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On the shiftâinvert Lanczos method for the buckling eigenvalue problem
- Source :
- International Journal for Numerical Methods in Engineering. 122:2751-2769
- Publication Year :
- 2021
- Publisher :
- Wiley, 2021.
-
Abstract
- We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem $Kx = \lambda K_Gx$, where $K$ is symmetric positive semi-definite, $K_G$ is symmetric indefinite, and the pencil $K - \lambda K_G$ 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 $(K - \sigma K_G)^{-1}$ 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 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 $K - \lambda K_G$ 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.
Details
- ISSN :
- 10970207 and 00295981
- Volume :
- 122
- Database :
- OpenAIRE
- Journal :
- International Journal for Numerical Methods in Engineering
- Accession number :
- edsair.doi...........91847b89956f7e6e0faf97aac5c84434