Supplementary procedures for the solution of stability problems using Krylov space methods

Ottmar Klaas, Rainer Niekamp, Erwin Stein University Hannover, Germany

Abstract

Classical implementations of stability computations are based on arc length methods or displacement controlled methods for solving the nonlinear load-displacement path. For each load step a Newton method is used to solve the nonlinear system of equation. In each Newton step a direct solver gives the solution of the linearized system of equations. Stability points on the load-displacement path can be found by checking the diagonal element of the decomposed matrix. If one diagonal element is zero a stability point has been found.

While solving the linearized system of equations with a Lanczos solver the information about zero or negative diagonal elements of the decomposed matrix is not available. Therefore the Krylov subspace, spanned by the Krylov vectors is investigated. As long as the preconditioning matrix is positiv definit this method gives the desired information. In the case of a not positiv definit preconditioning matrix (using a multi level method with a coarse grid solver for preconditioning, when the coarse grid has a negative eigenvalue) the Lanczos iteration has to be performed with complex numbers. It can be proved that the number of operations needed in each iteration step of the Lanczos method is the same compared to the preconditioning with a positiv definit matrix, since the scalars and vectors are either pure real or pure imaginary.

Examples for a stability analysis for a beam and a plate are shown to present the developed algorithms.