Publications

2012

• Constitutive Equation Gap
  
  • Pagano Stéphane
  • Bonnet Marc
  , 2012, pp.26p.. In this chapter we examine the concept of constitutive equation gap (CEG) as a tool for the identification of parameters associated with behavior models for solid materials. The concept of CEG was initially proposed for error estimation in the finite element method. It then turned out to be also a powerful tool for identification, especially with many applications in model updating. Essentially equivalent concepts have been proposed in other contexts for solving inversion problems, such as the elec- trostatic energy functionals of Kohn and Vogelius. Two important characteristics of CEG functionals are (i) their strong and clear physical meaning, and (ii) their additive character with respect to the structure, allowing the definition of local error indicators over substructures.
• Modeling and simulation of a grand piano.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  , 2012, pp.29. The purpose of this study is the time domain modeling and numerical simulation of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependant damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from horizontal. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model that will be discretized is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependant damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties. On the one hand, numerical stability of the discrete scheme can be compromised by nonlinear and coupling terms. A very efficient way to guarantee this stability is to construct a numerical scheme which ensures the conservation (or dissipation) of a discrete equivalent of the continuous energy, across time steps. A major contribution of this work has been to develop energy preserving schemes for a class of nonlinear systems of equations, in which enters the string model. On the other hand, numerical efficiency and computation time reduction require that the unknowns of each problem's part, for which time discretization is specific, hence different, be updated separately. To achieve this artificial decoupling, adapted Schur complements are performed after Lagrange multipliers are introduced. The potential of this time domain piano modeling is emphasized by realistic numerical simulations. Beyond greatly replicating the measurements, the program allows us to investigate the influence of physical phenomena (string stiffness or nonlinearity), geometry or materials on the general vibratory behavior of the piano, sound included. Spectral enrichment, ''phantom partials'' and nonlinear precursors are clearly revealed when large playing amplitudes are involved, highlighting how this approach can help better understand how a piano works.
• Remarks on the stability of Cartesian PMLs in corners
  
  • Bécache Eliane
  • Prieto Andrés
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2012, 62 (11), pp.1639-1653. This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure–velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results. (10.1016/j.apnum.2012.05.003)
  DOI : 10.1016/j.apnum.2012.05.003
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  , 2012, pp.32. We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.
• Étude de quelques problèmes de transmission avec changement de signe. Application aux métamatériaux.
  
  • Chesnel Lucas
  , 2012. Dans cette thèse, nous étudions quelques opérateurs présentant un changement de signe dans leur partie principale. Ces opérateurs apparaissent notamment en électromagnétisme lorsqu'on s'intéresse à la propagation des ondes dans des structures constituées de matériaux usuels et de matériaux négatifs en régime harmonique. Ici, nous appelons matériau négatif un matériau modélisé par une permittivité diélectrique et/ou une perméabilité magnétique négative(s). En raison du changement de signe des coefficients physiques, on ne peut utiliser les outils classiques pour étudier ce problème. Dans la première partie de ce mémoire, nous nous concentrons sur le problème de transmission scalaire auquel on peut réduire les équations de Maxwell lorsque la géométrie et les données présentent une invariance dans une direction. Avec la technique de la T-coercivité, basée sur des arguments géométriques, nous établissons des conditions nécessaires et suffisantes pour prouver le caractère bien posé de ce problème en domaine borné dans H^1. Nous montrons également comment on peut utiliser cette approche pour justifier la convergence des méthodes usuelles d'approximation par éléments finis. Dans un deuxième temps, au moyen de techniques différentes, issues de l'étude des équations elliptiques dans des domaines à géométrie singulière, nous définissons un nouveau cadre fonctionnel pour recouvrer le caractère Fredholm lorsque celui-ci est perdu dans H^1. Il apparaît alors un phénomène surprenant de trou noir. Tout se passe comme si des ondes étaient aspirées en un point. Nous réalisons ensuite une étude asymptotique par rapport à une petite perturbation de l'interface entre le matériau positif et le matériau négatif dans ce cadre fonctionnel. Au cours de notre analyse, nous mettons en évidence un curieux phénomène de valeur propre clignotante. La troisième partie de ce document est consacrée à l'étude des équations de Maxwell. Nous travaillons d'abord sur les équations de Maxwell 2D en exploitant les résultats obtenus pour le problème scalaire. Puis, nous nous intéressons aux équations de Maxwell 3D. Nous montrons qu'elles sont bien posées dès lors que les problèmes scalaires associés sont bien posés. Enfin, dans une quatrième partie, nous étudions le problème de transmission intérieur apparaissant en théorie de la diffraction. L'opérateur pour ce problème présente également un changement de signe dans sa partie principale. Nous abordons son étude en utilisant l'analogie existant avec le problème de transmission entre un matériau positif et un matériau négatif. Certaines configurations pour ce problème de transmission intérieur conduisent à considérer un problème de transmission du quatrième ordre avec changement de signe. Nous prouvons que cet opérateur présente des propriétés étonnamment différentes de celles de l'opérateur scalaire du second ordre.
• Time domain simulation of a piano. Part 1 : model description.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  , 2012. The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependant damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependant damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties.
• FaIMS: A fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equation
  
