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目录 contents

    摘要

    为缓解复杂工程产品设计优化中计算复杂度和计算精度之间的矛盾,结合最小二乘支持向量回归(least squares support vector regression,LSSVR)模型,提出一种基于差距映射的变可信度近似模型构建方法,即最小二乘支持向量回归差距映射(LSSVR with difference mapping framework,DMF-LSSVR)方法,以实现小样本条件下高精度近似模型的构建,并通过工程实例验证该方法的有效性。工程实例结果显示所提出的方法具有较高的预测精度,可为复杂工程产品的设计优化提供理论基础。

    Abstract

    In order to alleviate the conflict between computational complexity and accuracy in the design optimization of complex engineering products, a new difference mapping based variable-fidelity approximation modeling method based on least squares support vector regression was put forward, namely LSSVR with difference mapping framework (DMF-LSSVR), in order to achieve a highly accurate approximation model within a limited sample size. Its effectiveness was validated through several engineering cases. The results demonstrate that the proposed DMF-LSSVR achieves high predictive accuracy, which can provide theoretical basis for the design optimization of complex engineering products.

    近似模型通过对原始分析模型(也称源模型)数据进行拟合或插值计算后得到其输入与输出间的关系,用以替代原始分析模型。近似模型不需要知道源模型的黑箱特诊,因此几乎可以应用于任何领域,如:生物学、金融学、物理学以及地质工程学等。常用的近似模型有径向基函数(radial basis function,RBF)、响应面方法(response surface method, RSM)、克里金(Kriging)和支持向量回归(support vector regression,SVR)[1,2],但这些模型仅仅对高可信度(high fidelity,HF)源模型进行拟合或插值计算,属于单可信度近似模型。在现实情况中,同一物理规律存在多个不同的源模型,HF源模型可以更真实地描述实际情况。除了HF源模型,还有很多如工程估算公式等描述不那么准确的低可信度(low fidelity,LF)源模型。LF源模型可快速获取样本信息,但其精准度较低;HF源模型精准度较高,但其获取过程需要耗费大量人力、物力。

    在构建单可信度近似模型时,往往需要较多(几百个至几千个)样本点,这样会耗费大量的时间和成[3]。鉴于此,有学者提出了变可信度近似模[4],该模型充分结合了高、低两种可信度分析模型的优势,利用低可信度分析模型减少仿真时间与降低计算复杂程度,同时以少量的高可信度分析模型来确保近似精度。目前,学者们主要根据标度函数来建立变可信度近似模型。2006年,Gano等将加法和乘法相结合,开发出了混合标度模[5]。Marduel等对一阶、二阶标度函数进行对比研究,发现一阶标度函数的精确度比二阶的低,而二阶标度函数的计算成本较[6]。此外,很多学者将全局近似模型与标度函数进行结合,比如:Fischer等提出了梯度增强型Kriging(gradient-enhanced Kriging)标度模[7];Ghosh等开发了贝叶斯层次模型来集成高、低可信度模型信[8];Leary等针对热成型工艺的加热措施,开发了Kriging标度模[9];Han等将梯度增强型Kriging模型和低阶RSM进行一定的结[10];Qian等借助贝叶斯桥函数来提高仿真的准确度,以获得准确的仿真信息和参[11]。基于全局近似模型的标度函数可以准确地展现设计空间的全貌。

    另外一类使用到高、低可信度分析模型的常见方法为空间映射(space mapping,SM),其核心思想是借助映射参数建立高、低可信度分析模型间的关系,使得优化可以通过低可信度分析模型进[12,13]。Bandler等使用有限元方法和电磁模拟器,借助空间映射方法对波导滤波器进行优[14]。Wang等提出了基于再分析的空间映射方[15]。变可信度近似模型和空间映射方法最大的区别在于:前者是对LF分析模型的结果进行调整;而后者是对LF分析模型的设计空间参数进行“扭曲”,使得HF分析模型和LF分析模型在某些点(常为最优点)处的结果一致。在空间映射方法中,参数提取(parameter extraction)是寻求高、低可信度分析模型中设计参数间关系的关键步[16]

  • 1 最小二乘支持向量回归模型构建

    Vapnick在20世纪90年代提出的支持向量回归模型是将支持向量机应用于回归问[17],它是解决“局部极小点问题”“过学习问题”和“非线性和维度灾难问题”等的有力途[18,19] ,在多个领域获得了广泛的应用。

