非参数化概率盒下随机与认知不确定性的分离式 灵敏度分析

doi:10.7527/S1000-6893.2021.26658

Abstract

Abstract:

Sensitivity Analysis （SA） can identify the most important parameters affecting the complex system output to support robust design of a system. Non-parametric probability-box （P-box）， as a typical imprecise probabilistic model， can effectively quantify both aleatory and epistemic uncertainties， therefore extensively used in engineering practices. As aleatory and epistemic uncertainties are coupled in P-boxes， sensitivity analysis under the P-box framework is essential to evaluate their contributions in input P-box variables to output. This study develops a Separating Sensitivity Analysis （SSA） method for aleatory and epistemic uncertainties of non-parametric p-boxes. Two methods， i.e.， grid point method and expectation method， are introduced to separate the input aleatory and epistemic uncertainties in input P-box variables， respectively. A double loop procedure is utilized to propagate the input uncertainties and build the output P-box. Two uncertainty measures， namely， maximum variance metric and area metric， are proposed to evaluate the effects of input aleatory and epistemic uncertainties on the output aleatory and epistemic uncertainties， respectively. The lift-to-drag ratio prediction of the NACA0012 airfoil is exemplified to analyze the contributions of aleatory and epistemic uncertainties of income flow parameters and turbulence model parameters to those of the lift-to-drag ratio prediction result.

Key words: Separating Sensitivity Analysis (SSA), non-parametric probability-box, grid point method, maximum variance metric, area metric

CLC Number:

V211

Muchen WU, Jiangtao CHEN, Tangfan XIAHOU, Wei ZHAO, Yu LIU. Separating sensitivity analysis of aleatory and epistemic uncertainties in non-parametric probability-box[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023, 44(1): 226658-226658.

Figures/Tables 23

Fig. 1

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Fig. 4

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Fig. 8

Table 1

Boundary distributions of input non-parametric P-box variables

输入	Case 1：正态分布			Case 2：对数正态分布		Case 3：伽玛分布
输入	$F ¯ X (x)$	$F ̲ X (x)$		$F ¯ X (x)$	$F ̲ X (x)$	$F ¯ X (x)$		$F ̲ X (x)$
X₁	$N (- 1.5,0 . 252)$	$N (- 1,0 . 352)$		$L N (- 1.5,0 . 12)$	$L N (- 0.9,0 . 22)$	$Γ (1.2,0.5)$		$Γ (1.8,0.9)$
X₂	$N (0.5,0 . 32)$	$N (1,0 . 42)$		$L N (0.5,0 . 42)$	$L N (1.5,0 . 52)$	$Γ (1,2)$		$Γ (1.5,4)$
X₃	$N (0.7,0 . 352)$	$N (1.5,0 . 452)$		$L N (0.8,0 . 22)$	$L N (1.4,0 . 32)$	$Γ (2,1)$		$Γ (5,2)$
输入	Case 4：多峰分布				Case 5：混合边界分布
输入	$F ¯ X (x)$		$F ̲ X (x)$		$F ¯ X (x)$		$F ̲ X (x)$
X₁	$M N (0,0 . 052; 0.5,0 . 082)$		$M N (0.4,0 . 052; 0.7,0 . 22)$		$N (- 1.5,0 . 252)$		$N (- 1,0 . 352)$
X₂	$M N (0.8,0 . 22; 1,0 . 32)$		$M N (1.2,0 . 22; 1.5,0 . 32)$		$L N (0.5,0 . 42)$		$L N (1.5,0 . 52)$
X₃	$M N (1.2,0 . 052; 1.5,0 . 12)$		$M N (1.6,0 . 12; 2,0 . 22)$		$Γ (2,1)$		$Γ (5,2)$

