基于新型多可信度代理模型的多目标优化方法

doi:10.7527/S1000-6893.2022.26962

Abstract

Abstract:

Compared with a conventional helicopter， a compound helicopter with its rigid coaxial rotor， famous as an Advancing Blade Concept （ABC） rotor， has more stringent requirements for the aerodynamic performance of the rotor airfoil. And the 10 more index requirements， e.g.， the low-drag and high drag-divergence characteristics at higher Mach numbers， the high lift-drag characteristics at medium and low Mach numbers， and good pitching moment characteristics at all conditions， face the problem of many-objective optimization. To solve this issue， this paper first develops a novel nonlinear dimension-reduction technology based on Kernel Principal Component Analysis （KPCA）， then establishes an efficient many-objective robust optimization framework based on a novel Adaptive Multi-Fidelity Polynomial Chaos-Kriging （AMF-PCK） surrogate model. Moreover， the paper proposes a Variable-Fidelity Pseudo Expected Improvement Matrix （VF-PEIM） parallel in-filling method， which significantly improves the efficiency and ability of many-objective optimization. The novel AMF-PCK assisted multi-objective optimization method is used to optimize the 7% thickness rotor airfoil of such compound helicopter. The aerodynamic performances of the designed airfoils are compared with those of the classical OA407 airfoil at high， medium， and low Mach numbers comprehensively. Results demonstrate the effectiveness of the proposed many-objective global optimization method and a significant improvement of high-speed aerodynamic characteristics of the designed rotor airfoils.

Key words: many-objective global optimization, rotor airfoil, multi-fidelity surrogate model, nonlinear dimension-reduction, variable-fidelity pseudo expected improvement matrix

CLC Number:

V211.41

Huan ZHAO, Zhenghong GAO, Lu XIA. Novel multi-fidelity surrogate model assisted many-objective optimization method[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023, 44(6): 126962-126962.

Figures/Tables 19

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Fig. 5

Table 2

Eigenvalues of kernel matrix

编号	$e 1$	$e 2$	$e 3$	$e 4$	$e 5$	$e 6$	$e 7$
特征值	0.632 073	0.240 908	0.082 824	0.035 851	0.005 860	0.002 158	0.000 326

Table 2

Table 3

Eigenvectors （column vector） for corresponding eigenvalues

编号	$V 1$	$V 2$	$V 3$	$V 4$	$V 5$	$V 6$	$V 7$
1	-0.378 453	-0.379 864	0.192 506	-0.606 614	0.418 121	-0.358 839	0.062 019
2	0.338 424	-0.143 240	-0.877 325	-0.210 015	0.154 978	-0.151 503	-0.064 613
3	0.396 686	-0.391 887	0.144 435	0.341 405	0.226 711	-0.118 501	0.697 285
4	0.218 579	0.671 824	0.120 248	0.071 135	0.639 493	-0.262 522	-0.059 045
5	0.450 237	0.131 841	0.232 975	-0.317 968	-0.538 201	-0.578 670	0.002 027
6	0.365 525	-0.458 845	0.254 896	0.230 028	0.203 852	-0.035 977	-0.703 646
7	0.448 089	0.048 394	0.196 654	-0.558 818	0.097 934	0.655 109	0.084 646

Table 3

Table 4

Correlation analysis of reduced objectives

目标	$f 1$	$f 5$	$f 4$	$f 2$
$f 1$	1.000 00	-0.444 42	-0.781 39	-0.537 40
$f 5$	-0.444 42	1.000 00	-0.037 86	0.598 79
$f 4$	-0.781 39	-0.037 86	1.000 00	0.104 77
$f 2$	-0.537 40	0.598 79	0.104 77	1.000 00

