Publications

2010

• Simulations numériques de la multidiffusion acoustique en conduit, comparaison avec des modèles analytiques
  
  • Lunéville Éric
  • Mercier Jean-François
  , 2010. Nous nous intéressons à la caractérisation des effets de multi-diffusion dans les guides d'ondes. Nous considérons la propagation acoustique en régime harmonique dans un conduit horizontal 2D à parois rigides. Nous avons développé une approche numérique pour déterminer les propriétés effectives d'un milieu hétérogène aléatoire dans un conduit. A l'aide de simulations directes nous déterminons un champ cohérent en faisant la moyenne des champs sur de nombreuses réalisations différentes de désordre. En interprétant ce champ cohérent comme une onde se propageant dans un milieu homogène équivalent, les propriétés effectives de ce milieu sont extraites. Une comparaison avec des modèles analytiques de la littérature, développés en milieu infini et non en conduit, est effectuée. Une méthode d'éléments finis est choisie pour permettre de traiter des diffuseurs de formes arbitraires. Afin de réduire les temps de calcul, la méthode des éléments finis est couplée à une représentation intégrale du champ diffracté. Elle réduit la taille du maillage, mais nécessite l'évaluation de la fonction de Green du guide. Une réduction supplémentaire des temps de calcul est obtenue en considérant, non pas des configurations de diffuseurs complètement aléatoires, mais des configurations périodiques perturbées : les diffuseurs sont placés sur un réseau de référence périodique puis sont déplacés localement aléatoirement. Ceci permet de paralléliser les calculs, en divisant le domaine de calcul en tranches verticales. Pour chaque tranche, la matrice de diffusion est calculée. Enfin, la matrice de diffusion de la couche entière est obtenue par la combinaison des matrices de diffusion. Les calculs de transmissions effectives et de nombres d'ondes effectifs montrent un bon accord avec plusieurs modèles analytiques, sauf pour certaines fréquences, les fréquences de bandes interdites des réseaux périodiques sous-jacents. Dans ce cas, le réseau périodique perturbé se comporte en moyenne comme un réseau périodique.
• Transitoires de piano et non-linéarités des cordes : mesures et simulations
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  , 2010. Au cours de leur mouvement, les cordes du piano sont soumises à des variations de tension consécutives aux variations de longueur induites par le déplacement transversal. Ce phénomène est particulièrement prononcé au moment de l'attaque par le marteau, le déplacement moyen étant alors la plupart du temps d'un ordre de grandeur supérieur au diamètre de la corde. Il s'ensuit un couplage entre les ondes de flexion transversale et l'onde de compression longitudinale. Cette dernière se propage environ 10 à 20 fois plus rapidement que les ondes de flexion. Dans le domaine temporel, l'onde longitudinale apparaît sous la forme d'un précurseur qui excite l'ensemble de la structure de l'instrument avant l'arrivée des premières oscillations transversales. Elle joue donc un rôle crucial dans la composition du transitoire de piano. Dans le domaine spectral, le couplage transverse-longitudinal peut être vu comme une composition de non-linéarités quadratiques et cubiques. En conséquence, on observe l'apparition de combinaison de fréquences appartenant aux spectres respectifs des deux types de vibration. Afin de mieux comprendre ces phénomènes, nous avons entrepris des simulations numériques. Le modèle utilisé est un système non linéaire couplé mettant en jeu la vibration transversale et la vibration longitudinale ainsi que l'angle de flexion permettant de prendre en compte la raideur. L'énergie totale du système est conservée au cours du temps, impliquant la stabilité de la solution. Le schéma numérique proposé est un schéma innovant, non linéaire, implicite, qui conserve un équivalent discret de l'énergie totale à chaque pas de temps, et assure ainsi la stabilité numérique dans un cas non linéaire où cette dernière est difficile à obtenir. Les résultats des simulations sont examinés et discutés par comparaison avec des formes d'onde expérimentales obtenues sur la table d'harmonie cordée d'un piano droit.
• Opérateur DtN pour les guides cylindriques à paroi traitée en présence d'un écoulement uniforme
  
