利用连续多模态特性探索的自适应Kriging模型构建方法
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摘要
为了高效率地构建高精度的代理模型,提出一种利用连续多模态特性探索的自适应Kriging模型构建方法.首先在获得一个初始Kriging模型后,利用留一交叉验证策略计算出代理模型的精度;然后利用相对误差准则,从样本库中挑选出误差最大的样本点;再通过泰勒级数展开式快速获得新样本点及其近似响应值;如此不断地更新样本库及Kriging模型,最终使模型精度达到预定要求.对2个数学算例和1个工程实例进行了应用分析,结果表明,该方法能够以较快的速度得到较高精度的Kriging模型.
To build a highly accurate metamodel with high efficiency, an adaptive Kriging modeling method using the exploration of continuous and multi-modal characteristics was proposed. After obtaining an initial Kriging model, the leave-one-out cross-validation strategy was utilized to calculate the metamodel's accuracy. Then the sample point with the maximum error was selected from the sample database according to the relative error criterion. A new sample point and its approximate response value were acquired quickly by using the Taylor Series expansion. With the above procedures repeated, the sample database and the Kriging model were updated in order to make the Kriging model achieve the required accuracy. The method was applied to two mathematical problems and an engineering problem. Results show that the method can get a higher accurate Kriging model with higher efficiency.
To build a highly accurate metamodel with high efficiency, an adaptive Kriging modeling method using the exploration of continuous and multi-modal characteristics was proposed. After obtaining an initial Kriging model, the leave-one-out cross-validation strategy was utilized to calculate the metamodel's accuracy. Then the sample point with the maximum error was selected from the sample database according to the relative error criterion. A new sample point and its approximate response value were acquired quickly by using the Taylor Series expansion. With the above procedures repeated, the sample database and the Kriging model were updated in order to make the Kriging model achieve the required accuracy. The method was applied to two mathematical problems and an engineering problem. Results show that the method can get a higher accurate Kriging model with higher efficiency.
引文
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[13]Zou Linjun,Wu Yizhong,Mao Huping.Incremental Kriging model rebuilding method and its application in efficient global optimization[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(4):649-655(in Chinese)(邹林君,吴义忠,毛虎平.Kriging模型的增量构造及其在全局优化中的应用[J].计算机辅助设计与图形学学报,2011,23(4):649-655)
[14]Li G Z,Aute V,Azarm S.An accumulative error based adaptive design of experiments for offline metamodeling[J].Structural and Multidisciplinary Optimization,2010,40(1-6):137-155
[15]Viana F A C,Haftka R T,Steffen V,Jr.Multiple surrogates:how cross-validation errors can help us to obtain the best predictor[J].Structural and Multidisciplinary Optimization,2009,39(4):439-457
[16]Laurent L,Boucard P A,Soulier B.Generation of a cokriging metamodel using a multiparametric strategy[J].Computational Mechanics,2013,51(2):151-169
[17]Liu W Y,Batill S M.Gradient-enhanced response surface approximations using Kriging models[C]//Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization.Reston:American Institute of Aeronautics and Astronautics Inc,2002:108-117
[18]Zhu H G,Liu L,Long T,et al.A novel algorithm of maximin latin hypercube design using successive local enumeration[J].Engineering Optimization,2012,44(5):551-564
[19]Hazime R M,Dropps S H,Anderson D H,et al.Transient non-linear FEA and TMF life estimates of cast exhaust manifolds[C]//Proceedings of the SAE World Congress.Warrendale:SAE International,2003:1-10
[2]Sanchez E,Pintos S,Queipo N V.Toward an optimal ensemble of kernel-based approximations with engineering applications[J].Structural and Multidisciplinary Optimization,2008,36(3):247-261
