Publications

2012

• An elementary introduction to the construction and the analysis of Perfectly Matched Layers for time domain wave propagation
  
  • Joly Patrick
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2012, 57, pp.5-48.
• On the use of sampling methods to identify cracks in acoustic waveguides
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2012, 28 (10), pp.105011.1-105011.18. We consider the identification of cracks in an acoustic 2D/3D waveguide with the help of sampling methods such as the linear sampling method or the factorization method. A modal version of these sampling methods is used. Our paper emphasizes the fact that if one a priori knows the type of boundary condition which actually applies on the crack, then we shall adapt the formulation of our sampling method to such boundary conditions in order to improve the efficiency of the method. The need for such adaptation is proved theoretically and illustrated numerically with the help of 2D examples. We also show by using our modal formulation that the factorization method is applicable in a waveguide with the same data as the linear sampling method. © 2012 IOP Publishing Ltd. (10.1088/0266-5611/28/10/105011)
  DOI : 10.1088/0266-5611/28/10/105011
• Propagation of guided waves through weak penetrable scatterers
  
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Acoustical Society of America, Acoustical Society of America, 2012, 131 (3), pp.1874-1889. The scattering of a scalar wave propagating in a waveguide containing weak penetrable scatterers is inspected in the Born approximation. The scatterers are of arbitrary shape and present a contrast both in density and in wavespeed (or bulk modulus), a situation that can be translated in the context of SH waves, water waves, or transverse electric/transverse magnetic polarized electromagnetic waves. For small size inclusions compared to the waveguide height, analytical expressions of the transmission and reflection coefficients are derived, and compared to results of direct numerical simulations. The cases of periodically and randomly distributed inclusions are considered in more detail, and compared with unbounded propagation through inclusions. Comparisons with previous results valid in the low frequency regime are proposed. © 2012 Acoustical Society of America. (10.1121/1.3682037)
  DOI : 10.1121/1.3682037
• T-coercivity for scalar interface problems between dielectrics and metamaterials
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2012, 46, pp.1363-1387. Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of R^d, with d=2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria. (10.1051/m2an/2012006)
  DOI : 10.1051/m2an/2012006
• Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section
  
  • Imperiale Sébastien
  • Joly Patrick
  Advances in Applied Mechanics, New York ; London ; Paris [etc] : Academic Press, 2012, xx. This paper is the first contribution towards the rigorous justification of asymptotic 1D models for the time-domain simulation of the propagation of electromagnetic waves in coaxial cables. Our general objective is to derive error estimates between the "exact" solution of the full 3D model and the "approximate" solution of the 1D model known as the Telegraphist's equation. (10.4208/aamm.12-12S06)
  DOI : 10.4208/aamm.12-12S06
• Perfectly Matched Layer with Mixed Spectral Elements for the Propagation of Linearized Water Waves
  
  • Cohen Gary
  • Imperiale Sébastien
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.285-302. After setting a mixed formulation for the propagation of linearized water waves problem, we define its spectral element approximation. Then, in order to take into account unbounded domains, we construct absorbing perfectly matched layer for the problem. We approximate these perfectly matched layer by mixed spectral elements and show their stability using the 'frozen coefficient' technique. Finally, numerical results will prove the efficiency of the perfectly matched layer compared to classical absorbing boundary conditions. (10.4208/cicp.201109.261110s)
  DOI : 10.4208/cicp.201109.261110s
• Approximate Models for Wave Propagation Across Thin Periodic Interfaces
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 98 (1), pp.28-71. This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy. (10.1016/j.matpur.2012.01.003)
  DOI : 10.1016/j.matpur.2012.01.003
• A preconditioned 3-D multi-region fast multipole solver for seismic wave propagation in complex geometries
  
  • Chaillat Stéphanie
  • Semblat Jean-François
  • Bonnet Marc
  Communications in Computational Physics, Global Science Press, 2012, 11, pp.594-609. The analysis of seismic wave propagation and amplification in complex geological structures requires efficient numerical methods. In this article, following up on recent studies devoted to the formulation, implementation and evaluation of 3-D single- and multi-region elastodynamic fast multipole boundary element methods (FM-BEMs), a simple preconditioning strategy is proposed. Its efficiency is demonstrated on both the single- and multi-region versions using benchmark examples (scattering of plane waves by canyons and basins). Finally, the preconditioned FM-BEM is applied to the scattering of plane seismic waves in an actual configuration (alpine basin of Grenoble, France), for which the high velocity contrast is seen to significantly affect the overall efficiency of the multi-region FM-BEM. (10.4208/cicp.231209.030111s)
  DOI : 10.4208/cicp.231209.030111s
• Source point discovery through high frequency asymptotic time reversal
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  Journal of Computational Physics, Elsevier, 2012, 231, pp.4643-4661. (10.1016/j.jcp.2012.03.012)
  DOI : 10.1016/j.jcp.2012.03.012
• Complete Radiation Boundary Conditions for Convective Waves
  