    假设存在一组样本点集D=x1,y1,x2,y2,,xi,yi,,xn,yn,其中xi,yiRni=1,2,,nxiyi分别为设计变量和对应的响应值。利用SVR借助非线性映射ϕ·将样本集φx转换为高维空间集φ'x=φx1,φx2,,φxn,同时确定某一线性函数fx为目标函数,使得fxxi处的预测误差最小,即目标值fxiyi的最大偏差为ε,如此一来,原本的非线性问题被转换为高维度空间的线性问题。其中线性函数fx的表达式为:

    fx=wTϕx+b
    (1)

    式中:w为权重向量,w的范数值较小,则SVR的复杂度较小;ϕx为非线性转换基函数;b为偏差量。

    当式(1)中引入松弛变量ξiξi*,上述问题就变成式(2)所示的优化问题:

    minfx=12wTw+Ci=1nξi+ξi*s.t.yi-wTϕx-bε+ξiwTϕx+b-yiε+ξi*ξi0,ξi*0
    (2)

    其中:C是正规化参数,也称为惩罚参数,用以平衡模型复杂度和训练误差。

    在SVR中应用线性最小二乘规则,同时使用等式约束替代不等式约束,则上述优化问题可表示为:

    minfx=12wTw+12Ci=1nei2s.t.ei=yi-wTϕxi-b,i=1,2,,n
    (3)

    其中:12wTw是用来归一化大的权重值和惩罚值,12Ci=1nei2为回归误差。

    综上,LSSVR(least squares support vector regression,最小二乘支持向量回归)模型的构建是借助了线性方程,能够大幅度减小计算复杂度。与SVR相比,LSSVR训练速度快、稳定性高且易于控制。

  • 2 基于LSSVR的差距映射模型构建

    本文结合LSSVR模型,提出了基于差距映射的变可信度近似模型构建方法,即LSSVR差距映射法(LSSVR with difference mapping framework,DMF-LSSVR)。下面将详细介绍DMF-LSSVR的构建流程,如图1所示,主要包括基底函数创建、LSSVR差距映射和模型校验。

    图1
                            基于LSSVR的差距映射模型构建流程

    图1 基于LSSVR的差距映射模型构建流程

    Fig. 1 Construction process of difference mapping model based on LSSVR

    与传统可信度近似模型构建方法不同的是,LSSVR差距映射法中的样本点是高、低可信度分析模型上的实验设计点。根据这些样本数据集,首先创建一个基底函数,然后将该基底函数直接映射到高可信度分析模型,如果映射后的基底函数通过了模型校验,则将该基底函数作为近似高可信度分析模型。

  • 2.1 基底函数的创建

    当LF分析模型为工程经验公式等显性函数时,可以直接将LF分析模型ylx当成基底函数,或是利用Kriging、响应面函数等模型对LF分析模型进行近似得到基底函数。本文为了过滤部分噪音,对LF分析模型先进行近似再进行拉伸得到基底函数,可表示为:

    fbasex=scaledyˆlx=ρ0+ρ1yˆlx
    (4)

    式中:ρ0ρ1为基底函数的拉伸因子,可通过最小二乘方法得到,

    minLρ0,ρ1=j=1mρ0+ρ1yˆlxj-yhxj2s.t.l0ρ0u0,l1ρ1u1xjXH
    (5)

    式中:m为HF分析模型样本点的个数。

  • 2.2 LSSVR差距映射

    在上述基底函数的创建过程中,LF分析模型被拉伸到HF分析模型,但基底函数和HF分析模型间的容差依然存在,因此,采用LSSVR来近似两者之间的容差,从而创建差距映射模型。

    首先,利用拉伸因子ρ0ρ1计算基底函数和HF模型之间的容差δ=δx1,δx2,,δxm

    δxj=yhj-fbasexj=yhj-ρˆ0+ρˆ1yˆlxj,xjXH
    (6)

    按照一定的实验设计方法,产生一组样本数据:xj和对应的容差响应δxj,基于LSSVR的差距映射模型可以表示为:

    minfxj=12w2+C2j=1mej2s.t.yhj-δxj-wTϕxj-b=ejj=1,2,,m
    (7)

    经过拉格朗日变换后,式(7)可表达为:

    L=12w2+12Cj=1mei2-αjyj-wTϕxj-b-ej
    (8)

