Publications

2005

• Résolution des équations de Maxwell avec des éléments finis de Galerkin continus
  
  • Jamelot Erell
  , 2005. Les equations de Maxwell se resolvent aisement lorsque le domaine d'etude est regulier, mais lorsqu'il existe des singularites geometriques (coins rentrants en 2D, coins et aretes rentrants en 3D), le champ electromagnetique est localement non borne au voisinage de ces singularites. Nous nous interessons a la resolution des equations de Maxwell dans des domaines bornes, singuliers, a l'aide de methodes d'elements finis continus. En pratique, cela permet de modeliser des instruments de telecommunication comme les guides d'onde, les filtres a stubs. Nous analysons tout d'abord le probleme quasi-electrostatique 2D, afin de maitriser la discretisation en espace. Nous presentons trois methodes de calcul (formulations augmentees mixtes) qui donnent des resultats numeriques tres convaincants : - Une version epuree de la methode du complement singulier (conditions aux limites essentielles). - La methode de regularisation a poids : on introduit un poids qui depend des distances aux singularites geometriques (conditions aux limites essentielles). - La methode avec conditions aux limites naturelles. Nous etudions ensuite la generalisation de ces methodes aux domaines 3D. Nous detaillons la resolution des equations de Maxwell instationnaires en domaines singuliers 3D par la methode de regularisation a poids, et nous donnons des resultats numeriques inedits.
• Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées
  
  • Diaz Julien
  , 2005. Dans la première partie nous présentons deux méthodes numériques non conformes espace-temps pour la propagation d'ondes en interaction fluide-structure. Ces méthodes, robustes et précises, sont basées sur deux formulations mixtes dites duale-duale et primale-primale. Elles sont explicites, sauf à l'interface, et conservatives, ce qui en assure la stabilité. Nous les validons à l'aide de solutions analytiques calculées par la méthode de Cagniard-de Hoop (CdH). Dans la deuxième partie nous obtenons, via la méthode CdH, des estimations d'erreur pour l'utilisation de conditions aux limites absorbantes (CLA) ou couches absorbantes parfaitement adaptées (PML) pour la résolution de l'équation des ondes dans le demi-espace. La troisième partie est consacrée aux PMLs pour l'acoustique en écoulement: analyse (par CdH) de l'instabilité des PMLs classiques et construction de PMLs stabilisées. La dernière partie consiste en une présentation mathématique détaillée de la méthode CdH.
• Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques
  
  • Ezziani Abdelaâziz
  , 2005. Nous nous intéressons à la modélisation de la propagation d'ondes dans le sous sol. Nous présentons deux modèles de propagation : (i) une généralisation du modèle de Zener pour les milieux viscoélastiques, (ii) le modèle de Biot pour les milieux poroélastiques. Nous menons une analyse mathématique complète de ces modèles : résultat d'existence, d'unicité et de décroissance de l'énergie. Pour la résolution numérique nous construisons une méthode spécifique à chaque modèle, basée sur des approches variationnelles, une approximation par éléments finis mixtes en espace et différences finies en temps. Nous montrons pour chaque schéma, un résultat de décroissance d'énergie discrète qui conduit à une condition suffisante de stabilité. Pour simuler la propagation d'ondes dans les milieux ouverts, nous adaptons la technique de couches absorbantes parfaitement adaptées aux ondes viscoélastiques et poroélastiques. Enfin, nous présentons des validations numériques des méthodes développées.
• Augmented formulations for solving Maxwell equations
  
  • Ciarlet Patrick
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2005, 194 (2-5), pp.559-586. We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L2 scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation]. (10.1016/j.cma.2004.05.021)
  DOI : 10.1016/j.cma.2004.05.021
• Stabilité et résonances pour le problème du mouvement sur la houle
  
