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http://hdl.handle.net/10985/10242
Parametric solution of the Rayleigh-Benard convection model by using the PGD Application to nanofluids
AGHIGHI, Mohammad Saeid; AMMAR, Amine; METIVIER, Christel; CHINESTA, Francisco
Purpose – The purpose of this paper is to focus on the advanced solution of the parametric non-linear model related to the Rayleigh-Benard laminar flow involved in the modeling of natural thermal convection. This flow is fully determined by the dimensionless Prandtl and Rayleigh numbers. Thus, if one could precompute (off-line) the model solution for any possible choice of these two parameters the analysis of many possible scenarios could be performed on-line and in real time. Design/methodology/approach – In this paper both parameters are introduced as model extracoordinates, and then the resulting multidimensional problem solved thanks to the space-parameters separated representation involved in the proper generalized decomposition (PGD) that allows circumventing the curse of dimensionality. Thus the parametric solution will be available fast and easily. Findings – Such parametric solution could be viewed as a sort of abacus, but despite its inherent interest such calculation is at present unaffordable for nowadays computing availabilities because one must solve too many problems and of course store all the solutions related to each choice of both parameters. Originality/value – Parametric solution of coupled models by using the PGD. Model reduction of complex coupled flow models. Analysis of Rayleigh-Bernard flows involving nanofluids.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/102422015-01-01T00:00:00ZAGHIGHI, Mohammad SaeidAMMAR, AmineMETIVIER, ChristelCHINESTA, FranciscoPurpose – The purpose of this paper is to focus on the advanced solution of the parametric non-linear model related to the Rayleigh-Benard laminar flow involved in the modeling of natural thermal convection. This flow is fully determined by the dimensionless Prandtl and Rayleigh numbers. Thus, if one could precompute (off-line) the model solution for any possible choice of these two parameters the analysis of many possible scenarios could be performed on-line and in real time. Design/methodology/approach – In this paper both parameters are introduced as model extracoordinates, and then the resulting multidimensional problem solved thanks to the space-parameters separated representation involved in the proper generalized decomposition (PGD) that allows circumventing the curse of dimensionality. Thus the parametric solution will be available fast and easily. Findings – Such parametric solution could be viewed as a sort of abacus, but despite its inherent interest such calculation is at present unaffordable for nowadays computing availabilities because one must solve too many problems and of course store all the solutions related to each choice of both parameters. Originality/value – Parametric solution of coupled models by using the PGD. Model reduction of complex coupled flow models. Analysis of Rayleigh-Bernard flows involving nanofluids.An overview of the proper generalized decomposition with applications in computational rheology
http://hdl.handle.net/10985/8473
An overview of the proper generalized decomposition with applications in computational rheology
CHINESTA, Francisco; AMMAR, Amine; LEYGUE, Adrien; KEUNINGS, Roland
We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10985/84732011-01-01T00:00:00ZCHINESTA, FranciscoAMMAR, AmineLEYGUE, AdrienKEUNINGS, RolandWe review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates.A mesoscopic rheological model of moderately concentrated colloids
http://hdl.handle.net/10985/9962
A mesoscopic rheological model of moderately concentrated colloids
GRMELA, Miroslav; AMMAR, Amine; CHINESTA, Francisco; MAITREJEAN, Guillaume
We extend the Maffettone–Minale model by including non-elliptical shapes of dispersed particles, a new family of internal forces controlling particle deformations, and particle–particle interactions. The last extension is made by transposing the way the chain-chain interactions are mathematically expressed in the reptation theory to suspensions. The particle–particle interactions are regarded as a confinement to cages formed by surrounding particles and by introducing a new dissipative motion (an analog of the reptation motion) inside the cages. Nonlinear responses to imposed shear and elongational flows are found to be in qualitative agreement with available experimental data.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/99622014-01-01T00:00:00ZGRMELA, MiroslavAMMAR, AmineCHINESTA, FranciscoMAITREJEAN, GuillaumeWe extend the Maffettone–Minale model by including non-elliptical shapes of dispersed particles, a new family of internal forces controlling particle deformations, and particle–particle interactions. The last extension is made by transposing the way the chain-chain interactions are mathematically expressed in the reptation theory to suspensions. The particle–particle interactions are regarded as a confinement to cages formed by surrounding particles and by introducing a new dissipative motion (an analog of the reptation motion) inside the cages. Nonlinear responses to imposed shear and elongational flows are found to be in qualitative agreement with available experimental data.Review on the Brownian Dynamics Simulation of Bead-Rod-Spring Models Encountered in Computational Rheology
http://hdl.handle.net/10985/9991
Review on the Brownian Dynamics Simulation of Bead-Rod-Spring Models Encountered in Computational Rheology
CRUZ, Camilo; CHINESTA, Francisco; REGNIER, Gilles
