[pt] Este trabalho apresenta um estudo do cálculo de sensibilidades necessário para a análise de problemas inversos e de otimização, usando o método híbrido dos elementos de contorno. Com esta finalidade, é desenvolvida uma formulação que permite obter as sensibilidades à mudança de forma, por diferenciação implícita das integrais de contorno, de uma estrutura já discretizada. Demonstra-se que as sensibilidades das matrizes obtidas desta formulação apresentam propriedades espectrais definidas, que são derivadas da formulação básica do método híbrido dos elementos de contorno. Todo o desenvolvimento é feito para um problema da elastostática tridimensional, embora sejam apresentadas apenas aplicações de problemas bidimensionais e de potencial, como casos particulares. As singularidades que surgem na integração no cálculo das sensibilidades são facilmente solucionáveis a partir das integrais da formulação básica do método híbrido dos elementos de contorno. As implementações numéricas são feitas utilizando a linguagem de programação Maple V release 3. Para ambos os casos, de potencial e elasticidade bidimensional, são usados elementos lineares para a representação do contorno. São apresentadas comparações entre os resultados analíticos obtidos através desta formulação com os resultados obtidos usando a técnica de diferenças finitas (centradas), com o objetivo de demonstrar a eficiência e precisão da metodologia aqui desenvolvida. / [en] The present work describes a formulation for computing design sensitivities required in inverse problems and shape optimization of solid objects, in the frame of the hybrid boundary element method. The so-called direct differentiation method is applied in order to calculate the gradients, i.e. the implicit diferentiation of the discretized boundary is performed, resulting in a general and efficient analysis technique for shape design sensitivity analysis of all structural quantities. It is demonstrated that the resulting sensitivities matrices present some useful spectral properties, which are related to the matrix spectral properties of the basic hybrid formulation. This formulation is valid for tridimensional solids, although only potential and bidimensional applications are considered as particular cases. The singularities that appear in the resulting boundary integrals are exactly the same which have already been dealt with in the basic formulation. The analytical and numerical procedures were performed by using the mathematical package Maple V release 3. Linear boundary elements were used for both potential and elasticity problems. Numerical results obtained by the present procedure are compared to finite differences results to demonstrate the effectiveness of the present formulation. / [es] Este trabajo presenta un estudio del cálculo de sensibilidades, que tiene gran importancia en el análisis de problemas inversos y de optimización, usando el método híbrido de los elementos de contorno. Con esta finalidad, se desarrolla una formulación que permite obtener las sensibilidades al cambio de forma de una extructura ya discretizada, por diferenciación implícita de las integrales de contorno. Se demuestra que las sensibilidades de las matrices obtenidas por esta formulación presentan propriedades espectrales definidas, que son derivadas de la formulación básica del método híbrido de los elementos de contorno. El desarrollo de la formulación se realiza para un problema de elastostática tridimensional, aunque se presentan apenas las aplicaciones de problemas bidimensionales y de potencial, como casos particulares. Las singularidades que surgen en la integración en el cálculo de las sensibilidades pueden ser fácilmente resueltas a partir de las integrales de la formulación básica del método híbrido de los elementos de contorno. La implementación numérica utiliza el lenguaje de programación Maple V release 3. Para los casos de potencial y elasticidad bidimensional, se utilizan elementos lineales para la representación del contorno. Se comparan los resultados analíticos obtenidos a través de esta formulación con los resultados obtenidos usando la técnica de diferencias finitas (centradas), con el objetivo de demostrar la eficiencia y precisión de la metodología aqui desarrollada.

No description available.

