Submitted by Joyce Melo (joycemello79@gmail.com) on 2016-03-14T14:28:35Z No. of bitstreams: 1 DIssertação-Eder.pdf: 2150158 bytes, checksum: 370b81b52ed3a734cc7753e25cd017ab (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-03-15T14:52:32Z (GMT) No. of bitstreams: 1 DIssertação-Eder.pdf: 2150158 bytes, checksum: 370b81b52ed3a734cc7753e25cd017ab (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-03-15T15:15:56Z (GMT) No. of bitstreams: 1 DIssertação-Eder.pdf: 2150158 bytes, checksum: 370b81b52ed3a734cc7753e25cd017ab (MD5) / Made available in DSpace on 2016-03-15T15:15:56Z (GMT). No. of bitstreams: 1 DIssertação-Eder.pdf: 2150158 bytes, checksum: 370b81b52ed3a734cc7753e25cd017ab (MD5) Previous issue date: 2013-06-21 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we will do a study of one of the most beautiful theorems of geometry: Euler's theorem for polyhedra Convex, which lists the number of vertices V, edges A and F faces using the formula V -A + F = 2. The focus of the work is its use as a support to high school teacher. For this a little history of the theorem is presented and are given three statements distinct approaches. Moreover, it is made a brief analysis of how it is being done teaching spatial geometry, particularly the Euler's theorem in high school. For fi m, it is a suggestion for use of computer resources for the teaching of Euler's theorem using the software A Plethora of Polyhedra, open software, Universidade Federal Fluminense (UFF). / Neste trabalho fazemos um estudo de um dos mais belos teoremas da Geometria: o Teorema de Euler para Poliedros Convexos, que relaciona o número de vértices V , de arestas A e de faces F por meio da fórmula V −A+F = 2. O foco do trabalho é sua utilização como suporte ao professor do Ensino Médio. Para isto é apresentada um pouco da história do teorema e são dadas 3 demonstrações com abordagens distintas. Além disso, é feita uma breve análise de como está sendo feito o ensino de Geometria Espacial, particularmente o Teorema de Euler, no Ensino Médio. Por fim, faz-se uma sugestão para utilização de recursos computacionais para o ensino do Teorema de Euler através do software Uma Pletora de Poliedros, software aberto da Universidade Federal Fluminense (UFF).

La combustion d’hydrocarbures représente encore aujourd’hui une part très majoritaire de la production d’énergie mondiale, en particulier dans la propulsion aérospatiale. La plupart des brûleurs industriels sont alimentés par un carburant sous forme liquide, qui est injecté directement dans la chambre de combustion, ce qui génère une forte interaction entre le spray, l’écoulement turbulent et la zone de combustion. Cette interaction a déjà largement été étudiée, mais certaines questions restent ouvertes. En particulier, la prise en compte de la combustion de goutte isolée dans le cadre de la Simulation aux Grandes Echelles (‘Large Eddy Simulation’ LES) de géométries complexes reste un problème difficile. L’objectif de cette thèse est d’améliorer la modélisation de la combustion du spray dans le contexte de la LES de configurations complexes avec une approche Euler-Lagrange. Dans un premier temps, un modèle de combustion de gouttes incluant les différents régimes pour la LES, appelé MustARD pour « Multi-State Algorithm for Reacting Droplets », est proposé et validé dans plusieurs configurations académiques de complexité croissante. Dans un deuxième temps, MustARD est évalué sur une configuration de brûleur expérimental et comparé aux modèles classiques sans combustion de gouttes isolées. Cette étude montre que le régime de combustion de gouttes isolées n’est pas négligeable dans une telle configuration et qu’il modifie la structure de flamme. D’autre part, les comparaisons avec les résultats expérimentaux montrent que le modèle MustARD permet d’améliorer la précision des LES de sprays turbulents réactifs.

