Shape theory and mathematical design of a general geometric kernel through regular stratified objects

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This dissertation focuses on the mathematical design of a unified shape kernel for geometric computing, with possible applications to computer aided design (CAM) and manufacturing (CAM), solid geometric modelling, free-form modelling of curves and surfaces, feature-based modelling, finite element meshing, computer animation, etc. The generality of such a unified shape kernel grounds on a shape theory for objects in some Euclidean space. Shape does not mean herein only geometry as usual in geometric modelling, but has been extended to other contexts, e. g. topology, homotopy, convexity theory, etc. This shape theory has enabled to make a shape analysis of the current geometric kernels. Significant deficiencies have been then identified in how these geometric kernels represent shapes from different applications. This thesis concludes that it is possible to construct a general shape kernel capable of representing and manipulating general specifications of shape for objects even in higher-dimensional Euclidean spaces, regardless whether such objects are implicitly or parametrically defined, they have ‘incomplete boundaries’ or not, they are structured with more or less detail or subcomplexes, which design sequence has been followed in a modelling session, etc. For this end, the basic constituents of such a general geometric kernel, say a combinatorial data structure and respective Euler operators for n-dimensional regular stratified objects, have been introduced and discussed

Totally normal cellular stratified spaces and applications to the configuration space of graphs

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced by the third author by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.Comment: 44 pages. v2. Typos fixed. Accepted for publication by Topological Methods in Nonlinear Analysi

Conformal Metrics with Constant Q-Curvature

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Hamiltonian discontinuous Galerkin FEM for linear, stratified (in)compressible Euler equations: internal gravity waves

The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and energy. This required (i) the discretization of the Hamiltonian structure using alternating flux functions and symplectic time integration, (ii) the discretization of a divergence-free velocity field using Dirac's theory of constraints and (iii) the handling of large-scale computational demands due to the 3-dimensional nature of internal gravity waves and, in confined, symmetry-breaking fluid domains, possibly its narrow zones of attraction

Mimetic Coastal Ocean Modeling In General Coordinates And Using Machine Learning Based Predictions

Nonlinear internal waves are a ubiquitous and fundamental aspect of the coastal ecosystem understanding. However, they rely on extreme geographical conditions and precise dimensional equilibrium to be captured accurately. The General Curvilinear Coastal Ocean Model (GCCOM) was validated, serial and parallel versions for a set of experiments showcasing stratified and non-hydrostatic flow phenomena. Still, the 3D curvilinear capability has proven to be elusive. We apply cutting-edge numerical methods to improve upon the previously validated GCCOM, elevating it to field-scale capacity. This reformulation of the GCCOM equations uses novel 3D curvilinear mimetic operators, a buoyancy body force, and mimetic upwind and gradient-based momentum equations developed for this work. This model represents the most complete implementation of the 3D curvilinear mimetic operators utilizing the MOLE library or any other mimetic applications in literature to date. Results show it to be more physically accurate and better energy conserving than the validated GCCOM and other similar models, permitting the use of 3D curvilinear grids for arbitrary geometries, parallelizable arbitrary domain decomposition, and order-of-magnitude wider time steps. Additionally, we implement machine learning models to coastal ocean data to predict Dissolved Oxygen (DO) content with supervised methods; results show a Median Absolute Percentage Error (MAPE) of 2-6% for instantaneous indirect readings of DO and 0.18% for five days forecast of DO in coastal areas, using a previously predicted temperature of 1.60% MAPE. Dissolved Oxygen is known to be a critically important component to track in coastal environments but also expensive to measure and almost impossible to model with traditional methods due to high nonlinearity. The ML component of this thesis opens the possibility of high precision indirect estimates of biogeochemical quantities, along with highly accurate time series forecasts and a host of new applications of machine learning to environmental sciences

Research in Applied Mathematics, Fluid Mechanics and Computer Science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1998 through March 31, 1999