Frédérique Le Louër

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

A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems

\Gamma

A high order spectral algorithm for elastic obstacle scattering in three dimensions

. In this paper, following an idea developed by Kleinman and Martin [SIAM J. Appl. Math., 48 (1988), pp. 307–325] for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over

On the Fréchet derivative in elastic obstacle scattering

\Gamma

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

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 ener...

Short note: Spectrally accurate numerical solution of hypersingular boundary integral equations for three-dimensional electromagnetic wave scattering problems

The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Frechet differentiability of the boundary to far-field operator and give a characterization of the first Frechet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gaus Newton method and show numerical experiments in the special case of star-shaped obstacles.

Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

In this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developed by Ganesh and Hawkins 20 - for solving second kind boundary integral equations in electromagnetism - to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parametrised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles.

Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems

In this paper, we investigate the existence and characterizations of the Frechet derivative of solutions to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique---the factorization of the difference of the far-field pattern for two different scatterers---introduced by Kress and Paivarinta [SIAM J. Appl. Math., 59 (1999), pp. 1413--1426] to establish Frechet differentiability in acoustic scattering. For the Dirichlet boundary condition an alternative proof of a differentiability result due to Charalambopoulos is provided, and new results are proven for the Neumann and impedance exterior boundary value problems.

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

Abstract The 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.

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

In this paper we extend the spectrally accurate algorithms developed by Ganesh et al. in [2,3] to the numerical solution of a modified combined-field integral equation (M-CFIE) for electromagnetic wave scattering in three dimensions.