Publications

2015

• Refinements in the Jungle Universes
  
  • Simon-Petit Alicia
  • Yap Han Hoe
  • Perez Jérôme
  , 2016. How can effective barotropic matter emerge from the interaction of cosmological fluids in an isotropic and homogeneous cosmological model ? The dynamics of homogeneous and isotropic Friedmann-Lemaˆtre-Robertson-Walker universes is a natural special case of generalized Lotka-Volterra systems where each of the universes fluid components can be seen as a competitive species in a predator-prey model. (Jungle universe : [7]) In addition to numerical simulations illustrating this behaviour among the barotropic fluids filling the universe, we analytically pinpoint that effective time-dependent barotropic indices can arise from a physical coupling between those fluids whose dynamics could then look like that of another type of cosmic fluid, such as a cosmological constant. Since the nature of dark energy is still unknown, this dynamical approach could help understanding some of the properties of dark matter and dark energy at large cosmological scales.
• Theory of weakly nonlinear self-sustained detonations
  
  • Faria Luiz
  • Kasimov Aslan
  • Rosales Rodolfo
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2015, 784, pp.163-198. (10.1017/jfm.2015.577)
  DOI : 10.1017/jfm.2015.577
• Hamilton Jacobi Bellman Approach for some applied optimal control problems.
  
  • Assellaou Mohamed
  , 2015. The main objective of this thesis is to analyze the Hamilton Jacobi Bellman approach for some control problems of unusual forms. The first work is devoted to the numerical approximations of unbounded and discontinuous value functions associated with some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). The proof is based on classical shaking and regularization techniques. The second contribution concerns the probabilistic reachablility analysis. In particular, we characterize the chance-constrained backward reachable set by a level set of a discontinuous value function and we use the first theoretical results to derive the corresponding error estimates. In the second part of this thesis, we study a class of state constrained optimal control problem with maximum cost. We first describe the epigraph of the value function by an auxiliary optimal control problem whose value function is Lipschitz continuous. We show that the new value function is the unique Lipschitz continuous viscosity solution of a Hamilton Jacobi equation with a Dirichlet condition. Here, we give a review of the optimal trajectories and the associated feedback control for such control problems. In particular, we prove the convergence of a sequence of approximated optimal trajectories to the continuous one. We establish a link between the control problem and a viability kernel associated with an exit time function. The obtained results for the state constrained control problem with maximum cost are then extended to the state constrained control problem with Bolza cost. The study is motivated by a real application: the abort landing during low altitude wind-shears. Many algorithms of reconstruction of optimal feedback trajectories are studied and compared from numerical and theoretical points of view.
• Étude mathématique et numérique de structures plasmoniques avec coins
  
  • Carvalho Camille
  , 2015. Dans cette thèse, nous nous intéressons à la propagation d’ondes électromagnétiques dans des structures plasmoniques, composées d’un diélectrique et d’un métal. Les métaux exhibent aux fréquences optiques des propriétés électromagnétiques inhabituelles comme une permittivité diélectrique négative, alors que les diélectriques possèdent une permittivité positive. Ce changement de signe de permittivité a pour conséquence la propagation d’ondes de surface (plasmons de surface) à l’interface métal-diélectrique. Cette thèse concerne le cas où cette interface présente des coins. Des études théoriques ont été menées ces dernières années, combinant la méthode de la T-coercivité et l’analyse des singularités de coins. En particulier, il a été mis en évidence l’existence de deux régimes, selon les paramètres du problème (fréquence, matériau, géométrie). L’objectif de cette thèse est de développer, dans le cas bidimensionnel, une méthode numérique stable pour chacun de ces deux régimes, en apportant un traitement spécifique aux coins. Dans le premier régime (où les solutions appartiennent à l’espace "d’énergie classique"), nous développons des règles de maillages adaptées à la géométrie de l’interface pour garantir la convergence optimale des méthodes d’approximation par éléments finis : on parle de maillages T-conformes. Dans le second régime (où les solutions ne sont plus d’énergie finie), nous proposons une méthode numérique originale utilisant des PMLs (Perfectly Matched Layers) aux coins pour capturer les singularités, appelées ondes de trou noir car elles transportent de l’énergie absorbée par les coins. Nous appliquons ces techniques numériques à deux problèmes physiques : la diffraction par une onde plane d’une inclusion métallique polygonale, et la détermination des modes guidés d’un guide d’ondes plasmonique à section polygonale. Pour le problème de diffraction, nous montrons que les coins de l’inclusion métallique peuvent absorber de l’énergie, transportée par les ondes de trou noir, et nous calculons numériquement l’énergie absorbée par chaque coin. L’étude des modes guidés du guide plasmonique quant à elle s’écrit sous la forme d’un problème de théorie spectrale non classique. En présence d’ondes de trou noir, les valeurs propres associées aux modes guidés sont plongées dans le spectre essentiel. Pour les dévoiler, on utilise à nouveau des PMLs aux coins, ce qui revient à calculer les valeurs propres d’un opérateur étendu dont le spectre est discret.
• Chance constrained optimization of a three-stage launcher
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  , 2015.
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2015. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods.
• A Rellich type theorem for the Helmholtz equation in a conical domain
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Hazard Christophe
  • Tonnoir Antoine
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2015. We prove that there cannot exist square-integrable nonzero solutions to the Helmholtz equation in an axisymmetric conical domain whose vertex angle is greater than π. This implies in particular the absence of eigenvalues embedded in the essential spectrum of a large class of partial differential operators which coincide with the Laplacian in the conical domain. (10.1016/j.crma.2015.10.015)
  DOI : 10.1016/j.crma.2015.10.015
• Modèles asymptotiques et simulation numérique pour la diffraction d'ondes par des petites hétérogénéités
  
