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 quadrilateral meshes. In THU-spline construction, an mesh is taken as the input control mesh, where degenerated-patch [20] adopted in irregular regions to define $$C^1$$ -continuous bicubic splines, whereas regular only involve $$C^2$$ B-splines. Irregular are then smoothly joined with through truncation mechanism [31], leading globally smooth spline construction. Subsequently, refinement performed following B-spline construction [11] achieve flexible without propagating unanticipated regions. Challenges lie refining transition mixed types of play role. basis functions non-negative everywhere except near extraordinary vertices, slight negativity inevitable retain refinability defined using method. Such also have finite representation can be easily integrated existing element or isogeometric codes Bézier extraction.