Méthodes d'approximation par éléments finis et analyse a posteriori d'inéquations variationnelles modélisant des problèmes de fissures en élasticité linéaire

Abstract : In this PHD-Thesis, we perform the mathematical and the numerical analysis of models in the linearized elasticity for cracked materials arising in the mechanics of contact. The boundary conditions on the cracks faces are of unilateral contact type. From the mathematical point of view, such problems are formulated within the framework of the theory of the variational inequalities. We describe and study formulations which take into account the boundary conditions of the contact and, this the novelty in this work, which also take into account the geometry. The report of the dissertation is divided into two parts. The first part concerns a formulation known as "a regular domain formulation". It consists in extending the equations to the entire domain including the cracks. The conditions of contact on the crack are taken into account in the functional spaces. We carry out the mathematical analysis of this formulation and we consider two discretizations by the finite element method. The first discretization is written for the formulation with symmetry of the stress tensor, whereas the second discretization is based on a modified Hellinger-Reissner formulation with broken symmetry (of the stress tensor). For each discretization, we introduce the suitable finite elements. We obtain obtain optimal convergence rates for this type of variational inequalities . We explain the implementation and we give numerical results which confirm the theoretical results. The second part is dedicated to a fictitious domain method for such variational inequalities. We consider the case of the frictionless Signorini problem for the elasticity system. We carry out the numerical analysis to obtain optimal a priori estimates and results of convergence with respect to the discretization parameter. We also perform the a posteriori analysis by residual error estimates. We introduce new tools for such an analysis which are residual error indicators and we show their optimality with respect to the step size of discretization and their efficiency. These tools allow us to define an adaptive strategy to solve such problems. The adptivity in this context is in fact a multi-step strategy where the initial problem is first ``approximated in the infinite dimensional setting  before the discretization. The implementation details and some numerical results are given
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Souad Tahir. Méthodes d'approximation par éléments finis et analyse a posteriori d'inéquations variationnelles modélisant des problèmes de fissures en élasticité linéaire. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2006. Français. ⟨NNT : 2006METZ020S⟩. ⟨tel-01752445⟩

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