Certain applications that analyze damping effects require the solution of quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis (rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving desirable properties of maximum-continuity isogeometric analysis (IGA), reduces the interconnection between degrees of freedom by adding low-continuity basis functions. This connectivity reduction in rIGA’s algebraic system results in faster matrix LU factorizations when using multifrontal direct solvers. We compare computational costs of rIGA versus those of IGA when employing Krylov eigensolvers to solve quadratic eigenproblems arising in 2D vector-valued multifield problems. For large problem sizes, the eigencomputation cost is governed by the cost of LU factorization, followed by costs of several matrix–vector and vector–vector multiplications, which correspond to Krylov projections. We minimize the computational cost by introducing $C^0$ and $C^1$ separators at specific element interfaces for our rIGA generalizations of the curl-conforming Nédélec and divergence-conforming Raviart–Thomas finite elements. Let $p$ be the polynomial degree of basis functions; the LU factorization is up to $\mathcal{O}\left({(p-1)}^2\right)$ times faster when using rIGA compared to IGA in the asymptotic regime. Thus, rIGA theoretically improves the total eigencomputation cost by $\mathcal{O}\left({(p-1)}^2\right)$ for sufficiently large problem sizes. Yet, in practical cases of moderate-size eigenproblems, the improvement rate deteriorates as the number of computed eigenvalues increases because of multiple matrix–vector and vector–vector operations. Our numerical tests show that rIGA accelerates the solution of quadratic eigensystems by $\mathcal{O}(p-1)$ for moderately sized problems when we seek to compute a reasonable number of eigenvalues.

某些分析阻尼效应的应用需要求解二次特征值问题(QEP)。我们使用精细等几何分析 (rIGA) 来解决二次特征问题。rIGA 离散化，同时保持最大连续性等几何分析 (IGA) 的理想特性，通过添加低连续性基函数来减少自由度之间的互连。rIGA 代数系统中的这种连通性减少导致在使用多前沿直接求解器时更快的矩阵 LU 分解。我们比较了 rIGA 与 IGA 在使用 Krylov 特征求解器解决二维向量值多场问题中出现的二次特征问题时的计算成本。对于大型问题，特征计算成本由 LU 分解的成本控制，然后是几个矩阵-向量和向量-向量乘法的成本，它们对应于 Krylov 投影。我们通过引入最小化计算成本$C^0$和$C^1$特定元素界面处的分隔符，用于我们的 rIGA 推广符合 curl 的 Nédélec 和符合发散的 Raviart-Thomas 有限元。让$p$是基函数的多项式次数；LU 分解是$\mathcal{○}\left({(p-1)}^2\right)$与渐近状态下的 IGA 相比，使用 rIGA 时快几倍。因此，rIGA 理论上将总特征计算成本提高了$\mathcal{○}\left({(p-1)}^2\right)$对于足够大的问题大小。然而，在中等规模特征问题的实际情况下，由于多个矩阵-向量和向量-向量运算，随着计算的特征值数量的增加，改进率会下降。我们的数值测试表明，rIGA 通过以下方式加速了二次特征系统的求解$\mathcal{○}(p-1)$当我们寻求计算合理数量的特征值时，对于中等规模的问题。