Analysis-suitable T-splines: characterization, refineability, and approximation

TL;DR

This paper investigates the fundamental properties of analysis-suitable T-splines, including their characterization, refinement, and approximation capabilities, which are crucial for design and analysis in computational geometry.

Contribution

It provides a comprehensive characterization of analysis-suitable T-spline spaces, introduces a theory of local refinement, and establishes approximation results, advancing their application in analysis.

Findings

Analysis-suitable T-spline spaces contain smooth bicubic polynomial spaces.

Conditions for nested T-spline spaces during refinement are identified.

Basic approximation properties of T-splines are established.

Abstract

We establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended T-mesh of an analysis-suitable T-spline, is contained in the corresponding analysis-suitable T-spline space. This is accomplished through the theory of perturbed analysis-suitable T-spline spaces and a simple topological dimension formula. Second, we establish the theory of analysis-suitable local refinement and describe the conditions under which two analysis-suitable T-spline spaces are nested. Last, we demonstrate that these results can be used to establish basic approximation results which are critical for analysis.