Le nombre d’appareils électroniques et de systèmes de communication sans fil a considérablement augmenté au cours des 20 dernières années. Les boîtiers de blindage utilisés pour protéger les appareils électroniques contre les radiations externes, mais aussi pour limiter leurs émissions sont généralement conçus en alliages d’aluminium. Mais la nécessité de réduire le poids des aéronefs incite l’industrie aéronautique à l’utilisation de matériaux composites.La modélisation de boîtiers de blindage composés de matériaux homogènes est possible par l’utilisation d’outils numériques tels que la méthode des éléments finis. Mais la discrétisation de boîtiers constitués de matériaux composites impliquerait un nombre d’éléments trop important rendant impossible toute modélisation numérique. Le recours à l’homogénéisation semi-analytique est une possibilité pour s’affranchir de cette restriction. Les milieux homogènes équivalents obtenus avec ces méthodes peuvent être insérés dans des outils numériques pour simuler le comportement électromagnétique de boîtiers de blindage complexes. Les modèles d’homogénéisation existants, tel que le modèle de Maxwell-Garnett, sont toutefois limités a des applications quasi-statiques.La définition des propriétés effectives de matériaux composites illuminés par des ondes électromagnétiques est l’objectif principal de ce travail. Il en résulte deux méthodes d’homogénéisation dynamiques. La première introduit un effet de taille entre les fibres et la longueur d'onde. Elle permet ainsi d’étendre une méthode basée sur des problèmes d'inclusion aux micro-ondes. Mais elle reste limitée par l’apparition de l’effet de peau dans les renforts conducteurs. La seconde est basée sur la définition des pertes par effet Joule dans les fibres, permettant ainsi d’étendre la première méthode après l’apparition de l’effet de peau. Cette dernière est enfin utilisée pour modéliser le comportement électromagnétique d’un boîtier de blindage réaliste. / The number of electronic devices and wireless communication systems has significantly increased over the past 20 years. Shielding enclosures used to protect electronic devices against radiated waves and to limit their emissions are usually designed in aluminum alloys. But the need to reduce the weight of aircraft incites the aerospace industry to the use of composite materials.Modeling shielding enclosures composed of homogeneous materials is possible by the use of numerical tools such as the finite element method. But considering every details of the microstructure would involve a excessive number of unknowns preventing numerical modelings. The use of semi-analytical homogenization methods is a possibility to overcome this restriction. The equivalent homogeneous mediums obtained with these methods can be inserted into numerical tools to simulate the electromagnetic behavior of complex shielding enclosures. But classical homogenization models such as Maxwell-Garnett model, are limited to quasi-static applications.Calculating the effective properties of composite materials illuminated by electromagnetic waves is the main objective of this work. This leads to two dynamic homogenization methods. The first one introduces a size effect between the fibers and the wavelength. It allows to extend a method based on inclusion problems to microwave frequencies. However it is limited by the occurrence of the skin effect in conductive inclusions. The second consider Joule losses and extends the first method after the occurrence of the skin effect. This second homogenization method is finally used to model the behavior of a realistic shielding enclosure.

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No. of bitstreams: 2 Tese - Reinier Díaz Millán - 2015.pdf: 3568052 bytes, checksum: b4c892f77911a368e1b8f629afb5e66e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Nesta tese apresentamos v arios algoritmos para resolver os problemas de Desigualdade Variacional e Inclus~ao. Para o problema de desigualdade variacional propomos, no Cap tulo 2 uma generaliza c~ao do algoritmo cl assico extragradiente, utilizando vetores normais n~ao nulos do conjunto vi avel. Em particular, dois algoritmos conceituais s~ao propostos e cada um deles cont^em tr^es variantes diferentes de proje c~ao que est~ao relacionadas com algoritmos extragradientes modi cados. Duas buscas diferentes s~ao propostas, uma sobre a borda do conjunto vi avel e a outra ao longo das dire c~oes vi aveis. Cada algoritmo conceitual tem uma estrat egia diferente de busca e tr^es formas de proje c~ao especiais, gerando tr^es sequ^encias com diferente e interessantes propriedades. E feito a an alise da converg^encia de ambos os algoritmos conceituais, pressupondo a exist^encia de solu c~oes, continuidade do operador e uma condi c~ao mais fraca do que pseudomonotonia. No Cap tulo 4, n os introduzimos um algoritmo direto de divis~ao para o problema variacional em espa cos de Hilbert. J a no Cap tulo 5, propomos um algoritmo de proje c~ao relaxada em Espa cos de Hilbert para a soma de m operadores mon otonos maximais ponto-conjunto, onde o conjunto vi avel do problema de desigualdade variacional e dado por uma fun c~ao n~ao suave e convexa. Neste caso, as proje c~oes ortogonais ao conjunto vi avel s~ao substitu das por proje c~oes em hiperplanos que separam a solu c~ao da itera c~ao atual. Cada itera c~ao do m etodo proposto consiste em proje c~oes simples de tipo subgradientes, que n~ao exige a solu c~ao de subproblemas n~ao triviais, utilizando apenas os operadores individuais, explorando assim a estrutura do problema. Para o problema de Inclus~ao, propomos variantes do m etodo de divis~ao de forward-backward para achar um zero da soma de dois operadores, a qual e a modi ca c~ao cl assica do forwardbackward proposta por Tseng. Um algoritmo conceitual e proposto para melhorar o apresentado por Tseng em alguns pontos. Nossa abordagem cont em, primeramente, uma busca linear tipo Armijo expl cita no esp rito dos m etodos tipo extragradientes para desigualdades variacionais. Durante o processo iterativo, a busca linear realiza apenas um c alculo do operador forward-backward em cada tentativa de achar o tamanho do passo. Isto proporciona uma consider avel vantagem computacional pois o operador forward-backward e computacionalmente caro. A segunda parte do esquema consiste em diferentes tipos de proje c~oes, gerando sequ^encias com caracter sticas diferentes. / In this thesis we present various algorithms to solve the Variational Inequality and Inclusion Problems. For the variational inequality problem we propose, in Chapter 2, a generalization of the classical extragradient algorithm by utilizing non-null normal vectors of the feasible set. In particular, two conceptual algorithms are proposed and each of them has three di erent projection variants which are related to modi ed extragradient algorithms. Two di erent linesearches, one on the boundary of the feasible set and the other one along the feasible direction, are proposed. Each conceptual algorithm has a di erent linesearch strategy and three special projection steps, generating sequences with di erent and interesting features. Convergence analysis of both conceptual algorithms are established, assuming existence of solutions, continuity and a weaker condition than pseudomonotonicity on the operator. In Chapter 4 we introduce a direct splitting method for solving the variational inequality problem for the sum of two maximal monotone operators in Hilbert space. In Chapter 5, for the same problem, a relaxed-projection splitting algorithm in Hilbert spaces for the sum of m nonsmooth maximal monotone operators is proposed, where the feasible set of the variational inequality problem is de ned by a nonlinear and nonsmooth continuous convex function inequality. In this case, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which explores the structure of the problem. For the Inclusion Problem, in Chapter 3, we propose variants of forward-backward splitting method for nding a zero of the sum of two operators, which is a modi cation of the classical forward-backward method proposed by Tseng. The conceptual algorithm proposed here improves Tseng's method in many instances. Our approach contains rstly an explicit Armijo-type line search in the spirit of the extragradient-like methods for variational inequalities. During the iterative process, the line search performs only one calculation of the forward-backward operator in each tentative for nding the step size. This achieves a considerable computational saving when the forward-backward operator is computationally expensive. The second part of the scheme consists of special projection steps bringing several variants.