Table 1

Fig. 9

Fig. 10

Fig. 11

Table 2

Table 3

Boundary condition setup for flow over NACA0012 airfoil simulation

输入	非参数化概率盒边界分布	取值范围^［32］	试验标定^［32］
$M a ∞$	$F ¯ M a ∞ (x) = N (0.68,0 . 022) u (0.68 - x) + N (0.68,0 . 012) u (x - 0.68) F ̲ M a ∞ (x) = N (0.69,0 . 012) u (0.69 - x) + N (0.69,0 . 022) u (x - 0.69)$	［0.3，0.8］
$α$	$F ¯ α (x) = N (4.9,0 . 32) u (4.9 - x) + N (4.9,0 . 22) u (x - 4.9) F ̲ α (x) = N (5.1,0 . 22) u (5.1 - x) + N (5.1,0 . 32) u (x - 5.1)$	$[1 ∘, 13 ∘]$
$c b 1$	$F ¯ c b 1 (x) = N (0.132, (1 × 10 - 3) 2) u (0.132 - x) + N (0.132, (5 × 10 - 4) 2) u (x - 0.132) F ̲ c b 1 (x) = N (0.134, (5 × 10 - 4) 2) u (0.134 - x) + N (0.134, (1 × 10 - 3) 2) u (x - 0.134)$	［0.128 93，0.137］	0.135 5
$c v 1$	$F ¯ c v 1 (x) = N (7.05,0 . 032) u (7.05 - x) + N (7.05,0 . 022) u (x - 7.05) F ̲ c v 1 (x) = N (7.15,0 . 022) u (7.15 - x) + N (7.15,0 . 032) u (x - 7.15)$	［6.9，7.3］	7.1
$σ$	$F ¯ σ (x) = N (0.65,0 . 032) u (0.65 - x) + N (0.65,0 . 022) u (x - 0.65) F ̲ σ (x) = N (0.67,0 . 022) u (0.67 - x) + N (0.67,0 . 032) u (x - 0.67)$	［0.6，1］	2/3
$c w 2$	$F ¯ c w 2 (x) = N (0.29,0 . 032) u (0.29 - x) + N (0.29,0 . 022) u (x - 0.29) F ̲ c w 2 (x) = N (0.31,0 . 022) u (0.31 - x) + N (0.31,0 . 032) u (x - 0.31)$	［0.055，0.352 5］	0.3
$κ$	$F ¯ κ (x) = N (0.4,0 . 032) u (0.4 - x) + N (0.4,0 . 022) u (x - 0.4) F ̲ κ (x) = N (0.42,0 . 022) u (0.42 - x) + N (0.42,0 . 032) u (x - 0.42)$	［0.38，0.46］	0.41

Table 3

Fig. 12

Fig. 13

Fig. 14

Table 4

Separating sensitivity analysis results based on area metric

排序	$S ̂ e p s - e p$	$W C I S e p s - e p$
1	$M a ∞ {0.379 3}$	$M a ∞ {1.49 × 10 - 5}$
2	$α {0.236 1}$	$α {3.75 × 10 - 5}$
3	$κ {0.258 7}$	$κ {6.17 × 10 - 5}$
4	$c v 1 {0.045 2}$	$c v 1 {1.28 × 10 - 5}$
5	$c b 1 {0.040 9}$	$c b 1 {2.47 × 10 - 5}$
6	$c w 2 {0.010 1}$	$c w 2 {5.69 × 10 - 6}$
7	$σ {0.009 4}$	$σ {1.22 × 10 - 6}$

Table 4

Fig. 15

Table 5

Separating sensitivity analysis results based on maximum variance metric

排序	$S a l s - a l$
1	$M a ∞ {0.317 6}$
2	$α {0.105 3}$
3	$κ {0.090 1}$
4	$c w 2 {0.008 1}$
5	$c v 1 {0.007 8}$
6	$c b 1 {0.004 9}$
7	$σ {0.001 8}$

Table 5

Table 6

Comparison of lift-to-drag P-box’s area and maximum variance after and before setting inputs as constants

输入参数	面积（相对误差/%）	最大方差（相对误差/%）
$σ$	10.628 7 （1.02）	85.600 2 （1.52）
$c w 2$	10.615 3 （1.14）	85.466 5 （1.73）
$c b 1$	10.317 7 （3.91）	83.072 9 （4.49）
$c v 1$	10.249 0 （4.55）	82.597 9 （5.03）
$σ, c w 2$	10.502 5 （2.19）	84.508 1 （2.84）
$σ, c b 1$	10.206 3 （4.95）	82.006 0 （5.71）
$σ, c w 2, c b 1$	10.084 2 （6.09）	80.882 0 （7）
$σ, c b 1, c v 1$	9.711 7 （9.56）	78.082 9 （10.22）
$σ, c w 2, c b 1, c v 1$	9.590 6 （10.69）	77.069 8 （11.39）

Table 6

Table 7

Comparison of effects of aleatory and epistemic uncertainties on lift-to-drag ratio

输入参数	随机不确定性影响排序	认知不确定性影响排序
$M a ∞$	1	1
$α$	2	2
$κ$	3	3
$c w 2$	4	6
$c v 1$	5	4
$c b 1$	6	5
$σ$	7	7

Table 7

Table 8

Comparison of proposed method and reported method in Ref.［20］

排序	文献［20］	$S ̂ e p s - e p$	$S a l s - a l$
1	$M a ∞ {0.646 8}$	$M a ∞ {0.379 3}$	$M a ∞ {0.317 6}$
2	$α {0.568 9}$	$α {0.236 1}$	$α {0.105 3}$
3	$κ {0.489 2}$	$κ {0.258 7}$	$κ {0.090 1}$
4	$c v 1 {0.086 0}$	$c v 1 {0.045 2}$	$c w 2 {0.008 1}$
5	$c b 1 {0.081 1}$	$c b 1 {0.040 9}$	$c v 1 {0.007 8}$
6	$c w 2 {0.008 7}$	$c w 2 {0.010 1}$	$c b 1 {0.004 9}$
7	$σ {0.006 3}$	$σ {0.009 4}$	$σ {0.001 8}$

Table 8

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预测量	风洞试验	CFD模拟	相对误差/%
升力系数C_L	0.241^［27］	0.252 5	4.77
阻力系数C_D	0.007 9^［27］	0.008 5	7.59

Separating sensitivity analysis of aleatory and epistemic uncertainties in non-parametric probability-box

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