Publications

2005

• On numerical spectral analysis of the hydrodynamic equations
  
  • Hechme Grace
  • Nechepurenko Yuri
  • Sadkane Miloud.
  Russian Journal of Numerical Analysis and Mathematical Modelling, De Gruyter, 2005, 20 (2), pp.115-129. The aim of this paper is to gain insight into the spectral structure of the discrete analog of hydrodynamic equations linearized at a steady state and discuss how to compute its spectral characteristics connected with the main part of the spectrum. (10.1515/1569398054308685)
  DOI : 10.1515/1569398054308685
• 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.
• 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
• An efficient numerical method for the resolution of the Kirchhoff-Love dynamic plate equation
  
  • Bécache Eliane
  • Derveaux Grégoire
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2005, 21 (2), pp.323 - 348. We solve numerically the Kirchhoff-Love dynamic plate equation for an anisotropic heterogeneous material using a spectral method. A mixed velocity-moment formulation is proposed for the space approximation allowing the use of classical Lagrange finite elements. The benefit of using high order elements is shown through a numerical dispersion analysis. The system resulting from this spatial discretization is solved analytically. Hence this method is particularly efficient for long duration experiments. This time evolution method is compared with explicit and implicit finite differences schemes in terms of accuracy and computation time. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 (10.1002/num.20041)
  DOI : 10.1002/num.20041
• Sequences of L-infinity optimal control problems, Gamma-convergence and Hamilton-Jacobi equations,
  
  • Briani Ariela
  Asymptotic Analysis, IOS Press, 2005, 45 (3-4), pp.171-190. We consider the sequence of optimal control problems having as state equation y′(t)=an(t,y)+bn(t,u) (t∈(0,T], y(0)=x) and cost functional Jn(y,u)=esssup\nolimitst Î [0,T]fn(t,y(t),u(t)). We prove a Γ-convergence result and we study the entailed properties on the stability for the related Hamilton-Jacobi equations.
• NUMERICAL ANALYSIS OF TIME-DEPENDENT GALBRUN EQUATION IN AN INFINITE DUCT
  
  • Berriri Kamel
  • Bonnet-Ben Dhia Anne-Sophie
  • Joly Patrick
  , 2005, pp.6. In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized variational formulation of the problem, in the sub- sonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions.
• Numerical simulation of a guitar
  
  • Bécache Eliane
  • Chaigne Antoine
  • Derveaux Grégoire
  • Joly Patrick
  Computers & Structures, Elsevier, 2005, 83 (2-3), pp.107-126. The purpose of this study is to present a time-domain numerical modeling of the guitar. The model involves the transverse displacement of the string excited by a force pulse, the flexural motion of the soundboard and the sound radiation in the air. We use a specific spectral method for solving the Kirchhoff-Love's dynamic plate model for orthotropic material, a fictitious domain method for solving the fluid-structure interaction and a conservative scheme for the time discretization. One of the originality of the proposed scheme is a stable coupling method between a continuous time resolution and a discrete one. (10.1016/j.compstruc.2004.04.018)
  DOI : 10.1016/j.compstruc.2004.04.018
• The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
  
  • Ciarlet Patrick
  • Jung Beate
  • Kaddouri Samir
  • Labrunie Simon
  • Zou Jun
  Numerische Mathematik, Springer Verlag, 2005, 101, pp.423-450. This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. (10.1007/s00211-005-0621-6)
  DOI : 10.1007/s00211-005-0621-6
• 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
• 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
• 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
• 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.
• 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.
• 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
• 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
• 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
• 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
• 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
• 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