Este trabalho apresenta um modelo numérico para a análise da propagação de fissuras em estruturas bidimensionais não-homogêneas. O comportamento mecânico é simulado a partir da formulação elastostática do Método dos Elementos de Contorno (MEC) aplicada a materiais isotrópicos. O MEC é uma eficiente e robusta técnica numérica para análises de propagação de fissuras. A não exigência de uma malha de domínio pelo método permite uma representação precisa da concentração de tensão nas pontas. Além disso, a redução da dimensionalidade proporcionada pelo MEC facilita o processo de remalhamento durante o crescimento das fissuras. A formulação dual do MEC é adotada, na qual as equações integrais singular e hipersingular são aplicadas. A modelagem de domínios não-homogêneos é realizada a partir da técnica de sub-regiões. A Mecânica da Fratura Elástico-Linear (MFEL) é aplicada para a análise da fratura em materiais frágeis. Os fatores de intensidade de tensão são determinados a partir da integral-J e a teoria da máxima tensão circunferencial é adotada para definir a direção de propagação das fissuras e o fator de intensidade de tensão equivalente. Problemas envolvendo fraturamento hidráulico também são investigados a partir da aplicação da MFEL. A integral-J é modificada para a consideração da pressão hidrostática atuante sobre as faces da fissura. Estruturas sujeitas à fadiga de alto ciclo também são avaliadas. A lei de Paris é utilizada para a estimativa da taxa de crescimento das fissuras. O último tipo de problema considerado é a fratura em materiais quase-frágeis. O modelo de fissura coesiva é empregado para a representação do comportamento não-linear físico próximo à ponta. O sistema de equações não-linear obtido é resolvido a partir de um algoritmo iterativo denominado operador constante. O estado de tensão na ponta, determinado por extrapolação, é utilizado para a verificação da estabilidade à propagação e o caminho de crescimento é definido a partir da formulação da MFEL. São observadas boas correspondências entre os resultados obtidos e as respostas encontradas na literatura, indicando a eficiência e a robustez do código computacional proposto. Melhorias do modelo numérico implementado também são discutidas. / This work presents a numerical approach for crack propagation modelling in non-homogeneous two-dimensional structures. The mechanical structural behaviour is simulated using the elastostatic formulation of the Boundary Element Method (BEM) applied to isotropic materials. The BEM is an efficient and robust numerical technique for crack propagation analyses. The non-requirement of a domain mesh enables the BEM for accurately quantifying the stresses concentration at the crack tip. Moreover, the mesh dimension reduction provided by the BEM makes the remeshing procedures during crack growth a less complex task. The dual BEM formulation is adopted, in which singular and hypersingular integral equations are applied. The non-homogeneous domains are modelled using the sub-region technique. The Linear Elastic Fracture Mechanics (LEFM) is applied to analyze the fracture in brittle materials. The stress intensity factors are evaluated through the J-integral and the maximum circumferential stress theory is adopted to define the crack propagation angle and the equivalent stress intensity factor. Problems involving hydraulic fracture (fracking) are also investigated applying the LEFM. A modified J-integral scheme is implemented to consider the hydrostatic pressure acting at the crack faces. Structures subjected to high-cycle fatigue are also addressed. The Paris law is used to estimate the crack growth rate. The last type of problem considered is the fracture in quasi-brittle materials. The cohesive crack model is used to represent the material nonlinear behaviour next to the crack tip. The nonlinear system of equations obtained is solved by an iterative algorithm named constant operator. The state of stress at the tip, obtained by extrapolation, is used to verify crack growth stability and the crack path is defined by the LEFM formulation. Good agreement is observed among the results achieved by the BEM model and the responses available in literature, showing the efficiency and robustness of the proposed numerical scheme. Further improvements of the BEM code are also discussed.

