MATHICSE Technical Report: THU-Splines - Highly Localized Refinement on Smooth Unstructured Splines

We present a novel method named truncated hierarchical unstructured splines (THU-splines) that supports both local h-refinement and unstructured quadrilateral meshes. In a THU-spline construction, an unstructured quadrilateral mesh is taken as the input control mesh, where the degenerated-patch method is adopted in irregular regions to define C1-continuous bicubic splines, whereas regular regions only involve C2 B-splines. Irregular regions are then smoothly joined with regular regions through the truncation mechanism, leading to a globally smooth spline construction. Subsequently, local refinement is performed following the truncated hierarchical B-spline construction to achieve a flexible refinement without propagating to unanticipated regions. Challenges lie in refining transition regions where a mixed types of splines play a role. THU-spline basis functions are globally C1-continuous and are non-negative everywhere except near extraordinary vertices, where slight negativity is inevitable to retain refinability of the spline functions defined using the degenerated-patch method. THU-splines are piecewise polynomials that form a partition of unity. Such functions also have a finite representation that can be easily integrated with existing finite element or isogeometric codes through Bézier extraction.