Analysis-suitable unstructured T-splines: Multiple extraordinary points per face

TL;DR

This paper extends analysis-suitable T-splines to include multiple extraordinary points per face, enhancing geometric flexibility and maintaining high continuity, with demonstrated optimal convergence and practical application in automotive design.

Contribution

It introduces a generalized AST-spline framework with multiple extraordinary points per face, ensuring linear independence and partition of unity, and demonstrates improved efficiency over finite elements.

Findings

AST-splines with multiple extraordinary points achieve optimal convergence.

They require fewer degrees of freedom than finite element methods.

The framework is practically applied to automotive component design.

Abstract

Analysis-suitable T-splines (AST-splines) are a promising candidate to achieve a seamless integration between the design and the analysis of thin-walled structures in industrial settings. In this work, we generalize AST-splines to allow multiple extraordinary points within the same face. This generalization drastically increases the flexibility to build geometries using AST-splines; e.g., much coarser meshes can be generated to represent a certain geometry. The AST-spline spaces detailed in this work have $C^1$ inter-element continuity near extraordinary points and $C^2$ inter-element continuity elsewhere. We mathematically show that AST-splines with multiple extraordinary points per face are linearly independent and their polynomial basis functions form a non-negative partition of unity. We numerically show that AST-splines with multiple extraordinary points per face lead to optimal convergence rates for second- and fourth-order linear elliptic problems. To illustrate a possible isogeometric framework that is already available, we design the B-pillar and the side outer panel of a car using T-splines with the commercial software Autodesk Fusion360, import the control nets into our in-house code to build AST-splines, and import the B\'ezier extraction information into the commercial software LS-DYNA to solve eigenvalue problems. The results are compared with conventional finite elements. Good agreement is found, but conventional finite elements require significantly more degrees of freedom to reach a converged solution than AST-splines.