Numerical characteristics of constant average acceleration method in solution of nonlinear systems

Abstract

The currently available evaluation technique for the characteristics of an integration method is usually constructed from a whole step‐by‐step integration procedure for linear elastic systems. In this paper, a technique based on linearized analysis is proposed to evaluate the numerical properties of a step‐by‐step integration method for each time step for both linear and nonlinear systems. For this purpose, two parameters are introduced for each time step. One is used to describe the degree of nonlinearity (DON) and the other is employed to monitor the degree of convergence (DOC) as an iteration procedure is used. It seems that the proposed evaluation technique can reliably predict the numerical properties of an integration method in the solution of nonlinear systems. Furthermore, a guideline to have accurate solutions of nonlinear systems is proposed and confirmed with numerical examples. Although the evaluation technique is conducted on a time step, it is still indicative for a complete integration procedure since this procedure consists of each time step. It is verified that the constant average acceleration method (CAAM) is spectrally stable for nonlinear systems in a whole integration procedure based on a linearized stability analysis.