Modèles d'impédance généralisée en diffraction inverse

Nicolas Chaulet 1
1 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : The main objective of this thesis is to use asymptotic models for the resolution of inverse electromagnetic scattering problems. We consider in particular the case of generalized impedance conditions that are models for thin coatings or strongly absorbing media. The term "generalized impedance" signifies that the boundary condition involves a surface operator. The so-called classical impedance boundary conditions is a particular case of generalized impedance boundary condition where the operator reduces to the multiplication by a function. In the inverse problems context, the use of such approximate models simplifies the numerical resolution as well as the mathematical analysis. In the literature a lot of studies focus on inverse scattering with classical impedance boundary conditions, we extend them to the case of more complex surface operators containing surface derivatives for example. An important part of the thesis deals the application of optimization methods to find both a shape and parameters that characterize the boundary operator. Among others, we present the computation of the shape derivative for Helmholtz' and Maxwell's equations. Numerical illustrations of shape reconstruction and identification of boundary coefficients are also provided. We complement this work by studying a qualitative method - the factorization method - to retrieve a shape with a general form of generalized impedance boundary conditions. In relation with qualitative methods, we investigated the use of the so-called interior transmission eigenvalues associated with thin layer structures to obtain information about the layer thickness and properties. In this view, we derived and justified the full asymptotic development of the first interior transmission eigenvalue with respect to the small thickness of the layer. This development provides a simple procedure to compute the thickness of the layer from multi-static scattered field data.
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Nicolas Chaulet. Modèles d'impédance généralisée en diffraction inverse. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2012. Français. ⟨pastel-00761642⟩

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