[3]Shao W Z,Deng H S,Ma Y Z,et al.Extended Gaussian Kriging for computer experiments in engineering design[J].Engineering with Computers,2012,28(2):161-178
[4]Ishigami T,Shimada T.Study on application of Kriging to evaluation of radioactivity concentration for ensuring compliance with the criterion of site release[J].Journal of Nuclear Science and Technology,2015,52(9):1186-1204
[5]Guirguis D,Hamza K,Aly M,et al.Multi-objective topology optimization of multi-component continuum structures via a Kriging-interpolated level set approach[J].Structural and Multidisciplinary Optimization,2015,51(3):733-748
[6]Meng Wei,Tong Xiaoyan.Kriging model based on quantum particle swarm algorithm[J].Intelligence,2010,10(36):51-54(in Chinese)(孟伟,童晓艳.基于量子粒子群算法的Kriging近似模型[J].才智,2010,10(36):51-54)
[7]You Hailong,Jia Xinzhang.The construction and optimization of Kriging metamodel based on genetic algorithms[J].Journal of Computer-Aided Design&Computer Graphics,2007,19(1):64-68(in Chinese)(游海龙,贾新章.基于遗传算法的Kriging模型构造与优化[J].计算机辅助设计与图形学学报,2007,19(1):64-68)
[8]Aute V,Saleh K,Abdelaziz O,et al.Cross-validation based single response adaptive design of experiments for Kriging metamodeling of deterministic computer simulations[J].Structural and Multidisciplinary Optimization,2013,48(3):581-605
[9]He Donghai,Qi Rongbin,Qian Feng.Sequential optimization method based on Kriging surrogate model[J].Computers and Applied Chemistry,2014,31(11):1323-1328(in Chinese)(何东海,祁荣宾,钱锋.基于Kriging代理模型的序列优化[J].计算机与应用化学,2014,31(11):1323-1328)
[10]Wang Hongtao,Zhu Xiaocheng,Du Zhaohui.Application of adaptive Kriging approximation model in two dimensional diffuser aerodynamic optimization design[J].Chinese Journal of Computational Mechanics,2011,28(1):15-19(in Chinese)(王红涛,竺晓程,杜朝辉.自适应Kriging近似模型及其在二维扩压器优化设计中的应用[J].计算力学学报,2011,28(1):15-19)
[11]Chen Z Z,Qiu H B,Gao L,et al.A local adaptive sampling method for reliability-based design optimization using Kriging model[J].Structural and Multidisciplinary Optimization,2014,49(3):401-416
[12]Ulaganathan S,Couckuyt I,Ferranti F,et al.Performance study of multi-fidelity gradient enhanced Kriging[J].Structural and Multidisciplinary Optimization,2015,51(5):1017-1033
[13]Zou Linjun,Wu Yizhong,Mao Huping.Incremental Kriging model rebuilding method and its application in efficient global optimization[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(4):649-655(in Chinese)(邹林君,吴义忠,毛虎平.Kriging模型的增量构造及其在全局优化中的应用[J].计算机辅助设计与图形学学报,2011,23(4):649-655)
[14]Li G Z,Aute V,Azarm S.An accumulative error based adaptive design of experiments for offline metamodeling[J].Structural and Multidisciplinary Optimization,2010,40(1-6):137-155
[15]Viana F A C,Haftka R T,Steffen V,Jr.Multiple surrogates:how cross-validation errors can help us to obtain the best predictor[J].Structural and Multidisciplinary Optimization,2009,39(4):439-457
[16]Laurent L,Boucard P A,Soulier B.Generation of a cokriging metamodel using a multiparametric strategy[J].Computational Mechanics,2013,51(2):151-169
[17]Liu W Y,Batill S M.Gradient-enhanced response surface approximations using Kriging models[C]//Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization.Reston:American Institute of Aeronautics and Astronautics Inc,2002:108-117
[18]Zhu H G,Liu L,Long T,et al.A novel algorithm of maximin latin hypercube design using successive local enumeration[J].Engineering Optimization,2012,44(5):551-564
[19]Hazime R M,Dropps S H,Anderson D H,et al.Transient non-linear FEA and TMF life estimates of cast exhaust manifolds[C]//Proceedings of the SAE World Congress.Warrendale:SAE International,2003:1-10