CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Este trabalho à constituÃdo de duas partes. Na primeira parte estudamos mÃnimos nÃo negativos de funcionais elÃpticos degenerados, ∫ F (X, u, ∇u)dX → min, para nÃcleos variacionais F que sÃo descontÃnuos em u com descontinuidade de ordem ~ X{u>0}. A equaÃÃo de Euler-Lagrange à governada por uma equaÃÃo elÃptica degenerada e nÃo-homogÃnea, com fronteira livre entre as fases positiva e zero do mÃnimo. Mostraremos estimativa gradiente Ãtima e nÃo-degenerescÃncia do mÃnimo. TambÃm trataremos de propriedades de regularidade fracas e fortes de fronteira livre. Provaremos que o conjunto {u>0} tem localmente perÃmetro finito e que a fronteira livre reduzida ∂ red {u>0} tem medida Hn-1-total. Para problemas mais especÃficos que aparecem em Jet flows, provaremos que a fronteira livre reduzida à localmente o grÃfico de uma funÃÃo C1,y. Na segunda parte do trabalho forneceremos uma descriÃÃo bastante completa da teoria de regularidade Ãtima para uma famÃlia de problemas de fronteira livre de duas fases, heterogÃneos, y→ min, governados por operadores elÃpticos degenerados e nÃo-lineares. IncluÃdos em tal famÃlia estÃo os problemas de Jet flows heterogÃneos e os problemas de cavidades do tipo Prandtl-Batchelor, y = 0; equaÃÃes elÃpticas degeneradas singulares e sistemas do tipo obstÃculo y =1.VersÃes lineares destes problemas tÃm sido objeto de intensa pesquisa nas Ãltimas quatro dÃcadas ou mais. As contrapartidas nÃo-lineares tratadas neste trabalho introduzem novas e considerÃveis dificuldades, pois a maioria das teorias desenvolvidas anteriormente, tais como fÃrmulas de monotonicidade e de quase monotonicidade nÃo estÃo disponÃveis. Contudo, as soluÃÃes inovadoras desenvolvidas neste trabalho fornecem novas respostas mesmo no contexto clÃssico de equaÃÃes lineares e nÃo-degeneradas. / This work consists of two parts. In the first part we study nonnegative minimizers of general degenerate elliptic functionals, ∫ F (X, u, ∇u)dX → min, for variational kernels F that are discontinuous in ụ with discontinuity of order ~ X{u>0}. The Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate and nondegeneracy of minima. We also address weak and strong regularity properties of free boundary, ∂ red {u>0}, has H n-1- total measure. For more specific problems that arise in jet flows, we show the reduced free boundary is locally the graph of a C1,y function. In the second part of work we provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, y→ min, ruled by nonlinear, degenerate elliptic operators. Included in such family are heterogeneous jets and cavities problems of Prandtl-Batchelor type, y = 0; singular degenerate elliptic equations and obstacle type systems, y = 1. Linear versions of these problems have been subjects of intense research for the past four decades or so. The nonlinear counterparts treated in this present work introduce substantial new difficulties since the most of the classical theories developed earlier, such that as monotonicity and almost monotonicity formulae, are no longer available. Nonetheless, the innovative solutions designed in this work provide new answers even in the classical context of linear, nondegenerate equations.

