Generating high quality surface meshes is necessary as a prerequisite for many different numerical tasks. Given an input point cloud of a surface, implementations exist in software for the triangulations of the surface convex hull. Problems arise, however, when these surface meshes are not of sufficient quality, often leading to problems with stability and floating point errors for numerical methods. Concretely, we observe that this is often the case for Boundary Element Methods. In this thesis, I propose and implement an algorithm for the remeshing of closed, genus 0 surface meshes given an initial triangulation based on two parameters from the discrete Laplace Beltrami operator. I generate boundary conditions on this cut by adding new points on the surface mesh according to the gradient of one of the parameters, and then parameterizing the surface accordingly. After applying appropriate non-linear transforms on the parameters such that they become analogous to spherical coordinates on a unit sphere, we can obtain an interpolated point cloud which approximates the underlying surface. The resulting point cloud can then be triangulated to obtain a mesh of higher quality. I describe and detail a parallel sparse matrix solver, which mitigates one of the bottlenecks of the algorithm, and then conclude by 1) detailing mesh quality improvement on several example meshes to demonstrate the effectiveness of the algorithm, and 2) outlining modifications for surfaces with higher genus numbers and future research directions.

Examining Committee: 

 

                           Chair:              Dr. Ramani Duraiswami             
                          Members:        Dr. Nail Gumerov