La représentation implicite des volumes pour l'analyse par éléments finis avec XFEM et Level-sets

Abstract : The Finite Element Method (FEM) is widely used for numerical simulations of physical problems formulated in terms of partial differential equations (PDE). A crucial step in the process of analysis by this method is the discretization of the geometry to construct a mesh representing the approximation space of the problem. However, high quality mesh that conforms to the curved boundaries and sharp features, whose depends on the numerical results, still requires a significant amount of human time in the global process of analysis. The aim of this work is to implement a new approach that allows performing simulations on an object whose boundaries do not conform to the mesh, while retaining the benefits of FEM. For this purpose, the implicit representation of the domain (Level set) and the eXtended Finite Element Method (XFEM) are used. In the first step, the focus is to build objects by using Level sets independently of the spatial discretization (i.e. a simple mesh). Strategies have been developed to build implicit objects from the parametric representation (the most common in Computer Aided Design CAD), to preserve sharp features and correctly represent curved boundaries. In a second step, the focus lies on adapting XFEM to achieve a proper numerical integration and to preserve the stability of mixed formulations for managing Dirichlet constraints. The last part consists in verifying the accuracy and rate convergence in the case of implicit curved boundaries and of non-conforming objects to the mesh
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Mohammed Moumnassi. La représentation implicite des volumes pour l'analyse par éléments finis avec XFEM et Level-sets. Autre. Université Paul Verlaine - Metz, 2011. Français. ⟨NNT : 2011METZ033S⟩. ⟨tel-01749074⟩

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