O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de problemas de interação fluido-estrutura (IFE) em aplicações envolvendo corpos rígidos submersos, onde escoamentos incompressíveis de fluidos Newtonianos com superfície livre são considerados. Propõe-se o emprego da Análise Isogeométrica por permitir a unificação entre os procedimentos de pré-processamento e análise, melhorando assim as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação da conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, em uma formulação cinemática ALE (Arbitrary Lagrangean- Eulerian). A consideração da superfície livre no escoamento se dá tratando o fluido como um meio bifásico, através do método Level Set. O corpo rígido apresenta não linearidade na rotação e restrições representadas por vínculos elásticos e amortecedores viscosos, sendo a equação de equilíbrio dinâmico resolvida através do método de Newmark. O esquema de acoplamento sólido-fluido adotado é o particionado convencional, que impõe condições de compatibilidade cinemáticas e de equilíbrio sobre a interface sólido-fluido, analisando ambos os meios de maneira sequencial. A discretização das equações governantes é realizada através do esquema explícito de dois passos de Taylor-Galerkin, aplicado no contexto da Análise Isogeométrica. Por fim, são analisados alguns problemas da Dinâmica de Fluidos Computacional, de onde se concluiu que os resultados obtidos são bastante consistentes com os fenômenos envolvidos, com as ferramentas exclusivas da Análise Isogeométrica, como o refinamento k, melhorando a convergência dos resultados. Para escoamentos bifásicos, verificou-se que o método Level Set obteve resultados bastante promissores apresentando, entretanto, uma dissipação numérica excessiva. Propõe-se, para estudos futuros, a elaboração de esquemas numéricos que conservem melhor o volume da fase líquida do escoamento. / The present work aims to development of a numerical formulation based on Isogeometric Analysis for the study of Fluid-Structure Interaction problems in applications involving rigid bodies submerged, considering incompressible Newtonian flows with free surface. The use of the Isogeometric Analysis allows unification between the preprocessing and analysis steps, improving then the continuity of the base functions employed, both in the spatial discretization and approximation of the variables in the system of equations. The fundamental flow equations are formed by the Navier-Stokes and the mass conservation, described by de pseudo-compressibility hypothesis, in an ALE (Arbitrary Lagrangean-Eulerian) kinematic formulation. The free surface consideration of the flow is handled treating the fluid like a two- phase medium, using the Level Set method. The rigid body considers nonlinearity in rotation, and restrictions represented by elastic springs and viscous dampers, with the dynamic equilibrium equation being resolved using the Newmark’s method. The solid-fluid coupling scheme is the conventional partitioned, which imposes kinematics and equilibrium compatibility conditions on the solid-fluid interface, analyzing both mediums in a sequential manner. The governing equations are discretized using the explicit two step Taylor-Galerkin method, applied in an Isogeometric Analisys context. Finally, some Computational Fluid Dinamics problems are analysed, from which it was concluded that the results obtained are quite consistent with phenomena involved, with the unique tools of Isogeometric Analysis, such as k-refinement, improving the convergence of the results. For biphasic flows, it was verified that the Level Set method obtained very promising results, presenting, however, an excessive numerical dissipation. For future studies, it is proposed the elaboration of numerical schemes that better preserve the volume of the liquid phase of the flow.

