Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time

In this paper, we present a Θ(n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.published_or_final_versio

Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing

Motivated by the desire to cope with data imprecision, we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (<= k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Omega(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "standard" convex hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at SoCG 201

A Linear-Time Randomized Algorithm for the Bounded Voronoi Diagram of a Simple Polygon

For a polygon P, the bounded Voronoi diagram of P is a partition of P into regions assigned to the vertices of P: A point p inside P belongs to the region of a vertex v if and only if v is the closest vertex of P visible from p. We present a randomized algorithm that builds the bounded Voronoi diagram of a simple polygon in linear expected time. Among other applications, we can construct within the same time bound the generalized Delaunay triangulation of P and the minimal spanning tree on P &apos;s vertices that is contained in P

Coupled Experimentally-Driven Constraint Functions and Topology Optimization utilized in Design for Additive Manufacturing

Topology optimization (TO) is a structural optimization technique that searches for the proper material distribution inside a design space such that an objective function is maximized/ minimized. Rapid prototyping technologies such as additive manufacturing (AM) have allowed results from TO to be manufacturable. However, despite advancements in their ability to manufacture complex geometries, AM technologies still face certain constraints such as printing features at overhangs (unsupported features oriented at a certain angle from the axis normal to the build plate) and small feature sizes, amongst others. In the field of design for additive manufacturing (DfAM), it is common to only restrict one constraint to control the quality of the final parts. However, several studies have found that the final quality of a feature is heavily affected by at least two coupled constraints: the overhanging angle and the feature’s thickness. Modifying a structure’s layout while restricting only one constraint can uselessly increase the weight of a structure. To tackle this problem, the work done in this thesis considers the interplay between two geometrical constraints. The proposed research reviews some of the essential manufacturing constraints in topology optimization and emphasizes the need for coupling existing constraints. It first develops experiments to obtain a qualitative and a quantitative relationship between the design features’ surface qualities, orientation, and thickness. The relation between those parameters is used to update the layout of topologically optimized structures. The layout is changed by obtaining the medial axis of topologically optimized structures and then using implicit functions to conditionally thickening it. Throughout the analysis, it was observed that both the inclination and the thickness affect the surface quality. Furthermore, the effect of the parameters is more pronounced for low thicknesses and higher overhanging angles. The overhanging angle impacts the surface quality more than the thickness, which can be seen through ANOVA

New Results on Abstract Voronoi Diagrams

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To free oneself from these geometric notions, Klein introduced abstract Voronoi diagrams as a general construct covering many concrete Voronoi diagrams. Abstract Voronoi diagrams are based on a system of bisecting curves, one for each pair of abstract sites, separating the plane into two dominance regions, belonging to one site each. The intersection of all dominance regions belonging to one site p defines its Voronoi region. The system of bisecting curves is required to fulfill only some simple combinatorial properties, like Voronoi regions to be connected, the union of their closures cover the whole plane, and the bisecting curves are unbounded. These assumptions are enough to show that an abstract Voronoi diagram of n sites is a planar graph of complexity O(n) and can be computed in expected time O(n log n) by a randomized incremental construction. In this thesis we widen the notion of abstract Voronoi diagrams in several senses. One step is to allow disconnected Voronoi regions. We assume that in a diagram of a subset of three sites each Voronoi region may consist of at most s connected components, for a constant s, and show that the diagram can be constructed in expected time O(s2 n ∑3 ≤ j ≤ n mj / j), where mj is the expected number of connected components of a Voronoi region over all diagrams of a subset of j sites. The case that all Voronoi regions are connected is a subcase, where this algorithm performs in optimal O(n log n) time, because here s = mj =1. The next step is to additionally allow bisecting curves to be closed. We present an algorithm constructing such diagrams which runs in expected time O(s2 n log(max{s,n}) ∑2 ≤ j≤ n mj / j). This algorithm is slower by a log n-factor compared to the one for disconnected regions and unbounded bisectors. The extra time is necessary to be able to handle special phenomenons like islands, where a Voronoi region is completely surrounded by another region, something that can occur only when bisectors are closed. However, this algorithm solves many open problems and improves the running time of some existing algorithms, for example for the farthest Voronoi diagram of n simple polygons of constant complexity. Another challenge was to study higher order abstract Voronoi diagrams. In the concrete sense of an order-k Voronoi diagram points are collected in the same Voronoi region, if they have the same k nearest sites. By suitably intersecting the dominance regions this can be defined also for abstract Voronoi diagrams. The question arising is about the complexity of an order-k Voronoi diagram. There are many subsets of size k but fortunately many of them have an empty order-k region. For point sites it has already been shown that there can be at most O(k (n-k)) many regions and even though order-k regions may be disconnected when considering line segments, still the complexity of the order-k diagram remains O(k(n-k)). The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. Nevertheless, we were able to come up with proofs of purely topological and combinatorial nature of Jordan curves and certain permutation sequences, and hence we could show that also the order-k abstract Voronoi diagram has complexity O(k (n-k)), assuming that bisectors are unbounded, and the order-1 regions are connected. Finally, we discuss Voronoi diagrams having the shape of a tree or forest. Aggarwal et. al. showed that if points are in convex position, then given their ordering along the convex hull, their Voronoi diagram, which is a tree, can be computed in linear time. Klein and Lingas have generalized this idea to Hamiltonian abstract Voronoi diagrams, where a curve is given, intersecting each Voronoi region with respect to any subset of sites exactly once. If the ordering of the regions along the curve is known in advance, all Voronoi regions are connected, and all bisectors are unbounded, then the abstract Voronoi diagram can be computed in linear time. This algorithm also applies to diagrams which are trees for all subsets of sites and the ordering of the unbounded regions around the diagram is known. In this thesis we go one step further and allow the diagram to be a forest for subsets of sites as long as the complete diagram is a tree. We show that also these diagrams can be computed in linear time

