Publications

2008

• The linear sampling method in a waveguide: A modal formulation
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2008, 24 (1). This paper concerns the linear sampling method used to retrieve obstacles in a 2D or 3D acoustic waveguide. The classical mathematical results concerning the identifiability of the obstacle and the justification of the inverse method are established for this particular geometry. Our main concern is to derive a modal formulation of the linear sampling method that is well adapted to the waveguide configuration. In particular, thanks to such formulation, we highlight the fact that finding some obstacles from remote scattering data is more delicate in a waveguide than in free space. Indeed, the presence of evanescent modes increases the ill posedness of the inverse problem. However, we show that the numerical reconstruction of obstacles by using the far field is feasible, even by using a few incident waves. © 2008 IOP Publishing Ltd. (10.1088/0266-5611/24/1/015018)
  DOI : 10.1088/0266-5611/24/1/015018
• A new approach for approximating linear elasticity problems
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2008, 346 (5-6), pp.351-356. In this Note, we present and analyze a new method for approximating linear elasticity problems in dimension two or three. This approach directly provides approximate strains, i.e., without simultaneously approximating the displacements, in finite element spaces where the Saint Venant compatibility conditions are exactly satisfied in a weak form. To cite this article: P.G. Ciarlet, P. Ciarlet, Jr., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences. (10.1016/j.crma.2008.01.014)
  DOI : 10.1016/j.crma.2008.01.014
• An improved multimodal approach for non-uniform acoustic waveguides
  
  • Hazard Christophe
  • Lunéville Éric
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2008, 73 (4), pp.668-690. This paper explores from the point of view of numerical analysis a method for solving the acoustic time-harmonic wave equation in a locally non-uniform 2D waveguide. The multimodal method is based on a spectral description of the acoustic field in each transverse section of the guide, using Fourier-like series. In the case of sound-hard boundaries, the weak point of the method lies in the poor convergence of such series due to a poor approximation of the field near the boundaries. A remedy was proposed by Athanassoulis & Belibassakis (1999, J. Fluid Mech., 389, 275-301), where the convergence is improved thanks to the use of enhanced series. The present paper proposes a theoretical analysis of their idea and studies further improvements. For a general model which includes different kinds of non-uniformity (varying cross section, bend and inhomogeneous medium), a hybrid spectral/variational formulation is introduced. Error estimates are provided for a semi-discretized problem which concerns the effect of spectral truncation. These estimates are confirmed by numerical results. © The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. (10.1093/imamat/hxn006)
  DOI : 10.1093/imamat/hxn006
• Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates
  
  • Joly Patrick
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2008, 42 (2), pp.193--221. We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness $\epsilon$ is small with respect to the wavelength. In Part I [P. Joly and S. Tordeux, Multiscale Model. Simul. 5 (2006), no. 1, 304--336 (electronic); MR2221320 (2007e:35041)], we derived formally an asymptotic expansion of the solution with respect to $\epsilon$ using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order. (10.1051/m2an:2008004)
  DOI : 10.1051/m2an:2008004
• Identification of generalized impedance boundary conditions in inverse scattering problems
  
  • Bourgeois Laurent
  • Haddar Houssem
  , 2008, pp.27. In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact set. We also extend local stability results to the case of back-scattering data.
• Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2008. We present a new non conforming space-time mesh refinement method for symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method.
• Higher order time stepping for second order hyperbolic problems and optimal CFL conditions
  
  • Gilbert Jean Charles
  • Joly Patrick
  , 2008, 16, pp.67-93. We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. The corresponding results are quite promising and suggest various conjectures. (10.1007/978-1-4020-8758-5_4)
  DOI : 10.1007/978-1-4020-8758-5_4
• A spurious-free space-time mesh refinement for elastodynamics
  
  • Rodríguez Jerónimo
  International Journal for Multiscale Computational Engineering, Begell House, 2008, 6 (3), pp.263-279. We propose a generalization of the space-time mesh refinement technique for elastodynamics presented by 14 to the case where the discretization step (in space and time) on the fine grid is q N times finer than the one on the coarse grid. This method uses the conservation of a discrete energy to ensure the stability under the usual CFL condition. Some numerical examples show that the method is only first order accurate (and thus suboptimai with respect to the second-order interior scheme we have used) when the ratio of refinement is higher than 2. A Fourier analysis of the computed signals exhibits the presence of high-frequency waves (aliasing phenomena) polluting the fields on the fine grid. Those results provide valuable information with which to build a postprocessing by averaging that removes the spurious phenomena. Finally, we introduce a new numerical scheme, computing the postprocessed solution directly. This method is stable and second-order consistent, regardless of the ratio of refinement. Its performance is shown through a numerical simulation of the diffraction of elastic waves by small cracks. © 2008 by Begell House, Inc. (10.1615/intjmultcompeng.v6.i3.60)
  DOI : 10.1615/intjmultcompeng.v6.i3.60
• Spectral elements for the integral equations of time-harmonic Maxwell problems
  
  • Demaldent Édouard
  • Levadoux David
  • Cohen Gary
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2008, 56 (9), pp.3001-3010. We present a novel high-order method of moments (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, and at the same time it can be interpreted as a point-based scheme. We apply this method to field integral equations (FIEs) to solve time-harmonic electromagnetic scattering problems. Our approach is applied to the first and second Nédélec families of Hdiv conforming elements. It consists in a specific choice of the degrees of freedom (DOF), made in order to allow a fast integral evaluation. In this paper we describe these two sets of DOF and their corresponding quadrature rules. Sample numerical results on FIE confirm the good properties of our schemes: faster convergence rate and cheap matrix calculation. We also present observations on the choice of the discretization method, depending on the FIE selected. © 2008 IEEE. (10.1109/tap.2008.927551)
  DOI : 10.1109/tap.2008.927551
• A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (9), pp.1605-1631. We consider the time-harmonic Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic coefficients. We prove that, in the case of sign-shifting dielectric coefficients, the space of electric fields is compactly embedded in L 2. We build a three-field variational formulation equivalent to Maxwell system for sign-shifting magnetic coefficients and show that, under some suitable conditions, the formulation fits into the coercive plus compact framework. © 2008 World Scientific Publishing Company. (10.1142/s0218202508003145)
  DOI : 10.1142/s0218202508003145
• The linear sampling method in a waveguide: A formulation based on modes
  
  • Bourgeois Laurent
  • Lunéville Éric
  Journal of Physics: Conference Series, IOP Science, 2008, 135 (-), pp.012023. This paper concerns the Linear Sampling Method to retrieve obstacles in a 2D or 3D acoustic waveguide. We derive a modal formulation of the LSM which is suitable for the waveguide configuration. Despite the ill-posedness of the inverse problem is increased owing to the evanescent modes, numerical experiments show good reconstruction of obstacles by using the far field. © 2008 IOP Publishing Ltd. (10.1088/1742-6596/135/1/012023)
  DOI : 10.1088/1742-6596/135/1/012023