Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators

On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface $\Gamma$. In this paper, following an idea developed by Kleinman and Martin \cite{KlMa} for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over $\Gamma$ for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the well-posedness of the integral equations in the space of finite energy on smooth and non-smooth boundaries

Shape derivatives of boundary integral operators in electromagnetic scattering

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the G\^ateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles

Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

International audienceIt has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies

Optimisation de forme d'antennes lentilles intégrées aux ondes millimétriques

Integrated lens antennas are devices having electromagnetic waves as support and are composed of a primary source and a dielectric focusing system. The recent increase of applications in millimetric waves (example :adaptative cruise control radars), needs the design of lens antennas of a fex centimeters with specific requirements. One of the problems consists in determining the optimal shape of the lens given : (i) the primary source, (ii) the radiation pattern. This PhD thesis project aims to develop new tools for shape optimization using integral formulation of the problem.This PhD thesis is structured in two parts. In the first one wa have constructed sevral integral formulations for the dielectric scattering problem using boundary integral equation approach. In the second one we have studied the shape derivatives of the standard boundary integral operators in electromagnetism in order to incorporate this derivatives in shape optimization algorithms Les antennes lentilles sont des dispositifs ayant pour support les ondes électromagnétiques et sont constituées d'une source primaire et d'un système focalisant diélectrique. La montée en importance récente d'applications en ondes millimétriques (exemple : radars d'assistance et d'aide à la conduite), nécessite la construction d'antennes lentilles de quelques centimètres qui répondent à des cahiers des charges spécifiques à chaque cas. L'une des problématiques à résoudre consiste à déterminer la forme optimale de la lentille étant données : (i) les caractéristiques de la source primaire, (ii) les caractéristiques en rayonnement fixées. Ce projet de thèse vise à développer de nouveaux outils pour l'optimisation de forme en utilisant une formulation intégrale du problème.Cette thèse s'articule en deux parties. Dans la première nous avons construit plusieurs formulations intégrales pour le problème de diffraction diélectrique en utilisant une approche par équation intégrale surfacique. Dans la seconde nous avons étudié les dérivées de forme des opérateurs intégraux standard en électromagnétisme dans le but de les incorporer dans un algorithme d'optimisation de forme