Polycrystalline materials are widely used for industrial applications. These materials are highly anisotropic with different responses under different loading conditions. This dissertation uses a crystal plasticity scheme in the finite element framework (CPFEM) to study deformation mechanisms in alumina, aluminium and stainless steel – all polycrystalline. Four research cases in this dissertation have been presented in the form of manuscripts for publication. When possible, modelling predictions have been compared against various experimental techniques such as Diffraction Contrast Tomography (DCT), Neutron Diffraction (ND) and Electron Back Scatter Diffraction (EBSD). After an introduction (Chapter 1) and a literature review (Chapter 2) on plastic deformation and modelling techniques, the methodology and results are presented and discussed (Chapters 3 and 4). Measurements of elastic strains for individual grain families (ND) and local rotations (DCT and EBSD) are compared against corresponding predictions by the model following different loading modes. Each study reveals different degrees of agreement between predictions and measurements. The individual conclusions to each study are presented in Chapter 4. Some overall conclusions and suggestions for further work are presented in Chapter 5.

[pt] Apresenta-se uma formulação particular do método híbrido dos elementos de contorno para a análise de problemas planos de potencial e de elasticidade que, apesar de completamente geral, é apropriada a aplicações de mecânica da fratura. Funções do tipo de Westergaard são usadas como soluções fundamentais, Em uma generalização de uma proposta inicialmente feita por Tada et al. A formulação leva a conceitos de elementos de contorno de deslocamentos semelhante à apresentada por Crouch e Starfield, mas em um contexto variacional que permite interpretações mecânicas bem simples das equações matriciais resultantes. Problemas de topologia geral podem ser modelados, como no caso de domínios infinitos ou multiplamente conexos. A formulação, que é diretamente aplicável a placas com entalhes ou trincas curvas externas ou internas, permite a descrição adequada de altos gradientes de tensão e é uma ferramenta simples de avaliação de fatores de intensidade de tensão, com o que se podem verificar numericamente conceitos estabelecidos por Rice em 1968. Esta dissertação tem foco na fundamentação matemática da formulação para problemas de potencial e de elasticidade. Apresenta-se a implementação da formulação e são discutidos vários exemplos numéricos de validação. / [en] A particular implementation of the hybrid boundary element method is presented for the two dimensional analysis of potential and elasticity problems, which although general in concept, is suited for fracture mechanics applications. Generalized Westergaard stress functions, as proposed by Tada et al in 1993, are used as the problem‘s fundamental solutions. The proposed formulation leads to displacement-based concepts that resemble those presented by Crouch and Starfield, although in a variational framework that leads to matrix equations with sound mechanical meanings. Problems of general topology, such as in the case of unbounded and multiply-connected domains, may be modeled. The formulation, which is directly applicable to notches and generally curved, internal or external cracks, is specially suited for the description of the stress field in the vicinity of crack tips and is an easy means of evaluating stress intensity factors and of checking some basic concepts laid down by Rice in 1968. This dissertation focuses on the mathematical fundamentals of the formulation. Several validating numerical examples are presented.

The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to O(N_j) nonzero matrix entries without loss of accuracy of the underlying Galerkin scheme. Herein, O(N_j) denotes the number of unknowns. As we show in the present paper, the assembly of the compressed system matrix can be performed within optimal complexity. By numerical experiments we provide examples which corroborate the theory.

This paper is intended to present wavelet Galerkin schemes for the boundary element method. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators. This yields quasisparse system matrices which can be compressed to O(N_J) relevant matrix entries without compromising the accuracy of the underlying Galerkin scheme. Herein, O(N_J) denotes the number of unknowns. The assembly of the compressed system matrix can be performed in O(N_J) operations. Therefore, we arrive at an algorithm which solves boundary integral equations within optimal complexity. By numerical experiments we provide results which corroborate the theory.

No description available.

A parallel domain decomposition Laplace transform Boundary Element Method, BEM, algorithm for the solution of large-scale transient heat conduction problems will be developed. This is accomplished by building on previous work by the author and including several new additions (most note-worthy is the extension to 3-D) aimed at extending the scope and improving the efficiency of this technique for large-scale problems. A Laplace transform method is utilized to avoid time marching and a Proper Orthogonal Decomposition, POD, interpolation scheme is used to improve the efficiency of the numerical Laplace inversion process. A detailed analysis of the Stehfest Transform (numerical Laplace inversion) is performed to help optimize the procedure for heat transfer problems. A domain decomposition process is described in detail and is used to significantly reduce the size of any single problem for the BEM, which greatly reduces the storage and computational burden of the BEM. The procedure is readily implemented in parallel and renders the BEM applicable to large-scale transient conduction problems on even modest computational platforms. A major benefit of the Laplace space approach described herein, is that it readily allows adaptation and integration of traditional BEM codes, as the resulting governing equations are time independent. This work includes the adaptation of two such traditional BEM codes for steady-state heat conduction, in both two and three dimensions. Verification and validation example problems are presented which show the accuracy and efficiency of the techniques. Additionally, comparisons to commercial Finite Volume Method results are shown to further prove the effectiveness.