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de problemas de interação fluido-estrutura (IFE) em aplicações envolvendo corpos rígidos submersos, onde escoamentos incompressíveis de fluidos Newtonianos com superfície livre são considerados. Propõe-se o emprego da Análise Isogeométrica por permitir a unificação entre os procedimentos de pré-processamento e análise, melhorando assim as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação da conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, em uma formulação cinemática ALE (Arbitrary Lagrangean- Eulerian). A consideração da superfície livre no escoamento se dá tratando o fluido como um meio bifásico, através do método Level Set. O corpo rígido apresenta não linearidade na rotação e restrições representadas por vínculos elásticos e amortecedores viscosos, sendo a equação de equilíbrio dinâmico resolvida através do método de Newmark. O esquema de acoplamento sólido-fluido adotado é o particionado convencional, que impõe condições de compatibilidade cinemáticas e de equilíbrio sobre a interface sólido-fluido, analisando ambos os meios de maneira sequencial. A discretização das equações governantes é realizada através do esquema explícito de dois passos de Taylor-Galerkin, aplicado no contexto da Análise Isogeométrica. Por fim, são analisados alguns problemas da Dinâmica de Fluidos Computacional, de onde se concluiu que os resultados obtidos são bastante consistentes com os fenômenos envolvidos, com as ferramentas exclusivas da Análise Isogeométrica, como o refinamento k, melhorando a convergência dos resultados. Para escoamentos bifásicos, verificou-se que o método Level Set obteve resultados bastante promissores apresentando, entretanto, uma dissipação numérica excessiva. Propõe-se, para estudos futuros, a elaboração de esquemas numéricos que conservem melhor o volume da fase líquida do escoamento. / The present work aims to development of a numerical formulation based on Isogeometric Analysis for the study of Fluid-Structure Interaction problems in applications involving rigid bodies submerged, considering incompressible Newtonian flows with free surface. The use of the Isogeometric Analysis allows unification between the preprocessing and analysis steps, improving then the continuity of the base functions employed, both in the spatial discretization and approximation of the variables in the system of equations. The fundamental flow equations are formed by the Navier-Stokes and the mass conservation, described by de pseudo-compressibility hypothesis, in an ALE (Arbitrary Lagrangean-Eulerian) kinematic formulation. The free surface consideration of the flow is handled treating the fluid like a two- phase medium, using the Level Set method. The rigid body considers nonlinearity in rotation, and restrictions represented by elastic springs and viscous dampers, with the dynamic equilibrium equation being resolved using the Newmark’s method. The solid-fluid coupling scheme is the conventional partitioned, which imposes kinematics and equilibrium compatibility conditions on the solid-fluid interface, analyzing both mediums in a sequential manner. The governing equations are discretized using the explicit two step Taylor-Galerkin method, applied in an Isogeometric Analisys context. Finally, some Computational Fluid Dinamics problems are analysed, from which it was concluded that the results obtained are quite consistent with phenomena involved, with the unique tools of Isogeometric Analysis, such as k-refinement, improving the convergence of the results. For biphasic flows, it was verified that the Level Set method obtained very promising results, presenting, however, an excessive numerical dissipation. For future studies, it is proposed the elaboration of numerical schemes that better preserve the volume of the liquid phase of the flow.

O presente trabalho consiste no desenvolvimento de uma ferramenta computacional para análises de fratura e fadiga de componentes tridimensionais a partir de modelos geométricos de Desenho Assistido por Computador (CAD, acrônimo do inglês). Modelos de propagação de fissuras associados a leis empíricas de fadiga permitem a determinação da vida útil de peças mecânico-estruturais. Tais análises são de vital importância para garantir a segurança estrutural em diversos projetos de engenharia tais como os de pontes, plataformas off-shore e aeronaves. No entanto, a criação de modelos de análise a partir de modelos geométricos de CAD envolve diversas etapas intermediárias que visam a obtenção de malhas volumétricas adequadas. A grande maioria dos modelos de CAD trabalha com a representação de sólidos a partir de seu contorno utilizando superfícies paramétricas, dentre as quais se destacam as superfícies B-Splines Racionais Não Uniformes (NURBS, acrônimo do inglês). Para gerar malhas volumétricas é necessário que o conjunto de superfícies NURBS que descrevem o objeto seja \"estanque\", ou seja, sem lacunas e/ou superposições nas conexões das superfícies, o que não é possível garantir na grande maioria dos modelos constituídos por NURBS. As contribuições propostas no presente trabalho são aplicáveis a modelos baseados no Método dos Elementos de Contorno dual (MEC dual), os quais exigem apenas a discretização das superfícies do problema, ou seja, contorno mais fissuras. No intuito de criar os modelos de análise de maneira eficiente a partir dos modelos geométricos de CAD, desenvolveu-se uma estratégia de colocação que permite discretizar de maneira independente cada uma das superfícies NURBS que compõem os modelos geométricos sólidos. Com a estratégia proposta evitam-se as dificuldades no tratamento das conexões entre as superfícies sendo possível analisar modelos geométricos \"não estanques\". A implementação abrange superfícies NURBS, aparadas ou não, de ordens polinomiais quaisquer e elementos de contorno triangulares