O método dos elementos de contorno (MEC) surgiu como uma poderosa alternativa ao método dos elementos finitos (MEF) principalmente em casos como problemas de concentração de tensões ou onde o domínio se estende para o infinito. Em virtude das potencialidades já identificadas do MEC para a solução de problemas da geotecnia, em especial para problemas de túneis, este trabalho tem como objetivo desenvolver um programa que seja capaz de analisar as variáveis envolvidas na construção de túneis profundos através de um modelo numérico bidimensional baseado no MEC, implementando técnicas numéricas tais como: subelementação, técnica da sub-região e modelagem de inclusão e enrijecedores. O modelo numérico bidimensional foi calibrado para considerar o efeito tridimensional do problema de túneis no que se refere ao avanço da frente de escavação, para dois casos a saber: i) túneis sem suporte e ii) túneis com suporte. Os resultados mostraram grande precisão quando comparados com os resultados analíticos mesmo utilizando um número pequeno de elementos, provocando uma redução significativa no tempo de processamento se comparado com outros métodos. A técnica da subelementação produziu uma suavização nos resultados dos pontos internos localizados muito próximos do contorno. A técnica da sub-região, bem como a modelagem de inclusão e enrijecedores apresentaram resultados consistentes dando ao programa uma versatilidade maior. Na calibração dos parâmetros para a consideração do efeito tridimensional na escavação de túneis sem suporte, foi proposto o método da redução do carregamento com a construção do perfil de deformação longitudinal do túnel - LDP (Longitudinal Deformation Profile). Para a escavação de túneis com suporte foram propostos quatro métodos de análise: i) método da redução do carregamento sobre o túnel, ii) método da redução de rigidez do suporte, iii) método do acréscimo do carregamento sobre o túnel e iv) método do alívio de carga sobre o suporte. Todos esses métodos foram desenvolvidos a partir do modelo Kappa (\'kapa\'), elaborado neste trabalho a partir dos resultados encontrados na simulação numérica tridimensional realizado nos programas BEFE e BEFE++, e comparado com o modelo de Schwartz e Einstein. Por fim, o método para a construção do gráfico de deslocamento radial para túneis circulares suportados, considerando o atraso na instalação do suporte, utilizando um método numérico ou resultado analítico do estado plano de deformação se mostra como uma alternativa simples para análise do efeito tridimensional contido no problema de túneis. / The boundary element method (BEM) has appeared as a powerful alternative to the finite element method (FEM) mainly in the cases where a good accuracy is required, as for problems with strain or stress concentration and problems with domain extending to infinite. The objective of this work is to develop a formulation and the corresponding computational code to analyse the variables in a design of deep tunnels, using a improved BEM two-dimensional numerical model, in which the following techniques were implemented: sub-elementation, sub-region technique, reinforcements introduced by modifying locally the domain rigidity. The two-dimensional model was calibrated to take into account the three-dimensional effects appearing around the tunnel face advance for two cases: i) tunnel without support and ii) tunnel with support. The results showed good accuracy when compared with analytical results even when obtained by using coarse discretizations and therefore requiring less computer time in comparison with other numerical procedures. The sub-elementation technique has smoothed the results for internal points near the boundary. The sub-region technique and the reinforcement inclusions lead to accurate making the computer code reliable. For the parameter calibration to take into account the three-dimensional effects applied to non lined tunnels the method of loading reductions was proposed obtaining a tunnel longitudinal deformation profile - LDP. For the excavation of lined tunnels four methods of analysis were proposed: i) load reduction model, ii) reduction support stiffness model, iii) additional load model, and iv) decrease of lining load model. All these methods were developed from the kappa (\'kapa\') model, developed in this work using three-dimensional results obtained by using the computational systems BEFE and BEFE++ and compared with the Schwartz-Einstein method. Finally the method used to build the radial displacement graphic for lined circular tunnels, taking into account the support insertion delay, using either a numerical method or plane strain analytical solutions, was developed.

Lined silencers of various configurations are used to attenuate the noise from building HVAC equipment, gas turbines, and other machinery. First-mode analytical solutions are presented for sound attenuation along rectangular lined ducts, parallel-baffle silencers, and circular lined ducts. The sound absorptive lining is treated using a bulk property model. The analytical solutions entail solving a nonlinear characteristic equation in the transverse direction after the rigid-wall boundary condition is applied. The solution is compared to the boundary element solution and a local impedance analytical solution for several test cases.

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.

Abstract Electrostatic interactions are very important in biomolecular systems. Electrostatic forces have received a great deal of attention due to their long-range nature and the trade-off between desolvation and interaction effects. It remains a challenging task to study and to predict the effects of electrostatic interactions in biomolecular systems. Computer simulation techniques that account for such interactions are an important tool for the study of biomolecular electrostatics. This study is largely concerned with the role of electrostatic interactions in biomolecular systems and with developing novel models to estimate the strength of such interactions. First, a novel formulation based upon continuum electrostatics to compute the electrostatic potential in and around two biomolecules in a solvent with ionic strength is presented. Many, if not all, current methods rely on the (non)linear Poisson-Boltzmann equation to include ionic strength. The present formulation, however, describes ionic strength through the inclusion of explicit ions, which considerably extends its applicability and validity range. The method relies on the boundary element method (BEM) and results in two very similar coupled integral equations valid on the dielectric boundaries of two molecules, respectively. This method can be employed to estimate the total electrostatic energy of two protein molecules at a given distance and orientation in an electrolyte solution with zero to moderately high ionic strength. Secondly, to be able to study interactions between biomolecules and membranes, an alternative model partly based upon the analytical continuum electrostatics (ACE) method has been also formulated. It is desirable to develop a method for calculating the total solvation free energy that includes both electrostatic and non-polar energies. The difference between this model and other continuum methods is that instead of determining the electrostatic potential, the total electrostatic energy of the system is calculated by integrating the energy density of the electrostatic field. This novel approach is employed for the calculation of the total solvation free energy of a system consisting of two solutes, one of which could be an infinite slab representing a membrane surface.

Décrit comme l'un des algorithmes les plus prometteurs du 20ème siècle, le formalisme multipolaire appliqué à la méthode des éléments de frontière, permet de nos jours de traiter de larges problèmes encore inconcevables il y a quelques années. La motivation de ce travail de thèse est d'évaluer la capacité, ainsi que les avantages concernant les ressources numériques, de ce formalisme pour apporter une solution de référence aux problèmes de propagation sonore tri-dimensionnels en environnement urbain, dans l'objectif d'améliorer les algorithmes plus rapides déjà existants. Nous présentons la théorie nécessaire à l'obtention de l'équation intégrale de frontière pour la résolution de problèmes non bornés. Nous discutons également de l'équation intégrale de frontière conventionnelle et hyper-singulière pour traiter les artefacts numériques liés aux fréquences fictives, lorsque l'on résout des problèmes extérieurs. Nous présentons par la suite un bref aperçu historique et technique du formalisme multipolaire rapide et des outils mathématiques requis pour représenter la solution élémentaire de l'équation de Helmholtz. Nous décrivons les principales étapes, d'un point de vue numérique, du calcul multipolaire. Un problème de propagation sonore dans un quartier, composé de 5 bâtiments, nous a permis de mettre en évidence des problèmes d'instabilités dans le calcul par récursion des matrices de translations, se traduisant par des discontinuités sur le champs de pression de surface et une non convergence du solveur. Ceci nous a conduits à considérer le travail très récent de Gumerov et Duraiswamy en lien avec un processus récursif stable pour le calcul des coefficients des matrices de rotation. Cette version améliorée a ensuite été testée avec succès sur un cas de multi diffraction jusqu'à une taille dimensionnelle de problème de 207 longueur d'ondes. Nous effectuons finalement une comparaison entre un algorithme d'élément de frontière, Micado3D, un algorithme multipolaire et un algorithme basé sur le tir de rayons, Icare, pour le calcul de niveaux de pression moyennés dans une cour ouverte et fermée. L'algorithme multipolaire permet de valider les résultats obtenus par tir de rayons dans la cour ouverte jusqu'à 300 Hz (i.e. 100 longueur d'ondes), tandis que concernant la cour fermée, zone très sensible par l'absence de contribution directes ou réfléchies, des études complémentaires sur le préconditionnement de la matrice semblent requises afin de s'assurer de la pertinence des résultats obtenus à l'aide de solveurs itératifs / Described as one of the best ten algorithms of the 20th century, the fast multipole formalism applied to the boundary element method allows to handle large problems which were inconceivable only a few years ago. Thus, the motivation of the present work is to assess the ability, as well as the benefits in term of computational resources provided by the application of this formalism to the boundary element method, for solving sound propagation problems and providing reference solutions, in three dimensional dense urban environments, in the aim of assessing or improving fast engineering tools. We first introduce the mathematical background required for the derivation of the boundary integral equation, for solving sound propagation problems in unbounded domains. We discuss the conventional and hyper-singular boundary integral equation to overcome the numerical artifact of fictitious eigen-frequencies, when solving exterior problems. We then make a brief historical and technical overview of the fast multipole principle and introduce the mathematical tools required to expand the elementary solution of the Helmholtz equation and describe the main steps, from a numerical viewpoint, of fast multipole calculations. A sound propagation problem in a city block made of 5 buildings allows us to highlight instabilities in the recursive computation of translation matrices, resulting in discontinuities of the surface pressure and a no convergence of the iterative solver. This observation leads us to consider the very recent work of Gumerov & Duraiswamy, related to a ``stable'' recursive computation of rotation matrices coefficients in the RCR decomposition. This new improved algorithm has been subsequently assessed successfully on a multi scattering problem up to a dimensionless domain size equal to 207 wavelengths. We finally performed comparisons between a BEM algorithm, extit{Micado3D}, the FMBEM algorithm and a ray tracing algorithm, Icare, for the calculation of averaged pressure levels in an opened and closed court yards. The fast multipole algorithm allowed to validate the results computed with Icare in the opened court yard up to 300 Hz corresponding, (i.e. 100 wavelengths), while in the closed court yard, a very sensitive area without direct or reflective fields, further investigations related to the preconditioning seem required to ensure reliable solutions provided by iterative solver based algorithms