Due to legislative reasons and environmental concerns the automotive and transport sector are shifting their focus from traditional internal combustion engine (ICE) vehicles to development of battery electric vehicles (BEVs). This brings new challanges to design of cooling systems where axial fans are one of the key components. Axial fans are usually designed with regards to a certain operating condition and outside this region the efficiency of the fan drops drastically. Due to difficulty in specifying the exact operational parameters when placed in a car, post-design optimization may be necessary to ensure maximized performance. This thesis focuses on fan blade shape optimization through mesh morphing using the surrogate based optimization algorithm called Efficient Global Optimization (EGO). The target fans was a 9 bladed prototype fan by Johnson Electric with uneven blade spacing. The optimization uses steady state Reynolds-averaged Navier-Stokes (RANS) simulations to evaluate the fan designs and a Bezier curve parametrization in order to change the fan blade shape together with mesh morphing. The simulation setup was evaluated before peceding with the optimization, and showed good agreement close to intended operational conditions. Differences in turbulence modeling treatments were also evaluated in order to have a satisfactory agreement with measurement data. The EGO algorithm manages to provide fan designs with higher total-to-static efficiency at several different operational conditions. Evaluation of the optimized fan designs was limited to comparison with the provided measurement data and corrensponding simulations. Acoustic evaluation of selected fan designs is also attemped, but further work is required in order for the study to result in a quantitative comparison. / På grund av lagstiftning och miljöpåverkan har bil- och transportindustrin börjat skifta fokus från traditionella förbränningsfordon till utveckling av batteridrivna elbilar. Med detta medföljer nya utmaningar kring kylsystemsdesign där axiella fläktar är en av huvudkomponenterna hos systemet. Axiella fläktar är vanligtvis designade kring ett specifikt drifttillstånd och utanför detta har fläkten avsevärt lägre verkningsgrad. På grund av svårigheter att specificera detta drifttillstånd med hög precision, speciellt när fläkten monteras i en bil, kan efterdesigns-optimering vara nödvändigt för att uppnå maximal prestanda. Denna avhandling fokuserar på form-optimering av fläkt via mesh morphing med hjälp av den surrogat-baserade optimeringsalgoritmen Efficient Global Optimization (EGO). Fläkten som optimerades var en prototypfläkt designad av Johnson Electric med 9 fläktblad och icke-symmetriska mellanrum mellan bladen. I optimeringsprocessen användes icke-tidsberoende Reynolds-averaged Navier-Stokes (RANS) simuleringar för att utvärdera fläktdesignerna och parametrisering med hjälp av Bezier kurvor och mesh morphing för att ändra fläktbladen. Simulerings-uppställningen utvärderades innan optimeringen och bra överensstämning nära avsett driftstillstånd kunde påvisas. Skillnader i turbulens-modelering utvärderades även för att få en tillfredställande överensstämning med mätdata. EGO-algoritmen klarar att förse fläktdesigner med högre total-till-statisk verkningsgrad vid flera olika driftstillstånd. Utvärdering av fläktdesignerna var dock begränsad till jämförelse med mätdata och motsvarande simuleringsdata. En akustik utvärdering av utvalda fläkt-designer försöktes, men mer arbete krävs för att studien ska erhålla en kvantitativ jämförelse.

This dissertation consists of three studies of immersed finite element (IFE) methods for inter- face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems. First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of low-degree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates. Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary p-th degree IFE methods. In this framework, each IFE function is the extension of a p-th degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method. In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications. / Doctor of Philosophy / Interface problems arise from many science and engineering applications modeling the transmission of some physical quantities between multiple materials. Mathematically, these multiple materials in general are modeled by partial differential equations (PDEs) with discontinuous parameters, which poses challenges to developing efficient and reliable numerical methods and the related theoretical error analysis. The main contributions of this dissertation is on the development of a special finite element method, the so called immersed finite element (IFE) method, to solve the interface problems on a mesh independent of the interface geometry which can be advantageous especially when the interface is moving. Specifically, this dissertation consists of three projects of IFE methods: elasticity interface problems, higher-order IFE methods and interface inverse problems, including their design, analysis, and application.

