结构动力学中的广义多步显式积分算法

Generalized Multi-step Explicit Integration Method in Structural Dynamics

为了开发新的时间积分算法,通过对独立变量加速度的加权,提出了广义多步显式积分算法(GMEM).首先,在加速度显式法的基础上给出了通用的积分格式;其次,分析了所提算法的稳定性、数值耗散、数值色散和精度;最后,通过2个算例对3个广义多步显式积分算法(GMEM1、GMEM2和GMEM3-2)以及HHT-α法和Newmark法进行了对比分析.分析结果表明:本文所提算法是条件稳定的,在无阻尼系统中谱半径恒等于1;3步广义多步显式法最高具有3阶精度,在无阻尼系统中不存在数值耗散;GMEM2的均方根误差约为Newmark法的1/2,约为GMEM3-2的1.8倍.

In order to develop new time integration algorithms, a generalized multi-step explicit integration method （GMEM） was proposed by means of weighting independent variables, accelerations. Firstly, a general integration format was provided based on the acceleration explicit method. Furthermore, the stability, numerical dissipation, numerical dispersion and accuracy were analyzed. Finally, two numerical examples were employed to contrastively analyze three kinds of GMEMs （GMEM1, GMEM2 and GMEM3-2）, the HHT-a method and the Newmark method. The results indicate that the GMEM is conditionally stable. The spectral radius is identically equal to 1 in the system without damping. The GMEM of three steps can achieve the highest accuracy of three order. There is not numerical dissipation for the GMEM of three steps in undamped systems. The root mean square error of the GMEM2 is approximately half of that of the Newmark method, and 1.8 times that of the GMEM3-2 approximately.