Orientador: Euclides de Mesquita Neto / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-04T01:41:44Z (GMT). No. of bitstreams: 1 Menoni_JoseAntonio_D.pdf: 11918799 bytes, checksum: c09bbd80eae74f22092698eb851e1578 (MD5) Previous issue date: 2004 / Resumo: Este trabalho trata da formulação e da implementação da versão direta do Método dos Elementos de Contorno (MEC) para tratamento de problemas acústicos bidimensionais estacionários regidos pelo operador diferencial de Helrnholtz. São abordados tanto problemas internos, associados a domínios limitados, quanto problemas externos, associados a domínios ilimitados. A tese ainda aborda a solução de problemas diretos e inversos. A transformação da equação de Helrnholtz em Equação Integral de Contorno, bem como a síntese de sua Solução Fundamental é recuperada de forma detalhada no texto. Para o caso de problemas internos duas técnicas são estudadas para recuperação de grandezas modais de cavidades acústicas. A primeira é baseada na pesquisa direta das raÍzes do polinômio característico e a segunda é baseada na informação obtida a partir de Funções de Resposta em Freqüência sintetizadas pelo MEC. Os problemas da radiação e espalhamento acústico são formulados, implementados e validados. O trabalho apresenta ainda a solução de problemas inversos, no qual as variáveis acústicas em um contorno geométrico conhecido são determinadas a partir de medições em uma superficie fechada e que envolve o corpo radiante. Duas técnicas são utilizadas no processo inverso, a Decomposição em Valores Singulares e a técnica de regularização de Tikhonov. Discute-se a precisão e eficiência destas técnicas em função dos parâmetros que são variáveis presentes nestas técnicas / Abstract: The present Thesis reports a formulation and an implementation of the direct version of the Boundary Element Method (BEM) to model direct and indirect bidimensional stationary acoustic problems governed by the Helrnholz differential operator. Both internal and external problems, associated, respectively to bounded and unbounded domains, are treated in the analysis. The transformation of the Helmholtz differential equation into an equivalent Boundary Integral Equation (BIE) and the synthesis of its Fundamental Solution is recovered in detail. For internal problem two techniques are employed to obtain modal quantities of acoustic cavities. The fIrs is the direct search method of the characteristic polynomial roots. The second strategy is based on numerical Frequency Response Functions, synthesized by the BEM. Radiation and scatter problems are formulated, implemented and validated within the realm of the Boundary Element Method. The present work still addresses the solution of an inverse problem. The inverse problem consists of determining the acoustic variables on the boundary of a radiating or scattering body of known geometry, based on the acoustic fIelds measured over a c10sed surface which embodies the analized body. Two technique to solve the inversion problem are discussed. The fIrst is the Single Value Decomposition strategy and the other is the Tikhonov regularization strategy. The accuracy of this techniques are discussed as functions of the internal parameters which are intrinsic to those strategies / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

[pt] Esse trabalho propõe uma implementação numérica para otimização de forma em problemas bi-dimensionais de elasticidade. O objetivo principal é propor uma metodologia eficiente e robusta para solução de problemas de otimização de forma considerando a minimização de concentração de tensões. Na implementação proposta, a análise estrutural é realizada pelo Método dos Elementos de Contorno Simétrico de Galerkin (MECSG), evitando-se assim a dispendiosa etapa de geração da malha. A avaliação das tensões no contorno é obtida por meio de um método preciso, ideal para problemas com concentrações de tensões. Outro aspecto relevante na implementação é a adequada partição das equações do MECSG de forma a reduzir, consideravelmente, o esforço computacional associado à etapa da análise estrutural. O problema de otimização é resolvido utilizando-se um método de otimização moderno, conhecido como Programação Cônica de Segunda Orderm (PCSO). Especificamente, busca-se a reposta do problema de otimização não linear por meio da solução de uma sequência de subproblemas de PCSO. / [en] In this work a numerical implementation of shape optimization in two-dimensional linear elasticity problems is proposed. The main goal is to propose a robust and efficient methodology for the solution of shape optimization problems regarding the minimization of stress concentration effects. In the proposed implementation, the structural analysis is performed by the Symmetric Galerkin Boundary Element Method (SGBEM), thus disposing of the mesh generation burden. The boundary stress evaluation is carried out by an accurate approach which is ideally suited for problems with stress concentrations. Another relevant feature of the proposed implementation is a suitable partition of the SGBEM equations which aims at reducing the computational effort associated with the structural analysis stage. The solution for the optimization problem is obtained by means of a modern numerical optimization method, the so-called Second Order Conic Programming (SOCP). Specifically, the solution for the non-linear optimization is sought by solving a sequence of SOCP subproblems.

