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Implementation of the algorithm in a nonlinear elasticity software
The direct approach to numerical homogenization presented here can be used in a nonlinear elasticity solver by using the right hand sides of (5.76) and (5.77) as derivatives for the stress tensor and the stiffness matrix in (5.70). This method has the important advantage not to modify the structure of the existing solver. This method has been implemented within a classical finite element code (Modulef, INRIA, see [177] and [137]). The call of an analytical formula giving the stress tensor and the stiffness matrix at each Gauss point has been replaced by a subroutine solving itself a nonlinear elasticity cell problem (5.20) and providing the main program with (5.76) and (5.77). The global structure of the code remains therefore unchanged. Any sophisticated technique already used in the code directly adapts without modification: mixed finite elements, augmented Lagrangians, arc-length continuation and parallelization (see [177] and [125]). Numerical tests are reported on in the last section.
The major part of the computational cost comes from the computation of the homogenized constitutive law, as opposed to a classical nonlinear elasticity problem for which this is a simple evaluation of an analytical formula. For the computation of this homogenized constitutive relation itself, the main cost comes from solving the cell-problem (5.20). The resolution of linear systems is performed by direct inversions, such as Cholesky factorization, because of the large condition number and the lack of efficient preconditionners for nonlinear elasticity problems. Once the cell- problem is solved, the computation of (5.76) and (5.77) is obtained by solving a linear system with nine different right hand sides. This linear system is indeed the same as in the last iteration of the Newton algorithm solving the cell-problem, it is therefore already factorized.
Comparison of the two methods in the periodic setting
The analysis of the MsFEM requires a corrector result. In order to compare the two methods a monotone operator is considered in the setting of periodic homogenization of Section 5.2.4. The notation of Theorems 30 and 31 is used.
We suppose that the macroscopic mesh perfectly fits to the underlying periodic structure so that there is no mismatch (triangular mesh with equilateral triangles and periodic structure with equilateral triangle in 2D e.g.). This technical requirement allows to use a single periodic cell in order to compute the multiscale map (i.d. basis functions in the linear case), as pointed out in [72]. In this case, there is no resonance error due to the boundary conditions.
Setting of the problem and statement of the main results
For the numerical homogenization of elliptic problems, three major issues can be identified: the design of a numerical method, the convergence of the method and possibly the existence of a corrector, and the derivation of error estimates. The present paper focuses on the second issue in an abstract way which allows us to cover several methods proposed in the literature (multi- scale finite element method (MsFEM) and heterogeneous multiscale method (HMM)) in new and important settings. Depending on the applications, two types of assumptions are usually made: a space assumption (periodicity, stationary process, . . .) and a structure assumption (scalar- or vector-valued, linearity, monotonicity, . . .). One cannot expect the three questions to be answered positively with any assumptions. We address the second issue in the wide class of convex and quasiconvex energies, and the third issue in the more restricted class of strictly convex energies. Error estimates are made explicit in the periodic case. Our main achievements are a numerical corrector result in the convex case without space assumption, a complete error analysis in the pe- riodic and strictly convex case, and a convergence result in a general vector-valued case (nonlinear elasticity).
For the sake of simplicity and in order to relate the present work to existing approaches, we first consider a scalar elliptic PDE that is the Euler–Lagrange equation of a problem of minimization of an energy. The energy density is assumed to be convex and to vary at a scale small with respect to the size of the domain, which makes direct numerical simulations impossible to perform in practice. Our purpose is to introduce an averaged energy density which does not vary as much as the original energy density and prove that the associated minimization problem is a correct approximation of the original minimization problem in the sense of Γ-convergence. Although strict convexity ensures the existence and uniqueness of the solution to the Euler Lagrange equation and allows us to use a more direct approach (e.g., G-convergence), our arguments are based on variational principles. Most of the existing works (see, e.g., [72], [74], and [69]) actually treat the PDE. Our approach allows us to deal with the case of nonlinear elasticity, as will be seen. The first section is dedicated to the introduction of the averaged energy density in the framework of the homogenization theory for convex energies. Theorem 40 states the Γ- convergence of the averaged energy to the homogenized energy and Theorem 41 introduces a new general corrector result, which describes the fine scale features of the solution of the original minimization problem without assumption on the heterogeneities. This corrector is called a numerical corrector in reference to [74], where it was first derived in the stochastic and stationary case. In Propositions 5 and 6, we also introduce some error estimates. Section 6.2 is dedicated to the proofs of the main results.
