朱志辉, 刘禹兵, 高雪萌, 周高扬, 余志武. 概率密度演化方程差分格式的计算精度及初值条件改进[J]. 工程力学, 2022, 39(11): 13-21. DOI: 10.6052/j.issn.1000-4750.2021.06.0477
引用本文: 朱志辉, 刘禹兵, 高雪萌, 周高扬, 余志武. 概率密度演化方程差分格式的计算精度及初值条件改进[J]. 工程力学, 2022, 39(11): 13-21. DOI: 10.6052/j.issn.1000-4750.2021.06.0477
ZHU Zhi-hui, LIU Yu-bing, GAO Xue-meng, ZHOU Gao-yang, YU Zhi-wu. THE CALCULATION PRECISION OF PROBABILITY DENSITY EVOLUTION EQUATION DIFFERENCE SCHEME AND THE IMPROVEMENT OF INITIAL CONDITION[J]. Engineering Mechanics, 2022, 39(11): 13-21. DOI: 10.6052/j.issn.1000-4750.2021.06.0477
Citation: ZHU Zhi-hui, LIU Yu-bing, GAO Xue-meng, ZHOU Gao-yang, YU Zhi-wu. THE CALCULATION PRECISION OF PROBABILITY DENSITY EVOLUTION EQUATION DIFFERENCE SCHEME AND THE IMPROVEMENT OF INITIAL CONDITION[J]. Engineering Mechanics, 2022, 39(11): 13-21. DOI: 10.6052/j.issn.1000-4750.2021.06.0477

概率密度演化方程差分格式的计算精度及初值条件改进

THE CALCULATION PRECISION OF PROBABILITY DENSITY EVOLUTION EQUATION DIFFERENCE SCHEME AND THE IMPROVEMENT OF INITIAL CONDITION

  • 摘要: 由于随机荷载作用下工程结构响应的一阶时间导数往往大于结构响应自身,采用仅满足确定性结构响应分析计算精度的时间步长求解概率密度演化方程(Probability density evolution equation, PDEE)时,总变差减小(Total variation diminishing, TVD)性质差分法计算的结构响应标准差通常无法满足精度要求。该文分别对比了不同差分时间步长下单边差分、Lax-wendroff (L-W)双边差分和TVD差分三种PDEE数值求解方法的计算精度。为提升TVD差分的求解精度,该文提出了差分时间步长的合理选取方法和正态分布型初值条件,并通过数值算例验证了时间步长选取方法和正态分布型初值条件的准确性及普适性。数值算例结果表明:L-W双边差分法的样本离散性最小,误差允许范围内可采用的时间差分步长最大;当仅关注均值和标准差等统计指标时,建议使用L-W双边差分法;利用该文方法可有效降低TVD差分所带来的数值误差;正态分布型初值条件的标准差大小等于空间离散步长时,可以获得最小均值误差。

     

    Abstract: Since the first time derivative of engineering structure response under random load was often larger than the structure response itself, if the time step which only meets the precision requirements of deterministic structural response analysis was used to solve the probability density evolution equation (PDEE), the standard deviation of structural response calculated by total variation diminishing (TVD) property difference method cannot meet the precision requirements. The computational precision of three probability density evolution equation numerical solution methods, such as one-sided difference method, Lax-Wendroff (L-W) two-sided difference method and TVD property difference method are compared under different difference time steps. In order to improve the solving precision of TVD property difference method, a reasonable difference time step selection method and initial value condition of normal distribution type are proposed, and the precision and universality of the time step selection method and initial value condition of normal distribution type are verified by numerical examples. The results of numerical examples show that the sample discreteness of the two-sided difference method is the smallest, and the time difference step size is the largest within the allowable error range. When only the mean value and standard difference are concerned, the two-sided difference method is suggested. The numerical error caused by TVD property difference method can be effectively reduced by using this method. The minimum mean error can be obtained when the standard deviation of the initial condition of normal distribution is equal to the space step.

     

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