Dissertations / Theses on the topic 'Dual Boundary Element Method (DBEM)'

2016 - 2017
To comply with fatigue life requirements, it is often necessary to carry out fracture mechanics assessments of structural components undergoing cyclic loadings. Fatigue growth analyses of cracks is one of the most important aspects of the structural integrity prediction for components (bars, wires, bolts, shafts, etc.) in presence of initial or accumulated in‐service damage. Stresses and strains due to mechanical as well as thermal, electromagnetical, etc., loading conditions are typical for the components of engineering structures. The problem of residual fatigue life prediction of such type of structural elements is complex, and a closed form solution is usually not available because the applied loads not rarely lead to mixed-mode conditions. Frequently, engineering structures are modelled by using the Finite Element Method (FEM) due to the availability of many well‐known commercial packages, a widespread use of the method and its well-known flexibility when dealing with complex structures. However, modelling crack-growth with FEM involves complex remeshing processes as the crack propagates, especially when mixed‐mode conditions occur. Hence, extended FEMs (XFEMs) and meshless methods have been widely and successfully applied to crack propagation analyses in the last years. These techniques allow a mesh‐independent crack representation, and remeshing is not even required to model the crack growth. The drawbacks of such mesh independency consist of high complexity of the finite elements, of material law formulation and solver algorithm. On the other hand, the Dual Boundary Element Method (DBEM) both simplifies the meshing processes and accurately characterizes the singular stress fields at the crack tips (linear assumption must be verified). Furthermore, it can be easily used in combination with FEM and, such a combination between DBEM and FEM, allows to simulate fracture problems leveraging on the high accuracy of DBEM when working on fracture, and on the versatility of FEM when working on complex structural problems... [edited by Author]
XVI n.s. (XXX ciclo)

Orientador: Leandro Palermo Junior
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo
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Resumo: A avaliação da influêcia de um modelo coesivo de fratura no comportamento estrutural e a simulação de propagação de fraturas pré-existentes, com a Mecâica da Fratura Elástica Linear (MFEL), em problemas bidimensionais, usando o Método dos Elementos de Contorno Dual (DBEM), é o principal objetivo deste estudo. Problemas elásticos lineares em meio contínuo podem ser resolvidos com a equação integral de contorno de deslocamentos. O Método dos Elementos de Contorno Dual pode ser utilizado para resolver os problemas de fratura, onde a equação integral de contorno de forças de superfície é implementada em conjunto com a equação integral de contorno de deslocamentos. Elementos contínuos, descontínuos e mistos podem ser usados no contorno. Diferentes estrat?ias de posicionamento dos pontos de colocação são discutidas neste trabalho, onde os fatores de intensidade de tensão são avaliados com ténica de extrapolação de deslocamentos em fraturas existentes dos tipos: borda, inclinada e em forma de 'v¿. Um modelo coesivo é utilizado para avaliação de comportamento estrutural de um corpo de prova com fratura de borda segundo diferentes estratégias desenvolvidas: uma análise coesiva geral e uma análise coesiva iterativa, as quais são comparadas com o comportamento não coesivo. A força normal coesiva relaciona-se com o valor da abertura de fratura na direção normal na lei constitutiva na Zona de Processos Coesivos (ZPC). A simulação de propagação de uma fratura de borda existente e sua implementa?o num?ica no DBEM, sob deslocamento imposto, é realizada utilizando o critério da mínima tensão circunferencial. Palavras-chave: Método dos Elementos de Contorno; Métodos dos Elementos de Contorno Dual; Mecânica da Fratura Elástica Linear; Modelos Coesivos; Propagação de Fraturas
Abstract: An evaluation of the effect of the cohesive fracture model on the structural behavior and the crack propagation in pre-existing cracks with the Linear Elastic Fracture Mechanics (LEFM), for two dimensional problems, using the Dual Boundary Element Method (DBEM), is the main purpose of the present study. Linear elastic problems in continuum media can be solved with the boundary integral equation for displacements. The Dual Boundary Element Method can be used to solve fracture problems, where the traction boundary integral equation is employed beyond the displacement boundary integral equation. Conformal and non-conformal interpolations can be employed on the boundary. Different strategies for positioning the collocation points are discussed in this work, where the stress intensity factors are evaluated with the displacement extrapolation method to an existing single edge crack, central slant crack and central kinked crack. A cohesive model is used to evaluate the structural behavior of the specimen with a single edge crack under different strategies: a general cohesive analysis and an iterative cohesive analysis; which are compared with the non-cohesive behavior. The normal cohesive force is dependent of the crack opening value in the normal direction in the constitutive law of the Cohesive Process Zone (CPZ). A crack propagation of an existing single edge crack and its numerical implementation in DBEM, under constrained displacement, is analyzed using the maximum hoop stress criterion. Key Words: Boundary Element Method; Dual Boundary Element Method; Linear Elastic Fracture Mechanic; Cohesive Models; Propagation of Cracks
Mestrado
Estruturas
Mestre em Engenharia Civil

