Publications

2010

• A low Mach model for time harmonic acoustics in arbitrary flows
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  • Millot Florence
  • Pernet Sébastien
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6), pp.1868-1875. This paper concerns the finite element simulation of the diffraction of a time-harmonic acoustic wave in the presence of an arbitrary mean flow. Considering the equation for the perturbation of displacement (due to Galbrun), we derive a low-Mach number formulation of the problem which is proved to be of Fredholm type and is therefore well suited for discretization by classical Lagrange finite elements. Numerical experiments are done in the case of a potential flow for which an exact approach is available, and a good agreement is observed. (10.1016/j.cam.2009.08.038)
  DOI : 10.1016/j.cam.2009.08.038
• Energy Preserving Schemes for Nonlinear Hamiltonian Systems of Wave Equations. Application to the Vibrating Piano String.
  
  • Chabassier Juliette
  • Joly Patrick
  , 2010, pp.70. The problem of the vibration of a string is well known in its linear form, describing the transversal motion of a string, nevertheless this description does not explain all the observations well enough. Nonlinear coupling between longitudinal and transversal modes seams to better model the piano string, as does for instance the ''geometrically exact model'' (GEM). This report introduces a general class of nonlinear systems, ''nonlinear hamiltonian systems of wave equations'', in which fits the GEM. Mathematical study of these systems is lead in a first part, showing central properties (energy preservation, existence and unicity of a global smooth solution, finite propagation velocity \ldots). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of ''preserving schemes'' is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class.
• A quasi-reversibility approach to solve the inverse obstacle problem
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2010, 4 (3), pp.351-377. We introduce a new approach based on the coupling of the method of quasi-reversibility and a simple level set method in order to solve the inverse obstacle problem with Dirichlet boundary condition. We provide a theoretical justification of our approach and illustrate its feasibility with the help of numerical experiments in 2D. © 2010 American Institute of Mathematical Sciences. (10.3934/ipi.2010.4.351)
  DOI : 10.3934/ipi.2010.4.351
• La méthode des éléments finis : de la théorie à la pratique. Tome 2 : Compléments
  
  • Bécache Eliane
  • Ciarlet Patrick
  • Hazard Christophe
  • Lunéville Éric
  , 2010, pp.284. La méthode des éléments finis, apparue dans les années 50 pour traiter des problèmes de mécanique des structures, a connu depuis lors un développement continu et est utilisée, aujourd’hui, dans tous les domaines d’applications : mécanique, physique, chimie, économie, finance et biologie. Elle est maintenant intégrée à la plupart des logiciels de calcul scientifique, et de nombreux ingénieurs y sont confrontés dans le cadre de leur activité de modélisation et de simulation numérique. Cet ouvrage recouvre un cours d’éléments finis avancé dispensé à l’ENSTA Paris depuis plusieurs années et fait suite à un ouvrage introductif à la méthode des éléments finis paru dans la même collection. Le livre aborde les compléments indispensables à connaître dès lors qu’on aborde des problèmes plus réalistes. En particulier, les questions relatives à l’approximation par éléments finis des problèmes spectraux (éléments propres de problèmes elliptiques), des problèmes transitoires (équation de diffusion, équation des ondes) et des problèmes mixtes (équations de Stokes, équations de Maxwell). À l’instar du premier tome, nous présentons à la fois les bases théoriques des méthodes, les aspects de mise en œuvre et de nombreuses illustrations numériques.
• High-order Absorbing Boundary Conditions for anisotropic and convective wave equations
  
  • Bécache Eliane
  • Givoli Dan
  • Hagstrom Thomas
  Journal of Computational Physics, Elsevier, 2010, 229 (4), pp.1099-1129. High-order Absorbing Boundary Conditions (ABCs), applied on a rectangular artificial computational boundary that truncates an unbounded domain, are constructed for a general two-dimensional linear scalar time-dependent wave equation which represents acoustic wave propagation in anisotropic and subsonically convective media. They are extensions of the construction of Hagstrom, Givoli and Warburton for the isotropic stationary case. These ABCs are local, and involve only low-order derivatives owing to the use of auxiliary variables on the artificial boundary. The accuracy and well-posedness of these ABCs is analyzed. Special attention is given to the issue of mismatch between the directions of phase and group velocities, which is a potential source of concern. Numerical examples for the anisotropic case are presented, using a finite element scheme. © 2009 Elsevier Inc. All rights reserved. (10.1016/j.jcp.2009.10.012)
  DOI : 10.1016/j.jcp.2009.10.012
• Computation of light refraction at the surface of a photonic crystal using DtN approach
  
