CAPES / A resolução de equações polinomiais com coeficientes racionais consiste em parte significativa da história do desenvolvimento da álgebra. O problema era encontrar fórmulas que expressassem uma raiz por meio de operações aritméticas efetuadas sobre a equação original, isto é, determinar a resolubilidade por radicais da equação. O trabalho de vários matemáticos culminou, no século XVI, com a obtenção das fórmulas para a resolução de equações polinomiais de grau menor ou igual a 4. Três séculos depois, Niels Abel mostrou que não é possível obter uma fórmula para a equação geral de grau 5. Finalmente, Evariste Galois resolveu completamente o problema estudando o grupo de permutação das raízes e estabelecendo as condições exatas para a resolubilidade de uma equação polinomial. Neste trabalho apresentamos um breve histórico da obtenção de fórmulas para as raízes das equações de grau menor ou igual a 4 e a essência da matemática envolvida no estudo da resolubilidade por radiciais de equações polinomiais de grau maior ou igual a 5. / The solvability by radicals of polynomial equations with rational coefficients is an important part of the history of algebra. The problem was to express a root by means of basic arithmetic operations and radicals. Formulas to solve polynomial equations of degree lower than or equal to 4 were obtained in XVIth century. About three centuries later, Niels Abel showed that it is not possible to find a formula for the general equation of degree 5. Finally, Evariste Galois solved the problem by studying the permutations groups, establishing the exact conditions for the solvability of a polynomial equation. In this work we present a brief history of the classic formulas for the roots of equations with degree lower or equal to 4. Then we study solvability by radicals of polynomial equations of degree higher than or equal to 5.

Orientador: João Bosco Ribeiro do Val / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-14T11:15:46Z (GMT). No. of bitstreams: 1 Pinheiro_NathalieCarvalho_M.pdf: 2872807 bytes, checksum: 929137a3cc76534c1832b041f3175ff3 (MD5) Previous issue date: 2009 / Resumo: Os elevados custos de produção na extração de petróleo marítimo são reduzidos com um sistema inovador que a Petrobras está desenvolvendo, denominado VASPS (do inglês, Vertical Annular Separation and Pumping System), capaz de separar gás e líquido ainda no assoalho oceânico. Esta dissertação trata do ajuste do nível de l'iquido no reservatório do VASPS, via controle impulsional estocástico, que se distancia bastante de um regulador convencional. A vazão de entrada flutua e o controle consiste em alterar a velocidade da bomba de saída do tanque, mantendo o nível numa faixa de operação, na qual ele varia livremente sem prejuízo. Todavia, é necessário de um lado observar o risco de operar próximo aos extremos de nível e de outro minimizar o número de intervenções na velocidade da bomba para prolongar sua vida útil. Optou-se por modelar o sistema por meio de um processo de difusão e formular por controle impulsional. Para a sua solução, o controle impulsional é convertido em uma sequência de problemas de parada ótima iterados, resolvidos utilizando-se a Discretização do Valor Médio, MVS (do inglês, Mean Value Scheme). Esta dissertação introduz o uso do controle impulsional nesta aplicação além da técnica citada para resolver problemas de parada ótima. / Abstract: The high costs in offshore oil production are reduced with the use of an innovative system which has been developed by Petrobras, the Brazilian oil company. Called VASPS (Vertical Annular Separation and Pumping System), it consists of an undersea gas/liquid separator. This work presents a strategy for the liquid level adjustment in the VASPS tank, which is subject to uncertain liquid inflow. This is far from a strict control regulator problem, since the liquid level may drifts freely inside an operation range. Although, in one hand, it is necessary to account for a risky operation near the limits, on the other hand, acting freely and continuously in the controlled pump may drastically shorten the lifetime of the equipment. To prevent premature worn with halt in the oil production, the control input variations should be meager. We propose a stochastic impulse control for varying the outflow pump speed. This formulation is transformed in a sequence of iterated optimal stopping problems, which results in a sequence of variational inequalities. We employ the numerical method called Mean Value Scheme (MVS) to solve this type of problem. This monograph introduces impulse control to the resevoir level adjustment of VASPS, together with the application of the MVS to its solution. / Mestrado / Automação / Mestre em Engenharia Elétrica

