract
Uncertainties are a daily issue to deal with in aerospace engineering and applications. Robust optimization methods commonly use a random generation of the inputs and take [...]
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A monolithic FE formulation for the analysis of membranes in fluids
R. Rossi, P. Ryzhakov, E. Oñate

International Journal of Space Structures (2009). Vol. 24 (4), pp. 205-210

Abstract
We propose here an efficient approach for treating the interaction between membranes and fluids. Slight compressibility of the fluid is assumed. Classical total Lagrangian formulation including wrinkling is adopted for the membrane representation, whereas fluid is treated in an updated Lagrangian manner, developed in the current work. Assumption of slight compressibility of the fluid enables one to define the monolithic fluid-membrane system in a natural way. The displacements are the primary variables of both the fluid and the membrane domains. The formulation adopts the Particle Finite Element Method (PFEM) philosophy for free-surface identification and mesh regeneration. Three examples illustrate the functionality of the formulation in application to FSI problems involving motion of membranes in water
Abstract
We propose here an efficient approach for treating the interaction between membranes and fluids. Slight compressibility of the fluid is assumed. Classical total Lagrangian [...]
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Discrete element simulation of rock cutting
J. Rojek, E. Oñate, C. Labra, H. Kargl

Int. J. Rock Mechanics and Mining Sciences (2011). Vol. 48 (6), pp. 996-1010

Abstract
This paper presents numerical modelling of rock cutting processes. The model consists of a tool–rock system. The rock is modelled using the discrete element method, which is suitable to study problems of multiple material fracturing such as those involved in rock cutting. Both 2D and 3D models are considered in this work. The paper presents a brief overview of the theoretical formulation and calibration of the discrete element model by a methodology combining the dimensional analysis with simulation of the unconfined compressive strength (UCS) and indirect tension (Brazilian) tests. The rock cutting process with roadheader picks, which is typical for underground excavation, has been simulated. The results of the 2D and 3D analyses have been compared with one another, and numerical results have been compared with the available experimental data.
Abstract
This paper presents numerical modelling of rock cutting processes. The model consists of a tool–rock system. The rock is modelled using the discrete element method, [...]
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Numerical and Experimental Study of Overtopping and Failure of Rockfill Dams
A. Larese, R. Rossi, E. Oñate, M. Toledo, R. Morán Moya, H. Campos

International Journal of Geomechanics (2015). Vol. 15 (4), pp. 04014060-1 (published online)

Abstract
This paper aims to present and validate a numerical technique for the simulation of the overtopping and onset of failure in rockfill dams caused by mass sliding. This goal is achieved by coupling a fluid dynamic model for the simulation of the free surface and through-flow problems together with a numerical technique for the calculation of the rockfill response and deformation. Both the flow within the dam body and in its surroundings are taken into account. An extensive validation of the resulting computational method is performed by solving several failure problems on physical models of rockfill dams for which experimental results have been obtained by the authors.
Abstract
This paper aims to present and validate a numerical technique for the simulation of the overtopping and onset of failure in rockfill dams caused by mass sliding. This goal [...]
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A FIC‐based stabilized mixed finite element method with equal order interpolation for solid–pore fluid interaction problems
I. de-Pouplana, E. Oñate

Int. J. Numer. Anal. Meth. Geomech. (2017). Vol. 41 (1), pp. 110-134

Abstract
A new mixed displacement‐pressure element for solving solid–pore fluid interaction problems is presented. In the resulting coupled system of equations, the balance of momentum equation remains unaltered, while the mass balance equation for the pore fluid is stabilized with the inclusion of higher‐order terms multiplied by arbitrary dimensions in space, following the finite calculus (FIC) procedure. The stabilized FIC‐FEM formulation can be applied to any kind of interpolation for the displacements and the pressure, but in this work, we have used linear elements of equal order interpolation for both set of unknowns. Examples in 2D and 3D are presented to illustrate the accuracy of the stabilized formulation for solid–pore fluid interaction problems.
Abstract
A new mixed displacement‐pressure element for solving solid–pore fluid interaction problems is presented. In the resulting coupled system of equations, the balance [...]
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Numerical modelling of landslide‐generated waves with the particle finite element method (PFEM) and a non‐Newtonian flow model
F. Salazar, J. Irazábal González, A. Larese, E. Oñate

Int. J. Numer. Anal. Meth. Geomech. (2016). Vol. 40 (6), pp. 809-826

Abstract
Landslide‐generated impulse waves may have catastrophic consequences. The physical phenomenon is difficult to model because of the uncertainties in the kinematics of the mobilised material and to the intrinsic complexity of the fluid–soil interaction. The particle finite element method (PFEM) is a numerical scheme that has successfully been applied to fluid–structure interaction problems. It uses a Lagrangian description to model the motion of nodes (particles) in both the fluid and the solid domains (the latter including soil/rock and structures). A mesh connecting the particles (nodes) is re‐generated at every time step, where the governing equations are solved. Various constitutive laws are used for the sliding mass, including rigid solid and the Newtonian and non‐Newtonian fluids. Several examples of application are presented, corresponding both to experimental tests and to actual full‐scale case studies. The results show that the PFEM can be a useful tool for analysing the risks associated with landslide phenomena, providing a good estimate to the potential hazards even for full‐scale events.
Abstract
Landslide‐generated impulse waves may have catastrophic consequences. The physical phenomenon is difficult to model because of the uncertainties in the kinematics of the [...]
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Modelación del ensayo brasileño empleando modelación discreta
C. Recarey Morfa, E. Oñate, J. Rojek, J. Canet, F. Zárate, G. Cadoce, C. Labra

Ingeniería Civil (2005). Vol. 139, pp. 112-124

Abstract
Se presenta un modelo de elementos discretos o distintos (MED) en la modelizacion del ensayo de traccion indirecta o ensayo brasileno (Brazilian Test). Se realiza una comparacion entre los resultados obtenidos por el MED, con la de otros autores [26,35] que emplean el MEF. Se obtienen buenos resultados en el proceso de simulacion del ensayo brasileno, y la comparacion con el MEF (método de los elementos finitos) es muy efectiva. Se evidencia la similitud fisica que existe entre los resultados obtenidos por el MED y ensayos experimentales, lo que posibilita demostrar la fiabilidad y veracidad de la formulacion realizada. (A). English abstract: This work deals whit the use of the Distinct Element Method (DEM) to simulate an indirect traction test, also known as "the Brazilian test" A comparison is realized between results obtained with the DEM, and those obtained by other authors [26, 35] using the Finite Element Method (FEM). The Brazilian test's simulation gives good results and the comparison is very effective. There is a similarity between the results obtained using the DEM and those obtained from real experiments, similarity which proves the reliability and the veracity of this formulation.
Abstract
Se presenta un modelo de elementos discretos o distintos (MED) en la modelizacion del ensayo de traccion indirecta o ensayo brasileno (Brazilian Test). Se realiza una comparacion [...]
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Triangular prismatic solid-shell element with generalised deformation description
V. Mataix Ferrándiz, F. Flores, R. Rossi, E. Oñate

European Journal of Computational Mechanics (2018). Vol. 27 (1), pp. 1-32

Abstract
The solid-shells are an attractive kind of element for the simulation of f orming processes, due to the fact that any kind of generic 3D constitutive law can be employed without any kind of additional modification, besides the thermomechanic problem is formulated without additional assumptions. Additionally, this type of element allows the three-dimensional description of the deformable body, thus contact on both sides of the element can be treated easily. The present work consists in the development of a triangular prism element as a solid-shell, for the analysis of thin/thick shell, undergoing large deformations. The element is formulated in total Lagrangian formulation, and employs the neighbour (adjacent) elements to perform a local patch to enrich the displacement field. In the original formulation by Flores, a modified right Cauchy-Green deformation tensor () is obtained; in the present work a modified deformation gradient () is obtained, which allows to generalise the methodology and allows to employ a wide range of constitutive laws. The element is based in three modifications: (a) a classical assumed strain approach for transverse shear strains (b) an assumed strain approach for the in-plane components using information from neighbour elements and (c) an averaging of the volumetric strain over the element. The objective is to use this type of elements for the simulation of shells avoiding transverse shear locking, improving the membrane behaviour of the in-plane triangle and to handle quasi-incompressible materials or materials with isochoric plastic flow. Some examples have been evaluated to show the good performance of the element and results.
Abstract
The solid-shells are an attractive kind of element for the simulation of f orming processes, due to the fact that any kind of generic 3D constitutive law can be employed without [...]
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Interpretation of dam deformation and leakage with boosted regression trees
F. Salazar, M. Toledo, E. Oñate, B. Suarez