e quadrilaterais de aproximação linear. As equações integrais de deslocamentos e de forças de superfície são regularizadas e as integrais singulares e hipersingulares são tratadas pelo Método de Guiggiani. Fissuras de borda são inseridas nos modelos de análise a partir de um algoritmo de remalhamento simples baseado em tolerâncias dimensionais. O mesmo algoritmo é utilizado para as análises incrementais de propagação. Três técnicas de extração dos Fatores de Intensidade de Tensão (FIT) foram implementadas para os modelos baseados na Mecânica da Fratura Elástica Linear (MFEL), a saber, as técnicas de correlação, de extrapolação e de ajuste de deslocamentos. A extensão dessa última técnica para problemas tridimensionais é outra contribuição do presente trabalho. Os critérios da máxima taxa de liberação de energia e de Schöllmann foram utilizados para determinar o FIT equivalente e o caminho de propagação das fissuras. O ângulo de deflexão é determinado por um algoritmo de otimização e o ângulo de torção, definido para o critério de Schöllmann, é imposto no vetor de propagação a partir de uma formulação variacional unidimensional, definida sobre a linha de frente da fissura. Nos modelos de fadiga adota-se a MFEL e a equação de Paris-Erdogan para determinar a vida útil à propagação de defeitos preexistentes. Um procedimento iterativo foi desenvolvido para evitar a interpenetração da matéria após o contato das faces da fissura, permitindo análises de fadiga com carregamentos alternados. Como proposta para a continuidade da pesquisa propõe-se desenvolver formulações isogeométricas de elementos de contorno para analisar problemas de fratura e fadiga diretamente dos modelos geométricos de CAD, sem a necessidade de gerar as malhas de superfície. Um estudo numérico preliminar envolvendo uma versão isogeométrica do MEC dual baseada em NURBS e a versão convencional utilizando polinômios de Lagrange lineares e quadráticos foi realizado. A partir do estudo foi possível apontar as vantagens e desvantagens de cada formulação e sugerir melhorias para ambas. / The present work consists in the development of a computational tool for fracture and fatigue analysis of three-dimensional components obtained from geometrical models of Computer-Aided Design (CAD). Crack propagation models associated with empirical fatigue laws allow the determination of residual life for structural-mechanical pieces. These analyses are vital to ensure the structural safety in several engineering projects such as in bridges, offshore platforms and aircraft. However, the creation of the analysis models from geometrical CAD models requires several intermediary steps in order to obtain suitable volumetric meshes of the problems. The majority of CAD models represent solids with parametric surfaces to describe its boundaries, which is known as the Boundary representation (B-representation). The most common parametric surfaces are Non-Uniform Rational B-Splines (NURBS). To generate a volumetric mesh it is required that the set of surfaces that describe the object must be watertight, i.e., without gaps or superposition at the surfaces connections, which is not possible to unsure using NURBS. The contributions proposed at the present thesis are applicable to models based on the Dual Boundary Element Method (DBEM), which require only the discretization of the surfaces of the problems, i.e., boundary and cracks. A special collocation strategy was developed in order to create the analysis models efficiently from the geometrical CAD models. The collocation strategy allows discretizing independently each one of the NURBS surfaces that compose the geometrical solid models. Therefore, the difficulties in the treatment of the surface connections are avoided and it becomes possible to create analysis models from non-watertight geometrical models. The implementation covers trimmed and non-trimmed NURBS surfaces of any polynomial orders and also triangular and quadrilateral boundary elements of linear order. The displacement and traction boundary integral equations are regularized and the strong and hypersingular integrals are treated with the Guiggiani\'s method. Edge cracks are inserted in the models by a simple remeshing procedure based on dimensional tolerances. The same remeshing approach is adopted for the incremental crack propagation analysis. Three techniques were adopted to extract the Stress Intensity Factors (SIF) in the context of Linear Elastic Fracture Mechanics (LEFM), i.e., the displacement correlation, extrapolation and fitting techniques. The extension of this last technique to three-dimensional problems is another contribution of the present work. Both the general maximum energy realise rate and the Schöllmann\'s criteria were adopted to determine the equivalent SIF and the crack propagation path. The deflection angle is obtained by an optimization algorithm and the torsion angle, defined for the Schöllmann\'s criterion, is imposed in the propagation vector through a one-dimensional variational formulation defined over the crack front line. The concepts of LEFM are adopted together with the Paris-Erdogan equation in order to determine the fatigue life of pre-existing defects. An iterative procedure was developed to avoid the self-intersection of the crack surfaces allowing fatigue analysis with alternate loadings. Finally, as suggestion for future researches, it was started the study of isogeometric boundary element formulations in order to perform fracture and fatigue analysis directly from CAD geometries, without surface mesh generation. A preliminary numerical study involving an isogeometric version of the DBEM using NURBS and the conventional DBEM using linear and quadratic Lagrange elements was presented. From the study it was possible to point out the advantages and disadvantages of each approach and suggest improvements for both.