Table 4

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Table 5

References 38

1	DEB K， SAXEAN D. Searching for Pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems［C］∥ Piscataway： IEEE Press， 2006： 3352-3360.
2	ISHIBUCHI H， TSUKAMOTO N， NOJIMA Y. Evolutionary many-objective optimization： A short review［C］∥ 2008 IEEE Congress on Evolutionary Computation （IEEE World Congress on Computational Intelligence）. Piscataway： IEEE Press， 2008： 2419-2426.
3	PURSHOUSE R C， FLEMING P J. Evolutionary many-objective optimisation： An exploratory analysis［C］∥ The 2003 Congress on Evolutionary Computation. Piscataway： IEEE Press， 2003： 2066-2073.
4	SATO H， AGUIRRE H E， TANAKA K. Controlling dominance area of solutions and its impact on the performance of MOEAs［C］∥ Evolutionary Multi-Criterion Optimization. Berlin： Springer， 2007： 5-20.
5	DRECHSLER N， DRECHSLER R， BECKER B. Multi-objective optimisation based on relation favour［C］∥ Evolutionary Multi-Criterion Optimization. Berlin： Springer， 2001： 154-166.
6	KUKKONEN S， LAMPINEN J. Ranking-dominance and many-objective optimization［C］∥ 2007 IEEE Congress on Evolutionary Computation. Piscataway： IEEE， Press 2007： 3983-3990.
7	WAGNER T， BEUME N， NAUJOKS B. Pareto-， aggregation-， and indicator-based methods in many-objective optimization［C］∥ Evolutionary Multi-Criterion Optimization. Berlin： Springer， 2007： 742-756.
8	HUGHES E J. Evolutionary many-objective optimisation： Many once or one many？［C］∥ 2005 IEEE Congress on Evolutionary Computation. Piscataway： IEEE Press， 2005 ： 222-227.
9	ISHIBUCHI H， DOI T， NOJIMA Y. Incorporation of scalarizing fitness functions into evolutionary multiobjective optimization algorithms［C］∥ Parallel Problem Solving from Nature - PPSN IX. Berlin： Springer， 2006： 493-502.
10	ISHIBUCHI H， NOJIMA Y. Optimization of scalarizing functions through evolutionary multiobjective optimization ［C］∥ Evolutionary Multi-Criterion Optimization. Berlin： Springer， 2007： 51-65.
11	DEB K， SUNDAR J. Reference point based multi-objective optimization using evolutionary algorithms ［C］∥ Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation. New York： ACM， 2006： 635-642.
12	BROCKHOFF D， ZITZLER E. Are all objectives necessary？ On dimensionality reduction in evolutionary multiobjective Optimization［C］∥ Parallel Problem Solving from Nature - PPSN IX. Berlin： Springer， 2006： 533-542.
13	JAIMES A L， COELLO COELLO C A， CHAKRABORTY D. Objective reduction using a feature selection technique［C］∥ Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation. New York： ACM， 2008： 673-680.
14	DEB K， SAXENA D K. On finding pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems ［R］. Kanpur： Indian Institute of Technology， 2005.
15	SINGH H K， ISAACS A， RAY T. A Pareto corner search evolutionary algorithm and dimensionality reduction in many-objective optimization problems［J］. IEEE Transactions on Evolutionary Computation， 2011， 15（4）： 539-556.
16	ZHAO K， GAO Z H， HUANG J T， et al. Aerodynamic optimization of rotor airfoil based on multi-layer hierarchical constraint method［J］. Chinese Journal of Aeronautics， 2016， 29（6）： 1541-1552.
17	HUANG J T， ZHU Z， GAO Z H， et al. Aerodynamic multi-objective integrated optimization based on principal component analysis［J］. Chinese Journal of Aeronautics， 2017， 30（4）： 1336-1348.
18	SAXENA D K， DURO J A， TIWARI A， et al. Objective reduction in many-objective optimization： Linear and nonlinear algorithms［J］. IEEE Transactions on Evolutionary Computation， 2012， 17（1）： 77-99.
19	ZHAO H， GAO Z H， XU F， et al. Review of robust aerodynamic design optimization for air vehicles［J］. Archives of Computational Methods in Engineering， 2019， 26（3）： 685-732.
20	ZHAO H， GAO Z H. Uncertainty-based design optimization of NLF airfoil for high altitude long endurance unmanned air vehicles［J］. Engineering Computations， 2019， 36（3）： 971-996.
21	赵欢，高正红，夏露. 高速自然层流翼型高效气动稳健优化设计方法［J］. 航空学报， 2022， 43（1）： 124894.
	ZHAO H， GAO Z H， XIA L. Efficient robust aerodynamic design optimization method for high-speed NLF airfoil［J］. Acta Aeronautica et Astronautica Sinica， 2022， 43（1）： 124894 （in Chinese）.
22	ZHAO H， GUO Z H， XU F， et al. An efficient adaptive forward-backward selection method for sparse polynomial chaos expansion［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 355： 456-491.
23	PALAR P S， SHIMOYAMA K. On efficient global optimization via universal Kriging surrogate models［J］. Structural and Multidisciplinary Optimization， 2018， 57（6）： 2377-2397.
24	ZHAO H， GAO Z H， XU F， et al. Adaptive multi-fidelity sparse polynomial chaos-Kriging metamodeling for global approximation of aerodynamic data［J］. Structural and Multidisciplinary Optimization， 2021， 64（2）： 829-858.
25	SCHOBI R， SUDRET B， WIART J. Polynomial-chaos-based kriging［J］. International Journal for Uncertainty Quantification， 2015， 5（2）： 171-193.
26	TENENBAUM J B， DE SILVA V， LANGFORD J C. A global geometric framework for nonlinear dimensionality reduction［J］. Science， 2000， 290（5500）： 2319-2323.
27	WEINBERGER K Q， SAUL L K. Unsupervised learning of image manifolds by semidefinite programming［J］. International Journal of Computer Vision， 2006， 70（1）： 77-90.
28	SAXENA D K， DEB K. Non-linear dimensionality reduction procedures for certain large-dimensional multi-objective optimization problems： Employing correntropy and a novel maximum variance unfolding［C］∥ Evolutionary Multi-Criterion Optimization. Berlin： Springer， 2007： 772-787.
29	STURM J F. Using SeDuMi 1.02， a Matlab toolbox for optimization over symmetric cones［J］. Optimization Methods and Software， 1999， 11： 625-653.
30	KEANE A J. Statistical improvement criteria for use in multiobjective design optimization［J］. AIAA Journal， 2006， 44（4）： 879-891.
31	BAUTISTA D C T. A sequential design for approximating the pareto front using the expected pareto improvement function［M］. Columbus： The Ohio State University， 2009.
32	ZITZLER E. Evolutionary algorithms for multiobjective optimization： Methods and applications［M］. Ithaca： Shaker， 1999.
33	KNOWLES J. ParEGO： A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems［J］. IEEE Transactions on Evolutionary Computation， 2006， 10（1）： 50-66.
34	ZHAN D W， CHENG Y S， LIU J. Expected improvement matrix-based infill criteria for expensive multiobjective optimization［J］. IEEE Transactions on Evolutionary Computation， 2017， 21（6）： 956-975.
35	ZHAN D W， QIAN J C， LIU J， et al. Pseudo expected improvement matrix criteria for parallel expensive multi-objective optimization［C］∥ World Congress of Structural and Multidisciplinary. Cham： Springer， 2017： 175-190.
36	赵欢，高正红，夏露．基于新型高维代理模型的气动外形设计方法研究［J］．航空学报， 2023， 44（1）： 126924.
	ZHAO H， GAO Z H， XIA L， et al. Research on novel high-dimensional surrogate model-based aerodynamic shape design optimization method［J］. Acta Aeronautica et Astronautica Sinica， 2023， 44（1）： 126924 （in Chinese）.
37	ZHAO H， GUO Z H， XIA L. Efficient aerodynamic analysis and optimization under uncertainty using multi-fidelity polynomial chaos-Kriging surrogate model［J］. Computers & Fluids， 2022， 246： 105643
38	赵欢. 基于自适应多可信度多项式混沌-Kriging模型的高效气动优化方法［J］. 力学学报， 2023， 55（1）： 1-16.
	ZHAO H. Adaptive multi-fidelity polynomial chaos-kriging model-based efficient aerodynamic design optimization method［J］. Chinese Journal of Theoretical and Applied Mechanics， 2023， 55（1）： 1-16 （in Chinese）.

