Publications

2007

• Generalized eigenfunction expansions for conservative scattering problems with an application to water waves
  
  • Hazard Christophe
  • Loret François
  Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, Royal Society of Edinburgh, 2007, 137 (5), pp.995-1035. This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem. This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation. © 2007 The Royal Society of Edinburgh.
• Condition aux limites transparente pour la propagation acoustique dans un guide recouvert d'un matériau absorbant en présence d'un écoulement uniforme
  
  • Redon Emmanuel
  • Poernomo Sari S.
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  • Chambeyron Colin
  , 2007. L'objet de notre travail est de calculer par éléments finis le rayonnement acoustique dans un guide d'onde infini dont une paroi est recouverte d'un matériau absorbant, caractérisé par une impédance Z. Pour tronquer le domaine de calcul, on veut introduire des conditions aux limites transparentes déduites d'une décomposition modale, ce qui nécessite la détermination des modes du guide. En l'absence d'écoulement, la décomposition est aisée pour un guide à parois rigides. En présence d'un matériau absorbant, de nombreuses difficultés apparaissent même en l'absence d'écoulement : l'opérateur n'est plus auto-adjoint et les modes ne sont plus orthogonaux pour le produit scalaire usuel. On montre qu'on peut introduire un « produit scalaire particulier » qui conduit à une « pseudo orthogonalité des modes ». Toutefois, on montre qu'il existe une suite de valeurs « exceptionnelles » de l'impédance Zc pour lesquelles la décomposition modale n'est pas possible. La méthode est étendue au cas d'un écoulement uniforme. L'apparition de dérivées tangentielles de la pression sur la paroi traitée entraîne des difficultés supplémentaires. Un nouveau produit scalaire est introduit, incluant les valeurs de la pression sur la paroi traitée. On montre alors que les modes ne deviennent orthogonaux que si l'un des modes est d'indice élevé et que cette propriété suffit pour définir des conditions transparentes. Dans tous les cas, en dehors des valeurs critiques de l'impédance, on se ramène à la résolution d'une équation de dispersion complexe. La méthode de Newton Raphson est utilisée pour l'identification des modes dans le plan complexe. Nous avons développé une méthode numérique pour calculer les modes ou le rayonnement d'une source, basée sur le couplage entre des éléments finis et les conditions transparentes. Les simulations sont en bon accord, soit avec les résultats analytiques pour les modes, soit avec la pression rayonnée obtenue par une méthode alternative : en entourant le domaine de calcul par des couches absorbantes de type PML (Perfectly Matched Layers).
• Etude théorique et numérique de processus de retournement temporel.
  
  • Ben-Amar Chokri
  , 2007. La possibilité de focaliser l'énergie acoustique ou électromagnétique à la fois spatialement et temporellement constitue depuis quelques années un sujet de recherche intéressant en raison de la diversité et de l'importance des applications. Dans le domaine des télécommunications, en particulier en acoustique sous-marine, en téléphonie mobile et dans le domaine des réseaux locaux sans l, l'un des enjeux majeurs consiste à trouver des techniques sécurisées pour acheminer la plus grande quantité d'information possible entre un émetteur et différent utilisateur via l'utilisation d'ondes acoustiques ou électromagnétiques. Ces ondes sont susceptibles d'être réfléchies plusieurs fois entre l'émetteur et le récepteur compte tenu de la multitude des obstacles rencontres au cours de la propagation dans les milieux concernés. Ces obstacles pouvant être les parois des immeubles, les voitures ou les meubles dans le cas de téléphonie mobile ou bien les interfaces entre l'eau et l'air ou entre l'eau et le fond marin dans le cas de l'acoustique sous-marine. L'une des questions que se posent les chercheurs dans ce domaine, est de déterminer sous quelles conditions cette diffusion multiple peut influer positivement sur la concentration d'énergie au niveau du récepteur. D'autres applications concernent le domaine de l'imagerie et de la thérapie médicale. Il s'agit par exemple de visualiser certains organes ou tissus du corps humain au moyen d'ondes ultrasonores (fréquence allant de 1 MHz µ a 10 MHz). On cherche également à détecter, localiser et détruire les calculs rénaux et les tumeurs du sein ou du cerveau par concentration des ondes ultrasonores d'un fa» con non invasive. Le champ d'applications s'étend au contrôle non destructif des matériaux, on s'intéresse à l'observation des irrégularités dans les métaux ou encore a la géophysique pour la recherche pétrolière. Nous nous proposons dans cette thèse de donner différents modèles de retournement temporel et d'étudier et de simuler leurs propriétés de focalisation. Apres avoir dans un premier temps explique le principe général du retournement temporel et différentes approches de cette technique, nous décrivons plus précisément le contexte particulier de notre étude.
• Rayonnement acoustique dans un écoulement cisaille : une méthode d'éléments finis pour la simulation du régime harmonique.
  