  • Fliss Sonia
  • Cassan Eric
  • Bernier Damien
  Journal of the Optical Society of America B, Optical Society of America, 2010, 27 (7), pp.1492-1503. What we believe to be a new rigorous theoretical approach to the refraction of light at the interface of twodimensional photonic crystals is developed. The proposed method is based on the Dirichlet-to-Neumann (DtN) approach which consists of computing exactly the DtN operators associated with each half-space on both sides of the interface. It fully uses the properties of periodic optical media and takes naturally into account both the evanescent and propagative Bloch modes. Contrary to other proposed approaches, the new method is not based on modal expansions and their complicated electromagnetic field matching at the interfaces, but uses an operator vision. Intrinsically, each operator represents the effect along the interface of a particular medium independently of any medium and/or material that is placed in the other half-space. At the end, the whole computational effort to estimate DtN operators is restricted to the computation of a finite element problem in the periodicity cell of the photonic crystal. Field computations in arbitrary large part of the optical media can be then performed with a negligible computational effort. The method has been applied to the case of incoming plane waves as well as Gaussian beam profiles. It has successfully been compared with the standard plane wave expansion method and finite difference time domain (FDTD) simulations in the case of negative refraction, strongly dispersive, and lensing configurations. The proposed approach is amenable to the generalized study of dispersive phenomena in planar photonic crystals by a rigorous modeling approach avoiding the main drawbacks of FDTD. It is amenable to the study of arbitrary cascaded periodic optical media and photonic crystal heterostructures. © 2010 Optical Society of America. (10.1364/josab.27.001492)
  DOI : 10.1364/josab.27.001492
• About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Applicable Analysis, Taylor & Francis, 2010, 89 (11), pp.1745-1768. This article is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for Laplace's equation in domains with Lipschitz boundary. It completes the results obtained by Bourgeois [Conditional stability for ill-posed elliptic Cauchy problems: The case of C1,1 domains (part I), Rapport INRIA 6585, 2008] for domains of class C1,1. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired by Alessandrini et al. [Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Annali della Scuola Normale Superiore di Pisa 29 (2000), pp. 755-806]. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced by Lattés and Lions [Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967] to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates. © 2010 Taylor & Francis. (10.1080/00036810903393809)
  DOI : 10.1080/00036810903393809
• About stability and regularization of ill-posed elliptic Cauchy problems: The case of C1,1 domains
  
  • Bourgeois Laurent
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (4), pp.715-735. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C 1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV 9 (2003) 621-635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems. © EDP Sciences, SMAI, 2010. (10.1051/m2an/2010016)
  DOI : 10.1051/m2an/2010016
• Weighted regularization for composite materials in electromagnetism
  
  • Ciarlet Patrick
  • Lefèvre François
  • Lohrengel Stéphanie
  • Nicaise Serge
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (1), pp.75-108. In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of H(curl;Ω) whose fields u satisfy wα div(εu) ∈ L 2(Ω) and have vanishing tangential trace or tangential trace in L2(δΩ). The weight function w(x) is equivalent to the distance of x to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed. © EDP Sciences, SMAI 2009. (10.1051/m2an/2009041)
  DOI : 10.1051/m2an/2009041
• A kinetic mechanism inducing oscillations in simple chemical reactions networks
  
  • Coatléven Julien
  • Altafini Claudio
  Mathematical Biosciences and Engineering, AIMS Press, 2010, 7 (2), pp.301-312. It is known that a kinetic reaction network in which one or more secondary substrates are acting as cofactors may exhibit an oscillatory behavior. The aim of this work is to provide a description of the functional form of such a cofactor action guaranteeing the onset of oscillations in sufficiently simple reaction networks. (10.3934/mbe.2010.7.301)
  DOI : 10.3934/mbe.2010.7.301
• Analysis of Acoustic Wave Propagation in a Thin Moving Fluid
  