Engineering Structures (2016). Vol. 119, pp. 230-251

Abstract
Predictive models are essential in dam safety assessment. They have been traditionally based on simple statistical tools such as the hydrostatic-season-time (HST) model. These tools are well known to have limitations in terms of accuracy and reliability. In the recent years, the examples of application of machine learning and related techniques are becoming more frequent as an alternative to HST. While they proved to feature higher flexibility and prediction accuracy, they are also more difficult to interpret. As a consequence, the vast majority of the research is limited to prediction accuracy estimation. In this work, one of the most popular machine learning techniques (boosted regression trees), was applied to model 8 radial displacements and 4 leakage flows at La Baells Dam. The possibilities of model interpretation were explored: the relative influence of each predictor was computed, and the partial dependence plots were obtained. Both results were analysed to draw conclusions on dam response to environmental variables, and its evolution over time. The results show that this technique can efficiently identify dam performance changes with higher flexibility and reliability than simple regression models.
Abstract
Predictive models are essential in dam safety assessment. They have been traditionally based on simple statistical tools such as the hydrostatic-season-time (HST) model. These [...]
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Finite element modelling of fracture propagation in saturated media using quasi-zero-thickness interface elements
I. de-Pouplana, E. Oñate

Computers and Geotechnics (2018). Vol. 96, pp. 103-117

Abstract
A new computational technique for the simulation of 2D and 3D fracture propagation processes in saturated porous media is presented. A non-local damage model is conveniently used in conjunction with interface elements to predict the degradation pattern of the domain and insert new fractures followed by remeshing. FIC-stabilized elements of equal order interpolation in the displacement and the pore pressure have been successfully used under complex conditions near the undrained-incompressible limit. A bilinear cohesive fracture model describes the mechanical behaviour of the joints. A formulation derived from the cubic law models the fluid flow through the crack. Examples in 2D and 3D, using 3-noded triangles and 4-noded tetrahedra respectively, are presented to illustrate the accuracy and robustness of the proposed methodology.
Abstract
A new computational technique for the simulation of 2D and 3D fracture propagation processes in saturated porous media is presented. A non-local damage model [...]
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Numerical modelling of granular materials with spherical discrete particles and the bounded rolling friction model. Application to railway ballast
J. Irazábal González, F. Salazar, E. Oñate

Computers and Geotechnics (2017). Vol. 85, pp. 220-229

Abstract
The Discrete Element Method (DEM) was found to be an effective numerical method for the calculation of engineering problems involving granular materials. However, the representation of irregular particles using the DEM is a very challenging issue, leading to different geometrical approaches. This document presents a new insight in the application of one of those simplifications known as rolling friction, which avoids excessive rotation when irregular shaped materials are simulated as spheric particles. This new approach, called the Bounded Rolling Friction model, was applied to reproduce a ballast resistance test.
Abstract
The Discrete Element Method (DEM) was found to be an effective numerical method for the calculation of engineering problems involving granular materials. [...]
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Computationally optimized formulation for the simulation of composite materials and delamination failures
X. Martinez, F. Rastellini, S. Oller, F. Flores, E. Oñate

Composites Part B: Enginnering (2011). Vol. 42 (2), pp. 134-144

Abstract
The numerical simulation of complex failure modes of composite materials, such as delamination, can be computationally very demanding, as it requires special elements and/or numerical strategies to characterize damage onset and propagation. This work presents several formulations developed to optimize the computational performance of an explicit finite element code designed specifically for the simulation of large scale composite structures. The composite mechanical performance is obtained with the matrix-reinforced mixing theory, a simplified version of the serial/parallel mixing theory that does not require an iterative procedure or the calculation of the tangent stiffness matrix. The number of elements required to perform the simulation is reduced by stacking several layers inside a single finite element. This work also proposes a modification of the isotropic damage law, capable of taking into account the residual strength provided by friction in type II fracture modes. The ability of these formulations to successfully predict the mechanical performance of composite materials is assessed with the ply drop-off test. In this test a laminate with a change of thickness in its mid-span is loaded until it breaks due to a delamination process. The formulation proposed obtains a very accurate prediction of the experimental response of the test, as it provides a very good characterization of the initial laminate stiffness, the delamination onset, and its propagation along the specimen.
Abstract
The numerical simulation of complex failure modes of composite materials, such as delamination, can be computationally very demanding, as it [...]
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A hierarchical finite element for composite laminated beams using a refined zigzag theory
L. Nallim, S. Oller, E. Oñate, F. Flores

Composite Structures (2017). Vol. 163, pp. 168-184

Abstract
In this work a kinematics for laminated beams enriched with a refined formulation ZigZag (RZT), originally presented by Tessler et al. in 2007, introduced in a hierarchical one dimensional type “p” finite element is presented. The finite element employs Lagrange polynomials for the approximation of the degrees of freedom of the ends (nodes) and orthogonal Gram-Schmidt polynomials to the internal degrees of freedoms. This finite element allows a very low discretization, is free of shear locking and behaves very well when the analysis of laminated composites with accurate determination of local stresses and strains at laminar level is necessary. This element has been validated in the analysis of laminated beams with various sequences of symmetric and asymmetric stacking, studying in each case its accuracy and stability.
Abstract
In this work a kinematics for laminated beams enriched with a refined formulation ZigZag (RZT), originally presented by Tessler et al. in 2007, introduced in a hierarchical [...]
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Two-noded zigzag beam element accounting for shear effects based on an extended Euler Bernoulli theory
D. Di Capua, E. Oñate

Composite Structures (2015). Vol. 132, pp 1192-1205

Abstract
We present a new 2-noded beam element based on the refined zigzag theory and the classical Euler–Bernoulli beam theory for the static analysis of composite laminate and sandwich beams. The proposed element is able to take into account distortion effects due to shear elastic strains and can predict delamination. The element has four degrees of freedom per node. A C1 cubic Hermite interpolation is used for the vertical deflection while a C0 linear interpolation is employed for the other kinematics variables. The stiffness matrix and the load vector are calculated in explicit form using exact integration. The element is free from shear locking as confirmed with numerical tests on a wide range of the slenderness ratios. Numerical results show the ability of the EEBZ2 element to reproduce accurately the vertical deflection along the beam length and complex zigzag distributions of the axial displacement and the stresses across the thickness. Delamination effects are modeled by incorporating of an additional zigzag function corresponding to the kinematics of a zero thickness layer where delamination occurs. An example showing the capability of the new EEBZ2 element for accurately reproducing delamination effects is presented.
Abstract
We present a new 2-noded beam element based on the refined zigzag theory and the classical Euler–Bernoulli beam theory for the static analysis of composite laminate [...]
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Delamination in laminated plates using the 4-noded quadrilateral qlrz plate element based on the refined zigzag theory
A. Eijo, E. Oñate, S. Oller

Composite Structures (2014). Vol. 108, pp. 456-471

Abstract
A numerical method based on the Refined Zigzag Theory (RZT) to model delamination in composite laminated plate/shell structures is presented. The originality of this method is the use of 4-noded quadrilateral plate finite elements whit only seven variables per node to discretize the plate/shell geometry. The ability to capture the relative displacement between consecutive layers in fracture mode II and III is the more important advantage of this element, denoted QLRZ. A continuum isotropic damage model is used to model the mechanical behavior of the plies. The material non-lineal problem is solved with the modified Newton–Raphson method. The RZT plate theory, the QLRZ finite element and the isotropic damage model are described in this work. Also, the implicit integration algorithm is presented. The performance of the numerical model is analyzed by studying the delamination in a rectangular plate for two different laminates, using the 3D analysis as the reference solution
Abstract
A numerical method based on the Refined Zigzag Theory (RZT) to model delamination in composite laminated plate/shell structures is presented. The originality of this method [...]
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A numerical model of delamination in composite laminated beams using the LRZ beam element based on the refined zigzag theory
A. Eijo, E. Oñate, S. Oller

