In this thesis three different approaches for solving partial differential equa-tions with neural networks will be explored; namely Physics-Informed NeuralNetworks, Fourier Neural Operators and the Deep Ritz method. Physics-Informed Neural Networks and the Deep Ritz Method are unsupervised machine learning methods, while the Fourier Neural Operator is a supervised method. The Physics-Informed Neural Network is implemented on Burger’s equation,while the Fourier Neural Operator is implemented on Poisson’s equation and Darcy’s law and the Deep Ritz method is applied to several variational problems. The Physics-Informed Neural Network is also used for the inverse problem; given some data on a solution, the neural network is trained to determine what the underlying partial differential equation is whose solution is given by the data. Apart from this, importance sampling is also implemented to accelerate the training of physics-informed neural networks. The contributions of this thesis are to implement a slightly different form of importance sampling on the physics-informed neural network, to show that the Deep Ritz method can be used for a larger class of variational problems than the original publication suggests and to apply the Fourier Neural Operator on an application in geophyiscs involving Darcy’s law where the coefficient factor is given by exponentiated two-dimensional pink noise.

[pt] Algumas máquinas produzem solicitações dinâmicas que são transferidas às fundações por meio de movimentos vibratórios. É necessário analisar e estudar esses movimentos vibratórios durante o projeto da fundação para evitar danos aos equipamentos, à própria fundação e às estruturas vizinhas. Este tipo de analise também é muito importante para a segurança e saúde ocupacional de pessoas que venham a frequentar tais estruturas. Por esta razão as frequências naturais de vibração são importantes para o dimensionamento dinâmico de fundações, para se evitar efeitos de ressonância e amplificação de deslocamentos. Neste trabalho, estacas embutidas em fundações elásticas de Winkler foram estudadas. As vibrações longitudinais e transversais foram investigadas. As frequências naturais foram calculadas pelo método de Rayleigh-Ritz considerando estacas como elementos de barra embutidas em fundação elástica no caso da avaliação de vibrações longitudinais, e como vigas embutidas em base elástica no caso da avaliação das vibrações transversais. Na modelagem dos elementos de barra e viga utilizaram-se para aproximações dos deslocamentos funções convencionais enriquecidas com n funções adicionais. As constantes de mola de Winkler foram determinadas por provas de carga verticais e horizontais. Os resultados das vibrações longitudinais foram comparados com medições in situ com estacas reais. Os valores reais da vibração longitudinal das estacas foram obtidos pela aplicação da técnica Fast Fourier Transform nas acelerações resultantes de ensaios de carregamentos dinâmicos, que impõem uma vibração forçada no sistema estaca-solo. / [en] Some machines create dynamic efforts that are transferred to foundations by vibratory movements. Is necessary to analyze and study these movements during foundation design in order avoid damages in equipment, in foundation and in neighbor structures. This analysis is also important to safety and health of persons. For this reason natural vibration frequencies are important to dynamic design of foundations, to avoid resonance effects and displacements amplification. In this work, piles embedded in Winkler elastic foundations were studied. The longitudinal and transversal vibrations were investigated. The natural frequencies are calculated by Rayleigh-Ritz method modelling piles as bars embedded in elastic foundations in the case of longitudinal vibration evaluation, and beams embedded in elastic foundations in the case of transversal vibration evaluation. The bars and beams elements are modeled using conventional shape function enriched with n additional functions. The Winkler spring constants were evaluated by vertical and horizontal load tests. The longitudinal vibration results were compared with in situ measurements in real piles. The longitudinal vibration real values of piles were obtained by Fast Fourier Transform in acceleration data of dynamic load tests that impose a forced vibration in soil-pile system.

[pt] O presente trabalho mostra uma versão do método Rayleigh-Ritz com funções especializadas para a análise de placas circulares e anulares finas e espessas sujeitas a cargas fora do plano e em plano. As funções de aproximação para deslocamentos são polinómios em direção radial combinada com funções trigonométricas na direção circunferencial. Um recurso conveniente é o uso de funções nodais lineares, que permitem a fácil consideração de cargas nodais e condições de contorno (incluindo forças seguidoras), enriquecidos por polinômios de ordem superior, sem inclusão de nós adicionais. O modelo permite a variação da espessura e é aplicado em MAPLE18, possibilitando o cálculo de deslocamentos e tensões sob carregamento constante e de variação linear, as frequências de vibração, cargas de flambagem com alguns efeitos do nível de carga conservativa e não conservativa. Os exemplos mostram a eficácia desta abordagem na análise de tal estrutura e leva um novo enfoque a este problema clássico, que apresenta comparações interessantes e originais que descrevem o efeito de deformação de cisalhamento, no caso de vibrações o efeito das rotações inerciais e variação de espessura em placas circulares e anulares, incluindo deslocamentos, momentos e forças de cisalhamento, frequências de vibração, cargas de flambagem e uma análise de cargas seguidoras tangenciais não conservativas na estabilidade, utilizando o critério dinâmico é executada. / [en] The present work shows a version of the Rayleigh-Ritz method with specialized functions for the analysis of thin and thick circular and annular plates subjected to out-of-plane and in-plane loads. The approximation functions for displacements are polynomials in the radial direction combined with trigonometric functions in the circumferential direction. A convenient feature is the use of linear nodal functions, which allows for easy consideration of nodal loads and boundary conditions (including follower forces), enriched by higher order polynomials without inclusion of additional nodes. The model allows for thickness variation and was implemented in MAPLE18, enabling the calculation of displacements and stresses under constant and linearly varying load, frequencies of vibration, buckling loads with a few commands and the effect of the level of conservative and non-conservative on load the stability. The examples show the effectiveness of this approach in the analysis of such structures and bring new light to this classical problem, presenting interesting and novel comparisons illustrating the effect of shear deformation, in case of vibrations of the inertial rotations analysis and thickness variation in circular and annular plates, including displacements, moments and shear forces, vibration frequencies, buckling loads and a stability analysis of non-conservative tangential follower loads, using the dynamic criterion is performed.

