关于屈曲本征值问题的平移-反演Lanczos方法
On the shift-invert Lanczos method for the buckling eigenvalue problem
摘要：
考虑屈曲本征值问题$Kx = λK_Gx$的特征对提取问题，其中$K$是对称半正定的，$K_G$是对称不定的，束$K - λK_G$是奇异的，即$K$和$K_G$共享一个非平凡的公共零空间.此外，在实际结构的屈曲分析中，$K$的零空间以及$K$和$K_G$的公共零空间的基础是可用的。开发工业强度平移-反演Lanczos方法存在两个未决问题：（1）移位-反转算子$(K - σK_G)^{-1}$不存在或者是极端病态的，以及（2）由$K$引起的半内积的使用将Lanczos向量快速地驱动到$K$的零空间，这导致Lanczos向量在范数上的快速增长，并导致信息的永久信息的损失和方法的失败。在本文中，我们解决这两个问题，提出了广义屈曲谱变换的奇异束$K - λK_G$和正则化的内积通过低秩更新的半正定性的$K$。我们的方法的有效性证明了数值例子，包括一个工业屈曲分析。
英文摘要：
We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem $Kx = λK_Gx$, where $K$ is symmetric positive semi-definite, $K_G$ is symmetric indefinite, and the pencil $K - λK_G$ is singular, namely, $K$ and $K_G$ share a non-trivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace of $K$ and the common nullspace of $K$ and $K_G$ are available. There are two open issues for developing an industrial strength shift-invert Lanczos method: (1) the shift-invert operator $(K - σK_G)^{-1}$ does not exist or is extremely ill-conditioned, and (2) the use of the semi-inner product induced by $K$ drives the Lanczos vectors rapidly towards the nullspace of $K$, which leads to a rapid growth of the Lanczos vectors in norms and cause permanent loss of information and the failure of the method. In this paper, we address these two issues by proposing a generalized buckling spectral transformation of the singular pencil $K - λK_G$ and a regularization of the inner product via a low-rank updating of the semi-positive definiteness of $K$. The efficacy of our approach is demonstrated by numerical examples, including one from industrial buckling analysis.
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