Thesis (Ph.D.)--Joint Program in Physical Oceanography (Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences and the Woods Hole Oceanographic Institution), 2000. / Includes bibliographical references (p. 179-187). / A simple dynamic model is used with various observations to provide an approximate spectral description of low frequency oceanic variability. Such a spectrum has wide application in oceanography, including the optimal design of observational strategy for the deployment of floats, the study of Lagrangian statistics and the estimate of uncertainty for heat content and mass flux. Analytic formulas for the frequency and wavenumber spectra of any physical variable, and for the cross spectra between any two different variables for each vertical mode of the simple dynamic model are derived. No heat transport exists in the model. No momentum flux exists either if the energy distribution is isotropic. It is found that all model spectra are related to each other through the frequency and wavenumber spectrum of the stream-function for each mode, ... , where ... represent horizontal wavenumbers, w stands for frequency, n is vertical mode number, and ... are latitude and longitude, respectively. Given ... , any model spectrum can be estimated. In this study, an inverse problem is faced: ... is unknown; however, some observational spectra are available. I want to estimate ... if it exists. Estimated spectra of the low frequency variability are derived from various measurements: (i) The vertical structure of and kinetic energy and potential energy is inferred from current meter and temperature mooring measurements, respectively. (ii) Satellite altimetry measurements produce the geographic distributions of surface kinetic energy magnitude and the frequency and wavenumber spectra of sea surface height. (iii) XBT measurements yield the temperature wavenumber spectra and their depth dependence. (v) Current meter and temperature mooring measurements provide the frequency spectra of horizontal velocities and temperature. It is found that a simple form for ... does exist and an analytical formula for a geographically varying ... is constructed. Only the energy magnitude depends on location. The wavenumber spectral shape, frequency spectral shape and vertical mode structure are universal. This study shows that motion within the large-scale low-frequency spectral band is primarily governed by quasigeostrophic dynamics and all observations can be simplified as a certain function of ... The low frequency variability is a broad-band process and Rossby waves are particular parts of it. Although they are an incomplete description of oceanic variability in the North Pacific, real oceanic motions with energy levels varying from about 10-40% of the total in each frequency band are indistinguishable from the simplest theoretical Rossby wave description. At higher latitudes, as the linear waves slow, they disappear altogether. Non-equatorial latitudes display some energy with frequencies too high for consistency with linear theory; this energy produces a positive bias if a lumped average westward phase speed is computed for all the motions present. / by Xiaoyun Zang. / Ph.D.

<p align="justify">Une approche multidimensionelle pour la résolution numérique des équations d'Euler et de Navier-Stokes sur maillages non-structurés est proposée. Dans une première partie, un exposé complet des schémas de distribution, dits de "fluctuation-splitting" ,est décrit, comprenant une étude comparative des schémas décentrés, positifs et de 2ème ordre, pour résoudre l'équation de convection à coefficients constants, ainsi qu'une étude théorique et numérique de la précision des schémas sur maillages réguliers et distordus. L'extension à des lois de conservation non-linéaires est aussi abordée, et une attention particulière est portée au problème de la linéarisation conservative. Dans une deuxième partie, diverses discrétisations des termes visqueux pour l'équation de convection-diffusion sont développées, avec pour but de déterminer l'approche qui offre le meilleur compromis entre précision et coût. L'extension de la méthode aux systèmes des lois de conservation, et en particulier à celui des équations d'Euler de la dynamique des gaz, représente le noyau principal de la thèse, et est abordée dans la troisième partie. Contrairement aux schémas de distribution classiques, qui reposent sur une extension formelle du cas scalaire, l'approche développée ici repose sur une décomposition du résidu par élément en équations scalaires, modélisant le transport de variables caracteristiques. La difficulté vient du fait que les équations d'Euler instationnaires ne se diagonalisent pas, et admettent une infinité de solutions élémentaires (ondes simples) se propageant dans toutes les directions d'espace. En régime stationnaire, en revanche, les équations se diagonalisent complètement dans le cas des écoulements supersoniques, et partiellement dans le cas des écoulements subsoniques. Ainsi, les équations sous forme conservative peuvent être remplacées par un système équivalent comprenant deux équations totalement découplées, exprimant l'invariance de l'entropie et de l'enthalpie totale le long des lignes de courant, et deux autres équations, modélisant les effets purement acoustiques. En régime supersonique, celles-ci se découplent aussi, et expriment la convection le long des lignes de Mach d'invariants de Riemann généralisés. La discrétisation de ces équations par des schémas scalaires décentrés permet de simuler des écoulements continus et discontinus avec une grande précision et sans oscillations. Finalement, dans une dernière partie, l'extension aux équations de Navier-Stokes est abordée, et la discrétisation des termes visqueux par une approche éléments finis est proposée. Les résultats numériques confirment la précision et la robustesse de la méthode.</p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

