ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Kriging surrogate model and its application to design optimization: A review of recent progress
Received date: 2016-01-05
Revised date: 2016-03-15
Online published: 2016-03-29
Supported by
National Natural Science Foundation of China (11272265)
Over the past two decades, surrogate modeling has received much attention from the researchers in the area of aerospace science and engineering due to its capability of greatly improving the efficiency of design optimization when high-fidelity numerical analysis is employed. Design optimization via surrogate models is intensively researched and eventually leads to a new type of optimization algorithm which is called surrogate-based optimization (SBO). Among the available surrogate models, such as polynomial response surface model, radial-basis functions, artificial neutral network, support-vector regression, multivariate interpolation or regression, and polynomial chaos expansion, Kriging model is the most representative surrogate model which has great potential in engineering design and optimization. In the context of aircraft design, this paper reviews the theory, algorithm and recent progress for researches on the Kriging surrogate model. First, the fundamental theory and algorithm of Kriging model are briefly reviewed and the experience about how to improve the robustness and efficiency is presented. Second, three major breakthroughs of Kriging model in recent years are reviewed, including gradient-enhanced Kriging, CoKriging and hierarchical Kriging. Third, the optimization mechanism and framework of surrogate-based optimization using Kriging model are discussed. In the meanwhile, the concept of infill-sampling criterion and sub optimization is presented. Five infill-sampling criteria as well as the dedicated constraint handling methods are described. Furthermore, the newly developed local EI (expected improvement) method and termination criteria for SBO are introduced. Fourth, a number of test cases including benchmark optimization problems as well as aerodynamic and multidisciplinary design optimization problems are given to demonstrate the excellent performance and great potential of the surrogate-based optimization using Kriging model. At last, the key challenges as well as future directions about the theory, algorithm and applications are discussed.
HAN Zhonghua . Kriging surrogate model and its application to design optimization: A review of recent progress[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2016 , 37(11) : 3197 -3225 . DOI: 10.7527/S1000-6893.2016.0083
[1] HERNANDEZ S. Structural optimization:1960-2010 and beyond[J]. Computational Technology Reviews, 2010, 2:177-222.
[2] SCHMIT L A. Structural design by systematic synthesis[C]//Second Conference on Electronic Computation. Pittsburg:ASCE, 1960:105-132.
[3] HICKS R M, MUNNAN E M, VANDERPLAATS G N. An assessment of airfoil design by numerical optimization:NASA TM X-3092[R]. Washington, D.C.:NASA, 1974.
[4] HICKS R M, HENNE P A. Wing design by numerical optimization[J]. Journal of Aircraft, 1978, 15(7):407-412.
[5] SOBIESZCZANSKI-SOBIESKI J, HAFTKA R T. Multidisciplinary aerospace design optimization:Survey of recent developments[J]. Structural Optimization, 1997, 14(1):1-23.
[6] KROO I, ALTUS S, BRAUN R, et al. Multidisciplinary optimization methods for aircraft preliminary design:AIAA-1994-4325[R]. Reston:AIAA, 1994.
[7] AIAA MDO Technical Committee. Current state of the art:Multidisciplinary design optimization:AIAA White Paper[R]. Reston:AIAA, 1991.
[8] ZHANG K S, HAN Z H, LI W J, et al. Bilevel adaptive weighted sum method for multidisciplinary multi-objective optimization[J]. AIAA Journal, 2008, 46(10):2611-2622.
[9] 余雄庆. 飞机总体多学科设计优化的现状与发展方向[J]. 南京航空航天大学学报, 2008, 40(4):417-426. YU X Q. Multidisciplinary design optimization for aircraft conceptual and preliminary design:Status and directions[J]. Journal of Nanjing University of Aeronautics & Astronautics, 2008, 40(4):417-426(in Chinese).
[10] 穆雪峰, 姚卫星, 余雄庆, 等. 多学科设计优化中常用代理模型的研究[J]. 计算力学学报, 2005, 22(5):608-612. MU X F, YAO W X, YU X Q, et al. A survey of surrogate models used in MDO[J]. Chinese Journal of Computational Mechanics, 2005, 22(5):608-612(in Chinese).