  • Joly Patrick
  • Weder Ricardo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2010, 70, pp.2449-2472. We study the propagation of acoustic waves in a fluid that is contained in a thin two-dimensional tube and that it is moving with a velocity profile that depends only on the transversal coordinate of the tube. The governing equations are the Galbrun equations or, equivalently, the linearized Euler equations. We analyze the approximate model that was recently derived by Bonnet-Bendhia, Durufle, and Joly to describe the propagation of the acoustic waves in the limit when the width of the tube goes to zero. We study this model for strictly monotonic stable velocity profiles. We prove that the equations of the model of Bonnet-Bendhia, Durufle, and Joly are well posed, i.e., that there is a unique global solution, and that the solution depends continuously on the initial data. Moreover, we prove that for smooth profiles the solution grows at most as t(3) as t -> infinity, and that for piecewise linear profiles it grows at most as t(4). This establishes the stability of the model in a weak sense. These results are obtained by constructing a quasi-explicit representation of the solution. Our quasi-explicit representation gives a physical interpretation of the propagation of acoustic waves in the fluid and provides an efficient way to compute the solution numerically. (10.1137/09077237X)
  DOI : 10.1137/09077237X
• Generation of Higher-Order Polynomial Bases of Nédélec H(curl) Finite Elements for Maxwell's Equations
  
  • Bergot Morgane
  • Lacoste Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6). The goal of this study is the automatic construction of a vectorial polynomial basis for Nédélec mixed finite elements, particular, the generation of finite elements without the expression of the polynomial basis functions, using symbolic calculus: the exhibition of basis functions has no practical interest.
• Time harmonic wave diffraction problems in materials with sign-shifting coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6), pp.1912-1919. Some electromagnetic materials present, in a given frequency range, an effective dielectric permittivity and/or magnetic permeability which are negative. We are interested in the reunion of such a "negative" material and a classical one. More precisely, we consider here a scalar model problem for the simulation of a wave transmission between two such materials. This model is governed by a Helmholtz equation with a weight function in the ΔΔ principal part which takes positive and negative real values. Introducing additional unknowns, we have already proposed in Bonnet-Ben Dhia et al. (2006) [1] some new variational formulations of this problem, which are of Fredholm type provided the absolute value of the contrast of permittivities is large enough, and therefore suitable for a finite element discretization. We prove here that, under similar conditions on the contrast, the natural variational formulation of the problem, although not "coercive plus compact", is nonetheless suitable for a finite element discretization. This leads to a numerical approach which is straightforward, less costly than the previous ones, and very accurate. (10.1016/j.cam.2009.08.041)
  DOI : 10.1016/j.cam.2009.08.041
• Comparison of High-Order Absorbing Boundary Conditions and Perfectly Matched Layers in the Frequency Domain
  
  • Rabinovich Daniel
  • Givoli Dan
  • Bécache Eliane
  International Journal for Numerical Methods in Biomedical Engineering, John Wiley and Sons, 2010, 26, pp.1351-1369.
• Efficient computation of photonic crystal waveguide modes with dispersive material
  
  • Schmidt Kersten
  • Kappeler Roman
  Optics Express, Optical Society of America - OSA Publishing, 2010, 18 (7), pp.7307-7322. The optimization of PhC waveguides is a key issue for successfully designing PhC devices. Since this design task is computationally expensive, efficient methods are demanded. The available codes for computing photonic bands are also applied to PhC waveguides. They are reliable but not very efficient, which is even more pronounced for dispersive material. We present a method based on higher order finite elements with curved cells, which allows to solve for the band structure taking directly into account the dispersiveness of the materials. This is accomplished by reformulating the wave equations as a linear eigenproblem in the complex wave-vectors k. For this method, we demonstrate the high efficiency for the computation of guided PhC waveguide modes by a convergence analysis. © 2010 Optical Society of America. (10.1364/oe.18.007307)
  DOI : 10.1364/oe.18.007307