Structural systems have been evolving in terms of material properties and construction techniques, and their levels of protection against hazardous events have been the focus of different studies. For instance, the performance of the lateral force resisting systems has been investigated extensively to ensure that such systems would provide an adequate level of strength ductility capacity when subjected to seismic loading. However, with the increased occurrence of accidental and deliberate explosion incidents globally by more than three fold from 2004 to 2012, more studies have been focusing on the performance of such systems to blast loads and the different methods to quantify the inflicted damage. Although both blast and seismic design requires structures to sustain a level of ductility to withstand the displacement demands, the distributions of such demands from seismic ground excitation and blast loading throughout the structural system are completely different. Therefore, a ductile seismic force resisting system may not necessarily be sufficient to resist a blast wave. To address this concern, North American standards ASCE 59-11, CSA S850-12 provide response limits that define the different damage states that components may exhibit prior to collapse. Over the past ten years, a new configuration of reinforced masonry (RM) shear walls utilizing boundary elements (BEs) at the vertical edges of the wall has been investigated as an innovative configuration that enhances the wall’s in-plane performance. As such, they are included in the North American Masonry design standards, CSA S304-14 and TMS 402-16 as an alternative means to enhance the ductility of seismic force resisting systems. However, investigations regarding the out-of-plane performance of such walls are generally scarce in literature which hindered the blast design standards from providing unique response limits that can quantify the different damage states for RM walls with BEs. This dissertation has highlighted that some relevant knowledge gaps may lead to unconservative designs. Such gaps include (a) the RM wall with BEs out-of-plane behavior and damage sequence; and more specifically, (b) the BEs influence on the wall load-displacement response; as well as, (c) the applicability of using of the current response limits originally assigned for conventional RM walls to assess RM walls with BEs. Addressing these knowledge gaps is the main motivation behind this dissertation. In this respect, this dissertation reports an experimental program, that focuses on bridging the knowledge gap pertaining to the out-of-plane performance of seismically-detailed RM shear walls with BEs, which were not designed to withstand blast loads. Meanwhile, from the analytical perspective, plastic analyses were carried out taking into account the different mechanisms that the wall may undergo until peak resistance is achieved. This approach was adopted in order to quantify the resistance function of such walls and determine the contribution of the BEs and web to the overall wall resistance. In addition, the experimental results of the tested walls were used to validate a numerical finite element model developed to compare the resistance function of RM walls with and without BEs. Afterwards, the model was further refined to capture the walls’ performance under blast loads. The pressure impulse diagrams were generated to assess the capability of the current response limits in quantifying the different damage states for walls with different design parameters. Furthermore, new response limits were proposed to account for the out-of-plane ductility capacities of different wall components. Finally, a comparison between conventional rectangular walls and their counterparts with BEs using the proposed limits was conducted in the form of pressure-impulse diagram to highlight the major differences between both wall configurations. / Thesis / Doctor of Philosophy (PhD)