Kinetic theory is a mathematical framework intended to relate directly the most relevant characteristics of the molecular structure to the rheological behavior of the bulk system. In other words, kinetic theory is a micro-to-macro approach for solving the flow of complex fluids that circumvents the use of closure relations and offers a better physical description of the phenomena involved in the flow processes. Cornerstone models in kinetic theory employ beads, rods and springs for mimicking the molecular structure of the complex fluid. The generalized bead-rod-spring chain includes the most basic models in kinetic theory: the freely jointed bead-spring chain and the freely-jointed bead-rod chain. Configuration of simple coarse-grained models can be represented by an equivalent Fokker-Planck (FP) diffusion equation, which describes the evolution of the configuration distribution function in the physical and configurational spaces. FP equation can be a complex mathematical object, given its multidimensionality, and solving it explicitly can become a difficult task. Even more, in some cases, obtaining an equivalent FP equation is not possible given the complexity of the coarse-grained molecular model. Brownian dynamics can be employed as an alternative extensive numerical method for approaching the configuration distribution function of a given kinetic-theory model that avoid obtaining and/or resolving explicitly an equivalent FP equation. The validity of this discrete approach is based on the mathematical equivalence between a continuous diffusion equation and a stochastic differential equation as demonstrated by Itô in the 1940s. This paper presents a review of the fundamental issues in the BD simulation of the linear viscoelastic behavior of bead-rod-spring coarse grained models in dilute solution. In the first part of this work, the BD numerical technique is introduced. An overview of the mathematical framework of the BD and a review of the scope of applications are presented. Subsequently, the links between the rheology of complex fluids, the kinetic theory and the BD technique are established at the light of the stochastic nature of the bead-rod-spring models. Finally, the pertinence of the present state-of-the-art review is explained in terms of the increasing interest for the stochastic micro-to-macro approaches for solving complex fluids problems. In the second part of this paper, a detailed description of the BD algorithm used for simulating a small-amplitude oscillatory deformation test is given. Dynamic properties are employed throughout this work to characterise the linear viscoelastic behavior of bead-rod-spring models in dilute solution. In the third and fourth part of this article, an extensive discussion about the main issues of a BD simulation in linear viscoelasticity of diluted suspensions is tackled at the light of the classical multi-bead-spring chain model and the multi-bead-rod chain model, respectively. Kinematic formulations, integration schemes and expressions to calculate the stress tensor are revised for several classical models: Rouse and Zimm theories in the case of multi-bead-spring chains, and Kramers chain and semi-flexible filaments in the case of multi-bead-rod chains. The implemented BD technique is, on the one hand, validated in front of the analytical or exact numerical solutions known of the equivalent FP equations for those classic kinetic theory models; and, on the other hand, is control-set thanks to the analysis of the main numerical issues involved in a BD simulation. Finally, the review paper is closed by some concluding remarks.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/99912012-01-01T00:00:00ZCRUZ, CamiloCHINESTA, FranciscoREGNIER, GillesKinetic theory is a mathematical framework intended to relate directly the most relevant characteristics of the molecular structure to the rheological behavior of the bulk system. In other words, kinetic theory is a micro-to-macro approach for solving the flow of complex fluids that circumvents the use of closure relations and offers a better physical description of the phenomena involved in the flow processes. Cornerstone models in kinetic theory employ beads, rods and springs for mimicking the molecular structure of the complex fluid. The generalized bead-rod-spring chain includes the most basic models in kinetic theory: the freely jointed bead-spring chain and the freely-jointed bead-rod chain. Configuration of simple coarse-grained models can be represented by an equivalent Fokker-Planck (FP) diffusion equation, which describes the evolution of the configuration distribution function in the physical and configurational spaces. FP equation can be a complex mathematical object, given its multidimensionality, and solving it explicitly can become a difficult task. Even more, in some cases, obtaining an equivalent FP equation is not possible given the complexity of the coarse-grained molecular model. Brownian dynamics can be employed as an alternative extensive numerical method for approaching the configuration distribution function of a given kinetic-theory model that avoid obtaining and/or resolving explicitly an equivalent FP equation. The validity of this discrete approach is based on the mathematical equivalence between a continuous diffusion equation and a stochastic differential equation as demonstrated by Itô in the 1940s. This paper presents a review of the fundamental issues in the BD simulation of the linear viscoelastic behavior of bead-rod-spring coarse grained models in dilute solution. In the first part of this work, the BD numerical technique is introduced. An overview of the mathematical framework of the BD and a review