É bem sabido que métodos clássicos de elementos finitos e diferenças finitas para o problema de Helmholtz apresentam efeito de poluição, que pode deteriorar seriamente a qualidade da solução aproximada. Controlar o efeito de poluição é especialmente difícil quando são utilizadas malhas não uniformes. Para malhas uniformes com elementos quadrados são conhecidos métodos (p. e. o QSFEM, proposto por Babuska et al) que minimizam a poluição. Neste trabalho apresentamos inicialmente dois métodos de elementos finitos de Petrov-Galerkin com formulação relativamente simples, o RPPG e o QSPG, ambos com razoável robustez para certos tipos de distorções dos elementos. O QSPG apresenta ainda poluição mínima para elementos quadrados. Em seguida é formulado o QOFD, um método de diferenças finitas aplicável a malhas não estruturadas. O QOFD apresenta grande robustez em relação a distorções, mas requer trabalho extra para tratar problemas não homogêneos ou condições de contorno não essenciais. Finalmente é apresentado um novo método de elementos finitos de Petrov-Galerkin, o QOPG, que é formulado aplicando a mesma técnica usada para obter a estabilização do QOFD, obtendo assim a mesma robustez em relação a distorções da malha, com a vantagem de ser um método variacionalmente consistente. Resultados numéricos são apresentados ilustrando o comportamento de todos os métodos desenvolvidos em comparação com os métodos de Galerkin, GLS e QSFEM. / It is well known that classical finite elements or finite difference methods for Helmholtz problem present pollution effects that can severely deteriorate the quality of the approximate solution. To control pollution effects is especially difficult on non uniform meshes. For uniform meshes of square elements pollution effects can be minimized with the Quasi Stabilized Finite Element Method (QSFEM) proposed by Babusv ska el al, for example. In the present work we initially present two relatively simple Petrov-Galerkin finite element methods, referred here as RPPG (Reduced Pollution Petrov-Galerkin) and QSPG (Quasi Stabilized Petrov-Galerkin), with reasonable robustness to some type of mesh distortion. The QSPG also shows minimal pollution, identical to QSFEM, for uniform meshes with square elements. Next we formulate the QOFD (Quasi Stabilized Finite Difference) method, a finite difference method for unstructured meshes. The QOFD shows great robustness relative to element distortion, but requires extra work to consider non-essential boundary conditions and source terms. Finally we present a Quasi Optimal Petrov-Galerkin (QOPG) finite element method. To formulate the QOPG we use the same approach introduced for the QOFD, leading to the same accuracy and robustness on distorted meshes, but constructed based on consistent variational formulation. Numerical results are presented illustrating the behavior of all methods developed compared to Galerkin, GLS and QSFEM.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Apresenta-se uma abordagemnumérica para ummodelo que descreve a formação de padrões por sputtering iônico na superfície de ummaterial. Esse processo é responsável pela formação de padrões inesperadamente organizados, como ondulações, nanopontos e filas hexagonais de nanoburacos. Uma análise numérica de padrões preexistentes é proposta para investigar a dinâmica na superfície, baseada em ummodelo resumido em uma equação anisotrópica amortecida de Kuramoto-Sivashinsky, em uma superfície bidimensional com condições de contorno periódicas. Apesar de determinística, seu caráter altamente não-linear fornece uma rica gama de resultados, sendo possível descrever acuradamente diferentes padrões. Umesquema semi implícito de diferenças finitas com fatoração no tempo é aplicado na discretização da equação governante. Simulações foram realizadas com coeficientes realísticos relacionados aos parâmetros físicos (anisotropias, orientação do feixe, difusão). A estabilidade do esquema numérico foi analisada por testes de passo de tempo e espaçamento de malha, enquanto a verificação do mesmo foi realizada pelo Método das Soluções Manufaturadas. Ondulações e padrões hexagonais foram obtidos a partir de condições iniciais monomodais para determinados valores do coeficiente de amortecimento, enquanto caos espaço-temporal apareceu para valores inferiores. Os efeitos anisotrópicos na formação de padrões foramestudados, variando o ângulo de incidência. / A numerical approach is presented for amodel describing the pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots, and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model resumed in an anisotropic damped Kuramoto-Sivashinsky equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method ofManufactured Solutions has been used to verify the scheme. Ripples and hexagonal patterns were obtained from amonomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence.

