Publications

2001

• Study at high frequencies of a stratified waveguide
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Caloz Gabriel
  • Dauge Monique
  • Mahé Fabrice
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2001, pp.66 (2001) 231-257.
• Méthodes numériques: Éléments finis mixtes et méthode des domaines fictifs pour l'élastodynamique
  
  • Bécache Eliane
  • Joly Patrick
  • Tsogka Chrysoula
  , 2001.
• A limiting absorption principle for scattering problems with unbounded obstacles
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Tillequin Axel
  Mathematical Methods in the Applied Sciences, Wiley, 2001, 24 (14), pp.1089--1111. A generalized mode matching method that applies to a wide class of scattering problems is developed in the time harmonic two-dimensional Helmholtz case. This method leads by variational means to an integro-differential formulation whose unknown is the trace of the field on an unbounded one-dimensional interface. The well-posedness is proved after a careful study of the rather original functional framework. Owing to a fundamental density result--based upon some properties of a singular integral operator similar to the Hilbert transform--the limiting absorption principle related to this original formulation is established. Finally, two other situations are emphasized. Copyright © 2001 John Wiley & Sons, Ltd. (10.1002/mma.259)
  DOI : 10.1002/mma.259
• Analyse mathématique de l'équation de Galbrun en écoulement uniforme
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  • Lunéville Éric
  Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, Elsevier, 2001, 329 (8), pp.601-606. Nous considérons l'équation de Galbrun, utilisée en acoustique linéaire dans les écoulements. Dans un cas simple (conduit rigide avec écoulement uniforme) et en régime harmonique établi, nous montrons qu'une approche basée sur une formulation variationnelle régularisée du problème permet d'assurer la convergence d'une méthode d'éléments finis nodaux. (10.1016/S1620-7742(01)01373-3)
  DOI : 10.1016/S1620-7742(01)01373-3
• A fictitious domain method for unilateral contact problems in non-destructive testing
  
  • Bécache Eliane
  • Joly Patrick
  • Scarella Gilles
  , 2001. In this work, we present a numerical method for solving the diffraction of transient elastic waves by cracks of arbitrary shapes in complex media, with Signorini's boundary conditions on the crack. We use a fictitious domain method based on a mixed displacement-stress formulation for elastodynamics. We propose an off-centered time discretisation scheme for enforcing the stability.
• A Comparison of Methods for Calculating the Matrix Block Source Term in a Double Porosity Model
  
  • Alboin Clarisse
  • Jaffré Jérôme
  • Joly Patrick
  • Roberts Jean
  • Serres Christophe
  , 2001. Contaminant transport in a fractured porous medium can be modeled, under appropriate conditions, with a double porosity model. Such a model consists of a parabolic equation with a coupling term describing contaminant exchange between the fractures, which have high permeability, and the matrix block, which has low permeability. A locally conservative method based on mixed finite elements is used to solve the parabolic problem, and the calculation of the coupling term, which involves the solution of diffusion equations in the matrix blocks, is based on an analytic expression. Numerical experiment- s show that this analytic method for the coupling term compares favorably to several other methods.
• Stability of Perfectly Matched Layers, Group Velocities and Anisotropic Waves
  
  • Bécache Eliane
  • Fauqueux Sandrine
  • Joly Patrick
  , 2001. Perfectly Matched Layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagn- etic waves and extended, since then, to other models of wave propagation, including waves in elastic anisotropic media. In this last case, some numerical experiments have shown that the PMLs are not always stable. In this paper, we investigate this question from a theoretical point of view. In the first part, we derive a necessary condition for the stability of the PML model for a general hyperbolic system. This condition can be interpreted in terms of geometrical properties of the slowness diagrams and used for explaining instabilities observed with elastic waves but also with other propagation models (anisotropic Maxwell's equations, linearize- d Euler equations). In the second part, we specialize our analysis to orthotropic elastic waves and obtain separately a necessary stability condition and a sufficient stability condition that can be expressed in terms of inequalities on the elasticity coefficients of the model.
• On traces for functional spaces related to Maxwell's equations Part I: An integration by parts formula in Lipschitz polyhedra
  
  • Buffa Annalisa
  • Ciarlet Patrick
  Mathematical Methods in the Applied Sciences, Wiley, 2001, 24 (1), pp.9-30. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl, Omega), when Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces on the whole boundary and on a part of it (for partially vanishing fields and general ones.) In both cases it is possible to define ad hoc dualities among tangential trace and tangential components. In addition, the validity of two related integration by parts formulae is provided. Copyright (C) 2001 John Wiley & Sons, Ltd. (10.1002/1099-1476(20010110)24:1(9::aid-mma191)3.0.co;2-2)
  DOI : 10.1002/1099-1476(20010110)24:1(9::aid-mma191)3.0.co;2-2
• Higher order triangular finite elements with mass lumping for the wave equation.
  
