Christophe Hazard

On the solution of time-harmonic scattering problems for Maxwell's equations

This paper deals with the scattering of a monochromatic electromagnetic wave by a perfect conductor surrounded by a locally inhomogeneous medium. The direct numerical solution of this problem by a finite-element method requires special edge elements. The aim of the present paper is to give an equivalent formulation of the problem well suited for both easy theoretical investigation and numerical implementation. Following a well-known idea, this formulation is obtained by adding a regularizing term such as “grad div” in the time-harmonic Maxwell equations, which leads us to solve an elliptic problem similar to the vector Helmholtz equation instead of Maxwell’s equation. The numerical treatment of this new formulation requires only standard Lagrange finite elements.A unified approach, which is valid for the equations satisfied by either the electric or the magnetic field, is presented. It applies for a conductor with a Lipschitz-continuous boundary surrounded by a dissipative or nondissipative medium whose e...

A singular field method for the solution of Maxwell's equations in polyhedral domains

It is well known that in the case of a regular domain the solution of the time-harmonic Maxwells equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwells equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwells equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements.A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electri...

Variational formulations for the determination of resonant states in scattering problems

Consider the scattering of an acoustic wave by a rigid obstacle. The poles of the analytical continuation of the resolvent operator are called scattering frequencies. On their localization depend the time-decay of the solution and the location of the energy peaks of the steady-state solution.Two methods are proposed to construct explicitly the analytical continuation of the resolvent: the localized finite element method or the coupling between variational formulation and integral representation, which both rely upon the reduction of the exterior Helmholtz problem to a bounded domain. The determination of the scattering frequencies then amounts to solving a nonlinear eigenvalue problem for a completely continuous operator.Then, the expansion of the approximate steady-state solution in the vicinity of a scattering frequency is computed. Numerical results for a simple one-dimensional problem are presented.

Existence, uniqueness and analyticity properties for electromagnetic scattering in a two-layered medium

This paper is concerned with the solution of Maxwell equations in the modelling of the scattering of a time-harmonic electromagnetic wave by an obstacle located in a two-layered medium. The use of the Silver-Muller radiation condition in each layer is shown to provide a well-posed scattering problem. The analysis is based on the study of the Green tensor, which allows to relate the radiation condition to an integral representation formula. The analyticity properties of the scattering problem with respect to the frequency are then investigated. This gives rise to a limiting absorption principle and furnishes a characterization of the resonances.

SELECTIVE ACOUSTIC FOCUSING USING TIME-HARMONIC REVERSAL MIRRORS ∗

A mathematical study of the focusing properties of acoustic fields obtained by a time-reversal process is presented. The case of time-harmonic waves propagating in a nondissipative medium containing sound-soft obstacles is considered. In this context, the so-called D.O.R.T. method (decomposition of the time-reversal operator in French) was recently proposed to achieve selective focusing by computing the eigenelements of the time-reversal operator. The present paper describes a justification of this technique in the framework of the far field model, i.e., for an ideal time-reversal mirror able to reverse the far field of a scattered wave. Both cases of closed and open mirrors, that is, surrounding completely or partially the scatterers, are dealt with. Selective focusing properties are established by an asymptotic analysis for small and distant obstacles.

Determination of scattering frequencies for an elastic floating body

This paper studies the time-harmonic motions of a three-dimensional elastic floating body on the sea, in the case of finite and constant depth. In order to compute the resonant states of such a system, a variational formulation for the determination of the scattering frequencies of the problem is investigated, i.e., the poles of the analytic continuation of the solution operator. A practical method, based on a series expansion of the solution in a vicinity of infinity, is described. The scattering frequencies are shown to be the solutions of a nonlinear eigenvalue problem for a compact operator. Numerical results for a two-dimensional model are presented.

DIFFRACTION BY A DEFECT IN AN OPEN WAVEGUIDE: A MATHEMATICAL ANALYSIS BASED ON A MODAL RADIATION CONDITION ∗

We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholms alternative by reformulating the scattering problem as a Lippmann-Schwinger equation by means of the Greens function for the layered half-plane.

Generalized eigenfunction expansions for conservative scattering problems with an application to water waves

This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem. This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation.

Spectral Theory For an Elastic Thin Plate Floating on Water of Finite Depth

The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass.

Mathematical Analysis of the Junction of Two Acoustic Open Waveguides

The present paper concerns the scattering of a time-harmonic acoustic wave by the junction of two open uniform waveguides, where the junction is limited to a bounded region. We consider a two-dimensional problem for which wave propagation is described by the scalar Helmholtz equation. The main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave. We use here modal radiation conditions which extend the classical conditions used for closed waveguides. They are based on the generalized Fourier transforms which diagonalize the transverse contributions of the Helmholtz operator on both sides of the junction. We prove the existence and uniqueness of the solution, which seems to be the first result in this context. The originality lies in the proof of uniqueness, which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized Fourier variable.