Para uma análise mais exata do sistema estrutural, é necessário um estudo do comportamento interativo entre as diversas partes que o compõe, entre eles, destaca-se a interação entre os elementos de fundação e o maciço de solos. Neste trabalho foram desenvolvidas formulações numéricas para a análise da interação estaca-solo via acoplamento entre diferentes métodos numéricos: método dos elementos de contorno, método dos elementos finitos e método das diferenças finitas. As estacas podem estar submetidas a carregamentos horizontais, verticais e momentos aplicados em seu topo. Nestas formulações foram utilizadas, além das equações integrais de deslocamentos, as equações de suas derivadas, levando a um grau maior de singularidade, porém permitindo a adoção de aproximações mais refinadas para os deslocamentos e tensões ao longo da estaca. Todos os deslocamentos e suas derivadas referentes à estaca foram compatibilizados com os correspondentes do solo. Desenvolvidas as formulações, feito o devido acoplamento entre eles, foram analisados exemplos, que foram comparados com os resultados obtidos por outros pesquisadores, demonstrando sua validade. / For a more accurate analysis of the structural system, it is necessary to study the interactive behavior between the various parts that compose it, among them, there is the interaction between the foundation elements and massive soil. In this work, numerical formulations were developed for the analysis of pile-soil interaction by coupling between different numerical methods: the boundary element method, finite element method and finite difference method. Piles can be subjected to horizontal loads, vertical and moments applied on its top. In these formulations were used in addition to the displacement integral equations, the equations of their derivatives, leading to a higher degree of uniqueness but allowing the adoption of more sophisticated approaches to displacements and contact tractions along the pile. All displacements and their derivatives relating to the pile were matched with the corresponding soil. Developed formulations made due coupling between them were analyzed examples, which were compared with results obtained by other authors, demonstrating its validity.

Aplicações industriais envolvendo escoamentos multifásicos são inúmeras, sendo que, o aprimoramento de alguns desses processos pode resultar em um grande salto tecnológico com significativo impacto econômico. O estudo numérico dessas aplicações é imprescindível, pois fornece informações precisas e mais detalhadas do que a realização de testes experimentais. Um grande desafio é o estudo numérico de escoamentos viscoelásticos multifásicos envolvendo altas taxa de elasticidade, devido às instabilidades causadas por altas tensões elásticas, grandes deformações, e até mudanças topológicas na interface. Assim, a investigação numérica desse tipo de problema exige uma formulação precisa e robusta. No presente trabalho, um novo resolvedor de escoamentos bifásicos envolvendo fluidos complexos é apresentado, com particular interesse em escoamentos com altas taxas de elasticidade. A formulação proposta é baseada no método Volume-of-fluid (VOF) para representação da interface e no algoritmo Continuum Surface Force (CSF) para o balanço de forças na interface. A curvatura e advecção da interface são calculados via métodos geométricos para garantir a precisão dos resultados. Métodos de estabilização são utilizados quando números críticos de Weissenberg (Wi) são encontrados, devido ao famoso problema do alto número de Weissenberg (HWNP). O método da projeção, combinado com um método implícito para solução da equação da quantidade de movimento, são discretizados por um esquema de diferenças finitas em uma malha deslocada. Problemas de benchmarks foram resolvidos para acessar a precisão numérica da formulação em diferentes níveis de complexidade física, tal como representação e advecção da interface, influência das forças interfaciais, e características reológicas do fluido. A fim de demonstrar a capacidade do novo resolvedor, dois problemas bifásicos transientes, envolvendo fluidos viscoelásticos, foram resolvidos: o efeito de Weissenberg e o reômetro extensional (CaBER). O efeito de Weissenberg ou rod-climbing effect consiste em um bastão que gira dentro de um recipiente com fluido viscoelástico e, devido às forças elásticas, o fluido escala o bastão. Os resultados foram comparados com dados teóricos, numéricos e experimentais, encontrados na literatura para pequenas velocidades angulares. Além disso, resultados obtidos com altas velocidades angulares (alta elasticidade) são apresentados com o modelo Oldroyd-B, em que escaladas muito elevadas foram observadas. Valores críticos da velocidade angular foram identificados, e para valores acima foi observada a ocorrência de instabilidades elásticas, originadas pela combinação de tensões elásticas, curvatura interfacial, e escoamentos secundários. Até onde sabemos, numericamente, essas instabilidades