Introduction
Uncertainty propagation and sparse grid method
The proposed method for UP analysis
Extended Gauss integration
Tab.1 EGHI nodes and weights |
Level | Integration node | Integration weight | Algebraic precision |
---|---|---|---|
1 | 1 | ||
2 | 5 | ||
3 | 15 | ||
4 | 29 | ||
Transformation of extended integration nodes
Extended sparse grid technique
Tab.2 Information on multidimensional nodes |
3 | 1 | 2 | ||
2 | 1 | |||
4 | 1 | 3 | ||
2 | 2 | |||
3 | 1 |
Maximum entropy principle
Computational procedure
Numerical examples and discussions
Numerical Example 1
Case 1: Accuracy comparison
Tab.3 Results of first four-order moments of Case 1 in Example 1 |
Moments | MCS | UDRM (error) | SGNI (error) | Proposed method (error) |
---|---|---|---|---|
18.6192 | 18.6108 (0.05%) | 18.6132 (0.03%) | 18.6132 (0.030%) | |
6.1305 | 6.0184 (1.83%) | 6.1295 (0.02%) | 6.1303 (0.002%) | |
0.5976 | 0.2005 (66.45%) | 0.5645 (5.53%) | 0.5961 (0.240%) | |
3.5904 | 3.0546 (14.92%) | 2.9026 (19.16%) | 3.5548 (0.990%) |
Case 2: Efficiency comparison
Tab.4 Results of first four-order moments of Case 2 in Example 1 |
Moments | MCS | SGNI (error) | Proposed method (error) |
---|---|---|---|
18.6192 | 18.6132 (0.030%) | 18.6132 (0.030%) | |
6.1305 | 6.1304 (0.001%) | 6.1304 (0.001%) | |
0.5976 | 0.5978 (0.040%) | 0.5978 (0.040%) | |
3.5904 | 3.5964 (0.170%) | 3.5962 (0.160%) |
Case 3: Dimension comparison
Numerical Example 2
Tab.5 Distributions of input random variable in Example 2 |
Variables | Distribution | Parameter 1 | Parameter 2 |
---|---|---|---|
Normal | 1 | 0.12 | |
Normal | 5 | 0.5 | |
Weibull | 1 | 5 | |
Uniform | 2 | 6 | |
Lognormal | 2 | 0.2 | |
Beta | 2 | 5 | |
Normal | 1 | 0.12 | |
Normal | 1 | 0.12 | |
Normal | 1 | 0.12 | |
Normal | 1 | 0.12 |
Note: Parameter i (i=1, 2) denotes the i-th parameter of input random variable |
Case 1: Accuracy comparison
Tab.6 Results of first four-order moments of Case 1 in Example 2 |
Moments | MCS | UDRM (error) | SGNI (error) | Proposed method (error) |
---|---|---|---|---|
19.5144 | 20.5589 (5.35%) | 19.5616 (0.24%) | 19.5133 (0.01%) | |
7.6417 | 7.3465 (3.86%) | 7.5890 (0.69%) | 7.6449 (0.04%) | |
0.6202 | 0.2231 (64.04%) | 0.4481 (27.75%) | 0.6305 (1.65%) | |
3.7232 | 3.0687(17.58%) | 3.2924 (11.57%) | 3.7960 (1.96%) |
Case 2: Efficiency comparison
Tab.7 Results of first four-order moments of Case 2 in Example 2 |
Moments | MCS | SGNI (error) | Proposed method (error) |
---|---|---|---|
19.5144 | 19.5161 (0.01%) | 19.5129 (0.01%) | |
7.6417 | 7.6420 (0.004%) | 7.6418 (0.002%) | |
0.6202 | 0.6063 (2.24%) | 0.6095 (1.73%) | |
3.7232 | 3.7219 (0.04%) | 3.7256 (0.06%) |
An engineering application
Tab.8 Distributions of input random variables in Example 3 |
Variables | Distribution | Parameter 1/mm | Parameter 2 |
---|---|---|---|
Normal | 4.80 | 0.033 | |
Normal | 0.70 | 0.001 | |
Normal | 0.90 | 0.017 | |
Normal | 3.50 | 0.017 | |
Normal | 1.00 | 0.001 | |
Normal | 2.20 | 0.033 | |
Normal | 12.90 | 0.017 | |
Normal | 45.70 | 0.017 | |
Normal | 96.15 | 0.017 | |
Normal | 73.70 | 0.017 | |
Uniform | 114.00 | 115 mm | |
Uniform | 129.10 | 129.5 mm | |
Normal | 133.40 | 0.017 | |
Normal | 147.10 | 0.017 |
Note: Parameter i (i=1, 2) denotes the i-th parameter of input random variable |
Tab.9 First four-order moments of phase difference in Example 3 |
Moments | MCS | UDRM (error) | SGNI (error) | Proposed method (error) |
---|---|---|---|---|
−45.0458 | −45.3185 (0.60%) | −45.0591 (0.03%) | −45.0591 (0.03%) | |
2.6849 | 2.5567 (4.77%) | 2.6809 (0.15%) | 2.6856 (0.03%) | |
0.8739 | 0.2210 (74.71%) | 0.7780 (10.97%) | 0.8637 (1.17%) | |
4.6578 | 3.5320(24.17%) | 3.4343 (26.27%) | 4.4685 (4.06%) |