Composite Structures (2013). Vol. 104, pp. 270-280

Abstract
A method based on the Refined Zigzag Theory (RZT) to model delamination in composite laminated beam structures is presented. The novelty of this method is the use of one-dimensional finite elements to discretize the geometry of the beam. The key property of this beam element, named LRZ, is the possibility to capture the relative displacement between consecutive layers which occurs during delamination. The fracture mode that the LRZ element is capable to predict is mode II. In order to capture the relative displacement using the LRZ element it is necessary to adapt the RZT theory as presented in this paper. The mechanical properties of the layers are modeled using a continuum isotropic damage model. The modified Newton–Raphson method is used for solving the non-linear problem. The RZT theory, the LRZ finite element and the isotropic damage model are described in the paper. Also, the implicit integrations algorithm is presented. The performance of the LRZ element is analyzed by studying the delamination in a beam for two different laminates, using the plane stress solution as a reference.
Abstract
A method based on the Refined Zigzag Theory (RZT) to model delamination in composite laminated beam structures is presented. The novelty of this method is the use of one-dimensional [...]
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Robust design optimisation of advance hybrid (fiber-metal) composite structures
D. Lee, C. Morillo, S. Oller, G. Bugeda, E. Oñate

Composite Structures (2013). Vol. 99, pp. 181-192

Abstract
Hybrid Composite Structures (HCSs) are consisting of alternating layers of Fiber-Reinforced Polymer and metal sheets. Mechanical properties and responses for off-design conditions of HCSs can be improved using an innovative methodology coupling Multi-Objective Genetic Algorithm and robust design method. The concept of robust design approach ensures that a structure will be tolerant to unexpected loading and operating conditions. In this paper, two applications are considered; the first is to maximise the stiffness of the HCS while minimising its total weight through a Multi-Objective Design Optimisation. The second application considers a Robust Multi-Objective Design Optimisation (RMDO) to minimise total weight of HCS and to minimise both, the normalised mean displacement and the standard deviations of displacement, considering critical load cases. For the optimisation process, a distributed/parallel Multi-Objective Genetic Algorithm in robust multi-objective optimisation platform is used and it is coupled to a Finite Element Analysis based composite structure analysis tool to find the optimal combination of laminates sequences for HCSs. Numerical results show the advantages in mechanical properties of HCS over the metal structures, and also the use of RMDO methodology to obtain higher characteristics of HCS in terms of mechanical properties and its stability at the variability of load cases.
Abstract
Hybrid Composite Structures (HCSs) are consisting of alternating layers of Fiber-Reinforced Polymer and metal sheets. Mechanical properties and responses for off-design [...]
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Multilayered composite structure design optimisation using distributed/parallel multi-objective evolutionary algorithms
D. Lee, C. Morillo, G. Bugeda, S. Oller, E. Oñate

Composite Structures (2012). Vol. 94 (3), pp. 1087-1096

Abstract
This paper presents a research work on stacking sequence design optimisation for multilayered composite plate using a parallel/distributed evolutionary algorithm. The stacking sequence of fibres has a dramatic influence on the strength of multilayered composite plates. Multiple layers of fibre-reinforced material systems offer versatility in engineering material design due to the fact that the stacking sequence of each orthotropic layer can offer full advantage of superior mechanical properties. Numerical results show that the optimal composite structures have lower weight, higher stiffness and also affordable cost when compared to the extreme and intermediate composite structures. In addition, the benefits of using a parallel optimisation system are also presented.
Abstract
This paper presents a research work on stacking sequence design optimisation for multilayered composite plate using a parallel/distributed evolutionary algorithm. [...]
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Finite element modeling of free surface flow in variable porosity media
A. Larese, R. Rossi, E. Oñate

Archives of Comp. Meths. Engng. (2015). Vol. 22 (4), pp. 637-653

Abstract
The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm developed by the authors for solving this type of problems is presented. A modified form of the classical Navier–Stokes equations is proposed, with the principal aim of simulating in a unified way the seepage flow inside rockfill-like porous material and the free surface flow in the clear fluid region. The problem is solved using a semi-explicit stabilized fractional step algorithm where velocity is calculated using a 4th order Runge–Kutta scheme. The numerical formulation is developed in an Eulerian framework using a level set technique to track the evolution of the free surface. An edge-based data structure is employed to allow an easy OpenMP parallelization of the resulting finite element code. The numerical model is validated against laboratory experiments on small scale rockfill dams and is compared with other existing methods for solving similar problems.
Abstract
The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm [...]
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Data-Based Models for the Prediction of Dam Behaviour: A Review and Some Methodological Considerations
F. Salazar, R. Morán Moya, M. Toledo, E. Oñate

Archives of Comp. Meths. Engng. (2017). Vol. 24 (1), pp. 1-21

Abstract
Predictive models are an important element in dam safety analysis. They provide an estimate of the dam response faced with a given load combination, which can be compared with the actual measurements to draw conclusions about dam safety. In addition to numerical finite element models, statistical models based on monitoring data have been used for decades for this purpose. In particular, the hydrostatic-season-time method is fully implemented in engineering practice, although some limitations have been pointed out. In other fields of science, powerful tools such as neural networks and support vector machines have been developed, which make use of observed data for interpreting complex systems . This paper contains a review of statistical and machine-learning data-based predictive models, which have been applied to dam safety analysis . Some aspects to take into account when developing analysis of this kind, such as the selection of the input variables, its division into training and validation sets, and the error analysis, are discussed. Most of the papers reviewed deal with one specific output variable of a given dam typology and the majority also lack enough validation data. As a consequence, although results are promising, there is a need for further validation and assessment of generalisation capability. Future research should also focus on the development of criteria for data pre-processing and model application.
Abstract
Predictive models are an important element in dam safety analysis. They provide an estimate of the dam response faced with a given load combination, which can be compared [...]
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An Object-oriented Environment for Developing Finite Element Codes for Multi-disciplinary Applications
P. Dadvand, R. Rossi, E. Oñate

Archives of Comp. Meths. Engng. (2010). Vol. 17 (3), pp 253-297

Abstract
The objective of this work is to describe the design and implementation of a framework for building multi-disciplinary finite element programs. The main goals are generality, reusability, extendibility, good performance and memory efficiency. Another objective is preparing the code structure for team development to ensure the easy collaboration of experts in different fields in the development of multi-disciplinary applications. Kratos, the framework described in this work, contains several tools for the easy implementation of finite element applications and also provides a common platform for the natural interaction of different applications. To achieve this, an innovative variable base interface is designed and implemented. This interface is used at different levels of abstraction and showed to be very clear and extendible. A very efficient and flexible data structure and an extensible IO are created to overcome difficulties in dealing with multi-disciplinary problems. Several other concepts in existing works are also collected and adapted to coupled problems. The use of an interpreter, of different data layouts and variable number of dofs per node are examples of such approach. In order to minimize the possible conflicts arising in the development, a kernel and application approach is used. The code is structured in layers to reflect the working space of developers with different fields of expertise. Details are given on the approach chosen to increase performance and efficiency. Examples of application of Kratos to different multidisciplinary problems are presented in order to demonstrate the applicability and efficiency of the new object oriented environment.
Abstract
The objective of this work is to describe the design and implementation of a framework for building multi-disciplinary finite element programs. The main goals are generality, [...]
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Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element
G. Lomboy, S. Suthasupradit, K. Kim, E. Oñate

Archives of Comp. Meths. Engng. (2009). Vol. 16 (2), pp. 189-250

Abstract
The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented. The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening.
Abstract
The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming [...]
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Geometry optimization of the diffuser for the supersonic wind tunnel using genetic algorithm and adaptive mesh refinement technique
M. Kouhi, M. Manshadi, E. Oñate

Aerospace Science and Technology (2014). Vol. 36, pp. 64-74, 2014

Abstract
Design of two-dimensional supersonic diffusers as a part of the wind tunnel is investigated in this paper. A methodology based on the mixture of try-and-error method and optimization algorithm is developed to handle the design problem. In the first design step, using try-and-error approach, the main parameters related to the geometry of diffuser such as length, angle and area ratio between the throat and the outlet are determined assuming a diffuser with linear walls. The design criterion in this step is the fact that the shock wave should be created near the diffuser throat in order to benefit from the maximum efficiency of diffuser. In the second design step, considering the optimization methodology, it is tried to improve the optimum design obtained in the first step by modifying the wall point locations and keeping the rest of the geometry fixed. Hence, an optimization problem is defined to find the best curve for the diffuser wall instead of the a linear one used the in the first step. The objective function of this problem is to minimize the output Mach number using Genetic Algorithm (GA). The fluid flow is evaluated using the Euler equations in the conservative form where the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element scheme is used to discretize the flow equations. In order to capture the flow solution around the shock waves accurately, the adaptive mesh refinement technique is coupled to the flow solution. The demonstrated results show the efficiency of the proposed method for designing supersonic diffusers
Abstract
Design of two-dimensional supersonic diffusers as a part of the wind tunnel is investigated in this paper. A methodology based on the mixture of try-and-error method [...]
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Advances in discrete element modelling of underground excavations
C. Labra, J. Rojek, E. Oñate, F. Zárate