La présente thèse porte sur différentes questions émanant de la géométrie spectrale. Ce domaine des mathématiques fondamentales a pour objet d'établir des liens entre la géométrie et le spectre d'une variété riemannienne. Le spectre d'une variété compacte fermée M munie d'une métrique riemannienne $g$ associée à l'opérateur de Laplace-Beltrami est une suite de nombres non négatifs croissante qui tend vers l’infini. La racine carrée de ces derniers représente une fréquence de vibration de la variété. Cette thèse présente quatre articles touchant divers aspects de la géométrie spectrale. Le premier article, présenté au Chapitre 1 et intitulé « Superlevel sets and nodal extrema of Laplace eigenfunctions », porte sur la géométrie nodale d'opérateurs elliptiques. L’objectif de mes travaux a été de généraliser un résultat de L. Polterovich et de M. Sodin qui établit une borne sur la distribution des extrema nodaux sur une surface riemannienne pour une assez vaste classe de fonctions, incluant, entre autres, les fonctions propres associées à l'opérateur de Laplace-Beltrami. La preuve fournie par ces auteurs n'étant valable que pour les surfaces riemanniennes, je prouve dans ce chapitre une approche indépendante pour les fonctions propres de l’opérateur de Laplace-Beltrami dans le cas des variétés riemanniennes de dimension arbitraire. Les deuxième et troisième articles traitent d'un autre opérateur elliptique, le p-laplacien. Sa particularité réside dans le fait qu'il est non linéaire. Au Chapitre 2, l'article « Principal frequency of the p-laplacian and the inradius of Euclidean domains » se penche sur l'étude de bornes inférieures sur la première valeur propre du problème de Dirichlet du p-laplacien en termes du rayon inscrit d’un domaine euclidien. Plus particulièrement, je prouve que, si p est supérieur à la dimension du domaine, il est possible d'établir une borne inférieure sans aucune hypothèse sur la topologie de ce dernier. L'étude de telles bornes a fait l'objet de nombreux articles par des chercheurs connus, tels que W. K. Haymann, E. Lieb, R. Banuelos et T. Carroll, principalement pour le cas de l'opérateur de Laplace. L'adaptation de ce type de bornes au cas du p-laplacien est abordée dans mon troisième article, « Bounds on the Principal Frequency of the p-Laplacian », présenté au Chapitre 3 de cet ouvrage. Mon quatrième article, « Wolf-Keller theorem for Neumann Eigenvalues », est le fruit d'une collaboration avec Guillaume Roy-Fortin. Le thème central de ce travail gravite autour de l'optimisation de formes dans le contexte du problème aux valeurs limites de Neumann. Le résultat principal de cet article est que les valeurs propres de Neumann ne sont pas toujours maximisées par l'union disjointe de disques arbitraires pour les domaines planaires d'aire fixée. Le tout est présenté au Chapitre 4 de cette thèse. / The main topic of the present thesis is spectral geometry. This area of mathematics is concerned with establishing links between the geometry of a Riemannian manifold and its spectrum. The spectrum of a closed Riemannian manifold M equipped with a Riemannian metric g associated with the Laplace-Beltrami operator is a sequence of non-negative numbers tending to infinity. The square root of any number of this sequence represents a frequency of vibration of the manifold. This thesis consists of four articles all related to various aspects of spectral geometry. The first paper, “Superlevel sets and nodal extrema of Laplace eigenfunction”, is presented in Chapter 1. Nodal geometry of various elliptic operators, such as the Laplace-Beltrami operator, is studied. The goal of this paper is to generalize a result due to L. Polterovich and M. Sodin that gives a bound on the distribution of nodal extrema on a Riemann surface for a large class of functions, including eigenfunctions of the Laplace-Beltrami operator. The proof given by L. Polterovich and M. Sodin is only valid for Riemann surfaces. Therefore, I present a different approach to the problem that works for eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds of arbitrary dimension. The second and the third papers of this thesis are focused on a different elliptic operator, namely the p-Laplacian. This operator has the particularity of being non-linear. The article “Principal frequency of the p-Laplacian and the inradius of Euclidean domains” is presented in Chapter 2. It discusses lower bounds on the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplace operator in terms of the inner radius of the domain. In particular, I show that if p is greater than the dimension, then it is possible to prove such lower bound without any hypothesis on the topology of the domain. Such bounds have previously been studied by well-known mathematicians, such as W. K. Haymann, E. Lieb, R. Banuelos, and T. Carroll. Their papers are mostly oriented toward the case of the usual Laplace operator. The generalization of such lower bounds for the p-Laplacian is done in my third paper, “Bounds on the Principal Frequency of the p-Laplacian”. It is presented in Chapter 3. My fourth paper, “Wolf-Keller theorem of Neumann Eigenvalues”, is a joint work with Guillaume Roy-Fortin. This paper is concerned with the shape optimization problem in the case of the Laplace operator with Neumann boundary conditions. The main result of our paper is that eigenvalues of the Neumann boundary problem are not always maximized by disks among planar domains of given area. This joint work is presented in Chapter 4.

