On the shift‐invert Lanczos method for the buckling eigenvalue problem
Chao‐Ping Lin, Huiqing Xie, Roger G. Grimes, Zhaojun Bai · International Journal for Numerical Methods in Engineering · 2021
Abstract We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem , where K is symmetric positive semi‐definite, KG is symmetric indefinite, and the pencil 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 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 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.