The problem of packing ellipsoids consists in arranging a given collection of ellipsoids within a particular set. The ellipsoids can be freely rotated and translated, and must not overlap each other. A particular case of this problem arises when the ellipsoids are balls. The problem of packing balls has been the subject of intense theoretical and empirical research. In particular, many works have tackled the problem with optimization tools. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years. This problem appears in a large number of practical applications, such as the design of high-density ceramic materials, the formation and growth of crystals, the structure of liquids, crystals and glasses, the flow and compression of granular materials, the thermodynamics of liquid to crystal transition, and, in biological sciences, in the chromosome organization in human cell nuclei. In this work, we deal with the problem of packing ellipsoids within compact sets from an optimization perspective. We introduce continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space. We present two different models for the non-overlapping of ellipsoids. As these models have quadratic numbers of variables and constraints, we also propose an implicit variables models that has a linear number of variables and constraints. We also present models for the inclusion of ellipsoids within half-spaces and ellipsoids. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, we present illustrative numerical experiments that show the capabilities of the proposed models. / O problema de empacotamento de elipsoides consiste em arranjar uma dada coleção de elipsoides dentro de um determinado conjunto. Os elipsoides podem ser rotacionados e transladados e não podem se sobrepor. Um caso particular desse problema surge quando os elipsoides são bolas. O problema de empacotamento de bolas tem sido alvo de intensa pesquisa teórica e experimental. Em particular, muitos trabalhos têm abordado esse problema com ferramentas de otimização. O problema de empacotamento de elipsoides, por outro lado, começou a receber mais atenção apenas recentemente. Esse problema aparece em um grande número de aplicações práticas, como o projeto de materiais cerâmicos de alta densidade, na formação e crescimento de cristais, na estrutura de líquidos, cristais e vidros, no fluxo e compressão de materiais granulares e vidros, na termodinâmica e cinética da transição de líquido para cristal e em ciências biológicas, na organização de cromossomos no núcleo de células humanas. Neste trabalho, tratamos do problema de empacotamento de elipsoides dentro de conjuntos compactos do ponto de vista de otimização. Introduzimos modelos de programação não-linear contínuos e diferenciáveis e algoritmos para o empacotamento de elipsoides no espaço n-dimensional. Apresentamos dois modelos diferentes para a não-sobreposição de elipsoides. Como esses modelos têm números quadráticos de variáveis e restrições em função do número de elipsoides a serem empacotados, também propomos um modelo com variáveis implícitas que possui uma quantidade linear de variáveis e restrições. Também apresentamos modelos para a inclusão de elipsoides em semi-espaços e dentro de elipsoides. Através da aplicação de uma estratégia multi-start simples combinada com uma escolha inteligente de pontos iniciais e um resolvedor para otimização local de programas não-lineares, apresentamos experimentos numéricos que mostram as capacidades dos modelos propostos.

In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks. / QC 20101018

This paper is the last part of a three-fold article aimed at some efficient numerical methods for solving the Poisson problem in three-dimensional prismatic and axisymmetric domains. In the first and second parts, the Fourier singular complement method (FSCM) was introduced and analysed for prismatic and axisymmetric domains with reentrant edges, as well as for the axisymmetric domains with sharp conical vertices. In this paper we shall mainly conduct numerical experiments to check and compare the accuracies and efficiencies of FSCM and some other related numerical methods for solving the Poisson problem in the aforementioned domains. In the case of prismatic domains with a reentrant edge, we shall compare the convergence rates of three numerical methods: 3D finite element method using prismatic elements, FSCM, and the 3D finite element method combined with the FSCM. For axisymmetric domains with a non-convex edge or a sharp conical vertex we investigate the convergence rates of the Fourier finite element method (FFEM) and the FSCM, where the FFEM will be implemented on both quasi-uniform meshes and locally graded meshes. The complexities of the considered algorithms are also analysed.

<p>Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(<i>u</i>) ≡ <i>u</i>²_x<i>u</i>_xx+ 2<i>u</i>_x<i>u</i>_y<i>u</i>_xy+<i>u</i>²_y<i>u</i>_yy= 0. For numerical reasons ∆<i>q</i>(<i>u</i>) = div (|▼<i>u</i>|<i>q</i>▼<i>u</i>)<i> = </i>0, which (formally) approaches as ∆∞(<i>u</i>) = 0 as <i>q</i> → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(<i>u</i>) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.</p>

Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u2xuxx + 2uxuyuxy + u2yuyy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.

