Filtering Frequencies in a Shift-and-invert Lanczos Algorithm for the Dynamic Analysis of Structures

The shift-and-invert Lanczos algorithm is a commonly used solution procedure to compute the eigenpairs of large, sparse eigenvalue problems that arise when approximating the elastic dynamic response of large structures under the influence of seismic forces. Not all eigenvectors are equally important to the response when the structure is exposed to a mass-dependent external force of the form $g(t) Mb$, where $M$ is the mass matrix of the system and $b$ the rigid body vector. Structural engineers select eigenvectors $x_j$, $j=1,\ldots,\ell$, such that their cumulative mass participation, measured as $\sum_{j=1}^\ell (x_j^T Mb)^2/(b^TMb)$,is above a target threshold $\xi$. We show that when the starting vector for the unshifted Lanczos algorithm is the spatial distribution vector $b$, the Lanczos procedure can be used to provide an estimate of the cumulative mass participation. This allows us to identify intervals containing eigenvalues whose eigenvectors have a large contribution to the cumulative mass participation and filter out intervals containing eigenvalues whose eigenvectors have a negligible contribution. We use this information to devise a sequence of shifts $\sigma_1, \ldots,\sigma_p$ for the shift-and-invert Lanczos algorithm as well as a stopping criterion for the iteration with shift $\sigma_i$ so that the cumulative mass participation of the computed eigenvectors reaches the required level $\xi$. Numerical experiments on real engineering problems show that our approach computes up to $70\%$ fewer eigenvectors and requires fewer shifts, on average, than the more general shifting strategy proposed by Ericsson and Ruhe (Math. Comp., 35 (1980)) together with its modification presented in Grimes et al. (SIAM J. Matrix Anal. and Appl. 40(4), 1994)