of the scope of applications are presented. Subsequently, the links between the rheology of complex fluids, the kinetic theory and the BD technique are established at the light of the stochastic nature of the bead-rod-spring models. Finally, the pertinence of the present state-of-the-art review is explained in terms of the increasing interest for the stochastic micro-to-macro approaches for solving complex fluids problems. In the second part of this paper, a detailed description of the BD algorithm used for simulating a small-amplitude oscillatory deformation test is given. Dynamic properties are employed throughout this work to characterise the linear viscoelastic behavior of bead-rod-spring models in dilute solution. In the third and fourth part of this article, an extensive discussion about the main issues of a BD simulation in linear viscoelasticity of diluted suspensions is tackled at the light of the classical multi-bead-spring chain model and the multi-bead-rod chain model, respectively. Kinematic formulations, integration schemes and expressions to calculate the stress tensor are revised for several classical models: Rouse and Zimm theories in the case of multi-bead-spring chains, and Kramers chain and semi-flexible filaments in the case of multi-bead-rod chains. The implemented BD technique is, on the one hand, validated in front of the analytical or exact numerical solutions known of the equivalent FP equations for those classic kinetic theory models; and, on the other hand, is control-set thanks to the analysis of the main numerical issues involved in a BD simulation. Finally, the review paper is closed by some concluding remarks.On the Model Order Reduction of Confined Plasticity
http://hdl.handle.net/10985/10754
On the Model Order Reduction of Confined Plasticity
NASRI, Mohamed Aziz; AMMAR, Amine; CHINESTA, Francisco; ROBERT, Camille; EL AREM, Saber; MOREL, Franck
Forming processes usually involve irreversible plastic transformations. The calculation in that case becomes cumbersome when large parts and processes are considered. Recently Model Order Reduction techniques opened new perspectives for an accurate and fast simulation of mechanical systems. In some processes, plastic deformations remain very localized, for example in the immediate neighborhood of the surface. In that case, the in-plane characteristic dimension is several orders of magnitude higher than the one related to the deepness in which plasticity localizes. In those situations the use of standard mesh-based 3D discretization is challenging because the extremely different characteristic dimensions that to capture all the information requires the use of millions of nodes.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/107542016-01-01T00:00:00ZNASRI, Mohamed AzizAMMAR, AmineCHINESTA, FranciscoROBERT, CamilleEL AREM, SaberMOREL, FranckForming processes usually involve irreversible plastic transformations. The calculation in that case becomes cumbersome when large parts and processes are considered. Recently Model Order Reduction techniques opened new perspectives for an accurate and fast simulation of mechanical systems. In some processes, plastic deformations remain very localized, for example in the immediate neighborhood of the surface. In that case, the in-plane characteristic dimension is several orders of magnitude higher than the one related to the deepness in which plasticity localizes. In those situations the use of standard mesh-based 3D discretization is challenging because the extremely different characteristic dimensions that to capture all the information requires the use of millions of nodes.Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions
http://hdl.handle.net/10985/8467
Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions
AMMAR, Amine; CUETO, Elias; CHINESTA, Francisco
The numerical solution of the chemical master equation (CME) governing gene regulatory networks and cell signaling processes remains a challenging task owing to its complexity, exponentially growing with the number of species involved. Although most of the existing techniques rely on the use of Monte Carlo-like techniques, we present here a new technique based on the approximation of the unknown variable (the probability of having a particular chemical state) in terms of a finite sum of separable functions. In this framework, the complexity of the CME grows only linearly with the number of state space dimensions. This technique generalizes the so-called Hartree approximation, by using terms as needed in the finite sums decomposition for ensuring convergence. But noteworthy, the ease of the approximation allows for an easy treatment of unknown parameters (as is frequently the case when modeling gene regulatory networks, for instance). These unknown parameters can be considered as new space dimensions. In this way, the proposed method provides solutions for any value of the unknown parameters (within some interval of arbitrary size) in one execution of the program.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/84672012-01-01T00:00:00ZAMMAR, AmineCUETO, EliasCHINESTA, FranciscoThe numerical solution of the chemical master equation (CME) governing gene regulatory networks and cell signaling processes remains a challenging task owing to its complexity, exponentially growing with the number of species involved. Although most of the existing techniques rely on the use of Monte Carlo-like techniques, we present here a new technique based on the approximation of the unknown variable (the probability of having a particular chemical state) in terms of a finite sum of separable functions. In this framework, the complexity of the CME grows only linearly with the number of state space dimensions. This technique generalizes the so-called Hartree approximation, by using terms as needed in the finite sums decomposition for ensuring convergence. But noteworthy, the ease of the approximation allows for an easy treatment of unknown parameters (as is frequently the case when modeling gene regulatory networks, for instance). These unknown parameters can be considered as new space dimensions. In this way, the proposed method provides solutions for any value of the unknown parameters (within some interval of arbitrary size) in one execution of the program.On the solution of the multidimensional Langer’s equation using the proper generalized decomposition method for modeling phase transitions
