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Méthodes éléments finis mixtes robustes pour gérer l’incompressibilité en grandes déformations dans un cadre industriel

Abstract : Simulations in solid mechanics exhibit difficulties as dealing with incompressibility or nonlinearities due to finite strains, constitutive laws and contact. The basic motivation of our work is to propose efficient finite element methods capable of dealing with incompressibility in finite strain context, and using low order elements. Among the approaches in the literature, mixed formulations offer an interesting theoretical framework. In this work, a three-field mixed formulation (displacement, pressure, volumetric strain) is investigated. In some cases, this formulation can be condensed in a two-field formulation. However, it is well-known that the discrete problem given by the Galerkin finite element technique, does not inherit the “inf-sup” stability condition from the continuous problem: the finite elements used, and in particular the interpolation orders must be chosen so as to satisfy this stability condition. However, it is possible to circumvent it, by adding terms stabilizing the FE Galerkin formulation. The latter approach allows the use of equal order interpolation. In this work, stable finite elements of type P2/P1 are used as reference, and compared to a P1/P1 formulation, stabilized with a bubble function, or with a VMS method (Variational Multi-Scale) based on a sub-grid-space orthogonal to the FE space. Combined to a finite strain model based on logarithmic strain, these approaches are first validated on academic cases and then on industrial cases.
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Dina Al-Akhrass. Méthodes éléments finis mixtes robustes pour gérer l’incompressibilité en grandes déformations dans un cadre industriel. Autre. Ecole Nationale Supérieure des Mines de Saint-Etienne, 2014. Français. ⟨NNT : 2014EMSE0733⟩. ⟨tel-01077894⟩

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