L’analyse isogéométrique pour la résolution de problèmes de la mécanique du solide suscite de vifs intérêts depuis une dizaine d’année. En effet, cette méthode de discrétisation autorise la description exacte des géométries étudiées permettant ainsi de supprimer les erreurs dues à une mauvaise description du domaine spatial étudié. Cependant elle pose un problème théorique de propagation de raffinement lors de la localisation de maillage. Des méthodes pour contourner ce problème ont été proposée dans la littérature mais complexifient grandement la mise en œuvre de cette stratégie de résolution. Cette thèse propose une stratégie de raffinement localisé adaptatif en espace pour les problèmes de statique et en espace temps pour les problèmes de dynamique transitoire dans le cadre de l’analyse isogéométrique. Pour cela une méthode de localisation pour l’analyse isogéométrique en statique basée sur une résolution multigrille est tout d’abord développée pour des problèmes en deux dimensions. Elle présente l’avantage de contourner la problématique de propagation de raffinement de maillage due à l’analyse isogéométrique tout en étant plus simple à mettre en œuvre que les méthodes déjà existantes. De plus, l’utilisation de l’analyse isogéométrique permet de simplifier les procédures de raffinement lors de l’adaptation de maillage qui peuvent être complexes lors de l’utilisationd’éléments finis classiques. Une méthode de raffinement adaptatif espace temps basée sur une résolution multigrille est ensuite développée pour des problèmes en une dimension. Une étude sur la structure des opérateurs est proposée afin de choisir un intégrateur temporel adapté. Les performances de cette stratégies sont mises en évidence, puis une modification de la méthode de résolution est proposée afin de diminuer significativement les coûts de calculs associées à cette résolution. La méthode de raffinement adaptatif espace temps est appliquée à quelques exemples académiques afin de valider son bon comportement lors de la localisation. / Isogeometric analysis applied to structural mechanics problems is a topic of intense concerns for a decade. Indeed, an exact description of geometries studied is allowed by this discretization method suppressing errors due to a bad description of the spatial domain considered. However, a theoretical problem of refinement propagation appears during mesh localization. Local refinement methods for isogeometric analysis has been developed and implied a complexification of the implementation of such a resolution strategy. This PhD thesis expose a space adaptative refinement strategy for linear elastic problems and a space-time one for transient dynamic using isogeometric analysis. For this purpose, a localization method for isogeometric analysis based on a multigrid resolution is developed for 2D linear elastic problems. This method allow to circumvent mesh refinement propagation inherent to isogeometric analysis, and is easier to implement than existing methods. Moreover, the use of isogeometric analysis simplifies refinement procedures occuring during mesh adaptation and which can be really complex using classical finite element analysis. Then, a space-time adaptative refinement based on a multigrid resolution is developed for one dimensional in space problems. A study on operators structure is exposed in order to choose a well suited time integrator. This strategy's performances are highlighted, then an evolution of this method is set up in order to lower computational costs. The space-time adaptaptive refinement is applied to some academical examples to show it good behavior during localization.

Thesis deals with solving the problems of continuum mechanics by method of Isogeometric analysis. This relatively young method combines the advantages of precise NURBS geometry and robustness of the classical finite element method. The method is described on procedure of solving a plane Poissons boundary value problem. Solver is implemented in MatLab and algorithms are attached to the text.

