Uncertainty propagation analysis by an extended sparse grid technique

doi:10.1007/s11465-018-0514-x

Front. Mech. Eng.

2019, Vol. 14

Issue (1) : 33-46 https://doi.org/10.1007/s11465-018-0514-x

RESEARCH ARTICLE

Uncertainty propagation analysis by an extended sparse grid technique

X. Y. JIA¹, C. JIANG¹(

), C. M. FU¹, B. Y. NI¹, C. S. WANG², M. H. PING¹

¹. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
². Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China

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Abstract

In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.

Keywords uncertainty propagation analysis extended sparse grid maximum entropy principle extended Gauss integration Rosenblatt transformation high-order moments analysis

Corresponding Author(s): C. JIANG

Just Accepted Date: 18 May 2018 Online First Date: 10 July 2018 Issue Date: 30 November 2018

Cite this article:

X. Y. JIA,C. JIANG,C. M. FU, et al. Uncertainty propagation analysis by an extended sparse grid technique[J]. Front. Mech. Eng., 2019, 14(1): 33-46.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-018-0514-x
https://academic.hep.com.cn/fme/EN/Y2019/V14/I1/33

Level	Integration node	Integration weight	Algebraic precision
1	$V 11 = 0$	$A 11 = 1$	1
2	$V 12 = {− 1.7321, 0, 1.7321}$	$A 12 = {0.1667, 0.6667, 0.1667}$	5
3	$V 13 = {− 2.8613, − 0.7411, − 4.1850, − 1.7321, 0, ? 1.7321, 4.1850, 0.7411, 2.8613}$	$A 13 = {0.0080, 0.2701, 0.0001, 0.0949, 0.2540, ? 0.0949, 0.0001, 0.2701, 0.0080}$	15
4	$V 13 = {− 3.2053, − 2.5961, − 5.1870, − 1.2304, − 6.3634, − 2.8613, − 0.7411, − 4.1850, − 1.7321, 0, ? 1.7321, 4.1850, 0.7411, 2.8613, 6.3634, 1.2304, 5.1870, 2.5961, 3.2053}$	$A 14 = {0.0029, 0.0181, 0, 0.0612 0, 0.0063, 0.2083, 0.0001 0.0641, 0.3035, 0.0641, 0.0001 0.2083, 0.0063, 0, 0.0612, 0, 0.0181, 0.0029}$	29
$⋮$	$⋮$	$⋮$	$⋮$

Tab.1 EGHI nodes and weights

$\| i \|$	$i 1$	$i 2$	$X 1 i 1 ⊗ X 1 i 2$	$X 22$
3	1	2	${(0, − 1.732), (0, 0), (0, 1.732)}$	${(0, − 1.732), (0, 0), (0, 1.732) (− 1.732, 0), (1.732, 0), (0, − 2.861), (0, - 0.741), (0, − 4.185), (0, 2.861), (0, 0.741), (0, 4.185), (− 1.732, − 1.732), (− 1.732, 1.732), (1.732, − 1.732), (1.732, 1.732), (− 2.861, 0), (- 0.741, 0), (− 4.185, 0) (2.861, 0), (0.741, 0), (4.185, 0)}$
3	2	1	${(− 1.732, 0), (0, 0), (1.732, 0)}$
4	1	3	${(0, − 2.861), (0, - 0.741), (0, − 4.185), (0, − 1.732), (0, 0), (0, 1.732), (0, 2.861), (0, 0.741), (0, 4.185)}$
	2	2	${(− 1.732, − 1.732), (− 1.732, 0), (0, 0), (0, − 1.732), (− 1.732, 1.732), (0, 1.732), (1.732, − 1.732), (1.732, 0), (1.732, 1.732)}$
	3	1	${(− 2.861, 0), (- 0.741, 0), (− 4.185, 0), (− 1.732, 0), (0, 0), (1.732, 0), (2.861, 0), (0.741, 0), (4.185, 0)}$

Tab.2 Information on multidimensional nodes

X 22

Fig.1 Construction of multidimensional nodes

X 22

Fig.2 Computational flowchart of proposed method

Moments	MCS	UDRM (error)	SGNI (error)	Proposed method (error)
$μ$	18.6192	18.6108 (0.05%)	18.6132 (0.03%)	18.6132 (0.030%)
$σ$	6.1305	6.0184 (1.83%)	6.1295 (0.02%)	6.1303 (0.002%)
$τ$	0.5976	0.2005 (66.45%)	0.5645 (5.53%)	0.5961 (0.240%)
$κ$	3.5904	3.0546 (14.92%)	2.9026 (19.16%)	3.5548 (0.990%)