  • Duclairoir Eve-Marie
  , 2007. Les travaux de cette thèse concernent le rayonnement acoustique d'une source périodique en temps placée dans un conduit Infini, contenant un guide en écoulement parallèle cisaille. Le phénomène est modélise a l'aide de l'équation de Galbrun, dont l'inconnue u est la perturbation de déplacement. L'objectif de cette étude est de développer une méthode éléments Nis, susceptible d'être étendue à des géométries et des écoulements plus complexes. Cette thèse fait suite a celle de Guillaume Legendre qui a établi, dans le cas d'un ecoulement uniforme, une formulation dite régularisée de l'´equation de Galbrun afin de corriger un défaut d'ellipticité. Le but de ce manuscrit est détendre cette méthode à un écoulement non uniforme. La difficulte supplémentaire vient du fait que la vorticite ψ = rot u (qui intervient dans le terme de régularisation) ne peut plus être calculée a priori car le cisaillement induit un couplage entre acoustique et hydrodynamique. En régime dissipatif, nous avons explicite ψ en fonction de u à l'aide d'une convolution (le long des lignes de courant). Si l'ecoulement est lent, cette formule de convolution (qui devient une intégrale très oscillante) peut être approchée par une formule differentielle beaucoup plus simple dont l'utilisation conduit a un modèle ”faible Mach”. Des idées similaires ont ensuite été utilisées pour résoudre le problème non dissipatif, a l'aide de couches PML. Les deux approches (exacte et ”faible Mach”) ont été validées par des tests numériques en 2D et en 3D.
• Calcul de champs électromagnétiques et de répartition de charges surfaciques dans des domaines quasi-singulier.
  
  • Kaddouri Samir
  , 2007. La première partie de ce mémoire est consacrée à la résolution numérique du problème de Poisson avec conditions aux limites de Dirichlet dans un domaine prismatique ou axisymétrique, possédant une arête rentrante sur sa frontière. Nous présentons la Méthode de Fourier et du Complément Singulier consistant à combiner un développement en série (de Fourier) dans la direction parallèle à l'arête et la Méthode du Complément Singulier pour les problèmes bidimensionnels associés aux modes (de Fourier). L'analyse de la MFCS conduit à une vitesse de convergence optimale en O(h) lorsqu'on utilise les éléments finis de Lagrange P1 pour la discrétisation. La méthode ne requiert aucun raffinement de maillage au voisinage de la singularité. Nous nous intéressons ensuite au calcul de la densité de charge à la pointe d'une électrode lorsque celle-ci présente un faible rayon de courbureque nous abordons par la résolution du problème électrostatique. La relation entre le rayon de courbure et le champ électrique à la surface de la pointe est décrit par la loi empirique de Peek. Toutefois, celle-ci n'est valable que pour des électrodes minces à géométrie cylindriques ou sphériques. On justifie mathématiquement cette loi et on l'étend à d'autres géométries. A l'aide des développements asymptotiques multi-échelles, on établit explicitement le comportement de la densité de charge pour des géométries coincidant avec un cône à l'infini. Enfin, nous illustrons ce comportement asymptotique par des expériences numériques réalisées en dimension deux, et en dimension trois, pour des domaines axisymétriques. Les résultats sont comparés à ceux obtenue par une méthode intégrale.
• Resonances of an elastic plate in a compressible confined fluid
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2007, 60 (4), pp.397-421. We present a theoretical study of the resonances of a fluid-structure problem, an elastic plate placed in a duct in the presence of a compressible fluid. The case of a rigid plate has been largely studied. Acoustic resonances are then associated to resonant modes trapped by the plate. Due to the elasticity of the plate, we need to solve a quadratic eigenvalue problem in which the resonance frequencies k solve the equations γ(k) = k2, where γ are the eigenvalues of a self-adjoint operator of the form A + kB. First, we show how to study the eigenvalues located below the essential spectrum by using the min-max principle. Then, we study the fixed-point equations. We establish sufficient conditions on the characteristics of the plate and of the fluid to ensure the existence of resonances. Such conditions are validated numerically. © The author 2007. Published by Oxford University Press; all rights reserved. (10.1093/qjmam/hbm015)
  DOI : 10.1093/qjmam/hbm015
• Numerical Simulation of Acoustic Time Reversal Mirrors
  