Geração de malhas para domínios 2,5 dimensionais usando triangulação de delaunay restrita

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-graduação em Engenharia Mecânica, Florianópolis, 2001.Gerar uma malha consiste em discretizar um domínio geométrico em pequenos elementos de forma geométrica simplificada, como triângulos e/ou quadriláteros, em duas dimensões, e tetraedros e/ou hexaedros em três dimensões. Malhas são utilizadas em diversas áreas, como geologia, geografia e cartografia, onde elas fornecem uma representação compacta dos dados do terreno; em computação gráfica, a grande maioria dos objetos são mapeados e imagens; e, em matemática aplicada e computação científica, são essenciais na solução numérica de equações diferenciais parciais, resultantes do modelamento de problemas físicos. Este trabalho concentra-se no desenvolvimento de um gerador de malhas voltadas para esta última aplicação, mas que podem, também, ser empregadas nas outras áreas. Mais especificamente, o interesse está na geração de malhas de triângulos não-estruturadas, através do processo de triangulação de Delaunay, para aplicações na solução de problemas de transferência de calor em superfícies planas tridimensionais. Devido à utilização do método CVFEM (Control Volume based Finite Element Method) para a modelagem numérica, um paralelo entre a Triangulação de Delaunay e Diagramas de Voronoi é delineado, apresentando suas propriedades e aplicaçõe .São estudados os métodos de geração de triangulações de Delaunay para superfícies planas de inversão de aresta, divide-and-conquer e incremental. A estrutura de dados utilizada é a triangular, e o método de refino para garantia de qualidade de malha é baseado no algoritmo de Ruppert. Restrições geométricas são tratadas de forma que a malha gerada obedeça as intersecções e conexões entre diversas superfícies. A contribuição fundamental do presente trabalho está na extensão de métodos de riangulação de Delaunay e de refino de malha bidimensionais para domínios 2,5 dimensionais compostos, isto é múltiplos planos interconectados no espaço tridimensional tratados simultaneamente. Otimização de ângulos internos, tamanho e forma dos elementos através da especificação de parâmetros, conferem ao gerador desenvolvido versatilidade e generalidade

Développement de méthodes particulaires pour la résolution des écoulements à surface libre

Ce travail vise à développer des approches particulaires dans le but de simuler les écoulements à surface libre. Celles-ci s'inspirent des méthodes sans maillages, méthodes apparues durant ces deux dernières décennies, et présentant des avantages par rapport aux approches numériques standards. La première partie de la thèse est consacrée à présenter cette famille de méthodes, dont quelques unes des plus connues sont détaillées. Les principaux avantages de ces méthodes ainsi que les plus importants défis à leur encontre seront énumérés. Par la suite, la méthode SPH (Smoothed Particle Hydrodynamics) est utilisée pour simuler les écoulements à surface libre en utilisant le système de Saint-Venant homogène. Une étude mathématique variationnelle révèle que cette méthode aboutit à une formulation symétrique et donc numériquement instable. Le schéma obtenu est stabilisé par un décentrage (upwinding) qui consiste à introduire une viscosité artificielle. L'expression de cette viscosité est obtenue par une analogie avec les solveurs de Riemann. Cette technique de stabilisation conduit à des résultats probants oii les chocs sont bien captés. Toutefois, un effet de lissage est observé au niveau des discontinuités probablement dû à l'absence de technique de type MUSCL dans le décentrage introduit. La méthode SPH, comme la majorité des méthodes sans mailiage, possède une fonction de forme non-interpolante rendant difficile l'imposition des conditions aux limites. Ce problème est surmonté en adoptant une interpolation de type élément naturel. Une nouvelle méthode de type volumes finis a été présentée : Méthode des Volumes Naturels : MVN. Cette méthode s'inspire de l'application de la méthode des éléments naturels en formulation Lagrangienne particulaire. Les flux sont évalués sur les cellules de Voronoï en utilisant la méthode des éléments naturels. Le schéma obtenu est un schéma centré donc instable. La même procédure de stabilisation adoptée pour la méthode SPH a été appliquée pour la MVN. La MVN montre les mêmes avantages que la méthode SPH lorsqu'elle est appliquée en formulation Lagrangienne. De plus, le caractère interpolant de la fonction de forme de type éléments naturels, permet aisément d'imposer des conditions aux frontières de type Dirichlet. L'application de la MVN dans le cas des équations de Saint-Venant homogènes et ensuite non-homogènes (avec termes source) montre un bon potentiel de cette nouvelle méthode. Le terme source de type géométrique disparaît dans la formulation de type MVN Lagrangienne et les cas avec bathymétrie variable sont traités exactement comme les cas à bathymétrie nulle. Ainsi la profondeur d'eau est remplacée par le niveau de la surface libre. Le schéma obtenu vérifie la z-propriété et la C-propriété