Numerical methods and specifically the finite element method have improved significantly since their introduction in the 60's. These advances were mainly in: 1) introducing higher-order elements, 2) developing effective solution schemes, 3) developing sophisticated means of modeling the constitutive behavior of geotechnical materials, and 4) introducing iteration techniques to model material non-linearity. This thesis, on the other hand, deals with the topic of modeling the boundary conditions of the finite element problem. Typically, the boundary conditions will be approximated by specifying displacement constraints. such as restraining the bottom boundary of the finite element mesh against displacements in the horizontal and vertical directions (x- and y-directions). Where bedrock or dense residual soils underlie the soft foundation soil at a relatively shallow depth, this is a good assumption. However. when soft soil is encountered for large depths, the assumption of zero movement constraints for a mesh boundary at a shallower depth than the actual bedrock will result in a serious underestimation of stresses and displacements. By coupling boundary elements to the finite elements and using them to model the infinite extent of the foundation soil, a more realistic answer is obtained. Employing the coupled boundary element - finite element method, four cases were analyzed and the results compared to values of the pure finite element method. The results show that the coupled method indeed yielded higher stress- and displacement-values, indicating that the pure finite element method underestimates stresses and displacements when modeling very deep soils. / Master of Science

In this thesis the impact of inverted cones on a liquid surface is studied. It is known that with the right combination of velocity, geometry, and surface treatment, a cavity of air can be formed behind an impacting body and extended for a considerable distance. Other investigators have shown that the time and depth of the cone when this cavity collapses and seals follows a different power law for flat objects such as disks, then it does for slender objects such as cylinders. Intuitively it can be expected that a more slender body will have less drag and that the streamlined shape will not push the fluid out of it's way at impact to the same extent as a more blunt body, therefore forming a smaller cavity behind it. With a smaller initial cavity, the time and depth of it's eventual collapse can be expected to be less than that of a much more blunt object, such as a flat disk. To study this, a numerical model has been developed to simulate cones with the same base radius but different angles impacting on a liquid surface over a range of velocities, showing how the seal depth, time at cavity seal, and drag forces change. In order to ensure the numerical model is accurate, it is compared with experimental data including high speed video and measurements made of the force with time. It is expected that the results will fall inside the power law exponents reported by other authors for very blunt objects such as disks on one end of the spectrum, and long slender cylinders on the other. Furthermore, we expect that the drag force exerted on the cones will become lower as the L/D of the cone is increased. / Master of Science

Knowledge of the indoor propagation of sound has many important applications including acoustic prediction in homes, office buildings, stores, and schools, and the design of concert halls, auditoriums, classrooms, and factories. At low frequencies, interior acoustics are analyzed with the wave equation, but significant computational expense imposes an upper frequency limit. Thus, energy methods are often sought for high frequency analysis. However, conventional energy methods are significantly limited by vast simplifications or computational costs. Therefore, new improvements are still being sought. The basis of this dissertation is a recently developed mathematical model for interior acoustics known as the acoustic diffusion model. The model extends statistical methods in high frequency acoustics to predict the spatial distribution of acoustic energy in the volume over time as a diffusion process. Previously, solutions to the acoustic diffusion model have been limited to one dimensional (1-D) analytical solutions and to the use of the finite element method (FEM). This dissertation focuses on a new, efficient method for solving the acoustic diffusion model based on a boundary element method (BEM) solution using the Laplace transform. First, a Laplace domain solution to the diffusion model is obtained using the BEM. Then, a numerical inverse Laplace transform is used to efficiently compute the time domain response. The diffusion boundary element-Laplace transform solution (BE-LTS) is validated through comparisons with Sabine theory, ray tracing, and a diffusion FEM solution. All methods demonstrate excellent agreement for three increasingly complex acoustic volumes and the computational efficiency of the BE-LTS is exposed. Structural coupling is then incorporated in the diffusion BE-LTS using two methods. First, a simple transmission coefficient separating two acoustic volumes is implemented. Second, a structural power flow model represents the coupling partition separating acoustic volumes. The validation of these methods is successfully performed in an example through comparisons with statistical theory, a diffusion FEM solution, ray tracing, and experimental data. Finally, the diffusion model and the BE-LTS are shown to possess capabilities beyond that of room acoustics. The acoustic transmission through a heat exchanger, acoustic foam, and mufflers is successfully modeled using the diffusion BE-LTS and compared to experimental data. / Ph. D.

