This dissertation develops a variety of structure-preserving algorithms for mechanical systems with external forcing and also extends those methods to systems that evolve on non-Euclidean manifolds. The dissertation is focused on numerical schemes derived from variational principles – schemes that are general enough to apply to a large class of engineering problems. A theoretical framework that encapsulates variational integration for mechanical systems with external forcing and time-dependence and which supports the extension of these methods to systems that evolve on non-Euclidean manifolds is developed. An adaptive time step, energy-preserving variational integrator is developed for mechanical systems with external forcing. It is shown that these methods track the change in energy more accurately than their fixed time step counterparts. This approach is also extended to rigid body systems evolving on Lie groups where the resulting algorithms preserve the geometry of the configuration space in addition to being symplectic as well as energy and momentum-preserving. The advantages of structure-preservation in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching. / Doctor of Philosophy / Accurate numerical simulation of dynamical systems over long time horizons is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. In many of these applications, the governing physical equations derive from a variational principle and their solutions exhibit physically meaningful invariants such as momentum, energy, or vorticity. Unfortunately, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. In this dissertation, tools from geometric mechanics and computational methods are used to develop numerical integrators that respect the qualitative features of the physical system. The research presented here focuses on numerical schemes derived from variational principles– schemes that are general enough to apply to a large class of engineering problems. Energy-preserving algorithms are developed for mechanical systems by exploiting the underlying geometric properties. Numerical performance comparisons demonstrate that these algorithms provide almost exact energy preservation and lead to more accurate prediction. The advantages of these methods in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.

Carregamentos transientes termo hidráulicos do tipo Steam Hammer são eventos comuns em sistemas de tubulações de vapor com grandes potenciais de catástrofes em plantas de geração de energia. Uma vez iniciado o evento, ondas de pressões são geradas com amplitudes, geralmente, de grande magnitude ocasionando altas pressões no sistema, ruídos, deformações, fadiga, com possibilidade de danos materiais e econômicos e em casos extremos fatalidades. Os procedimentos da indústria para análise deste tipo de sistema consistem realização de análises estáticas equivalentes ou análise de espectro de resposta com carregamentos caracterizados por meio de métodos analíticos baseados em hipóteses simplificadoras do fluido e fluxo. Neste trabalho é proposta a analise de sistema de tubulações por meio do método de integração numérica com superposição modal e carregamento caracterizado por método numérico com base no método das características. Comparações foram efetuadas entre os resultados obtidos pela metodologia proposta e os procedimentos da indústria, demonstrando que, dado ao alto grau de conservadorismo, os procedimentos da indústria acarretam em superdimensionamento de estruturas e tubulações ocasionando custos adicionais de projeto, sendo a otimização do projeto obtida aplicando-se a metodologia proposta no trabalho. / Steam Hammer is a common transient thermo hydraulic event in steam piping systems with potential to cause serious damages in power generation plants. It generates pressure waves, generally of great amplitude and magnitude, causing high pressures in the system, associated with noise, with possible material and economic damages and, in extreme case, even fatalities. Industry procedures for analysis of these events consists of performing equivalent static analysis, or response spectrum analysis, with transient loadings calculated by analytical methods based on assumptions of fluid and flow. This paper proposes a new methodology for the piping system analysis by the numerical integration method with modal superposition and transient loadings calculated by a numerical method based on the method of characteristics. Comparisons were made between the results obtained by the proposed methodology and the stablished industry procedures, confirming that, due to the high degree of conservatism, industry procedures can lead to additional cost to the design, with the optimization of the design being obtained by applying the methodology proposed in this paper.

This thesis presents an automatic animation system for jellyfish that is based on a physical simulation of the organism and its surrounding fluid. Our goal is to explore the unusual style of locomotion, namely jet propulsion, which is utilized by jellyfish. The organism achieves this propulsion by contracting its body, expelling water, and propelling itself forward. The organism then expands again to refill itself with water for a subsequent stroke. We endeavor to model the thrust achieved by the jellyfish, and also the evolution of the organism's geometric configuration. <p> We restrict our discussion of locomotion to fully grown adult jellyfish, and we restrict our study of locomotion to the resonant gait, which is the organism's most active mode of locomotion, and is characterized by a regular contraction rate that is near one of the creature's resonant frequencies. We also consider only species that are axially symmetric, and thus are able to reduce the dimensionality of our model. We can approximate the full 3D geometry of a jellyfish by simulating a 2D slice of the organism. This model reduction yields plausible results at a lower computational cost. From the 2D simulation, we extrapolate to a full 3D model. To prevent our extrapolated model from being artificially smooth, we give the final shape more variation by adding noise to the 3D geometry. This noise is inspired by empirical data of real jellyfish, and also by work with continuous noise functions from the graphics community. <p> Our 2D simulations are done numerically with ideas from the field of computational fluid dynamics. Specifically, we simulate the elastic volume of the jellyfish with a spring-mass system, and we simulate the surrounding fluid using the semi-Lagrangian method. To couple the particle-based elastic representation with the grid-based fluid representation, we use the immersed boundary method. We find this combination of methods to be a very efficient means of simulating the 2D slice with a minimal compromise in physical accuracy.

