Submitted by Maria Cristina (library@lncc.br) on 2016-07-27T18:52:15Z No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2016-07-27T18:52:32Z (GMT) No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) / Made available in DSpace on 2016-07-27T18:52:42Z (GMT). No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) Previous issue date: 2016-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / In this work a new method for obstacles reconstruction from partial boundary measurements is proposed. For a given boundary excitation, we want to determine the quantity, locations and sizes of a number of obstacles embedding whiting a geometrical domain, from partial boundary measurements related to such an excitation. This problem is written in the form of ill-posed and over-determinated partial differential equation. The idea therefore is to rewrite it as an optimization problem where a shape functional measuring the misfit between the boundary measurement and the solution to an auxiliary boundary value problem is minimized with respect to a set of ball-shaped holes. The topological derivative concept is used for solving the resulting topology optimization problem, leading to a second-order reconstruction algorithm free of initial guess. The resulting algorithm is non-iterative and thus very robust with respect to noisy data. Finally, some numerical results are presented in order to demonstrate the effectiveness of proposed reconstruction algorithm. / O objetivo deste trabalho é apresentar um novo método de reconstrução de obstáculos. Mais precisamente, dada uma excitação deseja-se obter a solução de um problema inverso de reconstrução consistindo na determinação da quantidade, localização e tamanho de obstáculos no interior de um dado domínio geométrico a partir de leituras parciais da resposta à referida excitação. Este problema é escrito na forma de uma equação diferencial parcial sobredeterminada. Essa dificuldade é contornada reescrevendo o problema inverso na forma de um problema de otimização. A ideia básica consiste em minimizar um funcional de forma que mede a diferença entre o dado lido e o calculado numericamente em relação ao próprio domínio geométrico. Em particular o conceito de derivada topológica é utilizado, o que conduz a um algoritmo de reconstrução de segunda ordem e independente de qualquer chute inicial. Como o algoritmo resultante é não-iterativo, o processo de reconstrução torna-se extremamente robusto à presença de ruído. Vários exemplos numéricos de reconstrução são apresentados donde se verifica a validade dos resultados obtidos.

La Partie I présente l’obtention du développement asymptotique topologique pour une classe d’équations elliptiques quasilinéaires. Un point central réside dans la possibilité de définir la variation de l’état direct à l’échelle 1 dans R^N. Après avoir défini un cadre fonctionnel approprié faisant intervenir les normes L^p et L^2, et avoir justifié la classe d’équations considérée, la méthode se poursuit par l’étude du comportement asymptotique de la solution du problème d’interface non linéaire dans R^N et par une mise en dualité appropriée des états direct et adjoint aux différentes étapes d’approximation.La Partie II traite d’estimations et de développements asymptotiques de p-capacités de condensateurs, dont l’obstacle est d’intérieur vide et de codimension > ou = 2. Après les résultats préliminaires, les condensateurs équidistants permettent de donner deux illustrations de l’anisotropie engendrée par un segment dans l’équation de p-Laplace, puis d’établir une minoration de la p-capacité N-dimensionnelle d’un segment, qui fait intervenir les p-capacités d’un point, respectivement en dimensions N et (N-1). Les condensateurs elliptiques permettent d’établir que le gradient topologique de la 2-capacité n’est pas un outil approprié pour distinguer les courbes des obstacles d’intérieur non vide en 2D / Part I deals with obtaining topological asymptotic expansions for a class of quasilinear elliptic equations. A key point lies in the ability to define the variation of the direct state at scale 1 in R^N. After setting up an appropriate functional framework involving both the L^p and the L^2 norms, and then justifying the chosen class of equations, the approach goes on with the study of the asymptotic behavior of the solution of the nonlinear interface problem in R^N and by setting up an adequate duality scheme between the direct and adjoint states at each step of approximation. Part II deals with estimates and asymptotic expansions of condenser p-capacities and focuses on obstacles with empty interiors and with codimensions > ou = 2. After preliminary results, equidistant condensers are introduced to point out the anisotropy caused by a segment in the p-Laplace equation, and to provide a lower bound to the N-dimensional condenser p-capacity of a segment, by means of the N-dimensional and of the (N-1)-dimensional condenser p-capacities of apoint. Introducing elliptical condensers, it turns out that the topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with nonempty interior in 2D