﻿ 失效概率矩独立全局灵敏度指标的高效算法
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 力学学报  2016, Vol. 48 Issue (4): 1004-1012  DOI: 10.6052/0459-1879-15-411 0

### 引用本文 [复制中英文]

[复制中文]
Yun Wanying , Lü Zhenzhou , Jiang Xian . AN EFFICIENT METHOD FOR FAILURE PROBABILITY-BASED MOMENT-INDEPENDENT SENSITIVITY ANALYSIS[J]. Chinese Journal of Ship Research, 2016, 48(4): 1004-1012. DOI: 10.6052/0459-1879-15-411.
[复制英文]

### 文章历史

2015-11-11 收稿
2016-04-12 录用
2016-04-15网络版发表

1. 西北工业大学航空学院, 西安 710072 ;
2. 中国飞行试验研究院飞机所, 西安 710089

0 引言

1 基于失效概率的矩独立全局灵敏度指标的

Li等[16]证明了

 $\delta _i^P = V_{X_i } (E_{{ X}_{ - i} } (I_{ F} | X_i ))$ (4)

Wei等[18]通过添加$V(I_{ F})$这一常数项，使得失效概率矩独立全局灵敏度指标与基于方差的全局灵敏度指标在形式上完全统一，即

 $S_i = \dfrac{V_{X_i } (E_{{ X}_{ - i} } (I_{ F} | X_i )}{V(I_{ F} )}$ (5)

2 基于空间分割和重要抽样的Si计算新方法 2.1 基于空间分割的Si近似计算公式

 ${ E}_{X_i } (V_{{ X}_{ - i} } (I_{ F} | X_i )) =\\ \sum_{k = 1}^s {p_k (V(I_{ F} | X_i \in A_k ) - V_{X_i } (E(I_{ F} | X_i } ) | X_i \in A_k )) =\\ E_{A_k } (V(I_{ F} | X_i \in A_k )) - \sum_{k = 1}^s {p_k } V_{X_i } (E(I_{ F} | X_i ) | X_i \in A_k )$ (6)

 $f_{ X}^\ast ({ x}) = \left\{ \begin{array}{cl} \dfrac{f_{ X} ({ x})}{\int_{a_{k - 1} }^{a_k } f_{X_i } (x_i ) d x_i } ,& x_i \in [a_{k - 1} ,a_k] \\ 0 ,& x_i otin [a_{k - 1} ,a_k] \end{array} \right.$ (9)

 \eqalign{ & E({I_F}|{X_i} \in {A_k}) = {1 \over {\int_{{a_{k - 1}}}^{{a_k}} {{f_{{X_i}}}} ({x_i})d{x_i}}} \cdot \cr & \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } \cdots } \int_{{a_{k - 1}}}^{{a_k}} {{I_F}} (x){f_X}(x)d{x_i}\prod\limits_{j = 1,jei}^n d {x_j} \cr} (10)

 \eqalign{ & {E_{{A_k}}}(E({I_F}|{X_i} \in {A_k})) = \cr & \sum\limits_{k = 1}^s {P\{ {X_i} \in {A_k}\} \cdot E({I_F}|{X_i} \in {A_k})} = \cr & \sum\limits_{k = 1}^s {\int_{{a_{k - 1}}}^{{a_k}} {{f_{{X_i}}}({x_i})d{x_i} \cdot {1 \over {\int_{{a_{k - 1}}}^{{a_k}} {{f_{{X_i}}}} ({x_i})d{x_i}}}} } \cdot \cr & \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdots \int_{{a_{k - 1}}}^{{a_k}} {{I_F}(x){f_X}(x)d{x_i}} \prod\limits_{j = 1,jei}^n {d{x_j}} } } = \cr & \sum\limits_{k = 1}^s {\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdots \int_{{a_{k - 1}}}^{{a_k}} {{I_F}(x){f_X}(x)d{x_i}} } } } \prod\limits_{j = 1,jei}^n {d{x_j}} = \cr & \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdots \int_{{b_1}}^{{b_2}} {{I_F}(x){f_X}(x)} } } dx = E({I_F}) \cr} (11)

