We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems $(\mathbf{K}\mathbf{u}=\lambda \mathbf{M}\mathbf{u})$. rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces 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 factorization cost of the spectral transformation matrix $\mathbf{K}-{\sigma }_k\mathbf{M}$ controls the total computational cost of the eigensolution. Several multiplications of the operator matrix ${(\mathbf{K}-{\sigma }_k\mathbf{M})}^{-1}\mathbf{M}$ by vectors follow this factorization. Let $p$ be the polynomial degree of the basis functions and assume that IGA has maximum continuity of $p-1$. When using rIGA, we introduce $C^0$ separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to $\mathcal{O}\left(p^2\right)$ in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is $\mathcal{O}\left(p\right)$. In addition, rIGA improves the accuracy of every eigenpair of the first $N_0$ eigenvalues and eigenfunctions, where $N_0$ is the total number of modes of the original maximum-continuity IGA discretization.

我们使用精细的等几何分析（rIGA）解决广义的Hermitian特征问题 $（\mathbf{ķ}\mathbf{ü}=\lambda \mathbf{中号}\mathbf{ü}）$。rIGA保留了最大连续性等几何分析（IGA）的理想属性，同时它通过添加零连续性基函数来降低解决方案成本，从而降低了矩阵连通性。结果，rIGA丰富了近似空间并减少了自由度之间的相互联系。当使用带移位和反转频谱变换的Lanczos本征求解器时，我们比较了rIGA和IGA的计算成本。在给定间隔内所有本征对$[{\lambda }_s，{\lambda }_{Ë}]$ 很有意思，我们选择几个班次 ${\sigma }_{ķ}\in [{\lambda }_s，{\lambda }_{Ë}]$使用频谱切片技术。对于每个班次${\sigma }_{ķ}$，频谱变换矩阵的分解成本 $\mathbf{ķ}-{\sigma }_{ķ}\mathbf{中号}$控制本征解的总计算成本。运算符矩阵的几个乘法${（\mathbf{ķ}-{\sigma }_{ķ}\mathbf{中号}）}^{-\mathrm{1个}}\mathbf{中号}$向量的乘积遵循此因式分解。让$p$ 是基函数的多项式度，并假定IGA具有的最大连续性 $p-\mathrm{1个}$。在使用rIGA时，我们介绍$C^0$某些元素接口处的分隔符可最大程度地减少分解成本。对于这种设置，我们的理论估算值可以预测出可节省的计算量，以计算固定数量的特征对$\mathcal{Ø}（p^{\mathrm{2个}}）$在渐进状态下，即问题规模大。然而，我们的数值测试表明，对于中等大小的特征问题，观察到的总计算成本降低为$\mathcal{Ø}（p）$。此外，rIGA提高了第一个特征对的每个特征对的准确性${ñ}_0$ 特征值和特征函数，其中 ${ñ}_0$ 是原始最大连续性IGA离散化的模式总数。