Orientador: Sônia Maria Gomes, Jorge Stolfi / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T12:39:30Z (GMT). No. of bitstreams: 1 Castro_DouglasAzevedo_D.pdf: 8872633 bytes, checksum: a17b2761789c6a831631ac143fdf5ca7 (MD5) Previous issue date: 2011 / Resumo: O objetivo desta tese é desenvolver algoritmos baseados em malhas e bases funcionais inovadoras usando técnicas de multiescala para aproximação de funções e resolução de problemas de equações diferenciais. Para certas classes de problemas, é possível incrementar a eficiência dos algoritmos de multiescala usando bases adaptativas, associadas a malhas construídas de forma a se ajustarem com o fenômeno a ser modelado. Nesta abordagem, em cada nível da hierarquia, os detalhes entre a aproximação desse nível e a aproximação definida no próximo nível menos refinado pode ser usada como indicador de regiões que necessitam de mais ou menos refinamento. Desta forma, em regiões onde a solução é suave, basta utilizar os elementos dos níveis menos refinados da hierarquia, enquanto que o maior refinamento é feito apenas onde a solução tiver variações bruscas. Consideramos dois tipos de formulações para representações multiescala, dependendo das bases adotadas: splines diádicos e wavelets. A primeira abordagem considera espaços aproximantes por funções splines sobre uma hierarquia de malhas cuja resolução depende do nível. A outra abordagem considera ferramentas da analise wavelet para representações em multirresolução de médias celulares. O enfoque está no desenvolvimento de algoritmos baseados em dados amostrais d-dimensionais em malhas diádicas que são armazenados em uma estrutura de árvore binária. A adaptatividade ocorre quando o refinamento é interrompido em algumas regiões do domínio, onde os detalhes entre dois níveis consecutivos são suficientemente pequenos. Um importante aspecto deste tipo de representação é que a mesma estrutura de dados é usada em qualquer dimensão, além de facilitar o acesso aos dados nela armezenados. Utilizamos as técnicas desenvolvidas na construção de um método adaptativo de volumes finitos em malhas diádicas para a solução de problemas diferenciais. Analisamos o desempenho do método adaptativo em termos da compressão de memória e tempo de CPU em comparação com os resultados do esquema de referência em malha uniforme no nível mais refinado. Neste sentido, comprovamos a eficiência do método adaptativo, que foi avaliada levando-se em consideração os efeitos da escolha de diferentes tipos de fluxo numérico e dos parâmetros de truncamento / Abstract: The goal of this thesis is to develop algorithms based on innovative meshes and functional bases using multiscale techniques for function approximation and solution of differential equation problems. For certain classes of problems, one can increase the efficiency of multiscale algorithms using hierarchical adaptive bases, associated to meshes whose resolution varies according to the local features of the phenomenon to be modeled. In this approach, at each level of the hierarchy the details-differences between the approximation for that level and that of the next coarser level-can be used as indicators of regions that need more or less refinement. In this way, in regions where the solution is smooth, it suffices to use elements of the less refined levels of the hierarchy, while the maximum refinement is used only where the solution has sharp variations. We consider two classes of formulations for multiscale representations, depending on the bases used: dyadic splines and wavelets. The first approach uses approximation spaces consisting of spline functions defined over a mesh hierarchy whose resolution depends on the level. The other approach uses tools from wavelet analysis for multiresolu-tion representations of cell averages. The focus is on the development of algorithms based on sampled d-dimensional data on dyadic meshes which are stored in a binary tree structures. The adaptivity happens when the refinement is interrupted in certain regions of the domain, where the details between two consecutive levels are sufficiently small. This representation greatly simplifies the access to the data and it can be used in any dimension. We use these techniques to build an adaptive finite volume method on dyadic grids for the solution of differential problems. We analyze the performance of the method in terms of memory compression and CPU time, comparing it with the reference scheme (which uses a uniform mesh at the maximum refinement level). In these tests, we confirmed the efficiency of the adaptive method for various numeric flow formulas and various choices of the thresholding parameters / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: Eduardo Cardoso de Abreu / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T06:42:03Z (GMT). No. of bitstreams: 1 Vieira_Jardel_M.pdf: 2696940 bytes, checksum: 9517d68c44824f91bd411caf141d2ef1 (MD5) Previous issue date: 2015 / Resumo: O foco desta dissertação de mestrado