基于计算扰动修正的Wilson-θ法稳定性和精度分析

Stability and accuracy analyses of a modified Wilson-θ method based on calculation perturbation

Wilson-θ法作为一种隐式时域逐步积分方法,由于在迭代过程中不满足下一时刻的动力平衡方程,应用时会存在振幅衰减以及周期延长等问题,使得它在一些情况下难以得到准确解.针对该问题,本文基于计算扰动的概念,提出了一种改进的Wilson-θ法(Wilson-Newmark法),引入计算扰动的概念来量化Wilson-θ法中每一步中存在的计算误差,并利用Newmark-β的计算方法对得到的计算扰动进行再分配,使得改进后的计算方法可以同时满足动力学平衡方程和运动约束.为了考察Wilson-Newmark方法的适用性,对不同θ取值下该算法的稳定性进行了分析.此外,还从振幅衰减、周期延长以及超调3个方面对比了算法的精度.结合数值算例结果,可以认为该方法具有良好的稳定性,且与原始Wilson-θ法相比,该方法的计算精度有显著提高.

As an implicit direct time-integration method,the Wilson-θmethod endures an inevitable drawback in which the dynamic equilibrium equation of Wilson-θmethod is not satisfied at the next time step.Therefore,high amount of amplitude decay and period elongation will occur when Wilson-θmethod is applied,hindering the process of obtaining accurate answers in some cases.For the purpose of solving this problem,an improved Wilson-θmethod(the Wilson-Newmark method)is presented in accurate structural dynamic history analyses.The classical Wilson-θmethod is modified based on the concept of calculation perturbation,which is defined as the calculation error in each step during the calculation process.Consequently,the dynamic equilibrium equation and motion constraint are simultaneously satisfied by redistributing the calculation perturbation using Newmark-βmethod.To investigate the applicability of Wilson-Newmark method,we carry out the stability analysis so that the variation of the error divergence with different values ofθcan be followed.Then the algorithm precision is measured in terms of amplitude decay,period elongation,and overshoot.Combined with results of numerical examples,it can be concluded that this modified method secures high stability,and the calculation accuracy of this method is significantly improved comparing with the original method.