  • Hagstrom Thomas
  • Bécache Eliane
  • Givoli Dan
  • Stein Kurt
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.610-628. Local approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4]. (10.4208/cicp.231209.060111s)
  DOI : 10.4208/cicp.231209.060111s
• Solving the Homogeneous Isotropic Linear Elastodynamics Equations Using Potentials and Finite Elements. The Case of the Rigid Boundary Condition
  
  • Burel Aliénor
  • Imperiale Sébastien
  • Joly Patrick
  Numerical Analysis and Applications, Springer, 2012, 5 (2), pp.136-143. In this article, elastic wave propagation in a homogeneous isotropic elastic medium with rigid boundary is considered. A method based on the decoupling of pressure and shear waves via the use of scalar potentials is proposed. This method is adapted to a finite elements discretization, which is discussed. A stable, energy preserving numerical scheme is presented, as well as 2D numerical results. (10.1134/S1995423912020061)
  DOI : 10.1134/S1995423912020061
• Uniform controllability of scalar conservation laws in the vanishing viscosity limit
  
  • Léautaud Matthieu
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2012, 50 (3), pp.1661-1699. We deal with viscous perturbations of scalar conservation laws on a bounded interval with a general flux function f and a small dissipation coefficient Ɛ. Acting on this system on both endpoints of the interval, we prove global exact controllability to constant states with nonzero speed. More precisely, we construct boundary controls so that the solution is driven to the targeted constant state, and we moreover require these controls to be uniformly bounded as Ɛ → 0+ in an appropriate space. For general (nonconvex) flux functions this can be done for sufficiently large time, and for convex fluxes f, we have a precise estimate on the minimal time needed to control. © 2012 Society for Industrial and Applied Mathematics. (10.1137/100803043)
  DOI : 10.1137/100803043
• A low frequency model for acoustic propagation in a 2D flow duct: numerical computation
  
  • Joubert Lauris
  • Joly Patrick
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.508-524. In this paper we study a low frequency model for acoustic propagation in a 2D flow duct. For some Mach profile flow, we are able to give a well-posedness theorem. Its proof relies on a quasi-explicit expression of the solution which provides us an efficient numerical method. We give and comment numerical results for particular linear, tangent and quadratic profiles. Finally, we give a numerical validation of our asymptotic model.
• Giens 2011
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2012.
• Giens 2011
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2012.
• Application of the multi-level time-harmonic fast multipole BEM to 3-D visco-elastodynamics
  
  • Grasso Eva
  • Chaillat Stéphanie
  • Bonnet Marc
  • Semblat Jean-François
  Engineering Analysis with Boundary Elements, Elsevier, 2012, 36, pp.744-758. This article extends previous work by the authors on the single- and multi-domain time-harmonic elastodynamic multi-level fast multipole BEM formulations to the case of weakly dissipative viscoelastic media. The underlying boundary integral equation and fast multipole formulations are formally identical to that of elastodynamics, except that the wavenumbers are complex-valued due to attenuation. Attention is focused on evaluating the multipole decomposition of the viscoelastodynamic fundamental solution. A damping-dependent modification of the selection rule for the multipole truncation parameter, required by the presence of complex wavenumbers, is proposed. It is empirically adjusted so as to maintain a constant accuracy over the damping range of interest in the approximation of the fundamental solution, and validated on numerical tests focusing on the evaluation of the latter. The proposed modification is then assessed on 3D single-region and multi-region visco-elastodynamic examples for which exact solutions are known. Finally, the multi-region formulation is applied to the problem of a wave propagating in a semi-infinite medium with a lossy semi-spherical inclusion (seismic wave in alluvial basin). These examples involve problem sizes of up to about $3\,10^{5}$ boundary unknowns. (10.1016/j.enganabound.2011.11.015)
  DOI : 10.1016/j.enganabound.2011.11.015
• Mathematical and numerical modelling of piezoelectric sensors
  
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2012. The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element / finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.
• T-coercivity: Application to the discretization of Helmholtz-like problems
  
  • Ciarlet Patrick
  Computers & Mathematics with Applications, Elsevier, 2012, 64 (1), pp.22-34. To solve variational indefinite problems, a celebrated tool is the Banach-Ne?as-Babuka theory, which relies on the inf-sup condition. Here, we choose an alternate theory, T-coercivity. This theory relies on explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. We provide simple proofs to solve the exact and discrete problems, and to show convergence under fairly general assumptions. We also establish sharp estimates on the convergence rates. © 2012 Elsevier Ltd. All rights reserved. (10.1016/j.camwa.2012.02.034)
  DOI : 10.1016/j.camwa.2012.02.034