    根据KKT(Karush-Kuhn-Tucher)条件,对式(8)中的拉格朗日乘子求偏导,可以得到:

    Lb=j=1mαj=0Lw=w-j=1mαjϕxi=0Lej=αj-Cej=0Lαj=wTϕxj+b+ej-yhj=0
    (9)

    消去变量wej后,式(9)可以表示为:

    0eTeK+C-1eTbα=0δ
    (10)

    式中:α=[α1,α2,,αm]Te为单位向量,e=[1,1,,1]1×mKm×m方阵,其元素Kjk=φ([xhjyhj])Tφ([xhkyhk])=k([xhjyhj],[xhkyhk]),其中kxhjyhj,xhkyhk为核函数。

    求得αb之后,DMF-LSSVR模型可以表示为:

    fˆx=fbasex+δˆx=ρ0+ρ1ylx+j=1mαjKxxj+b
    (11)
  • 3 实例与分析

    为检验DMF-LSSVR模型的可行性,比较了它与普通单可信度近似模型以及典型的基于线性修正的变可信度近似模型的预测精度。采用了3种不同的预测精度指标,即相对均方根误差(relative root mean square error,RRMSE)、最大相对误差(relative maximum absolute error,RMAE)、R2,其中:RMSE、RMAE值越小表示模型的预测精度越高;R值越大表示模型预测精度越高。它们的计算公式如下:

    RRMSE=1STD1ni=1nyi-yˆi2RMAE=1STDmaxyi-yˆiR2=1-i=1nyi-yˆi2i=1nyi-y¯2,i=1,2,,nSTD=1n-1i=1nyi-y¯i2
    (12)

    式中:n为校验样本个数,y¯为校验样本观测处的平均值,yiyˆi分别为校验样本处的真实值和观测值。

  • 3.1 深孔模型的近似

    深孔模型被广泛用于比较不同计算机实验分析方法的性能,其表达式[20]

    fborehole=2πTuHu-Hllnrrw1+2LHulnrrwrw2Kw+TuTl
    (13)

    其中:

    rw[0.05,0.15],r[100,50000]Tu[63070,115600],Hu[990,1110]Tl[63.1,116],Hl[700,820]L[1120,1680],Kw[9855,12045]

    深孔模型中含有8个设计变量,用x1,x2,,x8表示,对应的低可信度分析模型表示为:

    yLow=0.4fborehole+0.07x12x8+x1x7/x3+x1x6/x2+x12x4
    (14)

    考虑不同实验设计样本大小对近似模型预测精度的影响。对于特定的样本大小,采用嵌套和非嵌套的实验设计来探索不同高低可信度样本间的关系对近似精度的影响。深孔模型的不同近似模型的预测精度见表1,其中,DMF-LSSVR代表本文提出的基于最小二乘支持向量回归的差距映射模型,VFP(variable fidelity polynomial,变保真多项式)代表文献[21]中的线性修正模型,这2种变可信度近似模型都采用了高低可信度样本数据,而其他单可信度模型仅采用了HF样本数据。表中,“Nested”代表嵌套样本,“Non-nested”代表非嵌套样本。

    表1 不同样本大小下深孔模型的不同近似模型的预测精度结果比较

    Table 1 Prediction accuracy comparison of different approximation models for deep hole model under different sample sizes

    近似模型样本大小RRMSERMAER2
    Nested DMF-LSSVR100.363 41.232 60.997 1
    200.331 01.132 10.997 3
    500.321 61.023 70.999 3
    Non-nested DMF-LSSVR100.386 51.423 10.989 8
    200.352 11.320 10.992 6
    500.331 61.231 90.996 4
    VFP100.501 21.632 10.387 9
    200.451 61.543 2-1.442 4
    500.412 31.265 80.978 3
    Kriging100.591 81.705 60.652 6
    200.493 21.592 50.766 8
    500.432 51.288 50.937 8
    RSM100.652 11.895 0-3.388 8
    200.562 31.785 0-8.826 4
    500.476 91.296 50.924 6
    RBF100.632 51.963 0-0.359 2
    200.533 91.812 3-0.219 1
    500.465 31.287 2-0.273 9