Acta Geotechnica (2009). Vol. 3 (4), pp. 317-322

Abstract
The paper presents advances in the discrete element modelling of underground excavation processes extending modelling possibilities as well as increasing computational efficiency. Efficient numerical models have been obtained using techniques of parallel computing and coupling the discrete element method with finite element method. The discrete element algorithm has been applied to simulation of different excavation processes, using different tools, TBMs and roadheaders. Numerical examples of tunnelling process are included in the paper, showing results in the form of rock failure, damage in the material, cutting forces and tool wear. Efficiency of the code for solving large scale geomechanical problems is also shown
Abstract
The paper presents advances in the discrete element modelling of underground excavation processes extending modelling possibilities as well as increasing computational efficiency. [...]
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A two-step monolithic method for the efficient simulation of incompressible flows
P. Ryzhakov, J. Cotela, R. Rossi, E. Oñate

Int. J. Numer. Meth. Fluids (2014). Vol 74 (12), pp. 919-934

Abstract
We propose a simple technique for improving computationally the efficiency of monolithic velocity–pressure solvers for incompressible flow problems. The idea consists in solving the discrete nonlinear system of governing equations in two steps: introducing ‘artificial’ compressibility first and afterwards correcting the solution by solving the original incompressible system. The speed‐up is obtained because of a better conditioning of the modified discrete system solved at the prediction step. The formulation can be easily implemented into existing monolithic codes requiring minor modification only. The paper concludes with two examples validating the formulation and facilitating the estimation of the obtained speed‐up. For the tests chosen, an average speed‐up is approximately double, suggesting that the method is a feasible approach for incompressible flows' simulation.
Abstract
We propose a simple technique for improving computationally the efficiency of monolithic velocity–pressure solvers for incompressible flow problems. The idea consists [...]
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Multi-objective aerodynamic shape optimization using MOGA coupled to advanced adaptive mesh refinement
M. Kouhi, D. Lee, G. Bugeda, E. Oñate

Computers and Fluids (2013). Vol. 88, pp. 298-312

Abstract
This paper demonstrates the big influence of the control of the mesh quality in the final solution of aerodynamic shape optimization problems. It aims to study the trade-off between the mesh refinement during the optimization process and the improvement of the optimized solution. This subject is investigated in the transonic airfoil design optimization using an Adaptive Mesh Refinement (AMR) technique coupled to Multi-Objective Genetic Algorithm (MOGA) and an Euler aerodynamic analysis tool. The methodology is implemented to solve three practical design problems; the first test case considers a reconstruction design optimization that minimizes the pressure error between a predefined pressure curve and candidate pressure distribution. The second test considers the total drag minimization by designing airfoil shape operating at transonic speeds. For the final test case, a multi-objective design optimization is conducted to maximize both the lift to drag ratio (L/D) and lift coefficient (Cl). The solutions obtained with and without adaptive mesh refinement are compared in terms of solution improvement and computational cost. Numerical results clearly show that the use of adaptive mesh refinement can improve the solution accuracy while reducing significant computational cost in both single- and multi-objective design optimizations.
Abstract
This paper demonstrates the big influence of the control of the mesh quality in the final solution of aerodynamic shape optimization problems. It aims to study the trade-off [...]
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An efficient edge‐based level set finite element method for free surface flow problems
R. Rossi, E. Oñate, A. Larese, P. Dadvand

Int. J. Numer. Meth. Fluids (2013). Vol. 71 (6), pp. 687-716

Abstract
We present an efficient technique for the solution of free surface flow problems using level set and a parallel edge‐based finite element method. An unstructured semi‐explicit solution scheme is proposed. A custom data structure, obtained by blending node‐based and edge‐based approaches is presented so to allow a good parallel performance. In addition to standard velocity extrapolation (for the convection of the level set function), an explicit extrapolation of the pressure field is performed in order to impose both the pressure boundary condition and the volume conservation. The latter is also improved with a modification of the divergence free constrain. The method is shown to allow an efficient solution of both simple benchmark cases and complex industrial examples
Abstract
We present an efficient technique for the solution of free surface flow problems using level set and a parallel edge‐based finite element method. An unstructured semi‐explicit [...]
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Robust active shock control bump design optimisation using hybrid parallel MOGA
D. Lee, G. Bugeda, J. Périaux, E. Oñate

Computers and Fluids (2013). Vol. 80, pp. 214 - 224

Abstract
The paper investigates a robust optimisation for detail design of active shock control bump on a transonic Natural Laminar Flow (NLF) aerofoil using a Multi-Objective Evolutionary Algorithm (MOEA) coupled to Computational Fluid Dynamics (CFDs) software. For MOEA, Robust Multi-Objective Optimisation Platform (RMOP) developed at CIMNE is used. For the active shock control bump design, two different optimisation methods are considered; the first method is a Pareto-Game based Genetic Algorithm in RMOP (denoted as RMOGA). The second method uses a Hybridised RMOGA with Game-Strategies and a parallel computation for high performance computation. Numerical results show not only how the concept of Shock Control Bump (SCB) coupled to CFD can improve aerodynamic performance of classic transonic aerofoil at the variability of flight conditions but also how high performance (parallel/distributed) computation with applying Hybrid-Game increases the efficiency of optimisation in terms of computational cost and results accuracy
Abstract
The paper investigates a robust optimisation for detail design of active shock control bump on a transonic Natural Laminar Flow (NLF) aerofoil using a Multi-Objective Evolutionary [...]
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An algorithm for the simulation of thermally coupled low speed flow problems
P. Ryzhakov, E. Oñate, R. Rossi

Int. J. Numer. Meth. Fluids (2012). Vol. 70' (1), pp. 1-19

Abstract
In this paper, we propose a computational algorithm for the solution of thermally coupled flows in subsonic regime. The formulation is based upon the compressible Navier–Stokes equations, written in nonconservation form. An efficient modular implementation is obtained by solving the energy equation separately and then using the computed temperature as a known value in the momentum-continuity system. If an explicit single-step time integration scheme for the energy equation is used, the decoupling results to be natural. Integration of the momentum-continuity system is carried out using a semi-explicit method, combining Runge–Kutta and Backward Euler schemes for the momentum and continuity equations, respectively. Implicit treatment of pressure leads to favorable time step estimates even in the low Mach number (Ma ≪ 1) regimes. The numerical dissipation introduced by the Backward Euler scheme ensures absence of the spurious high frequencies in the numerical solution. The key point of the method is the assumption of linear variation of the temperature within a time step. Combined with a fractional splitting of the momentum-continuity system, it allows to solve the continuity only once per time step. Omitting the necessity of solving for the pressure at every intermediate step of the Runge–Kutta scheme minimizes the computational cost associated to the implicit step and leads to an efficiency close to that of a purely explicit scheme. The method is tested using two benchmark examples.
Abstract
In this paper, we propose a computational algorithm for the solution of thermally coupled flows in subsonic regime. The formulation is based upon the compressible Navier–Stokes [...]
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Hybrid-Game Strategies for multi-objective design optimization in engineering
D. Lee, L. González, J. Periaux, A. Srinivas, E. Oñate

Computers and Fluids (2011). Vol. 47 (1), pp. 189-204

Abstract
A number of Game Strategies (GS) have been developed in past decades. They have been used in the fields of economics, engineering, computer science and biology due to their efficiency in solving design optimization problems. In addition, research in multi-objective (MO) and multidisciplinary design optimization (MDO) has focused on developing robust and efficient optimization methods to produce a set of high quality solutions with low computational cost. In this paper, two optimization techniques are considered; the first optimization method uses multi-fidelity hierarchical Pareto optimality. The second optimization method uses the combination of two Game Strategies; Nash-equilibrium and Pareto optimality. The paper shows how Game Strategies can be hybridised and coupled to Multi-Objective Evolutionary Algorithms (MOEA) to accelerate convergence speed and to produce a set of high quality solutions. Numerical results obtained from both optimization methods are compared in terms of computational expense and model quality. The benefits of using Hybrid-Game Strategies are clearly demonstrated
Abstract
A number of Game Strategies (GS) have been developed in past decades. They have been used in the fields of economics, engineering, computer science and biology due to their [...]
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Lift maximization with uncertainties for the optimization of high‐lift devices
Z. Tang, J. Périaux, G. Bugeda, E. Oñate

