65 research outputs found
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
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
Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle
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. The
latter are typically bounded on the space of tangential vector fields of mixed
regularity TH\sp{-1/2}(\Div_{\Gamma},\Gamma). Using Helmholtz decomposition,
we can base their analysis on the study of pseudo-differential 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. We also give a characterization of the first shape
derivative of the solution of the dielectric scattering problem as a solution
of a new electromagnetic scattering problem.Comment: arXiv admin note: substantial text overlap with arXiv:1002.154
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 . 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 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
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
When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging
We propose an automatic algorithm for 3D inverse electromagnetic scattering based on the combination of topological derivatives and regularized Gauss-Newton iterations. The algorithm is adapted to decoding digital holograms. A hologram is a two-dimensional light interference pattern that encodes information about three-dimensional shapes and their optical properties. The formation of the hologram is modeled using Maxwell theory for light scattering by particles. We then seek shapes optimizing error functionals which measure the deviation from the recorded holograms. Their topological derivatives provide initial guesses of the objects. Next, we correct these predictions by regularized Gauss-Newton techniques devised to solve the inverse holography problem. In contrast to standard Gauss-Newton methods, in our implementation the number of objects can be automatically updated during the iterative procedure by new topological derivative computations. We show that the combined use of topological derivative based optimization and iteratively regularized Gauss-Newton methods produces fast and accurate descriptions of the geometry of objects formed by multiple components with nanoscale resolution, even for a small number of detectors and non convex components aligned in the incidence direction
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
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