A modified Newmark/Newton-Raphson method with automatic differentiation for general nonlinear dynamics analysis

Abstract

The Newmark/Newton-Raphson (NNR) method is widely employed for solving nonlinear dynamic systems. However, the current NNR method exhibits limited applicability in complex nonlinear dynamic systems, as the acquisition of the Jacobian matrix required for Newton iterations incurs substantial computational costs and may even prove intractable in certain cases. To address these limitations, we integrate automatic differentiation (AD) into the NNR method, proposing a modified NNR method with AD (NNR-AD) to significantly improve its capability for effectively handling complex nonlinear systems. We have demonstrated that the NNR-AD method can directly solve dynamic systems with complex nonlinear characteristics, and its accuracy and generality have been rigorously validated. Furthermore, automatic differentiation significantly simplifies the computation of Jacobian matrices for such complex nonlinear dynamic systems. This improvement endows the NNR method with enhanced modularity, thereby enabling convenient and effective solutions for complex nonlinear dynamic systems.