http://hdl.handle.net/10985/8479
On the solution of the multidimensional Langer’s equation using the proper generalized decomposition method for modeling phase transitions
LAMARI, Hajer; AMMAR, Amine; LEYGUE, Adrien; CHINESTA, Francisco
The dynamics of phase transition in a binary mixture occurring during a quench is studied taking into account composition fluctuations by solving Langer’s equation in a domain composed of a certain number of micro-domains. The resulting Langer’s equation governing the evolution of the distribution function becomes multidimensional. Circumventing the curse of dimensionality the proper generalized decomposition is applied. The influence of the interaction parameter in the vicinity of the critical point is analyzed. First we address the case of a system composed of a single micro-domain in which phase transition occurs by a simple symmetry change. Next, we consider a system composed of two micro-domains in which phase transition occurs by phase separation, with special emphasis on the effect of the Landau free energy non-local term. Finally, some systems consisting of many micro-domains are considered.
http://dx.doi.org/10.1088/0965-0393/20/1/015007
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/84792012-01-01T00:00:00ZLAMARI, HajerAMMAR, AmineLEYGUE, AdrienCHINESTA, FranciscoThe dynamics of phase transition in a binary mixture occurring during a quench is studied taking into account composition fluctuations by solving Langer’s equation in a domain composed of a certain number of micro-domains. The resulting Langer’s equation governing the evolution of the distribution function becomes multidimensional. Circumventing the curse of dimensionality the proper generalized decomposition is applied. The influence of the interaction parameter in the vicinity of the critical point is analyzed. First we address the case of a system composed of a single micro-domain in which phase transition occurs by a simple symmetry change. Next, we consider a system composed of two micro-domains in which phase transition occurs by phase separation, with special emphasis on the effect of the Landau free energy non-local term. Finally, some systems consisting of many micro-domains are considered.On the multi‑scale description of electrical conducting suspensions involving perfectly dispersed rods
http://hdl.handle.net/10985/10253
On the multi‑scale description of electrical conducting suspensions involving perfectly dispersed rods
PEREZ, Marta; ABISSET-CHAVANNE, Emmanuelle; BARASINSKI, Anais; CHINESTA, Francisco; AMMAR, Amine; KEUNINGS, Roland
Nanocomposites allow for a significant enhancement of functional properties, in particular electrical conduction. In order to optimize materials and parts, predictive models are required to evaluate particle distribution and orientation. Both are key parameters in order to evaluate percolation and the resulting electrical networks. Many forming processes involve flowing suspensions for which the final particle orientation could be controlled by means of the flow and the electric field. In view of the multiscale character of the problem, detailed descriptions are defined at the microscopic scale and then coarsened to be applied efficiently in process simulation at the macroscopic scale. The first part of this work revisits the different modeling approaches throughout the different description scales. Then, modeling of particle contacts is addressed as they determine the final functional properties, in particular electrical conduction. Different descriptors of rod contacts are proposed and analyzed. Numerical results are discussed, in particular to evaluate the impact of closure approximations needed to derive a macroscopic description.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/102532015-01-01T00:00:00ZPEREZ, MartaABISSET-CHAVANNE, EmmanuelleBARASINSKI, AnaisCHINESTA, FranciscoAMMAR, AmineKEUNINGS, RolandNanocomposites allow for a significant enhancement of functional properties, in particular electrical conduction. In order to optimize materials and parts, predictive models are required to evaluate particle distribution and orientation. Both are key parameters in order to evaluate percolation and the resulting electrical networks. Many forming processes involve flowing suspensions for which the final particle orientation could be controlled by means of the flow and the electric field. In view of the multiscale character of the problem, detailed descriptions are defined at the microscopic scale and then coarsened to be applied efficiently in process simulation at the macroscopic scale. The first part of this work revisits the different modeling approaches throughout the different description scales. Then, modeling of particle contacts is addressed as they determine the final functional properties, in particular electrical conduction. Different descriptors of rod contacts are proposed and analyzed. Numerical results are discussed, in particular to evaluate the impact of closure approximations needed to derive a macroscopic description.Some Incipient Techniques For Improving Efficiency in Computational Mechanics
