Publications

2002

• Asymptotic analysis of shape functionals
  
  • Nazarov Serguei
  • Sokolowski Jan
  , 2002. A family of boundary value problems is considered in domains $\Omega(\varepsilon)=\Omega\setminus\overline\omega_\varepsilon \subset\mathbb{R}^n$, $n\ge 3$, with cavities $\omega_\varepsilon$ depending on a small parameter $\varepsilon\in (0,\varepsilon_0]$. An approximation ${\mathcal U}(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, of the solution $u(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Hölder spaces, i.e., pointwise estimates are established as well. The asymptotic solution ${\mathcal U}(\varepsilon, x)$ is used in order to derive the first term of the asymptotic expansion with respect to $\varepsilon$ for the shape functional ${\mathcal J}(\Xi(\varepsilon))={\mathbb{J}}_\varepsilon(u) \cong{\mathbb{J}}_\varepsilon({\mathcal U})$. In particular, we obtain the {\it topological derivative} $\cT(x)$ of the shape functional $\cJ(\Xi)$ at a point $x\in\O$. Volume and surface functionals are considered in the paper.
• Can we know how globular clusters form ?
  
  • Perez Jérôme
  , 2002.
• Modélisation numérique de la guitare acoustique.
  
  • Derveaux Grégoire
  , 2002. Le propos de cette étude est la modélisation numérique de la guitare acoustique dans le domaine temporel. La méthode consiste en l'élaboration d'un modèle qui s'attache à décrire les phénomènes vibratoires et acoustiques mis en jeu depuis le pincer de corde jusqu'au rayonnement 3D du son. La corde est modélisée par une équation des ondes amortie 1D. Elle est couplée à la table d'harmonie via le chevalet. Le mouvement de la table est régi par le modèle de plaque mince amortie de Kirchhoff--Love pour un matériau orthotrope et hétérogène, percée d'un trou et encastrée sur son bord externe. Le reste du corps de la guitare (fond, bords, manche...) est supposé rigide. La table rayonne à l'intérieur et à l'extérieur de la cavité. La modélisation complète du champ acoustique rayonné est une approche originale comparativement aux études antérieures portant sur la guitare. On obtient un système d'équations aux dérivées partielles que l'on résout numériquement dans le domaine temporel. On utilise une méthode spectrale spécifique pour la résolution de l'équation de plaque dynamique de Kirchhoff-Love. Pour l'équation de corde et l'équation des ondes acoustiques, on utilise une méthode mixte standard pour l'approximation spatiale et des différences finies centrées en temps. Le problème d'interaction fluide-structure est résolu par une méthode de domaines fictifs qui permet d'approcher finement la géométrie de la guitare tout en utilisant un maillage cubique régulier pour le calcul du champ sonore 3D. L'originalité du schéma de résolution du modèle est un couplage stable entre une méthode de résolution exacte en temps et une méthode discrète. Un nombre important de simulations numériques est réalisées, montrant la validité de la méthode et les très riches potentialités d'une telle approche.
• A non elliptic spectral problem related to the analysis of superconductive micro-strip lines
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ramdani Karim
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2002, 36 (3), pp.461-487. This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivity in modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvalues of A as the dielectric permittivity of the strip goes to -∞. (10.1051/m2an:2002021)
  DOI : 10.1051/m2an:2002021
• A hybrid approach for the computation of guided modes in integrated optics.
  
  • Bermúdez Alfredo
  • Gómez Pedreira Dolores
  • Joly Patrick
  Advances in Computational Mathematics, Springer Verlag, 2002, 16 (2-3), pp.229-261. In this work, we are interested in the numerical approximation of an eigenvalue problem arising from the computation of guided modes in integrated optics waveguides which are particular cases of open waveguides. (10.1023/A:1014454317209)
  DOI : 10.1023/A:1014454317209
• On the analysis of Bérenger's perfectly matched layers for Maxwell's equations
  
  • Bécache Eliane
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2002, 36 (1), pp.87-119. In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] 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. (10.1051/m2an:2002004)
  DOI : 10.1051/m2an:2002004
• Numerical simulation of corner singularities: a paradox in Maxwell-like problems
  
  • Hazard Christophe
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2002, 330 (1), pp.57-68. This paper sums up some recent studies related to the numerical solution of boundary value problems deriving from Maxwell's equations. These studies bring to light the theoretical origins of the 'corner paradox' pointed out by numerical experiments for years: In a domain surrounded by a perfect conductor, a 'nodal' discretization can approximate the electromagnetic field only if the domain has no reentrant corners or edges. The explanation lies in a mathematical curiosity: two different interpretations of the same variational equation, which are both well-posed and lead either to the physical or a spurious solution! Two strategies which were recently proposed to remedy this flaw of nodal elements are described. (10.1016/S1631-0721(02)01425-0)
  DOI : 10.1016/S1631-0721(02)01425-0
• Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings
  
