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Modèles asymptotiques et simulation numérique pour la diffraction d'ondes par des petites hétérogénéités

Simon Marmorat 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
CNRS - Centre National de la Recherche Scientifique : UMR7231, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France
Abstract : This work is dedicated to the study of the diffraction of acoustic waves by a set of small inclusions, as well as to the development of numerical methods for the simulation of such phenomenons. The main novelty of this work is that we deal with time-domain waves.The first part of this manuscript deals with the asymptotic analysis of the diffraction problem, which is carried out by matched asymptotics, the small parameter being the characteristic size of the defects $varepsilon$. This furnishes an asymptotic expansion of the acoustic field as a perturbation of the defect-free problem. We prove a consistency result between the total field and its $varepsilon$-asymptotic expansion.In the second part, using the results of the asymptotic analysis, we introduce two approximate models for the diffraction problem. These models are well-posed and their solution are precise approximations of the total acoustic field. One of the main features of these approximate models is that they both lie on a wave equation in the surrounding medium (without defects), coupled to auxiliary source terms which account for the presence of the inclusions. It is then possible to discretize these approximate models using a finite element method, leading to a numerical method which performs as fast as in the defect-free case, since the underlying wave operator is independent of the defects. We present several numerical results which validate both approximate models as well as some insights about numerical error analysis.
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Simon Marmorat. Modèles asymptotiques et simulation numérique pour la diffraction d'ondes par des petites hétérogénéités. Modélisation et simulation. Université Paris-Saclay, 2015. Français. ⟨NNT : 2015SACLY001⟩. ⟨tel-01243663v2⟩

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