Soil-structure interaction problems involve the solution of boundary value problems consisting of two domains: A near field finite domain representing the structure and adjacent soil, and a semi-infinite far-field domain representing the soil distant from the structure. Currently, the most used numerical method for solving such problems, the finite element method, considers only the near field, and neglects the effects of the far field. Depending on the domain size considered, this results in significant errors in the computed displacements and stress compared to closed form solutions. This research develops a numerical method in which both the near and far-field are considered. In this numerical procedure, the far field is assumed to be a homogenous elastic half-plane is modeled using boundary elements based on the Melan fundamental solution. A technique, called the substructure method, for coupling the boundary element method with finite element method is developed. Unlike other coupling techniques, the substructure method preserves the bandedness and symmetry of the system of equations resulting from the finite element method. The substructure method is implemented into a computer for program BEFEC for solving linear elastic and elasto-plastic plane strain problems. The proposed coupling technique is also incorporated into an existing finite element program SOILSTRUCT to perform soil-structure interaction analysis on U-frame lock structures. A series of analyses performed on the elastic strip footing problem indicate that significant errors occur in the predicted displacements and stresses when the effects of the the far field are ignored. These errors are unaffected by the boundary conditions assumed or the type of finite element used. The analyses demonstrated that the displacements and stresses obtained using the coupled BEM-FEM solution agree well with closed form solutions. Results of the soil-structure interaction analyses performed on U-frame lock structures indicates that neglecting the effects of the far field domain results in a significant underprediciton of the vertical displacements. These analyses also showed that there are significant differences in the computed shear stress and lateral pressures when the effects of the far field are considered. Results of the bearing capacity analyses of strip footing on elasto-plastic soils indicate that when the effects of the far field are taken into consideration, initial yield takes place at a higher load level. This in turn results is smaller plastic deformations as compared to the case when the far field is ignored. These analyses also shown that taking into consideration the far field results in significant differences in the computed stresses. These differences are diminished when the effects of self-weight are taken into consideration. The analyses performed on the Rankine earth pressure problem indicate that while the far field does not significantly affect the computed Rankine forces or lateral pressure distribution, much larger wall movements are required to reach both the active or passive States. / Ph. D.

Au cours de cette thèse, nous avons développé un outil numérique, basé sur une formulation intégrale en éléments de frontière, qui permet une analyse classique du comportement d'une fissure 3D soumise à des sollicitations mécaniques complexes. Cet outil industriel est destiné à être intégré dans un code de calcul à usage industriel. Dans le but d'appréhender l'impact de la microstructure sur le comportement en fissuration fragile, nous nous sommes intéressés aux modèles de comportement non local. Nous avons commencé par adopter le modèle de comportement élastique non local de Eringen, qui permet de décrire plus finement le comportement élastique au voisinage de la fissure en prenant en compte les interactions à longue distance au sein du matériau. Cette modélisation du comportement conduit, contrairement à l'approche classique, à un un champ de contrainte fini sur le front de la fissure et localement maximal en avant du front. Ces résultats montrent qu'il est possible de prévoir la stabilité et la direction de propagation de la fissure à l'aide d'un critère plus simple et plus naturel, basé sur les variations du champ de contrainte au voisinage du front de la fissure. La stratégie numérique adoptée permet de traiter indifféremment des cas de fissure en traction, compression, cisaillement ou sollicitation mixte. L'intérêt de l'approche non-locale étant clairement démontré, nous avons considéré la version améliorée du modèle de Eringen telle que proposée par Polizzotto. Cette modélisation est la plus appropriée pour les milieux finis et requiert une mise en oeuvre numérique particulière. Les bases d'une méthodologie numérique, initiée par R. Kouitat ont été établies. Cette méthode est fondée sur un couplage des éléments de frontière avec une méthode de collocation par points d'équations aux dérivées partielles. Les premiers résultats obtenus dans ce cadre sont très encourageants et montrent qu'il sera effectivement possible de traiter le phénomène irréversible de fissuration de la même façon que les problèmes de plasticité / In this thesis, we have developed a numerical tool, based on a classical boundary elements method, which allows a conventional analysis of a stationary crack in a 3D specimen under complex mechanical loading. In order to assess the impact of the microstructure on the brittle fracture, we were interested in non local models of behavior. First, we have adopted the non local elastic model due to Eringen. This refined constitutive equation allows to account for long range interactions in the description of the elastic behavior in the vicinity of the crack front. Unlike the traditional approach, this type of model leads to a finite stress field at the crack front. Moreover, the stress is locally maximal ahead of the front. These interesting results indicate that it is possible to predict the stability and direction of crack propagation in a simple and more naturel way by using stress based criteria. The implemented numerical strategy can handle cases of crack in tension or compression, under shear stress or mixed loadings. Having clearly highlighted the interest of non local models, we have adopted the improved version of Eringen elastic model as proposed by Polizzotto. This elastic model is applicable to finite domains and requires a specific numerical treatment. The basis of such a numerical strategy initiated by R. Kouitat has been established. The method couples the conventional boundary element method with local point interpolation of a strong form differential equation. Promising results are obtained and show that with such modeling of material behavior, it is possible to describe the irreversible process of fracturing in a similar way as plasticity