  • Chaillat Stéphanie
  • Biros George
  Journal of Computational Physics, Elsevier, 2012, 231 (12), pp.4403-4421. We propose an algorithm to compute an approximate singular value decomposition of least squares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency time-harmonic wave eqation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, non-destructive evaluation, and optical tomography). (10.1016/j.jcp.2012.02.006)
  DOI : 10.1016/j.jcp.2012.02.006
• Finite element computation of elastic propagation modes in open stratified waveguides
  
  • Treyssede Fabien
  • Nguyen Khac-Long
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2012, pp.1p.. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In several applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded. The physics of waves in open waveguides significantly differs from closed waveguides. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes (wavenumbers are then complex). From a numerical modeling point of view, the main difficulty lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. This behavior is seldom mentioned in the literature of elastic waveguides. Yet leaky modes have often been considered for NDT applications, which require waves of low attenuation in order to maximize the inspection range. A numerical approach is proposed for computing modes in open elastic waveguides, in the bidimensional case as a first step. The approach combines a semi-analytical finite element method with perfectly matched layers (PML). The technique of absorbing layers (AL) is also implemented, which consists in using large artificial layers of growing viscoelasticity. Numerical results are compared to analytical results. The efficiency of PML is compared to AL and parametric studies are briefly conducted in order to assess the convergence of both techniques. The physical meaning of leaky modes is also highlighted.
• A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials
  
  • Chung Eric T.
  • Ciarlet Patrick
  , 2012. Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit inf-sup operators, and numerical examples are provided to show the efficiency of the method.
• Propagation in a periodic succession of slabs with mixed negative/positive index
  
  • Maurel Agnes
  • Ourir Abdelwaheb
  • Mercier Jean-François
  • Pagneux Vincent
  , 2012. Metamaterials are artificial materials engineered using periodic inclusions of small inhomogeneities to enact effective macroscopic behavior. Until recently, most studies considered only ideal systems and did not address the possible effects of disorder.The first step in this direction was made in [Phys. Rev. B 70, 245102, 2004] where it was shown that the presence of a single defect led to the appearance of a localized mode. Since then, more general model of alternating sequences of right and left handed layers with random parameters have been studied, notably in [M. V. Gorkunov et al., Phys. Rev. E 73, 056605, 2006; Phys. Rev. Lett. 99, 193902, 2007]. The authors have shown that the localization properties differ dramatically from those exhibited by conventional disordered materials. We study wave propagation in such stratified media both experimentally and theoretically. Experiments confirm that the properties of the attenuation length differ dramatically from those exhibited by conventional alternated layer materials, notably in the intermediate value of the wavelength. Analytical prediction of the attenuation length is in good agreement with the observations.
• A complete FE simulation tools for NDT inspections with piezoelectric transducers
  
  • Imperiale Sebastien
  • Marmorat Simon
  • Leymarie Nicolas
  • Chatillon Sylvain
  , 2012. An ultrasonic inspection system involves the generation, propagation and reception of short transient signals. Piezoelectric transducers and particularly phased arrays are increasingly used in ultrasonic Non Destructive Testing (NDT) because of their ability to focus or deflect an ultrasonic beam in parts of complex geometries. To accurately model the sensitivity in transmission and reception of such sensors, a transient Finite Element (FE) model has been developed including not only piezoelectric effects but also all electrical elements such as pulser/receiver system and cabling. A particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. The definition of the inspection domain is made easier by a decomposition domain technique allowing, in the same time, local time stepping and efficient absorbing layers to optimize calculation cost. In order to illustrate all the capabilities of this simulation tool, several cases of NDT inspections are then presented through the analysis of the ultrasonic beam snapshots and the electrical signal read on the receiver.
• T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients
  
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2012. To solve variational indefinite problems, one uses classically the Banach-Necas-Babuška theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $div\sigma\nabla u = f$ set in $H^1_0$, with $\sigma$ exhibiting a sign change.
• Modélisation et simulation numérique d'un piano par modèles physiques
  
