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

• DOI: 10.1016/j.cma.2021.114494
• Publication Date: 2022-01-06T23:46:12Z
• Authors: Xiaodong Wei; Xin Li; Kuanren Qian; Thomas J.R. Hughes; Yongjie Jessica Z...; Hugo Casquero
• Keywords: FOS: Mathematics; 0211 other engineering and technologies; 0202 electrical engineering, electronic engineering, information engineering; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); 02 engineering and technology

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 Bézier 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.