极小失效概率估计的元模型二次重要抽样方法

doi:10.7527/S1000-6893.2025.32316

Abstract

Abstract:

Based on a surrogate model of the performance function with an adaptive learning strategy， the meta-model-based importance sampling （Meta-IS） method can approximate the optimal importance sampling probability density function （IS-PDF） for estimating failure probabilities， making it an efficient approach for reliability analysis. However， when dealing with extremely small failure probabilities， estimating the normalization factor in the IS-PDF becomes computationally expensive for Meta-IS. To mitigate the computational burden， a meta-model-based double importance sampling （Meta-IS₂） method for estimating extremely small failure probabilities is proposed. The hierarchical weighted clustering strategy is designed to construct an IS-PDF for estimating the normalization factor. The feasibility of the proposed method is verified with instants. The results show that， under equivalent accuracy， the computational efficiency of the proposed method is no less than that of the existing Meta-IS method. Furthermore， for the cases with extremely small failure probabilities， the proposed method significantly outperforms that of the existing Meta-IS method.

Key words: meta-model, importance sampling density, reliability analysis, extremely small failure probability, hierarchical weighted clustering strategy

CLC Number:

Xinglin LI, Zhenzhou LYU, Yizhou CHEN. Meta-model-based double importance sampling method for extremely small failure probability estimation[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(3): 232316.

Figures/Tables 18

Fig.1

Table 1

Distribution parameters （Case 1）

变量	分布类型	均值	标准差
$m$	正态分布	1	0.05
$c 1$	正态分布	1	0.10
$c 2$	正态分布	0.1	0.01
$r$	正态分布	0.79	0.05
$F 1$	正态分布	0.3	0.2
$t 1$	正态分布	1	0.2

Table 1

Table 2

Reliability analysis results （Case 1）

方法	重要样本池容量	估计 $P f ε$ 样本池容量	$P f$	$ε r$ /%	$C o v (P f)$ /%
MCS			$7.21 × 10 - 7$		3.7
Meta-IS	$5 × 103$	$1 × 109$	$6.95 × 10 - 7$	3.6	3.8
Meta-IS₂	$5 × 103$	$5 × 104$	$7.39 × 10 - 7$	2.5	1.9

Table 2

Table 3

Calculated cost statistics （Case 1）

方法	N_call	$t 1 / s$	$t 2 / s$	$t t o t a l / s$
MCS	$1 × 109$			106.56
Meta-IS	112+39	378.21	$1.37 × 104$	$1.41 × 104$
Meta-IS₂	112+39	378.21	3.13	381.34

Table 3

Fig.2

Table 4

Reliability analysis results （Case 2）

方法	重要样本池容量	估计 $P f ε$ 样本池容量	$P f$	$ε r$ /%	$C o v (P f)$ /%
MCS			$7.70 × 10 - 7$		3.6
Meta-IS	$3 × 103$	$1 × 109$	$7.67 × 10 - 7$	0.39	3.6
Meta-IS₂	$3 × 103$	$3 × 105$	$7.87 × 10 - 7$	2.20	4.1

Table 4

Table 5

Calculated cost statistics （Case 2）

方法	N_call	$t 1 / s$	$t 2 / s$	$t t o t a l / s$
MCS	$1 × 109$			262.57
Meta-IS	449+149	$3.39 × 103$	$1.93 × 105$	$1.97 × 105$
Meta-IS₂	449+149	$3.39 × 103$	88.88	$3.48 × 103$

Table 5

Fig.3

Table 6

Reliability analysis results （Case 3）

方法	重要样本池容量	估计 $P f ε$ 样本池容量	$P f$	$ε r$ /%	$C o v (P f)$ /%
MCS			$6.14 × 10 - 7$		4.0
Meta-IS	$2 × 103$	$1 × 109$	$6.10 × 10 - 7$	0.7	4.1
Meta-IS₂	$2 × 103$	$5 × 104$	$6.08 × 10 - 7$	1.0	1.2

Table 6

Table 7

Calculated cost statistics （Case 3）

方法	N_call	$t 1 / s$	$t 2 / s$	$t t o t a l / s$
MCS	$1 × 109$			49.78
Meta-IS	（13+15+9+8）+（1+0+0+0）	82.46	$7.39 × 103$	$7.46 × 103$
Meta-IS₂	（13+15+9+8）+（1+0+0+0）	82.46	1.56	84.02

Table 7

Fig.4

Fig.5

Table 8

Random variable distribution parameters of turbine shaft structure

变量	符号	分布类型	均值	标准差
转速 $ω$ /（rad∙s^-1）	$X 1$	正态分布	1 000	33.72
弹性模量 $E$ /MPa	$X 2$	正态分布	$1.46 × 105$	$4.38 × 103$
泊松比 $λ$	$X 3$	正态分布	0.325	$6.5 × 10 - 3$
扭矩 $T$ /（N·m）	$X 4$	正态分布	1 200	48
应力极限 $σ *$ /MPa	$X 5$	正态分布	480	17

Table 8

Fig.6

Fig.7

Table 9

Reliability analysis results （Case 4）

方法

重要

样本池

容量

估计 $P f ε$

样本池

容量

P f

C o v (P f)

Meta-IS

2 × 103

1 × 109

6.90 × 10 - 7

3.80

Meta-IS₂

2 × 103

2 × 104

6.86 × 10 - 7

3.24

Table 9

Table 10

Calculated cost statistics （Case 4）

方法	N_call	$t 1 / s$	$t 2 / s$	$t t o t a l / s$
Meta-IS	32+1	$2.99 × 103$	$2.81 × 103$	$5.80 × 103$
Meta-IS₂	32+1	$2.99 × 103$	0.33	$2.99 × 103$

Table 10

Fig.8

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Meta-model-based double importance sampling method for extremely small failure probability estimation

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