    为了更直观地展示各近似模型的性能,将不同近似模型(样本大小为20)的预测响应值与真实值作比较,结果如图2所示。其中“*”离直线 y=x 越近,代表近似模型的预测误差越小,当 “*”落在直线 y=x上时,代表近似模型的预测误差为零。由图2可以看出,本文所提出的DMF-LSSVR模型的预测误差最小。

    图2
                            深孔模型的不同近似模型的预测响应值比较

    图2 深孔模型的不同近似模型的预测响应值比较

    Fig. 2 Prediction response value comparison of different approximation model for deep hole model

  • 3.2 反向翼涡流发生器的设计

    飞行器运动到与地(水)面距离很近时会产生翼地效应,即飞行器机身的上下压力差会陡然增大,升力也随之增大。得益于翼地效应机制,在F1赛车和印地赛车等开放轮赛车中,反向翼被用来产生下向力以提升赛车的牵引力以及转弯速[22,23,24]

    本文采用Kuya[25]描述的一个高为 2 mm 的反向旋转接头边界层涡发生器装置的设计数据来检验DMF-LSSVR模型的性能。文中的风洞试验数据为归一化处理后的高可信度数据(见表2),模拟计算数据为低可信度信息。设计变量包括翼倾角α和底盘高度h/c,响应值为下向力CLs

    表2 风洞试验的高可信度数据

    Table 2 High fidelity data of wind tunnel test

    设计变量响应值CLs
    αh/c
    002.46
    0.501.79
    101.57
    00.51.81
    0.50.51.957 5
    10.51.33
    011.46
    0.511.8
    111.31

    3为4个校验点处的真实值和不同近似模型的预测值,以及对应的预测精度指标值。由表3可以看出:LF样本大小对模型的最终预测精度有影响,随着样本点增多,变可信度模型的预测精度也有所提高;无论是DMF-LSSVR模型还是Cokriging模[25],嵌套设计下的预测精度要比非嵌套设计下的预测预测精度低,这说明非嵌套实验设计更适合反向翼涡流发生器的设计。另外,本文所提出的DMF-LSSVR模型在预测精度方面要优于Kuya提出的Cokriging模型。

    表3 不同变可信度近似模型预测精度比较

    Table 3 Prediction accuracy comparison of different variable fidelity approximation models

    真实值与近似模型LF样本大小采样策略响应值RRMSERMAER2
    0.250.250.750.250.250.750.750.75
    真实值2.421.872.61.75---
    DMF-LSSVR9Nested1.949 91.587 91.920 51.858 60.850 94.110 70.186 1
    Non-nested1.975 11.610 71.864 41.670 70.837 64.349 80.254 8
    25Nested2.642 61.719 92.258 41.668 30.844 60.735 40.640 8
    Non-nested2.441 51.859 22.431 81.966 90.452 80.709 90.852 4

    Cokriging

    回归模型

    9Nested1.991.641.741.610.853 34.984 60.170 8
    Non-nested2.031.601.791.550.840 93.719 50.239 2
    25Nested2.371.691.981.550.786 81.711 10.417 9
    Non-nested2.511.562.081.510.783 41.099 30.426 4
    Cokriging 幅值模型25Nested2.481.691.981.480.808 61.431 10.358 0
    Non-nested2.651.572.091.480.822 10.944 40.314 3
  • 4 结论

    单可信度近似模型在对大型工程复杂产品设计优化时,所需的仿真时间较长和采样成本过高。为此,本文提出了一种新的变可信度近似模型构建方法,首先通过最小二乘方法构建基底函数,接着利用最小二乘支持向量回归模型创建基底函数和高可信度模型间的映射。最后采用工程实例对本文所提出的方法进行了验证,结果显示本文所提出的方法有较高的预测精度。

  • 参考文献

    • 1

      易平,程耿东.基于功能测度的概率结构优化设计的序列近似规划方法[C/OL]//中国力学学会学术大会,2005论文摘要集(下),北京:中国力学学会,2005:1097.[2018-01-01]. http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200508002423.htm.

      YI Ping, CHENG Geng-dong. Sequential approaximation planning method of the probability measure based structure optimization design[C/OL]//Academic Conference of Chinese Mechanics Society’ 2005 Abstract (Part Ⅱ), Beijing: China Institute of Mechanics Academic, 2005: 1097. [2018-01-01]. http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200508002423.htm.

    • 2

      张勇.基于代理模型的注塑成型工艺优化研究[D].宁波:宁波大学机械电子工程学院,2009:1-72.

