Research and Improvement of Kriging-HDMR Modeling Method

doi:10.3901/JME.2024.05.249

Abstract

Abstract: When dealing with high-dimensional problems in traditional representation model technology, the number of sample points required for modeling increases exponentially due to the increase in variable dimension, which will lead to a significant increase in computational cost. In order to build a representation model suitable for high-dimensional problems, based on the DIRECT optimization algorithm, the initial sample point position is improved, the initial sample set is expanded, and the Kriging modeling method and high-dimensional model representation (HDMR), which can avoid the established model from falling into local optimality, and proposes an improved Kriging-HDMR (iKriging-HDMR) modeling method. The iKriging-HDMR modeling method uses the advantages of HDMR to equate the response function of the high-dimensional problem to a series of low-dimensional function superpositions, taking advantage of the iKriging-HDMR modeling method to reduce the number of sample points required in the modeling process number. A new convergence condition is proposed to reduce the local error of the agent model to ensure that the established agent model has high accuracy. The effectiveness of the proposed method is verified by numerical examples and robot engineering applications. The results show that the proposed iKriging-HDMR modeling method can significantly reduce the number of sample points required for modeling, and has good calculation accuracy and efficiency.

Key words: high dimensional model representation, Kriging, DIRECT sampling, design of experiment, industrial robot

CLC Number:

TP242

MENG Yuan, SHI Baojun, ZHANG Dequan. Research and Improvement of Kriging-HDMR Modeling Method[J]. Journal of Mechanical Engineering, 2024, 60(5): 249-263.