Orientadores: Marco Antonio Teixeira, Alain Guy Jacquemard / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T23:50:08Z (GMT). No. of bitstreams: 1 Pereira_WeberFlavio_D.pdf: 1832947 bytes, checksum: 58bb202e90151fc6830fbc0cd1cf430e (MD5) Previous issue date: 2006 / Resumo: De acordo com a classificação feita por Anosov em 1959, obtemos diferentes tipos topológicos de sistemas "semi-lineares" descontínuos em JR4. Esta pré-classificação é feita através da apresentação das respectivas formas normais. Neste trabalho, consideramos perturbações não lineares de tais formas normais. As singularidades típicas são genericamente classificadas e o comportamento dos sistemas em torno destes pontos é analisado. Nosso foco é encontrar condições para a existência de uma família a l-parâmetro de órbitas periódicas terminando em singularidades no sentido do Teorema Centro de Lyapounov. As técnicas principais usadas são elementos do cálculo simbólico e da Teorida das Singularidades de Aplicações / Abstract: According to the classification made by Anosov in 1959, we derive several different topological types of semi-linear"discontinuous systems in R4. This pre-classification is done via pre-sentation of the respective normal forms. In this work, we consider non-linear perturbations of such normal forms. The typical singularities are generically classified and the behavior of the systems around these points is analyzed. Our focus is find conditions for the existence of 1-parameter family of periodic orbit terminating at the singularities in the sense of Lya- pounov Center Theorem. The main techniques used are elements of Symbolic Computation and Theory of Singularities of Mappings / Doutorado / Doutor em Matemática

O desenvolvimento de uma ferramenta de Dinâmica de Fluidos Computacional que utiliza Método de Elementos Finitos baseada na discretização de Galerkin descontínuo é apresentado neste trabalho com objetivo de resolver a equação de Euler linearizada para escoamento compressível em duas dimensões usando malhas estruturadas e não estruturadas. Procuramos utilizar esta ferramenta como um propagador de ondas sonoras para estudar fenômenos aeroacústicos. O problema de Riemann presente no fluxo convectivo da equação de Euler é tratado com um método upwind HLL e para o avanço da solução no tempo é usado o método de Runge-Kutta explícito de 4 estágios com segunda ordem de precisão. A eficiência computacional, a convergência do método e a precisão são testadas através de simulações de escoamentos já apresentadas na literatura. A taxa de convergência para altas ordens de aproximação é assintótica que é um resultado compatível com a formulação Galerkin descontínuo. / The development of a Computation Fluid Dynamic Tool based on the Finite Element Method with discontinuous Galerkin discretization is presented in this work. The aim of this study is to solve the compressible linearized Euler\'s equation in two dimensions on structured and non structured meshes. This tool has been used as a means to study aeroacoustics phenomena. The Riemann\'s problem presented on a convective flow in Euler\'s equation is tackled by a HLL\'s method and the time integration being used is the four-stage Runge-Kutta explicit method with second order of accuracy. The computational efficiency, the convergence of the method and the accuracy are tested by comparing our results for flow simulations with those that are available in the literature. The convergence rate to high approximation order is asymptotic and it shows a result which is compatible with a discontinuous Galerkin formulation.

[pt] Este trabalho é uma investigação da microestrutura e cinética dos fenômenos de precipitação que ocorrem na Liga 33 (Fe-Ni-Cr-Mo-Cu-N), sistema metálico desenvolvido pela Krupp VDM com o intuito de suportar altas temperaturas e ambiente corrosivo. A Liga 33 incorre precipitação contínua e descontínua simultaneamente, como resultado do tratamento de envelhecimento realizado numa faixa de temperatura correspondente àquela que o material atinge quando submetido a um processo de soldagem. A caracterização microestrutural foi realizada por microscopia ótica, microscopia eletrônica de varredura, e microscopia eletrônica de transmissão, incluindo microanálise, devido à ordem de grandeza nanométrica das fases precipitadas. A precipitação descontínua ocasiona uma estrutura lamelar no contorno dos grãos, resultado do crescimento cooperativo entre as lamelas, envolvendo átomos substitucionais (Cr, por exemplo) e intersticiais (N). A precipitação contínua ocorre no interior dos grãos gerando precipitados com diferentes morfologias. A microanálise revela que os produtos gerados em ambas as reações crescem competindo pelo Cr. Uma análise cinética- morfológica aponta para a natureza não estacionária da reação descontínua, que sofre gradativa diminuição de sua taxa de transformação, até ser totalmente paralisada. / [en] This work is an investigation of the microstructure and kinetics of the phenomena occurring inside Alloy 33 (Fe-Ni- Cr-Mo-Cu-N), a metallic system developed by Krupp VDM to endure high temperatures and corrosive environment. Alloy 33 incurs continuous and discontinuous precipitation simultaneously, as a result of the aging treatment induced in a temperature range correspondent to the one of a welding process in the referred material. The microstructural characterization was performed by optical microscopy, scanning electron microscopy and transmission electron microscopy, including microanalysis, due to the nanometric nature of the precipitated phases. Discontinuous precipitation produces a lamellar structure along grain boundaries as a result of a cooperative growth between the lamellae, involving substitucional and interstitial atoms, Cr and N respectively. Continuous precipitation occurs inside grains, generating precipitates with different morphologies. Microanalysis reveals that products of both precipitation reactions grow competing for Cr. A kinetic-morphological analysis points to the non-stationary characteristic of the discontinuous precipitation, where the transformation rate diminishes until it stops completely, as aging occurs.

