Publications

2013

• Time domain simulation of a piano. Part 1 : model description.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2013, 48 (05), pp.1241-1278. 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 dependent 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 dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties. (10.1051/m2an/2013136)
  DOI : 10.1051/m2an/2013136
• Enhanced transmission through gratings: Structural and geometrical effects
  
  • Maurel Agnès
  • Félix Simon
  • Mercier Jean-François
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2013, 88, pp.115416. Homogenization theory is used to derive the effective properties of gratings with complex subwavelength structures. Going beyond the effect of the filling fraction, geometrical effects are analyzed using a two-step homogenization process. An explicit expression for the transmission spectrum is derived, able to predict the Fabry-Perot resonances and the Brewster angle realizing broadband extraordinary transmission. With the same filling fraction, one expects from this analytical expression that gratings with different geometries may display very different transmission properties. This sensitivity to the microstructure geometry is exemplified in the case of gratings made of hard material and made of dielectric material. The analytical results are shown to be within a few percentage points as compared to full-wave numerical simulations, paving the way for transmission properties tuned by structural and geometrical manipulations. (10.1103/PhysRevB.88.115416)
  DOI : 10.1103/PhysRevB.88.115416
• Lipschitz stability estimate in the inverse Robin problem for the Stokes system
  
  • Egloffe Anne-Claire
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2013. We are interested in the inverse problem of recovering a Robin coefficient defined on some non accessible part of the boundary from available data on another part of the boundary in the nonstationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.
• Calcul des singularités dans les méthodes d’équations intégrales variationnelles
  
  • Salles Nicolas
  , 2013. La mise en œuvre de la méthode des éléments finis de frontière nécessite l'évaluation d'intégrales comportant un intégrand singulier. Un calcul fiable et précis de ces intégrales peut dans certains cas se révéler à la fois crucial et difficile. La méthode que nous proposons consiste en une réduction récursive de la dimension du domaine d'intégration et aboutit à une représentation de l'intégrale sous la forme d'une combinaison linéaire d'intégrales mono-dimensionnelles dont l'intégrand est régulier et qui peuvent s'évaluer numériquement mais aussi explicitement. L'équation de Helmholtz 3-D sert d'équation modèle mais ces résultats peuvent être utilisés pour les équations de Laplace et de Maxwell 3-D. L'intégrand est décomposé en une partie homogène et une partie régulière ; cette dernière peut être traitée par les méthodes usuelles d'intégration numérique. Pour la discrétisation du domaine, des triangles plans sont utilisés ; par conséquent, nous évaluons des intégrales sur le produit de deux triangles. La technique que nous avons développée nécessite de distinguer entre diverses configurations géométriques ; c'est pourquoi nous traitons séparément le cas de triangles coplanaires, dans des plans sécants ou parallèles. Divers prolongements significatifs de la méthode sont présentés : son extension à l'électromagnétisme, l'évaluation de l'intégrale du noyau de Green complet pour les coefficients d'auto-influence, et le calcul de la partie finie d'intégrales hypersingulières.
• Modeling and simulation of a grand piano
  
  • Chabassier Juliette
  • Joly Patrick
  • Chaigne Antoine
  Journal of the Acoustical Society of America, Acoustical Society of America, 2013, 134, pp.648. A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness and geometrical nonlinea- rity. The hammer-string interaction is governed by a nonlinear dissi- pative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers (PML). The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium and high regis- ter, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized. (10.1121/1.4809649)
  DOI : 10.1121/1.4809649
• FE heterogeneous multiscale method for long-time wave propagation
  
  • Abdulle Assyr
  • Grote Marcus J.
  • Stohrer Christian
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2013, 351, pp.495-499. A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our FE-HMM captures long-time dispersive effects of the true solution at a cost similar to that of a standard numerical homogenization scheme which, however, only captures the short-time macroscale behavior of the wave field. (10.1016/j.crma.2013.06.002)
  DOI : 10.1016/j.crma.2013.06.002
• Seismic elastic modeling for seismic imaging
  
