A-stable linear two-step time integration methods with consistent starting and their equivalent single-step methods in structural dynamics analysis
A-stable linear two-step time integration methods with consistent starting and their equivalent single-step methods in structural dynamics analysis
复制标题
结构动力学分析中一致起始的A稳定线性两步时间积分方法及其等效单步方法
DOI：
10.1002/nme.6623
复制
发表时间：
2021
影响因子：
2.9
通讯作者：
Jie Zhang
中科院分区：
文献类型：
--
作者：
Jie Zhang
针对结构动力分析中的一阶A稳定线性两步（LTS）时间积分方法，提出了一种谱一致的启动方法.基于一阶模型对结构动力学中的LTS方法进行了精度分析，使一阶瞬态系统中的算法能够在一致的框架下扩展到结构动力学中。结果表明，荷载的近似性对LTS方法与某些单步方法的等价性起着至关重要的作用。一阶系统中的两支广义α方法与A稳定LTS方法之间具有比谱等价更严格的算法等价性。将LTS方法中的误差常数和极限谱半径的度量推广到优化一类给定收敛阶的单步方法。利用单步方法的算法等价性，给出了A-稳定LTS方法的谱一致启动过程，从而得到了一种在启动步骤及其后步骤中具有一致谱半径和可控数值耗散的最优A-稳定LTS（OALTS）方法.与等效单步方法相比，OALTS方法不需要辅助变量，其位移、速度和加速度可同时达到二阶精度。本OALTS方法的性能进行了验证的数值例子，包括物理阻尼，外部负载，和/或非线性。
A spectral consistent starting procedure is proposed for the first‐order‐type A‐stable linear two‐step (LTS) time integration methods in structural dynamics analysis. The accuracy analysis for the LTS methods in structural dynamics is presented based on the first‐order model, which enables the algorithms in first‐order transient systems to be extended to structural dynamics under the umbrella of a consistent framework. It is indicated that the approximation of the loads plays an essential role in the equivalence between the LTS methods and some single‐step methods. An algorithmic equivalence feature, which is stricter than the spectral equivalence, is revealed between the two‐leg generalized‐α methods in first‐order systems and the A‐stable LTS methods. The measures of the error constant in the LTS methods and the ultimate spectral radius are extended to optimize a class of single‐step methods of a given convergence order. The spectral consistent starting procedure for the A‐stable LTS methods is developed by utilizing the algorithmic equivalence of the single‐step methods, which results in an optimal A‐stable LTS (OALTS) method possessing a consistent spectral radius with controllable numerical dissipation in the starting step and the steps thereafter. Comparing with the equivalent single‐step methods, the present OALTS method does not need the auxiliary variables, and its displacement, velocity and acceleration can achieve second‐order accuracy simultaneously. The performance of the present OALTS method is verified by numerical examples including physical damping, external loads, and/or nonlinearity.