  • Marmorat Simon
  , 2015. Cette thèse est consacrée à l'étude du problème de la diffraction d'une onde acoustique par un ensemble de petites hétérogénéités pénétrables ainsi qu'au développement de méthodes de simulation numérique dédiées à la résolution efficace de ce type de problèmes. La principale nouveauté de ces travaux provient du fait que nous traitons ce problème dans le domaine temporel.La première partie de ce manuscrit est consacrée à l'analyse asymptotique du problème de diffraction, menée à bien grâce à la méthode des développements asymptotiques raccordés, le petit paramètre étant la taille caractéristique des défauts ε. Ceci nous permet d'obtenir un développement du champ acoustique comme perturbation du problème sans défauts. Nous prouvons un résultat de consistance entre le champ exact et son développement asymptotique en ε.Dans la seconde partie, en s'appuyant sur les résultats de l'analyse asymptotique, nous proposons deux modèles approchés pour le problème de diffraction. Ces deux modèles sont bien-posés et leur solution sont chacune des approximations précises du champ total. La principale caractéristique de ces modèles approchés est qu'ils s'appuient tous deux sur une équation d'onde dans le milieu ambiant (sans défauts), couplée à des termes sources auxiliaires permettant de rendre compte de la présence des défauts. Il est ainsi envisageable, pour traiter ces problèmes approchés, d'utiliser une méthode de discrétisation par éléments finis présentant des performances de temps de calcul similaires au cas de la propagation d'une onde dans l'espace libre, puisque l'opérateur des ondes sous-jacent s'appuie sur une géométrie indépendante des petits défauts. Nous présentons un certain nombre de résultats numériques permettant de valider les deux modèles proposés ainsi qu'une analyse d'erreur numérique.
• Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion
  
  • Bayen Térence
  • Mairet Francis
  • Martinon Pierre
  • Sebbah Matthieu
  Optimal Control Applications and Methods, Wiley, 2015, 36 (6), pp.750-773. In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks. (10.1002/oca.2127)
  DOI : 10.1002/oca.2127
• Hausdorff volume in non equiregular sub-Riemannian manifolds
  
  • Ghezzi Roberta
  • Jean Frédéric
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2015, 126, pp.345–377. In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures). (10.1016/j.na.2015.06.011)
  DOI : 10.1016/j.na.2015.06.011
• Spectrum for a small inclusion of negative material
  
  • Chesnel Lucas
  • Claeys Xavier
  • Nazarov Sergueï A.
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2015, 66 (5), pp.2173-2196. We study a spectral problem (P δ) for a diffusion like equation in a 3D domain Ω. The main originality lies in the presence of a parameter σ δ , whose sign changes on Ω, in the principal part of the operator we consider. More precisely, σ δ is positive on Ω except in a small inclusion of size δ > 0. Because of the sign-change of σ δ , for all δ > 0 the spectrum of (P δ) consists of two sequences converging to ±∞. However, at the limit δ = 0, the small inclusion vanishes so that there should only remain positive spectrum for (P δ). What happens to the negative spectrum? In this paper, we prove that the positive spectrum of (P δ) tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of (P δ) behaves like δ −2 µ for some constant µ < 0. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.
• Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Le Louër Frédérique
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, 297 (1), pp.62–83. It has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies. (10.1016/j.cma.2015.08.013)
  DOI : 10.1016/j.cma.2015.08.013
• Underwater topography invisible for surface waves at given frequencies
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Nazarov Sergei
  • Taskinen Jari
  Wave Motion, Elsevier, 2015, 57. We consider scattering of surface waves modelled by the linear water wave equation in an unbounded two-dimensional domain of finite depth, at a given frequency and a given incidence. Using asymptotic analysis for small perturbations of the bottom shape, we build a fixed-point equation whose unique solution is a shape which cannot be detected by a distant observer. The method works at any incidence except π/4. (10.1016/j.wavemoti.2015.03.008)
  DOI : 10.1016/j.wavemoti.2015.03.008
• Stratified discontinuous differential equations and sufficient conditions for robustness
  
