Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles.

Patrice Hauret

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Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles.

Patrice Hauret¹

1:	CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

Le travail exposé dans ce mémoire consiste en une étude mathématique et numérique d'outils permettant la simulation de la dynamique de structures complexes non-linéaires, quasi-incompressibles, et présentant deux échelles de longueurs caractéristiques. Pour être plus précis concernant ce dernier point, les structures considérées sont supposées comporter des détails géométriques fins sur leur bord. Cette étude, réalisée en partenariat avec la Manufacture Française des Pneumatiques Michelin, est largement motivée par l'importance de calculs dynamiques en roulage du pneumatique afin de prédire la valeur de différentes grandeurs physiques : contraintes dans les matériaux, pressions de contact au sol ou encore rayonnement acoustique. Dans ce cadre, les difficultés d'obtention de simulations complètes et réalistes pour des coûts de calcul raisonnables sont liées à la complexité de la géométrie, au comportement des matériaux, au mode de sollicitation par contact, ou encore à l'intervention de différentes échelles de longueur, de temps ou de rigidité caractéristiques de la structure. Après une description de l'anatomie du pneumatique, nous mentionnons quelques uns des enjeux de la simulation numérique lors de la phase de conception. Nous soulignons ensuite les propriétés intrinsèques de la structure qui rendent ces études délicates. Enfin, nous délimitons les problèmes qui occupent le reste de ce mémoire, et esquissons la démarche adoptée. Référence est faite au contenu des chapitres et aux contributions apportées.

Defence date	2004-09-20
	Mathématiques et leurs applications
Library	Ecole Polytechnique
Keywords	Méthode numérique – Dynamique – Structures non-linéaires – Echelles
	: http://www.imprimerie.polytechnique.fr/Theses/Files/Hauret.pdf
English abstract	The work described in this thesis is a mathematical and numerical study tools to simulate the dynamics of complex nonlinear structures, quasi-incompressible, and has two characteristic length scales. To more specific about this last point, the structures considered are assumed to contain fine geometric details on their board. This study, conducted in partnership with the French Manufacturer of Tyres Michelin is largely motivated by the importance of dynamic calculations in the tire rolling to predict the value of different physical quantities: the constraints materials, ground contact pressure or acoustic radiation. In this context, difficulties in obtaining complete and realistic simulation for cost calculation are reasonably related to the complexity of the geometry, material behavior, the method of solicitation contact, or the intervention of different scales length, time or stiffness characteristics of the structure. After a description of the anatomy of the tire, we mention some issues of numerical simulation in the design phase. Then we underline the intrinsic properties of the structure that make these studies difficult. Finally, we delimit the issues that occupy the rest of this paper and outline the approach adopted. Reference is made to the content of chapters and contributions.

