A Lanczos-based method for structural dynamic reanalysis problems

In this paper, a new solution method for the modified eigenvalue problem with specific application to structural dynamic reanalysis is presented. The method, which is based on the block Lanczos algorithm, is developed for multiple low rank modifications to a system and calculates a few selected eigenpairs. Given the solution to the original system Ax = λx, procedures are developed for the modified standard eigenvalue Problem (A + ΔA)x? = λ x?, where

• 1 ΔA = Σ_jBS_jB^T, where S_j = S ∈ ?^{p × p}, p ? n and B ∈ ?^{n × p} is constant for all the perturbations S_j.
• 2 ΔA = Σ_iΣ_j B_iS_jB_i^T, where B_i ∈ ?^{n × p} may vary with the pertubations S_j.

The procedures are then extended for the reciprocal and generalized eigenvalue problems so that they are directly applicable to the structural dynamic reanalysis problem. Numerical examples are given to demonstrate the applications of the method.