nunca foram capturadas antes. O CaBER consiste no comportamento e colapso de um filamento de fluido viscoelástico, formado entre duas placas paralelas devido às forças capilares. Esse experimento envolve consideráveis dificuldades, dentre as quais podemos destacar a grande influência das forças capilares e a diferença de escalas de comprimento no escoamento. Em grande parte dos resultados encontrados na literatura, o CaBER é resolvido por modelos simplificados em uma dimensão. Resultados obtidos foram comparados com tais resultados da literatura e com soluções teóricas, apresentando admirável precisão. / Industrial applications involving multiphase flow are numerous. The improvement of some of these processes can result in a major technological leap with significant economic impact. The numerical study of these applications is essential because it provides accurate and more detailed information than conducting experiments. A challenge is the numerical study of high viscoelastic multiphase flows due to instabilities caused by the high elastic tension, large deformations and even topological changes in the interface. Thus the numerical investigation of this problem requires a robust formulation. In this study a new two-phase solver involving complex fluids is presented, with particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed formulation is based on the volume-of-fluid method (VOF) to interface representation and continuum surface force algorithm (CSF) for the balance of forces in the interface. The curvature and interface advection are calculated via geometric methods to ensure the accuracy of the results. Stabilization methods are used when critical Weissenberg numbers are found due to the famous high Weissenberg number problem (HWNP). The projection method combined with an implicit method for the solution of the momentum equation are discretized by a finite difference scheme in a staggered grid. Benchmark test problems are solved in order to access the numerical accuracy of different levels of physical complexities, such as the dynamic of the interface and the role of fluid rheology. In order to demonstrate the ability of the new resolver, two-phase transient problems involving viscoelastic fluids have been solved, theWeissenberg effect problem and the extensional rheometer (CaBER). The Weissenberg effect problem or rod-climbing effect consists of a rod that spins inside of a container with viscoelastic fluid and due to the elastic forces the fluid climbs the rod. The results were compared with numerical and experimental data from the literature for small angular velocities. Moreover results obtained for high angular velocities are presented using the Oldroyd-B model, which showed high climbing heights. Critical values of the angular speed have been identified. For values above a critical level were observed the occurrence of elastic instabilities caused by the combination of elastic tension, interfacial curvature and secondary flows. To our knowledge, numerically these instabilities were never captured before. The CaBER consists of the behavior and collapse of a viscoelastic fluid filament formed between two parallel plates due to capillary forces. This experiment involves considerable difficulties, among which we can highlight the great influence of the capillary forces and the difference of the length scales in the flow. In much of the results found in the literature, the CaBER is solved by simplified models. The results were compared with results reported in the literature and theoretical solutions, which showed remarkable accuracy.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-03-03T12:21:26Z No. of bitstreams: 1 alexandrebittencourtpigozzo.pdf: 866787 bytes, checksum: c6792b0cf7e2393b912ea7c1c8fdae2c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-03-06T20:06:30Z (GMT) No. of bitstreams: 1 alexandrebittencourtpigozzo.pdf: 866787 bytes, checksum: c6792b0cf7e2393b912ea7c1c8fdae2c (MD5) / Made available in DSpace on 2017-03-06T20:06:30Z (GMT). No. of bitstreams: 1 alexandrebittencourtpigozzo.pdf: 866787 bytes, checksum: c6792b0cf7e2393b912ea7c1c8fdae2c (MD5) Previous issue date: 2011-02-28 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O sistema imunológico humano (SIH) é composto por uma rede complexa de células, tecidos e órgãos especializados em defender o organismo contra doenças. Para atingir tal objetivo, o SIH identifica e extermina uma ampla gama de agentes patogênicos externos, como vírus e bactérias, além de células do próprio organismo que podem estar se comportando de forma anormal, e que poderiam dar origem a tumores, caso não fossem eliminadas. O SIH é ainda responsável pelo processo de eliminação de células mortas e renovação de algumas estruturas do organismo. A compreensão do SIH é, portanto, essencial. Entretanto a sua complexidade e a interação entre seus muitos componentes, nos mais diversos níveis, torna a tarefa extremamente complexa. Alguns de seus aspectos, no entanto, podem