  • Virieux Jean
  • Brossier Romain
  • Chaillat Stéphanie
  • Duchkov A. D.
  • Etienne Vincent
  • Lombard Bruno
  • Operto Stéphane
  , 2013.
• Parallel local time-stepping for elastodynamic equations
  
  • Dudouit Yohann
  • Giraud Luc
  • Millot Florence
  • Pernet Sébastien
  , 2013.
• Numerical modeling of nonlinear acoustic waves with fractional derivatives
  
  • Lombard Bruno
  • Mercier Jean-François
  , 2013.
• Modeling the grand piano
  
  • Chaigne Antoine
  • Chabassier Juliette
  • Joly Patrick
  , 2013. A global model of a piano is presented. Its aim is to reproduce the main vibratory and acoustic phenomena involved in the generation of a piano sound from the initial blow of the hammer against the strings to the radiation from soundboard to the air. One first originality of the work is due to the string model which takes both geometrical nonlinear effects and stiffness into account. Other significant improvements are due to the combined modeling of the three main couplings between the constitutive parts of the instrument: hammer-string, string-soundboard and soundboard-air coupling.
• Modeling the piano. Numerical Aspects.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Duruflé Marc
  • Joly Patrick
  , 2013. This paper deals with the discretization of our global piano model. We have to solve a complex system of coupled equations, where each subsystem has different spatial dimensions, which poses specific difficulties. The hammer-strings part is a 1D system governed by nonlinear equations. The soundboard is a 2D system with diagonal damping. The acoustic field is a 3D problem in an unbounded domain. Energy based methods allow to build an accurate and a priori stable scheme.
• A rigorous approach to the propagation of electromagnetic waves in co-axial cables
  
  • Beck Geoffrey
  • Joly Patrick
  • Imperiale Sébastien
  , 2013. We investigate the question of the electromagnetic propagation in thin electric cables from a mathemat- ical point of view via an asymptotic analysis with respect to the (small) transverse dimension of the ca- ble: as it has been done in the past in mechanics for the beam theory from 3D elasticity, we use such an approach for deriving simplified effective 1D models from 3D Maxwell’s equations.
• Computation of leaky modes in three-dimensional open elastic waveguides
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2013, pp.2p.. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. When the guiding structure is embedded into a solid matrix, waveguides are open and waves can be trapped or leaky. With numerical methods, one of the difficulty is that leaky modes attenuate along the axis (complex wavenumber) and exponentially grow along the transverse direction. The goal of this work is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) technique.
• Effective Transmission Conditions for Thin-Layer Transmission Problems in Elastodynamics
  
  • Bonnet Marc
  • Burel Aliénor
  • Joly Patrick
  , 2013. This research is motivated by the numerical modelling of ultrasonic non-destructive testing experiments. Some tested media feature thin layers which are difficult to handle in numerical computations due to the very small element size required for meshing them. To overcome these difficulties, one idea consists in using effective transmission conditions (ETCs) across the two interfaces bounding the layer. This work aims at establishing such ETCs by means of a formal asymptotic analysis with respect to the (small) layer thickness.
• Computation of Dispersion Curves in Elastic Waveguides of Arbitrary Cross-section embedded in Infinite Solid Media
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2013, pp.8p.. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, guiding structures are often buried in a large domain, considered as unbounded. Waveguides are then open and waves can be trapped or leaky. Analytical tools have been developed to model open solid waveguides but these tools are limited for simple geometries (plates, cylinders). With numerical methods, a difficulty is due to the unbounded geometry. Another issue is due to the presence of leaky modes, which grow exponentially along the transverse directions. The goal of this work is to implement a numerical approach to calculate modes in three dimensional elastic open waveguides, which combines the semi-analytical finite element method and the perfectly matched layers (PML) technique. Both Cartesian and cylindrical PML are implemented.
• Introduction to Identification Methods
  
  • Bonnet Marc
  , 2013 (1), pp.223-246. (10.1002/9781118578469.ch8)
  DOI : 10.1002/9781118578469.ch8
• Model for Shock Wave Chaos
  
