Algorithms for the calculation of limit and bifurcation points using Lanczos methods
Abstract
The computation of limit and bifurcation points in structural mechanics using iterative preconditioned Lanczos solvers is studied. Contrary to classical implementations of algorithms for the calculation of limit and bifurcation points, which depend in general strongly on observing the diagonal elements of the decomposed matrix  obtained by a Gauß or Cholesky decomposition  , we use an approach of determining limit and bifurcation points by examination of the subspace spanned by the iteration vectors of the Lanczos solver. Using a multilevel preconditioning with a coarse grid solver may result in a non positive definite preconditioning matrix if the coarse grid matrix is not positive definite in the postcritical solution branch. In that case the iteration has to be performed in the complex vector space. We prove by mathematical induction that all vectors and scalars are either purely real or purely imaginary. Therefore the generalized computation can be performed with about the same number of operations as in the case of a positive definite preconditioning matrix.
 Publication:

Computational Mechanics
 Pub Date:
 1997
 DOI:
 10.1007/s004660050242
 Bibcode:
 1997CompM..20..221K