      ZHANG Yong. Surrogate model based injection molding process optimization research[D]. Ningbo: Ningbo University, School of Mechatronics Engineering, 2009: 1-72.

    • 3

      彭科,胡凡,张为华,等.序列近似优化方法及其在火箭外形快速设计中的应用[J].国防科技大学学报,2016,38(1):129-136. doi:10.11887/j.cn.201601021

      PENG Ke, HU Fan, ZHANG Wei-hua, et al. Sequential approximate optimization method and its application in rapid design of rocket shape[J]. Journal of National University of Defense Technology, 2016,38(1): 129-136.

    • 4

      郑君.基于变可信度近似的设计优化关键技术研究[D].武汉:华中科技大学机械科学与工程学院,2014:1-141.

      ZHENG Jun. Research on key technologies of design optimization based on variable fidelity approximations[D]. Wuhan: Huazhong University of Science and Technology, School of Mechanical Science and Engineering, 2014: 1-141.

    • 5

      GANO S E, RENAUD J E, SANDERS B. Hybrid variable fidelity optimization by using a Kriging-based scaling function[J]. AIAA Journal, 2005, 43(11): 2422-2430. doi:10.2514/1.12466

    • 6

      MARDUEL X, TRIBES C, TREPANIER J Y. Variable-fidelity optimization: efficiency and robustness[J]. Optimization and Engineering, 2006, 7(4): 479-500. doi:10.1007/s11081-006-0351-3

    • 7

      FISCHER C C, GRANDHI R V. A surrogate-based adjustment factor approach to multi-fidelity design optimization[C]//17th AIAA Non-Deterministic Approaches Conference, Kissimmee, Florida, Jan.5-9, 2015. doi: 10.2514/6.2015-1375

    • 8

      GHOSH S, JACOBS R B, MAVRISY D N. Multi-source surrogate modeling with bayesian hierarchical regression[C]//17th AIAA Non-Deterministic Approaches Conference, Kissimmee, Florida, Jan.5-9, 2015. doi: 10.2514/6.2015-1817

    • 9

      LEARY S, BHASKER A, KEANE A. A knowledge-based approach to response surface modelling in multifidelity optimization[J]. Journal of Global Optimization, 2003, 26(3): 297-319. doi:10.1023/A:1023283917997

    • 10

      HAN Z H, GORTZ S. Hierarchical kriging model for variable-fidelity surrogate modeling[J]. AIAA Journal, 2012, 50(9): 1885-1896. doi:10.2514/1.J051354

    • 11

      ZHOU Q, SHAO X Y, JIANG P, et al. An adaptive global variable fidelity metamodeling strategy using a support vector regression based scaling function[J]. Simulation Modelling Practice and Theory, 2015, 59: 18-35. doi:10.1016/j.simpat.2015.08.002

    • 12

      BAKR M H, BANDLER J W, GEORGIEVA N, et al. A hybrid aggressive space-mapping algorithm for EM optimization[J]. IEEE Transactions on Microwave Theory and Techniques, 1999, 47(12): 2440-2449. doi:10.1109/22.808991

    • 13

      BANDLER J W, BIERNACKI R M, CHEN S H, et al. Space mapping technique for electromagnetic optimization[J]. IEEE Transactions on Microwave Theory and Techniques, 1994, 42(12): 2536-2544. doi:10.1109/22.339794

    • 14

      BANDLER J W, BIERNACKI R M, CHEN S H, et al. Space mapping optimization of waveguide filters using finite element and mode-matching electromagnetic simulators[J]. International Journal of RF and Microwave Computer-Aided Engineering, 1999, 9(1): 54-70. doi:10.1002/(sici)1099-047x(199901)9:1<54::aid-mmce8>3.0.co;2-8

    • 15

      WANG H, FAN T X, LI G Y. Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems[J]. Structural and Multidisciplinary Optimization, 2017, 55(6): 2143-2157. doi:10.1007/s00158-016-1633-9

    • 16

      BAKR M H, BANDLER J W, MADSEN K, et al. Review of the space mapping approach to engineering optimization and modeling[J]. Optimization and Engineering, 2000, 1(3): 241-276. doi: 10.1023/A:1010000106286

    • 17

      BACKLUND P B, SHAHAN D W, SEEPERSAD C C. A comparative study of the scalability of alternative metamodelling techniques[J]. Engineering Optimization, 2012, 44(7): 767-786. doi:10.1080/0305215x.2011.607817

    • 18

      WANG H, SHAN S Q, WANG G G, et al. Integrating least square support vector regression and mode pursuing sampling optimization for crashworthiness design[J]. Journal of Mechanical Design, 2011, 133(4): (04002-1)-(041002-10). doi:10.1115/1.4003840

    • 19

      SUYKENS J A K, GESTEL T V, BRABANTER J D, et al. Least squares support vector machines[M]. Singapore: World Scientific, 2002: 20-26.

