Safeguarded use of the implicit restarted lanczos technique for solving non‐linear structural eigensystems

Abstract

This paper presents a new algorithm for evaluating the eigenvalues and their corresponding eigenvectors for large-scale non-linear eigensystems in structural dynamics. The algorithm is based on solving a sequence of algebraic eigenproblems and updating the parameter 1. The implicitly restarted Lannos method had 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 1.The methodology of the new algorithm developed here is designed to evaluate a subset of the parametrized non-linear eigencurves at specific values of 1. Numerical experiments show that the new eigensolution technique is superior to existing approaches for large-scale problems and competitive for small-size problems. The main emphasis of this contribution is the derivation and analysis of this scheme for non-linear structural eigensystems.

A projected Hessian Gauss‐Newton algorithm for solving systems of nonlinear equations and inequalities

“…Recently Abdel-Aziz [2] proposed a new method for computing the projection using implicitly restarted Lanczos method which proved to be very successful in solving nonlinear structural eigensystems (see Abdel-Aziz [1]). …”

Bounds for the relative change of singular values of a real matrix

“…This will enhance the efficiency and will make the subproblems suitable for the design of accurate parallel divided-and-conquer algorithms. Deflation techniques are needed in large scale problems of structural eigensystems [7,1].…”