Comportement mécanique par homogénéisation de la dynamique des disclocations

La plasticité des métaux est principalement due à la présence de lignes de défauts appelées dislocations. En ce qui concerne la déformation plastique des cristaux, elle résulte principalement du déplacement de ces dislocations, dont l'ordre de longueur typique dans les métaux est 10^-6 m et l'épaisseur 10^-9 m. Dans cette étude, on considère la dynamique collective des dislocations en interactions. Dans ces conditions, les dislocations se regroupent en « murs » pour former des structures ordonnées à longue distance

Analyse Numérique et discrétisation par éléments spectraux avec joints des équations tridimensionnelles de l'électromagnétisme

Christine Bernardi, Tahar Zamène Boulmezaoud, Monique Dauge, Georges Delaunay, Yvon Maday, Jean Claude Nédélec (rapporteur), Pierre-Arnaud Raviart, Barbara Wohlmuth (rapporteur).The aim of this thesis is the numerical analysis and the discretisation of Maxwell's equations. The first purpose is to investigate time-harmonic Maxwell's equations in Lipschitz and multiply connected 3D bounded cavities. We prove the wellposedness of the current source problem by means of new formulation. The starting point is the curl-curl second order equation satisfied by the magnetic field. The use of an appropriate compact operator is the heart of the proof. Next, we propose a discretisation relying on spectral element and numerical integration. We prove the convergence of the discrete solution to the exact one and we derive errors estimates. Examples of the numerical solution are given and compared with those obtained by a finite element method in the case of a simple geometry. The last part is to propose a mortar spectral element method for solving heteregeneous Maxwell's equation in 3D bounded cavities. This part is mainly divided into two parts. The method is based on non-conforming decomposition of the domain into the union of non-overlapping parallelipeds. The first part is devoted to the presentation and the numerical analysis of the method. In the second part, we expose some numerical results which confirm the performance of the method.Cette thèse a pour objet l'analyse et la discrétisation numérique des équations tridimensionnelles de l'électromagnétisme. Ces travaux débutent par l'étude de ces équations dans un domaine b orné multiplement connexe. Un théorème d'existence général a été établi, en proposant une nouvelle approche du problème, en le reformulant à l'aide d'un opérateur approprié, tenant compte des omplexités géométriques du domaine. Dans la suite, après avoir donnée un résultat de régularité, on propose une approximation numérique de la solution par une méthode spectrale. La méthode est, d'une part, analysée numériquement dans le cas d'une décomposition conforme du domaine, et d'autre, implantée dans le cadre d'un code 3D. Des tests numériques illustrant les prévisions théoriques sont exposés et comparés à ceux obtenus par une méthode d'éléments finis de type P1 qu'on présentera sommairement. En outre, les quatre premières valeurs propres du problème discret sont calculées et comparées à celui du spectre exact. La dernière partie de cette thèse est consacrée à l'étude d'une décomposition de domaine par une méthode spectrale avec joints pour le problème de Maxwell. Il est utile de souligner que les paramètres physiques sont considérés dans cette partie comme pouvant être hétérogènes. On applique cette méthode à un problème type présenté. Ce dernier permet d'unifier deux approches qui habituellement sont distinguées: le problème d'évolution de Maxwell, et le problème de Maxwell en régime harmonique. Des estimations d'erreurs sont démontrées, elles reposent sur un lemme, qui est une variante du second lemme de Strang, permettant de décomposer l'erreur en la somme de trois erreurs principales: l'erreur sur la meilleure approximation, l'erreur de consistance et l'erreur d'intégration numérique. Cette dernière étant obtenue de ma ière classique, les deux autres erreurs ont nécessité une recherche plus approfondie, notamment, la définition d'opérateurs discrets et un Lemme d'augmentation de degré pour l'erreur sur la meilleure approximation. Enfin des courbes d'erreurs et des tests numériques sont exposés validant un code de calcul tridimensionnel développé pour l'approximation de la solution du problème type (pour des paramètres physiques homogènes et hétérogènes)

A mortar spectral element method for 3D Maxwell's equations

International audienceIn this paper we propose a mortar spectral element method for solving Maxwell's equations in 3D bounded cavities. The method is based on a non-conforming decomposition of the domain into the union of non-overlapping parallelepipeds. After proving an error estimate, we present some 3D computational results which confirm the performance of the method

Orthogonal projection algorithm for first and second order total variation denoising

