Publications

2010

• Modélisation par éléments finis mixtes spectraux de capteurs piézoélectriques
  
  • Imperiale Sébastien
  • Cohen Gary
  • Leymarie Nicolas
  • Joly Patrick
  , 2010. Généralement constitués de matériaux piézo-composites, les capteurs multi-éléments sont de plus en plus utilisés en contrôle non destructif par ultrasons en raison de leur capacité à focaliser ou dévier un faisceau ultrasonore pour des composants de formes canoniques ou complexes. Dans l'objectif de modéliser avec précision la sensibilité en émission et en réception de tels capteurs, une modélisation en domaine temporel par éléments finis a été développée. Les équations de la piézoélectricité reposent sur le couplage des équations de Maxwell et de l'élastodynamique. L'hypothèse électrostatique est utilisée, ramenant les équations à résoudre à des équations dynamiques couplées avec une équation statique. La résolution de l'équation statique est coûteuse et doit être faite à chaque pas de temps de la simulation. Ce travail présente le développement d'un modèle homogénéisé basé sur des éléments finis d'ordre élevé [1-2]. La discrétisation par éléments finis mixtes spectraux d'ordre élevé est particulièrement efficace lorsqu'on cherche à approcher des solutions régulières. Toutefois ces méthodes étant coûteuses sur des géométries complexes et sur-maillées, l'emploi de techniques d'homogénéisation applicables à la structure périodique des capteurs piézo-composites a pour but de réduire la complexité du modèle étudié, en simplifiant sa géométrie, bénéficiant ainsi des performances des méthodes d'ordre élevé. Un code éléments finis 2D basé sur ce concept original a été développé permettant de simuler des capteurs en mode émission (champ rayonné par le capteur dans le sabot et dans la pièce) mais aussi en réception (écho de sabot ou écho de surface). [1] Higher-order numerical method for transient wave equations, G. Cohen, Springer-Verlag, 2002. [2] Modèles asymptotiques en ferromagnétisme : couches minces et homogénéisation, H. Haddar, Thèse, 2000.
• 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.
• 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
• 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
• 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.
• Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data
  
  • Bokanowski Olivier
  • Megdich Nadia
  • Zidani Hasnaa
  Numerische Mathematik, Springer Verlag, 2010, 115 (1), pp.1--44. On étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le caractère anti-diffusif du schéma est illustré par quelques exemples numériques. (10.1007/s00211-009-0271-1)
  DOI : 10.1007/s00211-009-0271-1
• 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.
• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• 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.
• 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.
• 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
• 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
• 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
• 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