Es wurde die Optimierungssoftware SPC-Opt entwickelt, mit welcher sich Aufgaben aus den Bereichen der Formoptimierung sowie der Material- und Formidentifikation bearbeiten lassen. Zur Lösung von Identifikationsproblemen steht eine robuste Implementierung des Levenberg-Marquardt-Fletcher-Verfahrens zur Verfügung. Ergänzt wird dieses durch Line-Search- und Trust-Region-Verfahren, welche sich besonders für Aufgaben der Formoptimierung eignen. Es wurden effiziente Algorithmen zur Approximation der Hesse-Matrix sowie verschiedene Verfahren zur Startparametervariation integriert. Das Programm verfügt über Schnittstellen zur Nutzung von ABAQUS, ANSYS, MSC.MARC, eigenen FEM-Programmen sowie LUA-Skripten. Für Formoptimierungen können geometrische Konturen durch NURBS approximiert und deren Kontrollpunkte als Formparameter genutzt werden. Die Aktualisierung der FEM-Netze entsprechend der Formparameteränderung erfolgt durch ein analytisches Verfahren. Der zweite Schwerpunkt der Arbeit bezieht sich auf die Weiterentwicklung bestehender Verfahren zur Materialparameteridentifikation im Bereich der Gummiwerkstoffe. Hierbei wurde das Konzept der Anpassung anhand bauteilnaher Probekörper entwickelt. Dabei wurde am Beispiel einer Fahrwerksbuchse ein Probekörper entworfen, welcher dem originalen Bauteil zwar ähnlich sieht, jedoch eine deutlich einfachere Geometrie hat. Durch diesen konnte das Verhalten des Bauteils gut approximiert und sichergestellt werden, dass die im Rahmen der Parameteridentifikation durchgeführten FEM-Simulationen sicher konvergieren. Zudem wurden die Nutzerschnittstellen des inelastischen Morph-Stoffgesetz für MSC.MARC und ABAQUS weiterentwickelt, sodass diese nunmehr auch im industriellen Umfeld nutzbar sind. Es konnte nachgewiesen werden, dass die Verwendung bauteilnah identifizierter Parameter zu einer erheblich besseren Abbildung des Materialverhaltens führt als die Verwendung anhand von Standardprobekörpern identifizierter Parameter. Weiterhin zeigte sich, dass vor allem der Einsatz eines Stoffgesetzes mit der Möglichkeit zur Abbildung des charakteristischen Verhaltens von Elastomeren unbedingt erforderlich ist. / Within the scope of this work the optimization software SPC-Opt has been developed to successfully process tasks in the fields of shape optimization and parameter identification. The software includes a robust Levenberg-Marquardt-Fletcher algorithm, several line search and trust region algorithms as well as efficient methods for the approximation of the Hessian matrix. Additionally, procedures for the variation of initial parameters (Design Of Experiments) were implemented. The software includes interfaces to ABAQUS, ANSYS, MSC.MARC, in-house FEM programs and LUA scripts. Within shape optimization problems, geometric shapes are approximated by NURBS and the related control points are employed as design variables. For the update of the FE mesh during the variation of the design variables, a special analytical algorithm is used to preserve the mesh topology. Another focus is related to the further development of existing material parameter identification procedures for rubber materials. Therefor, the concept of component-oriented specimens was developed. Using the example of a bushing, a specimen was designed, which is similar to the original component but has a much simpler geometry. According to this, the behavior of the original component is approximated and the stability of necessary FE simulations is ensured. Additionally, the utilized Model of Rubber Phenomenology (MORPH) is improved in view of the industrial use. It is shown that the identification of material parameters using component-oriented specimens leads to a much better approximation of the original component behaviour than using standard specimens. Additionally, it is shown that the use of a material law which can consider characteritic properties of elastomers, is absolutely necessary.

