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 $\lambda$ ($\frac{1}{4}<\lambda<1$) 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 $\lambda=\frac{1}{2}$. While introducing $\lambda$ in hybrid subdivision significantly complicates the theoretical proof of $G^1$ continuity around extraordinary vertices, reducing $\lambda$ 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 $G^1$-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.

中文翻译：

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具有最佳收敛率的调谐混合非均匀细分曲面

本文提出了我们之前工作的增强版本，即混合非均匀细分曲面 [19]，以在等几何分析中实现最佳收敛速度。我们引入一个参数$\lambda$ ($\frac{1}{4}<\lambda<1$) 来控制不规则区域的收缩率，因此该方法称为调谐混合非均匀细分（tHNUS）。我们之前的工作对应于 $\lambda=\frac{1}{2}$ 的情况。虽然在混合细分中引入 $\lambda$ 使关于非凡顶点的 $G^1$ 连续性的理论证明显着复杂化，但当调整混合细分函数用作等几何分析的基础时，减少 $\lambda$ 可以恢复最佳收敛速度。从几何的角度来看，tHNUS 在非均匀参数化下保留了与 [19] 相当的形状质量。它的基函数是可细化的，并且几何映射在细化过程中保持不变。此外，我们证明了经过调整的混合细分曲面在全局范围内是 $G^1$-连续的。从分析的角度来看，tHNUS 基函数形成了一个非负的统一划分，全局线性无关，它们的样条空间是嵌套的。我们在数值上证明了 tHNUS 基函数可以通过非均匀参数化围绕异常顶点实现泊松问题的最佳收敛速度。并且它们的样条空间是嵌套的。我们在数值上证明了 tHNUS 基函数可以通过非均匀参数化围绕异常顶点实现泊松问题的最佳收敛速度。并且它们的样条空间是嵌套的。我们在数值上证明了 tHNUS 基函数可以通过非均匀参数化围绕异常顶点实现泊松问题的最佳收敛速度。