Publications

2007

• 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
• 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
• 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
• 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
• 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.
• 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
• Integrability of Bianchi Universes in scalar tensor theory of gravitation
  
  • Perez Jérôme
  • Larena Julien
  Classical and Quantum Gravity, IOP Publishing, 2007, 24 (11), pp.2901. In this paper, we develop a method based on the analysis of the Kovalewski exponents to study the integrability of anisotropic and homogeneous Universes. The formalism is developed in scalar-tensor gravity, the general relativistic case appearing as a special case of this larger framework. Then, depending on the rationality of the Kovalewski exponents, the different models, both in the vacuum and in the presence of a barotropic matter fluid, are classified, and their integrability is discussed. (10.1088/0264-9381/24/11/008)
  DOI : 10.1088/0264-9381/24/11/008
• 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
• Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations
  
  • Bokanowski Olivier
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2007, 30 (1), pp.1-33. We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe's Super-Bee scheme and of the "Ultra-Bee" scheme. We show how to apply these schemes to treat time-dependent first order Hamilton-Jacobi-Bellman equations with discontinuous initial data, possibly infinitely-valued. Numerical tests are proposed, in one and two space dimensions, in order to validate the methods. (10.1007/s10915-005-9017-0)
  DOI : 10.1007/s10915-005-9017-0
• Asymptotic expansion of highly conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  , 2007, 7 issue 1, pp.2040011-2040012. (10.1002/pamm.200700278)
  DOI : 10.1002/pamm.200700278
• 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.
• Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles.
  
  • Claeys Xavier
  • Collino Francis
  , 2007. Dans le contexte de la propagation des ondes electromagnétiques, nous nous intéressons au problème de diffraction par des fils minces parfaitement conducteurs. Si l'on suppose que leur épaisseur est bien plus petite que la longueur d'onde caractéritique de l'onde incidente, il n'est pas posible de prendre en compte des fils minces sans faire face à un problème de verrouillage numérique. Le modèle de Holland, largement utilisé dans les codes différences finis, fournit une solution pragmatique à ce problème, en modifiant le schéma numérique sur quelques noeuds du maillage avoisinant les fils. Jusqu'à présent ce modèle n'a pas re\c cu de justification théorique solide, et il implique un paramètre appelé l'inductance linéique, qu'il doit être choisi suivant des considértions heuristiques. Nous nous intéressons ici au problème modèle de la diffraction acourtique par un petit obstacle, avec condition de Dirichlet au bord, en deux dimensions dans un milieu homogène. Nous présentons et analysons un schéma numérique qui est compatible avec les méthodes éléments finis standards (sans raffinement de maillage) et ne souffre de verrouillage numérique. Ce schéma mélange des techniques d'analyse asymptotique avec une formulation de type domaine fictif. Suivant les résultats que nous démontrons sur ce schéma, nous aboutissons à une généralisation du modèle de Holland et à un calcul automatique de l'inductance linéique. Notre analyse amène, à notre connaissance, à la première justification théorique de ce type de modèle.
• Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries
  
  • Ciarlet Patrick
  • Jamelot Erell
  Journal of Computational Physics, Elsevier, 2007, 226 (1), pp.1122-1135. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D geometries with the help of a continuous approximation of the electromagnetic field. In this paper, we investigate how their framework can be adapted to compute the solution to the time-dependent Maxwell equations. In addition, we propose some extensions, such as the introduction of a mixed variational setting and its discretization, to handle the constraint on the divergence of the field. © 2007 Elsevier Inc. All rights reserved. (10.1016/j.jcp.2007.05.029)
  DOI : 10.1016/j.jcp.2007.05.029