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概率密度演化方程差分格式的计算精度及初值条件改进

朱志辉 刘禹兵 高雪萌 周高扬 余志武

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

doi: 10.6052/j.issn.1000-4750.2021.06.0477
基金项目: 国家自然科学基金项目(52078498);重庆市社会事业与民生保障科技创新专项项目(CSTC2016SHMSZX30014)
详细信息
    作者简介:

    刘禹兵(1998−),男,河南人,硕士生,主要从事车-桥耦合振动研究(E-mail: 1426007221@qq.com)

    高雪萌(1998−),女,河北人,博士生,主要从事车-桥耦合振动研究(E-mail: 874338175@qq.com)

    周高扬(1999−),男,江西人,博士生,主要从事车-桥耦合振动研究(E-mail: 806033364@qq.com)

    余志武(1955−),男,湖南人,教授,博士,博导,主要从事车-桥耦合振动研究(E-mail: zhwyu@csu.edu.cn)

    通讯作者:

    朱志辉(1979−),男,河南人,教授,博士,博导,主要从事车-桥耦合振动研究(E-mail: zzhh0703@163.com)

  • 中图分类号: O324

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。
  • 图  1  有限差分法求解过程示意图

    Figure  1.  Finite difference method solution process diagram

    图  2  基于正态分布函数的初值条件

    Figure  2.  Initial condition based on normal distribution function

    图  3  不同差分格式的单样本离散程度

    Figure  3.  The degree of single sample dispersion of different difference schemes

    图  4  不同差分时间步长的单样本均值

    Figure  4.  Single sample means with different difference time steps

    图  5  不同差分时间步长的单样本标准差

    Figure  5.  Single sample standard deviations with different difference time steps

    图  6  不同差分时间步长的多样本统计特征值

    Figure  6.  Multiple statistical eigenvalues of different difference time steps

    图  7  目标概率密度(精确解)

    Figure  7.  Target probability density (exact solution)

    图  8  时间步长0.0001 s的概率密度 (TVD差分数值解)

    Figure  8.  Probability density with time step of 0.0001 s (TVD difference numerical solution)

    图  9  与精确解的结果对比

    Figure  9.  The results are compared with the exact solution

    图  10  $\sigma = {\rm{d}}Z$时的概率密度

    Figure  10.  The probability density at $\sigma = {\rm{d}}Z$

    图  11  初值条件对累计误差的影响

    Figure  11.  Influence of initial condition on cumulative error

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出版历程
  • 收稿日期:  2021-06-24
  • 修回日期:  2021-10-12
  • 网络出版日期:  2021-10-21
  • 刊出日期:  2022-11-01

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