ser melhor compreendidos se modelados computacionalmente, permitindo a pesquisadores da área realizar um grande volume de experimentos e testar um grande número de hipóteses em um curto período de tempo. A longo prazo, pode-se vislumbrar um quadro onde todo o SIH poderá ser simulado, permitindo aos cientistas desenvolverem e testarem vacinas e medicamentos contra várias doenças, bem como contra a rejeição de órgãos e tecidos transplantados, diminuindo o uso de animais experimentais. Neste contexto, o presente trabalho visa implementar e simular computacionalmente um modelo matemático do SIH, sendo o objetivo principal reproduzir a dinâmica de uma resposta imune ao lipopolissacarídeo (LPS) em um pequena seção de um tecido. O modelo matemático é composto de um sistema de equações diferenciais parciais (EDPs) que incorpora a dinâmica de alguns tipos de células e moléculas do SIH durante uma resposta imune ao LPS no tecido. / The Human Immune System (HIS) consists of a complex network of cells, tissues and organs. The HIS plays an crucial role in defending the body against diseases. To achieve this goal, the immune system identifies and kills a wide range of external pathogens such as viruses and bacteria, and the body's own cells which are behaving abnormally. If these cells were not eliminated, they could give rise to tumors. The HIS is also responsible for removing dead cells and replacing some of the structures of the body. The understanding of the HIS is therefore essential. However, its complexity and the intense interaction among several components, in various levels, make this task extremely complex. Some of its aspects, however, may be better understood if a computational model is used, which allows researchers to test a large number of hypotheses in a short period of time. In the future we can envision a computer program that can simulate the entire HIS. This computer program will allow scientists to develop and test new drugs against various diseases, as well as to treat organ or tissue transplant rejection, without requiring animals experiments. In this scenario, our work aims to implement and simulate a mathematical model of the HIS. Its main objective is to reproduce the dynamics of a immune response to lipopoly-saccharides (LPS) in a microscopic section of a tissue. The mathematical model is composed of a system of partial differential equations (PDEs) that defines the dynamics of some tissues and molecules of the HIS during the immune response to the LPS.

Optimization problems involving uncertainties are common in a variety of engineering disciplines such as transportation systems, manufacturing, communication networks, healthcare and finance. The large number of input variables and the lack of a system model prohibit a precise analytical solution and a viable alternative is to employ simulation-based optimization. The idea here is to simulate a few times the stochastic system under consideration while updating the system parameters until a good enough solution is obtained. Formally, given only noise-corrupted measurements of an objective function, we wish to end a parameter which minimises the objective function. Iterative algorithms using statistical methods search the feasible region to improve upon the candidate parameter. Stochastic approximation algorithms are best suited; most studied and applied algorithms for funding solutions when the feasible region is a continuously valued set. One can use information on the gradient/Hessian of the objective to aid the search process. However, due to lack of knowledge of the noise distribution, one needs to estimate the gradient/Hessian from noisy samples of the cost function obtained from simulation. Simple gradient search schemes take much iteration to converge to a local minimum and are heavily dependent on the choice of step-sizes. Stochastic Newton methods, on the other hand, can counter the ill-conditioning of the objective function as they incorporate second-order information into the stochastic updates. Stochastic Newton methods are often more accurate than simple gradient search schemes. We propose enhancements to the Hessian estimation scheme used in two recently proposed stochastic Newton methods, based on the ideas of random directions stochastic approximation (2RDSA) [21] and simultaneous perturbation stochastic approximation (2SPSA-31) [6], respectively. The proposed scheme, inspired by [29], reduces the error in the Hessian estimate by (i) Incorporating a zero-mean feedback term; and (ii) optimizing the step-sizes used in the Hessian recursion. We prove that both 2RDSA and 2SPSA-3 with our Hessian improvement scheme converges asymptotically to the true Hessian. The key advantage with 2RDSA and 2SPSA-3 is that they require only 75% of the simulation cost per-iteration for 2SPSA with improved Hessian estimation (2SPSA-IH) [29]. Numerical experiments show that 2RDSA-IH outperforms both 2SPSA-IH and 2RDSA without the improved Hessian estimation scheme.