[pt] A modelagem matemática de microdispositivos, em que estrutura e microestrutura têm aproximadamente a mesma escala de magnitude, assim como de macroestruturas de natureza predominantemente granular ou cristalina, requer uma abordagem não-local de deformações e tensões. Há mais de cem anos os irmãos Cosserat já tinham desenvolvido uma teoria de grãos rígidos. No entanto, e sem detrimento de desenvolvimentos devidos a Toupin e outros pesquisadores, os trabalhos de Mindlin na década de 1960 podem ser considerados a base da chamada teoria gradiente de deformações, que se tornou recentemente objeto de um grande número de investigações analíticas e experimentais, motivadas pelo desenvolvimento de novos materiais estruturais e do crescente uso de dispositivos micro- e nanomecânicos na indústria. Mais recentemente, Aifantis e colaboradores conseguiram desenvolver uma teoria gradiente de deformações mais simplificada, com base somente em duas constantes elásticas adicionais, representativas de comprimentos característicos relacionados às energias de deformação superficial e volumétrica. Uma série de trabalhos recentes desenvolvidos por Beskos e colaboradores estendeu o campo de aplicações da proposta inicial de Aifantis e introduziu uma solução fundamental que de fato remonta aos trabalhos de Mindlin. A equipe de pesquisa de Beskos propôs as primeiras implementações 2D e 3D de elementos de contorno para análises de elasticidade gradiente tanto estáticas quanto no domínio da freqüência, inclusive para problemas da mecânica da fratura. Desde o tempo de Toupin e Mindlin procura-se estabelecer uma base variacional da teoria e uma formulação consistente das condições de contorno cinemáticas e de equilíbrio, o que parece ter tido êxito com os recentes trabalhos de Amanatidou e Aravas. Esta dissertação faz uma revisão da teoria gradiente da deformações e apresenta um estudo didático do problema mais simples que se possa conceber, que é o de uma barra sob diferentes tipos de ações axiais (Aifantis, Beskos). A solução fundamental para problemas 2D e 3D também é apresentada e estudada, tanto em termos de forças pontuais aplicadas, para uma implementação em termos de elementos de contorno, quanto de desenvolvimentos polinomiais (no caso estático), para implementação em termos de elementos finitos. Mostra-se que a teoria gradiente de deformação de Aifantis é adequada a uma formulação no contexto do potencial de Hellinger-Reissner, o que possibilita implementações híbridas de elementos finitos e de contorno. O presente trabalho de pesquisa objetiva o estudo do estado da arte no tema, com uma abordagem dos principais problemas de implementação computacional, inclusive em termos das integrais singulares que surgem. O desenvolvimento completo de programas de análise de elementos híbridos finitos e de contorno, para problemas estáticos e dinâmicos, está planejado para uma tese de doutorado em futuro próximo. / [en] The mathematical modeling of micro-devices in which structure and the microstructure are about the same scale of magnitude, as well as of macrostructure of markedly granular or crystal nature (microcomposites), demands a nonlocal approach for strains and stresses. More than one hundred years ago the Cosserat brothers had already developed a theory for rigid grains. However, and in no detriment due to Toupin and other researchers, Mindlin s work in the 1960s may be accounted the basis of the so-called strain gradient theory, which has recently become the subject of a large number of analytical and experimental investigations motivated by the development of news structural materials together with the increasing use of micro and nano-mechanical devices in the industry. More recently, Aifantis and coworkers managed to develop a simplified strain gradient theory based only on two additional elasticity constants that are representative of material lengths related to surface and volumetric strain energy. A series of very recent works done by Beskos and collaborators extended the field of applications of Aifantis propositions and introduced a fundamental solution that actually remounts to developments already laid down by Mindlin. Beskos workgroup may be regarded as the proponent of the first of the first boundary element 2D and 3D implementations on the subject for both statics and frequency-domain analyses, also including crack problems. Since Toupin and Mindlin`s time, investigations have been under development to establish the variational basis of the theory and to consistently formulate equilibrium and kinematic boundary conditions established by Amanatidou and Aravas. This dissertation makes a revision of the gradient strain elasticity theory and presents a didactic study of the simplest problem that can be conceived, i.e., a bar under different axial actions (Aifantis, Beskos). The fundamental solution for 2D and 3D problems is also presented and studied for an elastic medium submitted to a point force, for boundary methods developments, as well as submitted to polynomial stress fields (for static problems), as in the hybrid finite element method. It is shown that Aifantis strain gradient theory may be developed in the context of the Hellinger-Reissner potential, for the sake of hybrid finite and boundary element implementations. Goal of the present research work is as a detailed study of state art of the theme, which comprises an investigation of the singular integrals one must deal with in a computational implementation. The complete computational development for static and dynamic hybrid boundary/finite analyses is planned for a future doctoral thesis.