  • Hermosilla Cristopher
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.23. This paper is concerned with state-constrained discontinuous ordinary differential equations for which the corresponding vector field has a set of singularities that forms a stratification of the state domain. Existence of solutions and robustness with respect to external perturbations of the righthand term are investigated. Moreover, notions of regularity for stratifications are discussed. (10.3934/dcds.2015.35.4415)
  DOI : 10.3934/dcds.2015.35.4415
• Cournot Maps for Intercepting Evader Evolutions by a Pursuer
  
  • Aubin Jean-Pierre
  • Chen Luxi
  • Desilles Anna
  Dynamic Games and Applications, Springer Verlag, 2015, 5 (3), pp.275-296. Instead of studying evolutions governed by an evolutionary system starting at a given initial state on a prescribed future time interval, finite or infinite, we tackle the problem of looking both for a past interval [T − D, T ] of duration D and for the viable evolutions arriving at a prescribed terminal state at the end of the temporal window (and thus telescoping if more than one such evolutions exist). Hence, given time-duration dependent evolutionary system and viability constraints, as well as time dependent departure constraints, the Cournot map associates with any terminal time T and state x the durations D(T, x) of the intervals [T − D(T, x), T ], the starting (or initial) states at the beginning of the temporal window from which at least one viable evolution will reach the given terminal state x at T . Cournot maps can be used by a Pursuer to intercept an evader’s evolution in dynamic game theory. After providing some properties of Cournot maps are next investigated, above all, the regulation map piloting the viable evolutions at each time and for each duration from the beginning of the temporal window up to terminal time. The next question investigated is the selection of controls or regulons in the regulation map whenever several of them exist. Selection processes are either time dependent, when the selection operates at each time, duration, and state for selecting a regulon satisfying required properties (for instance, minimal norm, minimal speed), orintertemporal. In this case, viable evolutions are required to optimize some prescribed intertemporal functional, as in optimal control. This generates value functions, the topics of the second part of this study. An example is provided: the Pursuer is a security vehicle making the rounds along a predetermined path, the departure tube, for reaching any network location where and when alarms sound to signal the location (of the evader). The software of the Cournot algorithm computes the minimal duration and the moment when the Pursuer leaves its round to reach the detected location as soon as possible and how to proceed by embedding in the Pursuer system the graph of the feedback map governing the evolution of the Pursuer vehicle. (10.1007/s13235-014-0133-z)
  DOI : 10.1007/s13235-014-0133-z
• Error Estimates for Second Order Hamilton-Jacobi-Bellman Equations. Approximation of Probabilistic Reachable Sets
  
  • Assellaou Mohamed
  • Bokanowski Olivier
  • Zidani Hasnaa
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.3933 - 3964. This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-constrained reachability sets. The latters will be characterized as level sets of discontinuous value functions associated to adequate stochastic control problems. A precise analysis of the level-set approach is carried out and some numerical simulations are given to illustrate the approach. (10.3934/dcds.2015.35.3933)
  DOI : 10.3934/dcds.2015.35.3933
• Special issue on New trends in optimal control
  
  • Caillau Jean-Baptiste
  • Grüne Lars
  • de Pinho Maria Do Rosário
  • Trélat Emmanuel
  • Zidani Hasnaa
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.i - iv. This special volume gathers a number of new contributions addressing various topics related to the field of optimal control theory and sensitivity analysis. The field has a rich and varied mathematical theory, with a long tradition and a vibrant body of applications. It has attracted a growing interest across the last decades, with the introduction of new ideas and techniques, and thanks to various new applications. (10.3934/dcds.2015.35.9i)
  DOI : 10.3934/dcds.2015.35.9i
• Matched asymptotics approach to the construction and justification of reduced graph models for 3D Maxwell's equations in networks of thin co-axial cables
  