consiste em um estudo computacional de equações pseudo-parabólicas em mecânica dos fluidos e fenômenos de transporte de fluidos em meios porosos. Serão considerados problemas de valor de contorno e inicial associados a duas classes de modelos de equações de evolução pseudo-parabólicas: um modelo de advecção-difusão com termo pseudo-parabólico que exibe um certo caráter dispersivo e um outro modelo pseudo-parabólico "puro", i.e., sem a presença do termo de advecção. O primeiro modelo se relaciona com a modelagem física do fluxo de duas fases incompressíveis em dinâmica de fluidos em meios porosos, onde são considerados modelos de pressão capilar dinâmica, ou seja, em que os efeitos dinâmicos são também incluídos na diferença de pressão entre as fases fluidas. Uma discussão sobre a relevância física em aplicações e da importância matemática do sistema governante de equações para pressão capilar dinâmica em fenômenos de transporte de fluidos em meios porosos é também feita de modo a indicar algum suporte à escolha dos métodos estudados para aproximação numérica dos modelos consideradores. Além disso, um conjunto de experimentos numéricos é apresentado e discutido para avaliar a qualidade das soluções obtidas do estudo proposto, bem como para justificar variações dos métodos numéricos estudados. Especificamente, para o modelo pseudo-parabólico puro, os resultados são comparados com soluções analíticas para o caso linear. Para o modelo pseudo-parabólico com o termo de advecção, é avaliado se os resultados dos métodos numéricos empregados concordam qualitativamente com resultados da literatura / Abstract: The focus of this work consists of a computational study of pseudo-parabolic equations in fluid mechanics and transport phenomena in porous media. For concreteness, we consider initial-boundary value problems related to two classes of systems of evolution pseudo-parabolic equations: a advection-diffusion model, which in turn the pseudo-parabolic term exhibits a certain dispersive character, and a second of "purely" pseudo-parabolic nature, i.e., without the presence of advection term. The first model relates to the modeling of incompressible two-phase flow in porous media, which in turn takes into account the nonlinear dynamic capillary pressure effects, where the dynamic effects are also included into the pressure difference between the fluid phases. Further, a discussion of the physical and mathematical relevance of the governing system of equations for dynamic capillary pressure in porous media fluid transport phenomena is also made in order to drive the choice of the numerical approximations for the differential models under investigation. Moreover, a set of numerical experiments are presented and discussed to address the quality of the obtained solutions proposed study, as well as to justify variations of the numerical methods studied. Specifically, to the purely pseudo-parabolic model, the results are compared along with analytical solutions with respect to a linear case. On the other hand, to the nonlinear pseudo-parabolic model with advection term, it is performed numerical experiments in order to account the correct qualitative behavior of the computed solutions against the available results discussed in the recent literature / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

No presente trabalho estudamos e analisamos o método do Tubo de Trajetórias, um algoritmo conservativo, explícito, simples, fisicamente intuitivo, semi-Lagrangiano para equação de convecção. Mostramos que o método é incondicionalmente estável, essencialmente não-dispersivo, convergente e acurado de ordem 2 no tempo e no espaço. Soluções numéricas de sistemas e equações diferenciais ordinárias são testadas no contexto do método do Tubo de Trajetórias, com difíceis problemas clássicos. Aplicações são consideradas no âmbito do transporte oceânico e na advecção de frentes atmosféricas. A fim de testar as propriedades conservativas do método estudado, uma estimativa do erro de balanço de massa é usado aqui. Comparações com outras metodologias mostram a superioridade do método do Tubo de Trajetórias. / In the present work we studied and analyzed the Trajectories Tube method, a conservative, explicit, simple, physically intuitive, semi-Lagrangian algorithm for the convection equation. Kinematical aspects of the mechanics of continuous media are essentially the tools used for formulation and feasibility analysis. We showed that this method is unconditionally stable, essentially nondispersive, convergent and accurate of order two in time and space. Computational experiments with non-isochoric and isochoric motions show that the studied method can be used in compressible and incompressible flow. Numerical solutions of systems of ordinary differential equations (necessary conditions for acomplishment of the scheme) are tested in the Trajectories Tube method context, with classical difficult examples. Applications are considered in the ambit of oceanic transport and advection of atmospheric fronts, including the tracer problem within a Stommel gyre and the computation of the Dowell frontogenesis. Comparisions with other methodologies show the superiority of the Trajectories Tube method.

