Publications

2013

• Strongly oscillating singularities for the interior transmission eigenvalue problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  Inverse Problems, IOP Publishing, 2013, 19(10), pp.104004. In this paper, we investigate a two-dimensional interior transmission eigenvalue problem for an inclusion made of a composite material. We consider configurations where the difference between the parameters of the composite material and those of the background changes sign on the boundary of the inclusion. In a first step, under some assumptions on the parameters, we extend the variational approach of the T-coercivity to prove that the transmission eigenvalues form at most a discrete set. In the process, we also provide localization results. Then, we study what happens when these assumptions are not satisfied. The main idea is that, due to very strong singularities that can occur at the boundary, the problem may lose Fredholmness in the natural H1 framework. Using Kondratiev theory, we propose a new functional framework where the Fredholm property is restored. (10.1088/0266-5611/29/10/104004)
  DOI : 10.1088/0266-5611/29/10/104004
• A remark on Lipschitz stability for inverse problems
  
  • Bourgeois Laurent
  Comptes rendus hebdomadaires des séances de l'Académie des sciences, Gauthier-Villars, 2013, 351, pp.187--190. An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse medium problem for the Helmholtz equation. (10.1016/j.crma.2013.04.004)
  DOI : 10.1016/j.crma.2013.04.004
• Stability and dispersion analysis of the staggered discontinuous Galerkin method for wave propagation
  
  • Chang Hiu Ning
  • Chung Eric
  • Cohen Gary
  International Journal of Numerical Analysis and Modeling, Institute for Scientific Computing and Information, 2013, 10 (1), pp.233--256. Staggered discontinuous Galerkin methods have been developed recently and are adopted successfully to many problems such as wave propagation, elliptic equation, convection-diffusion equation and the Maxwells equations. For wave propagation, the method is proved to have the desirable properties of energy conservation, optimal order of convergence and blockdiagonal mass matrices. In this paper, we perform an analysis for the dispersion error and the CFL constant. Our results show that the staggered method provides a smaller dispersion error compared with classical finite element method as well as non-staggered discontinuous Galerkin methods.
• Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section
  
  • Imperiale Sébastien
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and elec- tric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain "homogenized" telegraphist's model. This model, which includes a non-local in time op- erator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross-section is not homogeneous. (10.1016/j.apnum.2013.03.011)
  DOI : 10.1016/j.apnum.2013.03.011
• Apposition of the topological sensitivity and linear sampling approaches to inverse scattering
  
  • Bellis Cédric
  • Bonnet Marc
  • Guzina B. B.
  Wave Motion, Elsevier, 2013, 50, pp.891-908. The focus of this study is the reconstruction of a penetrable obstacle in acoustic medium from the knowledge of incident time-harmonic waves and corresponding scattered fields. The problem is investigated by way of two competing approaches: the method of topological sensitivity and that of linear sampling, that have been successfully developed for a variety of physical settings (acoustic, electromagnetic, elastodynamic) as non-iterative tools for solving the inverse scattering problem. On adopting a particular scattering configuration -- plane waves impigning on a spherical obstacle -- that permits analytical treatment as the testing platform, a parallel is drawn between the two methods to evaluate their relative performance in reconstructing the obstacle from the scattered field data. For completeness, the comparison is made by considering a range of input parameters in terms of material properties of the scatterer, frequency of illuminating waves, and noise in the data. (10.1016/j.wavemoti.2013.02.013)
  DOI : 10.1016/j.wavemoti.2013.02.013
• Transparent boundary conditions for locally perturbed infinite hexagonal periodic media.
  
  • Besse Christophe
  • Coatléven Julien
  • Fliss Sonia
  • Lacroix-Violet Ingrid
  • Ramdani Karim
  Communications in Mathematical Sciences, International Press, 2013, 11 (4), pp.907-938. In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a non standard integral equation.
• Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics
  
  • Chaillat Stéphanie
  • Bonnet Marc
  Wave Motion, Elsevier, 2013, 50, pp.1090-1104. This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green's tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green's tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process. (10.1016/j.wavemoti.2013.03.008)
  DOI : 10.1016/j.wavemoti.2013.03.008
• Negative materials and corners in electromagnetism
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  • Claeys Xavier
  • Nazarov Serguei
  , 2013. no abstract
• T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients
  
  • Chesnel Lucas
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2013, 124 (1), pp.1-29. 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. (10.1007/s00211-012-0510-8)
  DOI : 10.1007/s00211-012-0510-8
• Numerical Microlocal analysis of 2-D noisy harmonic plane and circular waves
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  Asymptotic Analysis, IOS Press, 2013, 83 (1-2), pp.157--187. We present a mathematical and numerical analysis of the stability and accuracy of the NMLA (Numerical MicroLocal Analysis) method [J. Comput. Phys. 199(2) (2004), 717-741] and its discretization.We restrict to homogeneous space and focus on the two simplest cases: (1) Noisy plane wave packets, (2) Noisy point source solutions. A stability result is obtained through the introduction of a new "impedance" observable. The analysis of the point source case leads to a modified second order (curvature dependent) correction of the algorithm. Since NMLA is local, this second order improved version can be applied to general data (heterogeneous media). See [J. Comput. Phys. 231(14) (2012), 4643-4661] for a an application to a source discovery inverse problem. (10.3233/ASY-121157)
  DOI : 10.3233/ASY-121157
• Plasmonic cavity modes: black-hole phenomena captured by Perfectly Matched Layers
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2013. no abstract
• Scalar transmission problems between dielectrics and metamaterials: T-coercivity for the Discontinuous Galerkin approach.
  