Dans le contexte industriel actuel, où la simulation numérique joue un rôle majeur, de nombreux outils sont développés afin de rendre les calculs les plus performants et exacts possibles en utilisant les ressources numériques de façon optimale. Parmi ces outils, ceux non-intrusifs, c’est-à-dire ne modifiant pas les codes commerciaux disponibles mais permettant d’utiliser des méthodes de résolution avancées telles que l’analyse isogéométrique ou les couplages multi-échelles, apparaissent parmi les plus attirants pour les industriels. L’objectif de cette thèse est ainsi de coupler l’Analyse IsoGéométrique (AIG) et la Méthode des Éléments Finis (MEF) standard pour l’analyse de détails structuraux par une approche non-intrusive et certifiée. Dans un premier temps, on développe un lien global approché entre les fonctions de Lagrange, classiquement utilisées en éléments finis et les fonctions NURBS bases de l’AIG, ce qui permet d’implémenter des analyses isogéométriques dans un code industriel EF vu comme une boîte noire. Au travers d’exemples linéaires et non-linéaires implémentés dans le code industriel Code_Aster de EDF, nous démontrons l’efficacité de ce pont AIG\MEF et les possibilités d’applications industrielles. Il est aussi démontré que ce lien permet de simplifier l’implémentation du couplage non-intrusif entre un problème global isogéométrique et un problème local éléments finis. Ensuite, le concept de couplage non-intrusif entre les méthodes étant ainsi possible, une stratégie d’adaptation est mise en place afin de certifier ce couplage vis-à-vis d’une quantité d’intérêt. Cette stratégie d’adaptation est basée sur des méthodes d’estimation d’erreur a posteriori. Un estimateur global et des indicateurs d’erreur d’itération, de modèle et de discrétisation permettent de piloter la définition du problème couplé. La méthode des résidus est utilisée pour évaluer ces erreurs dans des cas linéaires, et une extension aux problèmes non-linéaires via le concept d’Erreur en Relation de Comportement (ERC) est proposée. / In the current industrial context where the numerical simulation plays a major role, a large amount of tools are developed in order to perform accurate and effective simulations using as less numerical resources as possible. Among all these tools, the non-intrusive ones which do not modify the existing structure of commercial softwares but allowing the use of advanced solving methods, such as isogeometric analysis or multi-scale coupling, are the more attractive to the industry. The goal of these thesis works is thus the coupling of the Isogeometric Analysis (IGA) with the Finite Element Method (FEM) to analyse structural details with a non-intrusive and certified approach. First, we develop an approximate global link between the Lagrange functions, commonly used in the FEM, and the NURBS functions on which the IGA is based. It’s allowed the implementation of isogeometric analysis in an existing finite element industrial software considering as a black-box. Through linear and nonlinear examples implemented in the industrial software Code_Aster of EDF, we show the efficiency of the IGA\FEM bridge and all the industrial applications that can be made. This link is also a key to simplify the non-intrusive coupling between a global isogeometric problem and a local finite element problem. Then, as the non-intrusive coupling between both methods is possible, an adaptive process is introduced in order to certify this coupling regarding a quantity of interest. This adaptive strategy is based on a posteriori error estimation. A global estimator and indicators of iteration, model and discretization error sources are computed to control the definition of the coupled problem. Residual base methods are performed to estimated errors for linear cases, an extension to the concept of constitutive relation errors is also initiated for non-linear problems.

Industrial development processes as well as research in physics, materials and engineering science rely on computer modelling and simulation techniques today. With increasing computer power, computations are carried out on multiple scales and involve the analysis of coupled problems. In this work, continuum modelling is therefore applied at different scales in order to facilitate a prediction of the effective material or structural behaviour based on the local morphology and the properties of the individual constituents. This provides valueable insight into the structure-property relations which are of interest for any design process. In order to obtain reasonable predictions for the effective behaviour, numerical models which capture the essential fine scale features are required. In this context, the efficient representation of discontinuities as they arise at, e.g. material interfaces or cracks, becomes more important than in purely phenomenological macroscopic approaches. In this work, two different approaches to the modelling of discontinuities are discussed: (i) a sharp interface representation which requires the localisation of interfaces by the mesh topology. Since many interesting macroscopic phenomena are related to the temporal evolution of certain microscopic features, (ii) diffuse interface models which regularise the interface in terms of an additional field variable and therefore