  • Beck Geoffrey
  • Imperiale Sebastien
  • Joly Patrick
  , 2015. We consider electromagnetic wave propagation in domains constituted by thin coaxial cables (made of a dielectric material which surrounds a metallic inner-wire) and a small junction. The goal is to trim down 3D Maxwell's equations in this complicated geometry to a quantum graph (see [3]) in which, along each edge, one is reduced to compute the electrical potential and current a by solving wave equations (the teleg-rapher's model) coupled by vertex conditions. In this work, using the method of matched asymp-totics, we propose improved Kirchhoff conditions and we give a rigorous justification of such a model reduction.
• Higher-order expansion of misfit functional for defect identification in elastic solids
  
  • Bonnet Marc
  • Cornaggia Rémi
  , 2015. In this work, least-squares functionals commonly used for defect identification are expanded in powers of the small radius of a trial inclusion, in the context of time-harmonic elastodynamics, generalizing to higher orders the concept of topological derivative. Such expansion , whose derivation and evaluation are facilitated by using an adjoint state, provides a basis for the quantitative estimation of flaws whereby a region of interest may be exhaustively probed at reasonable computational cost.
• Quantification of the model risk in finance and related problems
  
  • Laachir Ismail
  , 2015. The main objective of this thesis is the study of the model risk and its quantification through monetary measures. On the other hand we expect it to fit a large set of complex (exotic) financial products. The first two chapters treat the model risk problem both from the empirical and the theoretical point of view, while the third chapter concentrates on a theoretical study of another financial risk called basis risk. In the first chapter of this thesis, we are interested in the model-independent pricing and hedging of complex financial products, when a set of standard (vanilla) products are available in the market. We follow the optimal transport approach for the computation of the option bounds and the super (sub)-hedging strategies. We characterize the optimal martingale probability measures, under which the exotic option price attains the model-free bounds; we devote special interest to the case when the martingales are positive. We stress in particular on the symmetry relations that arise when studying the option bounds. In the second chapter, we approach the model risk problem from an empirical point of view. We study the optimal management of a natural gas storage and we quantify the impact of that risk on the gas storage value. As already mentioned, the last chapter concentrates on the basis risk, which is the risk that arises when one hedges a contingent claim written on a non-tradable but observable asset (e.g. the temperature) using a portfolio of correlated tradable assets. One hedging criterion is the mean-variance minimization, which is closely related to the celebrated Föllmer-Schweizer decomposition. That decomposition can be deduced from the resolution of a special Backward Stochastic Differential Equations (BSDEs) driven by a càdlàg martingale. When this martingale is a standard Brownian motion, the related BSDEs are strongly related to semi-linear parabolic PDEs. In that chapter, we formulate a deterministic problem generalizing those PDEs to the general context of martingales and we apply this methodology to discuss some properties of the Föllmer-Schweizer decomposition. We also give an explicit expression of such decomposition of the option payoff when the underlying prices are exponential of additives processes.
• Méthodes itératives de décomposition de domaine sans recouvrement avec convergence géométrique pour l'equation de Helmholtz
  
  • Lecouvez Matthieu
  , 2015. Cette thèse s’intéresse aux aspects mathématiques des méthodes itératives de résolution basées sur la décomposition de domaine et appliquées à la simulation numérique de propagation d’ondes harmoniques. Plus précisément, nous nous sommes intéressés à l’élaboration de conditions de transmission optimisées garantissant la convergence exponentielle de ce type de méthodes. Une telle convergence requiert l’utilisation d’opérateurs de transmission non locaux puisqu’ils doivent correspondre formellement à un opérateur pseudo-différentiel d’ordre 1. Une méthode de localisation des opérateurs est proposée pour réduire le coût engendré par ces opérateurs tout en conservant leurs propriétés et donc la convergence exponentielle de ces méthodes itératives. Dans un cadre général, la convergence des méthodes de décomposition de domaine est établie pour toute une classe d’opérateurs vérifiant certaines conditions de positivité et d’isomorphisme entre espaces de Sobolev. Nous proposons ensuite plusieurs opérateurs différents, dépendants de paramètres, qui vérifient les conditions nécessaires à la convergence exponentielle de la méthode. Un premier type d’opérateur se base sur les normes des espaces de Sobolev d’ordre fractionnaire, tandis qu’un second type d’opérateur découle des potentiels de Riesz (puissance fractionnaire de l’opérateur de Laplace-Beltrami). Nous proposons ensuite un schéma numérique permettant d’appliquer la théorie développée à une méthode d’éléments finis. Une analyse modale dans le cas de géométries simples vient tout d’abord valider les conclusions théoriques de convergence exponentielle, puis plusieurs expériences numériques mettent en évidence les avantages des conditions de transmission proposées, et particulièrement dans le cas où une précision très fine sur la solution est demandée.
• Optimization of the launcher ascent trajectory leading to the global optimum without any initialization: the breakthrough of the HJB approach
  
