On the shift-invert Lanczos method for the buckling eigenvalue problem

TL;DR

This paper develops a robust shift-invert Lanczos method for buckling eigenvalue problems with singular pencils, addressing ill-conditioning and nullspace issues through spectral transformation and regularization, with demonstrated industrial applications.

Contribution

It introduces a generalized spectral transformation and inner product regularization to improve the shift-invert Lanczos method for singular buckling eigenproblems.

Findings

Effective in handling singular pencils in buckling analysis

Numerical examples validate the method's robustness

Applicable to industrial buckling problems

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.