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On the shift‐invert Lanczos method for the buckling eigenvalue problem
Author(s) -
Lin ChaoPing,
Xie Huiqing,
Grimes Roger,
Bai Zhaojun
Publication year - 2021
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6640
Subject(s) - lanczos resampling , eigenvalues and eigenvectors , mathematics , buckling , pencil (optics) , lanczos algorithm , matrix pencil , regularization (linguistics) , mathematical analysis , computer science , structural engineering , physics , quantum mechanics , engineering , artificial intelligence
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, K G is symmetric indefinite, and the pencil K − λ K Gis singular, namely, K and K G 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 K G 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 ) − 1does 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 Gand 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.