    • 20

      ZHENG J, SHAO X Y, GAO L, et al. A prior-knowledge input LSSVR metamodeling method with tuning based on cellular particle swarm optimization for engineering design[J]. Expert Systems with Applications, 2014, 41(5): 2111-2125. doi:10.1016/j.eswa.2013.09.010

    • 21

      ZADEH P M, TOROPOV V V, WOOD A S. Metamodel-based collaborative optimization framework[J]. Structural and Multidisciplinary Optimization, 2009, 38(2): 103-115. doi:10.1007/s00158-008-0286-8

    • 22

      KATZ J. Aerodynamics of race cars[J]. Annual Review of Fluid Mechanics, 2006, 38(1): 27-63. doi:10.1146/annurev.fluid.38.050304.092016

    • 23

      ZHANG X, TOET W, ZERIHAN J. Ground effect aerodynamics of race cars[J]. Applied Mechanics Reviews, 2006, 59(1/6): 33-49. doi:10.1115/1.2110263

    • 24

      KUYA Y, TAKEDA K, ZHANG X. Computational investigation of a race car wing with vortex generators in ground effect[J]. Journal of Fluids Engineering-Transactions of the ASME, 2010, 132(2): 121102-121109. doi:10.1115/1.4000741

    • 25

      KUYA Y, TAKEDA K, ZHANG X, et al. Multifidelity surrogate modeling of experimental and computational aerodynamic data sets[J]. AIAA Journal, 2011, 49(2): 289-298. doi:10.2514/1.j050384

欧卫林

机 构:华中光电技术研究所 武汉光电国家实验室, 湖北 武汉 430223

Affiliation:Wuhan National Laboratory for Optoelectronics, Central China Institute of Optoelectronic Technology, Wuhan 430223, China

邮 箱:weil0@qq.com

作者简介:欧卫林(1974—),男,湖北武汉人,工程师,硕士,从事光电系统结构设计优化研究,E-mail:weil0@qq.com

郑君

机 构:中国地质大学 工程学院, 湖北 武汉 430074

Affiliation:Faculty of Engineering, China University of Geosciences, Wuhan 430074, China

角 色:通讯作者

Role:Corresponding author

邮 箱:zjlucia1209@126.com

作者简介:郑君(1987—),女,湖北黄岗人,博士生,从事复杂工程产品设计优化、地质钻探装备等研究,E-mail:zjlucia1209@126.com,https://orcid.org/0000-0002-7487-8540

1006-754X-2019-26-2-133/alternativeImage/ba57124c-6382-4164-b304-e29013818e9b-F001.jpg
近似模型样本大小RRMSERMAER2
Nested DMF-LSSVR100.363 41.232 60.997 1
200.331 01.132 10.997 3
500.321 61.023 70.999 3
Non-nested DMF-LSSVR100.386 51.423 10.989 8
200.352 11.320 10.992 6
500.331 61.231 90.996 4
VFP100.501 21.632 10.387 9
200.451 61.543 2-1.442 4
500.412 31.265 80.978 3
Kriging100.591 81.705 60.652 6
200.493 21.592 50.766 8
500.432 51.288 50.937 8
RSM100.652 11.895 0-3.388 8
200.562 31.785 0-8.826 4
500.476 91.296 50.924 6
RBF100.632 51.963 0-0.359 2
200.533 91.812 3-0.219 1
500.465 31.287 2-0.273 9
1006-754X-2019-26-2-133/alternativeImage/ba57124c-6382-4164-b304-e29013818e9b-F002.jpg
设计变量响应值CLs
αh/c
002.46
0.501.79
101.57
00.51.81
0.50.51.957 5
10.51.33
011.46
0.511.8
111.31
真实值与近似模型LF样本大小采样策略响应值RRMSERMAER2
0.250.250.750.250.250.750.750.75
真实值2.421.872.61.75---
DMF-LSSVR9Nested1.949 91.587 91.920 51.858 60.850 94.110 70.186 1
Non-nested1.975 11.610 71.864 41.670 70.837 64.349 80.254 8
25Nested2.642 61.719 92.258 41.668 30.844 60.735 40.640 8
Non-nested2.441 51.859 22.431 81.966 90.452 80.709 90.852 4

