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

We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem Kx=lambda KGx, where K is symmetric positive semi-definite, K-G is symmetric indefinite, and the pencil K-lambda KG is 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-sigma KG)-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-lambda KG 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.