Esta tese apresenta o desenvolvimento de um método de elementos de contorno (BEM) no domínio do tempo baseado em fontes de Rankine para analise linear de comportamento no mar de sistemas oceânicos. O método e formulado por dois problemas de valor inicial de contorno definidos para os potenciais de velocidade e aceleração, sendo este ultimo utilizado para calcular de maneira acurada a derivada temporal do potencial de velocidades. Testes de verificação são realizados para a solução dos problemas de difração, radiação e de corpo livre para flutuar. Uma vez verificada, a ferramenta e aplicada em dois problemas multicorpos considerados no estado-da-arte em termos de modelagem hidrodinâmica utilizando BEM. O primeiro trata do problema envolvendo duas embarcações atracadas a contrabordo. Este é um caso no qual os códigos baseados na teoria de escoamento potencial são conhecidos por apresentarem dificuldades na determinação das soluções, tendendo a superestimar as elevações de onda no vão entre as embarcações e a apresentar problemas de convergência numérica associados a efeitos ressonantes de onda. O problema e tratado por meio do método de damping lid e a convergência das series temporais e investigada avaliando diferentes níveis de amortecimento. Os resultados são comparados com dados experimentais. O segundo problema se refere a analise de sistemas multicorpos com grandes deslocamentos relativos. Neste problema, ferramentas no domínio da frequência nao podem ser utilizadas, por considerarem apenas malhas fixas. Deste modo, o presente método e estendido para considerar um gerador de malhas de paineis e um algoritmo de interpolação de ordem alta no laco de tempo do código, possibilitando a mudança de posições relativas entre os corpos durante a simulação. Os resultados são comparados com dados de experimentos executados especificamente para fins de verificação do código, apresentando uma boa concordância. De acordo com o conhecimento do autor, esta e a primeira vez que certas questões relativas a modelagem numérica destes dois problemas multicorpos são relatadas na literatura especializada em hidrodinâmica computacional. / The development of a time domain boundary elements method (BEM) based on Rankine\'s sources for linear seakeeping analysis of offshore systems is here addressed. The method is formulated by means of two Initial Boundary Value Problems defined for the velocity and acceleration potentials, the latter being used to ensure an accurate calculation of the time derivatives of the velocity potential. Verification tests for solving the difraction, radiation and free floating problems are presented. Once verified, the code is applied for two complex multi-body problems considered to be in the state-of-the-art for hydrodynamic modelling using BEM. The first is the seakeeping problem of two ships arranged in side-by-side, a problem in which all potential flow codes are known to have a poor performance, tending to provide unrealistic high wave elevations in the gap between the vessels and to present numerical convergence problems associated to resonant effects. The problem is here addressed by means of a damping lid method and the convergence of the time series with different damping levels is investigated. Results are compared to data measured in an experimental campaign. The second problem refers to the analysis of multi-body systems composed of bodies undergoing large relative displacements. This is a case that cannot be properly analyzed by frequency domain codes, since they only consider fixed meshes. For this application, the present numerical method is extended to consider a panel mesh generator in the time loop of the code, enabling the change of body relative positions during the computations. Furthermore, a higher order interpolation algorithm designed to recover the solutions of a previous time-step was also implemented, enabling the calculations to progress with reasonable accuracy in time. The numerical results are compared to data of experimental tests designed and executed for verification of the code, and presented a very good agreement. To the author\'s knowledge, this is the first time that certain issues concerning the numerical modelling of these two complex multi-body problems are reported in the literature specialized in hydrodynamic computations.

[pt] Este trabalho visa à implementação de um programa de elementos de contorno para problemas com milhões de graus de liberdade. Isto é obtido com a implementação do Método Fast Multipole (FMM), que pode reduzir o número de operações, para a solução de um problema com N graus de liberdade, de O(N(2)) para O(NlogN) ou O(N). O uso de memória também é reduzido, por não haver o armazenamento de matrizes de grandes dimensões como no caso de outros métodos numéricos. A implementação proposta é baseada em um desenvolvimento consistente do convencional, Método de colocação dos elementos de contorno (BEM) – com conceitos provenientes do Hibrido BEM – para problemas de potencial e elasticidade de larga escala em 2D e 3D. A formulação é especialmente vantajosa para problemas de topologia complicada ou que requerem soluções fundamentais complicadas. A implementação apresentada, usa um esquema para expansões de soluções fundamentais genéricas em torno de níveis hierárquicos de polos campo e fonte, tornando o FMM diretamente aplicável para diferentes soluções fundamentais. A árvore hierárquica dos polos é construída a partir de um conceito topológico de superelementos dentro de superelementos. A formulação é inicialmente acessada e validada em termos de um problema de potencial 2D. Como resolvedores iterativos não são necessários neste estágio inicial de simulação numérica, podese acessar a eficiência relativa à implementação do FMM. / [en] This work aims to present an implementation of a boundary element solver for problems with millions of degrees of freedom. This is achieved through a Fast Multipole Method (FMM) implementation, which can lower the number of operations for solving a problem, with N degrees of freedom, from O(N(2)) to O(NlogN) or O(N). The memory usage is also very small, as there is no need to store large matrixes such as required by other numerical methods. The proposed implementations are based on a consistent development of the conventional, collocation boundary element method (BEM) - with concepts taken from the variationally-based hybrid BEM - for large-scale 2D and 3D problems of potential and elasticity. The formulation is especially advantageous for problems of complicated topology or requiring complicated fundamental solutions. The FMM implementation presented in this work uses a scheme for expansions of a generic fundamental solution about hierarchical levels of source and field poles. This makes the FMM directly applicable to different kinds of fundamental solutions. The hierarchical tree of poles is built upon a topological concept of superelements inside superelements. The formulation is initially assessed and validated in terms of a simple 2D potential problem. Since iterative solvers are not required in this first step of numerical simulations, an isolated efficiency assessment of the implemented fast multipole technique is possible.