  • Ben-Amar Chokri
  • Gmati Nabil
  • Hazard Christophe
  • Ramdani Karim
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2007, 67 (3), pp.777-791. We study the time reversal phenomenon in a homogeneous and non-dissipative medium containing sound-hard obstacles. We propose two mathematical models of time reversal mirrors in the frequency domain. The first one takes into account the interactions between the mirror and the obstacles. The second one provides an approximation of these interactions. We prove, in both cases, that the time reversal operator $T$ is selfadjoint and compact. The D.O.R.T method (french acronym for Decomposition of the Time Reversal Operator) is explored numerically. In particular, we show that selective focusing, which is known to occur for small and distant enough scatterers, holds when the wavelength and the size of these scatterers are of the same order of magnitude (medium frequency situation). Moreover, we present the behaviour of the eigenvalues of $T$ according to the frequency and we show their oscillations due to the interactions between the mirror and the obstacles and between the obstacles themselves. (10.1137/060654542)
  DOI : 10.1137/060654542
• Conservative coupling between finite elements and retarded potentials. Application to vibroacoustics
  
  • Grob Pascal
  • Joly Patrick
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2007, 29 (3), pp.1127-1159. The numerical simulation of vibroacoustics is concerned with the radiation of sound emitted by thin vibrating mechanical structures. We present a numerical method that combines boundary elements (integral equation) for the acoustic wave equation with standard finite elements for the mechanics. The originality of our work is that we consider the time domain problem and use retarded potentials for writing the integral equation. We establish a nonstandard variational formulation of this new problem, in space-time for the acoustic equation, and in space only for the mechanic equation. The basic ideas for the discretization are the following: (i) Space finite elements and finite differences with time step $\Delta t$ (a $\theta$-scheme, $ 0 \leq \theta \leq 1/2$) are used for the discretization of the mechanic equation. (ii) Space-time finite elements are used for the discretization of the acoustic equation, which is "projected" two times on two staggered time grids of time step $2 \Delta t$. The use of staggered twice larger time grids for the discretization of the acoustic equation (see (ii) above) plays a key role in the cancellation of the "coupling terms" (between the two equations), which is crucial in the energy analysis. Copyright © 2007 Society for Industrial and Applied Mathematics (10.1137/050647141)
  DOI : 10.1137/050647141
• Mixed Higher Order Spectral Finite Elements for Reissner-Mindlin Equations in the Time Domain
  
  • Cohen Gary
  • Grob Pascal
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2007, 29 (3), pp.986-1005. We construct a class of high order numerical approximations for the Reissner-Mindlin plate model in the time domain, based on mixed spectral finite elements with mass lumping. In this way we obtain explicit time-stepping schemes. We first compare the Reissner-Mindlin model to three-dimensional (3D) solutions to validate our method. Then, we show the advantages of the schemes in terms of accuracy and computational time. (10.1137/050642332)
  DOI : 10.1137/050642332
• Well-posedness of the Drude-Born-Fedorov model for chiral media
  