[11] VIANA F A C, SIMPSON T W, BALABANOV V, et al. Metamodeling in multidisciplinary design optimization:how far have we really come?[J]. AIAA Journal, 2014, 52(4):670-690.
[12] 张健, 李为吉. 飞机多学科设计优化中的近似方法分析[J]. 航空计算技术, 2005, 35(3):5-8. ZHANG J, LI W J. Approximation methods analysis in multidisciplinary design optimization[J]. Aeronautical Computer Technique, 2005, 35(3):5-8(in Chinese).
[13] HAN Z H, GOERTZ S. A hierarchical Kriging model for variable-fidelity surrogate modeling[J]. AIAA Journal, 2012, 50(3):1885-1896.
[14] HAN Z H, ZIMMERMANN R, GOERTZ S. An alternative CoKriging model for variable-fidelity surrogate modeling[J]. AIAA Journal, 2012, 50(5):1205-1210.
[15] HAN Z H, ZHANG K S. Surrogate-based optimization[M]. InTech Book, 2012:343-362.
[16] SCHMIT L A, FARSHI B. Some approximation concepts for structural synthesis[J]. AIAA Journal, 1974, 12(5):692-699.
[17] GIUNTA A A, WATSON L T. A Comparison of approximation modeling techniques:Polynomial versus interpolation models:AIAA-1998-4758[R]. Reston:AIAA, 1998.
[18] KOCH P N, SIMPSON T W, ALLEN J K, et al. Statistical approximations for multidisciplinary design optimization:the problem of the size[J]. Journal of Aircraft, 1999, 36(1):275-286.
[19] SEVANT N E, BLOOR M I G, WILSON M J. Aerodynamic design of a flying wing using response surface methodology[J]. Journal of Aircraft, 2000, 37(4):562-569.
[20] JEONG S, MURAYAMA M, YAMAMOTO K. Efficient optimization design method using Kriging model[J]. Journal of Aircraft, 2005, 42(2):413-420.
[21] VAVALLE A, QIN N. Iterative response surface based optimization scheme for transonic airfoil design[J]. Journal of Aircraft, 2007, 44(2):365-376.
[22] KRIGE D G. A statistical approach to some basic mine valuations problems on the witwatersrand[J]. Journal of the Chemical, Metallurgical and Mining Engineering Society of South Africa, 1951, 52(6):119-139.
[23] SACKS J, WELCH W J, MITCHELL T J, et al. Design and analysis of computer experiments[J]. Statistical Science, 1989, 4(4):409-423.
[24] POWELL M J D. Algorithms for approximation[M]. New York:Oxford University Press, 1987:141-167.
[25] BUHMANN M D. Acta numerica[M]. New York:Cambridge University Press, 2000:1-38.
[26] KRISHNAMURTHY T. Response surface approximation with augmented and compactly supported radial basis functions:AIAA-2003-1748[R]. Reston:AIAA, 2003.
[27] MULLUR A A, MESSAC A. Extended radial basis functions:more flexible and effective metamodeling[J]. AIAA Journal, 2005, 43(6):1306-1315.
[28] PARK J, SANDBERG I W. Universal approximation using radial-basis-function networks[J]. Neural Computation, 1991, 3(2):246-257.
[29] ELANAYAR S V T, SHIN Y C. Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems[J]. IEEE Transactions on Neural Networks, 1994, 5(4):594-603.
[30] SMOLA A J, SCHÖLKOPF B. A tutorial on support vector regression[J]. Statistics and Computing, 2004, 14(3):199-222.
[31] FORRESTER A I J, KEANE A J. Recent advances in surrogate-based optimization[J]. Progress in Aerospace Sciences, 2009, 45(1):50-79.
[32] WANG Q, MOIN P, IACCARINO G. A rational interpolation scheme with super-polynomial rate of convergence[J]. SIAM Journal of Numerical Analysis, 2010, 47(6):4073-4097.
[33] WANG Q, MOIN P, IACCARINO G. A high-order multi-variate approximation scheme for arbitrary data sets[J]. Journal of Computational Physics, 2010, 229(18):6343-6361.
