Publications

2014

• Propagation in waveguides with varying cross-section and curvature: A new light on the role of supplementary modes in multimodal methods
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470 (2166), pp.20130743. We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems. (10.1098/rspa.2014.0008)
  DOI : 10.1098/rspa.2014.0008
• Extraordinary transmission through subwavelength dielectric gratings in the microwave range
  
  • Ahmed Akarid
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  Optics Letters, Optical Society of America - OSA Publishing, 2014, 39 (13), pp.3752-3755. We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry–Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime. (10.1364/OL.39.003752)
  DOI : 10.1364/OL.39.003752
• On the absence of trapped modes in locally perturbed open waveguides
  
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2014, pp.14. This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the three-dimensional scalar wave equation is considered. (10.1093/imamat/hxu046)
  DOI : 10.1093/imamat/hxu046
• Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects
  
  • Abdulle Assyr
  • Grote Marcus J.
  • Stohrer Christian
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2014, 12 (3), pp.1230–1257. 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 new FE-HMM-L method captures not only the short-time behavior of the wave field, well described by classical homogenization theory, but also more subtle long-time dispersive effects, both at a computational cost independent of the microscale. Optimal error estimates in the energy norm and the $L^2$-norm are proved over finite time intervals, which imply convergence to the solution from classical homogenization theory when both the macro- and the microscale are refined simultaneously. Numerical experiments illustrate the usefulness of the FE-HMM-L method and corroborate the theory. (10.1137/13094195X)
  DOI : 10.1137/13094195X
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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