Formoptimierung bezeichnet die Bestimmung der Geometrischen Gestalt eines Gebietes auf dem eine partielle Differentialgleichung (PDE) wirkt, sodass bestimmte gegebene Zielgrößen, welche von der Lösung der PDE abhängen, Extrema annehmen. Bei der Diskret Adjungierten Methode wird der Gradient einer Zielgröße bezüglich einer beliebigen Anzahl von Formparametern mit Hilfe der Lösung einer adjungierten Gleichung der diskretisierten PDE effizient ermittelt. Dieser Gradient wird dann in Verfahren der numerischen Optimierung verwendet um eine optimale Lösung zu suchen. Da sowohl die Zielgröße als auch der Gradient für die diskretisierte PDE ermittelt werden, sind beide zunächst vom verwendeten Netz abhängig. Bei groben Netzen sind sogar Unstetigkeiten der diskreten Zielfunktion zu erwarten, wenn bei Änderungen der Formparameter sich das Netz unstetig ändert (z.B. Änderung Anzahl Knoten, Umschalten der Konnektivität). Mit zunehmender Feinheit der Netze verschwinden jedoch diese Unstetigkeiten aufgrund der Konvergenz der Diskretisierung. Da im Zuge der Formoptimierung Zielgröße und Gradient für eine Vielzahl von Iterierten der Lösung bestimmt werden müssen, ist man bestrebt die Kosten einer einzelnen Auswertung möglichst gering zu halten, z.B. indem man mit nur moderat feinen oder adaptiv verfeinerten Netzen arbeitet. Aufgabe dieser Diplomarbeit ist es zu untersuchen, ob mit gängigen Methoden adaptiv verfeinerte Netze hinreichende Genauigkeit der Auswertung von Zielgröße und Gradient erlauben und ob eventuell Anpassungen der Optimierungsstrategie an die adaptive Vernetzung notwendig sind. Für die Untersuchungen sind geeignete Modellprobleme aus der Festigkeitslehre zu wählen und zu untersuchen. / Shape optimization describes the determination of the geometric shape of a domain with a partial differential equation (PDE) with the purpose that a specific given performance function is minimized, its values depending on the solution of the PDE. The Discrete Adjoint Method can be used to evaluate the gradient of a performance function with respect to an arbitrary number of shape parameters by solving an adjoint equation of the discretized PDE. This gradient is used in the numerical optimization algorithm to search for the optimal solution. As both function value and gradient are computed for the discretized PDE, they both fundamentally depend on the discretization. In using the coarse meshes, discontinuities in the discretized objective function can be expected if the changes in the shape parameters cause discontinuous changes in the mesh (e.g. change in the number of nodes, switching of connectivity). Due to the convergence of the discretization these discontinuities vanish with increasing fineness of the mesh. In the course of shape optimization, function value and gradient require evaluation for a large number of iterations of the solution, therefore minimizing the costs of a single computation is desirable (e.g. using moderately or adaptively refined meshes). Overall, the task of the diploma thesis is to investigate if adaptively refined meshes with established methods offer sufficient accuracy of the objective value and gradient, and if the optimization strategy requires readjustment to the adaptive mesh design. For the investigation, applicable model problems from the science of the strength of materials will be chosen and studied.

Šiame baigiamajame darbe optimizuojama plokščiojo kūno forma esant žinomai apkrovai ir kraštinėms sąlygoms. Aptariami pagrindiniai optimizavimo tipai, globaliojo optimizavimo ir kontinuumo diskretizavimo metodai. Forma optimizuojama trimis skirtingais neadaptyviais genetiniais algoritmais ir trimis adaptyviais, taikant migracijos strategijas. Poslinkiai ir von Mises įtempimai skaičiuojami baigtinių elementų metodu. Darbo užduotis realizuota C++ kalba. Darbą sudaro 9 dalys: įvadas, kūno formos modeliavimo metodų apžvalga, baigtinių elementų metodas, genetiniai algoritmai, kūno formos optimizavimo uždavinys, rezultatų palyginimas, išvados, literatūros sąrašas, priedai. Darbo apimtis - 62 p. teksto be priedų, 39 iliustr., 5 lent., 24 bibliografinių šaltinių. / The task of Master thesis is to optimize shape of plane body with given loading and boundary conditions. The classes of structural optimization problems, the methods of global optimization and discretization of continual structures are discussed. For shape optimization three different not adaptive and three adaptive genetic algorithms with migration strategy are used, displacements and von Mises stresses are calculated using finite element method. The program was created using C++ language. The work consists of 9 parts: introduction, overview of methods used for shape optimization, finite element method, genetic algorithms, shape optimization of plane body, results, conclusions, references, appendixes. Work consist of 62 p. text without appendixes, 39 pictures, 5 tables, 24 bibliographical entries. Appendixes are included.