[pt] Este estudo apresenta um método computacional para modelar campos eletromagnéticos em guias de onda retangulares curvados com seção transversal uniforme, usando o método variacional de Rayleigh-Ritz. Potenciais aplicações desta pesquisa em engenharia incluem o projeto de alimentadores para antenas, conversores de modais na faixa de micro-ondas, filtros, entre outros. Embora vários modelos tenham sido propostos para resolver este problema, as técnicas numéricas convencionais baseadas em elementos finitos, diferenças finitas e volumes finitos requerem altos custos computacionais. Para superar esses problemas, foi desenvolvida uma formulação variacional para resolver as equações de Maxwell em um sistema de coordenadas toroidal local, por meio de um novo funcional introduzido neste trabalho. O funcional foi adaptado para domínios uniformemente curvados com seção transversal arbitrária, e investigações analíticas foram conduzidas para confirmar suas características estacionárias. O formalismo Rayleigh-Ritz foi utilizado para converter o funcional em um problema equivalente de autovalores e autovetores, usando uma expansão em harmônicas retangulares de um guia de onda reto como funções de base para modelar um guia de onda retangular curvo. Um algoritmo numérico foi desenvolvido em Matlab para validar nosso modelo, e os resultados foram comparados com soluções perturbacionais e numéricas de referência, demonstrando alta precisão e menor custo computacional. / [en] This study presents a computational method for modeling electromagnetic fields in curved rectangular waveguides with uniform cross-section, using the variational Rayleigh-Ritz method. The potential applications of this research in engineering include the design of feeders for antennas, microwave mode converter devices, filters, among others. While various models have been proposed to solve this problem, conventional numerical techniques based on finite elements, finite differences, and finite volumes require high computational costs. To overcome these issues, a variational formulation for solving Maxwell s equations in a local toroidal coordinate system was developed via a novel functional introduced in this work. The functional was adapted to handle uniformly bend domains with arbitrary cross-section, and analytical investigations were conducted to confirm its stationary characteristics. The Rayleigh-Ritz formalism was employed to convert the functional into an equivalent problem of eigenvalues and eigenvectors using an expansion in terms of rectangular harmonics of a straight waveguide as basis functions for modeling a bend rectangular waveguide. A numerical algorithm was developed in Matlab to validate our model, and the results were compared against reference perturbational and numerical solutions, demonstrating high accuracy and lower computational costs.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.

[pt] Nos últimos anos, o considerável crescimento da demanda por outras fontes de energia justifica os investimentos realizados na construção de parques eólicos, pois a energia eólica tem se mostrado como a que traz o menor impacto ao meio ambiente. A energia eólica é gerada a partir da ação do vento, uma fonte de energia renovável, totalmente limpa e inesgotável. As torres eólicas são construídas em grandes parques com uma distância definida entre elas. Quanto mais altas são as torres eólicas, melhor é o desempenho delas, visto que temos o aproveitamento máximo do vento, sendo mais forte e menos turbulento em grandes alturas. Os materiais utilizados na fabricação das torres eólicas são basicamente o aço, o concreto armado e o concreto protendido. As torres na maioria dos casos, têm seção circular ou anular. No entanto, devemos atentar de maneira especial para a fundação da torre eólica, que deverá ser dimensionada e detalhada a fim de garantir a integridade da estrutura como um todo. Neste contexto, este trabalho tem como objetivo apresentar os cálculos necessários para projetos preliminares e a verificação do dimensionamento da estrutura de fundação de geradores eólicos. Anteprojetos e projetos estruturais foram revisados e avaliados para a obtenção de um maior conhecimento sobre o funcionamento de tais elementos estruturais. Uma planilha foi criada a partir do programa Excel, na qual se efetuam os cálculos da respectiva fundação por meio de soluções analíticas. Para efeitos de comprovação da veracidade da planilha, alguns exemplos serão apresentados e calculados a partir da planilha criada, assim como verificados e comparados com a mesma através da modelagem da fundação em questão em programa de análise em elementos finitos. Aspectos de estabilidade e dinâmica de placas circulares também são discutidos. / [en] In recent years, considerable growth in demand for alternative energy sources justifies the investments made in the construction of wind farms, since wind energy appears to bring the least impact on the environment. Wind energy is generated from wind action, a source of renewable energy, totally clean and inexhaustible. Wind towers are built in huge parks with a defined distance between them. The higher the wind towers are, the better their performance, since we have the maximum use of the wind which is stronger and less turbulent at great heights. The materials used in the manufacture of wind towers are basically steel, reinforced concrete and prestressed concrete. The towers are usuallyin most cases, with circular or annular section. However, we should pay attention in a special way on the foundation of the wind tower, which should be designed and detailed to ensure the integrity of the structure as a whole. In this context, this dissertation aims to present the calculations required for the preliminary design or design verification of the foundation of wind turbines. Preliminary design and design evaluations were reviewed and evaluated to obtain a better understanding of the functioning of such structural elements. A spreadsheet was created using the Excel program, which performs the calculations of its foundation through analytical solutions. For the purpose of proving the accuracy of the worksheet results, some examples are presented and calculated from the created spreadsheet, and checked and compared to results obtained from an independent finite element analysis program. Stability and dynamics aspects of circular plates are also discussed.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.