  • Chabassier Juliette
  , 2012. Cette étude porte sur la modélisation et la simulation numérique d'un piano, en domaine temporel, par modèles phy- siques. Nous souhaitons rendre compte du comportement vibratoire et acoustique du piano, en prenant en compte les éléments principaux qui contribuent à la production du son. La table d'harmonie est modélisée par une équation bidimensionnelle de plaque épaisse, le système de Reissner Mindlin, pour un matériau orthotrope et hétérogène, dont l'amortissement dépend de la fréquence. Grâce aux équations de la vibroacoustique, la table rayonne dans l'air, dans lequel on souhaite calculer le champ acoustique complet autour de la ceinture du piano, que l'on suppose rigide. La table d'harmonie est d'autre part sollicitée par les cordes, à travers le chevalet où elles présentent un léger angle par rapport au plan horizontal. Chaque corde est modélisée par un système d'équations monodimensionnelles amorties dans lequel on prend en compte non seulement les ondes transversales excitées par le marteau, mais aussi la raideur à travers les ondes de cisaillement, ainsi que le couplage avec les ondes longi- tudinales provenant de la prise en compte des non linéarités géométriques. Le marteau est lancé avec une vitesse initiale vers un chœur de cordes, contre lequel il s'écrase avant d'être repoussé par les cordes. La force d'interaction dépend de façon non linéaire de l'écrasement du marteau.Le modèle complet de piano, que l'on souhaite résoudre numériquement, consiste donc en un système couplé d'équations aux dérivées partielles, dont chacune revêt des difficultés de nature différente : la corde est régie par un système d'équations non linéaires, la table d'harmonie est soumise à un amortissement dépendant de la fréquence, la propagation acoustique requiert un très grand nombre d'inconnues; auxquelles s'ajoute la difficulté inhérente aux couplages. D'une part, la stabilité numérique du schéma discret peut être compromise par la présence d'équations non linéaires et de nombreux couplages. Une méthode efficace pour garantir cette stabilité a priori est de construire un schéma qui conserve, ou dissipe, un équivalent discret de l'énergie physique d'un pas de temps au suivant. Une contribution majeure de ce travail a été de développer des schémas préservant une énergie discrète pour une classe de systèmes non linéaires dans laquelle s'inscrit le modèle de corde. D'autre part, afin d'augmenter l'efficacité de la méthode et de réduire le coût des calculs numériques, il est souhaitable de mettre à jour de façon découplée les inconnues liées aux différentes parties du problème, sur lesquelles la discrétisation en temps est faite de façon différente, afin de s'adapter aux spécificités de chacune. L'introduction de multiplicateurs de Lagrange nous permet de réaliser ce découplage artificiel grâce à des compléments de Schur adaptés. L'utilisation du code de calcul en situation réaliste montre le potentiel d'une telle modélisation d'un piano complet en domaine temporel. Au delà de très bien reproduire les mesures, il est possible d'étudier l'influence de certains phénomènes physiques (corde raide, non linéaire), de la géométrie ou encore des matériaux utilisés sur le comportement vibratoire général du piano, et sur le son en particulier. L'enrichissement spectral, ainsi que l'apparition des " partiels fantômes " et du précurseur non linéaire sont clairement mis en évidence pour les grandes amplitudes de jeu, soulignant l'intérêt de notre approche dans la compréhension du fonctionnement de l'instrument.
• T-coercivity: application to the discretization of Helmholtz-like problems
  
  • Ciarlet Patrick
  , 2012. To solve variational indefinite problems, a celebrated tool is the Banach-Nečas- Babuška 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.
• Wave propagation in locally perturbed periodic media (case with absorption): Numerical aspects
  
  • Fliss Sonia
  • Joly Patrick
  Journal of Computational Physics, Elsevier, 2012, 231 (4), pp.1244-1271. We are interested in the numerical simulation of wave propagation in media which are a local perturbation of an infinite periodic one. The question of finding artificial boundary conditions to reduce the actual numerical computations to a neighborhood of the perturbation via a DtN operator was already developed in at the continuous level. We deal in this article with the numerical aspects associated to the discretization of the problem. In particular, we describe the construction of discrete DtN operators that relies on the numerical solution of local cell problems, non stationary Ricatti equations and the discretization of non standard integral equations in Floquet variables. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.10.007)
  DOI : 10.1016/j.jcp.2011.10.007
• Time-Harmonic Acoustic Scattering in a Complex Flow: a Full Coupling Between Acoustics and Hydrodynamics
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  • Millot Florence
  • Pernet Sébastien
  • Peynaud Emilie
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.555-572. For the numerical simulation of time harmonic acoustic scattering in a complex geometry, in presence of an arbitrary mean flow, the main difficulty is the coexistence and the coupling of two very different phenomena: acoustic propagation and convection of vortices. We consider a linearized formulation coupling an augmented Galbrun equation (for the perturbation of displacement) with a time harmonic convection equation (for the vortices). We first establish the well-posedness of this time harmonic convection equation in the appropriate mathematical framework. Then the complete problem, with Perfectly Matched Layers at the artificial boundaries, is proved to be coercive + compact, and a hybrid numerical method for the solution is proposed, coupling finite elements for the Galbrun equation and a Discontinuous Galerkin scheme for the convection equation. Finally a 2D numerical result shows the efficiency of the method. (10.4208/cicp.221209.030111s)
  DOI : 10.4208/cicp.221209.030111s
• 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
• 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.
• Operator Factorization for Multiple-Scattering Problems and an Application to Periodic Media
  
  • Coatléven Julien
  • Joly Patrick
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.303-318. This work concerns multiple-scattering problems for time-harmonic equations in a reference generic media. We consider scatterers that can be sources, obstacles or compact perturbations of the reference media. Our aim is to restrict the computational domain to small compact domains containing the scatterers. We use Robin-to-Robin (RtR) operators (in the most general case) to express boundary conditions for the interior problem. We show that one can always factorize the RtR map using only operators defined using single-scatterer problems. This factorization is based on a decomposition of the diffracted field, on the whole domain where it is defined. Assuming that there exists a good method for solving single-scatterer problems, it then gives a convenient way to compute RtR maps for a random number of scatterers.
• 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