Efficient numerical methods for option pricing is an active field of research. This project has the goal to examine possible ways to improve an established method of numerical pricing. The method is based on an adaptive finite difference method in price and uses the backwards differentiation formula of order 2, BDF2, in time. The project will focus on improvements to the time integration through implementation of discontinuous Galerkin methods, dG. Empirical convergence and accuracy results are obtained for equidistant dG-methods up to order 3 and performance is compared to BDF2. The dG-methods do not succeed in outperforming the BDF2-method when comparing accuracy to time for computation, but they do match the performance. Possible ways for improvements are suggested.

Combating drug resistant malaria has been historically challenging, and remains so today. Recent reports from Southeast Asia show that Plasmodium falciparum is developing resistance to even our best defenses; artemisinin-based therapies. This development threatens to become a significant challenge in controlling malaria infections worldwide, making research into developing and characterizing new antimalarial drugs increasingly important. The purpose of this study was to characterize the resistance potential of novel antimalarial compound JPC-3210 in vitro using P. falciparum clones. JPC-3210 is a new long acting drug with potential to be used in combination with fast-acting drugs like artemisinins to cure drug resistant malaria. In this study several methods were used to characterize the efficacy and resistance potential of JPC-3210. To determine the frequency of resistance generation in P. falciparum clones, parasites were kept under continuous drug pressure for thirty days, at which point drug pressure was removed and cultures were observed for signs of recrudescence. P. falciparum clones also were exposed to increasing levels of intermittent drug pressure that involved 3-4 days of drug exposure followed by a recovery period. The step-wise experiment was conducted over three months with drug pressure being increased step-wise until a maximal concentration of 700 ng/ml of JPC-3210; resistance was measured phenotypically in drug susceptibility assays at multiple time points. Additionally, the ability of JPC-3210 to induce dormant stage parasites, and its effect on dihydroartemisinin (DHA)-induced dormant stages was assessed in both a chloroquine resistant parasite (W2) and in an artemisinin resistant clone (4G). Results showed that the frequency of resistance against JPC-3210 in W2 clones was less when compared to that of atovaquone. The step-wise pulse exposure of JPC-3210 induced resistance in W2 clones, however, resistance proved unstable. Dormant stage parasites were not induced by JPC-3210, even at high concentrations in W2 or 4G clones, furthermore, the effect of JPC-3210 on dormant-induced parasites was found to be dose dependent, yet the drug did not kill DHA-induced dormant rings. JPC-3210 appears to be a good drug to use in combination with other antimalarial compounds for treatment of P. falciparum, but further research is needed. Future studies to assess the field performance of new antimalarial compounds by investigating resistance and dormancy profiles in vitro, and thereby maximizing out understanding of such drugs and their optimal implementation, are of the utmost importance.