[34] WIENER N. The homogeneous chaos[J]. American Journal of Mathematics, 1938, 60(4):897-936.
[35] XIU D. Numerical methods for stochastic computations:A spectral method approach[M]. Princeton:Princeton University Press, 2010:152.
[36] QUEIPO N V, HAFTKA R T, SHYY W, et al. Surrogate-based analysis and optimization[J]. Progress in Aerospace Sciences, 2005, 41(1):1-28.
[37] FORRESTER A I J, SÓBESTER A, KEANE A. Engineering design via surrogate modeling:A practical guide[M]. Chichester:John Wiley & Sons, 2008.
[38] KEANE A J, NAIR P B. Computational approaches for aerospace design:The pursuit of excellence[M]. Chichester:John Wiley & Sons, 2005.
[39] GIUNTA A A, WOJTKIEWICZ J S F, ELDRED M S. Overview of modern design of experiments methods for computational simulations:AIAA-2003-649[R]. Reston:AIAA, 2003.
[40] FANG K T, LIN D, WINKER P, et al. Uniform design:Theory and application[J]. Technometrics, 2000, 42(3):237-248.
[41] MATHERON G M. Principles of geostatistics[J]. Economic Geology, 1963, 58(8):1246-1266.
[42] MATHERON G. Theory of regionalized variable and its applications[M]. Fontainebleau:Ecole des Mines, 1971.
[43] SIMPSON T W, MAUERY T M, KORTE J J, et al. Kriging models for global approximation in simulation-based multidisciplinary design optimization[J]. AIAA Journal, 2001, 39(12):2233-2241.
[44] MARTIN J D, SIMPSON T W. Use of Kriging models to approximate deterministic computer models[J]. AIAA Journal, 2005, 43(4):853-863.
[45] TOAL D J J, BRESSLOFF N W, KEAN A J. Kriging hyperparameter tuning strategies[J]. AIAA Journal, 2008, 46(5):1240-1252.
[46] 王晓锋, 席光. 基于Kriging模型的翼型气动性能优化设计[J]. 航空学报, 2005, 26(5):545-549. WANG X F, XI G. Aerodynamic optimization design for airfoil based on Kriging model[J]. Acta Aeronautica et Astronautica Sinica, 2005, 26(5):545-549(in Chinese).
[47] 许瑞飞, 宋文萍, 韩忠华. 改进Kriging模型在翼型气动优化设计中的应用研究[J]. 西北工业大学学报, 2010, 28(4):503-510. XU R F, SONG W P, HAN Z H. Application of improved kriging-model-based optimization method in airfoil aerodynamic design[J]. Journal of Northwestern Polytechnical University, 2010, 28(4):503-510(in Chinese).
[48] LIU J, HAN Z H, SONG W P. Efficient kriging-based optimization design of transonic airfoils:Some key issues:AIAA-2012-0967[R]. Reston:AIAA, 2012.
[49] 孙俊峰, 刘刚, 江雄, 等. 基于Kriging模型的旋翼翼型优化设计研究[J]. 空气动力学学报, 2013, 31(4):437-441. SUN J F, LIU G, JIANG X, et al. Research of rotor airfoil design optimization based on the Kriging model[J]. Acta Aerodynamica Sinica, 2013, 31(4):437-441(in Chinese).
[50] HAN Z H, LIU J, SONG W P, et al. Surrogate-based aerodynamic shape optimization with application to wind turbine airfoils:AIAA-2013-1108[R]. Reston:AIAA, 2013.
[51] 白俊强, 王波, 孙智伟, 等. 基于松散式代理模型管理框架的亚音速机翼优化设计方法研究[J]. 西北工业大学学报, 2011, 29(4):515-519. BAI J Q, WANG B, SUN Z W, et al. Developing optimization design of subsonic wing with losse type of agent model[J]. Journal of Northwestern Polytechnical University, 2011, 29(4):515-519(in Chinese).
[52] 孙美建, 詹浩. Kriging模型在机翼气动外形优化中的应用[J]. 空气动力学学报, 2011, 29(6):759-764. SUN M J, ZHAN H. Application of Kriging surrogate model for aerodynamic shape optimization of wing[J]. Acta Aerodynamica Sinica, 2011, 29(6):759-764(in Chinese).
