Publications

2006

• A justification of Peek's empirical law in electrostatics [Justification de la loi de Peek en électrostatique]
  
  • Ciarlet Patrick
  • Kaddouri Samir
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2006, 343 (10), pp.671-674. We consider the computation of the electrostatic charge density at the tip of a rounded corner. The relation between the curvature radius and the electrostatic field is given by Peek's empirical law which is valid only for thin, cylindrical or spherical, geometries. In this Note, we justify mathematically this law and extend it to other geometries. With the help of multiscaled asymptotic expansions, we derive an expression for the charge density for geometries which coincide at infinity with a cone. A numerical illustration is provided. To cite this article: P. Ciarlet Jr., S. Kaddouri, C. R. Acad. Sci. Paris, Ser. I 343 (2006). © 2006 Académie des sciences. (10.1016/j.crma.2006.10.009)
  DOI : 10.1016/j.crma.2006.10.009
• Exact boundary conditions for periodic waveguides containing a local perturbation
  
  • Joly Patrick
  • Li Jing-Rebecca
  • Fliss Sonia
  Communications in Computational Physics, Global Science Press, 2006, 1 (6), pp.945-973. We consider the solution of the Helmholtz equation $-\Delta u({\bf x}) - n({\bf x})^2\omega^2 u({\bf x}) = f({\bf x})$, ${\bf x}=(x,y)$, in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f({\bf x})$ is supported in $\Omega^0:={{\bf x}\in {\Omega} \; | a^
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2006. Cet exposé porte sur la modélisation de la diffraction d'ondes en régime harmonique dans des milieux bidimensionnels comportant des fentes minces. Nous utilisons la technique des développements asymptotiques raccordés pour obtenir et justifier le développement asymptotique de la solution à tout ordre en fonction de l'épaisseur de la fente.
• High Order Generalized Impedance Boundary Conditions in Electromagnetic Scattering Problems
  
  • Duruflé Marc
  • Haddar Houssem
  • Joly Patrick
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.533-542. We briefly review the use and the derivation of Generalized Impedance Boundary Conditions (GIBC) in the case of thin dielectric coating and in the case of strongly absorbing medium, within the context of electromagnetic scattering problem at a fixed frequency. We then numerically test the validity and accuracy of these boundary conditions in the case of high absorption. A numerical treatment of the corner singularity is proposed to recover the accuracy of the GIBC for singular geometries.
• Discontinuous Galerkin methods for Maxwell's equations in the time domain
  
  • Cohen Gary
  • Ferrieres Xavier
  • Pernet Sébastien
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.494-500. In this article, we describe a new high-order Discontinuous Galerkin approach to Maxwell's equations in the time domain. This approach is based on hexahedral meshes and uses a mass-lumping technique. Thanks to the orthogonality of the basis functions and a judicious choice of the approximation spaces, it provides an efficient solver for these equations in terms of storage and CPU time. (10.1016/j.crhy.2006.03.004)
  DOI : 10.1016/j.crhy.2006.03.004
• Perfectly matched layers for time-harmonic acoustics in the presence of a uniform flow
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 44 (3), pp.1191-1217. This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included. © 2006 Society for Industrial and Applied Mathematics. (10.1137/040617741)
  DOI : 10.1137/040617741
• Etude d'un problème modèle pour la diffraction par des fils minces par développements asymptotiques raccordés Cas 2D
  
  • Claeys Xavier
  • Haddar Houssem
  • Joly Patrick
  , 2006, pp.52. Dans ce rapport, nous analysons un problème modèle pour l'étude de la diffraction d'une onde par des fils minces. Nous nous intéressons, en deux dimensions, à la solution sortante de l'équation de Helmholtz à l'extérieur d'un obstacle de petit diamètre (vis-à-vis de la longueur d'onde) sur la frontière duquel est imposée une condition de Dirichlet homogène ou une condition de Neumann homogène. Un développement à tout ordre de cette solution par rapport au diamètre de l'obstacle est obtenu.
• Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion
  
  • Joly Patrick
  • Tordeux Sébastien
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2006, 5 (1), pp.304--336 (electronic). In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter ε/λ, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of ε/λ: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series. (10.1137/05064494X)
  DOI : 10.1137/05064494X
• Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots
  
  • Joly Patrick
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2006, 40 (1), pp.63--97. The authors study the asymptotic properties of the solution to the Helmoholtz equation with Neumann boundary conditions in a dumbbell-type domain in the regime when the "handle'' is thin and tightening to a curve. A mathematical analysis is done for the model problem posed in the half-plane with an infinite thin straight channel. It is proved that the solution of such a perturbed problem converges to the solution of the limiting problem for the same equation posed in the whole half-plane. Optimal estimates for the convergence rate are obtained in various norms. The authors also construct one more approximation for the solution of the perturbed problem which takes into account the presence of the channel. It is shown that this approximation is better in the sense that the estimates for the difference between this approximation and the "perturbed'' solution are smaller in order than similar estimates for the limiting solutions. The authors also conjecture that the last mentioned estimates are optimal; this conjecture is supported by a series of numerical results. (10.1051/m2an:2006008)
  DOI : 10.1051/m2an:2006008