Esse trabalho trata da análise de corpos multi-fraturados utilizando o método dos elementos de contorno. Este método numérico é conhecido por ser robusto e preciso neste tipo de problema e também por requerer pequeno esforço computacional na criação da malha de elementos para o crescimento das fissuras. Duas metodologias para a análise do comportamento das fissuras são consideradas. A primeira é a já consagrada metodologia dual. Por meio desta técnica equações integrais distintas são aplicadas às faces da fissura. Estas equações integrais são escritas em termos de deslocamentos e forças de superfície. A segunda metodologia é a que emprega a formulação singular onde a fissura é considerada como um vazio no domínio sendo as faces da fissura separadas por uma pequena distância. No tocante ao crescimento das fissuras foi desenvolvido um procedimento especial para a determinação da direção de crescimento das fissuras o qual mostrou-se muito eficiente levando a resultados precisos. O crescimento das fissuras é efetuado considerando o fator de intensidade de tensão atuante na extremidade de cada fissura. Dessa forma, as fissuras mais solicitadas apresentam maior comprimento de propagação tornando a análise mais realista. Os fatores de intensidade de tensão são calculados por meio de duas técnicas. A primeira é a já conhecida técnica de correlação de deslocamentos a qual relaciona os deslocamentos atuantes nas faces da fissura. Uma técnica alternativa é também utilizada a qual emprega o campo de tensões presente na extremidade da fissura. Após a determinação dos fatores de intensidade de tensão quatro diferentes teorias de interação de modos podem ser utilizadas para a determinação do ângulo de propagação. Foram analisadas estruturas sendo os resultados comparados aos previstos analiticamente e também numericamente. As respostas obtidas foram satisfatórias validando assim a metodologia proposta neste trabalho / This work deals with analysis of multi-fractured bodies using boundary element method. This numerical method is known to be robust and accurate in this kind of problem and by small computational effort to create elements mesh of crack growth. Two methodologies to analyze of crack behavior are considerate. The first is consecrated dual methodology. Through this technique different kind of integral equations are applied to crack boundaries. These integrals equations are written in displacements and traction variables. Second methodology is singular formulation. Through this technique crack is represented like a hole in body and the crack boundaries is separated by a small gap. For crack growth was created a special proceeding to determination crack growth direction. This method is very efficient and your results are accurate. Crack growth is made through the stress intensity factor performed in crack tip. Then the cracks more requested going to propagate with a larger length’s growth turning this model very realistic. The stress intensity factors are calculated through two techniques in this work. First is the known correlation displacement technique which related displacement in crack boundaries. An alternative technique is also used which consider stress field in crack tip. After determination of stress intensity factors four different theories are used to calculate the crack growth angle. In this work were analyzed structures with results are compared with analytical and numerical answers. The results obtained went very satisfactory validating the methodology proposed