Le calcul des efforts hydrodynamiques de premier ordre sur un ou plusieurs corps perçant la surface libre est aujourd'hui bien maîtrisé, et plusieurs codes de calcul implémentant la méthode des singularités (dite BEM ou méthode d'élément frontière) ont été développés. Le cadre est la théorie linéarisée des écoulements potentiels à une surface libre. Dans ces codes BEM, les singularités utilisées ont la propriété intrinsèque de satisfaire à la fois l'équation de Laplace dans le domaine fluide ainsi que la condition linéarisée de surface libre. Ces singularités, dites fonctions de Green à surface libre, dans le domaine fréquentiel en profondeur infinie et sans vitesse d'avance constituent le point focal de cette thèse. Tout d'abord, les expressions mathématiques existantes pour la fonction de Green de surface libre sont examinées. Douze expressions différentes sont passées en revue et analysées. Plusieurs méthodes numériques existantes sont comparées par rapport à leur temps de calcul et leur précision. Ensuite, une série d'équations différentielles ordinaires (ODEs) pour les fonctions de Green de surface libre dans le domaine temporel et le domaine fréquentiel et leur gradient est établie. Ces ODEs peuvent être utilisées pour mieux comprendre les propriétés de la fonction de Green et peuvent constituer un moyen alternatif de calculer ces fonctions de Green et leurs dérivées. Cependant, il est difficile de résoudre numériquement ces ODEs à cause de l'existence d'une singularité à l'origine. Cette difficulté est éliminée en modifiant les ODEs par l'utilisation de nouvelles fonctions sans singularité. Les nouvelles ODEs sont ensuite écrites sous forme canonique en utilisant une nouvelle définition de la fonction vectorielle. La forme canonique peut être résolue avec les conditions initiales à l'origine puisque tous les termes impliqués sont finis. Une méthode d'expansion basée sur une série de fonctions logarithmiques et de polynômes ordinaires, très efficace pour les problèmes de basse fréquence, a également été développée pour obtenir des solutions analytiques. Enfin, la méthode basée sur les ODE pour calculer la fonction de Green est implémentée et un nouveau solveur BEM est obtenu. L'élimination des fréquences irrégulières est incluse. Le nouveau solveur est validé par comparaison des coefficients hydrodynamiques à des solutions analytiques pour une hémisphère, ainsi qu'à des résultats numériques obtenus avec un solveur commercial pour un chaland parallèlépipédique et le porte-conteneurs KCS. / The boundary element method (BEM) with constant panels is a common approach for wave-structure interaction problems. It is based on the linear potential-flow theory. It relies on the frequency-domain free-surface Green function, which is the focus of this thesis. First, the mathematical expressions and numerical methods for the frequency-domain free-surface Green function are investigated. Twelve different expressions are reviewed and analyzed. Several existing numerical methods are compared including their computational time and accuracies. Then, a series of ordinary differential equations (ODEs) for the time-domain and frequency-domain free-surface Green functions and their derivatives are derived. These ODEs can be used to better understand the properties of the Green function and can be an alternative way to calculate the Green functions and their derivatives. However, it is challenging to solve the ODEs for the frequency-domain Green function with initial conditions at the origin due to the singularity. This difficulty is removed by modifying the ODEs by using new functions free of singularity. The new ODEs are then transformed in their canonic form by using a novel definition of the vector functions. The canonic form can be solved with the initial conditions at the origin since all involved terms are finite. An expansion method based on series of logarithmic function together with ordinary polynomials which is very efficient for low frequency problems is also developed to obtain analytical solutions. Finally, the ODE-based method to calculate the Green function is implemented and an efficient BEM solver is obtained. The removal of irregular frequencies is included. The new solver is validated by comparison of hydrodynamic coefficients to analytical solutions for a heaving and surging hemisphere, and to numerical results obtained with a commercial solver for a box barge and the KCS container ship.

The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited also in three-dimensional case for implementation on a MIMD computer and that they are much more efficient than usual direct solution techniques.