  • Chung Eric T.
  • Ciarlet Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2013, 239, pp.189--207. no abstract
• A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides
  
  • Fliss Sonia
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (2), pp.B438 - B461. This work deals with one-dimensional infinite perturbation---namely, line defects---in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. Contrary to existing methods, this one is exact, but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. © 2013, Society for Industrial and Applied Mathematics (10.1137/12086697X)
  DOI : 10.1137/12086697X
• Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: Mathematical justification of the Foldy-Lax model
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2013, 50 (1), pp.18-28. We are concerned with a two-dimensional problem which models the scattering of a time-harmonic acoustic wave by an arbitrary number of sound-soft circular obstacles. Assuming that their radii are small compared to the wavelength, we propose a mathematical justification of different levels of asymptotic models available in the physical literature, including the so-called Foldy-Lax model. © 2012 Elsevier B.V. (10.1016/j.wavemoti.2012.06.001)
  DOI : 10.1016/j.wavemoti.2012.06.001
• Analysis of the factorization method for a general class of boundary conditions
  
  • Chamaillard Mathieu
  • Chaulet Nicolas
  • Haddar Houssem
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013. We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the testing function. (10.1515/jip-2013-0013)
  DOI : 10.1515/jip-2013-0013
• Domain decomposition for the neutron SPN equations
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Baudron Anne-Marie
  • Lautard Jean-Jacques
  , 2013. Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method applied to the neutron SPN equations, which are an approximation of the transport neutron equation. This method is based on the Schwarz iterative algorithm with optimized Robin interface conditions to handle communications. From a computational point of view, this method is rather easy to implement. We give some numerical results in highly heterogeneous 3D configurations. Computations are carried out with the MINOS solver, which is a multigroup SPN solver of the APOLLO3® neutronics code. Numerical experiments show that the method is robust and efficient, and that our choice of the Robin parameters is satisfactory.no abstract
• Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Yu Tang Fei
  Journal of Computational Physics, Elsevier, 2013, 235, pp.14--31. In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell's equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee's scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. (10.1016/j.jcp.2012.10.019)
  DOI : 10.1016/j.jcp.2012.10.019
• On the Well-Posedness , Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.
• A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials
  
  • Chung Eric T.
  • Ciarlet Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2013, 239, pp.189-207. 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. (10.1016/j.cam.2012.09.033)
  DOI : 10.1016/j.cam.2012.09.033
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2013, pp.075012. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods. (10.1088/0266-5611/29/7/075012)
  DOI : 10.1088/0266-5611/29/7/075012
• Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces
  
  • Ciarlet Patrick
  Journal of Numerical Mathematics, De Gruyter, 2013, 21 (3), pp.173-180. Since it was originally designed, the Scott-Zhang interpolation operator has been very popular. Indeed, it possesses two keys features: it can be applied to fields without pointwise values and it preserves the boundary condition. However, no approximability properties seem to be available in the literature when the regularity of the field is weak. In this Note, we provide some estimates for such weakly regular fields, measured in Sobolev spaces with fractional order between 0 and 1 (10.1515/jnum-2013-0007)
  DOI : 10.1515/jnum-2013-0007
• Obstacles in acoustic waveguides becoming "invisible" at given frequencies
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Nazarov Sergei
  Acoustical Physics / Akusticheskii zhurnal, MAIK Nauka/Interperiodica, 2013, 59(6), pp.633--639. We prove the existence of gently sloping perturbations of walls of an acoustic two-dimensional waveguide, for which several waves at given frequencies pass by the created obstacle without any distortion or with only a phase shift. (10.1134/S1063771013050047)
  DOI : 10.1134/S1063771013050047
• Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
  
  • Jamelot Erell
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2013, 241, pp.445--463. no abstract
• On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems
  
  • Blanc F.
  • Touzé Cyril
  • Mercier Jean-François
  • Ege Kerem
  • Bonnet-Ben Dhia Anne-Sophie
  Mechanical Systems and Signal Processing, Elsevier, 2013, 36 (2), pp.520-539. Numerical computation of Nonlinear Normal Modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three di erent mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the center manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve. (10.1016/j.ymssp.2012.10.016)
  DOI : 10.1016/j.ymssp.2012.10.016
• 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