A novel numerical method is presented for applications to general fracture mechanics problems in engineering. The coupled dual boundary element-scaled boundary finite element method (DBE-SBFEM) incorporates the numerical accuracy of the SBFEM and the geometric versatility of the DBEM. Background theory, detailed derivations and literature reviews accompany the extensions made to the methods constituents necessary for their coupling as part of the present work. The coupled DBE-SBFEM, its constituent components and their application to linear elastic fracture mechanics are critically assessed and presented with numerical examples to demonstrate both method convergence and improvements over previous work. Further, a proof of concept demonstrates an alternative formation of the DBEM that both negates the need for hyper-singular integration and lends itself to a wider variety of imposed boundary conditions. Conclusions to this work are drawn and further recommendations for research in this area are made.

This thesis deals with the implementation of the dual reciprocity boundary element method for problems governed by the Helmholtz equation. The main emphasis is the simulation of acoustic cavities, such as passenger compartments of aeroplanes and of cars. The boundary element method is a numerical analysis scheme which only needs a discretisation of the boundary of the domain of interest. The dual reciprocity boundary element method is a variation of this technique. The advantage of applying this special scheme for the Helmholtz equation is that in the final equation, unlike with the 'classical' boundary element method, the matrices are independent of the wave number. The Helmholtz problems in this research involve different geometries, properties and dimensions. A total of seven codes are employed, ranging from one- to three-dimensional problem solvers. The one-dimensional code is capable of solving related eigenfrequency and wave generation problems. Three of the two-dimensional codes are applied to eigenfrequency analysis: one is based on the constant element, one uses linear elements and one employs quadratic elements. A final two-dimensional code is used for wave propagation problems and uses the constant element. The three-dimensional codes are applied to eigenfrequency analysis only; one uses constant triangular elements, and another employs linear triangular elements. The kernel integrations in the two- and three-dimensional codes are all based on analytical solutions, given in this thesis. A new method of quadratic element kernel integration is described in detail. The three-dimensional triangular boundary element kernel integrations are based on improved analytical formulations.

In this thesis, the two-dimensional, transient, laminar flow of viscous and incompressible fluids is solved by using the dual reciprocity boundary element method (DRBEM). Natural convection and mixed convection flows are also solved with the addition of energy equation. Solutions of natural convection flow of nanofluids and micropolar fluids in enclosures are obtained for highly large values of Rayleigh number. The fundamental solution of Laplace equation is used for obtaining boundary element method (BEM) matrices whereas all the other terms in the differential equations governing the flows are considered as nonhomogeneity. This is the main advantage of DRBEM to tackle the nonlinearities in the equations with considerably small computational cost. All the convective terms are evaluated by using the DRBEM coordinate matrix which is already computed in the formulation of nonlinear terms. The resulting systems of initial value problems with respect to time are solved with forward and central differences using relaxation parameters, and the fourth-order Runge-Kutta method. The numerical stability analysis is developed for the flow problems considered with respect to the choice of the time step, relaxation parameters and problem constants. The stability analysis is made through an eigenvalue decomposition of the final coefficient matrix in the DRBEM discretized system. It is found that the implicit central difference time integration scheme with relaxation parameter value close to one, and quite large time steps gives numerically stable solutions for all flow problems solved in the thesis. One-and-two-sided lid-driven cavity flow, natural and mixed convection flows in cavities, natural convection flow of nanofluids and micropolar fluids in enclosures are solved with several geometric configurations. The solutions are visualized in terms of streamlines, vorticity, microrotation, pressure contours, isotherms and flow vectors to simulate the flow behaviour.