This thesis presents an automatic animation system for jellyfish that is based on a physical simulation of the organism and its surrounding fluid. Our goal is to explore the unusual style of locomotion, namely jet propulsion, which is utilized by jellyfish. The organism achieves this propulsion by contracting its body, expelling water, and propelling itself forward. The organism then expands again to refill itself with water for a subsequent stroke. We endeavor to model the thrust achieved by the jellyfish, and also the evolution of the organism's geometric configuration. <p> We restrict our discussion of locomotion to fully grown adult jellyfish, and we restrict our study of locomotion to the resonant gait, which is the organism's most active mode of locomotion, and is characterized by a regular contraction rate that is near one of the creature's resonant frequencies. We also consider only species that are axially symmetric, and thus are able to reduce the dimensionality of our model. We can approximate the full 3D geometry of a jellyfish by simulating a 2D slice of the organism. This model reduction yields plausible results at a lower computational cost. From the 2D simulation, we extrapolate to a full 3D model. To prevent our extrapolated model from being artificially smooth, we give the final shape more variation by adding noise to the 3D geometry. This noise is inspired by empirical data of real jellyfish, and also by work with continuous noise functions from the graphics community. <p> Our 2D simulations are done numerically with ideas from the field of computational fluid dynamics. Specifically, we simulate the elastic volume of the jellyfish with a spring-mass system, and we simulate the surrounding fluid using the semi-Lagrangian method. To couple the particle-based elastic representation with the grid-based fluid representation, we use the immersed boundary method. We find this combination of methods to be a very efficient means of simulating the 2D slice with a minimal compromise in physical accuracy.

Carregamentos transientes termo hidráulicos do tipo Steam Hammer são eventos comuns em sistemas de tubulações de vapor com grandes potenciais de catástrofes em plantas de geração de energia. Uma vez iniciado o evento, ondas de pressões são geradas com amplitudes, geralmente, de grande magnitude ocasionando altas pressões no sistema, ruídos, deformações, fadiga, com possibilidade de danos materiais e econômicos e em casos extremos fatalidades. Os procedimentos da indústria para análise deste tipo de sistema consistem realização de análises estáticas equivalentes ou análise de espectro de resposta com carregamentos caracterizados por meio de métodos analíticos baseados em hipóteses simplificadoras do fluido e fluxo. Neste trabalho é proposta a analise de sistema de tubulações por meio do método de integração numérica com superposição modal e carregamento caracterizado por método numérico com base no método das características. Comparações foram efetuadas entre os resultados obtidos pela metodologia proposta e os procedimentos da indústria, demonstrando que, dado ao alto grau de conservadorismo, os procedimentos da indústria acarretam em superdimensionamento de estruturas e tubulações ocasionando custos adicionais de projeto, sendo a otimização do projeto obtida aplicando-se a metodologia proposta no trabalho. / Steam Hammer is a common transient thermo hydraulic event in steam piping systems with potential to cause serious damages in power generation plants. It generates pressure waves, generally of great amplitude and magnitude, causing high pressures in the system, associated with noise, with possible material and economic damages and, in extreme case, even fatalities. Industry procedures for analysis of these events consists of performing equivalent static analysis, or response spectrum analysis, with transient loadings calculated by analytical methods based on assumptions of fluid and flow. This paper proposes a new methodology for the piping system analysis by the numerical integration method with modal superposition and transient loadings calculated by a numerical method based on the method of characteristics. Comparisons were made between the results obtained by the proposed methodology and the stablished industry procedures, confirming that, due to the high degree of conservatism, industry procedures can lead to additional cost to the design, with the optimization of the design being obtained by applying the methodology proposed in this paper.