An easy-to-use spreadsheet version of a finite difference method for progressive landslide analysis has been developed. The finite difference method was originally developed by Dr. Stig Bernander, earlier adjunct professor at Luleå University of Technology and head of the Design Department of Skanska AB in Gothenburg, Sweden.. The so called Muskrat Falls Project consists in the ongoing construction of a hydroelectric power plant in Churchill River Valley, Labrador, Canada. The site hosting the project includes a land ridge which is supposed to be used as a natural dam and thus be submitted to important water pressures. Yet, previous landslides in the area have shown that a stability analysis is worth to be carried out in order to ensure the safety of the facility. Until now, investigations have only been carried out using the traditional limit equilibrium method and related elastic-plastic theory. For the sake of simplicity, this approach does not take into account deformations outside and inside the sliding body. However, because of the soil features in Churchill River Valley and particularly its ‘deformation softening’ behavior, there is increasing evidence that the conventional analysis is not relevant in this situation. Further, when analyzing the total stability of the ridge, only a horizontal failure surface has been used and not an inclined one, which is very optimistic and rather unrealistic.. In order to provide a more reliable study, a progressive failure analysis has been performed according to the finite difference method of Dr. Stig Bernander. The development of a spreadsheet adapted to this particular problem has allowed getting quickly and easily numerical results for several cases of study and assumptions. For assumed material properties and geometries of failure, the critical load-carrying capacity is below 1000 kN/m whereas a rise of the water level with 21 m will give an increased load of Nq  = 2420 kN/m. This is more than twice of the what the ridge may stand with the assumed properties. The investigation has led to the conclusion that the situation will be risky for many combinations of soil properties if the water level is raised as high as initially planned. The investigation also shows that more material tests are necessary and that stabilization work may be needed to eliminate the risk for a landslide.

Philosophiae Doctor - PhD / In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis. / South Africa

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

L'oxydation sélective conduit à un appauvrissement du substrat qui contribue aux phénomènes de corrosion sous contrainte. Afin de prédire cet appauvrissement en élément oxydable, un modèle basé sur la diffusion dans le substrat et l'équilibre des flux au niveau de l'interface alliage/oxyde a été développé. La cinétique d'oxydation du matériau est utilisée comme donnée d'entrée du modèle et permet de piloter le déplacement de l'interface alliage/oxyde.Le modèle d'appauvrissement développé a été validé dans la cas de l'oxydation de l'alliage binaire Pt-Ni à 850 °C, en reproduisant la solution analytique du modèle de Wagner. L'utilisation de la méthode des différences finies a permis de lever les hypothèses d'une concentration à l'interface alliage/oxyde et d'un coefficient d'interdiffusion constants.L'appauvrissement initial en élément oxydable d'un alliage binaire (passivation, pré-oxydation) a peu d'influence sur l'évolution de l'appauvrissement de l'alliage lors de son oxydation à haute température. En revanche, lorsque le coefficient d'interdiffusion de l'alliage est plus grand en subsurface qu'en volume de cet alliage (écrouissage), la concentration en élément oxydable au niveau de l'interface alliage/oxyde décroit au cours de l'oxydation. Les profils d'appauvrissement présentent alors un point d'inflexion dû à une diffusion plus rapide au voisinage de l'interface.Enfin, le modèle a été appliqué à l'oxydation en milieu aqueux à une température de 290 °C pour étudier la déchromisation de l'alliage 690 en milieu primaire des REP. Ceci a permis de conforter la cinétique d'oxydation logarithmique directe proposée dans la littérature pour cet alliage. / High temperature selective oxidation of alloys usually results in substrate depletion. This phenomenon can be at the origin of stress corrosion cracking (SCC) initiation. To predict the depletion of the oxidizable element, a model based on diffusion in the substrate and fluxes balance at the oxide/alloy interface has been developed. The oxidation kinetics is used as input data of the model in order to drive the oxide/alloy interface displacement when the oxidizable element is "consumed" from the alloy.The depletion model presented in this work has been validated in the case of the oxidation at 850 °C of the Pt-Ni binary alloy by reproducing the Wagner's analytical solution. The two assumptions of a fixed concentration at the oxide/alloy interface, and of a diffusion coefficient homogeneous in the substrate and constant during oxidation, have been relaxed using finite difference method.An initial depletion of a binary alloy in oxidizable element (passivation, pre-oxidation) has a low impact on the substrate depletion evolution during high temperature oxidation. However, when the interdiffusion coefficient in the alloy is greater in subsurface than in the bulk (work-hardening), the oxidizable element concentration at the oxide/alloy interface decreases during the oxidation. Depletion profiles show an inflexion point due to faster diffusion close to the interface.Finally, the model has been applied to oxidation in aquaous media at a temperature of 290 °C to study Cr-depletion of alloy 690 in PWR primary water. The direct logarithmic oxidation kinetics proposed in the literature has been validated