Safeguarded Use of the Implicit Restarted Lanczos Technique for Solving Nonlinear Structural Eigensystems

This paper presents a new algorithms for evaluating the eigenvalues and their corresponding eigenvectors for large scale nonlinear eigensystems in structural dynamics. The algorithm is based on solving a sequence of algebraic eigenproblems and updating the parameter, lambda. The Implicitly Restarted Lanczos method has been determined to be well suited for solving the linear eigenproblems that arise in this context. A zero-finder approach that uses rational interpolation to approximate the generalized eigenvalues has been developed to update lambda. The methodology of the new algorithm developed here is designed to evaluate a subset of the parameterized nonlinear eigencurves at specific values of lambda. Numerical experiments show that the new eigensolution technique is superior to the pre-existing approaches for the large scale problems and competitive for the small ones. The main emphasis of this contribution is the derivation and analysis of this scheme for eigensystems that are based on the frequency dependent shape functions