[pt] Malhas são amplamente usadas na discretização de domínios geométricos em aplicações na engenharia, como simulações de fluxo, transmissão de calor e deformação mecânica. O problema de geração de malhas é bem conhecido e estudado, mas a geração automática de malhas para um domínio físico com geometrias complexas, criando elementos que obedeçam a forma do objeto, e de tamanho e qualidade adequados, ainda é um desafio. Neste trabalho, foram estudados e implementados métodos para gerar malhas com restrições arbitrárias. O gerador implementado é do tipo de Delaunay, que constrói malhas Delaunay com restrições, e utiliza as propriedades da malha para inserir novos vértices e melhorar a qualidade dos elementos. / [en] Meshes are widely used in the discretization of geometric domains for engineering applications such as fluid flow simulator, heat transfer simulations and mechanical deformation. The mesh generation problem is well known and studied, nevertheless the automatic generation of meshes to domains with complex geometry, creating elements that conform to the forms, and of adequate size and quality, is still a challenge. In this work, mesh generation methods capable of generation mesh of arbitrary restrictions were studied and implemented. The implemented generator is a Delaunay generator, which constructs constrained Delaunay meshes, and utilizes the properties of the mesh to insert new vertices and improve the quality of the elements.

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.

It is not where in the world you are that matters. It is where you are with respect to a reference point whether on land or at sea. That is the basis behind Differencing GPS. Utilizing the carrier wave and Gold Code (GC) signal transmitted by GPS satellites, this project uses two GPS receivers and a system integration manager utilizing neural networks and expert systems to determine a user position and speed relative to a fixed point on earth. Two methods of determining the user position are employed: classic triangulation and measuring the difference in the Doppler shift of the carrier wave between the user and the reference receiver. The idea is for the user to know where they are in relationship to a designated fixed point and navigate with respect to that fixed point. The user could range from a farmer or an aircraft out at sea attempting to land on the deck of a carrier.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23684号 / 理博第4774号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 小野 薫, 教授 玉川 安騎男, 教授 望月 拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

A height field is a set of height distance values sampled at a finite set of sample points in a two-dimensional parameter domain. A height field usually contains a lot of redundant information, much of which can be removed without a substantial degradation of its quality. A common approach to reducing the size of a height field representation is to use a piecewise polygonal surface approximation. This consists of a mesh of polygons that approximates the surfaces of the original data at a desired level of accuracy. Polygonal surface approximation of height fields has numerous applications in the fields of computer graphics and computer vision. Triangular mesh approximations are a popular means of representing three-dimensional surfaces, and multiresolution analysis (MRA) is often used to obtain compact representations of dense input data, as well as to allow surface approximations at varying spatial resolution. Multiresolution approaches, particularly those moving from coarse to fine resolutions, can often improve the computational efficiency of mesh generation as well as can provide easy control of level of details for approximations. This dissertation concerns the use of wavelet-based MRA methods to produce a triangular-mesh surface approximation from a single height field dataset. The goal of this study is to obtain a fast surface approximation for a set of height data, using a small number of approximating elements to satisfy a given error criterion. Typically, surface approximation techniques attempt to balance error of fit, number of approximating elements, and speed of computation. A novel aspect of this approach is the direct evaluation of wavelet coefficients to assess surface shape characteristics within each triangular element at a given scale. Our approach hierarchically subdivides and refines triangles as the resolution level increases. / Ph. D.

Monte Carlo Markov chains (MCMCs) are used in many areas as a way to model a system’s behavior. By running a probabilistic simulation on a system’s state space, we can estimate properties of the system that could be untenable to directly compute. It is of interest to determine how quickly a Markov chain mixes\textemdash that is, settles into its stationary distribution. One such chain is induced by taking a binary search tree and performing a rotation or flip on one of its edges. We know that this chain eventually settles into the uniform distribution, but the time complexity bounds on the number of steps it takes to do so are not tight. We showcase an MCMC experiment suggesting that the true mixing time is likely higher than $\Omega(n^{\frac{3}{2}})$, the known lower bound. We also discuss choosing heuristics to approximate the total variation distance from the uniform distribution when a direct calculation is computationally infeasible\textemdash this calculation takes time proportional to the size of the state space, which for the binary tree chain is $\Theta\left(\frac{1}{n^{\frac{3}{2}}}4^n\right)$. Additionally, we discuss scaling the MCMC simulation as a whole to accommodate large state spaces. These findings serve to guide future studies on the direction of theoretical research on mixing times, as well as providing a framework for similar MCMC experiments.