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Tuned Hybrid Non-Uniform Subdivision Surfaces with Optimal Convergence Rates
This paper presents an enhanced version of our previous work, hybrid
non-uniform subdivision surfaces [19], to achieve optimal convergence rates in
isogeometric analysis. We introduce a parameter
() to control the rate of shrinkage of irregular
regions, so the method is called tuned hybrid non-uniform subdivision (tHNUS).
Our previous work corresponds to the case when . While
introducing in hybrid subdivision significantly complicates the
theoretical proof of continuity around extraordinary vertices, reducing
can recover the optimal convergence rates when tuned hybrid
subdivision functions are used as a basis in isogeometric analysis. From the
geometric point of view, the tHNUS retains comparable shape quality as [19]
under non-uniform parameterization. Its basis functions are refinable and the
geometric mapping stays invariant during refinement. Moreover, we prove that a
tuned hybrid subdivision surface is globally -continuous. From the
analysis point of view, tHNUS basis functions form a non-negative partition of
unity, are globally linearly independent, and their spline spaces are nested.
We numerically demonstrate that tHNUS basis functions can achieve optimal
convergence rates for the Poisson's problem with non-uniform parameterization
around extraordinary vertices