Int. J. Numer. Meth. Fluids (2010). Vol. 64 (2), pp. 119-135

Abstract
In this paper, the aerodynamic shape optimization problems with uncertain operating conditions have been addressed. After a review of robust control theory and the possible approaches to take into account uncertainties, the use of Taguchi robust design methods in order to overcome single point design problems in aerodynamics is proposed. Under the Taguchi concept, a design with uncertainties is converted into an optimization problem with two objectives which are the mean performance and its variance, so that the solutions are as less sensitive to the uncertainty of the input parameters as possible. Furthermore, the modified non‐dominated sorting genetic algorithms are used to capture a set of compromised solutions (Pareto front) between these two objectives. The flow field is analyzed by Navier–Stokes computation using an unstructured mesh. In order to reduce the number of expensive evaluations of the fitness function a response surface modeling is employed to estimate the fitness value using the polynomial approximation model. During the solution of the optimization problem, a semi‐torsional spring analogy is used for the adaption of the computational mesh to all the obtained geometrical configurations. The proposed approach is applied to the robust optimization of the 2D high‐lift devices of a business aircraft by maximizing the mean and minimizing the variance of the lift coefficients with uncertain free‐stream angle of attack at landing flight condition.
Abstract
In this paper, the aerodynamic shape optimization problems with uncertain operating conditions have been addressed. After a review of robust control theory and the possible [...]
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Simulation of flows with violent free surface motion and moving objects using unstructured grids
R. Löhner, C. Yang, E. Oñate

Int. J. Numer. Meth. Fluids (2007). Vol. 53 (8), pp 1315-1338

Abstract
A volume of fluid (VOF) technique has been developed and coupled with an incompressible Euler/Navier–Stokes solver operating on adaptive, unstructured grids to simulate the interactions of extreme waves and three‐dimensional structures. The present implementation follows the classic VOF implementation for the liquid–gas system, considering only the liquid phase. Extrapolation algorithms are used to obtain velocities and pressure in the gas region near the free surface. The VOF technique is validated against the classic dam‐break problem, as well as series of 2D sloshing experiments and results from SPH calculations. These and a series of other examples demonstrate that the ability of the present approach to simulate violent free surface flows with strong nonlinear behaviour.
Abstract
A volume of fluid (VOF) technique has been developed and coupled with an incompressible Euler/Navier–Stokes solver operating on adaptive, unstructured grids to simulate [...]
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Optimum aerodynamic shape design including mesh adaptivity
G. Bugeda, E. Oñate

Int. J. Numer. Meth. Fluids (1995). Vol. 20 (8-9), pp. 915-934

Abstract
This paper presents a methodology for solving shape optimization problems in the context of fluid flow problems including adaptive remeshing. The method is based on the computation of the sensitivities of the geometrical design parameters, the mesh, the flow variables and the error estimator to project the refinement parameters from one design to the next. This sensitivity analysis is described for the incompressible potential equations and the Euler equations. The efficiency of the proposed method is checked by means of two 2D inverse problems.
Abstract
This paper presents a methodology for solving shape optimization problems in the context of fluid flow problems including adaptive remeshing. The method is based on the computation [...]
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A general procedure for deriving stabilized space–time finite element methods for advective–diffusive problems
E. Oñate, M. Manzán

Int. J. Numer. Meth. Fluids (1999). Vol. 31 (1), pp. 203-221

Abstract
A procedure to derive stabilized space–time finite element methods for advective–diffusive problems is presented. The starting point is the stabilized balance equation for the transient case derived by Oñate [Comput. Methods Appl. Mech. Eng., 151, 233–267 (1998)] using a finite increment calculus approach. A description of the new stabilization method and a procedure for computing the stabilization parameter of the space–time solution is given. The efficiency of the stabilization approach is shown in the solution of some transient advective–diffusive problems, including the non‐linear Burger's equation.
Abstract
A procedure to derive stabilized space–time finite element methods for advective–diffusive problems is presented. The starting point is the stabilized balance [...]
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Optimum aerodynamic shape design for fluid flow problems including mesh adaptivity
G. Bugeda, E. Oñate

Int. J. Numer. Meth. Fluids (1999). Vol. 30 (2), pp. 161-178

Abstract
This paper presents a methodology for solving shape optimization problems in the context of fluid flow problems including adaptive remeshing. The method is based on the computation of the sensitivities of the geometrical design parameters, the mesh, the flow variables and the error estimator to project the refinement parameters from one design to the next. The efficiency of the proposed method is checked out in two 2D optimization problems using a full potential model coupled with a boundary layer model. The first one corresponds to an internal flow problem and the second one corresponds to an external flow problem. The work presented here can be considered as a continuation of previous work (Bugeda and Oñate, Int. J. Numer. Methods Fluids, 20, 915–924 (1995)). In that paper, the sensitivity analysis corresponding to both an incompressible potential flow and a Euler flow were derived together with a strategy for the adaptive remeshing. In the present paper, the sensitivity analysis is derived for a full potential flow coupled with a boundary layer model, and a new error estimator is employed.
Abstract
This paper presents a methodology for solving shape optimization problems in the context of fluid flow problems including adaptive remeshing. The method is based on the computation [...]
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A Phenomenological Model for the Solidification of Eutectic and Hypoeutectic Alloys Including Recalescence and Undercooling
M. Chiumenti, M. Cervera, E. Salsi, A. Zonato

J. of Heat Transfer (2018). (preprint) Vol. 140 (8), pp. 12

Abstract
In this work, a novel phenomenological model is proposed to study the liquid-to-solid phase change of eutectic and hypoeutectic alloy compositions. The objective is to enhance the prediction capabilities of the solidification models based on a-priori definition of the solid fraction as a function of the temperature field. However, the use of models defined at the metallurgical level is avoided to minimize the number of material parameters required. This is of great industrial interest because, on the one hand, the classical models are not able to predict recalescence and undercooling phenomena, and, on the other hand, the complexity as well as the experimental campaign necessary to feed most of the microstructure models available in the literature make their calibration difficult and very dependent on the chemical composition and the treatment of the melt. Contrarily, the proposed model allows for an easy calibration by means of few parameters. These parameters can be easily extracted from the temperature curves recorded at the hot spot of the quick cup test, typically used in the differential thermal analysis (DTA) for the quality control of the melt just before pouring. The accuracy of the numerical results is assessed by matching the temperature curves obtained via DTA of eutectic and hypoeutectic alloys. Moreover, the model is validated in more complex casting experiments where the temperature is measured at different thermocouple locations and the metallurgical features such as grain size and nucleation density are obtained from an exhaustive micrography campaign. The remarkable agreement with the experimental evidence validates the predicting capabilities of the proposed model.
Abstract
In this work, a novel phenomenological model is proposed to study the liquid-to-solid phase change of eutectic and hypoeutectic alloy compositions. The objective is to enhance [...]
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Numerical modelling of microstructure evolution in Friction Stir Welding (FSW)"
N. Dialami, M. Cervera, M. Chiumenti

Metals (2018). (preprint) Vol. 8 (3), pp. 183

Abstract
This work studies the metallurgical and microstructural aspects of Friction Stir Welding (FSW) in terms of grain size and microhardness. The modelling is based on the combination of an apropos kinematic framework for the local simulation of FSW processes and a material particle tracing technique for tracking the material flow during the weld. The resulting grain size and microhardness values are validated with experimental observations from an identical processed sample. A Sheppard-Wright constitutive relation is adopted to describe the mechanical behavior of AZ31 Mg alloy considered in this work. The strain rate and temperature histories obtained from the numerical model are stored on the tracers. The relationship among the grain size, microhardness, strain rate, and temperature is obtained using Zener-Hollomon parameter and Hall-Petch relationship. A linear description relates the logarithm of average grain size to the logarithm of Zener-Hollomon parameter. The relationship between microhardness and average grain size stands away from the linear trend.
Abstract
This work studies the metallurgical and microstructural aspects of Friction Stir Welding (FSW) in terms of grain size and microhardness. The modelling is based on the combination [...]
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Finite element analysis and experimental validation of the thermomechanical behavior in laser solid forming of Ti-6Al-4V
X. Lu, X. Lin, M. Chiumenti, M. Cervera, J. Li, L. Ma, L. Wei, Y. Hu, W. Huang