Cokriging

回归模型

9Nested1.991.641.741.610.853 34.984 60.170 8
Non-nested2.031.601.791.550.840 93.719 50.239 2
25Nested2.371.691.981.550.786 81.711 10.417 9
Non-nested2.511.562.081.510.783 41.099 30.426 4
Cokriging 幅值模型25Nested2.481.691.981.480.808 61.431 10.358 0
Non-nested2.651.572.091.480.822 10.944 40.314 3

图1 基于LSSVR的差距映射模型构建流程

Fig. 1 Construction process of difference mapping model based on LSSVR

表1 不同样本大小下深孔模型的不同近似模型的预测精度结果比较

Table 1 Prediction accuracy comparison of different approximation models for deep hole model under different sample sizes

图2 深孔模型的不同近似模型的预测响应值比较

Fig. 2 Prediction response value comparison of different approximation model for deep hole model

表2 风洞试验的高可信度数据

Table 2 High fidelity data of wind tunnel test

表3 不同变可信度近似模型预测精度比较

Table 3 Prediction accuracy comparison of different variable fidelity approximation models

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  • 参考文献

    • 1

      易平,程耿东.基于功能测度的概率结构优化设计的序列近似规划方法[C/OL]//中国力学学会学术大会,2005论文摘要集(下),北京:中国力学学会,2005:1097.[2018-01-01]. http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200508002423.htm.

      YI Ping, CHENG Geng-dong. Sequential approaximation planning method of the probability measure based structure optimization design[C/OL]//Academic Conference of Chinese Mechanics Society’ 2005 Abstract (Part Ⅱ), Beijing: China Institute of Mechanics Academic, 2005: 1097. [2018-01-01]. http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200508002423.htm.

    • 2

      张勇.基于代理模型的注塑成型工艺优化研究[D].宁波:宁波大学机械电子工程学院,2009:1-72.

      ZHANG Yong. Surrogate model based injection molding process optimization research[D]. Ningbo: Ningbo University, School of Mechatronics Engineering, 2009: 1-72.

    • 3

      彭科,胡凡,张为华,等.序列近似优化方法及其在火箭外形快速设计中的应用[J].国防科技大学学报,2016,38(1):129-136. doi:10.11887/j.cn.201601021

      PENG Ke, HU Fan, ZHANG Wei-hua, et al. Sequential approximate optimization method and its application in rapid design of rocket shape[J]. Journal of National University of Defense Technology, 2016,38(1): 129-136.

    • 4

      郑君.基于变可信度近似的设计优化关键技术研究[D].武汉:华中科技大学机械科学与工程学院,2014:1-141.

      ZHENG Jun. Research on key technologies of design optimization based on variable fidelity approximations[D]. Wuhan: Huazhong University of Science and Technology, School of Mechanical Science and Engineering, 2014: 1-141.

    • 5

      GANO S E, RENAUD J E, SANDERS B. Hybrid variable fidelity optimization by using a Kriging-based scaling function[J]. AIAA Journal, 2005, 43(11): 2422-2430. doi:10.2514/1.12466

    • 6

      MARDUEL X, TRIBES C, TREPANIER J Y. Variable-fidelity optimization: efficiency and robustness[J]. Optimization and Engineering, 2006, 7(4): 479-500. doi:10.1007/s11081-006-0351-3

    • 7

      FISCHER C C, GRANDHI R V. A surrogate-based adjustment factor approach to multi-fidelity design optimization[C]//17th AIAA Non-Deterministic Approaches Conference, Kissimmee, Florida, Jan.5-9, 2015. doi: 10.2514/6.2015-1375

    • 8

      GHOSH S, JACOBS R B, MAVRISY D N. Multi-source surrogate modeling with bayesian hierarchical regression[C]//17th AIAA Non-Deterministic Approaches Conference, Kissimmee, Florida, Jan.5-9, 2015. doi: 10.2514/6.2015-1817

