时域逐步积分算法稳定性与精度的对比分析

 选取8种时域逐步积分算法，系统对比分析算法的稳定性和计算精度。介绍时域逐步积分算法理论精度和计算精度的概念：计算时间步长趋于0时算法表现出来的精度为理论精度，而在实际计算中一般选取在满足稳定性条件下尽可能大的时间步长，此时算法表现出来的精度为计算精度。分析结果表明，算法的计算精度与理论精度是不一致的，算法的计算精度与其振幅衰减率和周期延长率相一致。对算法的计算精度从理论上进行推导分析，得到无阻尼情况下振幅衰减率与周期延长率的显式表达公式。给出线弹性和弹塑性情况下的单自由度结构振动算例，对算法在弹塑性情况下的表现进行初步研究。结合算例计算结果与理论分析，可以清楚地揭示不同算法在实际计算中的表现，从而为在实际计算中选取合适的积分算法提供参考。

COMPARISON ANALYSIS OF STABILIZATION AND ACCURACY OF STEP-BY-STEP TIME-INTEGRATION METHODS

Eight step-by-step time-integration methods are introduced，and their stability and computational accuracy are analyzed and compared systemically. The definitions of theoretical accuracy and computational accuracy of step-by-step time-integration methods are introduced. The theoretical accuracy of the method is the accuracy when step time approaches to be zero. While in practical computation，the step time is always chosen as large as possible under the precondition of stability demand；and the accuracy the method shows at this time is defined as computational accuracy. The analysis shows that the computational accuracy of a method deviates from its theoretical accuracy and is consistent with its amplitude decay rate and period elongation. Until now，few attention has been paid to the study of computational accuracy. The computational accuracy of the methods is analyzed theoretically；and the amplitude decay and period elongation of the methods are formulated under undamped case. Numerical examples including linear-elastic and elastoplastic vibrations of structure with single degree of freedom are computed；and the behaviors of the methods in elastoplastic condition are analyzed preliminarily. Combining calculation results with theoretical analysis，the eight methods¢ behaviors in practical computation are clearly demonstrated. Thus，references are provided to select an appropriate step-by-step time-integration method in practical computation.