A problem encountered in the boundary element method is the difficulty caused by corners and/or discontinuous boundary conditions. An existing code using standard linear continuous elements is modified to overcome such problems using the multiple node method with an auxiliary boundary collocation approach. Another code is implemented applying the gradient approach as an alternative to handle such problems. Laplace problems posed on variety of domain shapes have been introduced to test the programs. For Poisson problems the programs have been developed using a transformation to a Laplace problem. This method cannot be applied to solve Poissontype equations. The dual reciprocity boundary element method (DRM) which is a generalised way to avoid domain integrals is introduced to solve such equations. The gradient approach to handle corner problems is co-opted in the program using DRM. The program is modified to solve non-linear problems using an iterative method. Newton's method is applied in the program to enhance the accuracy of the results and reduce the number of iterations. The program is further developed to solve coupled Poisson-type equations and such a formulation is considered for the biharmonic problems. A coupled pair of non-linear equations describing the ohmic heating problem is also investigated. Where appropriate results are compared with those from reference solutions or exact solutions. v

In this research project, the Boundary Element Method (BEM) is developed and formulated for the solution of two-dimensional convection-diffusion-reaction problems. A combined approach with the dual reciprocity boundary element method (DRBEM) has been applied to solve steady-state problems with variable velocity and transient problems with constant and variable velocity fields. Further, the radial integration boundary element method (RIBEM) is utilised to handle non-homogeneous problems with variable source term. For all cases, a boundary-only formulation is produced. Initially, the steady-state case with constant velocity is considered, by employing constant boundary elements and a fundamental solution of the adjoint equation. This fundamental solution leads to a singular integral equation. The conservation laws, usually applied to avoid this integration, do not hold when a chemical reaction is taking place. Then, the integrals are successfully computed using Telles' technique. The application of the BEM for this particular equation is discussed in detail in this work. Next, the steady-state problem for variable velocity fields is presented and investigated. The velocity field is divided into an average value plus a perturbation. The perturbation is taken to the right-hand-side of the equation generating a non-homogeneous term. This nonhomogeneous equation is treated by utilising the DRM approach resulting in a boundary-only equation. Then, an integral equation formulation for the transient problem with constant velocity is derived, based on the DRM approach utilising the fundamental solution of the steady-state case. Therefore, the convective terms will be encompassed by the fundamental solution and lie within the boundary integral after application of Greens's second identity, leaving on the right-hand-side of the equation a domain integral involving the time-derivative only. The proposed DRM method needs the time-derivative to be expanded as a series of functions that will allow the domain integral to be moved to the boundary. The expansion required by the DRM uses functions which take into account the geometry and physics of the problem, if velocity-dependent terms are used. After that, a novel DRBEM model for transient convection-diffusion-reaction problems with variable velocity field is investigated and validated. The fundamental solution for the corresponding steady-state problem is adopted in this formulation. The variable velocity is decomposed into an average which is included into the fundamental solution of the corresponding equation with constant coefficients, and a perturbation which is treated using the DRM approximation. The mathematical formulation permits the numerical solution to be represented in terms of boundary-only integrals. Finally, a new formulation for non-homogeneous convection-diffusion-reaction problems with variable source term is achieved using RIBEM. The RIM is adopted to convert the domain integrals into boundary-only integrals. The proposed technique shows very good solution behaviour and accuracy in all cases studied. The convergence of the methods has been examined by implementing different error norm indicators and increasing the number of boundary elements in all cases. Numerical test cases are presented throughout this research work. Their results are sufficiently encouraging to recommend the use of the techniques developed for solution of general convection-diffusion-reaction problems. All the simulated solutions for several examples showed very good agreement with available analytical solutions, with no numerical problems of oscillation and damping of sharp fronts.

The standard approach to modelling fluid flow through a porous medium was developed decades ago, when computational resources were insufficient to feasibly simulate the flow directly. In this thesis, the feasibility of such flow simulation with modern computing power is demonstrated via the development of three accurate and efficient dual-scale models of porous media flow. An important outcome of the research is that the new dual-scale modelling framework accurately and efficiently simulates flows with a range of Reynolds numbers through a variety of heterogeneous porous media.

