Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates

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 lambda (14<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 lambda=12. While introducing lambda in hybrid subdivision significantly complicates the theoretical proof of G(1) continuity around extraordinary vertices, reducing lambda 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.

Automated summary

This article introduces tuned hybrid nonuniform subdivision (tHNUS) surfaces, which achieve optimal convergence rates in isogeometric analysis by controlling the rate of shrinkage. tHNUS retains comparable shape quality as HNUS, with refinable basis functions and global continuity. Through numerical demonstrations, it is shown that tHNUS basis functions can achieve optimal convergence rates for specific problems.