http://hdl.handle.net/10985/6477
Some Incipient Techniques For Improving Efficiency in Computational Mechanics
AMMAR, Amine; CHINESTA, Francisco
This contribution presents a review of different techniques available for alleviating simulation cost in computational mechanics. The first one is based on a separated representation of the unknown fields; the second one uses a model reduction based on the Karhunen-Loève decomposition within an adaptive scheme, and the last one is a mixed technique specially adapted for reducing models involving local singularities. These techniques can be applied in a large variety of models.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10985/64772008-01-01T00:00:00ZAMMAR, AmineCHINESTA, FranciscoThis contribution presents a review of different techniques available for alleviating simulation cost in computational mechanics. The first one is based on a separated representation of the unknown fields; the second one uses a model reduction based on the Karhunen-Loève decomposition within an adaptive scheme, and the last one is a mixed technique specially adapted for reducing models involving local singularities. These techniques can be applied in a large variety of models.On the space-time separated representation of integral linear viscoelastic models
http://hdl.handle.net/10985/9958
On the space-time separated representation of integral linear viscoelastic models; Représentation séparée espace-temps pour des comportements viscoélastiques linaires intégraux
AMMAR, Amine; ZGHAL, Ali; MOREL, Franck; CHINESTA, Francisco
The analysis of materials mechanical behavior involves many computational challenges. In this work, we are addressing the transient simulation of the mechanical behavior when the time of interest is much larger than the characteristic time of the mechanical response. This situation is encountered in many applications, as for example in the simulation of materials aging, or in structural analysis when small-amplitude oscillatory loads are applied during a long period, as it occurs for example when characterizing viscoelastic behaviors by calculating the complex modulus or when addressing fatigue simulations. Moreover, in the case of viscoelastic behaviors, the constitutive equation is many times expressed in an integral form avoiding the necessity of using internal variables, fact that results in an integro-differential model. In order to efficiently simulate such a model, we explore in this work the use of a space-time separated representation.; L'analyse du comportement mécanique des matériaux entraîne de nombreuses difficultés du point de vue numérique. Dans ce travail, nous allons nous focaliser sur l'une d'entre elles, celle associée à la simulation transitoire du comportement mécanique quand l'intervalle temporel d'intérêt est substantiellement plus long que le temps caractéristique associé à la réponse mécanique. Cette situation est fréquemment retrouvée dans la caractérisation rhéologique des matériaux viscoélastiques (pour la détermination du module complexe) ainsi que quand on s'attaque à la simulation de la fatigue. De plus, dans le cas des matriaux viscoélastiques, le comportement est généralement décrit par une loi de comportement intégrale qui évite le besoin d'utiliser des variables internes, donnant lieu à un modèle mécanique integro-différentiel. Pour une résolution efficace, nous analysons ici l'utilisation d'une représentation séparée en espace-temps.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/99582015-01-01T00:00:00ZAMMAR, AmineZGHAL, AliMOREL, FranckCHINESTA, FranciscoThe analysis of materials mechanical behavior involves many computational challenges. In this work, we are addressing the transient simulation of the mechanical behavior when the time of interest is much larger than the characteristic time of the mechanical response. This situation is encountered in many applications, as for example in the simulation of materials aging, or in structural analysis when small-amplitude oscillatory loads are applied during a long period, as it occurs for example when characterizing viscoelastic behaviors by calculating the complex modulus or when addressing fatigue simulations. Moreover, in the case of viscoelastic behaviors, the constitutive equation is many times expressed in an integral form avoiding the necessity of using internal variables, fact that results in an integro-differential model. In order to efficiently simulate such a model, we explore in this work the use of a space-time separated representation.
L'analyse du comportement mécanique des matériaux entraîne de nombreuses difficultés du point de vue numérique. Dans ce travail, nous allons nous focaliser sur l'une d'entre elles, celle associée à la simulation transitoire du comportement mécanique quand l'intervalle temporel d'intérêt est substantiellement plus long que le temps caractéristique associé à la réponse mécanique. Cette situation est fréquemment retrouvée dans la caractérisation rhéologique des matériaux viscoélastiques (pour la détermination du module complexe) ainsi que quand on s'attaque à la simulation de la fatigue. De plus, dans le cas des matriaux viscoélastiques, le comportement est généralement décrit par une loi de comportement intégrale qui évite le besoin d'utiliser des variables internes, donnant lieu à un modèle mécanique integro-différentiel. Pour une résolution efficace, nous analysons ici l'utilisation d'une représentation séparée en espace-temps.