Table of contents :
Partie I Introduction
1 Présentation des problématiques et des modèles
1.1 Motivation (in English below)
1.2 Quelques problématiques de la modélisation multiéchelle (in English below)
1.2.1 Dérivation mathématique de modèles effectifs
1.2.2 Couplage discret-continu
1.2.3 Couplage continu-continu
1.2.4 Objectifs de la thèse
1.3 Modélisation en mécanique des milieux continus
1.3.1 Energie, déformation et contraintes
1.3.2 Equations de l’élastostatique et de l’élastodynamique tridimensionnelle
2 Analyse mathématique et numérique des équations de l’élastostatique
2.1 Analyse mathématique
2.1.1 Elasticité linéaire
2.1.2 Application du théorème des fonctions implicites
2.1.3 Méthode directe du calcul des variations
2.1.4 Quasiconvexité et semi-continuité inférieure des fonctionnelles intégrales
2.1.5 Polyconvexité et théorie d’existence en élasticité non linéaire
2.1.6 Γ-convergence, application aux fonctionnelles intégrales et à la dérivation de modèles
2.2 Méthodes numériques en élasticité
2.2.1 Résultat d’approximation et estimation d’erreur
2.2.2 Méthode des éléments finis en élasticité
2.2.3 Méthodes de décomposition de domaine
2.2.4 Discrétisation temporelle pour l’élastodynamique
3 Homogénéisation – approches théoriques et numériques
3.1 Théorie de l’homogénéisation
3.1.1 Développement formel à échelles multiples
3.1.2 Homogénéisation périodique et fonctions oscillantes de Tartar
3.1.3 Homogénéisation périodique par convergence à deux échelles
3.1.4 Homogénéisation des opérateurs elliptiques linéaires par H-convergence
3.1.5 Homogénéisation des intégrales multiples par Γ-convergence
3.1.6 Principe de localisation et problèmes de G-fermeture
3.2 Méthodes numériques pour l’homogénéisation et homogénéisation numérique
3.2.1 Eléments finis mutiéchelles dans le cas linéaire
3.2.2 Approximation de l’opérateur homogénéisé
4 Contributions de la thèse (in English below)
4.1 Méthodes numériques en homogénéisation
4.1.1 Homogénéisation périodique en élasticité non linéaire
4.1.2 Cadre général pour l’analyse des méthodes d’homogénéisation numérique
4.2 Modélisation multiéchelle
4.2.1 Sur un problème de G-fermeture pour un passage discret-continu
4.2.2 Passage discret-continu pour des énergies d’interaction de spin
4.2.3 Dérivation variationnelle d’une énergie caoutchoutique à partir d’un modèle discret
4.2.4 Application aux problématiques de la mécanique
4.3 Méthodes partitionnées en interaction fluide-structure
Part II Numerical homogenization of elliptic equations
5 A direct approach to numerical homogenization in finite elasticity
5.1 Physical motivation
5.2 A quick review of periodic homogenization theory
5.2.1 Convexity and minimization problems
5.2.2 Basic homogenization result
5.2.3 Homogenization for connected media
5.2.4 Homogenization of elliptic operators in divergence form
5.2.5 Some open issues
5.3 Approximation result for the standard homogenization problem
5.3.1 Approximation theory for standard energy densities
5.3.2 Approximation result for a homogenized energy density
5.3.3 Error estimates in the convex case
5.4 Numerical method
5.4.1 Presentation
5.4.2 Computation of the stress tensor and the stiffness matrix
5.4.3 Implementation of the algorithm in a nonlinear elasticity software
5.5 Alternative method: multiscale finite elements (MsFEM)
5.5.1 Description of the method
5.5.2 Comparison of the two methods in the periodic setting
5.6 Numerical tests
5.6.1 The convex case
5.6.2 Buckling of the cell-problem in the standard case
5.6.3 Shear band instabilities
5.6.4 Tests on a wider class of energies
6 Analytical framework for numerical homogenization
6.1 Setting of the problem and statement of the main results
6.1.1 Homogenization of convex energy densities
6.1.2 Main results
6.1.3 Extension to the quasiconvex case
6.2 Proof of the main results
