Combination of optimal three-step composite time integration method with multi-point iterative methods for geometric nonlinear structural dynamics

Abstract

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.