  • Haddar Houssem
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2002, 143 (2), pp.201--236. (10.1016/S0377-0427(01)00508-8)
  DOI : 10.1016/S0377-0427(01)00508-8
• Régularité spatio-temporelle de la solution des équations de Maxwell dans des domaines non-convexes
  
  • Garcia Emmanuelle
  • Labrunie Simon
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2002, 334 (4), pp.293-298. La méthode du complément singulier, développée afin de résoudre les équations de Maxwell dans des domaines non convexes (cf. [5,2] pour des domaines bidimensionnels en absence et en présence de charges, [3] pour des domaines axisymétriques), est basée sur une décomposition orthogonale de l'espace des solutions. Après avoir rappelé les résultats classiques de régularité dans des domaines lipschitziens, nous donnons plusieurs résultats de régularité en espace et en temps de la solution et de ses composantes, qui sont valables dans plusieurs géométries effectivement utilisées en calcul numérique. (10.1016/S1631-073X(02)02221-5)
  DOI : 10.1016/S1631-073X(02)02221-5
• Méthodes asymptotiques pour la propagation des ondes acoustiques dans les milieux comportant des fentes
  
  • Tordeux Sébastien
  , 2002.
• Condition inf-sup pour l'élément fini de Taylor-Hood P2-iso-P1, 3D; application aux équations de Maxwell
  
  • Ciarlet Patrick
  • Girault Vivette
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2002, 335 (10), pp.827-832. On considère la discrétisation des équations de Maxwell, telle qu'elle a été proposée dans [3,2,1]. Les approximations numériques du champ électromagnétique et du multiplicateur de Lagrange associé à la divergence du champ sont réalisées à l'aide de l'élément fini de Taylor-Hood P2-iso-P1, et complétées de fonctions-test singulières, lorsque le domaine de calcul est non convexe, à bord non régulier. Le but de la Note est de prouver l'existence d'une condition inf-sup discrète. On peut également appliquer ce résultat à la discrétisation du système de Stokes en vitesse-pression (10.1016/S1631-073X(02)02564-5)
  DOI : 10.1016/S1631-073X(02)02564-5
• Asymptotical models for wave propagation in media including slots
  
  • Tordeux Sébastien
  , 2002.
• A new family of mixed finite elements for the linear elastodynamic problem
  
  • Bécache Eliane
  • Joly Patrick
  • Tsogka Chrysoula
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2002, 39 (6), pp.2109-2132. We construct and analyze a new family of quadrangular (in two dimensions) or cubic (in three dimensions) mixed finite elements for the approximation of elastic wave equations. Our elements lead to explicit schemes (via mass lumping), after time discretization, including in the case of anisotropic media. Error estimates are given for these new elements. Copyright © 2002 Society for Industrial and Applied Mathematics (10.1137/S0036142999359189)
  DOI : 10.1137/S0036142999359189
• A method for computing guided waves in integrated optics. Part (II) Numerical approximation
  
  • Joly Patrick
  • Pedreira Dolores
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2002, 39 (5), pp.1684-1711. In this paper, we are interested in the numerical approximation of the method for computing guided modes in integrated optics introduced in our earlier paper [Gómez Pedreira and Joly, SIAM J. Numer. Anal., 39 (2001), pp. 596--623]. We saw that the main step of the computation of the guided modes consists of computing the eigenvalues of a compact operator ${\rm K}$ depending on the frequency $\omega$ and the wavenumber $\beta$. This second paper is devoted to the discretization of this problem and to the corresponding convergence analysis. In particular, we derive error estimates in the eigenvalue approximation, which will be of exponential accuracy. Copyright © 2002 Society for Industrial and Applied Mathematics (10.1137/S0036142900377711)
  DOI : 10.1137/S0036142900377711
• On the long-time behavior of unsplit Perfectly Matched Layers
  
  • Bécache Eliane
  • Petropoulos Peter G.
  • Gedney Stephen
  , 2002. Some recent work \cite{jsc} have shown that the «classical» models of Perfectly Matched Layers (PML), typically used as Absorbing Boundary Condition- s in Computational Electromagnetics codes, could lead to long-time linear growth of the solution. We propose here new PML which eliminate this undesirab- le long-time behavior. For these new PML equations, we give energy arguments that show the fields in the layer are bounded by a time-independent constant hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields.
• Acoustic diffraction by a rigid plate in a uniform flow
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Dahi Lynda
  • Lunéville Éric
  • Pagneux Vincent
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2002, 12 (5), pp.625-647. We present a mathematical framework for the problem of the acoustic diffraction by a plate located in a duct, in the presence of a uniform flow. The model is based on a velocity potential representation outside a vortex sheet which is induced by the trailing edge of the plate. Using limiting absorption techniques and singular functions decompositions, we show how to get a theoretical and numerical well-suited formulation. (10.1142/S0218202502001829)
  DOI : 10.1142/S0218202502001829
• On the Validation of the Linear Sampling Method in Electromagnetic Inverse Scattering Problems
  
