Toric Parameterization Based Isogeometric Collocation Method for Planar Multi-Sided Physical Domains

Isogeometric collocation method (IGC) shows high computational efficiency compared with isogeometric Galerkin method (IGG) when solving partial differential equations (PDEs). However, few studies about IGC have focused on multi-sided physical domains. In this paper, the authors propose a new IGC method based on toric parameterization (IGCT) for the multi-sided planar physical domains. Due to the high order continuity of toric basis functions, the IGCT method shows more accurate numerical approximation. Moreover, the authors generalize the adaptive w-refinement method into IGCT (IGCT-w), in which the weights of basis functions in physical domains are optimized independently for geometry representation. The numerical accuracy of IGCT-w is significantly improved by an order of magnitude in comparison with IGCT method. To save the computational cost of IGCT-w, the authors devise a selection of weights scheme according to relative residuals. Finally, several numerical examples demonstrate the effectiveness and robustness of the proposed method.