    • 9

      LEARY S, BHASKER A, KEANE A. A knowledge-based approach to response surface modelling in multifidelity optimization[J]. Journal of Global Optimization, 2003, 26(3): 297-319. doi:10.1023/A:1023283917997

    • 10

      HAN Z H, GORTZ S. Hierarchical kriging model for variable-fidelity surrogate modeling[J]. AIAA Journal, 2012, 50(9): 1885-1896. doi:10.2514/1.J051354

    • 11

      ZHOU Q, SHAO X Y, JIANG P, et al. An adaptive global variable fidelity metamodeling strategy using a support vector regression based scaling function[J]. Simulation Modelling Practice and Theory, 2015, 59: 18-35. doi:10.1016/j.simpat.2015.08.002

    • 12

      BAKR M H, BANDLER J W, GEORGIEVA N, et al. A hybrid aggressive space-mapping algorithm for EM optimization[J]. IEEE Transactions on Microwave Theory and Techniques, 1999, 47(12): 2440-2449. doi:10.1109/22.808991

    • 13

      BANDLER J W, BIERNACKI R M, CHEN S H, et al. Space mapping technique for electromagnetic optimization[J]. IEEE Transactions on Microwave Theory and Techniques, 1994, 42(12): 2536-2544. doi:10.1109/22.339794

    • 14

      BANDLER J W, BIERNACKI R M, CHEN S H, et al. Space mapping optimization of waveguide filters using finite element and mode-matching electromagnetic simulators[J]. International Journal of RF and Microwave Computer-Aided Engineering, 1999, 9(1): 54-70. doi:10.1002/(sici)1099-047x(199901)9:1<54::aid-mmce8>3.0.co;2-8

    • 15

      WANG H, FAN T X, LI G Y. Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems[J]. Structural and Multidisciplinary Optimization, 2017, 55(6): 2143-2157. doi:10.1007/s00158-016-1633-9

    • 16

      BAKR M H, BANDLER J W, MADSEN K, et al. Review of the space mapping approach to engineering optimization and modeling[J]. Optimization and Engineering, 2000, 1(3): 241-276. doi: 10.1023/A:1010000106286

    • 17

      BACKLUND P B, SHAHAN D W, SEEPERSAD C C. A comparative study of the scalability of alternative metamodelling techniques[J]. Engineering Optimization, 2012, 44(7): 767-786. doi:10.1080/0305215x.2011.607817

    • 18

      WANG H, SHAN S Q, WANG G G, et al. Integrating least square support vector regression and mode pursuing sampling optimization for crashworthiness design[J]. Journal of Mechanical Design, 2011, 133(4): (04002-1)-(041002-10). doi:10.1115/1.4003840

    • 19

      SUYKENS J A K, GESTEL T V, BRABANTER J D, et al. Least squares support vector machines[M]. Singapore: World Scientific, 2002: 20-26.

    • 20

      ZHENG J, SHAO X Y, GAO L, et al. A prior-knowledge input LSSVR metamodeling method with tuning based on cellular particle swarm optimization for engineering design[J]. Expert Systems with Applications, 2014, 41(5): 2111-2125. doi:10.1016/j.eswa.2013.09.010

    • 21

      ZADEH P M, TOROPOV V V, WOOD A S. Metamodel-based collaborative optimization framework[J]. Structural and Multidisciplinary Optimization, 2009, 38(2): 103-115. doi:10.1007/s00158-008-0286-8

    • 22

      KATZ J. Aerodynamics of race cars[J]. Annual Review of Fluid Mechanics, 2006, 38(1): 27-63. doi:10.1146/annurev.fluid.38.050304.092016

    • 23

      ZHANG X, TOET W, ZERIHAN J. Ground effect aerodynamics of race cars[J]. Applied Mechanics Reviews, 2006, 59(1/6): 33-49. doi:10.1115/1.2110263

    • 24

      KUYA Y, TAKEDA K, ZHANG X. Computational investigation of a race car wing with vortex generators in ground effect[J]. Journal of Fluids Engineering-Transactions of the ASME, 2010, 132(2): 121102-121109. doi:10.1115/1.4000741

    • 25

      KUYA Y, TAKEDA K, ZHANG X, et al. Multifidelity surrogate modeling of experimental and computational aerodynamic data sets[J]. AIAA Journal, 2011, 49(2): 289-298. doi:10.2514/1.j050384