avoid topological mesh updates are considered as an alternative. With the two combinations (i) Extended Finite Elemente Method (XFEM) + sharp interface model, and (ii) Isogeometric Analysis (IGA) + diffuse interface model, two fundamentally different approaches to the modelling of discontinuities are investigated in this work. XFEM reduces the continuity of the approximation by introducing suitable enrichment functions according to the discontinuity to be modelled. Instead, diffuse models regularise the interface which in many cases requires even an increased continuity that is provided by the spline-based approximation. To further increase the efficiency of isogeometric discretisations of diffuse interfaces, adaptive mesh refinement and coarsening techniques based on hierarchical splines are presented. The adaptive meshes are found to reduce the number of degrees of freedom required for a certain accuracy of the approximation significantly. Selected discretisation techniques are applied to solve a coupled magneto-mechanical problem for particulate microstructures of Magnetorheological Elastomers (MRE). In combination with a computational homogenisation approach, these microscopic models allow for the prediction of the effective coupled magneto-mechanical response of MRE. Moreover, finite element models of generic MRE microstructures are coupled with a BEM domain that represents the surrounding free space in order to take into account finite sample geometries. The macroscopic behaviour is analysed in terms of actuation stresses, magnetostrictive deformations, and magnetorheological effects. The results obtained for different microstructures and various loadings have been found to be in qualitative agreement with experiments on MRE as well as analytical results. / Industrielle Entwicklungsprozesse und die Forschung in Physik, Material- und Ingenieurwissenschaft greifen in einem immer stärkeren Umfang auf rechnergestützte Modellierungs- und Simulationsverfahren zurück. Die ständig steigende Rechenleistung ermöglicht dabei auch die Analyse mehrskaliger und gekoppelter Probleme. In dieser Arbeit kommt daher ein kontinuumsmechanischer Modellierungsansatz auf verschiedenen Skalen zum Einsatz. Das Ziel der Berechnungen ist dabei die Vorhersage des effektiven Material- bzw. Strukturverhaltens auf der Grundlage der lokalen Werkstoffstruktur und der Eigenschafen der konstitutiven Bestandteile. Derartige Simulationen liefern interessante Aussagen zu den Struktur-Eigenschaftsbeziehungen, deren Verständnis entscheidend für das Material- und Strukturdesign ist. Um aussagekräftige Vorhersagen des effektiven Verhaltens zu erhalten, sind numerische Modelle erforderlich, die wesentliche Eigenschaften der lokalen Materialstruktur abbilden. Dabei kommt der effizienten Modellierung von Diskontinuitäten, beispielsweise Materialgrenzen oder Rissen, eine deutlich größere Bedeutung zu als bei einer makroskopischen Betrachtung. In der vorliegenden Arbeit werden zwei unterschiedliche Modellierungsansätze für Unstetigkeiten diskutiert: (i) eine scharfe Abbildung, die üblicherweise konforme Berechnungsnetze erfordert. Da eine Evolution der Mikrostruktur bei einer derartigen Modellierung eine Topologieänderung bzw. eine aufwendige Neuvernetzung nach sich zieht, werden alternativ (ii) diffuse Modelle, die eine zusätzliche Feldvariable zur Regularisierung der Grenzfläche verwenden, betrachtet. Mit der Kombination von (i) Erweiterter Finite-Elemente-Methode (XFEM) + scharfem Grenzflächenmodell sowie (ii) Isogeometrischer Analyse (IGA) + diffuser Grenzflächenmodellierung werden in der vorliegenden Arbeit zwei fundamental verschiedene Zugänge zur Modellierung von Unstetigkeiten betrachtet. Bei der Diskretisierung mit XFEM wird die Kontinuität der Approximation durch eine Anreicherung der Ansatzfunktionen gemäß der abzubildenden Unstetigkeit reduziert. Demgegenüber erfolgt bei einer diffusen Grenzflächenmodellierung eine Regularisierung. Die dazu erforderliche zusätzliche Feldvariable führt oft zu Feldgleichungen mit partiellen Ableitungen höherer Ordnung und weist in ihrem Verlauf starke Gradienten auf. Die daraus resultierenden Anforderungen an den Ansatz werden durch eine Spline-basierte Approximation erfüllt. Um die Effizienz dieser isogeometrischen Diskretisierung weiter zu erhöhen, werden auf der Grundlage hierarchischer Splines adaptive Verfeinerungs- und Vergröberungstechniken entwickelt. Ausgewählte Diskretisierungsverfahren werden zur mehrskaligen Modellierung des gekoppelten magnetomechanischen Verhaltens von Magnetorheologischen Elastomeren (MRE) angewendet. In Kombination mit numerischen Homogenisierungsverfahren, ermöglichen die Mikrostrukturmodelle eine Vorhersage des effektiven magnetomechanischen Verhaltens von MRE. Außerderm wurden Verfahren zur Kopplung von FE-Modellen der MRE-Mikrostruktur mit einem Randelement-Modell der Umgebung vorgestellt. Mit Hilfe der entwickelten Verfahren kann das Verhalten von MRE in Form von Aktuatorspannungen, magnetostriktiven Deformationen und magnetischen Steifigkeitsänderungen vorhergesagt werden. Im Gegensatz zu zahlreichen anderen Modellierungsansätzen, stimmen die mit den hier vorgestellten Methoden für unterschiedliche Mikrostrukturen erzielten Vorhersagen sowohl mit analytischen als auch experimentellen Ergebnissen überein.