This thesis presents a systematic numerical modeling framework to simulate the stress-deformation and coupled stress-deformation-flow processes by performing uniaxial and biaxial compressive tests on fractured rock models with considering the effects of different loading conditions, different loading directions (anisotropy), and coupled hydro-mechanical processes for evaluating strength and deformability behavior of fractured rocks. By using code UDEC of discrete element method (DEM), a series of numerical experiments were conducted on discrete fracture network models (DFN) at an established representative elementary volume (REV), based on realistic geometrical and mechanical data of fracture systems from field mapping at Sellafield, UK. The results were used to estimate the equivalent Young’s modulus and Poisson’s ratio and to fit the Mohr-Coulomb and Hoek-Brown failure criteria, represented by equivalent material properties defining these two criteria. The results demonstrate that strength and deformation parameters of fractured rocks are dependent on confining pressures, loading directions, water pressure, and mechanical and hydraulic boundary conditions. Fractured rocks behave nonlinearly, represented by their elasto-plastic behavior with a strain hardening trend. Fluid flow analysis in fractured rocks under hydro-mechanical loading conditions show an important impact of water pressure on the strength and deformability parameters of fractured rocks, due to the effective stress phenomenon, but the values of stress and strength reduction may or may not equal to the magnitude of water pressure, due to the influence of fracture system complexity. Stochastic analysis indicates that the strength and deformation properties of fractured rocks have ranges of values instead of fixed values, hence such analyses should be considered especially in cases where there is significant scatter in the rock and fracture parameters. These scientific achievements can improve our understanding of fractured rocks’ hydro-mechanical behavior and are useful for the design of large-scale in-situ experiments with large volumes of fractured rocks, considering coupled stress-deformation-flow processes in engineering practice. / <p>QC 20141111</p>

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.

Les travaux présentés portent sur l'étude de modèles numériques multi-fidèles, déterministes ou stochastiques. Plus précisément, les modèles considérés disposent d'un paramètre réglant la qualité de la simulation, comme une taille de maille dans un modèle par différences finies, ou un nombre d'échantillons dans un modèle de Monte-Carlo. Dans ce cas, il est possible de lancer des simulations basse fidélité, rapides mais grossières, et des simulations haute fidélité, fiables mais coûteuses. L'intérêt d'une approche multi-fidèle est de combiner les résultats obtenus aux différents niveaux de fidélité afin d'économiser du temps de simulation. La méthode considérée est fondée sur une approche bayésienne. Le simulateur est décrit par un modèle de processus gaussiens multi-niveaux développé dans la littérature que nous adaptons aux cas stochastiques dans une approche complètement bayésienne. Ce méta-modèle du simulateur permet d'obtenir des estimations de quantités d'intérêt, accompagnés d'une mesure de l'incertitude associée. L'objectif est alors de choisir de nouvelles expériences à lancer afin d'améliorer les estimations. En particulier, la planification doit sélectionner le niveau de fidélité réalisant le meilleur compromis entre coût d'observation et gain d'information. Pour cela, nous proposons une stratégie séquentielle adaptée au cas où les coûts d'observation sont variables. Cette stratégie, intitulée "Maximal Rate of Uncertainty Reduction" (MRUR), consiste à choisir le point d'observation maximisant le rapport entre la réduction d'incertitude et le coût. La méthodologie est illustrée en sécurité incendie, où nous cherchons à estimer des probabilités de défaillance d'un système de désenfumage. / The presented works focus on the study of multi-fidelity numerical models, deterministic or stochastic. More precisely, the considered models have a parameter which rules the quality of the simulation, as a mesh size in a finite difference model or a number of samples in a Monte-Carlo model. In that case, the numerical model can run low-fidelity simulations, fast but coarse, or high-fidelity simulations, accurate but expensive. A multi-fidelity approach aims to combine results coming from different levels of fidelity in order to save computational time. The considered method is based on a Bayesian approach. The simulator is described by a state-of-art multilevel Gaussian process model which we adapt to stochastic cases in a fully-Bayesian approach. This meta-model of the simulator allows estimating any quantity of interest with a measure of uncertainty. The goal is to choose new experiments to run in order to improve the estimations. In particular, the design must select the level of fidelity meeting the best trade-off between cost of observation and information gain. To do this, we propose a sequential strategy dedicated to the cases of variable costs, called Maximum Rate of Uncertainty Reduction (MRUR), which consists of choosing the input point maximizing the ratio between the uncertainty reduction and the cost. The methodology is illustrated in fire safety science, where we estimate probabilities of failure of a fire protection system.

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.

Práce je zaměřená na numerickou simulaci dvoufázového proudění. Je studován matematický model a numerická aproximace toku dvou nemísitelných nestlačitelných tekutin. Rozhraní mezi tekutinami je popsáno pomocí pomocí tzv. level set metody. Představena je diskretizace problému v prostoru a v čase. Metoda konečných prvk· se zpětnou Eulerovou metodou je aplikována na Navierovy-Stokesovy rovnice a časoprostorová nespojitá Galerkinova metoda je použita k řešení transportního problému. D·raz je kladen na analýzu chyby nespojité Galerkinovy metody přímek a časoprostorové nespojité Galerkinovy metody pro transportní problém. Jsou prezentovány numerické výsledky. 1