Publications

2004

• Nodal finite element methods for Maxwell's equations [Eléments finis nodaux pour les équations de Maxwell]
  
  • Jamelot Erell
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2004, 339 (11), pp.809-814. An original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: E=ER+λxP, where ER∈H1(ω)², λ depends on the data and domain and xP is known explicitly. The same splitting can used for the magnetic field. No cut-off function is needed and improved error estimates are derived. © 2004 Académie des sciences. Publié par Elsevier SAS. Tous droits réservés. (10.1016/j.crma.2004.10.020)
  DOI : 10.1016/j.crma.2004.10.020
• Selective Acoustic Focusing Using Time-Harmonic Reversal Mirrors
  
  • Hazard Christophe
  • Ramdani Karim
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2004, 64 (3), pp.1057-1076. A mathematical study of the focusing properties of acoustic fields obtained by a time-reversal process is presented. The case of time-harmonic waves propagating in a nondissipative medium containing sound-soft obstacles is considered. In this context, the so-called D.O.R.T. method (decomposition of the time-reversal operator in French) was recently proposed to achieve selective focusing by computing the eigenelements of the time-reversal operator. The present paper describes a justification of this technique in the framework of the far field model, i.e., for an ideal time-reversal mirror able to reverse the far field of a scattered wave. Both cases of closed and open mirrors, that is, surrounding completely or partially the scatterers, are dealt with. Selective focusing properties are established by an asymptotic analysis for small and distant obstacles. (10.1137/S0036139903428732)
  DOI : 10.1137/S0036139903428732
• Scattering of an elastic wave by a single dislocation
  
  • Maurel Agnès
  • Mercier Jean-François
  • Lund Fernando
  Journal of the Acoustical Society of America, Acoustical Society of America, 2004, 115 (6), pp.2773-2780. The scattering amplitude for the scattering of anti-plane shear waves by screw dislocations, and of in-plane shear and acoustic waves by edge dislocations are computed within the framework of elasticity theory. The former case reproduces well-known results obtained on the basis of an electromagnetic analogy. The latter case involves four scattering amplitudes in order to fully take into account mode conversion, and an adequately generalized optical theorem for vector waves is provided. In contrast to what happens for scattering by obstacles, the scattering amplitude increases with wavelength, and, in general, mode conversion in the forward direction does not vanish. (10.1121/1.1687735)
  DOI : 10.1121/1.1687735
• Solving Maxwell equations in 3D prismatic domains [Résolution des équations de Maxwell dans des domaines prismatiques tridimensionnels]
  
  • Ciarlet Patrick
  • Garcia Emmanuelle
  • Zou Jun
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2004, 339 (10), pp.721-726. In this Note, we introduce the Fourier Singular Complement Method, for solving Maxwell equations in a 3D prismatic domain. The numerical implementation of this method provides a continuous approximation of the electromagnetic field. It can be applied to the computation of propagating and evanescent modes in prismatic stub filters, thus generalizing 2D approaches. © 2004 Académie des Sciences. Published by Elsevier SAS. All rights reserved. (10.1016/j.crma.2004.09.032)
  DOI : 10.1016/j.crma.2004.09.032
• Mathematical modeling for acoustic waves in media including thin slot
  
  • Tordeux Sébastien
  , 2004.
• Nonhomogeneous nilpotent approximations for systems with singularities
  
  • Vendittelli Marilena
  • Oriolo Giuseppe
  • Jean Frédéric
  • Laumond Jean-Paul
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2004, 49 (2), pp.261-266. Nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability, such as nonholonomic mechanisms. However, conventional homogeneous approximations exhibit a drawback: in the neighborhood of singular points (where the system growth vector is not constant) the vector fields of the approximate dynamics do not vary continuously with the approximation point. The geometric counterpart of this situation is that the sub-Riemannian distance estimate provided by the classical Ball-Box Theorem is not uniform at singular points. With reference to a specific family of driftless systems, we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points. It is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance. (10.1109/TAC.2003.822872)
  DOI : 10.1109/TAC.2003.822872
• On the Fortet-Mourier metric for the stability of Stochastic Optimization Problems, an example
  