Ce travail est consacre au developpement et a l'application d'une methode d'integrales de frontiere (dual reciprocity boundary element method) pour resoudre des problemes de propagation acoustique dans des milieux fermes ou d'etendue infinie dans lesquels siegent des gradients de temperature et/ou de vitesse. L'equation de propagation a resoudre est une equation d'helmholtz ou les inhomogeneites du milieu sont traitees comme des termes sources et placees dans le membre droite de l'equation. Ces termes sources sont interpoles par des fonctions d'approximation pour lesquelles des solutions particulieres regulieres sont obtenues. L'integrale volumique apparaissant dans la formulation integrale du probleme est ainsi convertie en une somme finie d'integrales de surface. Puisque la precision de la methode est fortement dependante de la nature des fonctions d'approximation, les proprietes de convergence et le comportement des fonctions utilisees dans cette etude sont examines numeriquement. L'application de la methode sur un cas aeroacoustique tres simple permet ensuite de dresser des criteres de convergence. Des applications a des milieux fermes (propagation d'onde plane dans un tube a ondes stationnaires en presence d'un gradient longitudinal de temperature) et ouverts (etudes de la refraction des ondes acoustiques a travers un panache thermique et dans un jet libre subsonique axisymmetrique) sont effectuees et comparees aux resultats experimentaux. Les bons accords observes montrent l'interet de la methode principalement en milieu ouvert. Finalement, nous etudions les effets du champ thermique d'une flamme turbulente sur la propagation acoustique. A partir d'un modele aerothermoacoustique, le champ acoustique source est decompose en une serie de monopoles equivalents. Les resultats numeriques mettent en evidence des phenomenes d'amplification acoustique a basses frequences.

No contexto do método dos elementos de contorno, este trabalho apresenta comparativamente três formulações em distintos aspectos. Visando a análise de sólidos bidimensionais no campo da mecânica da fratura, primeiramente é estudada a equação singular ou em deslocamentos. Em seguida, a formulação hiper-singular ou em forças de superfície é avaliada. Por último, a formulação dual, que emprega ambas equações é analisada. Para esta análise, elementos contínuos e descontínuos são empregados, equações numéricas e analíticas com ponto fonte dentro e fora do contorno são testadas, usando aproximação linear. A formulação é inicialmente empregada a problemas da mecânica da fratura elástica linear e em seguida extendida a problemas não-lineares, especialmente o modelo coesivo. Exemplos numéricos diversos averiguam as formulações, comparando com resultados analíticos ou disponíveis na literatura.
In this work three boundary elment formulations applied to fracture mechanics are studied. Aiming the analysis of two-dimensional solids with emphasis on the crack problem, the first considered method is the one based on using displacement equations only (singular formulation). The second scheme discussed in this work is a formulation based on the use of traction equations (hyper-singular formulation). Finally the dual boundary element method that uses singular and hyper-singular equations is considered. The numerical schemes have been implemented using continuous and discontinuous linear boundary and crack elements. The boundary and crack integral were all carried out by using analytical expressions, therefore increasing the accuracy of the algebraic system obtained for each one of the studied schemes. The developed numerical programs were applied initially to elastic fracture mechanics and then extended to analyze cohesive cracks. Several numerical examples were solved to verify the accuracy of each one of the studied models, comparing the results with the analytical solutions avaliable in the literature.

O presente trabalho desenvolve uma formulação do Método dos Elementos de Contorno para análise de problemas tridimensionais de fraturamento no regime transiente. Utilizam-se as soluções fundamentais da elastostática para obter a matriz de massa, empregando-se o Método da Reciprocidade Dual e a discretização do domínio por células tridimensionais. Para a integração no tempo são utilizados os algoritmos de Newmark e Houbolt. O fenômeno do fraturamento é abordado através da consideração de um campo de tensões iniciais, introduzindo-se o conceito de dipolos de tensão. Os tensores desenvolvidos que se relacionam aos dipolos, derivados das soluções fundamentais, são também apresentados. É utilizado o modelo de fratura coesiva. O contorno é discretizado utilizando-se elementos triangulares planos com aproximação linear, e elementos constantes para a superfície fictícia de fraturamento. São feitas várias aplicações cujos resultados obtidos confirmam a importância e a adequação da formulação apresentada para os problemas propostos.
This work presents a Boundary Element Method (BEM) formulation for analysis of three-dimensional fracture mechanics transient problems. Elastostatics fundamental solutions are considered in order to obtain the mass matrix, using both Dual Reciprocity Method and three-dimensional cell discretization. Newmark and Houbolt algorithms are employed to evaluate the time integrals. The fracture effects are captured by using dipoles of stresses, derived from an initial stress field. The tensors related to those dipoles, developed in the present work, are presented. The cohesive crack is the adopted model. Body boundary is discretized though linear flat triangular elements and the fracture surfaces are approximated by constant flat triangular elements. Some applications are processed to show the efficiency of presented BEM formulations.