  • Hazard Christophe
  • Lenoir Marc
  , 2005.
• Numerical simulation of a guitar
  
  • Bécache Eliane
  • Chaigne Antoine
  • Derveaux Grégoire
  • Joly Patrick
  Computers & Structures, Elsevier, 2005, 83 (2-3), pp.107-126. The purpose of this study is to present a time-domain numerical modeling of the guitar. The model involves the transverse displacement of the string excited by a force pulse, the flexural motion of the soundboard and the sound radiation in the air. We use a specific spectral method for solving the Kirchhoff-Love's dynamic plate model for orthotropic material, a fictitious domain method for solving the fluid-structure interaction and a conservative scheme for the time discretization. One of the originality of the proposed scheme is a stable coupling method between a continuous time resolution and a discrete one. (10.1016/j.compstruc.2004.04.018)
  DOI : 10.1016/j.compstruc.2004.04.018
• The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
  
  • Ciarlet Patrick
  • Jung Beate
  • Kaddouri Samir
  • Labrunie Simon
  • Zou Jun
  Numerische Mathematik, Springer Verlag, 2005, 101, pp.423-450. This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. (10.1007/s00211-005-0621-6)
  DOI : 10.1007/s00211-005-0621-6
• An efficient numerical method for the resolution of the Kirchhoff-Love dynamic plate equation
  
  • Bécache Eliane
  • Derveaux Grégoire
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2005, 21 (2), pp.323 - 348. We solve numerically the Kirchhoff-Love dynamic plate equation for an anisotropic heterogeneous material using a spectral method. A mixed velocity-moment formulation is proposed for the space approximation allowing the use of classical Lagrange finite elements. The benefit of using high order elements is shown through a numerical dispersion analysis. The system resulting from this spatial discretization is solved analytically. Hence this method is particularly efficient for long duration experiments. This time evolution method is compared with explicit and implicit finite differences schemes in terms of accuracy and computation time. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 (10.1002/num.20041)
  DOI : 10.1002/num.20041
• NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT
  
  • Berriri Kamel
  • Bonnet-Ben Dhia Anne-Sophie
  • Joly Patrick
  , 2005, pp.6. In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized variational formulation of the problem, in the sub- sonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions.
• Matching of asymptotic expansions for the wave propagation in media with thin slot
  
  • Tordeux Sébastien
  • Joly Patrick
  , 2005. This 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.
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2005. 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.
• Realistic numerical modeling of human head tissues exposure to electromagnetic waves from mobiles phones
  
  • Scarella Gilles
  • Clatz Olivier
  • Lanteri Stéphane
  • Beaume Grégory
  • Oudot Steve
  • Pons Jean-Philippe
  • Piperno Serge
  • Joly Patrick
  • Wiart Joe
  , 2005. The ever-rising diffusion of cellular phones has brought about an increased concern for the possible consequences of electromagnetic radiation on human health. Possible thermal effects have been investigated, via experimentation or simulation, by several research projects in the last decade. Concerning numerical modeling, the power absorption in a user's head is generally computed using discretized models built from clinical MRI data. The vast majority of such numerical studies have been conducted using Finite Differences Time Domain methods, although strong limitations of their accuracy are due to heterogeneity, poor definition of the detailed structures of head tissues (staircasing effects), etc. In order to propose numerical modeling using Finite Element or Discontinuous Galerkin Time Domain methods, reliable automated tools for the unstructured discretization of human heads are also needed. Results presented in this article aim at filling the gap between human head MRI images and the accurate numerical modeling of wave propagation in biological tissues and its thermal effects.
• Diffraction of an acoustic wave by a plate in a uniform flow: A numerical approach
  
