Neste trabalho é analisado o problema elástico tridimensional através do método dos elementos de contorno empregando a solução fundamental de Kelvin. São utilizadas duas formulações principais: a formulação clássica e a formulação hiper-singular. A primeira utiliza a solução fundamental de Kelvin clássica e a segunda aplica uma derivada direcional da solução fundamental de Kelvin. O contorno é discretizado utilizando-se elemento triangular plano com aproximações constante, linear e quadrática. As integrais singulares são desenvolvidas analiticamente para o elemento constante, e semi-analiticamente para os elementos linear e quadrático. São apresentadas técnicas de integração de contorno considerando-se a eficiência e a precisão para a integral quase singular. São apresentados vários exemplos numéricos, inclusive problemas esbeltos, e seus resultados são comparados com valores conhecidos pela teoria de elasticidade, ou ainda, comparados com valores disponíveis na literatura. / In this work the three-dimensional elastic problem is analyzed by the boundary element method using the Kelvin fundamental solution. Two main formulations are applied. The first one uses the classical Kelvin fundamental solution and the other, hyper-singular, uses a derivative of the Kelvin fundamental solution. The boundary is discretized by flat triangular elements with constant, linear and quadratic approximations. The singular integrals are analytically developed for constant elements, while for linear and quadratic elements a semi-analytical process is employed. Different techniques to perform quasi-singular boundary integrals are presented and their efficiency and accuracy are compared. Several numerical examples are presented, including slender problems. The results are compared with known solutions given by the theory of elasticity, or with other results found in the literature.

Neste trabalho é analisado o problema elástico tridimensional através do método dos elementos de contorno empregando a solução fundamental de Kelvin. São utilizadas duas formulações principais: a formulação clássica e a formulação hiper-singular. A primeira utiliza a solução fundamental de Kelvin clássica e a segunda aplica uma derivada direcional da solução fundamental de Kelvin. O contorno é discretizado utilizando-se elemento triangular plano com aproximações constante, linear e quadrática. As integrais singulares são desenvolvidas analiticamente para o elemento constante, e semi-analiticamente para os elementos linear e quadrático. São apresentadas técnicas de integração de contorno considerando-se a eficiência e a precisão para a integral quase singular. São apresentados vários exemplos numéricos, inclusive problemas esbeltos, e seus resultados são comparados com valores conhecidos pela teoria de elasticidade, ou ainda, comparados com valores disponíveis na literatura. / In this work the three-dimensional elastic problem is analyzed by the boundary element method using the Kelvin fundamental solution. Two main formulations are applied. The first one uses the classical Kelvin fundamental solution and the other, hyper-singular, uses a derivative of the Kelvin fundamental solution. The boundary is discretized by flat triangular elements with constant, linear and quadratic approximations. The singular integrals are analytically developed for constant elements, while for linear and quadratic elements a semi-analytical process is employed. Different techniques to perform quasi-singular boundary integrals are presented and their efficiency and accuracy are compared. Several numerical examples are presented, including slender problems. The results are compared with known solutions given by the theory of elasticity, or with other results found in the literature.

Les structures en forme de tige sont omniprésentes dans le monde aujourd'hui. Désormais, prédire avec précision leur comportement pour l'ingénierie et les environnements virtuels est indispensable pour de nombreuses industries, notamment l'infographie, l'animation par ordinateur et la conception informatique. Dans ce mémoire, nous explorons un nouveau modèle de calcul pour les tiges élastiques qui exploite les données de simulation pour reproduire les effets de coque et de solide présents dans les tiges qui brisent les hypothèses de la théorie classique de la tige de Kirchhoff, présentant ainsi une voie d'amélioration possible pour de nombreux états de l'art techniques. Notre approche consiste à prendre un ensemble de données de simulations à partir de solides volumétriques ou de coques pour former un nouveau modèle d'énergie définie positive polynomiale d'ordre élevé pour une tige élastique. Cette nouvelle énergie élargit la gamme des comportements des matériaux qui peuvent être modélisés pour la tige, permettant ainsi de capturer une plus large gamme de phénomènes. Afin de proposer et tester ce modèle, nous concevons un pipeline expérimental pour tester les limites de la théorie linéaire des tiges et étudier les géométries d'interface entre les cas coque à tige et volume à coque pour observer les effets d'un modèle de matériau non linéaire et une section transversale non elliptique dans la déformation de la tige. Nous étudions également la relation entre la courbure de la tige et la déformation de la section transversale et la courbure pour introduire une modification sur le terme de flexion de l'énergie. Cela nous permet de reproduire à la fois le comportement de flexion asymétrique présent dans les poutres volumétriques minces et les poutres à coque avec des sections transversales non convexes. Des suggestions pour de nouvelles améliorations des modèles et des techniques expérimentales sont également données. / Rod-like structures are ubiquitous in the world today. Henceforth accurately predicting their behavior for engineering and virtual environments are indispensable for many industries including computer graphics, computer animation, and computational design. In this thesis we explore a new computational model for elastic rods that leverages simulation data to reproduce shell and solid-like effects present in rods that break the assumptions of the classical Kirchhoff rod theory, thus presenting a possible improvement avenue to many states-of-the-art techniques. Our approach consists of taking a data set of simulations from both volumetric solids or shells to train a novel high-order polynomial positive-definite energy model for an elastic rod. This new energy increases the range of material behaviors that can be modeled for the rod, thus allowing for a larger range of phenomena to be captured. In order to propose and test this model, we design an experimental pipeline to test the limits of the linear theory of rods and investigate the interface geometries between the Shell-Rod and Volume-Shell cases to observe the effects of a nonlinear material model and a non-elliptical cross-section in the rod deformation. We also investigate the relation between rod curvature and deformation of the cross-section and curvature to introduce a modification on the bending term of the energy. This allows us to reproduce both the asymmetric bending behavior present in thin volumetric solid and shell beams with non-convex cross-sections. Suggestions for further improvements in models and experimental techniques are also given.