Shape-preserving meshes and generalized Morse-Smale complexes

Abstract

Discrete representation of a surface, especially the triangle mesh, is ubiquitous in numerical

simulation and computer graphics. Compared with isotropic triangle meshes,

anisotropic triangle meshes provide more accurate results in numerical simulation by

capturing anisotropic features more faithfully. Furthermore, emerging applications in

computer graphics and geometric modeling require reliable differential geometry information

estimated on these anisotropic meshes. The first part of this thesis proposes

a special type of anisotropic meshes, called shape-preserving meshes, provides guaranteed

convergence of discrete differential operators on these meshes and devises an

algorithm for generating shape-preserving meshes on free-form surfaces based on the

mesh optimization framework with centroidal Voronoi tessellation (CVT). To improve

the numerical stability in simulation, we discuss how to reduce the number of obtuse

triangles in the mesh. The second part of the thesis discusses the non-uniqueness

of anisotropic meshes to represent the same anisotropy defined on a domain, shows

that of all anisotropic meshes, there exists one instance minimizing the number of

obtuse triangles, and proposes a variational approach to suppressing obtuse triangles

in anisotropic meshes by introducing a Minkowski metric in the CVT framework.

On a complex shape, its topological information is also highly useful to guide the

mesh generation. To extract topology properties, the Morse-Smale complex (MSC) is

a classical tool and widely used in computer graphics. However, on a manifold with

boundary, its MSC is not well defined. The final part of this thesis generalizes the MSC

to manifolds with boundaries. Based on this generalized MSC (GMSC), an operator to

merge n GMSCs of manifolds partitioning a large manifold is proposed. The merging

operator is used in a divide-and-conquer approach on a massive data set, providing

the potential to employ the computational power in a parallel manner.