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On the shift‐invert Lanczos method for the buckling eigenvalue problem
International Journal for Numerical Methods in Engineering ( IF 2.9 ) Pub Date : 2021-02-01 , DOI: 10.1002/nme.6640
Chao‐Ping Lin 1 , Huiqing Xie 2 , Roger Grimes 3 , Zhaojun Bai 1, 4
Affiliation  

We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem K x = λ K G x , where K is symmetric positive semi‐definite, KG is symmetric indefinite, and the pencil K λ K G is singular, namely, K and KG share a nontrivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace of K and the common nullspace of K and KG are available. There are two open issues for developing an industrial strength shift‐invert Lanczos method: (1) the shift‐invert operator ( K σ K 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 toward the nullspace of K, which leads to a rapid growth of the Lanczos vectors in norms and causes 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 λ K 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.

中文翻译:

关于屈曲特征值问题的平移Lanczos方法

我们考虑提取屈曲特征值问题的几个所需特征对的问题 ķ X = λ ķ G X ,其中K是对称正半定数,K G是对称不定数,并且铅笔 ķ - λ ķ G 是单数,即KK G共享一个非平凡的公共零空间。此外,在结构的实际屈曲分析中,可以使用K的零空间以及KK G的公共零空间的基础。开发工业强度平移-反转Lanczos方法有两个未解决的问题:(1)平移-反转算子 ķ - σ ķ G - 1个 不存在或状况极差,并且(2)使用由K诱导的半内积使Lanczos向量迅速朝向K的零空间,这导致Lanczos向量在范数上迅速增长,并导致信息的永久丢失和方法的失败。在本文中,我们通过提出奇异铅笔的广义屈曲谱变换来解决这两个问题 ķ - λ ķ G 通过对K的半正定性进行低阶更新来对内积进行正则化。数值示例(包括来自工业屈曲分析的示例)证明了我们方法的有效性。
更新日期:2021-02-01
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