Publications

2006

• 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.
• 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
• 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.
• 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
• On the convergence of the fictitious domain method for wave equation problems
  
  • Bécache Eliane
  • Rodríguez Jerónimo
  • Tsogka Chrysoula
  , 2006, pp.37. This paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis is presented in the acoustic case and numerical convergence is shown both for acoustic and elastic waves.
• Computing reducing subspaces of a large linear matrix pencil
  
  • Hechme Grace
  • Nechepurenko Yuri.
  Russian Journal of Numerical Analysis and Mathematical Modelling, De Gruyter, 2006, 21 (3), pp.185-198. This paper deals with the computation of the reducing subspace associated with the rightmost part of the spectrum of a large matrix pencil A-λ B with B = diag(I,0). Two variants of the Jacobi-Davidson method are discussed and developed. One is based on the Euclidean inner product and the second on the semi-inner product induced by B. Both versions use real arithmetics and incorporate an efficient deflation procedure. Numerical results are reported. (10.1515/156939806777320359)
  DOI : 10.1515/156939806777320359