A otimização topológica de estruturas está relacionada à concepção de projetos que executem suas funções com nível de segurança adequado empregando a quantidade mínima de material. Neste trabalho, determina-se a geometria ótima de estruturas planas por meio do acoplamento do Método dos Elementos de Contorno (MEC) ao Método Level Set (MLS). O algoritmo é composto por 3 etapas: problema mecânico, otimização topológica e reconstrução da estrutura. O problema mecânico é resolvido pelas equações algébricas do MEC. A otimização topológica é determinada pelo MLS, este representa a geometria do corpo e suas evoluções por meio da função Level Set (LS) avaliada em seu nível zero. Na reconstrução realiza-se o remalhamento, pois a cada iteração a estrutura é modificada. O acoplamento proposto resulta na geometria ótima da estrutura sem a necessidade da aplicação de filtros. Os exemplos analisados mostram que algoritmo desenvolvido capta adequadamente a geometria ótima das estruturas. Com esse trabalho, avança-se no campo das aplicações do acoplamento MEC-MLS e no desenvolvimento de soluções inovadoras para problemas complexos de engenharia. / In general, the topological optimization of structures is related to design projects that perform their functions with appropriate security levels using the minimum amount of material. This research determines the optimal geometry of 2D structures by coupling the Boundary Blement Method (BEM) to Level Set Method (LSM). The algorithm consists of 3 steps: mechanical model, topology optimization and structure reconstruction. The mechanical model is solved by BEM algebraic equations. The topology optimization is determined using the MLS, the geometry of the body is determined by the Level Set (LS) function evaluated at the zero level. The reconstruction achieves the remeshing, because for each iteration of the structure is modified. The proposed coupling results in the optimal geometry of the structure without the filters application. The examples show that the algorithm developed captures adequately the optimal geometry of the structures. With this dissertation, it is possible advance in the field of applications of the BEM - LSM and develop innovative solutions to complex engineering problems.

O presente trabalho trata da análise do comportamento mecânico da madeira, material anisotrópico, submetida ao crescimento de fissuras e aos processos mecânicos que causam sua ruptura, utilizando os conceitos de Mecânica da Fratura. A avaliação do colapso de peças de madeira, pela da mecânica da fratura, foi realizada por meio de análises experimentais. Para a determinação das propriedades de fratura da madeira foram determinadas propriedades mecânicas da madeira, tais com: módulo de ruptura a flexão estática, módulo de elasticidade à flexão estática, resistência à tração paralela às fibras, resistência ao cisalhamento, resistência ao fendilhamento paralela às fibras e resistência à tração normal às fibras. Foram efetuados ainda ensaios experimentais referentes à mecânica da fratura em corpos de prova tipo SENB (Single Edge Notched Beam). A modelagem numérica foi realizada utilizando-se um código computacional baseado no método de elementos de contorno. Para as simulações foram utilizados os valores das propriedades físicas, de resistência e de elasticidade obtidas na caracterização experimental dos elementos de madeira. Os resultados experimentais foram comparados com as análises numéricas para determinação de uma formulação específica para situações de colapso de elementos de madeira. / The present work deals with the analysis of the mechanical behavior of wood, anisotropic material, submitted to the growth of cracks and the mechanical processes that cause its rupture, using the concepts of Fracture Mechanics. The evaluation of the collapse of pieces of wood, by the fracture mechanics, was carried out by means of experimental analyzes. To determine the fracture properties of the wood, mechanical properties of the wood were determined, such as: modulus of static bending, modulus of elasticity at static bending, tension resistance parallel to grain, shearing resistance, cleavage strength parallel to grain and tension resistance normal to grain. Experimental tests were performed on fracture mechanic in Single Edge Notched beams (SENB). Numerical modeling was performed using a computational code based on the boundary element method. For the simulations, the values of the physical properties, strength and elasticity obtained in the experimental characterization of the wood elements were used. The experimental results were compared with the numerical analyzes to determine a specific formulation for situations of collapse of wood elements.