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1 Introduction 11
1.1 Position du problème . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Le pneumatique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 Quelques enjeux pour la simulation . . . . . . . . . . . . . . . . . . . 13
1.1.3 Sources de difficultés . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Travail de thèse et contributions . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Eléments de mécanique des milieux continus 23
2.1 Dynamique des milieux continus . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Cinématique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Système de l'élastodynamique . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 Problème mixte en déplacement-pression . . . . . . . . . . . . . . . . 26
2.2 Hyperélasticité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Energie emmagasinée . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Forme indépendante du referentiel . . . . . . . . . . . . . . . . . . . 29
2.2.3 Isotropie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Viscoelasticite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Contact sans frottement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Time integration in nonlinear elastodynamics 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Quasi-incompressible elastodynamics . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 The incompressible model . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Variational quasi-incompressible formulation . . . . . . . . . . . . . 43
3.2.3 Conservation properties . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Eciency and semi-explicit strategies . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 A centered explicit scheme . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 A semi-implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Computational complexity of the semi-implicit scheme . . . . . . . . 55
3.4 Conservation analysis for some usual schemes . . . . . . . . . . . . . . . . . 56
3.4.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.2 Midpoint based schemes . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.3 Trapezoidal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.4 Midpoint scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.5 Exactly conservative schemes . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Dissipative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.1 Conservation analysis for the HHT scheme . . . . . . . . . . . . . . 72
3.5.2 A new dissipative scheme in the nonlinear framework . . . . . . . . 78
3.6 Extensions of the conservative approach . . . . . . . . . . . . . . . . . . . . 80
3.6.1 Frictionless contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.6.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.7 Numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7.1 A simple cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7.2 Ball impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 A stabilized discontinuous mortar formulation 103
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Nonconforming setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.1 Position of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.3 Approximate problem . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.1 Inf-sup condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 Local rigid motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 Minimal Lagrange multipliers spaces . . . . . . . . . . . . . . . . . . 117
4.3.4 Standard result of coercivity . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Uniform coercivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.1 Fundamental assumptions . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4.2 Generalized Korn's inequality . . . . . . . . . . . . . . . . . . . . . . 124
4.4.3 A Scott & Zhang like interpolation operator for mortar methods . . 125
4.4.4 Uniform coercivity result . . . . . . . . . . . . . . . . . . . . . . . . 138
4.4.5 Existence result for problem (4.7) . . . . . . . . . . . . . . . . . . . . 139
4.5 Error estimates in elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.5.1 Approximation of displacements . . . . . . . . . . . . . . . . . . . . 140
4.5.2 Approximation of
uxes . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.6 Generalization to elastodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 148
4.6.1 Position of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.6.2 A midpoint nonconforming fully discrete approximation. . . . . . . . 150
4.6.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.7 Analysis of discontinuous mortar spaces . . . . . . . . . . . . . . . . . . . . 167
4.7.1 Stabilized first order elements . . . . . . . . . . . . . . . . . . . . . . 167
4.7.2 A counter example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.7.3 Numerical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.7.4 A useful lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.7.5 Second order stabilized interface elements . . . . . . . . . . . . . . . 177
4.8 Some numerical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.8.1 Penalized formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.8.2 Exact integration of the constraint . . . . . . . . . . . . . . . . . . . 188
4.9 Numerical tests for discontinuous mortar-elements . . . . . . . . . . . . . . 189
4.10 Appendix A : Mesh-dependent norms. . . . . . . . . . . . . . . . . . . . . . 202
4.11 Appendix B : Dependence of the constant in Korn's inequalities . . . . . . . 205
4.11.1 Poincaré-Friedrichs inequalities . . . . . . . . . . . . . . . . . . . . . 206
4.11.2 Dependence of the constant in Korn's second inequality . . . . . . . 207
4.11.3 Semi-norm estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5 Mortiers : contributions industrielles 219
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.2 Formulation mortier sur une surface courbe . . . . . . . . . . . . . . . . . . 221
5.2.1 Construction des espaces tangents . . . . . . . . . . . . . . . . . . . 222
5.2.2 Construction du carreau. . . . . . . . . . . . . . . . . . . . . . . . . 223
5.2.3 Projection sur la surface . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.2.4 Contrainte mortier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.3 Algorithme d'assemblage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.3.1 Algorithme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.4 Essais numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.4.1 Recollements au tour de roue . . . . . . . . . . . . . . . . . . . . . . 232
5.4.2 Recollement d'un pain unique . . . . . . . . . . . . . . . . . . . . . . 235
5.4.3 Mortiers et dynamique . . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
6 Two-scale Dirichlet-Neumann preconditioners 241
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.2 A mortar formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.2.1 Continuous problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6.3 Two-scale preconditioners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.3.2 Two possible denitions for ^D0 . . . . . . . . . . . . . . . . . . . . . 252
6.4 Condition number analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
6.4.1 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
6.4.2 Spectral equivalence for the simple Dirichlet-Neumann . . . . . . . . 257
6.4.3 Spectral equivalence for the enhanced Dirichlet Neumann . . . . . . 262
6.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
6.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.6.1 A basic two-scale model . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.6.2 Extension to a quasi-Newton method . . . . . . . . . . . . . . . . . . 276
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7 Conclusion 281

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