On Isogeometric Subdivision Methods for PDEs on Surfaces
For researchers in geometric modeling and computational PDEs, this work provides a practical quadrature strategy that improves robustness near irregular vertices, though it is an incremental improvement over existing isogeometric subdivision methods.
This paper presents an isogeometric discretization approach using subdivision surfaces for solving PDEs on surfaces, focusing on Loop's subdivision scheme. Through numerical experiments, a mid-edge quadrature scheme is identified as the most robust and efficient choice for handling extraordinary vertices.
Subdivision surfaces are proven to be a powerful tool in geometric modeling and computer graphics, due to the great flexibility they offer in capturing irregular topologies. This paper discusses the robust and efficient implementation of an isogeometric discretization approach to partial differential equations on surfaces using subdivision methodology. Elliptic equations with the Laplace-Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problems are considered. Thereby, efficiency relies on the proper choice of a numerical quadrature scheme which preserves the expected higher order consistency. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. In this paper, the focus is on Loop's subdivision scheme on triangular meshes. Based on a series of numerical experiments, different quadrature schemes are compared and a mid-edge quadrature, which is easy-to-implement via lookup tables, turns out to be a preferable choice due to its robustness and efficiency.