  • Ciarlet Patrick
  • Legendre Guillaume
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (3), pp.461-484. We consider a chiral medium in a bounded domain, enclosed in a perfectly conducting material. We solve the transient Maxwell equations in this domain, when the medium is modeled by the Drude-Born-Fedorov constitutive equations. The input data is located on the boundary, in the form of given surface current and surface charge densities. It is proved that, except for a countable set of chirality admittance values, the problem is mathematically well-posed. This result holds for domains with non-smooth boundaries. © World Scientific Publishing Company. (10.1142/s0218202507001991)
  DOI : 10.1142/s0218202507001991
• A fictitious domain method with mixed finite elements for elastodynamics
  
  • Bécache Eliane
  • Rodríguez Jerónimo
  • Tsogka Chrysoula
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2007, 29 (3), pp.1244-1267. We consider in this paper the wave scattering problem by an object with Neumann boundary conditions in an anisotropic elastic body. To obtain an efficient numerical method (permitting the use of regular grids) we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics. We first observe that the method does not always converge when the Qdiv1 - Q0 finite element is used. In particular, the method converges for some scattering object geometries but not for others. Note that the convergence of the Qdiv1 - Q0 finite element method was shown in [E. Bécache, P. Joly, and C. Tsogka, SIAM J. Numer. Anal., 39 (2002), pp. 2109-2132] for the elastodynamic problem in the absence of a scattering object (i.e., without the coupling of the mixed finite elements with the fictitious domain method). Therefore we propose here a modification of the Q div1 - Q0 element following the approach in [E. Bécache, J. Rodríguez, and C. Tsogka, On the convergence of the fictitious domain method for wave equation problems, Technical report 5802, INRIA, 2006], where the simpler acoustic case was considered. To study the numerical properties of the new element we carry out a dispersion analysis. Several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution. © 2007 Society for Industrial and Applied Mathematics. (10.1137/060655821)
  DOI : 10.1137/060655821
• Characterization of the kernel of the operator CURL CURL
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  • Geymonat Giuseppe
  • Krasucki Françoise
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 344 (série I), pp.305-308. In a simply-connected domain Ω in R3, the kernel of the operator CURLCURL acting on symmetric matrix fields from L2s (Ω) to H−2 s (Ω) coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in L2s (Ω). To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. (10.1016/j.crma.2007.01.001)
  DOI : 10.1016/j.crma.2007.01.001
• Scattering from infinite rough tubular surfaces
  
  • Claeys Xavier
  • Haddar Houssem
  Mathematical Methods in the Applied Sciences, Wiley, 2007, 30 (4), pp.389--414. (10.1002/mma.787)
  DOI : 10.1002/mma.787
• On the numerical solution of the heat equation I: Fast solvers in free space
  
  • Li Jing-Rebecca
  • Greengard Leslie
  Journal of Computational Physics, Elsevier, 2007, 226 (2), pp.1891--1901. We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain. It relies on a recently developed spectral approximation of the free-space heat kernel coupled with the non-uniform fast Fourier transform. Unlike finite difference and finite element techniques, there is no need for artificial boundary conditions on a finite computational domain. The method is explicit, unconditionally stable, and requires an amount of work of the order O(NMlogN), where N is the number of discretization points in physical space and M is the number of time steps. We refer to the approach as the fast recursive marching (FRM) method. (10.1016/j.jcp.2007.06.021)
  DOI : 10.1016/j.jcp.2007.06.021
• Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai Minh
  , 2007, pp.62. This report is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) used in electromagnetic scattering problems from imperfect conductors as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth.
• Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2 SPEC. ISS.), pp.408-417. We consider a simplified scalar model problem related to Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic constants. We build two variational formulations equivalent to the model problem. We show that, under some suitable conditions, both formulations are well-posed since they fit into the coercive plus compact framework. Advantages over previous studies is the validity of the formulations in the general case of Lipschitz interface between the two media and L∞ dielectric and magnetic constants. An interesting feature of these formulations is that they allow a simple finite element numerical implementation. © 2006 Elsevier B.V. All rights reserved. (10.1016/j.cam.2006.01.046)
  DOI : 10.1016/j.cam.2006.01.046
• Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners
  