  • Job Stéphane
  • Lunéville Éric
  • Mercier Jean-François
  Journal of Computational Acoustics, World Scientific Publishing, 2005, 13 (4), pp.689-709. We study the diffraction in time harmonic regime of an acoustic wave by a rigid plate in the presence of a uniform flow in a duct. Contrary to prior analytical studies, using Wiener-Hopf techniques and thus restricted to semi-infinite plates, we use a, finite elements method which allows us to deal with plates of finite length. To take into account irrotational perturbations induced by the trailing edge of the plate, a potential formulation requires the introduction of a vortex sheet behind the plate. The key point of the method is to get access at the singular coefficient of the velocity potential near the trailing edge, in order to cancel it using the so-called Kutta-Joukowski condition. This approach leads to an efficient finite elements method, and numerical computations are presented: we show the amplitude of the vortex sheet versus the Mach number and the plate length and the dissipated acoustic power versus the Mach number and the frequency. This method is extended to the case of two aligned plates to analyze the influence of the choice of the boundary condition on the downstream plate which interacts with a vortex sheet. © IMACS. (10.1142/s0218396x05002840)
  DOI : 10.1142/s0218396x05002840
• A dual-primal coupling technique with local time step for wave propagation problems
  
  • Bécache Eliane
  • Joly Patrick
  • Rodriguez Jeronimo
  , 2005. We are interested in space-time refinement methods for linear wave propagation. In 1,2 , some stable numerical schemes using non-conforming grids in space and time have been proposed. These methods use a Lagrange multiplier to cope with the interface conditions. The choice of the discretization space of this additional unknown can be in some cases not trivial. In the present paper we propose an alternative method. The main new idea is to use different variational formulations in the fine and in the coarse grids. We present a time discretization that leads to the conservation of a discrete energy and provide a complete stability and error analysis in the case where the time step is twice smaller in one domain than in the other one.
• Matching of asymptotic expansions for the wave propagation in media with thin slot
  
  • Tordeux Sébastien
  • Joly Patrick
  , 2005. In this talk we will use the matching of asymptotic expansion to derive new slot models.This models will be mathematically validated via some error estimates.
• On numerical spectral analysis of the hydrodynamic equations
  
  • Hechme Grace
  • Nechepurenko Yuri
  • Sadkane Miloud.
  Russian Journal of Numerical Analysis and Mathematical Modelling, De Gruyter, 2005, 20 (2), pp.115-129. The aim of this paper is to gain insight into the spectral structure of the discrete analog of hydrodynamic equations linearized at a steady state and discuss how to compute its spectral characteristics connected with the main part of the spectrum. (10.1515/1569398054308685)
  DOI : 10.1515/1569398054308685
• Non-stationary elastic wavefields from an apodized normal transducer. Near-field asymptotics and numerics
  
  • Bécache Eliane
  • Kiselev Aleksei
  Acta Acustica united with Acustica, Hirzel Verlag, 2005, 91 (5), pp.822-830. We simulate non-stationary radiating near-field of a normal transducer acting at the surface of an isotropic homogeneous elastic half-space. The transducer is assumed large compared to the characteristic wavelength. Effects of non-constance of distribution of pressure over the aperture of the transducer on the wavefield are considered in detail. These are i) excitation of a plane S-wave, ii) anomalous polarization in the plane P-wave, and iii) suppression of edge waves by an apodization of the pressure distribution. Asymptotic formulas are tested against a numerical method based on new mixed finite elements. The agreement is found excellent within the bounds of the asymptotic theory.
• A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation
  
  • Bourgeois Laurent
  Inverse Problems, IOP Publishing, 2005, 21 (3), pp.1087-1104. This work concerns the use of the method of quasi-reversibility to solve the Cauchy problem for Laplace's equation. We describe a mixed formulation of the method and its relationship with a classical formulation. A discretized formulation using finite elements of class C0 is derived from the mixed formulation, and convergence of the solution of this discretized problem with noisy data to the exact solution is analysed. Finally, a simple numerical example is implemented in order to show the feasibility of the method. © 2005 IOP Publishing Ltd. (10.1088/0266-5611/21/3/018)
  DOI : 10.1088/0266-5611/21/3/018
• Mixed spectral finite elements for the linear elasticity system in unbounded domains
  