References

[1] 魏昕，吴义忠，陈立平. 基于径向基函数的DIRECT全局优化方法[J]. 华中科技大学学报，2012，40(5)：6-9. WEI Xin，WU Yizhong，CHEN Liping. DIRECT global optimization method using radial basis functions[J]. Journal of Huazhong University of Science and Technology，2012，40(5)：6-9.
[2] 崔宝珍，孔维娜，马恺. 多项式响应面代理模型在立柱结构优化中的应用[J]. 机械设计，2017，34(4)：44-48. CUI Baozhen，KONG Weina，MA Kai. Application of polynomial response surface surrogate model in the large column structure optimization[J]. Journal of Machine Design，2017，34(4)：44-48.
[3] 李铁柱，李光耀，陈涛，等. 基于Kriging近似模型的汽车乘员约束系统稳健性设计[J]. 机械工程学报，2010，46(22)：123-129. LI Tiezhu，LI Guangyao，CHEN Tao，et al. Robustness design of occupant restraint system based on Kriging model[J]. Journal of Mechanical Engineering，2010，46(22)：123-129.
[4] 李恩颖，王琥，李光耀. 基于支持向量机回归的材料参数反求方法[J]. 机械工程学报，2012，48(6)：90-95. LI Enying，WANG Hu，LI Guangyao. Material parameter inverse technique based on support vector regression[J]. Journal of Mechanical Engineering，2012，48(6)：90-95.
[5] 韩忠华. Kriging模型及代理优化算法研究进展[J]. 航空学报，2016，37(11)：3197-3225. HAN Zhonghua. Kriging surrogate model and its application to design optimization：a review of recent progress[J]. Acta Aeronautica et Astronautica Sinica，2016，37(11)：3197-3225.
[6] FORRESTER A，KEANE A. Recent advances in surrogate-based optimization[J]. Progress in Aerospace Sciences，2009，45(1)：50-79.
[7] 彭磊，刘莉，龙腾. 基于动态径向基函数代理模型的优化策略[J]. 机械工程学报，2011，47(7)：164-170. PENG Lei，LIU Li，LONG Teng. Optimization strategy using dynamic radial basis function metamodel[J]. Journal of Mechanical Design，2011，47(7)：164-170.
[8] 龙腾，郭晓松，彭磊，等. 基于信赖域的动态径向基函数代理模型优化策略[J]. 机械工程学报，2014，50(7)：184-190. LONG Teng，GUO Xiaosong，PENG Lei ，et al. Optimization strategy using dynamic radial basis function metamodel based on trust region[J]. Journal of Mechanical Engineering，2014，50()：184-190.
[9] CHOWDHURY R，RAO B，PRASAD A. High dimensional model representation for structural reliability analysis[J]. Communications in Numerical Methods in Engineering，2009，25(4)：301-337.
[10] WU Xiaojing，PENG Xuhao，CHEN Weisheng，et al. A developed surrogate-based optimization framework combining HDMR-based modeling technique and TLBO algorithm for high-dimensional engineering problems[J]. Structural and Multidisciplinary Optimization，2019，60(2)：663-680.
[11] 汤龙，李光耀，王琥. Kriging-HDMR非线性近似模型方法[J]. 力学学报，2011，43(4)：780-784. TANG Long，LI Guangyao，WANG Hu. Kriging-HDMR metemodeling technique for nonlinear problems[J]. Chinese Journal of Theoretical and Applied Mechanics，2011，43(4)：780-784.
[12] 黄志远，邱浩波，蔡习文. IDIRECT-HDMR高维近似模型方法及工程应用[J]. 机械制造与自动化，2015，44(3)：100-103. HUANG Zhiyuan，QIU Haobo，CAI Xiwen. IDIRECT-HDMR approximation model for high dimensional problems and engineering application[J]. Machine Building & Automation，2015，44(3)：100-103.
[13] WANG Hu，TANG Long，LI Guangyao. Adaptive MLS-HDMR metamodeling techniques for high dimensional problems[J]. Expert Systems with Applications，2011，38(11)：14117-14126.
[14] SHAN Songqing，WANG G. Metamodeling for high dimensional simulation-based design problems[J]. Journal of Mechanical Design，2010，132(5)：051009.
[15] CAI Xiwen，GAO Liang，LI Xinyu. Efficient generalized surrogate-assisted evolutionary algorithm for high-dimensional expensive problems[J]. IEEE Transactions on Evolutionary Computation，2020，24(2)：365-379.
[16] LIU Jie，CAI Heng，JIANG Chao，et al. An interval inverse method based on high dimensional model representation and affine arithmetic[J]. Applied Mathematical Modelling，2018，63：732-743.
[17] YUE Xinxin，ZHANG Jian，GONG Weijie，et al. An adaptive PCE-HDMR metamodeling approach for high-dimensional problems[J]. Structural and Multidisciplinary Optimization，2021，64(1)：141-162.
[18] LIU Jie，ZHAO Yue，LEI Fei，et al. Net-HDMR metamodeling method for high-dimensional problems[J]. Journal of Mechanical Design，2023，145(9)：1-25.
[19] RABITZ H，ALIS O. General foundations of high-dimensional model representations[J]. Journal of Mathematical Chemistry，1999，25(2)：197-233.
[20] HUANG Zhiyuan，QIU Haobo，ZHAO Ming，et al. An adaptive SVR-HDMR model for approximating high dimensional problems[J]. Engineering Computations，2015，32(3)：643-667.
[21] WANG Qian，FANG Hongbing. An adaptive high‐dimensional model representation method for reliability analysis of geotechnical engineering problems[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2020，44(12)：1705-1723.
[22] ZHANG Qi，WU Yizhong，LU Li，et al. An adaptive Dendrite-HDMR metamodeling technique for high-dimensional problems[J]. Journal of Mechanical Design，2022，144(8)：081701.
[23] JONES D，PERTTUNEN C，STUCKMAN B. Lipschitzian optimization without the Lipschitz constant[J]. Journal of Optimization Theory and Applications，1993，79(1)：157-181.
[24] 汤龙. 多参数非线性优化方法关键技术研究及应用[D].长沙：湖南大学，2013. TANG Long. The key of research and application of multi-parameter nonlinear optimization[D].Changsha：Hunan University，2013.
[25] MENG Zeng，ZHANG Zhuohui，ZHANG Dequan，et al. An active learning method combining Kriging and accelerated chaotic single loop approach (AK-ACSLA) for reliability-based design optimization[J]. Computer Methods in Applied Mechanics and Engineering，2019，357：112570.
[26] MENG Zeng，ZHANG Zhuohui，LI Gang，et al. An active weight learning method for efficient reliability assessment with small failure probability[J]. Structural and Multidisciplinary Optimization，2019，61(3)：1157-1170.
[27] LOPHAVEN S，NIELSEN H，SONDERGAARD J. DACE：a matlab kriging toolbox[M]. Lyngby：IMM，Informatics and Mathematical Modelling，The Technical University of Denmark，2002.
[28] WANG G，SHAN Songqing. Review of metamodeling techniques in support of engineering design optimization[J]. Journal of Mechanical Design，2007，129(4)：370-380.
[29] JIN R，CHEN W，SIMPSON T W. Comparative studies of metamodelling techniques under multiple modelling criteria[J]. Structural and Multidisciplinary Optimization，2001，23(1)：1-13.
[30] 周炜. 飞机自动化装配工业机器人精度补偿方法与实验研究[D]. 南京：南京航空航天大学，2012. ZHOU Wei. Compensation Method of industrial robot accuracy and experimental research for aircraft automated assembly[D]. Nanjing：Nanjing University of Aeronautics and Astronautics，2012.
[31] ZHANG Dequan，SHEN Shuohuo，WU Jinhui，et al. Kinematic trajectory accuracy reliability analysis for industrial robots considering intercorrelations among multi-point positioning errors[J]. Reliability Engineering & System Safety，2023，229：108808.
[32] ZHANG Dequan，HAN Zhonghao，WANG Fang，et al. Proficiency of statistical moment-based methods for analysis of positional accuracy reliability of industrial robots[J]. International Journal of Mechanics and Materials in Design，2021，17(2)：403-418.