Méthodes variationnelles, Domaines fictifs et conditions aux limites artificielles pour des problèmes hyperboliques linéaires. Applications aux ondes dans les solides

Eliane Bécache 1
1 ONDES - Modeling, analysis and simulation of wave propagation phenomena
Inria Paris-Rocquencourt, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR2706
Abstract : This manuscript describes my research on the mathematical and numerical analysis of wave propagation problems. The first chapter is devoted to numerical methods for the propagation or the diffraction of elastic waves in solids : (i) retarded potentials in homogeneous isotropic media, (ii) tomography for seismic imaging, (iii) paraxial equations, (iv) mixed finite elements for elastodynamics. This last point, the most detailed here, is part of a general strategy to design an efficient numerical method which can deal with complex media (anisotropic, heterogeneous) including obstacles of arbitrary shape. It has been developed in the idea of using the fictitious domain method presented in the second chapter. After a short description of the method for a model scalar problem, it is first presented for the diffraction of elastic waves by a crack modeled either with a free surface boundary condition or with a unilateral contact boundary condition, and then for a problem of musical acoustics (modelization of the guitar). The third chapter is concerned with the question of artificial boundary conditions to bound the computational domain. Methods of Perfectly Matched Layers (PML) are analyzed for transient problems (electromagnetism, acoustics, elastodynamics, general first order hyperbolic systems) and for a time harmonic problem of aeroacoustics in an infinite waveguide. The manuscript ends with a brief review of the work in progress and on new perspectives.
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Habilitation à diriger des recherches
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Eliane Bécache. Méthodes variationnelles, Domaines fictifs et conditions aux limites artificielles pour des problèmes hyperboliques linéaires. Applications aux ondes dans les solides. Mathématiques [math]. Université Paris Dauphine - Paris IX, 2003. ⟨tel-00004880v2⟩

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