  • Cohen Gary
  • Joly Patrick
  • Roberts Jean
  • Tordjman Nathalie
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2001, 38 (6), pp.2047-2078. In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown. (10.1137/S0036142997329554)
  DOI : 10.1137/S0036142997329554
• Equations intégrales pour l'équation des ondes
  
  • Bécache Eliane
  , 2001. On s'intéresse à des problèmes de diffraction des ondes par un obstacle dans un milieu homogène. Ces problèmes se situent souvent dans un domaine extérieur (infini). Il peut s'agir d'ondes acoustiques, élastiques ou électromagnétiques. Typiquement, on va considérer un domaine dans lequel une onde se propage. La question est de savoir comment se propage cette onde après s'être diffractée sur l'obstacle. La solution vérifie une Equation aux Dérivées Partielles dans le domaine considéré avec des conditions initiales et des conditions aux limites sur le bord. Il existe plusieurs méthodes permettant de résoudre ce problème numériquement qui sont essentiellement: <ul> <li> les méthodes de différences finies/éléments finis</li> <li>les méthodes d'équations intégrales.</li></ul>
• On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications
  
  • Buffa Annalisa
  • Ciarlet Patrick
  Mathematical Methods in the Applied Sciences, Wiley, 2001, 24 (1), pp.31-48. Hedge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L-2 tangential fields and then the attention is focused on some particular Sobolev spaces of order - 1/2. In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Gamma is the boundary of a polyhedron Omega, these spaces are important in the analysis of tangential trace mappings for vector fields in H(curl, Omega) on the whole boundary or on a part of it. By means of,these Hedge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright (C) 2001 John Wiley & Sons, Ltd. (10.1002/1099-1476(20010110)24:1(31::aid-mma193)3.0.co;2-x)
  DOI : 10.1002/1099-1476(20010110)24:1(31::aid-mma193)3.0.co;2-x
• On the analysis of Bérenger's Perfectly Matched Layers for Maxwell's equations
  
  • Bécache Eliane
  • Joly Patrick
  , 2001. In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger \citeber:pml for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last technique allows us to prove the stability of the Yee's scheme for discretizing PML's.
• A generalized mode matching method for scattering problems with unbounded obstacles
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Tillequin Axel
  Journal of Computational Acoustics, World Scientific Publishing, 2001, 9 (4), pp.1611-1631. The two-dimensional time-harmonic acoustic scattering by a semi-infinite waveguide composed of two parallel rigid plates is considered. An original mode matching method is developed to avoid the use of the Wiener-Hopf technique, generalizing usual mode matching methods to the case of unbounded media. A brief description of the method is given by means of Fourier decomposition leading to a well-posed variational problem with unknown being the trace of the solution on a well-chosen interface. From the numerical point of view, a local approximation is first considered by using Lagrange P1 finite elements on a segment of the interface. This leads to the computation of oscillatory integrals involving Fourier transform and complex square root functions. As a matter of accuracy, a special function is added to the finite element space, in order to take into account the asymptotic behavior of the solution. Finally, this method is extended to deal with local perturbations of the media by coupling the previous method to a classical integral one. (10.1142/S0218396X01001005)
  DOI : 10.1142/S0218396X01001005
• Effective boundary conditions for thin ferromagnetic coatings. Asymptotic analysis of the 1D model
  
  • Haddar Houssem
  • Joly Patrick
  Asymptotic Analysis, IOS Press, 2001, 27 (2), pp.127--160.
• A method for computing guided waves in integrated optics. I. Mathematical analysis.
  
  • Pedreira Dolores Gómez
  • Joly Patrick
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2001, 39 (2), pp.596-623. Electromagnetic waveguides in integrated optics are propagation structures which are invariant under translation in one space direction and whose cross section is a local perturbation of a stratified medium. In this paper, we propose a new method for computing the guided modes of such devices under the weak guiding assumption. The method results from a combination of analytical techniques which take into account the unbounded and stratified character of the propagation medium so that numerical computations can be reduced to a neighborhood of the perturbation. Copyright © 2001 Society for Industrial and Applied Mathematics (10.1137/S0036142900368277)
  DOI : 10.1137/S0036142900368277
• An hybrid Approach for the Computation of Guided Modes in Integrated Optics
  
  • Bermudez A.
  • Pedreira Gómez
  • Joly Patrick
  , 2001. In this work, we are interested in the numerical approximation of an eigenvalu- e problem posed in $\Rd$ arising from the computation of guided modes in integrated optics waveguides, which are particular cases of open waveguides- . We consider a stratified waveguide translationally invariant in the infinite propagation direction. Its cross-section is also supposed to be an unbounded and stratified medium where an appropiate perturbation of the refraction index has been introduced to ensure the existence of guided modes. Under the weak guiding assumption, a method to compute the guided modes has been proposed and analyzed in [11]. This method appears as a combination of analytical methods which take into account the unbounded and stratified character of the propagation medium and numerical computations which can be reduced to a neighborhood of the perturbation. In this paper, we are concerned with the numerical implementation of the method and we present some numerical results.
• On a Convolution Operator Arising in a Double Porosity Model
  
  • Alboin Clarisse
  • Jaffré Jérôme
  • Joly Patrick
  • Roberts Jean
  , 2001. Contaminant transport in a fractured porous medium can be modeled, under appropriate conditions, with a double porosity model. Such a model consists of a parabolic equation with a coupling term describing contaminant exchange between the fractures, which have high permeability, and the matrix block, which has low permeability. Interpreting the coupling term as an operator, pseudo-differential in time, we obtain a method for calculating this term which is rapid in comparison with standard discretization methods.