Made available in DSpace on 2014-12-17T14:57:51Z (GMT). No. of bitstreams: 1 SidneyOL.pdf: 4301786 bytes, checksum: 9f7a7a0d30a925198ccebaa046c885a4 (MD5) Previous issue date: 2009-02-09 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This work proposes a computational methodology to solve problems of optimization in structural design. The application develops, implements and integrates methods for structural analysis, geometric modeling, design sensitivity analysis and optimization. So, the optimum design problem is particularized for plane stress case, with the objective to minimize the structural mass subject to a stress criterion. Notice that, these constraints must be evaluated at a series of discrete points, whose distribution should be dense enough in order to minimize the chance of any significant constraint violation between specified points. Therefore, the local stress constraints are transformed into a global stress measure reducing the computational cost in deriving the optimal shape design. The problem is approximated by Finite Element Method using Lagrangian triangular elements with six nodes, and use a automatic mesh generation with a mesh quality criterion of geometric element. The geometric modeling, i.e., the contour is defined by parametric curves of type B-splines, these curves hold suitable characteristics to implement the Shape Optimization Method, that uses the key points like design variables to determine the solution of minimum problem. A reliable tool for design sensitivity analysis is a prerequisite for performing interactive structural design, synthesis and optimization. General expressions for design sensitivity analysis are derived with respect to key points of B-splines. The method of design sensitivity analysis used is the adjoin approach and the analytical method. The formulation of the optimization problem applies the Augmented Lagrangian Method, which convert an optimization problem constrained problem in an unconstrained. The solution of the Augmented Lagrangian function is achieved by determining the analysis of sensitivity. Therefore, the optimization problem reduces to the solution of a sequence of problems with lateral limits constraints, which is solved by the Memoryless Quasi-Newton Method It is demonstrated by several examples that this new approach of analytical design sensitivity analysis of integrated shape design optimization with a global stress criterion purpose is computationally efficient / Neste trabalho prop?e-se uma metodologia computacional para resolver problemas de Otimiza??o de Forma para projeto estrutural. A aplica??o ? particularizada para problemas bidimensionais em estado plano de tens?es, de modo a minimizar a massa atendendo um crit?rio de tens?o. Para atender ao crit?rio param?trico de tens?es ? proposto um crit?rio global de tens?o de von Mises, dessa maneira, amplia-se o crit?rio local de tens?es sobre o dom?nio, visando ? obten??o de programas mais seguros. O problema ? aproximado pelo M?todo dos Elementos Finitos utilizando elementos triangulares da base Lagrangiana padr?o com seis n?s, tendo uma estrat?gia de gera??o autom?tica de malhas baseada em um crit?rio geom?trico do elemento. O modelo geom?trico do contorno material ? definido por curvas param?tricas B-splines. Estas curvas possuem caracter?sticas vantajosas para implementa??o do processo de otimiza??o de forma, que se utiliza dos pontos-chave para determinar o m?nimo do problema. A formula??o do problema de otimiza??o faz uso do M?todo Lagrangiano Aumentado, que transforma o problema de otimiza??o com restri??o, em problema irrestrito. A solu??o da fun??o Lagrangiana Aumentada ? alcan?ada pela determina??o da an?lise das sensibilidades anal?ticas em rela??o aos pontos-chave da curva B-spline. Como conseq??ncia, o problema de otimiza??o reduz-se ? solu??o de uma seq??ncia de problemas de limites laterais do tipo caixa, o qual ? resolvido por um m?todo de proje??o de segunda ordem que usa o m?todo de Quase-Newton projetado sem mem?ria. S?o demonstrados v?rios exemplos para o M?todo de Otimiza??o de Forma integrado a An?lise da Sensibilidade Anal?tica sob o crit?rio global de tens?o de von Mises

