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<p>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. The proposed NNR-AD method directly solves 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 enhancement simplifies, improves the efficiency, and modularizes the existing NNR method, enabling it to address the analysis of complex nonlinear dynamic systems. Such improvements facilitate the extension of the NNR method to broader applications in analyzing complex nonlinear dynamic systems. Our work will simplify the numerical implementation of the NNR method for complex nonlinear dynamic problems.</p>

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

  • 期刊论文

  • 发表日期：

    2026年1月30日

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  • Yifan Jiang,
  • Yuhong Jin,
  • Lei Hou,
  • Yi Chen,
  • Shijun Wang,
  • Andong Cong,
  • Qingye Meng

摘要

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. The proposed NNR-AD method directly solves 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 enhancement simplifies, improves the efficiency, and modularizes the existing NNR method, enabling it to address the analysis of complex nonlinear dynamic systems. Such improvements facilitate the extension of the NNR method to broader applications in analyzing complex nonlinear dynamic systems. Our work will simplify the numerical implementation of the NNR method for complex nonlinear dynamic problems.