Additive Manufacturing (2018). (preprint) Vol. 21, pp. 30-40

Abstract
A three-dimensional (3D) thermomechanical coupled model for Laser Solid Forming (LSF) of Ti-6Al-4V alloy has been calibrated through experiments of 40-layers metal deposition using different scanning strategies. The sensitivity analysis of the mechanical parameters shows that the thermal expansion coefficient as well as the elastic limit of Ti-6Al-4V have a great impact on the mechanical behavior. Using the validated model and optimal mechanical parameters, the evolution of thermo-mechanical fields in LSF has been analyzed. It has been found that the stresses and distortions develop in two stages, after the deposition of the first layer and during the cooling phase after the manufacturing of the component. The cooling phase is the responsible of 70% of the residual stresses and 60% of the total distortions. The analyses indicate that by controlling the initial substrate temperature (pre-heating phase) and the final cooling phase it is possible to mitigate both distortion and residual stresses. Hence, the influence of different pre-heating procedures on the mechanical fields has been analyzed. The results show that increasing the pre-heating temperature of the substrate is the most effective way to reduce the distortions and residual stresses in Additive Manufacturing.
Abstract
A three-dimensional (3D) thermomechanical coupled model for Laser Solid Forming (LSF) of Ti-6Al-4V alloy has been calibrated through experiments of 40-layers metal [...]
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Appraisement of planar, bending and twisting cracks in 3D with isotropic and orthotropic damage models
G. Barbat, M. Cervera, M. Chiumenti

International Journal of Fracture (2018). (preprint) Vol. 210 (1-2), pp. 45–79

Abstract
This paper discusses the modeling of cracking in quasi-brittle materials using isotropic and orthotropic damage constitutive laws. A mixed strain/displacement finite element formulation is used, taking advantage of its enhanced precision and its enforced interelemental strain continuity. On the one hand, this formulation avoids the spurious mesh dependency of the computed solution associated to standard elements and does not require the use of tracking techniques. On the other hand, it greatly alleviates the spurious stress locking associated to the use of orthotropic models on standard finite elements. The performance of several isotropic and orthotropic damage constitutive laws is assessed through an extensive comparison with analytical solutions, numerical tests and experimental evidence reported in the literature. The behavior of the different damage models in terms of crack surface, collapse mechanism and force displacement curves is investigated performing 3D analyses in several conditions including Mode I, Mixed Mode and Mode III fracture. When performing the appraisement of planar, bending and twisting cracks, the enhanced accuracy of the mixed formulation allows for a distinct assessment of the several damage models considered. Aspects related to the behavior of damage models, such as the influence of Poisson’s ratio, the shape of the damage surface and the adoption of isotropic and orthotropic models are investigated and noteworthy conclusions are drawn
Abstract
This paper discusses the modeling of cracking in quasi-brittle materials using isotropic and orthotropic damage constitutive laws. A mixed strain/displacement finite element [...]
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A novel positive/negative projection in energy norm for the damage modeling of quasi-brittle solids
J. Wu, M. Cervera

Int. J. Solids and Structures (2018). (preprint) Vol. 139–140, pp. 250–269

Abstract
The asymmetric tensile/compressive material behavior and microcracks closure-reopening (MCR) effects exhibited by quasi-brittle solids are of significant importance to the nonlinear responses of engineering structures under cyclic loading, e.g., earthquake excitations. Based on our previous work (Cervera et al., 1995; Faria et al., 1998; Wu et al., 2006) this work addresses a novel thermodynamically consistent unilateral damage model for concrete. In particular, the positive/negative projection (PNP) of the effective stress tensor and the additive bi-scalar damage constitutive relation are maintained owing to the conceptual simplicity and computational efficiency. It is found that the classical PNP widely adopted in the literature is not optimal for this damage model, since the resulting stiffness is not always of major symmetry. Consequently, a well-defined free energy potential does not exist in general cases and the model cannot be cast into the framework of thermodynamics with internal variables. Furthermore, the damage induced anisotropy cannot be captured, exhibiting excessive lateral deformationsunder uniaxial tension. To overcome the above issues, a novel PNP, variationally interpreted as the closest point projection of the effective stress in energy norm, is proposed with closed-form solution. With the novel PNP, the secant stiffness tensorof the proposed unilateral damage model always possesses major symmetry and exhibits orthotropic behavior under uniaxial tension and mixed tension/compression. The corresponding thermodynamics framework is then given, resulting in an energy release rate based rounded-Rankine type damage criterion appropriate for tensile failure in quasi-brittle solids. Several numerical examples of single-point verifications and benchmark tests are presented. It is demonstrated that the proposed model is capable of characterizing localized failureof concrete under proportional and non-proportional static loading, as well as the MCR effects under seismic cyclic loading.
Abstract
The asymmetric tensile/compressive material behavior and microcracks closure-reopening (MCR) effects exhibited by quasi-brittle solids are of significant [...]
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Cracking of quasi-brittle structures under monotonic and cyclic loadings: A d+/d− damage model with stiffness recovery in shear
M. Cervera, C. Tesei, G. Ventura

Int. J. Solids and Structures (2017). (preprint) Vol. 135, pp. 148–171

Abstract
In the present paper, a new d+/d− damage model apt for quasi-brittle materials is described and its validation is carried out considering unreinforced concrete, reinforced concrete and masonry elements. Two independent scalar damage variables, d+ and d−, in combination with the split of the reversible strain tensor into its positive and negative counterparts, are adopted in order to simulate the pronounced dissimilar response under tension and compression, typical of these materials. An energy-equivalent framework is considered for representing the orthotropy induced in the material by the degradation process, with the consequence that a thermodynamically consistent constitutive operator, positive definite, symmetric and strain-driven, is derived. In addition to the degradation parameters, the permanent strain tensor is also contemplated by the model and a modification of the exponential softening modulus is proposed in order to treat the evolution of the two causes of dissipation, damage and irreversible deformations, in a coupled way. The formulation is integrated with a multidirectional damage procedure, addressed to extend the microcrack closure-reopening (MCR) capabilities of the model to shear cyclic conditions, characterized by orthogonal (or however intersecting) sets of cracks. Maintaining unaltered the dependence of the constitutive law from two scalar indeces, d+ and d−, this approach activates or deactivates a tensile(compressive) damage value on the base of the current maximum (minimum) principal strain direction. In correspondence with damage activation (crack opening) or deactivation (crack closure), a smooth transition is introduced, in order to avoid abrupt changes in stiffness and enhance the numerical performance and robustness of the multidirectional procedure. The adequacy of the proposed constitutive model in reproducing experimental results has been proven for both monotonic and cyclic loading conditions. The two examples of application involving cyclic loads, dominated by shear, constitute a validation of the multidirectional damage approach, showing how the suitable representation of unilateral effects and permanent deformations is essential to model the observed structural response in terms of maximum resistance, evolution of stiffness degradation and dissipation capacity.
Abstract
In the present paper, a new d+/d− damage model apt for quasi-brittle materials is described and its validation is carried out considering unreinforced [...]
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Experimental validation of a FSW model with an enhanced friction model: application to a threaded cylindrical pin tool
N. Dialami, M. Cervera, M. Chiumenti, A. Segatori, W. Osikowicz

Metals (2017). (preprint) Vol. 7 (11), pp. 491

Abstract
This work adopts a fast and accurate two-stage computational strategy for the analysis of FSW (Friction stir welding) processes using threaded cylindrical pin tools. The coupled thermo-mechanical problem is equipped with an enhanced friction model to include the effect of non-uniform pressure distribution under the pin shoulder. The overall numerical strategy is successfully validated by the experimental measurements provided by the industrial partner (Sapa). The verification of the numerical model using the experimental evidence is not only accomplished in terms of temperature evolution but also in terms of torque, longitudinal, transversal and vertical forces.
Abstract
This work adopts a fast and accurate two-stage computational strategy for the analysis of FSW (Friction stir welding) processes using threaded cylindrical pin tools. The coupled [...]
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Enhanced friction model for Friction Stir Welding (FSW) analysis: Simulation and experimental validation
N. Dialami, M. Chiumenti, M. Cervera, A. Segatori, W. Osikowicz

