Weighted Isogeometric Collocation based on Subdivision Surfaces

This paper explores the isogeometric collocation method based on subdivision surfaces. Subdivision basis functions offer high-order continuity within elements and smoothness across the entire domain, making them well-suited for isogeo-metric collocation in complex domains. The challenge in isogeometric collocation with subdivision surfaces lies in selecting an appropriate set of collocation points on irregular patches and imposing boundary conditions correctly to ensure consistency in computational accuracy with regular regions. This paper employs the weighted isogeometric collocation method by integrating boundary conditions and interior PDE equations in a weighted manner. Specifically, weights are assigned to each equation corresponding to each collocation point. For regular patches, face centroids are utilized, while each irregular patch is subdivided n times around extraordinary points, with a face centroid positioned within each subpatch. Using the weighted isogeometric collocation method, stable collocation solutions can be achieved. Numerical experiments demonstrate that the proposed isogeometric collocation method with modified Loop subdivision achieves the same convergence order as observed in regular regions, whereas the Loop subdivision fails to achieve it.