Ce travail de thèse traite les domaines du rayonnement d'antennes basses fréquences et de la propagation du champ électromagnétique, en présence du sol et d'environnement réalistes. Des éléments qui diffèrent du cas canonique du sol plan homogène et infini sont considérés par le mot "réalistes". Les méthodes numériques classiques pour l'électromagnétisme, en particulier pour la modélisation du rayonnement d'antennes, sont plutôt adaptées à des fréquences au-delà des hautes fréquences et au rayonnement en espace libre. Deux axes de recherche ont été définis comme suit: les effets d'environnements situés en zone proche sur le rayonnement d'antennes filaires basses fréquences et l'étude de systèmes éloignés de la source par la création d'une méthode hybride. En ce qui concerne le rayonnement en zone proche, une méthode des différences finies dans le domaine temporel est utilisée. Celle-ci est lourde d'un point de vue des temps de calculs et de l'occupation de la mémoire, cependant elle est effective du fait du maillage volumique de la scène. Dans le cas de la propagation lointaine, la solution parfaite du rayonnement d'un dipôle infinitésimal électrique vertical au-dessus d'un sol plan homogène infini a été traitée par la création d'un algorithme adaptatif d'intégration numérique. Ce dernier permet d'obtenir les valeurs du champ électromagnétique dans tout l'espace avec une précision numérique contrôlée. Une technique originale d'hybridation a été réalisée entre l'algorithme d'intégration et la méthode des différences finies pour rendre compte efficacement des effets d'éléments lointains sur la propagation du champ. / This work is about the radiation of low frequency antennas and the propagation of the electromagnetic field, in the presence of soil and realistic environments. Elements that differ from the canonical case of homogeneous infinite and planar ground are considered by the word realistic. Conventional numerical methods for electromagnetism, in particular for modeling the radiation of antennas are rather suitable for frequencies beyond the high frequencies and in free space. Two research areas were defined as follows : the effects of environments located near the low frequency wire antennas and the study of systems far from the source through the creation of a hybrid method. Regarding the near-field radiation, a method of finite-difference in the time-domain is used. It needs a large computer memory size and long solution times, however it is effective because the entire scene is included in a volumetric mesh. In the case of field propagation, the radiation of a vertical electric current element above a homogeneous flat ground has been computed with the creation of an adaptive algorithm for numerical inte- gration. This allows to obtain the values of the electromagnetic field in the whole space with controlled numerical accuracy. An original technique of hybridization was performed between the integration algorithm and the method of finite differences to effectively observ the effects of distant elements on the field propagation.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The process modeling and simulation is a greatly important procedure for many chemical and biotechnological processes. The process simulation allows to predict elementary behavior of the state variables of the process, leading to many economical and process advantages, such as the avoidance of losses of time and materials for not knowing the process particularities, the safety guarantee, the product quality, and, mostly the process optimization, permitting to study and to reach the best conditions of a process, which shall yield in more products with quality produced with less effort and expenditures. In this work, the modeling and simulation of two biological processes enzymatic hydrolysis and solid state fermentation were performed in order to develop models and estimate parameters that enable an engineer to predict the process behavior and to make decisions about the process. The modeling procedure also involves the computing of differential equations, and algebraic-differential equations; in this manner, the engineer must be able to use different numerical integration methods. Mainly two parameters estimation procedures were used Particle-Swarm Optimization and Levenberg-Marquardt -, and two numerical integration methods were also resorted Runge-Kutta and Dormand-Prince. Experimental data from previous works were used to perform several tests in order to assure that the models were predicting correctly the state variables of the process and, in this manner, were reliable and useful. In the first paper, several mechanistic and empirical models are tested to fit the enzymatic hydrolysis experimental data; statistical tests were performed to verify which of those models would best describe the process, which was found to be and entirely empirical non-autonomous model. The following papers are about the modeling of the solid-state fermentation process. The model was found to be very accurate and adequate to be used for predictions, mainly for the bed temperature of the packed-bed bioreactor, since it could predict the temperature gradients along the time and height of the bed. Also, some numerical procedures such as parameters identifiability, to realize which were the most important parameters to be estimated, and model reparametrization, to reduce the total number of parameters to be estimated and avoid magnitude problems of the model, were successfully performed. This work has shown that the modeling and simulation of processes holds an enormous importance for industry, and different techniques can be applied with more or less effort and success. Further, hopefully, this work has contributed to the state of the art of modeling, in a general way, for biological processes. / A modelagem e a simulação de processos consistem em um recurso de grande importância para diversos processos químicos e biotecnológicos. A simulação de processos nos permite predizer o comportamento das variáveis de estado do processo, levando-nos a vantagens técnicas e econômicas, como, por exemplo, a prevenção de perdas de tempo e insumos por não conhecer particularidades do processo, a garantia de