Este texto trata do Método dos Elementos de Contorno Dual empregando a solução fundamental anisotrópica. As integrais impróprias que surgem nesta formulação são regularizadas pela técnica da subtração de singularidade. Aplica-se a transformação de coordenadas auto-adaptativa de Telles para a avaliação das integrais quase-singulares. Apresenta-se o programa computacional desenvolvido utilizando os paradigmas da programação orientada a objetos e processamento em paralelo. Foram analisados diversos problemas e os resultados obtidos comparados àqueles da solução analítica. Os resultados alcançados mostraram-se satisfatórios validando a formulação proposta.
This text deals with the Dual Boundary Element Formulation Method using the fundamental solution for anisotropic body. The improper integrals that arise in this formulation are regularized using the singularity subtraction technique. The self-adaptive coordinate transformation developed by Telles is used to evaluate the near-singular integrals. The computer program developed using the paradigms of object-oriented programming and parallel processing is presented. Several problems were analyzed and its results compared with those proposed by analytical solution. The results achieved were satisfactory therefore validating the proposed formulation.

Neste trabalho, apresenta-se a formulação do método dos elementos de contorno dual (MECD) aplicada a análise de problemas da Mecânica da Fratura Elástica Linear (MFEL). O objetivo da pesquisa consiste em avaliar o fator de intensidade de tensão (FIT) de sólidos bidimensionais fraturados, por meio de três técnicas distintas, quais são: a técnica da correlação dos deslocamentos, a técnica com base no estado de tensão na extremidade da fratura e a técnica da integral. As análises são realizadas utilizando o código computacional desenvolvido durante a pesquisa, que incorpora as formulações diretas em deslocamento e em força de superfície, do método dos elementos de contorno (MEC), com destaque para a utilização dos elementos de contorno curvos de ordem qualquer. No MECD as equações integrais singulares do tipo O(\'R POT.-1\') e O(\'R POT.-2\') são avaliadas satisfatoriamente com o Método da Subtração de Singularidade (MSS). Dessas integrais resultam termos analíticos, os quais são avaliados por meio do Valor Principal de Cauchy (VPC) e da Parte Finita de Hadamard (PFH). Compara-se o código desenvolvido com as soluções analíticas encontradas na literatura inclusive na análise de sólidos com fraturas predefinidas e para a avaliação do FIT, que produziram bons resultados.
This work presents the dual boundary element formulation applied to linear crack problem. The goal of this research is the evaluation of stress intensity factor for two-dimensional crack problem using three different techniques, which are: the technique of correlation of displacements, the technique based on the state of tension at the crack tip and J integral. The analysis is performed using the computational code developed during the research, which incorporates the direct formulations related to displacement and traction boundary element equation. A greater emphasis is given to the use of curved boundary element of any order. In the dual boundary element method the singular integral equations with singular others O(\'R POT.-1\') and O(\'R POT.-2\') are assessed satisfactorily with the application of the singularity subtraction method. The results of these singular integrals are evaluated by the Cauchy Principal Value and the Hadamard Finite Part. The code developed is compared with the analytical solutions found in the literature including the analysis of solids with fractures default and evaluation of stress intensity factor, which produced good results.

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In this work is implemented a numerical model to simulate computationally the distribution of pressures, velocities, temperatures and heat flows in two-dimensional stationary control volumes. The relation between temperatures and velocities is established by the advective-diffusive Equation, using the Dual Reciprocity Boundary Element Method formulation...
Neste trabalho é desenvolvido um modelo numérico para simular computacionalmente a distribuição de pressões, velocidades, temperaturas e fluxos de calor estacionários em volumes de controle bidimensionais. A relação do campo de temperaturas e velocidades é governada pela equação da Difusão-Advecção, resolvida através da formulação com Dupla Reciprocidade do Método dos Elementos de Contorno. Admite-se a lei de Darcy para associar pressão e velocidade, resultando num modelo matemático dado pela Equação de Laplace, no caso linear. Na análise não-linear insere-se a dependência entre do campo de velocidades e as temperaturas, resultando num campo matematicamente representado pela Equação de Poisson. Os resultados da solução desse problema são então implementados no modelo difusivo-advectivo, gerando temperaturas e fluxos de calor

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.

Thesis (M.S.)--University of Wisconsin--Madison, 1991.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 89-94).

