Weighted isogeometric collocation based on subdivision surfaces
摘要
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 isogeometric 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 twice around extraordinary points, with a face centroid placed within each subpatch. Using the weighted isogeometric collocation method, stable collocation solutions can be achieved. A tuned Loop subdivision scheme is proposed here to enhance subdivision surface quality while maintaining favorable convergence properties. Numerical experiments demonstrate that the proposed isogeometric collocation method with tuned Loop subdivision achieves the same convergence order as in regular regions, whereas Loop subdivision fails to achieve it.