[53] 何欢, 朱广荣, 何成, 等. 基于Kriging模型的结构耐撞性优化[J]. 南京航空航天大学学报, 2014, 46(2):297-303. HE H, ZHU G R, HE C, et al. Crashworthiness optimization based on Kriging metamodeling[J]. Journal of Nanjing University of Aeronautics & Astronautics, 2014, 46(2):297-303(in Chinese).
[54] 刘克龙, 姚卫星, 穆雪峰. 基于Kriging代理模型的结构形状优化方法研究[J]. 计算力学学报, 2006, 23(3):344-347, 362. LIU K L, YAO W X, MU X F. A method of structural shape optimization based on Kriging model[J]. Chinese Journal of Computational Mechanics, 2006, 23(3):344-347, 362(in Chinese).
[55] 杨军, 刘勇琼, 艾春安, 等. 改进Kriging模型在固冲发动机导弹气动优化设计中的应用[J]. 固体火箭技术, 2013, 36(5):590-593. YANG J, LIU Y Q, AI C A, et al. Application of improved Kriging-model-based optimization method in solid rocket-ramjet missile's aerodynamic design[J]. Journal of Solid Rocket Technology, 2013, 36(5):590-593(in Chinese).
[56] 孙凯军, 宋文萍, 韩忠华. 基于Kriging模型的高超声速舵面优化设计[J]. 航空计算技术, 2012, 42(2):9-12. SUN K J, SONG W P, HAN Z H. Optimization design of hypersonic control surface based on Kriging model[J]. Aeronautical Computing Technique, 2012, 42(2):9-12(in Chinese).
[57] 郑侃, 廖文和, 张翔. 基于近似模型管理的微小卫星结构多目标优化设计[J]. 中国机械工程, 2012, 23(6):655-659. ZHENG K, LIAO W H, ZHANG X. Multi-objective optimization design for microsatellite structure based on approximation model management[J]. China Mechanical Engineering, 2012, 23(6):655-659(in Chinese).
[58] 姚拴宝, 郭迪龙, 孙振旭, 等. 基于Kriging代理模型的高速列车头型多目标优化设计[J]. 中国科学, 2013, 43(2):186-200. YAO S B, GUO D L, SUN Z X, et al. Multi-objective optimization of the streamlined head of high-speed trains based on the kriging model[J]. Science China, 2013, 43(2):186-200(in Chinese).
[59] 韩永志, 高行山, 李立州. 基于Kriging模型的涡轮叶片多学科设计优化[J]. 航空动力学报, 2007, 22(7):1055-1059. HAN Y Z, GAO X S, LI L Z. Kriging model-based multidisciplinary design optimization for turbine blade[J]. Journal of Aerospace Power, 2007, 22(7):1055-1059(in Chinese).
[60] 张科施, 韩忠华, 李为吉, 等. 基于近似技术的高亚声速运输机机翼气动/结构优化设计[J]. 航空学报, 2006, 27(5):810-815. ZHANG K S, HAN Z H, LI W J, et al. Multidisciplinary aerodynamic/structural design optimization for high subsonic transport wing using approximation technique[J]. Acta Aeronautica et Astronautica Sinica, 2006, 27(5):810-815(in Chinese).
[61] SUN W Y, YUAN Y X. Optimization theory and methods:nonlinear programming[M]. New York:Springer Ebooks, 2006.
[62] HOLLAND J H. Adaptation in natural and artificial systems[M]. Ann Arbor:The University of Michigan Press, 1975.
[63] SLOTNICK J, KHODADOUST A, ALONSO J, et al. CFD vision 2030 study:A path to revolutionary computational aerosciences:NASA/CR-2014-218178[R]. Washington, D.C.:NASA Langley Research Center, 2014.
[64] 吴宽展, 刘学军, 吕宏强. 超临界翼型设计中的多响应代理模型[J]. 航空计算技术, 2014, 44(4):17-22. WU K Z, LIU X J, LV H Q. Multi-output surrogate model in supercritical airfoil design[J]. Aeronautical Computing Technique, 2014, 44(4):17-22(in Chinese).
