THU-Splines: Highly Localized Refinement on Smooth Unstructured Splines

TL;DR

This paper introduces THU-splines, a new local refinement method for unstructured quadrilateral meshes that maintains smoothness and compatibility with finite element and isogeometric analysis.

Contribution

The paper proposes a novel THU-spline construction combining irregular and regular regions with local refinement capabilities on unstructured meshes.

Findings

Supports both local $h$-refinement and unstructured quadrilateral meshes.

Ensures global $C^1$ continuity with smooth joins between regions.

Compatible with finite element and isogeometric analysis through Bézier extraction.

Abstract

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 [18] is adopted in irregular regions to define $C^1$-continuous bicubic splines, whereas regular regions only involve $C^2$ B-splines. Irregular regions are then smoothly joined with regular regions through the truncation mechanism [29], leading to a globally smooth spline construction. Subsequently, local refinement is performed following the truncated hierarchical B-spline construction [10] 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 $C^1$-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. Such functions also have a finite representation that can be easily integrated with existing finite element or isogeometric codes through B\'{e}zier extraction.