The Pareto set of a multi-objective design problem is a piecewise continuous k-dimensional manifold, and this fact has always been neglected by traditional multi-objective genetic algorithms. A multi-objective optimization algorithm based on manifold reconstruction is proposed in this paper. The manifold reconstruction algorithm is employed for building the mapping between the design space and the objective space, and the probability distribution of the solution set is built. Then the manifold structure in the objective space is extended, enabling the advancing of the solution set in the objective space to optimize the algorithm. The analytic design cases show that the proposed algorithm is adaptive to problems with diverse Pareto structure features, and the optimization efficiency is improved significantly. The proposed algorithm is also verified by multi-objective aerodynamic design problems. The results demonstrated that about 80% computational cost can be saved compared with traditional multi-objective genetic algorithms. The proposed algorithm has the ability to significantly shorten the aerodynamic design cycle.
SONG Chao
,
LI Weibin
,
ZHOU Zhu
,
LIU Hongyang
,
LAN Qingsheng
. Multi-objective aerodynamic optimization algorithm based on manifold reconstruction[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020
, 41(5)
: 623687
-623687
.
DOI: 10.7527/S1000-6893.2019.23687
[1] 黄江涛, 高正红, 余婧, 等. 大型民用飞机气动外形典型综合设计方法[J]. 航空学报, 2019, 40(2):522369. HUANG J T, GAO Z H, YU J, et al. A typical integrated design method for aerodynamic shape optimization of large civil aircraft[J]. Acta Aeronautica et Astronautica Sinica, 2019, 40(2):522369(in Chinese).
[2] 杨慧, 宋文萍, 韩忠华, 等. 旋翼翼型多目标多约束气动优化设计[J]. 航空学报, 2012, 33(7):1218-1226. YANG H, SONG W P, HAN Z H, et al. Multi-objective and multi-constrained optimization design for a helicopter rotor airfoil[J]. Acta Aeronautica et Astronautica Sinica, 2012, 33(7):1218-1226(in Chinese).
[3] 夏露, 高正红, 李天. 飞行器外形多目标多学科综合优化设计方法研究[J]. 空气动力学学报, 2003, 21(3):275-281. XIA L, GAO Z H, LI T. Investigation of integrated multi-disciplinary and multi-objective optimization of the aircraft configuration design method[J]. Acta Aerodynamica Sinica, 2003, 21(3):275-281(in Chinese).
[4] FONSECA C M, FLEMING P J. Genetic algorithms for multi objective optimization:Formulation discussion and generalization[C]//Proceeding of the Fifth International Conference, 1993:416-423.
[5] SRINIVAS N, DEB K. Multi objective optimization using nondominated sorting in genetic algorithms[J]. Evolutionary Computation, 1994, 2(3):221-248.
[6] DEB K, PRATAP A, AGARWAL S, et al. A fast and elitist multiobjective genetic algorithm:NSGA-II[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2):182-197.
[7] 李倩, 詹浩, 朱军. 基于Pareto遗传算法的机翼多目标优化设计研究[J]. 西北工业大学学报, 2010, 28(1):134-137. LI Q, ZHAN H, ZHU J. Exploring combination of Pareto Genetic Algorithm(GA) with aerodynamic analysis software for multi-objective optimization of wing[J]. Journal of Northwestern Polytechnical University, 2010, 28(1):134-137(in Chinese).
[8] 李权, 郭兆电, 雷武涛, 等. 基于工程环境的气动多目标优化设计平台研究[J]. 航空学报, 2016, 37(1):255-268. LI Q, GUO Z D, LEI W T, et al. Engineering environment-based multi-objective optimization platform for aerodynamic design[J]. Acta Aeronautica et Astronautica Sinica, 2016, 37(1):255-268(in Chinese).
[9] MARLER R T, ARORA J S. Survey of multi-objective optimization methods for engineering[J]. Structural and Multidisciplinary Optimization, 2004, 26(6):369-395.
[10] TAN K C, LEE T H, KHOR E F. Evolutionary algorithms for multi-objective optimization:Performance assessments and comparisons[J]. Artificial Intelligence Review, 2002, 17(4):251-290.
[11] KIM H J, LIOU M S. Aerodynamic optimization using a hybrid moga-local search method[C]//51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Reston:AIAA, 2010:1-23.
[12] LAUMANNS M, RUDOLPH G, SCHWEFEL H P. Mutation control and convergence in evolutionary multi-objective optimization[C]//Proceedings of the 7th International Conference on Soft Computing, 2001:24-29.
[13] LARRANAGA P, LOZANO J A. Estimation of distribution algorithms:A new tool for evolutionary computat-ion[M]. Berlin:Springer Science & Business Media, 2001.
[14] ZHOU A, ZHANG Q, JIN Y, et al. A model-based evolutionary algorithm for bi-objective optimization[C]//2005 IEEE Congress on Evolutionary Computation. Piscataway, NJ:IEEE Press, 2005:2568-2575.
[15] ZHOU A, JIN Y, ZHANG Q, et al. Combining model-based and genetics-based offspring generation for multi-objective optimization using a convergence criterion[C]//2006 IEEE International Congress on Evolutionary Computation. Piscataway, NJ:IEEE Press, 2006:3234-3241.
[16] ZHANG Q, ZHOU A, JIN Y. RM-MEDA:A regularity model-based multi-objective estimation of distribution algorithm[J]. IEEE Transactions on Evolutionary Computation, 2008, 12(1):41-63.
[17] KUHN H W, TUCKER A W. Nonlinear programming[M]. Boston:Birk-Hauser, 2014:247-258.
[18] KAMBHATLA N, LEEN T K. Dimension reduction by local principal component analysis[J]. Neural Computation, 19979(7):1493-1516.
[19] 詹炜. 求解高维多目标优化问题的流形学习算法研究[D]. 武汉:中国地质大学, 2013. ZHAN W. Research on manifold learning algorithm for high-dimensional multi-objective optimization problems[D]. Wuhan:China University of Geosciences, 2013(in Chinese).
[20] YANG D D, JIAO L C, GONG M G. Hybrid multi-objective estimation of distribution algorithm by Local Linear Embedding and an immune inspired algorithm[C]//2009 IEEE Congress on Evolutionary Computation. Piscataway, NJ:IEEE Press, 2009:463-470.
[21] LARRANAGA P, LOZANO J A. Estimation of distribution algorithms. A new tool for evolutionary computat-ion[M]. Boston:Kluwer Academic Publishers, 2002.
[22] 孟德宇, 徐晨, 徐宗本. 基于ISOMAP的流形结构重建方法[J]. 计算机学报, 2010, 33(3):545-555. MENG D Y, XU C, XU Z B. A new manifold reconstruction method based on ISOMAP[J]. Chinese Journal of Computers, 2010, 33(3):545-555(in Chinese).
[23] REYES S M, COELLO C A C. A study of fitness inheritance and approximation techniques for multi-objective particle swarm optimization[C]//2005 IEEE Congress on Evolutionary Computation. Piscataway:IEEE Press, 2005, 1:65-72.
[24] KULFAN B M. Universal parametric geometry representation method[J]. Journal of Aircraft, 2008, 45(1):142-158.
[25] SELIG M, DETERS R, WILIAMSON G. Wind tunnel testing airfoils at low Reynolds numbers[C]//49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Reston:AIAA, 2011:1-32.
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