[65] ZIMMERMANN R. On the condition number anomaly of gaussian correlation matrices[J]. Linear Algebra and its Applications, 2015, 466:521-526.
[66] VIANA F A, HAFTKA R T, STEFFEN V. Multiple surrogates:How cross-validation errors can help us to obtain the best predictor?[J]. Structural and Multidisciplinary Optimization, 2009, 39(4):439-457.
[67] KOBLEWALIK J, OSBORNE M R. Methods for unconstrained optimization problems[M]. New York:Elsevier, 1968.
[68] NELDER J A, MEAD R. A simple method for function minimization[J]. The Computer Journal, 1965, 7(4):308-313.
[69] LOPHAVEN S N, NIELSEN H B, SØNDERGAARD J. DACE-A MATLAB Kriging toolbox (version 2.0):IMM-REP-2002-12[R]. Lyngby:Informatics and Mathematical Modelling, Technical University of Denmark, 2002.
[70] YIN J, NG S H, NG K M. Kriging meta model with modified nugget-effect:The heteroscedastic variance case[J]. Computers Industrial Engineering, 2011, 61(3):760-777.
[71] FORRESTER A I J, KEANE A J, BRESSLOFF N W. Design and analysis of noisy computer experiments[J]. AIAA Journal, 2006, 44(10):2331-2339.
[72] ETMAN L F P. Design and analysis of computer experiments:The Method of Sacks et al:Engineering Mechanics report WFW 94-098[R]. Eindhoven, The Netherlands:Eindhoven University of Technology, 1994.
[73] OSIO I G, AMON C H. An engineering design methodology with multistage bayesian surrogate and optimal sampling[J]. Research in Engineering Design, 1996, 8(4):189-206.
[74] JONES D R, SCHONLAU M, WELCH W J. Efficient global optimization of expensive black-box functions[J]. Journal of Global Optimization, 1998, 13(4):455-492.
[75] COX D D, JOHN S. SDO:A statistical method for global optimization[C]//IEEE International Conference on Systems, Man & Cybernetics, 1992:1241-1246.
[76] TORCZON V, TROSSET M W. Use approximation to accelerate engineering design optimization:AIAA-1998-4800[R]. Reston:AIAA, 1998.
[77] BOOKER A J. Design and analysis of computer experiments:AIAA-1998-4757[R]. Reston:AIAA, 1998.
[78] MECKESHEIMER M, BARTON R R, SIMPSON T, et al. Metamodeling of combined discrete/continuous responses[J]. AIAA Journal, 2001, 39(10):1950-1959.
[79] CHEN S K, XIONG Y, CHE W. Multiresponse and multistage metamodeling approach for design optimization[J]. AIAA Journal, 2009, 47(1):206-218.
[80] HAN Z H, ZHANG K S, SONG W P, et al. Optimization of active flow control over an airfoil using a surrogate-management framework[J]. Journal of Aircraft, 2010, 47(2):603-612.
[81] ZHANG K S, HAN Z H, LI W J, et al. Coupled aerodynamic/structural optimization of a subsonic transport wing using a surrogate model[J]. Journal of Aircraft, 2008, 45(6):2167-2171.
[82] CHUNG J J, ALONSO H S. Using gradients to construct cokriging approximation models for high-dimensional design optimization problems:AIAA-2002-0317[R]. Reston:AIAA, 2002.
[83] CHUNG J J, ALONSO H S. Design of a low-boom supersonic business jet using cokriging approximation models:AIAA-2002-5598[R]. Reston:AIAA, 2002.
[84] LIU W, BATILL S M. Gradient-enhanced response surface approximations using kriging models:AIAA-2002-5456[R]. Reston:AIAA, 2002.
[85] LAURENCEAU J, SAGAUT P. Building efficient response surfaces of aerodynamic functions with Kriging and Cokriging[J]. AIAA Journal, 2008, 46(2):498-507.
[86] JAMESON A. Optimum aerodynamic design using cfd and control theory:AIAA-1995-1729[R]. Reston:AIAA, 1995.
[87] DWIGHT R, HAN Z H. Efficient uncertainty quantification using gradient enhanced Kriging:AIAA-2009-2276[R]. Reston:AIAA, 2009.
[88] HAN Z H, GOERTZ S, ZIMMERMANN R. Improving variable-fidelity surrogate modeling via gradient-enhanced Kriging and a generalized hybrid bridge function[J]. Aerospace Science and Technology, 2013, 25(1):177-189.
[89] YAMAZAKI W, RUMPFKEIL M P, MAVRIPLIS D J. Design optimization utilizing gradient/hessian enhanced surrogate model:AIAA-2010-4363[R]. Reston:AIAA, 2010.
[90] HAN Z H. Improving adjoint-based aerodynamic optimization via gradient-enhanced Kriging:AIAA-2012-0670[R]. Reston:AIAA, 2012.
[91] DAVID M. Geostatistical ore reserve estimation[M]. Amsterdam:Elsevier, 1977:1-364.
[92] JOURNEL A G, HUIJBREGTS J C. Mining geostatistics[M]. New York:Academic Press, 1978:1-600.
[93] MYERS D E. Matrix formulation of Cokriging[J]. Mathematical Geology, 1982, 14(3):249-257.
[94] GOOVAERTS P. Ordinary CoKriging revisited[J]. Mathematical Geology, 1997, 30(1):21-42.
[95] PAPRITZ A. Standardized vs customary ordinary CoKriging:some comments on the article "the geostatistical analysis of experiments at the landscape-scale" by T.F.A. Bishop and R.M., Lark[J]. Geoderma, 2008, 146(1):391-396.
[96] KENNEDY M C, O'HAGAN A. Predicting the output from a complex computer code when fast approximations are available[J]. Biometrika, 2000, 87(1):1-13.
[97] QIAN Z, WU C F J. Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments[J]. Technometrics, 2008, 50(2):192-204.
[98] FORRESTER A I J, SÓBESTER A, KEANE A J. Multi-fidelity optimization via surrogate modeling[J]. Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, 2007, 463(2088):3251-3269.
[99] KUYA Y, TAKEDA K, ZHANG X, et al. Multifidelity surrogate modeling of experimental and computational aerodynamic data sets[J]. AIAA Journal, 49(2):289-298.
[100] ZIMMERMANN R, HAN Z H. Simplified cross-correlation estimation for multi-fidelity surrogate cokriging models[J]. Advance and Applications in Mathematical Sciences, 2010, 7(2):181-201.
[101] HAN Z H, ZIMMERMANN R, GOERTZ S. A new CoKriging method for variable-fidelity surrogate modeling of aerodynamic data:AIAA-2010-1225[R]. Reston:AIAA, 2010.
[102] JOSEPH V R, HUNG Y, SUDJIANTO A. Blind Kriging:A new method for developing metamodels[J]. Journal of Mechanical Design, 2008, 130(3):350-353.
[103] ZHAO L, CHOI K K, LE I. Metamodeling method using dynamic Kriging for design optimization[J]. AIAA Journal, 2011, 49(9):2034-2046.
[104] LIANG H Q, ZHU M, WU Z. Using cross-validation to design trend function in Kriging surrogate modeling[J]. AIAA Journal, 2014, 52(10):2313-2327.
[105] ANKENMAN B E, NELSON B L, STAUM J. Stochastic Kriging for simulation metamodeling[J]. Operations Research, 2010, 58(2):371-382.
[106] LIU J, HAN Z H, SONG W P. Comparison of infill sampling criteria in kriging-based aerodynamic optimization[C]//28th Congress of the International Council of the Aeronautical Sciences, 2012.
[107] JONES D R. A taxonomy of global optimization methods based on response surfaces[J]. Journal of Global Optimization,2001, 21(4):345-383.
[108] BOOKER A J, DENNIS J J, FRANK P D, et al. A rigorous framework for optimization of expensive functions by surrogates[J]. Structural Optimization, 1998, 17(1):1-13.
[109] SASENA M J, PAPALAMBROS P Y, GOOVAERTS P. Exploration of metamodeling sampling criteria for constrained global optimization[J]. Engineering Optimization, 2002, 34(34):263-278.
