Basis construction for polynomial spline spaces over arbitrary T-meshes

TL;DR

This paper introduces a novel method for constructing basis functions for polynomial spline spaces over arbitrary T-meshes, ensuring stability, linear independence, and broad applicability.

Contribution

It presents the first general approach to basis construction for PT-splines on any T-mesh, including a technique to eliminate redundant edges and improve stability.

Findings

Constructed basis functions are proven to be linearly independent and complete.

The PT-spline basis is more versatile than LR B-splines across various T-meshes.

PT-splines outperform HB-splines in certain hierarchical T-mesh scenarios.

Abstract

This paper presents the first method for constructing bases for polynomial spline spaces over an arbitrary T-meshes (PT-splines for short). We construct spline basis functions for an arbitrary T-mesh by first converting the T-mesh into a diagonalizable one via edge extension, ensuring a stable dimension of the spline space. Basis functions over the diagoalizable T-mesh are constructed according to the three components in the dimension formula corresponding to cross-cuts, rays, and T $l$-edges in the diagonalizable T-mesh, and each component is assigned some local tensor product B-splines as the basis functions. We prove this set of functions constitutes a basis for the diagonalizable T-mesh. To remove redundant edges from extension, we introduce a technique, termed Extended Edge Elimination (EEE) to construct a basis for an arbitrary T-mesh while reducing structural constraints and unnecessary refinements. The resulting PT-spline basis ensures linear independence and completeness, supported by a dedicated construction algorithm. A comparison with LR B-splines, which may lack linear independence and are limited to LR-meshes, highlights the PT-spline's versatility across any T-mesh. Examples are also provided to demonstrate that dimensional instability in spline spaces is related with basis function degradation and that PT-splines are advantageous over HB-splines for certain hierarchical T-meshes.