Publications

2014

• Mathematical modeling of a discontinuous Myers condition
  
  • Lunéville Éric
  • Mercier Jean-François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2014, 48 (5), pp.1529-1555. (10.1051/m2an/2014008)
  DOI : 10.1051/m2an/2014008
• Generalized method for retrieving effective parameters of anisotropic metamaterials
  
  • Mercier Jean-François
  • Castanié Aurore
  • Félix Simon
  • Maurel Agnes
  Optics Express, Optical Society of America - OSA Publishing, 2014, 22 (24), pp.29977-29953. Electromagnetic or acoustic metamaterials can be described in terms of equivalent effective, in general anisotropic, media and several techniques exist to determine the effective permeability and permittivity (or effective mass density and bulk modulus in the context of acoustics). Among these techniques, retrieval methods use the measured reflection and transmission coefficients (or scattering coefficients) for waves incident on a metamaterial slab containing few unit cells. Until now, anisotropic effective slabs have been considered in the literature but they are limited to the case where one of the axes of anisotropy is aligned with the slab interface. We propose an extension to arbitrary orientations of the principal axes of anisotropy and oblique incidence. The retrieval method is illustrated in the electromagnetic case for layered media, and in the acoustic case for array of tilted elliptical particles. (10.1364/OE.22.029937)
  DOI : 10.1364/OE.22.029937
• Wood's anomalies for arrays of dielectric scatterers
  
  • Maurel Agnès
  • Félix Simon
  • Mercier Jean-François
  • Ourir Abdelwaheb
  • Djeffal Zine Eddine
  Journal of the European Optical Society : Rapid publications, European Optical Society, 2014, 9, pp.14001. The Rayleigh Wood anomalies refer to an unexpected repartition of the electromagnetic energy between the several interference orders of the light emerging from a grating. Since Hessel and Oliner (Appl. Opt. 4, 1275-1297 (1965)), several studies have been dedicated to this problem, focusing mainly on the case of metallic gratings. In this paper, we derive explicit expressions of the reflection coefficients in the case of dielectric gratings using a perturbative approach. This is done in a multimodal description of the field combined with the use of the admittance matrix, analog to the so-called electromagnetic impedance. Comparisons with direct numerical calculations show a good agreement with our analytical prediction. (10.2971/jeos.2014.14001)
  DOI : 10.2971/jeos.2014.14001
• Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Optical Society of America, Optical Society of America, 2014, 31 (10), pp.2249-2255. We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain. (10.1364/JOSAA.31.002249)
  DOI : 10.1364/JOSAA.31.002249
• The finite element method in solid mechanics
  
  • Bonnet Marc
  • Frangi Attilio
  • Rey Christian
  , 2014, pp.365. The book focuses on topics that are at the core of the Finite Element Method (FEM) for the mechanics of deformable solids and structures.Its main objective is to provide the reader, who is assumed to be familiar with standard continuum solid mechanics, with a clear grasp of the essentials, sufficient background for reading and exploiting the research literature on computational solid mechanics, and a working knowledge of the main implementational issues of the FEM.This book arises from a course taught since 2004 to last-year students of Ecole Polytechnique (France). It is intended for Master and PhD students, as well as scientists and engineers looking for a rigorous introduction to FEM theory and programming for linear and non-linear analyses in solid mechanics.As a distinguishing feature, in addition to sections devoted to theory and concepts presented in general terms, each chapter also features other sections (interspersed with the former) devoted to detailed description of specific features (e.g. the construction of a specific finite element), annotated Matlab code and/or numerical examples produced with it, or worked-out analytical examples.
• The "exterior approach" to solve the inverse obstacle problem for the Stokes system
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2014, pp.Pages: 23 - 51. We apply an "exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming infite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in 2D show the effciency of the two mixed formulations and of the exterior approach based on one of them. (10.3934/ipi.2014.8.23)
  DOI : 10.3934/ipi.2014.8.23
• Wave propagation through penetrable scatterers in a waveguide and through a penetrable gratings
  
  • Maurel Agnès
  • Mercier Jean-François
  • Félix Simon
  Journal of the Acoustical Society of America, Acoustical Society of America, 2014, 135 (1), pp.165-174. A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances. (10.1121/1.4836075)
  DOI : 10.1121/1.4836075
• High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls
  
  • Schmidt Kersten
  • Anastasia Thöns-Zueva
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, pp.1823. (10.1142/S0218202514500080)
  DOI : 10.1142/S0218202514500080
• Quasi-local transmission conditions for non-overlapping domain decomposition methods for the Helmholtz equation
  
  • Collino Francis
  • Joly Patrick
  • Lecouvez Matthieu
  • Stupfel Bruno
  Comptes Rendus. Physique, Académie des sciences (Paris), 2014, 15 (5), pp.403-414. In this article, we present new transmission conditions for a domain decomposition method, applied to a scattering problem. Unlike other conditions used in the literature, the conditions developed here are non-local, but can be written as an integral operator (as a Riesz potential) on the interface between two domains. This operator, of order View the MathML source12, leads to an exponential convergence of the domain decomposition algorithm. A spectral analysis of the influence of the operator on simple cases is presented, as well as some numerical results and comparisons. (10.1016/j.crhy.2014.04.005)
  DOI : 10.1016/j.crhy.2014.04.005
• Space-time focusing of acoustic waves on unknown scatterers
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2014, pp.19. Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit-receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy-Lax model. In this context, several mathematical and numerical arguments that support our idea are explored. (10.1016/j.wavemoti.2014.07.009)
  DOI : 10.1016/j.wavemoti.2014.07.009
• Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz resonators
  
  • Lombard Bruno
  • Mercier Jean-François
  Journal of Computational Physics, Elsevier, 2014, 259, pp.421-443. Acoustic wave propagation in a one-dimensional waveguide connected with Helmholtz resonators is studied numerically. Finite amplitude waves and viscous boundary layers are considered. The model consists of two coupled evolution equations: a nonlinear PDE describing nonlinear acoustic waves, and a linear ODE describing the oscillations in the Helmholtz resonators. The thermal and viscous losses in the tube and in the necks of the resonators are modeled by fractional derivatives. A diffusive representation is followed: the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations. A splitting method is then applied to the evolution equations: their propagative part is solved using a standard TVD scheme for hyperbolic equations, whereas their diffusive part is solved exactly. Various strategies are examined to compute the coefficients of the diffusive representation; finally, an optimization method is preferred to the usual quadrature rules. The numerical model is validated by comparisons with exact solutions. The properties of the full nonlinear solutions are investigated numerically. In particular, the existence of acoustic solitary waves is confirmed. (10.1016/j.jcp.2013.11.036)
  DOI : 10.1016/j.jcp.2013.11.036
• Mathematical modelling of multi conductor cables
  
  • Beck Geoffrey
  • Imperiale Sebastien
  • Joly Patrick
  Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 2014, pp.26. This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell’s equations. In essence, we extend and complete previous results to the multi-wires case. (10.3934/dcdss.2015.8.521)
  DOI : 10.3934/dcdss.2015.8.521