Publications

2010

• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• 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
• Electrowetting of a 3D drop: Numerical modelling with electrostatic vector fields
  
  • Ciarlet Patrick
  • Scheid Claire
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (4), pp.647-670. The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case. © EDP Sciences, SMAI, 2010. (10.1051/m2an/2010014)
  DOI : 10.1051/m2an/2010014
• A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems, IOP Publishing, 2010, 26 (9), pp.095016. In this paper, we introduce a new version of the method of quasi-reversibility to solve the ill-posed Cauchy problems for the Laplace's equation in the presence of noisy data. It enables one to regularize the noisy Cauchy data and to select a relevant value of the regularization parameter in order to use the standard method of quasi-reversibility. Our method is based on duality in optimization and is inspired by the Morozov's discrepancy principle. Its efficiency is shown with the help of some numerical experiments in two dimensions. © 2010 IOP Publishing Ltd. (10.1088/0266-5611/26/9/095016)
  DOI : 10.1088/0266-5611/26/9/095016
• Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string
  
  • Chabassier Juliette
  • Joly Patrick
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2010, 199 (45-48), pp.2779-2795. This paper considers a general class of nonlinear systems, "nonlinear Hamiltonian systems of wave equations". The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). 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 trivial. 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. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the "geometrically exact model", or approximations of this model. Numerical results are presented. (10.1016/j.cma.2010.04.013)
  DOI : 10.1016/j.cma.2010.04.013
• Finite Element Modeling of Airflow During Phonation
  
  • Sidlof Petr
  • Lunéville Éric
  • Chambeyron Colin
  • Doaré Olivier
  • Chaigne A
  • Horáček J
  Journal of Computational and Applied Mechanics, Miskolci Egyetemi Kiadó, 2010, 4, pp.121-132. In the paper a mathematical model of airflow in human vocal folds is presented. The geometry of the glottal channel is based on measurements of excised human larynges. The airflow is modeled by nonstationary incompressible Navier-Stokes equations in a 2D computational domain, which is deformed in time due to vocal fold vibration. The paper presents numerical results and focuses on flow separation in glottis. Quantitative data from numerical simulations are compared to results of measurements by Particle Image Velocimetry (PIV), performed on a scaled self-oscillating physical model of vocal folds.
• Weak vector and scalar potentials. Applications to Poincaré's theorem and Korn's inequality in Sobolev spaces with negative exponents.
  
  • Amrouche Chérif
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Analysis and Applications, World Scientific Publishing, 2010, 8 (1), pp.1-17. In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality. (10.1142/s0219530510001497)
  DOI : 10.1142/s0219530510001497