segurança, a qualidade do produto e, principalmente, a otimização do processo, permitindo estudar e alcançar as melhores condições para o referido processo, o que deve culminar em mais produto produzido com melhor qualidade e com menos esforços e custos. Neste trabalho, a modelagem e a simulação de dois processos biotecnológicos hidrólise enzimática e fermentação em estado sólido foram feitas com a validação dos modelos propostos com dados experimentais através da técnica de estimação de parâmetros para permitir ao engenheiro prever o comportamento do processo e tomar decisões. O procedimento de modelagem também envolve a avaliação de equações diferenciais e de equações algébricas, dessa forma, o engenheiro deve estar apto a usar diferentes métodos de integração numérica. Dois procedimentos principais de estimação de parâmetros foram utilizados Otimização por Enxame de Partículas e Levenberg-Marquardt -, e se lançou mão de dois métodos de integração numérica, Runge-Kutta e Dormand-Prince. Dados experimentais de trabalhos anteriores foram utilizados para realizar diversos testes para assegurar a precisão dos modelos em predizer as variáveis de estado do processo e, portanto, serem modelos confiáveis e úteis. No primeiro artigo apresentado, diversos modelos mecanicísticos e empíricos foram testados para se ajustarem aos dados experimentais da hidrólise enzimática; testes estatísticos foram realizados para verificar qual dos modelos melhor descreveria o processo, de forma que o melhor modelo se mostrou ser um modelo totalmente empírico não-autônomo. Os demais artigos tratam da modelagem de um processo de fermentação em estado sólido. Verificou-se que o modelo é bastante preciso e adequado para o uso em predições, principalmente para o perfil de temperatura no leito do biorreator, uma vez que o modelo prevê os gradientes de temperatura ao longo do tempo e da altura do leito. Também, procedimentos numéricos, tais como a análise de identifiabilidade dos parâmetros, para a percepção de quais são os parâmetros mais importantes para a estimação, e a reparametrização do modelo, para reduzir o número total de parâmetros a serem estimados e evitar problemas de magnitude no modelo, foram empregadas com sucesso. Este trabalho mostrou que a modelagem e a simulação de processos possuem enorme importância para a indústria, e diferentes técnicas podem ser aplicadas com maior ou menor esforço e sucesso. Além disso, espera-se que o trabalho tenha contribuído para o estado da arte em modelagem, de uma maneira geral, na área de bioprocessos.

The calculation of ellipsoidal areas in Geodesy is a not trivial task. By this motive, is common the calculation of plane areas, whose difficulty degree is enough small. For short areas, the simple abstraction of the real format of the Land is sufficient. For bigger areas, however, the Cartography uses the equivalent projections that, for being indirect forms, present a minor reliability in relation the direct numerical methods. More grave still it is the case of Brazil, where the areas send for certification together the Instituto Nacional de Colonização e Reforma Agrária (INCRA), second to the Brazilian Law 10267 of Cadastro Nacional de Imóveis Rurais (CNIR), are calculated as function of the coordinates of the projection system Universal Transverse Mercator (UTM). This projection system is conformal, that is, presenting distortions in relation to the ellipsoidal. The majority of the used commercial programs does not present more efficient forms of calculation. The present work of research is directed in this direction. We searchs a executable program that incorporates fundamental characteristics to the calculation of areas and to programming, generating a product that can present effective results consumer market. A methodology was chosen that considers solution for the calculation of any areas or polygon form that they desires to calculate. The methodology was codified in different forms and was generated an executable in C language. The results had been compared with the same integration carried through in environment of calculator HP42S and in the Spring system through the Albers Equivalent Conical Projection. The research shows that the proposal methodology possess ample potential to computational employment, having presented great efficiency in the calculation of ellipsoidais areas. / O cálculo das áreas elipsoidais em Geodésia não é uma tarefa trivial. Por esse motivo, é comum o cálculo de áreas planas, cujo grau de dificuldade é bastante pequeno. Para áreas pequenas, a simples abstração do formato real da Terra é suficiente. Para áreas maiores, entretanto, a Cartografia se vale das chamadas projeções equivalentes que, por serem formas indiretas, apresentam uma confiabilidade menor em relação a métodos numéricos diretos. Mais grave ainda é o caso do Brasil, em que as áreas enviadas para certificação junto ao Instituto Nacional de Colonização e Reforma Agrária (INCRA), segundo a Lei 10267/2001 do Cadastro Nacional de imóveis Rurais (CNIR), são calculadas em função das coordenadas do sistema de projeção Universal Transverse Mercator (UTM). Esse sistema de projeção é conforme, ou seja, apresenta distorções em relação à área elipsoidal. Isso se deve ao fato de que a maioria dos programas comerciais usados não apresentam formas de cálculo mais eficientes. O presente trabalho de pesquisa é dirigido nesse sentido. Busca-se chegar a um programa executável que incorpore características fundamentais ao cálculo de áreas e à programação, gerando um produto que possa apresentar resultados efetivos para o mercado consumidor. Escolheu-se uma metodologia que propõe solução para qualquer que seja a forma das poligonais que constituem o polígono cuja área deseja-se calcular. A metodologia foi codificada de diferentes formas e foi gerado um executável final em linguagem C. Os resultados foram comparados com a mesma integração realizada em ambiente de calculadora HP42S e no sistema Spring através da Projeção Cônica Equivalente de Albers. A pesquisa mostra que a metodologia proposta possui amplo potencial para emprego em meio computacional, tendo apresentado grande eficiência no cálculo de áreas elipsoidais.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.