Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates

• Published in: International journal for numerical methods in engineering Vol. 122; no. 9; pp. 2117 - 2144
• Main Authors: Wei, Xiaodong, Li, Xin, Zhang, Yongjie J., Hughes, Thomas J. R.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.05.2021; Wiley Subscription Services, Inc
• Subjects: Apexes; Basis functions; Convergence; extraordinary vertex; isogeometric analysis; nonuniform subdivision; optimal convergence rates; Parameterization
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.6608

This article presents an enhanced version of our previous work, hybrid nonuniform subdivision (HNUS) surfaces, to achieve optimal convergence rates in isogeometric analysis (IGA). We introduce a parameter λ (14<λ<1) to control the rate of shrinkage of irregular regions, so the method is called tuned hybrid nonuniform subdivision (tHNUS). Thus, HUNS is a special case of tHNUS when λ=12. While introducing λ in hybrid subdivision significantly complicates the theoretical proof of G1 continuity around extraordinary vertices, reducing λ can recover optimal convergence rates when tHNUS functions are used as a basis in IGA. From the geometric point of view, tHNUS retains comparable shape quality as HNUS under nonuniform parameterization. Its basis functions are refinable and the geometric mapping stays invariant during refinement. Moreover, we prove that a tHNUS surface is globally G1‐continuous. From the analysis point of view, tHNUS basis functions form a nonnegative partition of unity, are globally linearly independent, and their spline spaces are nested. In the end, we numerically demonstrate that tHNUS basis functions can achieve optimal convergence rates for the Poisson's problem with nonuniform parameterization around extraordinary vertices.