We use the refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems $({Ku=\lambda Mu})$. The rIGA framework conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) discretizations while reducing the computation cost of the solution through partitioning the computational domain by adding zero-continuity basis functions. As a result, rIGA enriches the approximation space and decreases the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval ${[\lambda_s,\lambda_e]}$ are of interest, we select several shifts ${\sigma_k\in[\lambda_s,\lambda_e]}$ using a spectrum slicing technique. For each shift $\sigma_k$, the cost of factorization of the spectral transformation matrix ${K-\sigma_k M}$ drives the total computational cost of the eigensolution. Several multiplications of the operator matrices ${(K-\sigma_k M)^{-1} M}$ by vectors follow this factorization. Let $p$ be the polynomial degree of basis functions and assume that IGA has maximum continuity of ${p-1}$, while rIGA introduces $C^0$ separators to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to ${O(p^2)}$ in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderately-sized eigenproblems, the total computational cost reduction is $O(p)$. Nevertheless, rIGA improves the accuracy of every eigenpair of the first $N_0$ eigenvalues and eigenfunctions. Here, we allow $N_0$ to be as large as the total number of eigenmodes of the original maximum-continuity IGA discretization.

中文翻译：

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广义 Hermitian 特征问题的精细等几何分析

我们使用精细等几何分析 (rIGA) 来解决广义 Hermitian 特征问题 $({Ku=\lambda Mu})$。rIGA 框架保留了最大连续性等几何分析 (IGA) 离散化的理想特性，同时通过添加零连续性基函数来划分计算域，从而降低了解决方案的计算成本。因此，rIGA 丰富了近似空间并减少了自由度之间的互连。我们比较了 rIGA 与 IGA 的计算成本，在使用 Lanczos 特征求解器和平移和反转光谱变换时。当给定区间 ${[\lambda_s,\lambda_e]}$ 内的所有特征对都感兴趣时，我们使用频谱切片技术选择几个位移 ${\sigma_k\in[\lambda_s,\lambda_e]}$。对于每个班次 $\sigma_k$，谱变换矩阵 ${K-\sigma_k M}$ 的分解成本决定了特征解的总计算成本。运算符矩阵 ${(K-\sigma_k M)^{-1} M}$ 与向量的几次乘法遵循这种分解。设 $p$ 是基函数的多项式次数，并假设 IGA 具有 ${p-1}$ 的最大连续性，而 rIGA 引入了 $C^0$ 分隔符以最小化因式分解成本。对于这种设置，我们的理论估计可以预测计算节省，以在渐近机制中计算固定数量的高达 ${O(p^2)}$ 的特征对，即大问题。然而，我们的数值测试表明，对于中等大小的特征问题，总的计算成本降低是 $O(p)$。尽管如此，rIGA 提高了前 $N_0$ 个特征值和特征函数的每个特征对的准确性。这里，