The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modified Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI methods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small perturbation from the sinusoid motion problem and satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomials to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem. MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain. Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implementation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions.

The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis.

Une variété horosphérique est une variété algébrique normale dans laquelle un groupe algébrique réductif opère avec une orbite ouverte fibrée en tores sur une variété de drapeaux. La dimension de ces tores est appelée le rang de la variété horosphérique. En particulier, les variétés toriques et les variétés de drapeaux sont horosphériques. Dans cette thèse, on classifie les variétés horosphériques de Fano en termes de certains polytopes rationnels qui généralisent les polytopes réflexifs considérés par V.Batyrev. Puis on obtient une majoration du degré des variétés horosphériques lisses de Fano, analogue à celle donnée par O.Debarre dans le cas torique. On étend un résultat récent de C.Casagrande : les variétés horosphériques Q-factorielles de Fano ont leur nombre de Picard majoré par deux fois la dimension. On donne aussi de nombreux exemples en rang 2.

We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

Neste trabalho é desenvolvida uma solução semi-analítica para a Equação de Langevin assintótica (Equação de Deslocamento Aleatório) aplicada à dispersão de poluentes na Camada Limite Convectiva (CLC). A solução tem como ponto de partida uma equação diferencial de primeira ordem para o deslocamento aleatório, sobre a qual é aplicado o Método Iterativo de Picard. O novo modelo é parametrizado por um coeficiente de difusão obtido a partir da Teoria de Difusão Estatística de Taylor e de um modelo para o espectro de turbulência, assumindo a supersposição linear dos efeitos de turbulência térmica e mecânica. A avaliação do modelo é realizada através da comparação com dados de concentração medidos durante o experimento de dispersão de Copenhagen e com resultados obtidos por outros quatro modelos: modelo de partículas estocástico para velocidade aleatória (Modelo de Langevin), solução analítica da equação difusão-advecção, solução numérica da equação difusão-advecção e modelo Gaussiano. Uma análise estatística revela que o modelo proposto simula satisfatoriamente os valores de concentração observados e apresenta boa concordância com os resultados dos outros modelos de dispersão. Além disso, a solução através do Método Iterativo de Picard pode apresentar algumas vantagem em relação ao método clássico de solução.

Neste trabalho é desenvolvida uma solução semi-analítica para a Equação de Langevin assintótica (Equação de Deslocamento Aleatório) aplicada à dispersão de poluentes na Camada Limite Convectiva (CLC). A solução tem como ponto de partida uma equação diferencial de primeira ordem para o deslocamento aleatório, sobre a qual é aplicado o Método Iterativo de Picard. O novo modelo é parametrizado por um coeficiente de difusão obtido a partir da Teoria de Difusão Estatística de Taylor e de um modelo para o espectro de turbulência, assumindo a supersposição linear dos efeitos de turbulência térmica e mecânica. A avaliação do modelo é realizada através da comparação com dados de concentração medidos durante o experimento de dispersão de Copenhagen e com resultados obtidos por outros quatro modelos: modelo de partículas estocástico para velocidade aleatória (Modelo de Langevin), solução analítica da equação difusão-advecção, solução numérica da equação difusão-advecção e modelo Gaussiano. Uma análise estatística revela que o modelo proposto simula satisfatoriamente os valores de concentração observados e apresenta boa concordância com os resultados dos outros modelos de dispersão. Além disso, a solução através do Método Iterativo de Picard pode apresentar algumas vantagem em relação ao método clássico de solução.

Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthazar / Banca: Elso Drigo Filho / Resumo: Neste trabalho estuda-se alguns sistemas dinâmicos utilizando um novo método para aproximar a matriz de transição de estados (STM) para sistemas periódicos no tempo. Este método é baseado na transformação de Lyapunov-Floquet (L-F), e utiliza a expansão polinomial de Chebyshev para aproximar o termo periódico. O método iterativo de Picard é usado para aproximar a STM. Os multiplicadores de Floquet, determinados através deste método, permitem construir o diagrama de estabilidade do sistema dinâmico. Esta técnica é aplicada para analisar a estabilidade e os pontos de bifurcação do sistema dinâmico formado por um pêndulo elástico com excitação vertical periódica no suporte. Além dessa aplicação, é analisada também a equação de Mathieu e a estabilidade do sistema dinâmico constituído por partículas carregadas e imersas em um campo magnético perturbado. / Abstract: In this work some dynamic systems are studied using a new method to approach state transition matrix (STM) for time-periodic systems. This method is based on Lyapunov- Floquet transformation (transformation L-F) and uses the Chebyshev polynomial expansion to approach the periodical term. The Picard iterative method is used to approach the STM. The Floquet multipliers determined through this method, allow to draw the stability diagram of the dynamic system. This technique is applied to analyze the stability and bifurcation points of the dynamic system formed by an elastic pendulum with periodic vertical excitation on support. Besides this application, the Mathieu equation is analyzed and also the stability of the dynamical system constituted by charged particle in a perturbed magnetic field is discussed. / Mestre

Neste trabalho é desenvolvida uma solução semi-analítica para a Equação de Langevin assintótica (Equação de Deslocamento Aleatório) aplicada à dispersão de poluentes na Camada Limite Convectiva (CLC). A solução tem como ponto de partida uma equação diferencial de primeira ordem para o deslocamento aleatório, sobre a qual é aplicado o Método Iterativo de Picard. O novo modelo é parametrizado por um coeficiente de difusão obtido a partir da Teoria de Difusão Estatística de Taylor e de um modelo para o espectro de turbulência, assumindo a supersposição linear dos efeitos de turbulência térmica e mecânica. A avaliação do modelo é realizada através da comparação com dados de concentração medidos durante o experimento de dispersão de Copenhagen e com resultados obtidos por outros quatro modelos: modelo de partículas estocástico para velocidade aleatória (Modelo de Langevin), solução analítica da equação difusão-advecção, solução numérica da equação difusão-advecção e modelo Gaussiano. Uma análise estatística revela que o modelo proposto simula satisfatoriamente os valores de concentração observados e apresenta boa concordância com os resultados dos outros modelos de dispersão. Além disso, a solução através do Método Iterativo de Picard pode apresentar algumas vantagem em relação ao método clássico de solução.

Ce mémoire examine le syntagme nominal de l’ancien français et sa déclinaison en re- gardant deux œuvres, l’une représentant l’anglo-normand et l’autre représentant le pi- card. Le mémoire vise à déterminer si le système casuel est stable ou en train de dispa- raître, si les formes casuelles sont en train de se confondre, et si le picard est plus con- servateur que l’anglo-normand, comme on l’affirme souvent.  L’œuvre anglo-normande est La Vie de Saint Gilles par Guillaume de Berneville et l’œuvre picarde est La Vie de Sainte Cristine par Gautier de Coincy. Les deux œuvres sont des hagiographies, détaillant la vie de ces deux saints.  Les résultats montrent qu’il y a plus de confusion dans l’œuvre anglo-normande en ce qui concerne le système casuel, où l’on remplace plus souvent les formes du cas sujet par les formes du cas régime. Pour confirmer plus fermement ces différences quant au système, il faudrait étudier par exemple d’autres constructions grammaticales que le syntagme nominal ou des fonctions syntaxiques qui sont fortement liées au système casuel. / The paper examines the noun phrase in Old French and its declination by studying two literary works, one representing Anglo-Norman and the other representing Picard. The paper aims to determine if the case system is stable or in the process of disappearing, if the case forms are distinct and not easily confused, and if Picard is more conservative than Anglo-Norman, as is expected.  The Anglo-Norman literary work is La Vie de Saint Gilles by Guillaume de Berneville and the Picard literary work is La Vie de Sainte Cristine by Gautier de Coin- cy. The two works are hagiographies, describing the life of these two saints.  The results show that there is a greater confusion in the Anglo-Norman work when it comes to the case system, as it more often replaces the cas sujet forms with cas régime forms. In order to confirm the differences concerning the case system more firmly, one would have to study, for example, other grammatical constructions than the noun phrase or clause elements that are closely linked to the case system.

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