Int. J. Mechanical Sciences (2017). (preprint) Vol. 133, pp. 555-567

Abstract
Friction is one of the main heat generation mechanisms in Friction Stir Welding (FSW). This phenomenon occurs between the pin and the workpiece as the rotating tool moves along the weld line. An accurate friction model is essential for obtaining realistic results in a FSW simulation in particular temperature, forces and torque. In this work, a modified Norton's friction law is developed. The suggested enhanced friction model aims at providing not only the realistic temperature field but also the forces and torque. This model does not exclusively relate the frictional shear stress to the sliding velocity; conversely it takes into account the effect of non-uniform pressure distribution under the shoulder, as this latter has an important role in the process of heat generation. Longitudinal, transversal and vertical forces and torque are numerically calculated. The effect of the enhanced friction model is reflected in these forces. In particular, it leads to a more realistic estimation of the transversal and longitudinal forces in comparison with the results obtained using former models. The friction model is successfully validated by the experimental measurements provided by the industrial partner (Sapa). The experimental analysis is performed for the material characterization, the calibration of the friction model and, more generally, the assessment of the overall numerical strategy proposed for the FSW simulation.
Abstract
Friction is one of the main heat generation mechanisms in Friction Stir Welding (FSW). This phenomenon occurs between the pin and the workpiece as the rotating tool moves [...]
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Numerical modelling and experimental validation in Selective Laser Melting
M. Chiumenti, E. Neiva, E. Salsi, M. Cervera, E. Badia, J. Moya, Z. Chen, C. Lee, C. Davies

Additive Manufacturing (2017). (preprint) Vol. 18, pp. 171-185

Abstract
In this work a finite-element framework for the numerical simulation of the heat transfer analysis of additive manufacturing processes by powder-bed technologies, such as Selective Laser Melting, is presented. These kind of technologies allow for a layer-by-layer metal deposition process to cost-effectively create, directly from a CAD model, complex functional parts such as turbine blades, fuel injectors, heat exchangers, medical implants, among others. The numerical model proposed accounts for different heat dissipation mechanisms through the surrounding environment and is supplemented by a finite-element activation strategy, based on the born-dead elements technique, to follow the growth of the geometry driven by the metal deposition process, in such a way that the same scanning pattern sent to the numerical control system of the AM machine is used. An experimental campaign has been carried out at the Monash Centre for Additive Manufacturing using an EOSINT-M280 machine where it was possible to fabricate different benchmark geometries, as well as to record the temperature measurements at different thermocouple locations. The experiment consisted in the simultaneous printing of two walls with a total deposition volume of 107 cm3 in 992 layers and about 33,500 s build time. A large number of numerical simulations have been carried out to calibrate the thermal FE framework in terms of the thermophysical properties of both solid and powder materials and suitable boundary conditions. Furthermore, the large size of the experiment motivated the investigation of two different model reduction strategies: exclusion of the powder-bed from the computational domain and simplified scanning strategies. All these methods are analysed in terms of accuracy, computational effort and suitable applications
Abstract
In this work a finite-element framework for the numerical simulation of the heat transfer analysis of additive manufacturing processes by powder-bed technologies, [...]
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Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy
M. Cervera, G. Barbat, M. Chiumenti

Computational Mechanics (2017). (preprint) Vol. 60 (5), pp. 767–796

Abstract
This paper discusses the finite element modeling of cracking in quasi-brittle materials. The problem is addressed via a mixed strain/displacement finite element formulation and an isotropic damage constitutive model. The proposed mixed formulation is fully general and is applied in 2D and 3D. Also, it is independent of the specific finite element discretization considered; it can be equally used with triangles/tetrahedra, quadrilaterals/hexahedra and prisms. The feasibility and accuracy of the method is assessed through extensive comparison with experimental evidence. The correlation with the experimental tests shows the capacity of the mixed formulation to reproduce the experimental crack path and the force–displacement curves with remarkable accuracy. Both 2D and 3D examples produce results consistent with the documented data. Aspects related to the discrete solution, such as convergence regarding mesh resolution and mesh bias, as well as other related to the physical model, like structural size effect and the influence of Poisson’s ratio, are also investigated. The enhanced accuracy of the computed strain field leads to accurate results in terms of crack paths, failure mechanisms and force displacement curves. Spurious mesh dependency suffered by both continuous and discontinuous irreducible formulations is avoided by the mixed FE, without the need of auxiliary tracking techniques or other computational schemes that alter the continuum mechanical problem.
Abstract
This paper discusses the finite element modeling of cracking in quasi-brittle materials. The problem is addressed via a mixed strain/displacement finite element formulation [...]
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Tracking multi-directional intersecting cracks in numerical modelling of masonry shear walls under cyclic loading
S. Salaustros, M. Cervera, L. Pelà

Meccanica (2017). (preprint) Vol. 53 (7), pp. 1757–1776

Abstract
In-plane cyclic loading of masonry walls induces a complex failure pattern composed of multiple diagonal shear cracks, as well as flexural cracks. The realistic modelling of such induced localized cracking necessitates the use of costly direct numerical simulations with detailed information on both the properties and geometry of masonry components. On the contrary, computationally efficient macro-models using standard smeared-crack approaches often result in a poor representation of fracture in the simulated material, not properly localized and biased by the finite element mesh orientation. This work proposes a possible remedy to these drawbacks of macro-models through the use of a crack-tracking algorithm. The macro-modelling approach results in an affordable computational cost, while the tracking algorithm aids the mesh-bias independent and localized representation of cracking. A novel methodology is presented that allows the simulation of intersecting and multi-directional cracks using tracking algorithms. This development extends the use of localized crack approaches using tracking algorithms to a wider field of applications exhibiting multiple, arbitrary and interacting cracking. The paper presents also a novel formulation including into an orthotropic damage model the description of irreversible deformations under shear loading. The proposed approach is calibrated through the comparison with an experimental test on a masonry shear wall against in-plane cyclic loading.
Abstract
In-plane cyclic loading of masonry walls induces a complex failure pattern composed of multiple diagonal shear cracks, as well as flexural cracks. The realistic modelling [...]
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An enhanced finite element macro-model for the realistic simulation of localized cracks in masonry structures: A large-scale application
S. Salaustros, L. Pelà, M. Cervera, P. Roca

International Journal of Architectural Heritage (2017). (preprint) Vol. 12 (3), pp. 432-447

Abstract
Finite element macro-modeling approaches are widely used for the analysis of large-scale masonry structures. Despite their efficiency, they still face two important challenges: the realistic representation of damage and a reasonable independency of the numerical results to the used discretization. In this work, the classical smeared crack approach is enhanced with a crack-tracking algorithm, originating from the analysis of localized cracking in quasi-brittle materials. The proposed algorithm is for the first time applied to a large-scale wall exhibiting multiple shear and flexural cracking. Discussion covers structural aspects, as the response of the structure under different assumptions regarding the floor rigidity, but also numerical issues, commonly overlooked in the simulation of large structures, such the mesh-dependency of the numerical results.
Abstract
Finite element macro-modeling approaches are widely used for the analysis of large-scale masonry structures. Despite their efficiency, they still face two important challenges: [...]
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Strain localization analysis of elastic-damaging frictional-cohesive materials: Analytical results and numerical verification
J. Wu, M. Cervera

Materials (2017). (preprint) Vol. 10 (4), pp. 434

Abstract
Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell’s kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for the “Simó and Ju 1987” isotropic damage model, the localization angles and the resulting cohesive model heavily depend on lateral deformations usually ignored in classical crack models for quasi-brittle solids. To remedy this inconsistency, a modified damage model is proposed. Its strain localization analysis naturally results in a consistent frictional-cohesive crack model of damage type, which can be regularized as a classical smeared crack model. The analytical results are numerically verified by the recently-proposed mixed stabilized finite element method, regarding a singly-perforated plate under uniaxial tension. Remarkably, for all of the damage models discussed in this work, the numerically-obtained localization angles agree almost exactly with the closed-form results. This agreement, on the one hand, consolidates the strain localization analysis based on Maxwell’s kinematics and, on the other hand, illustrates versatility of the mixed stabilized finite element method.
Abstract
Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of [...]
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3D numerical modelling of twisting cracks under bending and torsion of skew notched beams
L. Benedetti, M. Cervera, M. Chiumenti

