﻿ 一种新的矩独立重要性测度分析方法及高效算法<sup>*</sup>
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1. 西安现代控制技术研究所, 西安 710065;
2. 西北工业大学 航空学院, 西安 710072

A new moment-independent importance measure analysis method and its efficient algorithm
GONG Xiangrui1,2, LYU Zhenzhou2, SUN Tianyu1, ZHANG Leilei1, FENG Lei1
1. Xi'an Institute of Modern Control Technology, Xi'an 710065, China;
2. College of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Received: 2018-03-15; Accepted: 2018-09-14; Published online: 2018-10-18 11:09
Corresponding author. GONG Xiangrui, E-mail: gxrui1991@126.com
Abstract: In order to analyze the effect of input random variables on the failure probability of structural systems more reasonably, a new moment-independent importance measure analysis method is proposed in this paper. The traditional importance measure index can only estimate the influence of input random variables on the output response of structural systems at fixed points, while the new index proposed in this paper can fully reflect the average influence of input random variables on the output response of structural systems when they change in all reduced intervals of their distribution areas, which is more in line with engineering practice. Seeking to find the new index, this paper presents two algorithms:the conventional double-loop-repeated-set Monte Carlo (DLRS MC) method and adaptive radial-based importance sampling (ARBIS) method. The results of DLRS MC method can be used as a reference solution, yet its calculation process is slow and strenuous. Under the condition of the precision of solving the new index is met, the calculation efficiency of ARBIS method is greatly improved. Finally, a numerical example and an engineering example are given to illustrate the significance of the new index and the efficiency of the proposed algorithm.
Keywords: global sensitivity analysis     importance measure     failure probability     adaptive strategy     impor-tance sampling method

1 新的矩独立重要性测度指标 1.1 新指标的定义

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1.2 输入变量缩减区间获取方法

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Hk(k=1, 2, …, N)组成一个(3N×2)的分位数矩阵Q

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2 新指标与基于方差的指标的关系

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3 新指标的高效求解算法

DLRS MC方法可以获得高精度的结果值，但其计算量很大，计算效率较低。本节提出了一种新的求解算法——ARBIS方法，旨在对新的矩独立重要性测度指标进行高效求解。

3.1 自适应超球重要抽样方法

ARBIS方法最初是用来计算失效概率的，而计算新的基于失效概率的矩独立重要性测度指标的核心就是计算非条件失效概率和条件失效概率。因此，本节基于ARBIS方法构建新指标的求解算法。

ARBIS方法[18-19]的基本思想是自适应地寻找到一个落在功能函数安全域内的最大的超球。当抽样点落入超球内部时，就将该抽样点归类为安全点，不必再代入结构系统功能函数进行求解。因此，该方法在计算结构系统输出响应的失效概率时，计算量将大幅度降低。同时，ARBIS方法采用自适应的方法来获取满足条件的最大超球半径，相比于传统的梯度搜索算法，效率更高，稳健性也更好。

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3.2 采用自适应超球重要抽样方法求解新指标

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4 算例分析 4.1 数值算例

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 随机变量 DLRS MC方法 ARBIS方法 δiP/10-2 SD/10-2 δiP/10-2 SD/10-2 X1 0.061 0 0.010 0.058 2 0.048 X2 0.226 4 0.007 0.216 7 0.013 X3 1.257 2 0.035 1.217 4 0.039 计算量 446 000 2 619

DLRS MC方法的计算结果可以看作参考精确解。从表 1可以看出，ARBIS方法计算得到的结果与DLRS MC方法计算结果基本相同。为了说明2种算法计算过程的收敛性，图 2给出了采用2种算法计算δiP值的标准差。可以看出，当2种算法计算结果均收敛时，ARBIS方法计算量远小于DLRS MC方法，计算效率提高很多。

 图 2 DLRS MC方法和ARBIS方法计算的新的矩独立重要性测度指标的收敛曲线 Fig. 2 Convergence curves of new moment-independent importance measure indices of numerical example computed by DLRS MC and ARBIS methods

4.2 工程算例

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 图 3 汽车前轴结构示意图 Fig. 3 Schematic of automobile front axle structure

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 随机变量 均值 标准差 a/mm 12 0.60 b/mm 65 3.25 t/mm 14 0.70 h/mm 85 4.25 M/(N·mm) 3.5×106 7.5×105 T/(N·mm) 3.1×106 1.5×105

 随机变量 DLRS MC方法 ARBIS方法 δiP SD/10-3 δiP SD/10-3 a 0.004 9 0.276 0.004 7 0.249 b 0.006 4 0.394 0.006 2 0.333 t 0.016 3 0.546 0.015 7 0.608 h 0.001 4 0.173 0.001 3 0.149 M 6.49×10-5 0.092 1.95×10-5 0.012 T 0.009 5 0.426 0.009 2 0.321 计算量 446 000 4 475

 图 4 DLRS MC方法和ARBIS方法计算汽车前轴的收敛曲线 Fig. 4 Convergence curves of automobile front axle computed by DLRS MC and ARBIS methods

2种算法计算的新的矩独立重要性测度指标的排序均相同：tTbahM。这表明当分别固定6个输入随机变量在其各自的分布区域内缩减变化时，t对汽车前轴结构的失效概率影响程度最大，而M对其失效概率的影响几乎可以忽略不计。

5 结论

1) 建立了一种新的区间划分技术来等可能地获得输入随机变量所有可能的缩减区间，并给出了相应证明。

2) 引入了自适应超球重要抽样方法来进行新指标的求解，提高了计算效率。

3) 给出了一个数值算例和一个工程算例，说明了本文所提新指标的意义，同时也验证了新算法的高效性。

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#### 文章信息

GONG Xiangrui, LYU Zhenzhou, SUN Tianyu, ZHANG Leilei, FENG Lei

A new moment-independent importance measure analysis method and its efficient algorithm

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(2): 283-290
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0130