  • Cohen Gary
  • Fauqueux Sandrine
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2005, 26 (3), pp.864-884. In this paper, we present a mixed formulation of a spectral element approximation of the linear elasticity system. After studying the main features of this approach, we construct perfectly matched layers (PMLs) for modeling unbounded domains. Then, algorithmic issues are discussed and numerical results are given. Copyright © 2005 Society for Industrial and Applied Mathematics (10.1137/S1064827502407457)
  DOI : 10.1137/S1064827502407457
• Modèles asymptotiques pour la propagation des ondes dans des milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2005, pp.54. Dans ce rapport, nous nous intéressons à la propagation d'ondes acoustiques dans des milieux comportant des fentes minces. Nous proposons un modèle approché permettant de ramener la fente mince à sa surface moyenne et nous menons une analyse détaillée de ce modèle approché (stabilité et estimation d'erreur), dans un cas académique particulier.
• On the use of the reciprocity-gap functional in inverse scattering from planar cracks
  
  • Ben Abda Amel
  • Delbary Fabrice
  • Haddar Houssem
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (10), pp.1553-1574. (10.1142/S0218202505000819)
  DOI : 10.1142/S0218202505000819
• Robust high order non-conforming finite element formulation for time domain fluid-structure interaction
  
  • Diaz Julien
  • Joly Patrick
  Journal of Computational Acoustics, World Scientific Publishing, 2005, 13 (3), pp.403-431. In this paper we present various numerical methods for solving time-dependent fluid-structure interaction problem in two or three dimensions that we claim to be efficient, robust and highly accurate. These methods, based on mixed variational formulations, are explicit and conservative and can be of arbitrary high order in space. Their accuracy will be illustrated via a comparison with analytical solutions in simple configuration. (10.1142/S0218396X05002736)
  DOI : 10.1142/S0218396X05002736
• AN ANALYSIS OF HIGHER ORDER BOUNDARY CONDITIONS FOR THE WAVE EQUATION
  
  • Diaz Julien
  • Joly Patrick
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2005, 65 (5), pp.1547-1575. Thanks to the use of the Cagniard–De Hoop method, we derive an analytic solution in the time domain for the half-space problem associated with the wave equation with Engquist– Majda higher order boundary conditions. This permits us to derive new convergence results when the order of the boundary condition tends to infinity, as well as error estimates. The theory is illustrated by numerical results. (10.1137/S0036139903436145)
  DOI : 10.1137/S0036139903436145
• An Error Analysis of Conservative Space-Time Mesh Refinement Methods for the 1D Wave Equation
  
  • Joly Patrick
  • Rodríguez Jerónimo
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2005, 43, pp.825-859. We study two space-time mesh refinement methods as the one introduced in [F. Collino, T. Fouquet, and P. Joly, Numer. Math., 95 (2003), pp. 197-221].The stability of such methods is guaranteed by construction through the conservation of a discrete energy. In this paper, we show the L 2 convergence of these schemes and provide optimal error estimates. The proof is based on energy techniques and bootstrap arguments. Our results are validated with numerical simulations and compared with results from plane wave analysis [F. Collino, T. Fouquet, and P. Joly, Numer. Math., 95 (2003), pp. 223-251]. (10.1137/040603437)
  DOI : 10.1137/040603437
• High Spatial Order Finite Element Method to Solve Maxwell's Equations in Time Domain
  
  • Pernet Sébastien
  • Ferrieres Xavier
  • Cohen Gary
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2005, 53 (9), pp.2889 - 2899. This paper presents a finite element method with high spatial order for solving the Maxwell equations in the time domain. In the first part, we provide the mathematical background of the method. Then, we discuss the advantages of the new scheme compared to a classical finite-difference time-domain (FDTD) method. Several examples show the advantages of using the new method for different kinds of problems. Comparisons in terms of accuracy and CPU time between this method, the FDTD and the finite-volume time-domain methods are given as well. (10.1109/TAP.2005.856046)
  DOI : 10.1109/TAP.2005.856046