TetGen, towards a quality tetrahedral mesh generator

TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation

State of the Art: Updating Delaunay Triangulations for Moving Points

This paper considers the problem of updating efficiently a two-dimensional Delaunay triangulation when vertices are moving. We investigate the three current state-of-the-art approaches to solve this problem: --1-- the use of kinetic data structures, --2-- the possibility of moving points from their initial to final position by deletion and insertion and --3-- the use of "almost" Delaunay structure that postpone the necessary modifications. Finally, we conclude with a global overview of the above-mentioned approaches while focusing on future works

Nature of the learning algorithms for feedforward neural networks

The neural network model (NN) comprised of relatively simple computing elements, operating in parallel, offers an attractive and versatile framework for exploring a variety of learning structures and processes for intelligent systems. Due to the amount of research developed in the area many types of networks have been defined. The one of interest here is the multi-layer perceptron as it is one of the simplest and it is considered a powerful representation tool whose complete potential has not been adequately exploited and whose limitations need yet to be specified in a formal and coherent framework. This dissertation addresses the theory of generalisation performance and architecture selection for the multi-layer perceptron; a subsidiary aim is to compare and integrate this model with existing data analysis techniques and exploit its potential by combining it with certain constructs from computational geometry creating a reliable, coherent network design process which conforms to the characteristics of a generative learning algorithm, ie. one including mechanisms for manipulating the connections and/or units that comprise the architecture in addition to the procedure for updating the weights of the connections. This means that it is unnecessary to provide an initial network as input to the complete training process.After discussing in general terms the motivation for this study, the multi-layer perceptron model is introduced and reviewed, along with the relevant supervised training algorithm, ie. backpropagation. More particularly, it is argued that a network developed employing this model can in general be trained and designed in a much better way by extracting more information about the domains of interest through the application of certain geometric constructs in a preprocessing stage, specifically by generating the Voronoi Diagram and Delaunav Triangulation [Okabe et al. 92] of the set of points comprising the training set and once a final architecture which performs appropriately on it has been obtained, Principal Component Analysis [Jolliffe 86] is applied to the outputs produced by the units in the network's hidden layer to eliminate the redundant dimensions of this space

A computational geometric approach for an ensemble-based topological entropy calculation in two and three dimensions

From the stirring of dye in viscous fluids to the availability of essential nutrients spreading over the surface of a pond, nature is rife with examples of mixing in two-dimensional fluids. The long-time exponential growth rate of a thin filament of dye stretched by the fluid is a well-known proxy for the quality of mixing in two dimensions. This growth rate in turn gives a lower bound on the flow's topological entropy, a measure quantifying the complexity of chaotic dynamics. In the real-world study of mixing, topological entropy may be hard to compute; the velocity field may not be known or may be expensive to recover or approximate, thus limiting our knowledge of the governing system and underlying mechanics driving the mixing. Central to this study are two questions: \emph{How can stretching rates in two-dimensional planar flows best be computed using only trajectory data?}, and \emph{Can a method for computing stretching rates in higher dimensions from only trajectory data be developed?}. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation (E-tec), a method to derive a lower-bound on topological entropy that requires only finite number of system trajectories, like those obtained from ocean drifters, and no detailed knowledge of the velocity field. E-tec is demonstrated to be computationally more efficient than other competing methods in two dimensions that accommodate trajectory data. This is accomplished by considering the evolution of a ``rubber band" wrapped around the data points and evolving with their trajectories. E-tec records the growth of this band as the collective motion of trajectories strike, deform, and stretch it. This exponential growth rate acts as a lower bound on the topological entropy. In this manuscript, I demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time. Driving the efficiency of E-tec in two dimensions is the use of computational geometry tools. Not only this, by computing stretching rates in this new computational geometry framework, I extend E-tec to three dimensions using two methods. First, I consider a two-dimensional rubber sheet stretched around a collection of points in a three-dimensional flow. Similar to the band-stretching component of two-dimensional E-tec, a three-dimensional triangulation is used to record the growth of the sheet as it is stretched and deformed by points evolving in time. Second, I calculate the growth rates of one-dimensional rubber strings as they are stretched by the edges of this dynamic, moving triangulation

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains