基于多项式混沌法的翼型不确定性分析及梯度优化设计

doi:10.7527/S1000-6893.2022.27446

Abstract

Abstract:

An efficient and reliable uncertainty gradient optimization design method is developed by coupling the adjoint method and the non-intrusive polynomial chaos method. Using the adjoint equation method to solve the derivative of the objective function with respect to the uncertain variable， we develop a gradient-enhanced polynomial chaos expansion method. Various examples at subsonic and transonic speeds prove that this method can improve the efficiency and accuracy of uncertainty analysis. Meanwhile， the sensitivity of uncertain variables is quantified using a global sensitivity analysis method based on variance decomposition. A statistical moment gradient solution method for the coupled adjoint equation of polynomial chaos is established， and an uncertain gradient optimization design system built by combining the gradient-enhanced polynomial chaos expansion method. Based on the optimization design system， the deterministic and uncertain optimization design research of two-dimensional low subsonic and transonic airfoils is conducted. The optimization results show that， compared with the deterministic optimal design， the uncertain optimal design can improve the ability to resist the uncertainty perturbation of Mach number and angle of attack by reasonably balancing the deterministic performance and the uncertain performance， and optimize the average performance and performance robustness. The mean value and the standard deviation of the drag coefficient can be reduced by up to 17% and 80%， respectively. The deterministic optimization design may lead to a decrease in performance robustness.

Key words: polynomial chaos expansion, adjoint equations, uncertainty analysis, gradient optimization design, airfoil

CLC Number:

V224

Yifu CHEN, Yuhang MA, Qingsheng LAN, Weiping SUN, Yayun SHI, Tihao YANG, Junqiang BAI. Uncertainty analysis and gradient optimization design of airfoil based on polynomial chaos expansion method[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023, 44(8): 127446.

Figures/Tables 36

Fig. 1

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Table 1

Table 2

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 3

Subsonic airfoil： Monte Carlo and PCE calculation results

Method	CFD evaluations	$m e a n (C L)$	$m e a n (C D)$	$m e a n (C m z)$	$s t d (C L)$	$s t d (C D)$	$s t d (C m z)$
MC	5 000	0.196 4	0.009 151	-3.138×10^-4	0.025 96	5.642×10^-5	1.117×10^-4
MC	10 000	0.196 4	0.009 151	-3.138×10^-4	0.026 03	5.668×10^-5	1.117×10^-4
PCE（p=1）	4	0.196 3	0.009 148	-3.061×10^-4	0.026 28	5.438×10^-5	1.172×10^-4
PCE（p=2）	9	0.196 4	0.009 152	-3.127×10^-4	0.026 34	5.704×10^-5	1.119×10^-4
PCE（p=3）	16	0.196 4	0.009 152	-3.061×10^-4	0.026 32	6.409×10^-5	1.256×10^-4
GPCE（p=1）	4	0.196 3	0.009 147	-3.146×10^-4	0.026 31	5.720×10^-5	1.085×10^-4
GPCE（p=2）	9	0.196 4	0.009 151	-3.142×10^-4	0.026 31	5.669×10^-5	1.127×10^-4
GPCE（p=3）	16	0.196 4	0.009 151	-3.141×10^-4	0.025 94	5.720×10^-5	1.118×10^-4

Table 3

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Table 4

Transonic airfoils： Monte Carlo and PCE calculation results

Method	CFD evaluations	$m e a n (C L)$	$m e a n (C D)$	$m e a n (C m z)$	$s t d (C L)$	$s t d (C D)$	$s t d (C m z)$
MC	5 000	0.823 7	0.023 11	0.106 9	0.047 81	0.007 812	0.009 361
MC	10 000	0.823 3	0.023 12	0.106 9	0.047 77	0.007 860	0.009 247
PCE（p=1）	4	0.846 1	0.022 48	0.107 9	0.042 38	0.009 672	0.013 290
PCE（p=2）	9	0.815 3	0.022 58	0.105 1	0.062 18	0.009 673	0.013 620
PCE（p=3）	16	0.821 6	0.023 24	0.106 9	0.074 70	0.006 973	0.009 014
PCE（p=4）	25	0.821 9	0.023 14	0.106 8	0.051 86	0.007 205	0.008 989
GPCE（p=1）	4	0.844 4	0.022 58	0.107 9	0.041 37	0.009 283	0.011 230
GPCE（p=2）	9	0.817 3	0.023 35	0.106 7	0.060 30	0.009 216	0.011 810
GPCE（p=3）	16	0.816 5	0.022 97	0.106 9	0.062 20	0.007 078	0.009 289
GPCE（p=4）	25	0.827 7	0.023 12	0.107 2	0.047 89	0.007 809	0.009 438

Table 4

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Fig. 20

Fig. 21

Table 5

Low subsonic airfoil optimization results （Drag coefficient in counts）

Configuration	$C D$	$C D p$	$C D v$	$C L$	$C m$	$α$ /（°）	$μ C D$	$σ C D$
Initial	98.79	27.38	71.41	0.299	-0.070 1	0.597 5	110.65	26.52
DeOpt	88.76	16.85	71.91	0.300	-0.042 9	0.597 5	93.04	8.40
UMOpt	89.46	17.20	72.26	0.299	-0.059 0	0.597 5	91.98	5.43

Table 5

Fig. 22

Fig. 23

Fig. 24

Fig. 25

Fig. 26

Table 6

Transonic airfoil optimization results （Drag coefficient in counts）

Configuration	$C D$	$C D p$	$C D v$	$C L$	$C m$	$α$ /（°）	$μ C D$	$σ C D$
Initial	94.46	32.57	61.89	0.379	-0.094	0.387	94.72	1.56
DeOpt	88.82	27.74	61.79	0.379	-0.044	0.387	91.44	5.11
UMOpt	88.96	27.17	61.79	0.379	-0.045	0.387	89.19	1.60

Table 6

Fig. 27

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Fig. 29

Fig. 30

References 45

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设计变量	伴随方程梯度	有限差分梯度	相对误差	最优步长
2	1.031 81	1.031 37	4.17×10^-4	10^-4
10	4.517 75	4.517 47	6.19×10^-4	10^-3
15	1.274 37	1.274 12	1.96×10^-4	10^-3
22	0.067 760	0.067 761	1.47×10^-5	10^-3

设计变量	伴随方程梯度	有限差分梯度	相对误差	最优步长
2	-1.711×10^-4	-1.718×10^-4	4.07×10^-3	10^-5
10	3.218×10^-4	3.215×10^-4	9.33×10^-4	10^-4
15	8.548×10^-3	8.555×10^-3	8.18×10^-4	10^-3
22	-7.370×10^-4	-7.331×10^-4	5.32×10^-3	10^-3

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Uncertainty analysis and gradient optimization design of airfoil based on polynomial chaos expansion method

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