We use backward error analysis for differential equations to obtain modified or distorted equations describing the behaviour of the Newmark scheme applied to the transient structural dynamics equation. Based on the newly derived distorted equations, we give expressions for the numerically or algorithmically distorted stiffness and damping matrices of a system simulated using the Newmark scheme. Using these results, we show how to construct compensation terms from the original parameters of the system, which improve the performance of Newmark simulations. The required compensation terms turn out to be slight modifications to the original system parameters (e.g. the damping or stiffness matrices), and can be applied without changing the time step or modifying the scheme itself. Two such compensations are given: one eliminates numerical damping, while the other achieves fourth-order accurate calculations using the traditionally second-order Newmark method. The performance of both compensation methods is evaluated numerically to demonstrate their validity, and they are compared to the uncompensated Newmark method, the generalized-$\alpha$ method and the 4th-order Runge--Kutta scheme.

翻译：本文运用微分方程的后向误差分析技术，推导出描述Newmark格式应用于瞬态结构动力学方程时的修正方程或畸变方程。基于新推导的畸变方程，我们给出了采用Newmark格式模拟系统时产生的数值或算法畸变刚度矩阵与阻尼矩阵的表达式。利用这些结果，我们展示了如何根据系统原始参数构造补偿项以提升Newmark模拟的性能。所需的补偿项表现为对原始系统参数（如阻尼矩阵或刚度矩阵）的轻微修正，且无需改变时间步长或修改算法本身即可实施。本文提出两种补偿方案：一种可消除数值阻尼，另一种则能使传统二阶精度的Newmark方法实现四阶精度计算。通过数值实验评估了两种补偿方法的性能以验证其有效性，并与未补偿的Newmark方法、广义-$\alpha$方法及四阶Runge--Kutta格式进行了对比分析。