Publications

2014

• 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
• 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
• Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws
  
  • Ciarlet Patrick
  • Wu Haijun
  • Zou Jun
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (2), pp.779-807. In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of finite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of the Gauss' laws in some appropriate norm, in addition to the standard optimal convergence in energy-norm, under the general weak regularity assumptions that hold for both convex and non-convex polyhedral domains and for the discontinuous coefficients that may have large jumps across the interfaces between different media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes. (10.1137/120899856)
  DOI : 10.1137/120899856
• A new Fast Multipole formulation for the elastodynamic half-space Green's tensor
  
  • Chaillat Stéphanie
  • Bonnet Marc
  Journal of Computational Physics, Elsevier, 2014, 258, pp.787-808. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the full-space Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about $N=6 10^5$ degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. (10.1016/j.jcp.2013.11.010)
  DOI : 10.1016/j.jcp.2013.11.010
• Study of a Model Equation in Detonation Theory
  
  • Faria Luiz
  • Kasimov Aslan
  • Rosales Rodolfo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2014, 74 (2), pp.547-570. (10.1137/130938232)
  DOI : 10.1137/130938232
• 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
• 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
• 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
• 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
• 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