Apresentam-se neste trabalho formulações do método dos elementos de contorno (MEC), visando sua utilização em problemas de fratura e também de modelos de confiabilidade e otimização aplicados na análise de problemas de fadiga. Com relação aos progressos e avanços nas formulações do MEC, apresentam-se modelos que representam o processo de crescimento de fissuras em domínios planos constituídos por materiais frágeis, quase-frágeis e dúcteis. Considerando esses diferentes tipos de materiais, a formulação numérica adotada na análise descreve o comportamento estrutural não linear decorrente do processo de propagação das fissuras e conseqüente degradação estrutural. Nos modelos de fratura é empregada a formulação MEC dual, a qual é mais adequada para a análise da propagação aleatória de fissuras. São também apresentadas as expressões dos operadores tangente para as formulações não lineares que tratam os problemas de fratura elástico linear e coesiva, problemas de contato e os problemas de domínios enrijecidos. Com relação às análises de confiabilidade estrutural, o modelo mecânico de fadiga é acoplado a algoritmos de confiabilidade para a determinação do índice de confiabilidade e do conjunto de valores aleatórios com maior probabilidade de ocorrência. São testados alguns algoritmos de confiabilidade, podendo-se claramente definir um deles como mais eficiente para a análise de problemas de fadiga. A esse modelo é acoplado um algoritmo de otimização para a determinação das dimensões do elemento estrutural e dos intervalos para os procedimentos de manutenção e inspeção, que levam ao mínimo custo estrutural com base nas incertezas determinadas pelo modelo de confiabilidade. São apresentados vários exemplos validando e mostrando a eficiência das formulações desenvolvidas. / This work deals with the development of boundary element method (BEM) formulations to be used in engineering problems. Particular attention is given to using these formulations in development of reliability and optimization models applied to fatigue problems. Contributions to BEM formulations are developed, particularly, models that deal with crack growth in plane domains composed by brittle, quasi-brittle and ductile materials. Taking into account these different types of materials, the proposed formulation properly represents the nonlinear structural behaviour induced by crack growth and the resulting structural damage. The dual BEM formulation is adopted here for the proposed crack model and to analyse random crack propagation. In this thesis tangent operators are used in the non-linear BEM formulations, in order to deal with cohesive crack, contact problems and debonding problems in reinforced domains. Regarding structural reliability analysis, the fatigue mechanical model was coupled with appropriate reliability algorithms to compute the reliability index and other important random values. Several reliability algorithms were tested for this coupled model, in order to find the most efficient in the analysis of fatigue problems. An optimization model was also coupled with the fatigue reliability model, in order to evaluate the optimal structural element dimensions and also to schedule the intervals for maintenance and inspection procedures, taking into account the minimum cost and problem uncertainties. Many examples are presented in order to show the efficiency and accuracy of the proposed formulations in dealing with crack propagation, fatigue reliability analysis and optimization problems.

A aplicação da análise inversa é objeto de estudo nos mais diversos campos da ciência e da engenharia. A motivação para o tratamento de tais problemas se deve ao fato de que em muitas aplicações dessas áreas do conhecimento, há a necessidade da identificação de parâmetros físicos e geométricos a partir de dados do domínio medidos experimentalmente, já que tais parâmetros de entrada são desconhecidos para uma análise direta do problema. Neste tipo de análise o problema principal está na quantidade e qualidade dos dados experimentais obtidos, que são na maioria das vezes insuficientes para garantir que o sistema gerado apresente solução única, gerando com isto um problema essencialmente mal-posto. Assim, de forma geral o emprego confiável da análise inversa implica na utilização de ferramentas eficientes de aquisição de dados experimentais aliada a técnicas numéricas de regularização que buscam a minimização da função objetiva gerada por algum método numérico, como por exemplo, o método dos elementos de contorno (MEC). Sendo assim, o presente trabalho tem por objetivo apresentar uma formulação para resolução de problemas inversos de valor de contorno e estimativa dos parâmetros do modelo coesivo, através de medidas de campos de deslocamentos, em sólidos bidimensionais com domínio formado por multi-regiões via (MEC), utilizando-se de técnicas tais como: mínimos quadrados, regularização de Tikhonov, decomposição em valor singular (SVD) e filtro de Tikhonov, para regularização do problema. Além disto, são apresentados alguns exemplos de aplicação da formulação desenvolvida. / The application of inverse analysis is nowadays subject of research of many fields in engineering and science. The motivation to consider this problem is due to the fact that in many applications of these knowledge areas, physical and geometric parameters, that are not directly known, can be identified using domain data measured experimentally. In this kind of analysis the main problem is the quantity and the quality of the obtained experimental data, which, many times, are not sufficient to guarantee that the generated system of equations has only one solution, leading therefore to an ill-posed problem. Thus, in general the reliable use of the inverse analysis requires using efficient tools for experimental data acquisition together with the numerical techniques of regularization needed to impose the minimization of the objective function written by using any numerical method, as the boundary element method (BEM) for instance. In this context, the objective of the present work is to derive a formulation for resolution of boundary-value inverse problems and to estimate the material parameters of the cohesive model, by using measured displacements fields, in multi-region two-dimensional solid by BEM, using techniques such as: least squares, Tikhonov regularization, singular value decomposition (SVD) and Tikhonov filtering, for the problem regularization. Some application examples are presented using the developed formulation to illustrate its performance.