  • Ouedraogo Boureima
  • Redon Emmanuel
  • Mercier Jean-François
  , 2010. On s'intéresse au problème de rayonnement acoustique d'une source dans une conduite cylindrique axisymétrique infinie dont la paroi est recouverte d'un matériau absorbant en présence d'un écoulement uniforme. Dans le but d'utiliser la méthode des éléments finis, le domaine infini doit être tronqué par une frontière artificielle sur laquelle une condition limite transparente est introduite. La méthode exposée dans ce travail consiste à écrire la condition transparente sous la forme d'un opérateur Dirichlet to Neumann (DtN) obtenu par la généralisation de travaux réalisés dans le cas d'un guide 2D infini. Cet opérateur est basé sur une décomposition modale qu'il est facile d'expliciter si le guide est rigide: le problème aux valeurs propres associé est alors auto-adjoint. En présence d'un matériau absorbant, modélisé par une impédance locale Z sur la paroi, des difficultés apparaissent car le problème aux valeurs propres n'est plus auto-adjoint. Néanmoins, une relation de bi-orthogonalité existe et permet de définir l'opérateur DtN. Dans le cas plus général d'un guide traité en présence d'un écoulement uniforme, le problème aux valeurs propres est encore non auto-adjoint et il n'existe plus de relation de bi-orthogonalité exacte pour la pression seule. Toutefois, nous montrons qu'il est possible d'écrire une relation d'orthogonalité valide asymptotiquement qui permet d'exprimer une bonne approximation de l'opérateur DtN cherché. La méthode est présentée dans le cas académique d'un guide cylindrique axisymétrique rectiligne, mais elle s'étend à des configurations plus complexes à condition que la partie étudiée soit comprise entre deux tronçons de guide rectiligne dans lesquels l'écoulement est uniforme. En particulier, le cas d'un guide non rectiligne en présence d'un écoulement potentiel sera présenté comme illustration.
• Higher-Order Finite Elements for Hybrid Meshes Using New Nodal Pyramidal Elements
  
  • Bergot Morgane
  • Cohen Gary
  • Duruflé Marc
  Journal of Scientific Computing, Springer Verlag, 2010, 42 (3), pp.345--381. We provide a comprehensive study of arbitrarily high-order finite elements defined on pyramids. We propose a new family of high-order nodal pyramidal finite element which can be used in hybrid meshes which include hexahedra, tetrahedra, wedges and pyramids. Finite elements matrices can be evaluated through approximate integration, and we show that the order of convergence of the method is conserved. Numerical results demonstrate the efficiency of hybrid meshes compared to pure tetrahedral meshes or hexahedral meshes obtained by splitting tetrahedra into hexahedra. (10.1007/s10915-009-9334-9)
  DOI : 10.1007/s10915-009-9334-9
• On the use of graphs in discrete tomography
  
  • de Werra Dominique
  • Costa Marie-Christine
  • Picouleau Christophe
  • Ries Bernard
  Annals of Operations Research, Springer Verlag, 2010, 175, pp.287-307. In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases. We show how various classical techniques of operations research like matching, 2−SAT, network flows are applied to derive some of these results. (This paper is an updated version of a tutorial published in 4'OR in 2008.) (10.1007/s10479-009-0649-6)
  DOI : 10.1007/s10479-009-0649-6
• Approximate Models for Wave Propagation Across Thin Periodic Interfaces
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  , 2010. This work deals with the scattering of acoustic waves by a thin ring which contains many regularly-spaced heterogeneties. We provide a complete description of the asymptotic of the solution with respect to the period and the thickness of the heterogeneities. Then, we build a simplified model replacing the thin perforated ring by an effective transmission condition. We pay particular attention to the stabilization of the effective transmission condition. Error estimates and numerical simulations are carried out to validate the accuracy of the model.
• A comparison of sample-based Stochastic Optimal Control methods
  
  • Girardeau Pierre
  , 2010. In this paper, we compare the performance of two scenario-based numerical methods to solve stochastic optimal control problems: scenario trees and particles. The problem consists in finding strategies to control a dynamical system perturbed by exogenous noises so as to minimize some expected cost along a discrete and finite time horizon. We introduce the Mean Squared Error (MSE) which is the expected $L^2$-distance between the strategy given by the algorithm and the optimal strategy, as a performance indicator for the two models. We study the behaviour of the MSE with respect to the number of scenarios used for discretization. The first model, widely studied in the Stochastic Programming community, consists in approximating the noise diffusion using a scenario tree representation. On a numerical example, we observe that the number of scenarios needed to obtain a given precision grows exponentially with the time horizon. In that sense, our conclusion on scenario trees is equivalent to the one in the work by Shapiro (2006) and has been widely noticed by practitioners. However, in the second part, we show using the same example that, by mixing Stochastic Programming and Dynamic Programming ideas, the particle method described by Carpentier et al (2009) copes with this numerical difficulty: the number of scenarios needed to obtain a given precision now does not depend on the time horizon. Unfortunately, we also observe that serious obstacles still arise from the system state space dimension.
• Shape derivatives of boundary integral operators in electromagnetic scattering
  