碩士
國立臺灣海洋大學
河海工程學系
97
The scattering of large-scale ocean structure wave forces by an array of vertical circular cylinders. Au and Brebbia (1983) used the boundary element method (BEM) to calculate wave height and wave force. However, these studies are required as the basis for constant water depth. In practical engineering applications were restricted. Berkhoff (1972) used the linear wave theory to derive for mild-slope equation (MSE), However, MSE was non-homogeneous Helmholtz equation. If calculated by BEM, it will have to complex points of the field. So, Zhu (1993a) application of dual reciprocity boundary element method (DRBEM) calculated for wave diffraction and refraction problem. But MSE derived process ignored bottom curvature term and bottom slope squared term. Bed may not be able to respond to impact of the wave disturbance. Thus, Chamberlain and Porter (1995) derived for modified mild-slope equation (MMSE), which includes the bottom curvature term and bottom slope squared term. The studies showed that the MMSE was more accurate than the MSE calculated on the relative wave field in the wave height. Therefore, this paper application of DRBEM to calculated large-scale structures surrounding the wave force from MSE and MMSE equation.Study results of this paper, the calculated results were compared with those by Liu and Lin (2007), MacCamy and Fuchs (1954) and Linton and Evans (1990) are conducted, in order to show the applicability of this model. Good agreements were obtained. So the numerical calculations apply to regular waves pass through cylinders making wave force in constant depth、varying depth; Effect with the bottom curvature term and bottom slope squared term, MSE and MMSE wave field have more obvious changes in intermediate water depth. And the dimensionless wave force due to the terrain slope, incident wave direction and position of pile groups. When the incident wave direction and configuration of pile groups in the same direction, Wave force changes range ( value) also became smaller; the bed slope which more steep, MSE and MMSE dimensionless wave force had the greater difference.

碩士
國立臺灣海洋大學
機械與機電工程學系
100
Degenerate scales for elliptical, regular N-gon and half-disc domains are studied by using the boundary element method (BEM) and conformal mapping. For solving two-dimensional Laplace problems by using the BEM, the special size of geometry results in a rank-deficient matrix, and is different from the size effect problem in physics. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal mapping of unit logarithmic capacity in the complex variables. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM in this thesis. Analytical formula of an ellipse for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that a zero eigenvalue results in a degenerate scale. By using the conformal mapping technique, it is interesting to find that the absolute value of the logarithmic capacity equals to 1 in the case of degenerate scale. Based on the singular value decomposition, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left singular vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right singular vector in the dual integral formulation corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators. To deal with the problem of non-uniqueness solution, four regularization techniques, the hypersingular BIE, the constraint of boundary flux equilibrium, addition of a rigid body term in the fundamental solution and the Combined Helmholtz Exterior integral Equation Formulation (CHEEF) approach, are employed to promote the rank of influence matrices to be full rank. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the ordinary scale while the null field for the interior domain and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is also found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of the conformal mapping. Besides, no failure CHEEF point outside the domain can be found due to the nonzero field of the complementary domain in the case of degenerate scale. Only one trial in the BEM is required to determine the degenerate scale. Both analytical and numerical results agree well in the demonstrative examples.

We report a Hessian-implicit optimization method to quickly solve the charge optimization problem over protein molecules: given a ligand and its complex with a receptor, determine the ligand charge distribution that minimizes the electrostatic free energy of binding. The new optimization couples boundary element method (BEM) and primal-dual interior point method (PDIPM); initial results suggest that the method scales much better than the previous methods. The quadratic objective function is the electrostatic free energy of binding where the Hessian matrix serves as an operator that maps the charge to the potential. The unknowns are the charge values at the charge points, and they are limited by equality and inequality constraints that model physical considerations, i.e. conservation of charge. In the previous approaches, finite-difference method is used to model the Hessian matrix, which requires significant computational effort to remove grid-based inaccuracies. In the novel approach, BEM is used instead, with precorrected FFT (pFFT) acceleration to compute the potential induced by the charges. This part will be explained in detail by Shihhsien Kuo in another talk. Even though the Hessian matrix can be calculated an order faster than the previous approaches, still it is quite expensive to find it explicitly. Instead, the KKT condition is solved by a PDIPM, and a Krylov based iterative solver is used to find the Newton direction at each step. Hence, only Hessian times a vector is necessary, which can be evaluated quickly using pFFT. The new method with proper preconditioning solves a 500 variable problem nearly 10 times faster than the techniques that must find a Hessian matrix explicitly. Furthermore, the algorithm scales nicely due to the robustness in number of IPM iterations to the size of the problem. The significant reduction in cost allows the analysis of much larger molecular system than those could be solved in a reasonable time using the previous methods.
Singapore-MIT Alliance (SMA)

In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation.