Localized adaptive radiation condition for coupling boundary with finite element methods applied to wave propagation problems

first published online October 3, 2013 doi:10.1093/imanum/drt038International audienceThe wave propagation problems addressed in this paper involve a relatively large and impenetrable surface on which is posed a comparatively small penetrable heterogeneous material. Typically the numerical solution of such kinds of problems is solved by coupling boundary and finite element methods. However, a straightforward application of this technique gives rise to some difficulties which mainly are related to the solution of a large linear system whose matrix consists of sparse and dense blocks. To face such difficulties, the adaptive radiation condition technique is modified by localizing the truncation interface only around the heterogeneous material. Stability and error estimates are established for the underlying approximation scheme. Some alternative methods are recalled or designed making it possible to compare the numerical efficiency of the proposed approach

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

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

Numerical study of acoustic multiperforated plates

International audienceIt is rather classical to model multiperforated plates by approximate impedance boundary conditions. In this article we would like to compare an instance of such boundary conditions obtained through a matched asymptotic expansions technique to direct numerical computations based on a boundary element formulation in the case of linear acoustic

Formulation courants et charges pour la résolution par équations intégrales des équations de l'électromagnétisme

Cette thèse a consisté à élaborer une méthode qui permet de résoudre l équation intégrale comportant comme inconnues les courants et les charges introduite récemment par Taskinen et Ylä-Oijala par une méthode d éléments frontière sans aucune contrainte de continuité au niveau des interfaces des éléments aussi bien pour les courants que pour les charges. Nous avons d abord montré comment on pouvait construire cette équation de façon simple et similaire à celle des formulations intégrales usuelles en imposant au problème intérieur relatif au système de Picard, qui est en fait une extension du système de Maxwell, des conditions aux limites adéquates. Pour des géométries régulières de l objet diffractant, nous avons établi de façon théorique la stabilité et la convergence des schémas numériques ci-dessus en montrant que cette équation peut être décomposée sous la forme d un système elliptique coercif et d un opérateur compact dans le cadre des fonctions de carré intégrable.Toute cette étude a été confirmée par des tests numériques tridimensionnels. Comme pour les équations intégrales usuelles de seconde espèce, le cadre théorique valable pour des surfaces régulières ne l est plus pour des surfaces avec des singularités. L utilisation formelle de cette équation,pour des surfaces singulières, a donné des résultats entachés d erreur. Nous avons mis en évidence l origine des instabilités numériques à l origine de ces erreurs lorsque les géométries sont singulières en développant une version bidimensionnelle de cette équation. Cette version nous a permis en particulier de montrer que les instabilités étaient dues à des oscillations parasites concentrées autour des singularités de la géométrie. Dans ce cadre nous avons pu mettre en oeuvre plus aisément des approches pour supprimer ou atténuer ces oscillations parasites ou leur effet sur les calculs en champ lointain. Nous avons montré qu un procédé d augmentation des degrés de liberté pour la charge par rapport au courant pouvait sensiblement réduire ces instabilités. A la suite de l amélioration observée sur les résultats dans le cas 2D, nous avons transposé cette procédure au cas tridimensionnel. A travers divers tests, nous avons constaté l amélioration de la qualité de l approximation amenée par la procédure de stabilisationThe objective of this thesis was to develop a method that solves the integral equation whose unknowns are the currents and the charges, recently introduced by Taskinen and Ylä-Oijala, by a boundary element method without any continuity constraint at the interfaces of the elements,for both the unknowns. We first show how to construct this equation in a simple way, similar tothe usual integral formulations, through imposing to the internal problem related to the Picard system,which is an extension of the Maxwell system, appropriate boundary conditions. For regular geometries, we have established a theoretical background ensuring the stability and the convergence of numerical scheme, by proving that this equation can be decomposed in a coercive elliptic and a compact parts in the context of square integrable functions. Our study was validated by three-dimensional numerical tests. In the case of usual integral equations of the second kind, the theoretical background for smooth surfaces is no longer valid when the surfaces is singular. The formal use of this equation for singular surfaces gave erroneous results. We pointed out the origin of numerical instabilities bydeveloping a two-dimensional version of this equation. This version has allowed us to show that the instabilities were due to parasitic oscillations accumulating on the geometrical singularities. In this context, we have implemented some approaches to reduce this parasitic oscillations on the calculations in the far field.We have shown that the method of increasing the freedom degrees for the charges relatively to the current could significantly reduces these instabilities. As a result, we have implemented this procedure in three-dimensional case. Throughout various tests, we noted the improvement on the approximation brough bay to the stabilization procedureTOULOUSE-INSA-Bib. electronique (315559905) / SudocSudocFranceF

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

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

International audienceOne 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 thin slot. 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 eigenvectors of this problem. The results are compared with direct computations

Lower and upper bounds for the Rayleigh conductivity of a perforated plate

International audienceLower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for inclined perforations. The main techniques are a proper use of the variational principles of Dirichlet and Kelvin in the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in the two-dimensional axisymmetric case and the full three-dimensional one