  • Ciarlet Patrick
  • Kaddouri Samir
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (6), pp.845-876. We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments. © World Scientific Publishing Company. (10.1142/s0218202507002133)
  DOI : 10.1142/s0218202507002133
• Non-Spurious Spectral Like Element Methods for Maxwell's equations
  
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2007, pp.282-304. In this paper, we give the state of the art for the so called "mixed spectral elements" for Maxwell's equations. Several families of elements, such as edge elements and discontinuous Galerkin methods (DGM) are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is a novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.
• A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners
  
  • Bourgeois Laurent
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 345 (7), pp.385-390. We prove in this Note a stability estimate for ill-posed elliptic Cauchy problems in a domain with corners. This result completes an earlier result obtained for a smooth domain. To cite this article: L. Bourgeois, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences. (10.1016/j.crma.2007.09.014)
  DOI : 10.1016/j.crma.2007.09.014
• Locating an obstacle in a 3D finite depth ocean using the convex scattering support
  
  • Bourgeois Laurent
  • Chambeyron Colin
  • Kusiak Steven
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2 SPEC. ISS.), pp.387-399. We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525-1548]. A few examples are presented to show the feasibility of the method. © 2006 Elsevier B.V. All rights reserved. (10.1016/j.cam.2006.01.045)
  DOI : 10.1016/j.cam.2006.01.045
• The singularity expansion method applied to the transient motions of a floating elastic plate
  
  • Hazard Christophe
  • Loret François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2007, 41 (5), pp.925-943. In this paper we propose an original approach for the simulation of the time-dependent response of a floating elastic plate using the so-called Singularity Expansion Method. This method consists in computing an asymptotic behaviour for large time obtained by means of the Laplace transform by using the analytic continuation of the resolvent of the problem. This leads to represent the solution as the sum of a discrete superposition of exponentially damped oscillating motions associated to the poles of the analytic continuation called resonances of the system, and a low frequency component associated to a branch point at frequency zero. We present the mathematical analysis of this method for the two-dimensional sea-keeping problem of a thin elastic plate (ice floe, floating runway, ...) and provide some numerical results to illustrate and discuss its efficiency. © EDP Sciences, SMAI 2007. (10.1051/m2an:2007040)
  DOI : 10.1051/m2an:2007040
• Convergence analysis of the Jacobi-Davidson method applied to a generalized eigenproblem
  
  • Hechme Grace
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 345 (5), pp.293-296. In this Note we consider the Jacobi-Davidson method applied to a nonsymmetric generalized eigenproblem. We analyze the convergence behavior of the method when the linear systems involved, known as the correction equations, are solved approximately. Our analysis also exhibits quadratic convergence when the corrections are solved exactly. To cite this article: G. Hechme, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences. (10.1016/j.crma.2007.07.003)
  DOI : 10.1016/j.crma.2007.07.003
• Spectral theory for an elastic thin plate floating on water of finite depth
  
  • Hazard Christophe
  • Meylan Michael H.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2007, 68 (3), pp.629-647. The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass. © 2007 Society for Industrial and Applied Mathematics. (10.1137/060665208)
  DOI : 10.1137/060665208
• Generalized formulations of Maxwell's equations for numerical Vlasov-Maxwell simulations
  
  • Ciarlet Patrick
  • Barthelmé Régine
  • Sonnendrücker Eric
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (5), pp.657--680. (10.1142/S0218202507002066)
  DOI : 10.1142/S0218202507002066
• Asymptotic analysis for the solution to the Helmholtz problem in the exterior of a finite thin straight wire
  
  • Claeys Xavier
  , 2007. In this document we are interested in the solution of the Helmholtz equation with Dirichlet boundary condition in the exterior of a thin elongated body. We suppose that the geometry is well described in ellipsoidal coordinates. We propose an asymptotic analysis of this problem, using matched expansions. This leads to the construction of an approximate field with more explicit expression. The approximate field is composed of the first terms of the asymptotic expansion of the exact solution. Our study also leads to a validation of an acoustic version of the Pocklington's equation.