Interface balance laws are derived to describe transport across a phase interface. This is used to derive generalized conditions for phase nucleation and growth, valid even for solids with inhomogeneous surface stress.<div><br></div><div>An explicit interface tracking approach called Enriched Isogeometric Analysis (EIGA) is used to simulate phase evolution. Algebraic level sets are used as a measure of distance and for point projection, both necessary operations in EIGA. Algebraic level sets are observed to often fail for surfaces. Rectification measures are developed to make algebraic level sets more robust and applicable for general surfaces. The proposed methods are demonstrated on electromigration problems. The simulations are validated by modeling electromigration experiments conducted on Cu-TiN line structures.</div><div><br></div><div>To model topological changes, common in phase evolution problems, Boolean operations are performed on the algebraic level sets using R-functions. This is demonstrated on electromigration simulations on solids with multiple voids, and on a bubble coalescence problem. </div>

<p>Moving boundary problems such as solidification, crack propagation, multi-body contact or shape optimal design represent an important class of engineering problems. Common to these problems are one or more moving interfaces or boundaries. One of the main challenges associated with boundary evolution is the difficulty that arises when the topology of the geometry changes. Other geometric issues such as distance to the boundary, projected point on the boundary and intersection between surfaces are also important and need to be efficiently solved. In general, the present thesis is concerned with the geometric arrangement and behavioral analysis of evolving parametric boundaries immersed in a domain. </p> <p>The first problem addressed in this thesis is the packing of ellipsoidal fillers in a regular domain and to estimate their effective physical behavior. Particle packing problem arises when one generates simulated microstructures of particulate composites. Such particulate composites used as thermal interface materials (TIMs) motivates this work. The collision detection and distance calculation between ellipsoids is much more difficult than other regular shapes such as spheres or polyhedra.  While many existing methods address the spherical packing problems, few appear to achieve volume loading exceeding 60%. The packing of ellipsoidal particles is even more difficult than that of spherical particles due to the need to detect contact between the particles. In this thesis, an efficient and robust ultra-packing algorithm termed Modified Drop-Fall-Shake is developed. The algorithm is used to simulate the real mixing process when manufacturing TIMs with hundreds of thousands ellipsoidal particles. The effective thermal conductivity of the particulate system is evaluated using an algorithm based on Random Network Model. </p> <p><br></p> <p>In problems where general free-form parametric surfaces (as opposed to the ellipsoidal fillers) need to be evolved inside a regular domain, the geometric distance from a point in the domain to the boundary is necessary to determine the influence of the moving boundary on the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge,  an algebraic procedure is presented here to find the level sets of rational parametric surfaces commonly utilized by commercial CAD systems. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bezier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain can also be used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G₀ continuity---problems where the Newton--Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively. </p> <p><br></p> <p>Next, ALS is also extended for boundary representation (B-rep) models that are popularly used in CAD systems for modeling solids. B-rep model generally contains multiple NURBS patches due to the trimming feature used to construct such models, and as a result are not ``watertight" or mathematically compatible at patch edges. A time consuming geometry clean-up procedure is needed to preprocess geometry prior to finite element mesh generation using a B-rep model, which can take up to 70% of total analysis time according to literature. To avoid the need to clean up geometry and directly provide link between CAD and CAE integration,  signed algebraic level sets using novel inner/outer bounding box strategy is proposed for point classification of B-rep model. Several geometric examples are demonstrated, showing that this technique naturally models single patch NURBS geometry as well, and can deal with multiple patches involving planar trimming feature and Boolean operation. During the investigation of algebraic level sets, a complex self-intersection problem is also reported, especially for three-dimensional surface. The self-intersection may occur within an interval of interest during implicitization of a curve or surface since the implicitized curve or surface is not trimmed and extends to infinity. Although there is no robust and universal solution the problem, two potential solutions are provided and discussed in this thesis.