Publications

2005

• 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
• 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.
• How to Mask the Structure of Codes for a Cryptographic Use
  
  • Berger Thierry Pierre
  • Loidreau Pierre
  Designs, Codes and Cryptography, Springer Verlag, 2005, 35, pp.63-79.
• 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.
• Monge Solutions for discontinuous Hamiltonians
  
  • Briani Ariela
  • Davini Andrea
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2005, 11 (2), pp.229-251. We consider an Hamilton-Jacobi equation of the form H ( x , D u ) = 0 x ∈ Ω ⊂ ℝ N , ( 1 ) where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed. (10.1051/cocv:2005004)
  DOI : 10.1051/cocv:2005004
• Pour quelques bits d'information
  
  • Loidreau Pierre
  MISC - Le journal de la sécurité informatique, Lavoisier, 2005, 20. A vulgairement parler, nous vivons actuellement dans ce que d'aucuns nomment la société de l'information, qui nous envahit par tous les pores. Nous sommes à l'avènement de l'ère du tout numérique. Les zéros et les uns sont les seigneurs, circulent de ci de là, et ils repasseront par ici s'il ne sont pas déjà passés par-là, dans les communications filaires, par fibre optique, par ondes radio.
• 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
• Space-time mesh refinement for elastodynamics. Numerical results
  
  • Bécache Eliane
  • Joly Patrick
  • Rodríguez Jerónimo
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2005, 194, pp.355-366. We propose here a generalization to the elastodynamic equations of the spacetime mesh refinement technique introduced in [Raffinement de maillage spatio-temporel pour les eŽquations de Maxwell, Ph.D. thesis, Université de Dauphine, Paris, 2000] for the Maxwell's equations. This method uses a discrete energy conservation to ensure the stability. The method is presented in a variational way applicable to other type of hyperbolic systems. Several numerical experiments are provided to show the efficiency of this approach. (10.1016/j.cma.2004.02.023)
  DOI : 10.1016/j.cma.2004.02.023
• Another approach to linearized elasticity and a new proof of Korn's inequality
  
  • Ciarlet Patrick
  • Ciarlet Philippe G.
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (02), pp.259-271. We describe and analyze an approach to the pure traction problem of three-dimensional linearized elasticity, whose novelty consists in considering the linearized strain tensor as the "primary" unknown, instead of the displacement itself as is customary. This approach leads to a well-posed minimization problem, constrained by a weak form of the St Venant compatibility conditions. Interestingly, it also provides a new proof of Korn's inequality. (10.1142/S0218202505000352)
  DOI : 10.1142/S0218202505000352
• Perfectly matched layers for time-harmonic acoustics in the presence of a uniform flow
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  , 2005, pp.35. This paper is devoted to the resolution of the time-harmonic linearized Galbrun's equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting amplitude process, we characterize the outgoing solution radiated by a compactly supported source. Then, we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included.
• Mathematical analysis of the acoustic diffraction by a muffler containing perforated ducts
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Drissi Dora
  • Gmati Nabil
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (7), pp.1059-1090. We consider the three-dimensional scalar problem of acoustic propagation in a muffler. We develop and analyze a Fredholm-type formulation for a stationary fluid in the timeharmonic setting. We prove a homogenization result for a muffler containing periodically perforated ducts. Essentially, the perforated boundaries become completely transparent when the period of perforations, which is assumed to be of the same order as the size of perforations, tends to zero. We also derive a homogenized impedance condition when the perforated duct is coated by an absorbing material. We present numerical examples in two dimensions, obtained from coupling finite elements in the muffler with modal decompositions in the inlet and outlet ducts, which demonstrate the limiting validity of the theoretical results. © World Scientific Publishing Company. (10.1142/s0218202505000649)
  DOI : 10.1142/s0218202505000649
• A globally convergent steering algorithm for regular nonholonomic systems
  
  • Jean Frédéric
  • Oriolo G.
  • Vendittelli M.
  , 2005, pp.7514-7519. (10.1109/CDC.2005.1583374)
  DOI : 10.1109/CDC.2005.1583374
• 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.
• Singular electromagnetic fields: Inductive approach [Singularités électromagnétiques: Une approche inductive]
  
  • Assous Franck
  • Ciarlet Patrick
  • Garcia Emmanuelle
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2005, 341 (10), pp.605-610. In a non-convex polyhedral domain, we describe the local trace (i.e. defined on a face) of the normal derivative of an L2 function, with L2 Laplacian. We then provide generalized integration by parts formulae for the Laplace, divergence and curl operators. Finally, these results allow us to split electromagnetic fields into regular and singular parts, which can be characterized. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. (10.1016/j.crma.2005.09.034)
  DOI : 10.1016/j.crma.2005.09.034
• Generalized Impedance Impedance Boundary Conditions For Scattering by Strongly Absorbing Obstacles: The Scalar Case
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai-Minh
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (08), pp.1273-1300. (10.1142/S021820250500073X)
  DOI : 10.1142/S021820250500073X