No presente trabalho estudamos e analisamos o método do Tubo de Trajetórias, um algoritmo conservativo, explícito, simples, fisicamente intuitivo, semi-Lagrangiano para equação de convecção. Mostramos que o método é incondicionalmente estável, essencialmente não-dispersivo, convergente e acurado de ordem 2 no tempo e no espaço. Soluções numéricas de sistemas e equações diferenciais ordinárias são testadas no contexto do método do Tubo de Trajetórias, com difíceis problemas clássicos. Aplicações são consideradas no âmbito do transporte oceânico e na advecção de frentes atmosféricas. A fim de testar as propriedades conservativas do método estudado, uma estimativa do erro de balanço de massa é usado aqui. Comparações com outras metodologias mostram a superioridade do método do Tubo de Trajetórias. / In the present work we studied and analyzed the Trajectories Tube method, a conservative, explicit, simple, physically intuitive, semi-Lagrangian algorithm for the convection equation. Kinematical aspects of the mechanics of continuous media are essentially the tools used for formulation and feasibility analysis. We showed that this method is unconditionally stable, essentially nondispersive, convergent and accurate of order two in time and space. Computational experiments with non-isochoric and isochoric motions show that the studied method can be used in compressible and incompressible flow. Numerical solutions of systems of ordinary differential equations (necessary conditions for acomplishment of the scheme) are tested in the Trajectories Tube method context, with classical difficult examples. Applications are considered in the ambit of oceanic transport and advection of atmospheric fronts, including the tracer problem within a Stommel gyre and the computation of the Dowell frontogenesis. Comparisions with other methodologies show the superiority of the Trajectories Tube method.

Indiana University-Purdue University Indianapolis (IUPUI) / Our ability to engineer materials is limited by our capacity to tailor the material’s microstructure morphology and predict resulting properties. The insufficient knowledge on microstructure-property relationship is due to complexity and randomness in all materials at different scales. The objective of this research is to establish a design optimization methodology for microstructured materials. The material design problem is stated as finding the optimum microstructure to maximize the desired performance satisfying material processing constrains. This problem has been solved in this thesis by means of numerical techniques through four main steps: microstructure characterization, model reconstruction, property evaluation, and optimization. Two methods of microstructure characterizations have been investigated along with the advantages and disadvantages of each method. The first microstructure characterization method is a statistical method which utilizes correlation functions to extract the microstructural information. Algorithms for calculating these correlations functions have been developed and optimized based on their computational cost using MATLAB software. The second microstructure characterization method is physical characterization which works based on evaluation of physical features in microstructured domain. These features have been measured by means of MATLAB codes. Three model reconstruction techniques are proposed based on these characterization methods and employed to generate material models for further evaluation. The first reconstructing algorithm uses statistical functions to reconstruct the statistical equivalent model through simulating annealing optimization method. The second algorithm uses cellular automaton concepts to simulate the grain growth utilizing physical descriptors, and the third one generates elliptical inclusions in a material matrix using physical characteristic of microstructure. The finite element method is used to analysis the mechanical behavior of material models. Several material samples with different microstructural characteristics have been generated to model the micro-scale design domain of AZ31 magnesium alloy and magnesium matrix composite with silicon carbide fibers. Then, surrogate models have been created based on these samples to approximate the entire design domain and demonstrate the sensitivity of the desired mechanical property to two independent microstructural features. Finally, the optimum microstructure characteristics of material samples for fracture strength maximization have been obtained.