  • Bourgeois Eric
  • Bokanowski Olivier
  • Desilles Anna
  • Zidani Hasnaa
  , 2015 (Paper 410), pp.12 pages. We investigate the resolution of the launcher ascent trajectory problem by the so-called Hamilton- Jacobi- Bellman (HJB) approach, relying on the Dynamic Programming Principle. The method gives a global optimum and does not need any initialization procedure. Despite these advantages, this approach is seldom used, because of the difficulties of computing the solution of the HJB equation for high dimension problems. The present study shows that an efficient resolution is found. An illustration of the method is proposed on a heavy class launcher, for a typical GEO mission. This study has been performed in the frame of the CNES Launchers Research & Technology program.
• Global optimization approach for the climbing problem of multi-stage launchers
  
  • Desilles Anna
  • Zidani Hasnaa
  • Bokanowski Olivier
  • Bourgeois Eric
  , 2015. This paper deals with a problem of trajectory optimization of the flight phases of a threestage launcher. The aim of this optimization problem is to minimize the consumption of ergols that is need to steer the launcher from the Earth to the GEO. Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman (HJB) approach. An interesting by-product of the HJB approach is the synthesis of the optimal control in feedback form. Once the HJB equation is solved, for any starting point, the reconstruction of the optimal trajectory can be performed in real time. We aim at showing that combining several new techniques for the HJB approach we can obtain efficient solutions to a fully nonlinear control problem. The global optimization procedure proposed here takes also into account parametric optimisation that appears in the flight phases. Several recent advanced numerical techniques for HJB equations (high order finite difference schemes, parallel computing) are used to solve a 6-dimensional optimal control problem within a reasonable CPU time. This work has been carried out in the frame of CNES Launchers’ Research and Technology program
• Conditions transparentes pour la diffraction d'ondes en milieu élastique anisotrope
  
  • Tonnoir Antoine
  , 2015. Cette thèse est motivée par la simulation numérique du Contrôle Non Destructif par ultrasons. Elle vise à concevoir une méthode de calcul par éléments finis (EF) de la diffraction d’ondes élastiques harmoniques en temps par un défaut borné dans une plaque anisotrope infinie. L'objectif est de tenir compte du caractère non borné de la plaque tout en restreignant les calculs EF à une zone bornée autour du défaut. Ce point est difficile en raison de l'anisotropie, et, en particulier, les méthodes de type couches absorbantes parfaitement adaptées sont inopérantes. Dans cette thèse, nous avons considéré principalement des cas bidimensionnels plus simples qui nous ont permis de mettre en place les ingrédients essentiels d'une méthode destinée au cas tridimensionnel de la plaque. La première partie traite du problème de diffraction dans une bande infinie. L'approche classique consiste à écrire des conditions transparentes en raccordant sur une frontière le déplacement et la contrainte axiale exprimés à l'aide des modes de la plaque dans les parties saines d'une part, et des EF dans la zone perturbée d'autre part. Nous avons mis en évidence l'intérêt d'écrire ces raccords sur deux frontières séparées en introduisant un recouvrement entre le domaine modal et EF. Nous pouvons ainsi exploiter les relations de bi-orthogonalité valables pour une anisotropie arbitraire, et également accélérer la convergence des méthodes itératives de résolution. Dans la seconde partie, qui constitue le cœur de la thèse, nous avons étudié le problème de diffraction dans un milieu anisotrope infini dans les deux directions. L'idée clé est que l'on peut exprimer (via la transformée de Fourier) la solution dans un demi-plan en fonction de sa trace sur son bord. Ainsi, l'approche consiste à coupler plusieurs représentations analytiques de la solution dans des demi-plans entourant le défaut (au moins 3) avec la représentation EF. La difficulté est d'assurer la compatibilité de ces représentations, en particulier dans les intersections infinies des demi-plans. Cela nous conduit à une reformulation couplant, via des opérateurs intégraux, à la fois la solution dans un domaine borné contenant le défaut, et ses traces sur les bords des demi-plans. Numériquement, une troncature et une discrétisation dans les variables d'espace et de Fourier sont nécessaires. Pour chacune de ces deux parties, les méthodes ont été implémentées et validées à l'aide d'un code C++ développé pendant la thèse, d'abord dans le cas scalaire acoustique plus simple, puis dans le cas de l'élasticité.
• Optimisation d’un lanceur
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  , 2015.