Browse Theses

A major challenge facing the community of computational scientists is the ability to approximate geometry. The geometry of an object may be represented in many ways: explicitly, as a mathematical relationship or a geometric model, or implicitly, as an implicit equation or even data that may have underlying geometric structure. Whatever the case, in order to analyze, view, or compute with the object, its geometry often requires approximation.

There are many practical applications available to illustrate this point. For example, computer images from medical CAT or MR scans are generated from volume data, and the image is extracted by building a surface (i.e., approximating geometry) corresponding to some data value. Or consider the numerical solution of partial differential equations. Of the various approaches to solving such equations, such as finite difference, finite volume, and finite element, the basic procedure of each is to discretize the domain (i.e., approximate geometry) where local approximations to the solution result in large systems of linear and/or non-linear equations. These equations can then be automatically processed by large computer systems to produce results limited mainly by the degree of refinement of the initial numerical grid.

The process of geometric approximation, or mesh generation, has seen many approaches. Many have been novel and extremely useful, but most have been tailored for a particular class of geometry. Achieving the ultimate goal, that of automatically generating geometric approximations to arbitrary geometry, is the subject of this thesis. Other topics addressed in this thesis include: the representation of geometry, prior art, adaptive methods for local mesh refinement, controlling mesh refinement, the overall place of geometric approximation for computation, and algorithms for geometric approximation. In addition, the most important question addressed is "What is a valid mesh__ __"

To address these topics, a new form of triangulation, the geometric triangulation is defined. The geometric triangulation forms a valid approximation to geometry because it satisfies the conditions of topological compatibility (i.e., topologically equivalent) and geometric similarity (i.e., a valid approximation). Efficient algorithms based on the octree spatial decomposition and Delaunay triangulation techniques are also described. In particular, the method of successive triangulation is proposed as the algorithm for fully automatic mesh generation.