Investigation of WBZ‐α method for solving nonlinear systems

Abstract

Purpose

It has been verified that the WBZ‐α method of Wood, Bossak and Zienkiewicz can have unconditional stability and numerical dissipation for linear elastic systems. However, it is still unclear about its performance in the solution of nonlinear systems analytically. Hence, this study proposes to analytically investigate its numerical characteristics for solving nonlinear systems.

Design/methodology/approach

Two parameters are introduced to facilitate the basic analysis for nonlinear systems. One is the step degree of nonlinearity, which describes the stiffness change within a time step, and the other is the step degree of convergence, which describes the convergence error due to an iteration procedure.

Findings

It is theoretically proved that the sub‐family of WBZ‐α method of −1≤α<0, β=(1/4)(1−α)² and γ=(1/2)−α is unconditionally stable and has desired numerical dissipation for any nonlinear systems even with the presence of convergence error. These theoretical results are confirmed by numerical examples.

Originality/value

This analytical study reveals that the performance of the WBZ‐α method for nonlinear systems is in general the same as that for linear elastic systems.

Citation

Chang, S. (2007), "Investigation of WBZ‐α method for solving nonlinear systems", Engineering Computations, Vol. 24 No. 4, pp. 384-406. https://doi.org/10.1108/02644400710748706