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Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates
Author(s) -
Wei Xiaodong,
Li Xin,
Zhang Yongjie J.,
Hughes Thomas J. R.
Publication year - 2021
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6608
Subject(s) - subdivision , mathematics , subdivision surface , isogeometric analysis , basis function , partition (number theory) , rate of convergence , basis (linear algebra) , partition of unity , convergence (economics) , invariant (physics) , surface (topology) , mathematical analysis , geometry , combinatorics , computer science , finite element method , polygon mesh , economics , economic growth , computer network , channel (broadcasting) , physics , archaeology , mathematical physics , history , thermodynamics
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 (1 4 < 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 = 1 2. While introducing λ in hybrid subdivision significantly complicates the theoretical proof of G 1 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 G 1 ‐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.