广义Hermitian特征值问题的精化等几何分析
Refined isogeometric analysis for generalized Hermitian eigenproblems
摘要：
我们使用精化等几何分析（RIGA）来求解广义厄米特特征值问题$({Ku=λMu})$ . RIGA框架保留了最大连续性等几何分析（IGA）离散化的理想属性，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，减少了自由度之间的互连。我们比较的计算成本的RIGA与IGA时，采用Lanczos本征解与移位和反转谱变换。当给定间隔${[λ_s,λ_e]}$内的所有本征对都感兴趣时，我们使用频谱切片技术选择几个移位${σ_k\in[λ_s,λ_e]}$。对于每个移位$σ_k$，谱变换矩阵${K-σ_k M}$的因式分解的成本驱动本征解的总计算成本。在这种因式分解之后，运算符矩阵${(K-σ_k M)^{-1} M}$与向量的几次乘法。令$p$为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$C^0$分离器以最小化因子分解成本。对于这种设置，我们的理论估计预测计算节省计算一个固定数量的特征对高达${O(p^2)}$在渐近制度，即大的问题大小。然而，我们的数值试验表明，对于中等规模的特征值问题，总的计算成本减少是$O(p)$。然而，rIGA提高了第一$N_0$特征值和特征函数的每个特征对的精度。在这里，我们允许$N_0$是一样大的原始最大连续性IGA离散化的本征模的总数。
英文摘要：
We use the refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems $({Ku=λMu})$. 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 ${[λ_s,λ_e]}$ are of interest, we select several shifts ${σ_k\in[λ_s,λ_e]}$ using a spectrum slicing technique. For each shift $σ_k$, the cost of factorization of the spectral transformation matrix ${K-σ_k M}$ drives the total computational cost of the eigensolution. Several multiplications of the operator matrices ${(K-σ_k M)^{-1} M}$ 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.
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