On the shift-invert Lanczos method for the buckling eigenvalue problem
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We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem Kx = λ K_Gx, where K is symmetric positive semi-definite, K_G is symmetric indefinite, and the pencil K - λ 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 - σ 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 - λ 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.
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