Treffer: High-Order Surface Reconstruction and its Applications to Surface Integrals and Surface Remeshing

High-order surface reconstruction is a numerical technique to obtain high-order approximations of both geometry and its differential quantities such as normals, curvatures, etc. | over a discrete surface mesh. Its computational framework is based on local polynomial fittings using a weighted least squares approach. In this dissertation, we complete the scope of this framework to compute high-order approximations of surface integrals and demonstrate the application of the complete framework to various mesh-based numerical computations for high-order numerical methods. The computational framework relies on an efficient underlying mesh data structure for various traversal queries. For this purpose, an array-based mesh data structure was developed to represent the mesh for efficient mesh query and modification operations. Our methods are mainly developed for applications with high-order methods in mind. Surface integration is a fundamental operation in many scientific and engineering applications. The standard methods for numerical computation are generally limited to second-order of accuracy due to lower-order approximations to geometry and integrand. This limitation is overcome by extending the computational framework for high-order surface reconstruction to a function defined over the surface and coupling it with high-order quadrature rules. We theoretically analyze the accuracy of our method and prove that it can achieve high-order of accuracy and verify it with numerical experiments as well. A widely used operation by many applications is the modification of the surface mesh by vertex redistribution, edge flipping, refinement or coarsening, such that the resulting mesh improves certain properties such as mesh quality, error distribution, etc. It is vital to preserve the geometric accuracy of the mesh as it undergoes the modification operations. Our computational framework provides an efficient high-order point projection strategy that can be easily coupled with various mesh quality improving techniques. We ...