  • Strugarek Cyrille
  Stochastic Programming E-Print Series (SPEPS), 2004, 2004 (25). #HTML We consider the use of the Fortet-Mourier metric between two probability measures to bound the error term made by an approximated solution of a stochastic program. After a short analysis of usual stability arguments, we propose a simple example of stochastic program which enlightens the importance of the information structure. As a conclusion, we underline the need to take into account both the probability measure and the information structure in the discretization of a stochastic program.
• Elastic wave propagation through a random array of dislocations
  
  • Maurel Agnès
  • Mercier Jean-François
  • Lund Fernando
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2004, 70, pp.024303. A number of unsolved issues in materials physics suggest there is a need for an improved quantitative understanding of the interaction between acoustic (more generally, elastic) waves and dislocations. In this paper we study the coherent propagation of elastic waves through a two dimensional solid filled with randomly placed dislocations, both edge and screw, in a multiple scattering formalism. Wavelengths are supposed to be large compared to a Burgers vector and dislocation density is supposed to be small, in a sense made precise in the body of the paper. Consequently, the basic mechanism for the scattering of an elastic wave by a line defect is quite simple ("fluttering"): An elastic wave will hit each individual dislocation, causing it to oscillate in response. The ensuing oscillatory motion will generate outgoing (from the dislocation position) elastic waves. When many dislocations are present, the resulting wave behavior can be quite involved because of multiple scattering. However, under some circumstances, there may exist a coherent wave propagating with an effective wave velocity, its amplitude being attenuated because of the energy scattered away from the direction of propagation. The present study concerns the determination of the coherent wavenumber of an elastic wave propagating through an elastic medium filled with randomly placed dislocations. The real part of the coherent wavenumber gives the effective wave velocity and its imaginary part gives the attenuation length (or elastic mean free path). The calculation is performed perturbatively, using a wave equation for the particle velocity with a right hand side term, valid both in two and three dimensions, that accounts for the dislocation motion when forced by an external stress. In two dimensions, the motion of a dislocation is that of a massive particle driven by the incident wave; both screw and edge dislocations are considered. The effective velocity of the coherent wave appears at first order in perturbation theory, while the attenuation length appears at second order. (10.1103/PhysRevB.70.024303)
  DOI : 10.1103/PhysRevB.70.024303
• Mathematical justification of simplified models for acoustics wave in media including thin slots
  
  • Tordeux Sébastien
  , 2004.
• On the long-time behavior of unsplit Perfectly Matched Layers
  
  • Bécache Eliane
  • Petropoulos Peter
  • Gedney Stephen
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2004, 52 (5). This paper shows how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit perfectly matched layers (PML) typically used as absorbing boundary conditions in computational electromagnetics codes. For the 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. (10.1109/TAP.2004.827253)
  DOI : 10.1109/TAP.2004.827253
• Perfectly matched layers for the convected Helmholtz equation
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2004, 42 (1), pp.409-433. In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Bérenger's perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for time-dependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented. © 2004 Society for Industrial and Applied Mathematics. (10.1137/s0036142903420984)
  DOI : 10.1137/s0036142903420984
• Mathematical justification of simplified models for acoustics wave in media including thin slots
  
  • Tordeux Sébastien
  , 2004.
• Sur la reconstruction des polynômes linéaires : un nouvel algorithme de décodage des codes de Gabidulin
  
  • Loidreau Pierre
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2004.
• Attacks on Public-Key Cryptosystems Based on Free Partially Commutative Monoids and Groups
  
  • Levy-Dit-Vehel Françoise
  • Perret Ludovic
  Progress in Cryptology - INDOCRYPT 2004, 2004, 3348, pp.275-289. At indocrypt 2003, Abisha, Thomas and Subramanian have proposed a public key encryption scheme and a zero-knowledge authentication protocol based on the word problem on monoids, as well as a group variant of these systems. We here present a total break attack on each of the two encryption schemes. The complexity bounds of our algorithms show that these schemes are insecure for practical parameter sizes. In the monoid setting, we go one step further by proposing an algorithm that breaks the NP-hard problem underlying both the encryption scheme and the zero-knowledge protocol, as well as an upper bound on its complexity. (10.1007/978-3-540-30556-9_22)
  DOI : 10.1007/978-3-540-30556-9_22