 $V_{A_k } (E(I_{ F} | X_i \in A_k )) =\\ E_{A_k } (E^2(I_{ F} | X_i \in A_k )) - E_{A_k }^2 (E(I_{ F} | X_i \in A_k ))= \\ E_{A_k } (E^2(I_{ F} | X_i \in A_k )) - E^2(I_{ F} )$ (12)
 $E_{A_k } (V(I_{ F} | X_i \in A_k )) =\\ E_{A_k } (E(I_{ F}^2 | X_i \in A_k ) - E^2(I_{ F} | X_i \in A_k )) =\\ E(I_{ F} ^2) - E_{A_k } (E^2(I_{ F} | X_i \in A_k ))$ (13)

 $E_{A_k } (V(I_{ F} | X_i \in A_k ) + V_{A_k } (E(I_{ F} | X_i \in A_k )) = V(I_{ F} )$ (14)
2.3 基于连续区间上的全方差公式和重要抽样的 Si的近似计算

 $S_i \approx 1 - \dfrac{E_{A_k } (V(I_{ F} | X_i \in A_k ))}{V(I_{ F} )} =\\ 1 - \dfrac{V(I_{ F} ) - V_{A_k } (E(I_{ F} | X_i \in A_k ))}{V(I_{ F} )} =\\ \dfrac{V_{A_k } (E(I_{ F} | X_i \in A_k ))}{V(I_{ F} )}$ (15)

 $P_{X_i }^H (A_k ) = \int_{a_{k - 1} }^{a_k } {H_{X_i } (x_i )d x_i }$ (20)
 $P_{{X_i}}^f({A_k}) = \int_{{a_{k - 1}}}^{{a_k}} {{f_{{X_i}}}({x_i})d{x_i}}$ (21)

 $B_k = \{I_{ F} | X_i \in A_k \} ,1 ≤ k ≤ s$ (22)

 $g = D_0 - D(E,X,Y,w,t,L)$ (27)

 图 3 屋架结构计算结果随样本变化图 Fig. 3 Results of the proposed method varying with the increase of sample size for roof truss

 图 4 屋架结构100次重复计算结果标准差随样本变化图 Fig. 4 Standard derivation of the proposed method by 100 iterations varying with increase of sample size

4 总结

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AN EFFICIENT METHOD FOR FAILURE PROBABILITY-BASED MOMENT-INDEPENDENT SENSITIVITY ANALYSIS
Yun Wanying1, Lü Zhenzhou1, Jiang Xian2
1. School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China ;
2. Aircraft Flight Test Technology Institute, Chinese Flight Test Establishment, Xi'an 710089, China
Abstract: The failure probability-based moment-independent sensitivity index well analyzes how uncertainty in the failure probability of a model can be apportioned to different sources of uncertainty in the model inputs. At present, the existing sampling-based methods to estimate this index can not make full use of samples. Therefore, in this paper, we mainly concern how to improve the utilization of samples to accurately estimate this index. Based on the law of total variance in the successive intervals without overlapping proved in this paper, we propose an efficient method to estimate the failure probability-based moment-independent sensitivity index by combining the idea of space-partition and importance sampling, which only requires one set of input-output samples and the computational cost is independent of the dimensionality of inputs. The proposed method firstly uses importance sampling density function which can promise that a large number of samples will drop into the failure domain to generate a set of samples and then simultaneously obtain the sensitivity indices for all the input variables by repeatedly using this single set of samples. It is because of this that proposed method greatly improves the utilization of samples. Examples in this paper illustrate that our proposed method has higher efficiency, accuracy, convergence and robustness than the existing ones, and demonstrate its good prospect in engineering applications.
Key words: failure probability    global sensitivity index    law of total variance    space-partition    importance sampling