  • Kasimov Aslan
  • Faria Luiz
  • Rosales Rodolfo
  Physical Review Letters, American Physical Society, 2013, 110 (10). (10.1103/PhysRevLett.110.104104)
  DOI : 10.1103/PhysRevLett.110.104104
• Topological derivative for qualitative inverse scattering
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2013.
• Radiation condition for a non-smooth interface between a dielectric and a metamaterial
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Claeys Xavier
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in $H^1$. This is due to the degeneration of the two dual singularities which then behave like $r^{\pm i\eta}=e^{\pm i\eta\ln\,r}$ with $\eta\in\mathbb{R}^{\ast}$. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to $H^1$ one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  Applicable Analysis, Taylor & Francis, 2013, pp.1-22. We consider inverse obstacle scattering problems for the wave equation with Robin or Neu- mann 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 so- lutions 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. (10.1080/00036811.2013.772583)
  DOI : 10.1080/00036811.2013.772583
• Higher-Order Discontinuous Galerkin Method for Pyramidal Elements using Orthogonal Bases
  
  • Bergot Morgane
  • Duruflé Marc
  Numerical Methods for Partial Differential Equations, Wiley, 2013, 29 (1), pp.144-169. We study arbitrarily high-order finite elements defined on pyramids on discontinuous Galerkin methods. We propose a new family of high-order pyramidal finite element using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non-affine elements. Different strategies for the inversion of mass matrix are also considered and discussed. Numerical experiments are conducted for 3-D Maxwell's equations. (10.1002/num.21703)
  DOI : 10.1002/num.21703
• Qualitative identification of cracks using 3D transient elastodynamic topological derivative: formulation and FE implementation
  
  • Bellis Cédric
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2013, 253, pp.89-105. A time-domain topological derivative (TD) approach is developed for transient elastic-wave imaging of buried cracks. The TD, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack, is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape. Following previous studies on cavity identification in similar conditions, the TD field is here considered as a natural and computationally efficient approach for defining a crack location indicator function. This study emphasizes the implementation and exploitation of TD fields using the standard displacement-based FEM, a straightforward exploitation of the relevant sensitivity formulation established here. Results on several numerical experiments on 3D elastodynamic and acoustic configurations are reported and discussed, allowing to assess and highlight many features of the proposed TD-based fast qualitative crack identification, including its ability to identify multiple cracks and its robustness against data noise. (10.1016/j.cma.2012.10.006)
  DOI : 10.1016/j.cma.2012.10.006
• Two-dimensional Maxwell's equations with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. We consider the theoretical study of time harmonic Maxwellʼs equations in presence of sign-changing coefficients, in a two-dimensional configuration. Classically, the problems for both the Transverse Magnetic and the Transverse Electric polarizations reduce to an equivalent scalar Helmholtz type equation. For this scalar equation, we have already studied consequences of the presence of sign-changing coefficients in previous papers, and we summarize here the main results. Then we focus on the alternative approach which relies on the two-dimensional vectorial formulations of the TM or TE problems, and we exhibit some unexpected effects of the sign-change of the coefficients. In the process, we provide new results on the scalar equations. (10.1016/j.apnum.2013.04.006)
  DOI : 10.1016/j.apnum.2013.04.006
• An hp-finite element approximation of guided modes in photonic crystal waveguides using transparent boundary conditions
  
  • Klindworth Dirk
  • Schmidt Kersten
  • Fliss Sonia
  Computers & Mathematics with Applications, Elsevier, 2013. no abstract
• Large Scale Parameter Estimation Problems in Frequency-Domain Elastodynamics Using an Error in Constitutive Equation Functional
  
  • Banerjee Biswanath
  • Walsh Timothy
  • Aquino Wilkins
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2013, 253, pp.60-72. This paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L2 minimization using quasi-Newton methods. (10.1016/j.cma.2012.08.023)
  DOI : 10.1016/j.cma.2012.08.023