6.2.1 The one-dimensional linear case
6.2.2 Proof of Theorem
6.2.3 Proof of Theorem
6.2.4 Proof of Theorem
6.2.5 Proof of Proposition
6.2.6 Proof of Proposition
6.3 Relation to some existing numerical approaches
6.3.1 HMM
6.3.2 MsFEM
6.3.3 Summary of some numerical analysis results
6.4 Conclusion
6.5 Appendix
6.5.1 Γ-limit and boundary conditions
6.5.2 Proof of Lemma 6.7
6.5.3 Proof of Lemma 6.12
6.5.4 Proof of Lemma 6.13
6.5.5 Sketch of proof of Proposition
7 Numerical homogenization with oversampling
7.1 Introduction
7.2 Numerical homogenization methods
7.2.1 Main notations
7.2.2 Minimization problem
7.2.3 Averaged energy densities
7.2.4 Numerical corrector and fine scale features
7.2.5 HMM
7.2.6 MsFEM
7.3 Windowing in the periodic case
7.3.1 Setting of the problem
7.3.2 Mathematical formulation
7.3.3 A remark on boundary conditions
7.3.4 A remark on the volume element C(x, η)
7.3.5 Interpretation of the MsFEM in Petrov-Galerkin formulation
7.4 Windowing for general heterogeneities
7.4.1 Scaling of the windowing
7.4.2 Convergence results
7.4.3 Fine scale reconstruction
7.5 Conclusion
Part III Discrete to continuum limits
8 Exact bounds for the effective behaviour of a ‘discrete’ polycrystal
8.1 Introduction
8.2 Main result and derivation of bounds in two dimensions
8.2.1 Optimal bounds
8.2.2 Simple bounds
8.2.3 Bounds from arithmetic means
8.2.4 Bounds from harmonic means
8.3 Optimality of the bounds
8.3.1 Arithmetic bound
8.3.2 Harmonic bound
8.3.3 Optimal bounds
8.4 Extension to higher dimension
8.4.1 Bounds from arithmetic means
8.4.2 Bounds from harmonic means
8.4.3 Optimality of the bounds
8.5 Interpretation in terms of G-closure
8.5.1 Derivation of the trivial bounds
8.5.2 Interpretation in terms of quasiconvexification
8.5.3 Extensions to any number of conductivities and dimensions
9 Variational description of spin systems
9.1 Introduction
9.2 Notation and preliminary results
9.3 Compactness and integral representation results for spin systems
9.3.1 Pairwise-interaction energies
9.3.2 Case 1 < p < ∞
9.3.3 Case p = ∞
9.3.4 Proof in Lp, 1 < p < ∞
9.3.5 Proof in L∞
9.4 Minimum problems
9.5 Homogenization
9.5.1 Homogenization in Lp, 1 < p < ∞
9.5.2 Homogenization in L∞
9.6 Ferromagnetic-antiferromagnetic systems: existence of the bulk limit
9.7 Non-pairwise-interaction energies
10 Mathematical derivation of a rubber-like stored energy functional
10.1 Mesoscopic model
10.1.1 Free energy of a polymeric chain
10.1.2 Mesoscopic modeling of a rubber-like polymer
10.1.3 Stochastic networks and mesoscopic energies
10.2 Derivation of a macroscopic model
10.2.1 Main result
10.2.2 Sketch of the proof
10.3 Mechanical properties of the macroscopic model
10.3.1 Objectivity
10.3.2 Isotropy
10.3.3 Behaviour at small strains
10.3.4 Compressibility issues
10.3.5 Physical setting
10.4 Numerical tests and further issues
10.4.1 Numerical validation of the model
10.4.2 Comments and further developments
Part IV Fluid-structure interaction problems
11 Domain decomposition based Newton methods for fluid-structure interaction problems
11.1 Introduction
11.1.1 Equations de Navier-Stokes
11.1.2 Théorie des coques
11.1.3 Couplage fluide-structure
11.1.4 Aim of the study
11.2 Classical solution methods
11.2.1 Monolithic formulation
11.2.2 Dirichlet to Neumann formulations
11.2.3 Symmetric Steklov-Poincaré formulation
11.3 Mechanical setting
11.3.1 The coupled problem
11.3.2 Weak formulation
11.4 Semi-discretized weak formulation
11.4.1 Implicit coupling scheme
11.4.2 Abstract formulations
11.4.3 Steklov-Poincaré operators
11.5 A partitioned Newton method
11.5.1 Weak state operators derivatives
11.5.2 Domain decomposition method
11.5.3 Complexity analysis
11.6 Description of a 3D shell for fluid-structure interaction
11.7 Preliminary numerical results
11.8 Conclusion
Bibliographie