  • Collino Francis
  • Fares M'B.
  • Haddar Houssem
  , 2002. We present in this study some three dimensional numerical results that validate the use of the Linear Sampling Method as an inverse solver in electromagnetic scattering problems. We recall that this method allows the reconstruction of the shape of an obstacle from the knowledge of multisati- c radar data at a fixed frequency. It does not require any a priori knowledge on the physical properties of the scatterer nor any nonlinear optimization scheme. This study contains also some analytical results in the simplified case of a spherical scatterer that somehow make the link between known abstract theoretical results and the numerical scheme. Special attention has been made to point out the influence of the frequency on the inversion accuracy.
• A fully Discrete Approximation for a Control Problem of Parabolic Variational Inequalities
  
  • Bergounioux Maïtine
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2002, 39 (6), pp.2014-2033. In this work we consider a numerical approximation of an optimal control problem governed by variational inequalities. We use a total discretization scheme: implicit Euler discretization with respect to the time variable and finite element method for the space variable, and we give convergence results. Copyright © 2002 Society for Industrial and Applied Mathematics (10.1137/S0036142900373403)
  DOI : 10.1137/S0036142900373403
• Heat transfer associated to natural convection flow in a partly porous cavity
  
  • Mercier Jean-François
  • Weisman Catherine
  • Firdaouss M
  • Le Quéré Patrick
  Journal of Heat Transfer, American Society of Mechanical Engineers, 2002, 124 (1), pp.130-143. The object of this study is the heat transfer associated to a buoyancy-induced flow developing in a rectangular cavity, partly filled with porous medium, with given heat flux from the sides. Numerical experiments show the existence of a quasi parallel solution away from the top and bottom end walls, characterized by a linear vertical temperature profile. Analytical expressions are derived for the velocity and temperature which, combined with an energy balance, enable us to relate the stratification to the other governing parameters. Influence of the porous layer's permeability and width, and of the Rayleigh number on the flow structure and heat transfer is thoroughly investigated, and summarized in simple relationships (10.1115/1.1418372)
  DOI : 10.1115/1.1418372
• Higher-Order Numerical Methods for Transient Wave Equations
  
  • Cohen Gary
  , 2002. Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given. (10.1007/978-3-662-04823-8)
  DOI : 10.1007/978-3-662-04823-8
• Le transfert inconscient
  
  • Loidreau Pierre
  MISC - Le journal de la sécurité informatique, Lavoisier, 2002, 2.
• A comparison of methods for calculating the matrix block source term in a double porosity model for contaminant transport.
  
  • Alboin Clarisse
  • Jaffré Jérôme
  • Joly Patrick
  • Roberts Jean E.
  • Serres Christophe
  Computational Geosciences, Springer Verlag, 2002, 6 (3-4), pp.523-543. 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 experiments show that this semi-analytic method for the coupling term is accurate and faster than several other methods but at a small expense of computer memory.
• Diffraction by an acoustic grating perturbed by a bounded obstacle
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ramdani Karim
  Advances in Computational Mathematics, Springer Verlag, 2002, 16 (2-3), pp.113-138. An original approach to solve 2D time harmonic diffraction problems involving locally perturbed gratings is proposed. The propagation medium is composed of a periodically stratified half-space and a homogeneous half-space containing a bounded obstacle. Using Fourier and Floquet transforms and integral representations, the diffraction problem is formulated as a coupled problem of Fredholm type with two unknowns: the trace of the diffracted field on the interface separating the two half-spaces on one hand, and the restriction of the diffracted field to a bounded domain surrounding the obstacle, on the other hand. (10.1023/A:1014437813575)
  DOI : 10.1023/A:1014437813575
• A Direct Study in a Hilbert-Schmidt Framework of the Riccati Equation Appearing in a Factorization Method of Second Order Elliptic Boundary Value Problems
  
  • Henry Jacques
  • Ramos Angel M.
  , 2002. In this report we come back to the method of factorization of a second order elliptic boundary value problem presented in . In this paper, it was shown that, in the case of a cylinder, the boundary value problem can be factorized in two uncoupled first order initial value problems. This factorization utilizes the Dirichlet to Neumann operator which satisfies a Riccati equation. Here we consider Hilbert-Schmidt operators, a framework already used by R. Temam which provides tools for a direct study of this Riccati equation.
• L'identification à divulgation nulle de connaissance
  
  • Loidreau Pierre
  MISC - Le journal de la sécurité informatique, Lavoisier, 2002, 1.