Airfoil	Ma_dd0	Ma=0.87		Ma=0.60		Ma=0.50		Ma=0.40		Ma=0.30
Airfoil	Ma_dd0	C_d₀	C_m0	C_l_max	K_max	C_l_max	K_max	C_l_max	K_max	C_l_max	K_max
OA407	0.854	0.010 84	-0.029 5	0.911 1	75.77	1.083 7	90.77	1.182 2	86.18	1.262 1	80.64
OPT0701	0.861	0.008 92	-0.023 2	0.946 7	76.81	1.098 3	91.89	1.193 5	87.31	1.285 6	82.55
OPT0702	0.869	0.008 05	-0.015 2	0.923 3	76.01	1.075 4	82.14	1.053 5	79.77	1.120 4	76.56
OPT0703	0.875	0.006 73	-0.003 5	0.874 5	73.06	0.909 8	64.50	0.947 2	61.44	0.936 5	59.15

[1]	Jiaqi LIU, Rongqian CHEN, Jinhua LOU, Xu HAN, Hao WU, Yancheng YOU. Aerodynamic shape optimization of high-speed helicopter rotor airfoil based on deep learning [J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(9): 529828-529828.
[2]	Huan ZHAO, Zhenghong GAO, Lu XIA. Aerodynamic shape design optimization method based on novel high⁃dimensional surrogate model [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023, 44(5): 126924-126924.
[3]	YANG Hui, SONG Wenping, HAN Zhonghua, XU Jianhua. Multi-objective and Multi-constrained Optimization Design for a Helicopter Rotor Airfoil [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, 33(7): 1218-1226.

Novel multi-fidelity surrogate model assisted many-objective optimization method

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 19

References 38

Related Articles 3

Recommended Articles

Metrics

Comments