Le but de cette thèse est de définir une méthode d'optimisation de formes qui conjugue l'utilisation de la dérivée seconde de forme et la méthode des lignes de niveaux pour la représentation d'une forme.On considèrera d'abord deux cas plus simples : un cas d'optimisation paramétrique et un cas d'optimisation discrète.Ce travail est divisé en quatre parties.La première contient le matériel nécessaire à la compréhension de l'ensemble de la thèse.Le premier chapitre rappelle des résultats généraux d'optimisation, et notamment le fait que les méthodes d'ordre deux ont une convergence quadratique sous certaines hypothèses.Le deuxième chapitre répertorie différentes modélisations pour l'optimisation de formes, et le troisième se concentre sur l'optimisation paramétrique puis l'optimisation géométrique.Les quatrième et cinquième chapitres introduisent respectivement la méthode des lignes de niveaux (level-set) et la méthode des éléments-finis.La deuxième partie commence par les chapitres 6 et 7 qui détaillent des calculs de dérivée seconde dans le cas de l'optimisation paramétrique puis géométrique.Ces chapitres précisent aussi la structure et certaines propriétés de la dérivée seconde de forme.Le huitième chapitre traite du cas de l'optimisation discrète.Dans le neuvième chapitre on introduit différentes méthodes pour un calcul approché de la dérivée seconde, puis on définit un algorithme de second ordre dans un cadre général.Cela donne la possibilité de faire quelques premières simulations numériques dans le cas de l'optimisation paramétrique (Chapitre 6) et dans le cas de l'optimisation discrète (Chapitre 7).La troisième partie est consacrée à l'optimisation géométrique.Le dixième chapitre définit une nouvelle notion de dérivée de forme qui prend en compte le fait que l'évolution des formes par la méthode des lignes de niveaux, grâce à la résolution d'une équation eikonale, se fait toujours selon la normale.Cela permet de définir aussi une méthode d'ordre deux pour l'optimisation.Le onzième chapitre détaille l'approximation d'intégrales de surface et le douzième chapitre est consacré à des exemples numériques.La dernière partie concerne l'analyse numérique d'algorithmes d'optimisation de formes par la méthode des lignes de niveaux.Le Chapitre 13 détaille la version discrète d'un algorithme d'optimisation de formes.Le Chapitre 14 analyse les schémas numériques relatifs à la méthodes des lignes de niveaux.Enfin le dernier chapitre fait l'analyse numérique complète d'un exemple d'optimisation de formes en dimension un, avec une étude des vitesses de convergence / The main purpose of this thesis is the definition of a shape optimization method which combines second-order differentiationwith the representation of a shape by a level-set function. A second-order method is first designed for simple shape optimization problems : a thickness parametrization and a discrete optimization problem. This work is divided in four parts.The first one is bibliographical and contains different necessary backgrounds for the rest of the work. Chapter 1 presents the classical results for general optimization and notably the quadratic rate of convergence of second-order methods in well-suited cases. Chapter 2 is a review of the different modelings for shape optimization while Chapter 3 details two particular modelings : the thickness parametrization and the geometric modeling. The level-set method is presented in Chapter 4 and Chapter 5 recalls the basics of the finite element method.The second part opens with Chapter 6 and Chapter 7 which detail the calculation of second-order derivatives for the thickness parametrization and the geometric shape modeling. These chapters also focus on the particular structures of the second-order derivative. Then Chapter 8 is concerned with the computation of discrete derivatives for shape optimization. Finally Chapter 9 deals with different methods for approximating a second-order derivative and the definition of a second-order algorithm in a general modeling. It is also the occasion to make a few numerical experiments for the thickness (defined in Chapter 6) and the discrete (defined in Chapter 8) modelings.Then, the third part is devoted to the geometric modeling for shape optimization. It starts with the definition of a new framework for shape differentiation in Chapter 10 and a resulting second-order method. This new framework for shape derivatives deals with normal evolutions of a shape given by an eikonal equation like in the level-set method. Chapter 11 is dedicated to the numerical computation of shape derivatives and Chapter 12 contains different numerical experiments.Finally the last part of this work is about the numerical analysis of shape optimization algorithms based on the level-set method. Chapter 13 is concerned with a complete discretization of a shape optimization algorithm. Chapter 14 then analyses the numerical schemes for the level-set method, and the numerical error they may introduce. Finally Chapter 15 details completely a one-dimensional shape optimization example, with an error analysis on the rates of convergence