We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space

Cette thèse s'inscrit dans un projet d'innovation duale RAPID financé par DGA/DS/MRIS et appelé GREAT faisant intervenir la société Axessim, l'ONERA, INRIA, l'IRMA et le CEA. Ce projet a pour but la mise en place d'une solution industrielle de simulation électromagnétique basée sur une méthode Galerkin Discontinue (GD) parallèle sur maillage hexaédrique. Dans un premier temps, nous établissons un schéma numérique adapté à un système de loi de conservation. Nous pouvons ainsi appliquer cette approche aux équations de Maxwell, mais également à tout système hyperbolique. Dans un second temps, nous mettons en place une parallélisation à deux niveaux de ce schéma. D'une part, les calculs sont parallélisés sur carte graphique au moyen de la bibliothèque OpenCL. D'autre part, plusieurs cartes graphiques peuvent être utilisées, chacune étant pilotée par un processus MPI. De plus, les communications MPI et les calculs OpenCL sont asynchronisés permettant d'obtenir une forte accélération. / This thesis is part of a dual innovation project funded by RAPID DGA/DS/MRIS and called GREAT involving Axessim company, ONERA, INRIA, IRMA and the CEA. This project aims at the establishment of an industrial solution of electromagnetic simulation based on a method Discontinuous Galerkin (DG) on parallel hexahedral mesh. First, we establish a numerical scheme adapted to a conservation law system. We can apply this approach to the Maxwell equations but also to any hyperbolic system. In a second step, we set up a two-level parallelization of this scheme. On the one hand, the calculations are parallelized on graphics card using the OpenCL library. On the other hand, multiple graphics cards can be used, each driven by a MPI process. In addition, MPI communications and OpenCL computations are launched asynchronously in order to obtain a strong acceleration.

Avian influenza is a contagious viral disease caused by influenza virus type A. Avian influenza can be disastrous (if it occurs), due to the short incubation period (about 1--4 days). Thus it is important to study this disease so that we are more prepared to manage it in the future. A classical system of differential equations (the half-saturated incidence model) and three Filippov models --- an avian-only model with culling of infected birds, an SIIR (Susceptible-Infected-Infected-Recovered) model with quarantine of infected humans and an avian-only model with culling both susceptible and infected birds --- that are governed by ordinary differential equations with discontinuous right-hand sides (i.e., differential inclusion) are proposed to study the transmission of avian influenza. The effect of half-saturated incidence is investigated, and the outcome of this model is compared with the bilinear incidence model. Both models remain endemic whenever their respective basic reproduction numbers are greater than one. The half-saturated incidence model generates more infected individuals than the bilinear incidence model. This may be because the bilinear incidence model is underestimating the number of infected individuals at the outbreak. For the Filippov models, the number of infected individuals is used as a reference in applying control strategies. If this number is greater than a threshold value, a control measure has to be employed immediately to avoid a more severe outbreak. Otherwise, no action is necessary. We perform dynamical system analysis for all models. The existence of sliding modes and the flow on the discontinuity surfaces are determined. In addition, numerical simulations are conducted to illustrate the dynamics of the models. Our results suggest that if appropriate tolerance thresholds are chosen such that all trajectories of the Filippov models are converging to an equilibrium point that lies in the region below the infected tolerance threshold or on the discontinuity surface, then no control strategy is necessary as we consider the outbreak is tolerable. Otherwise, we have to apply control strategies to contain the outbreak. Hence a well-defined threshold policy is crucial for us to combat avian influenza effectively.

A software program was written that implements a finite element analysis (FEA) solution as the basis of an optimization function used for guiding the inverse design problem of aligning fibres, minimizing displacements in a fibre-reinforced polymer composite part in response to a given loading condition, for various part geometries. The FEA solution makes use of the superlinear RGNTet4 element, which includes 3 displacement and 3 rotational degrees of freedom at 4 nodes. Convergence testing verified the accuracy of the solver versus symbolic results for simple cases. Multivariate optimization over fibre orientations and volume fractions was carried out for a simple test case using the NLOpt nonlinear optimization library. Both derivative-free and gradient-based algorithms were tested. Low-Storage Broyden-Fletcher-Goldfarb-Shannon was the most effective algorithm. Four more complex cases were examined, and by varying fibre orientations, reductions of 48%, 66%, 58% and 32% were achieved in displacements at the loaded nodes.