Uncertainty propagation analysis by an extended sparse grid technique

doi:10.1007/s11465-018-0514-x

Frontiers of Mechanical Engineering

2019, Vol. 14

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

本期目录

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.

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

收稿日期: 2017-10-26 出版日期: 2018-11-30

Corresponding Author(s): C. JIANG

引用本文:

. [J]. Frontiers of Mechanical Engineering, 2019, 14(1): 33-46.
X. Y. JIA, C. JIANG, C. M. FU, B. Y. NI, C. S. WANG, M. H. PING. Uncertainty propagation analysis by an extended sparse grid technique. Front. Mech. Eng., 2019, 14(1): 33-46.

链接本文:

https://academic.hep.com.cn/fme/CN/10.1007/s11465-018-0514-x
https://academic.hep.com.cn/fme/CN/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

$\| i \|$	$i 1$	$i 2$	$X 1 i 1 ⊗ X 1 i 2$	$X 22$
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

Fig.1

Fig.2

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

Fig.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

Fig.4

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

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

Fig.5

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

Fig.6

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

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

Fig.7

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