Engineering Fracture Mechanics (2017). (preprint) Vol. 176, pp. 235–256

Abstract
The testing of mode III and mixed mode failure is every so often encountered in the dedicated literature of mechanical characterization of brittle and quasi-brittle materials. In this work, the application of the mixed strain displacement ε-u finite element formulation to three examples involving skew notched beams is presented. The use of this FE technology is effective in problems involving localization of strains in softening materials. The objectives of the paper are: (i) to test the mixed formulation in mode III and mixed mode failure and (ii) to present an enhancement in terms of computational time given by the kinematic compatibility between irreducible displacement-based and the mixed strain-displacement elements. Three tests of skew-notched beams are presented: firstly, a three point bending test of a PolyMethyl MethaAcrylate beam; secondly, a torsion test of a plain concrete prismatic beam with square base; finally, a torsion test of a cylindrical beam made of plain concrete as well. To describe the mechanical behavior of the material in the inelastic range, Rankine and Drucker-Prager failure criteria are used in both plasticity and isotropic continuum damage formats. The proposed mixed formulation is capable of yielding results close to the experimental ones in terms of fracture surface, peak load and global loss of carrying capability. In addition, the symmetric secant formulation and the compatibility condition between the standard irreducible method and the strain-displacement one is exploited, resulting in a significant speedup of the computational procedure.
Abstract
The testing of mode III and mixed mode failure is every so often encountered in the dedicated literature of mechanical characterization of brittle and quasi-brittle [...]
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A fast and accurate two-stage strategy to evaluate the effect of the pin tool profile on metal flow, torque and forces during friction stir welding
N. Dialami, M. Cervera, M. Chiumenti, C. Agelet de Saracibar

Int. J. Mechanical Sciences (2017). (preprint) Vol. 122, pp. 215–227

Abstract
Pin geometry is a fundamental consideration in friction stir welding (FSW). It influences the thermal behaviour, material flow and forces during the weld and reflects on the joint quality. This work studies four pin tools with circular, triflute, trivex, and triangular profiles adopting a validated model of FSW process developed by the authors. The effect of the rotating tool geometry on the flow behaviour and process outcomes is analysed. Additionally, longitudinal and transversal forces and torque are numerically calculated and compared for the different pin shapes. The study is carried out for slip and stick limiting friction cases between pin and workpiece. The main novelties of the paper are a “speed-up” two-stage simulation methodology and a piecewise linear version of the constitutive model, both of them conceived for the use in real case industrial applications, where the achievement of accuracy with affordable simulation times is of importance. The Norton-Hoff constitutive model is adopted to characterize the material behaviour during the weld. The piecewise linear version of the model developed by the authors greatly facilitates the convergence of the numerical solution ensuring both computational efficiency and accuracy. A two-stage computational procedure is applied. In the first stage, a forced transient is carried out; in the second one, the magnitudes of interest are computed. The study shows that the proposed modelling approach can be used to predict and interpret the FSW behaviour for a specific pin geometry. Moreover, the reduction of the simulation time using the two-stage strategy can be up to 90%, compared to a standard single stage strategy.
Abstract
Pin geometry is a fundamental consideration in friction stir welding (FSW). It influences the thermal behaviour, material flow and forces during the weld and [...]
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An energy-equivalent d+/d− damage model with enhanced microcrack closure-reopening capabilities for cohesive-frictional materials
M. Cervera, C. Tesei

Materials (2017). (preprint) Vol. 10 (4), pp. 433

Abstract
In this paper, an energy-equivalent orthotropic d+/d− damage model for cohesive-frictional materials is formulated. Two essential mechanical features are addressed, the damage-induced anisotropy and the microcrack closure-reopening (MCR) effects, in order to provide an enhancement of the original d+/d−model proposed by Faria et al. 1998, while keeping its high algorithmic efficiency unaltered. First, in order to ensure the symmetry and positive definiteness of the secant operator, the new formulation is developed in an energy-equivalence framework. This proves thermodynamic consistency and allows one to describe a fundamental feature of the orthotropic damage models, i.e., the reduction of the Poisson’s ratio throughout the damage process. Secondly, a “multidirectional” damage procedure is presented to extend the MCR capabilities of the original model. The fundamental aspects of this approach, devised for generic cyclic conditions, lie in maintaining only two scalar damage variables in the constitutive law, while preserving memory of the degradation directionality. The enhanced unilateral capabilities are explored with reference to the problem of a panel subjected to in-plane cyclic shear, with or without vertical pre-compression; depending on the ratio between shear and pre-compression, an absent, a partial or a complete stiffness recovery is simulated with the new multidirectional procedure.
Abstract
In this paper, an energy-equivalent orthotropic d+/d− damage model for cohesive-frictional materials is formulated. Two essential mechanical features are addressed, [...]
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Finite element modelling of internal and multiple localized cracks
S. Salaustros, L. Pelà, M. Cervera, P. Roca

Computational Mechanics (2017). (preprint) Vol. 59 (2), pp. 299–316

Abstract
Tracking algorithms constitute an efficient numerical technique for modelling fracture in quasi-brittle materials. They succeed in representing localized cracks in the numerical model without mesh-induced directional bias. Currently available tracking algorithms have an important limitation: cracking originates either from the boundary of the discretized domain or from predefined “crack-root” elements and then propagates along one orientation. This paper aims to circumvent this drawback by proposing a novel tracking algorithm that can simulate cracking starting at any point of the mesh and propagating along one or two orientations. This enhancement allows the simulation of structural case-studies experiencing multiple cracking. The proposed approach is validated through the simulation of a benchmark example and an experimentally tested structural frame under in-plane loading. Mesh-bias independency of the numerical solution, computational cost and predicted collapse mechanisms with and without the tracking algorithm are discussed.
Abstract
Tracking algorithms constitute an efficient numerical technique for modelling fracture in quasi-brittle materials. They succeed in representing localized cracks in the numerical [...]
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Numerical simulation and experimental calibration of Additive Manufacturing by blown powder technology. Part I: thermal analysis
M. Chiumenti, X. Lin, M. Cervera, W. Lei, Y. Zheng, W. Huang

Rapid Prototyping Journal (2017). (preprint) Vol. 23 (2), pp. 448-463

Abstract
In this work the numerical simulation of Additive Manufacturing (AM) processes is addressed. The numerical results are compared with the experimental campaign carried out at SKLSP laboratories, where a Laser Solid Forming (LSF) machine, also referred to as Laser Engineered Net Shaping (LENS), is used to fabricate metal parts directly from CAD models. Ti-6Al-4V metal powder is injected into the molten pool created by a focused, high-energy laser beam and a layer of added material is sinterized according to the laser scanning pattern speciÖed by the user. The objectives of this paper are twofold: Örstly, this work is intended to calibrate the software for the numerical simulation of the AM process, to achieve high accuracy. Secondly, the sensitivity of the numerical model to the process parameters and modelling data is analysed. The numerical model adopts an apropos FE activation technology, which reproduces the same scanning pattern set for the numerical control system of the AM machine. This consists of a complex sequence of polylines, used to deÖne the contour of the component, and hatches patterns to Öll the inner section. The full sequence is given through the Common Layer Interface (CLI) format, a standard format for di§erent manufacturing processes such as Rapid Prototyping (RP), Shape Metal Deposition (SMD) or machining processes, among others. The result is a layer-by-layer metal deposition which can be used to build-up complex structures for components such as turbine blades, aircraft sti§eners, cooling systems, or medical implants, among others.
Abstract
In this work the numerical simulation of Additive Manufacturing (AM) processes is addressed. The numerical results are compared with the experimental campaign carried out [...]
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Local–global strategy for the prediction of residual stresses in FSW processes

N. Dialami, M. Cervera, M. Chiumenti, C. Agelet de Saracibar

International Journal of Advanced Manufacturing Technology (2017). (preprint) Vol. 88 (9), pp. 3099–3111

Abstract
This work describes the local–global strategy proposed for the computation of residual stresses in friction stir welding (FSW) processes. A coupling strategy between the analysis of the process zone nearby the pin tool (local level analysis) and the simulation carried out for the entire structure to be welded (global level analysis) is implemented to accurately predict the temperature histories and, thereby, the residual stresses in FSW. As a first step, the local problem solves the material stirring as well as the heat generation induced by the pin and shoulder rotation at the heat affected zone. The Arbitrary Lagrangian Eulerian (ALE) formulation is adopted to deal with the rotation of complex pin shapes. A thermo-rigid-viscoplastic constitutive law is employed to characterize the viscous flow of the material, driven by the high-strain rates induced by the FSW process. A mixed temperature–velocity–pressure finite element technology is use