International audienceThe denoising problem is the process of removing the noise from a degraded image. As we know, the Rodin Osher Fatemi (ROF) denoising model based on total variation is a robust approach for solving the ill-posed problem. To avoid the staircasing effects caused by the first order total variation, the second order one is proposed. In this work, we present an orthogonal projection algorithm for solving the ROF model with first and second order total variation. The efficiency and robustness against noise of the proposed model are illustrated and compared with the classical methods through numerical simulations

Blind deconvolution using bilateral total variation regularization: a theoretical study and application

International audienceBlind image deconvolution recovers a deblurred image and the blur kernel from a blurred image. From a mathematical point of view, this is a strongly ill-posed problem and several works have been proposed to address it. One successful approach proposed by Chan and Wong, consists in using the total variation (TV) as a regularization for both the image and the kernel. These authors also introduced an Alternating Minimization (AM) algorithm in order to compute a physical solution. Unfortunately, Chanâs approach suffers in particular from the ringing and staircasing effects produced by the TV regularization. To address these problems, we propose a new model based on Bilateral Total Variation (BTV) regularization of the sharp image keeping the same regularization for the kernel. We prove the existence of a minimizer of a proposed variational problem in a suitable space using a relaxation process. We also propose an AM algorithm based on our model. The efficiency and robustness of our model are illustrated and compared with the TV method through numerical simulations

A tool for scanning document-images with a photophone or a digicam

International audienceIn this work, we propose a tool to scan a document-image acquired with a cameraphone. Firstly, we try to reduce the noise in the document-image. Then we build a new image by cropping or by perspective rectifying the denoised one. From this step, we can expect the document to a real quadrangle. The new document is analyzed and we try to find images, logo or non text element in the document-image with the aid of an image segmentation. At this stage, we provide deux parts of the document image: the text part and the " non text " part of the document-image (images, logos, non ...). The text part of the document-image is enhanced by an original pde's based model that we proposed. The " non text " document is enhanced by classical methods such as retinex processing. Then, we merge both parts of the document image by a poisson image editing. The effectiveness and the robustness of the proposed process are shown on numerical examples in real-world situation (images acquired from cameraphones)

An Evolutionary Approach For Blind Deconvolution Of Barcode Images With Nonuniform Illumination

International audience—This paper deals with a joint nonuniform illumination estimation and blind deconvolution for barcode signals by using evolutionary algorithms. Indeed, such optimization problems are highly non convex and a robust method is needed in case of noisy and/or blurred signals and nonuniform illumination. Here, we present the construction of a genetic algorithm combining discrete and continuous optimization which is successfully applied to decode real images with very strong noise and blur

Kinetic BGK model for a crowd: Crowd characterized by a state of equilibrium

International audienceThis article focuses on dynamic description of the collective pedestrian motion based on the kinetic model of Bhatnagar-Gross-Krook. The proposed mathematical model is based on a tendency of pedestrians to reach a state of equilibrium within a certain time of relaxation. An approximation of the Maxwellian function representing this equilibrium state is determined. A result of the existence and uniqueness of the discrete velocity model is demonstrated. Thus, the convergence of the solution to that of the continuous BGK equation is proven. Numerical simulations are presented to validate the proposed mathematical model

Estimating contact forces and pressure in a dense crowd: Microscopic and macroscopic models

International audienceThis paper deals with the estimation of pressure at collisions times during the movement of a dense crowd. Through the non-smooth contact dynamics approach for rigid and deformable solids, proposed by Frémond and his collaborators, the value of pressure and contact forces at collisions points, generated through congestion or panic situation are estimated. Firstly, we propose a second-order microscopic model, in which the crowd is treated as a system of rigid solids. Contact forces are rigorously defined by taking into account multiple simultaneous contacts and the non-overlapping condition between pedestrians. We show that for a dense crowd, percussions can be seen as contact forces. Secondly, in order to overcome the restrictive hypothesis related to the geometric form adapted to model the pedestrian, a continuous equivalent approach is proposed where the crowd is modeled as a deformable solid, the pressure is then defined by the divergence of the stress tensor and calculated according to volume and surface constraints. This approach makes it possible to retain an admissible right-velocity, including both the non-local interactions between non-neighbor pedestrians and the choice of displacement strategy of each pedestrian. Finally, the comparison between the two proposed approaches and some other existing approaches are presented on several illustrative examples to estimate the contact forces between pedestrians