</p> <p><br></p> <p>In order to improve the computational efficiency of analysis in immersed boundary problems, an efficient local refinement technique for both mesh and quadrature  using the kd-tree data structure is further proposed. The kd-tree sub-division is theoretically proved to be more efficient against traditional quad-/oct-tree subdivision methods. In addition, an efficient local refinement strategy based on signed algebraic level sets is proposed to divide the cells. The efficiency of kd-tree based mesh refinement and adaptive quadrature is later shown through numerical examples comparing with oct-tree subdivision, revealing significant reduction of degrees of freedom and quadrature points.</p> <p><br></p> <p>Towards analysis of moving boundaries problems, an explicit interface tracking method termed enriched isogeometric analysis (EIGA) is adopted in this thesis. EIGA utilizes NURBS shape function for both geometry representation and field approximation. The behavior field is modeled by a weighted blending of the underlying domain approximation and enriching field, allowing high order continuity naturally. Since interface is explicitly represented, EIGA provides direct geometric information such as normals and curvatures. In addition, the blending procedure ensures strong enforced boundary conditions. An important moving boundary problem, namely, reaction-diffusion problem, is investigated using EIGA. In reaction-diffusion problems, the phase interfaces evolve due to chemical reaction and diffusion under multi-physics driven forces, such as mechanical, electrical, thermal, etc. Typical failure phenomenon due to reaction-diffusion problems include void formation and intermetallic compound (IMC) growth. EIGA is applied to study factors and behavior patterns in these failure phenomenon, including void size, current direction, current density, etc. A full joint simulation is also conducted to study the degradation of solder joint under thermal aging and electromigration. </p>

[pt] A Análise Isogeométrica (IGA) é um método de análise numérica de estruturas que surge com a proposta de unificação entre projeto e simulação, permitindo a criação de modelos computacionais que preservam a geometria exata do problema. Essa abordagem é possível por meio de uma classe de funções matemáticas denominadas NURBS (Non-Uniform Rational B-Splines), amplamente utilizadas em sistemas CAD (Computer-Aided Design) para modelagem de curvas e superfícies. Na análise isogeométrica, as mesmas funções que representam a geometria aproximam as variáveis de campo. Neste contexto, foi desenvolvido este trabalho que tem como objetivo fornecer uma ferramenta no âmbito da mecânica computacional para análise isogeométrica bidimensional de problemas de elasticidade linear, incluindo as etapas de modelagem, análise e visualização de resultados. O sistema é composto por dois softwares: FEMEP (Finite Element Method Educational Computer Program), desenvolvido em Python e responsável pela etapa de modelagem geométrica, e FEMOOLab (Finite Element Method Object-Oriented Laboratory), software MATLAB para análise e exibição de resultados. A ferramenta proposta apresenta uma interface gráfica de usuário (GUI) que permite a visualização e manipulação intuitiva de curvas NURBS com recursos avançados de modelagem, como interseção de curvas e recursos de reconhecimento de região que agilizam e simplificam o processo. Uma contribuição significativa deste trabalho reside na capacidade de gerar malhas isogeométricas não estruturadas, utilizando T-Splines baseadas em um algoritmo de decomposição de domínio. O sistema de código aberto permite a colaboração e o desenvolvimento contínuo pela comunidade de usuários e desenvolvedores. / [en] Isogeometric Analysis (IGA) is a numerical analysis method for structures that arises with the proposal of unification between design and simulation, allowing the creation of computational models that preserve the exact geometry of the problem. This approach is possible by a class of mathematical functions called NURBS (Non-Uniform Rational B-Splines), widely used in CAD (Computer-Aided Design) systems for modeling curves and surfaces. In isogeometric analysis, the same functions representing the geometry approximate the field variables. In this context, this work was developed to provide a tool within the scope of computational mechanics for two-dimensional isogeometric analysis of linear elasticity problems, including the steps of modeling, analysis, and visualization of results. The system consists of two software programs: FEMEP (Finite Element Method Educational Computer Program), developed in Python and responsible for the geometric modeling stage, and FEMOOLab (Finite Element Method Object-Oriented Laboratory), a MATLAB software for analysis and display of results. The proposed tool features a graphical user interface (GUI) that allows intuitive visualization and manipulation of NURBS curves with advanced modeling features such as curve intersection and region recognition features that streamline and simplify the process. A significant contribution of this work lies in the ability to generate non-structured isogeometric meshes, using T-Splines based on a domain decomposition algorithm. The open-source system allows collaboration and continuous development by the community of users and developers.