[110] PARR J M, KEANE J A, FORRESTER I J, et al. Infill sampling criteria for surrogate-based optimization with constraint handling[J]. Engineering Optimization, 2012, 44(10):1147-1166.
[111] DEB K. An efficient constraint handling method for genetic algorithms[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 186(2-4):311-338
[112] ONG Y S, NAIR P B, KEANE A J. Evolutionary optimization of computationally expensive problems via surrogate modeling[J]. AIAA Journal, 2003, 41(4):687-696.
[113] 刘俊. 基于代理模型的高效气动优化设计方法及应用[D]. 西安:西北工业大学, 2015. LIU J. Efficient surrogate-based optimization method and its application in aerodynamic design[D]. Xi'an:Northwestern Polytechnical University, 2015(in Chinese).
[114] 高月华, 王希诚. 基于Kriging代理模型的多点加点序列优化方法[J]. 工程力学, 2012, 29(4):90-95. GAO Y H, WANG X C. A sequential optimization method with multi-point sampling criterion based on kriging surrogate mode[J]. Engineering Mechanics, 2012, 29(4):90-95(in Chinese).
[115] 刘俊, 宋文萍, 韩忠华, 等. 梯度增强的Kriging模型与Kriging模型在优化设计中的比较研究[J]. 西北工业大学学报, 2015, 3(3):819-826. LIU J, SONG W P, HAN Z H, et al. Comparative study of gek (gradient-enhanced Kriging) and Kriging when applied to design optimization[J]. Journal of Northwestern Polytechnical University, 2015, 3(3):819-826(in Chinese).
[116] JAMIL M, YANG X S. A literature survey of benchmark functions for global optimization problems[J]. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4(2):150-194.
[117] ROSENBROCK H H. An automatic method for finding the greatest or least value of a function[J]. The Computer Journal, 1960, 3(3):175-184.
[118] MICHALEWICZ Z. Genetic algorithm, numerical optimization, and constraints[C]//Proceedings of the Sixth International Conference on Genetic Algorithms, 1995:151-158.
[119] TESFAHUNEGN Y A, KOZIEL S, GRAMANZINI J R, et al. Application of direct and surrogate-based optimization to two-dimensional benchmark aerodynamic problems:A comparative study:AIAA-2015-0265[R]. Reston:AIAA, 2015.
[120] ZHANG Y, HAN Z H, SHI L X, et al. Multi-round surrogate-based optimization for benchmark aerodynamic design problems:AIAA-2016-1545[R]. Reston:AIAA, 2016.
[121] REN J, THELEN A, AMRIT A, et al. Application of multi-fidelity optimization techniques to benchmark aerodynamic design problems:AIAA-2016-1542[R]. Reston:AIAA, 2016.
[122] LYU Z, KENWAY G KW, MARTINS J R R A. Aerodynamic shape optimization studies on the common research model wing benchmark[J]. AIAA Journal, 2015, 53(4):968-985.
[123] KENWAY G K W, MARTINS J R R A. Multipoint aerodynamic shape optimization investigations of the common research model wing[J]. AIAA Journal, 2016, 54(1):112-128.
[124] KENWAY G K W, MARTINS J R R A. Aerodynamic shape optimization of the CRM configuration including buffet-onset conditions:AIAA-2016-1294[R]. Reston:AIAA, 2016.
[125] LIEM R, KENWAY G K W, MARTINS J R R A. Multi-mission aircraft fuel burn minimization via multipoint aerostructural optimization[J]. AIAA Journal, 2015, 53(1):104-122.
[126] 张科施, 韩忠华, 李为吉, 等. 一种考虑气动弹性的运输机机翼多学科优化方法[J]. 空气动力学学报, 2008, 26(1):1-7. ZHANG K S, HAN Z H, LI W J, et al. A method of coupled aerodynamic/structural integration optimization for transport-wing design[J]. Acta Aerodynamica Sinica, 2008, 26(1):1-7(in Chinese).
[127] KENWAY G K W, KENNEDY G J, MARTINS J R R A. Scalable parallel approach for high-fidelity steady-state aeroelastic analysis and derivative computations[J]. AIAA Journal, 2014, 52(5):935-951.
/
| 〈 |
|
〉 |
CJA