Computing Curvature and Curvature Normals on Smooth Logically Cartesian Surface Meshes

This thesis describes a new approach to computing mean curvature and mean curvature normals on smooth logically Cartesian surface meshes. We begin by deriving a finite-volume formula for one-dimensional curves embedded in two- or three- dimensional space. We show the exact results on curves for specific cases as well as second-order convergence in numerical experiments. We extend this finite-volume formula to surfaces embedded in three-dimensional space. Exact results are again derived for special cases and second-order convergence is shown numerically for more general cases. We show that our formula for computing curvature is an improvement over using the “cotan” formula on a triangulated quadrilateral mesh and is conceptually much simpler than the formula proposed by Liu et al. (“A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes”, Computers and Mathematics with Applications, 2008), and is equivalent in performance