  • Costabel Martin
  • Le Louër Frédérique
  , 2010. We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem.
• 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
• Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint
  
  • Hermant Audrey
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2010, 61 (1), pp.85-127. This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points. (10.1007/s00245-009-9076-y)
  DOI : 10.1007/s00245-009-9076-y
• 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.
• Explicit polyhedral approximation of the Euclidean ball
  
  • Bonnans J. Frederic
  • Lebelle M.
  RAIRO - Operations Research, EDP Sciences, 2010, 44 (1), pp.45-60. We discuss the problem of computing points of IRn whose convex hull contains the Euclidean ball, and is contained in a small multiple of it. Given a polytope containing the Euclidean ball, we introduce its successor obtained by intersection with all tangent spaces to the Euclidean ball, whose normals point towards the vertices of the polytope. Starting from the L-infinity ball, we discuss the computation of the two first successors, and give a complete analysis in the case when n = 6. (10.1051/ro/2010003)
  DOI : 10.1051/ro/2010003
• Optimal control of a parabolic equation with time-dependent state constraints
  
  • Bonnans J. Frederic
  • Jaisson Pascal
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (7), pp.4550-4571. In this paper we study the optimal control problem of the heat equation by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied at each time. Using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations, we show that both the control and multiplier are continuous in time. Under some natural geometric hypotheses, we can prove that extended polyhedricity holds, allowing to obtain no-gap second-order optimality conditions, that characterize quadratic growth. An expansion of the value function and of approximate solutions can be computed for a directional perturbation of the r.h.s. of the state equation.
• 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
• An efficient data structure to solve front propagation problems
  
  • Bokanowski Olivier
  • Cristiani Emiliano
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2010, 42 (2), pp.251--273. In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an adapted anti-diffusive Ultra-Bee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d=2,3,4. (10.1007/s10915-009-9329-6)
  DOI : 10.1007/s10915-009-9329-6
• Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide
  
  • Baronian Vahan
  • Bonnet-Ben Dhia Anne-Sophie
  • Lunéville Éric
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6), pp.1945-1952. This work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries, etc.) located in a 3D infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on some artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non-standard properties of elastic modes. However, a biorthogonality relation allows us to build an operator, relating hybrid displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the 2D case. (10.1016/j.cam.2009.08.045)
  DOI : 10.1016/j.cam.2009.08.045
• Finite element simulations of multiple scattering in acoustic waveguides
  
  • Lunéville Éric
  • Mercier Jean-François
  Waves in Random and Complex Media, Taylor & Francis, 2010, 20 (4), pp.615-633. We develop a numerical method to characterize multiple-scattering effects in a duct. To reduce time computations, a FEM method coupled to an integral representation of the scattered pressure field, requiring the evaluation of the duct Green's function, is used, combined with the consideration of random scatterer configurations in the form of perturbed periodic arrays. This strategy reduces the mesh size and allows parallel computations. Application to the computation by an averaging process of effective transmissions is presented. When sending a plane wave, although waveguides involve mode conversion, the mean field is found to be a plane wave. Good agreement is found when comparing to analytical models, not initially designed for a duct, except for some frequencies, recognized to be the band gap frequencies of the underlying periodic arrays. © 2010 Taylor & Francis. (10.1080/17455031003753000)
  DOI : 10.1080/17455031003753000
• Identification of generalized impedance boundary conditions in inverse scattering problems
  
  • Bourgeois Laurent
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2010, 4 (1), pp.19-38. In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact subset of a finite dimensional space. (10.3934/ipi.2010.4.19)
  DOI : 10.3934/ipi.2010.4.19
• 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
• Asymptotic modelling of conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2010, 61 (4), pp.603-626. We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H1-modelling error for an expansion with N terms is bounded by O(ε^{N+1}) in the exterior of the sheet and by O(ε^{N+1/2}) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results. (10.1007/s00033-009-0043-x)
  DOI : 10.1007/s00033-009-0043-x
• 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