Un problème de Laplace non standard en milieu non borné

Projet POEMSDans le cadre des problèmes elliptiques en dimension deux, nous nous intéressons à un domaine constitué d'un demi-espace connecté à une bande infinie. Un résultat d'existence et unicité est obtenu pour un problème de Laplace inhomogène muni de comportements asymptotiques à l'infini

Self-similar perturbation near a corner: matching versus multiscale expansions for a model problem

International audienc

Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions

International audienceThe cavity model is a wide-spread powerful empirical approach for the numerical simulation of microstrip antennas. It is based on several hypotheses assumed a priori: a dimension reduction in the cavity, that is, the zone limited by a metallic patch and the ground plane in which is fed the antenna, supplied by the additional condition that the open sides of the cavity act as magnetic walls. An additional important assumption of this model consists in an adequate description of the singular field behavior in the proximity of the edge of the patch. A simplified two-dimensional problem incorporating the main features of the field behavior near the edge of the patch and inside the cavity is addressed. The method of matched asymptotic expansions is used to carry out a two-scale asymptotic analysis of the field relatively to the thickness of the cavity. All the empirical hypotheses at the basis of the derivation of the cavity model can thus be recovered. Proved error estimates are given in a simplified framework where the dielectric constants of the substrate are assumed to be 1 in order to avoid some unimportant technical difficulties

Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the $${\wedge}$$ -product of 1-forms on $${\partial D}$$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discusse

Modélisation par sources ponctuelles multipolaires équivalentes de petites sphères pour le calcul rapide et précis de la diffraction des ondes électromagnétiques

International audienc

Approximation by multipoles of the multiple acoustic scattering by small obstacles and application to the Foldy theory of isotropic scattering.

50 (avec 1,5 interligne)International audienceThe asymptotic analysis, carried out in this paper, for the problem of a multiple scattering of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy's method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and taking into account of self-interactions

Matching of asymptotic expansions for the wave propagation in media with thin slot

International audienceThis talk concerns the modelizing of scattering in the harmonic regime in two dimensional domains with thin slots. We use the technique of matching asymptotic expansions to obtain and justify the asymptotic expansion of the solution to any order with respect to the width of the slot

An efficient truncated SVD of large matrices based on the low-rank approximation for inverse geophysical problems

International audienceIn this paper, we propose a new algorithm to compute a truncated singular value decomposition (T-SVD) of the Born matrix based on a low-rank arithmetic. This algorithm is tested in the context of acoustic media. Theoretical background to the low-rank SVD method is presented: the Born matrix of an acoustic problem can be approximated by a low-rank approximation derived thanks to a kernel independent multipole expansion. The new algorithm to compute T-SVD approximation consists of four steps, and they are described in detail. The largest singular values and their left and right singular vectors can be approximated numerically without performing any operation with the full matrix. The low-rank approximation is computed due to a dynamic panel strategy of cross approximation (CA) technique. At the end of the paper, we present a numerical experiment to illustrate the efficiency and precision of the algorithm proposed

Matching of asymptotic expansions for the wave propagation in media with thin slot

International audienceThis talk concerns the modelizing of scattering in the harmonic regime in two dimensional domains with thin slots. We use the technique of matching asymptotic expansions to obtain and justify the asymptotic expansion of the solution to any order with respect to the width of the slot

Matching of Asymptotic Expansions for a 2-D eigenvalue problem with two cavities linked by a narrow hole

One question of interest in an industrial conception of air planes motors is the study of the deviation of the acoustic resonance frequencies of a cavity which is linked to another one through a narrow hole. These frequencies have a direct impact on the stability of the combustion in one of these two cavities. In this work, we aim is analyzing the eigenvalue problem for the Laplace operator with Dirichlet boundary conditions. Using the Matched Asymptotic Expansions technique, we derive the asymptotic expansion of this eigenmodes. Then, these results are validated through error estimates. Finally, we show how we can design a numerical method to compute the eigenvalues of this problem. The results are compared with direct computations