Tab.3 Results of first four-order moments of Case 1 in Example 1

Fig.3 PDF of system response in Example 1. (a) PDF of Case 1; (b) PDF of Case 2; (c) PDF of Case 3

Moments	MCS	SGNI (error)	Proposed method (error)
$μ$	18.6192	18.6132 (0.030%)	18.6132 (0.030%)
$σ$	6.1305	6.1304 (0.001%)	6.1304 (0.001%)
$τ$	0.5976	0.5978 (0.040%)	0.5978 (0.040%)
$κ$	3.5904	3.5964 (0.170%)	3.5962 (0.160%)

Tab.4 Results of first four-order moments of Case 2 in Example 1

Fig.4 The relative error of the first-order moments varies with the dimension n. The relative error of (a) the mean, (b) the standard deviation, (c) the skewness, and (d) the kurtosis varies with the dimension n

Variables	Distribution	Parameter 1	Parameter 2
$X 1$	Normal	1	0.12
$X 2$	Normal	5	0.5
$X 3$	Weibull	1	5
$X 4$	Uniform	2	6
$X 5$	Lognormal	2	0.2
$X 6$	Beta	2	5
$X 7$	Normal	1	0.12
$X 8$	Normal	1	0.12
$X 9$	Normal	1	0.12
$X 10$	Normal	1	0.12

Tab.5 Distributions of input random variable in Example 2

Moments	MCS	UDRM (error)	SGNI (error)	Proposed method (error)
$μ$	19.5144	20.5589 (5.35%)	19.5616 (0.24%)	19.5133 (0.01%)
$σ$	7.6417	7.3465 (3.86%)	7.5890 (0.69%)	7.6449 (0.04%)
$τ$	0.6202	0.2231 (64.04%)	0.4481 (27.75%)	0.6305 (1.65%)
$κ$	3.7232	3.0687(17.58%)	3.2924 (11.57%)	3.7960 (1.96%)

Tab.6 Results of first four-order moments of Case 1 in Example 2

Fig.5 PDF of system response in Example 2. (a) PDF of Case 1; (b) PDF of Case 2

Moments	MCS	SGNI (error)	Proposed method (error)
$μ$	19.5144	19.5161 (0.01%)	19.5129 (0.01%)
$σ$	7.6417	7.6420 (0.004%)	7.6418 (0.002%)
$τ$	0.6202	0.6063 (2.24%)	0.6095 (1.73%)
$κ$	3.7232	3.7219 (0.04%)	3.7256 (0.06%)

Tab.7 Results of first four-order moments of Case 2 in Example 2

Fig.6 Finite element model of power amplifier link

Variables	Distribution	Parameter 1/mm	Parameter 2
$X 1$	Normal	4.80	0.033
$X 2$	Normal	0.70	0.001
$X 3$	Normal	0.90	0.017
$X 4$	Normal	3.50	0.017
$X 5$	Normal	1.00	0.001
$X 6$	Normal	2.20	0.033
$X 7$	Normal	12.90	0.017
$X 8$	Normal	45.70	0.017
$X 9$	Normal	96.15	0.017
$X 10$	Normal	73.70	0.017
$X 11$	Uniform	114.00	115 mm
$X 12$	Uniform	129.10	129.5 mm
$X 13$	Normal	133.40	0.017
$X 14$	Normal	147.10	0.017

Tab.8 Distributions of input random variables in Example 3

Moments	MCS	UDRM (error)	SGNI (error)	Proposed method (error)
$μ$	−45.0458	−45.3185 (0.60%)	−45.0591 (0.03%)	−45.0591 (0.03%)
$σ$	2.6849	2.5567 (4.77%)	2.6809 (0.15%)	2.6856 (0.03%)
$τ$	0.8739	0.2210 (74.71%)	0.7780 (10.97%)	0.8637 (1.17%)
$κ$	4.6578	3.5320(24.17%)	3.4343 (26.27%)	4.4685 (4.06%)

Tab.9 First four-order moments of phase difference in Example 3

Fig.7 PDF of phase difference in Example 3

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