Adaptive Refinement for Unstructured T-Splines with Linear Complexity

TL;DR

This paper introduces an adaptive refinement algorithm for unstructured 2D T-spline meshes that maintains key mathematical properties and achieves linear complexity, enabling efficient and flexible mesh refinement.

Contribution

It proposes a novel refinement scheme using direction indices and edge subdivision, extending structured T-spline refinement to unstructured meshes with linear complexity.

Findings

Preserves global linear independence and analysis-suitability.

Ensures sparsity of the system matrix.

Achieves linear complexity in mesh refinement operations.

Abstract

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.