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

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

朱志辉^{1, 2, ,},
刘禹兵^1,,
高雪萌^1,,
周高扬^1,,
余志武^{1, 2,}

1.
中南大学土木工程学院，湖南，长沙 410075
2.
中南大学高速铁路建造技术国家工程研究中心，湖南，长沙 410075

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
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- HTML全文浏览量: 92
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- 被引次数: 0
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

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

ZHU Zhi-hui^{1, 2
, ,},
LIU Yu-bing^1
,,
GAO Xue-meng^1
,,
ZHOU Gao-yang^1
,,
YU Zhi-wu^{1, 2
,}

1.
School of Civil Engineering, Central South University, Changsha, Hu’nan 410075, China
2.
National Engineering Research Center for High Speed Railway Construction Technology, Changsha, Hu’nan 410075, China

摘要

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。
- 概率密度演化理论 /
- 随机振动 /
- 有限差分法 /
- 计算精度 /
- 差分稳定性
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.
- probability density evolution theory /
- random vibration /
- finite difference method /
- calculation precision /
- differential stability

HTML全文

图 1 有限差分法求解过程示意图

Figure 1. Finite difference method solution process diagram

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图 2 基于正态分布函数的初值条件

Figure 2. Initial condition based on normal distribution function

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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

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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

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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

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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

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图 7 目标概率密度(精确解)

Figure 7. Target probability density (exact solution)

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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)

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图 9 与精确解的结果对比

Figure 9. The results are compared with the exact solution

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图 10 $\sigma = {\rm{d}}Z$时的概率密度

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

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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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