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Cette thèse est composée de deux parties. La première partie utilise l’analyse stochastique pour fournir des bornes pour la probabilité de surcharge de différents systèmes grâce aux inégalités de concentration. Bien qu’ils soient généraux, nous appliquons ces résultats à des réseaux sans-fil réels tels que le WiMax et le traffic utilisateur multi-classe dans un système OFDMA. Dans la seconde partie, nous trouvons des liens entre la topologie de la couverture dans un réseau de capteur et celle du complexe simplicial correspondant. Cette analogie met en valeur de nouvelles facettes des certains objets mathématiques comme les nombres de Betti, le nombre de k-simplexes, et la caractéristique d’Euler. Puis, nous utilisons conjointement la topologie algébrique et l’analyse stochastique, en considérant que les positions des capteurs sont une réalisation d’un processus ponctuel de Poisson. Nous en déduisons les statistiques du nombre de k-simplexe et de la caractéristique d’Euler, ainsi que des bornes supérieures pour la distribution des nombres de Betti, le tout en d dimen- sions. Nous démontrons aussi que le nombre de k-simplexes converge vers une distribution Gaussienne quand la densité de capteurs tend vers l’infini à une vitesse de convergence connue. Enfin, nous nous limitons au cas unidimensionnel. Dans ce cas, le problème devient équivalent à résoudre une file M/M/1/1 préemptive. Nous obtenons ainsi des résultats analytiques pour des quantités telles que la distribution du nombre de composantes connexes et la probabilité de couverture totale. / This thesis has two main parts. Part I uses stochastic anlysis to provide bounds for the overload probability of different systems thanks to concentration inequalities. Although the results are general, we apply them to real wireless network systems such as WiMax and mutliclass user traffic in an OFDMA system. In part I I, we find more connections between the topology of the coverage of a sensor network and the topology of its corresponding simplicial complex. These connections highlight new aspects of Betti numbers, the number of k-simplices, and Euler characteristic. Then, we use algebraic topology in conjunction with stochastic analysis, after assuming that the positions of the sensors are points of a Point point process. As a consequence we obtain, in d dimensions, the statistics of the number of k-simplices and of Euler characteristic, as well as upper bounds for the distribution of Betti numbers. We also prove that the number of k-simplices tends to a Gaussian distribution as the density of sensors grows, and we specify the convergence rate. Finally, we restrict ourselves to one dimension. In this case, the problem becomes equivalent to solving a M/M/1/1 preemptive queue. We obtain analytical results for quantites such as the distribution of the number of connected components and the probability of complete coverage.

In this essay the heuristic method of proofs and refutations, as pre- sented in the book Proofs and refutations by Imre Lakatos, is reviewed and discussed. Some background is given of heuristic methodology in contrast to the deductivist method and then Euler’s polyhedron for- mula is presented. Examples of both local and global mathematical monsters are introduced and handled through the application of the historical progression of Cauchy’s proof of Euler’s polyhedron formula. This leads to show the method of lemma-incorporation as the superior approach. The method is presented in the light of Lakatos’ heuristic philosophy, followed by criticism to his method. It is suggested to use Lakatos’ method as an addition to today’s formal mathematics to increase creativity and expand mathematical theorems further. The conclusion is drawn that Lakatos’ method of proofs and refutations is not prefect, but should utilize the existing criticism to be perfected through its own practice.

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An inconsistency exists in bubble force models used in the standard Euler-Euler simulations. The bubble force models are typically developed by assuming that the forces act on the bubbles' centers of mass. However, in the standard Euler-Euler model, each bubble force is a function of the local gas volume fraction because the phase-averaging method is used. This inconsistency can lead to gas over-concentration in the center or near the wall of a channel when the bubble diameter is larger than the computational cell size. Besides, a mesh-independent solution may not exist in such cases. In addition, the bubble deformation is not fully considered in the standard Euler-Euler model. In this thesis, a particle-center-averaging method is used to represent the bubble forces as forces that act on the bubbles' centers of mass. A particle-center-averaged Euler-Euler approach for bubbly flow simulations is developed by combining the particle-center-averaged Euler-Euler framework with a Gaussian convolution method. The convolution method is used to convert the phase-averaged and the particle-center-averaged quantities. The remediation of the inconsistency in the standard Euler-Euler model by the particle-center-averaging method is demonstrated using a simplified two-dimensional test case. Bubbly flows in different vertical pipes are used to validate the particle-center-averaged Euler-Euler approach. The bubbly flow simulation results for the particle-center-averaged Euler-Euler model and the standard Euler-Euler model are compared with experimental data. For monodisperse simulations, the particle-center-averaging method alleviates the over-predictions of the gas volume fraction peaks for wall-peaking cases and for finely dispersed flow case. Whereas, no improvement is found in the simulated gas volume fraction for center-peaking cases because the over-prediction caused by the inconsistency has been smoothed by the turbulent dispersion. Moreover, the axial gas and liquid velocities simulated with both Euler-Euler models are similar, which proves that the closure models for bubble forces and turbulence are correctly applied in the particle-center-averaged Euler-Euler model. For fixed polydisperse simulations, the particle-center-averaging method can also alleviate the over-prediction of the gas volume fraction peak in the center or near the wall of a pipe. The axial gas velocities simulated with both Euler-Euler models are about the same. Comparisons are also made for the simulation results of bubbly flows in a cylindrical bubble column and the experimental data. The gas volume fractions and the axial gas velocities simulated with both Euler-Euler models almost coincide with each other, which indicates that the sink and source terms for the continuity equations and the degassing boundary are set correctly in the particle-center-averaged Euler-Euler model. An oblate ellipsoidal bubble shape is considered in the particle-center-averaged Euler-Euler simulations by an anisotropic diffusion. The influence of bubble shape on the simulation results of bubbly pipe flows is investigated. The results show that considering the oblate ellipsoidal bubble shape in simulations can further alleviate the over-predictions of the gas volume fraction peaks for wall peaking cases, but it has little influence on the gas volume fractions of center-peaking cases and the axial gas velocities.

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Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.