An extension to VORO++ for multithreaded computation of Voronoi cells

VORO++ is a software library written in C++ for computing the Voronoi tessellation, a technique in computational geometry that is widely used for analyzing systems of particles. VORO++ was released in 2009 and is based on computing the Voronoi cell for each particle individually. Here, we take advantage of modern computer hardware, and extend the original serial version to allow for multithreaded computation of Voronoi cells via the OpenMP application programming interface. We test the performance of the code, and demonstrate that we can achieve parallel efficiencies greater than 95% in many cases. The multithreaded extension follows standard OpenMP programming paradigms, allowing it to be incorporated into other programs. We provide an example of this using the VoroTop software library, performing a multithreaded Voronoi cell topology analysis of up to 102.4 million particles.Comment: Fix typo and section number

Polyhedral Voronoi diagrams for additive manufacturing

International audienceA critical advantage of additive manufacturing is its ability to fabricate complex small-scale structures. These microstructures can be understood as a metamaterial: they exist at a much smaller scale than the volume they fill, and are collectively responsible for an average elastic behavior different from that of the base printing material making the fabricated object lighter and/or flexible along specific directions. In addition, the average behavior can be graded spatially by progressively modifying the microstructure geometry.The definition of a microstructure is a careful trade-off between the geometric requirements of manufacturing and the properties one seeks to obtain within a shape: in our case a wide range of elastic behaviors. Most existing microstructures are designed for stereolithography (SLA) and laser sintering (SLS) processes. The requirements are however different than those of continuous deposition systems such as fused filament fabrication (FFF), for which there is currently a lack of microstructures enabling graded elastic behaviors.In this work we introduce a novel type of microstructures that strictly enforce all the requirements of FFF-like processes: continuity, self-support and overhang angles. They offer a range of orthotropic elastic responses that can be graded spatially. This allows to fabricate parts usually reserved to the most advanced technologies on widely available inexpensive printers that also benefit from a continuously expanding range of materials

Otimização de sistemas de atendimento emergenciais utilizando a teoria das filas e diagrama de voronoi ponderado

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-graduação em Engenharia de ProduçãoEste trabalho trata do problema de localização ótima de Estações de Emergência, localizadas no Corpo de Bombeiros, em que é necessário calcular o número de ambulâncias em cada estação, para atender a população de uma determinada região. Esse procedimento é realizado por meio da teoria das filas, devendo-se, para isso, considerar: o tempo médio de deslocamento da ambulância até a ocorrência, o número de chamadas diárias extremas para cada estação e também o tempo de espera na fila. O tempo médio de deslocamento é calculado pela teoria dos valores extremos, em que se considera somente o maior valor observado em cada dia. O tempo de espera, na fila, deve ser atribuído pelos administradores, podendo ser maior ou menor, dependendo, muitas vezes, de recursos financeiros. Neste trabalho, é considerado um tempo de espera, na fila, de um minuto e, para o deslocamento, de quatro minutos, num total de cinco minutos, podendo esse valor ser alterado para outras situações. Pelo diagrama de Voronoi multiplicativo por pesos, é realizada a divisão da área em estudo, nesse caso, a cidade de Curitiba-PR, onde há nove estações de emergência, sendo que cada ponto de estação será considerado como centro gerador de regiões. A otimização das colocações das estações é realizada com os modelos contínuos minmax e minmaxmin por meio do método simplex, em que é otimizado cada ponto de localização da Estação do Corpo de Bombeiros. A abordagem permite calcular, para um determinado conjunto de dados, o número suficiente de ambulâncias sem que haja limitação com relação à quantidade de unidades, o que dispensa o uso de simulação. Sendo assim, a metodologia apresentada é um instrumento que pode ser aplicado para qualquer região, desde que se tenham os dados relativos às ocorrências e os tempos de deslocamento, registrados num período considerado. Os algoritmos e software utilizados demonstraram uma boa performance para os modelos usados, contribuindo para a melhoria do sistema e rapidez no atendimento emergencial

Skeletal representations of orthogonal shapes

Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods