Skip to main content
SpringerLink
Log in

Hybrid meta-model-based design space exploration method for expensive problems

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

Multiple meta-models used together in the search process at least can offer an insurance against the poorly fitted meta-models and can improve robustness of the predictions, compared with the single meta-model based methods. In this work, a hybrid meta-model-based design space exploration (HMDSE) method is proposed. In the proposed method, a part of the current expensive points which are evaluated by the expensive problems to be solved are used firstly to construct a so-called important region. And then, three representative meta-models, kriging, radial basis functions (RBF), and quadratic function (QF), are used in the search of the obtained important region. To avoid the local minima, the remaining region will be searched simultaneously. In addition, the whole design space will also be searched to further demonstrate the global optimum. Through test by six benchmark math functions with design variables ranging from 10 to 24, the proposed HMDSE method shows great accuracy, efficiency, and robustness compared with the efficient global optimization (EGO). Then, it is applied in a practical vehicle lightweight design problem with 30 design variables, achieving desired results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294

    Article  Google Scholar 

  • Adorio EP (2005) MVF - multivariate test functions library in C for unconstrained global optimization. www.geocities.ws/eadorio/mvf.pdf

  • Byrd RH, Schnabel RB, Shultz GA (1987) A trust region algorithm for nonlinearly constrained optimization. SIAM J Numer Anal 24(5):1152–1170

    Article  MathSciNet  MATH  Google Scholar 

  • Cai Y, Zhang L, Gu J, Yue Y, Wang Y (2018) Multiple meta-models based design space differentiation method for expensive problems. Struct Multidiscip Optim 57(6):2249–2258

    Article  MathSciNet  Google Scholar 

  • Celis M, Dennis J, Tapia R (1985) A trust region strategy for nonlinear equality constrained optimization. In: Boggs PT, Byrd RH, Schnabel RB (eds) Numerical optimization 1984. SIAM, Philadelphia, pp 71–82

    Google Scholar 

  • Clarke SM, Griebsch JH, Simpson TW (2005) Analysis of support vector regression for approximation of complex engineering analyses transactions of ASME. J Mech Des 127(6):1077–1087

    Article  Google Scholar 

  • Cressie N (1988) Spatial Prediction and Ordinary Kriging. Math Geol 20(4):405–421

    Article  MathSciNet  MATH  Google Scholar 

  • Dyn N, Levin D, Rippa S (1986) Numerical procedures for surface fitting of scattered data by radial basis functions. SIAM J Sci Stat Comput 7(2):639–659

    Article  MATH  Google Scholar 

  • Fadel GM, Cimtalay S (1993) Automatic evaluation of move-limits in structural optimization. Structural Optimization 6(4):233–237

    Article  Google Scholar 

  • Fadel GM, Riley MF, Barthelemy JM (1990) Two point exponential approximation method for structural optimization. Structural Optimization 2(2):117–124

    Article  Google Scholar 

  • Fang HB, Horstemeyer MF (2006) Global response approximation with radial basis functions. Eng Optim 38(4):407–424

    Article  MathSciNet  Google Scholar 

  • Fang H, Rais-Rohani M, Liu Z, Horstemeyer MF (2005) A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput Struct 83(25–26):2121–2136

    Article  Google Scholar 

  • Fang KT, Li R, Sudjianto A (2006) Design and modeling for computer experiments. Taylor & Francis Group, LLC, London

    MATH  Google Scholar 

  • Ferreira W, Serpa A (2016) Ensemble of metamodels: the augmented least squares approach. Struct Multidiscip Optim 53(5):1–28

    Article  MathSciNet  Google Scholar 

  • Ferreira W, Serpa A (2018) Ensemble of metamodels: extensions of the least squares approach to efficient global optimization. Struct Multidiscip Optim 57(1):131–159

    Article  MathSciNet  Google Scholar 

  • Friedman JH (1991) Multivariate adaptive regression splines. Ann Stat 19(1):1–67

    Article  MathSciNet  MATH  Google Scholar 

  • Goel T, Haftka RT, Shyy W, Queipo NV (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216

    Article  Google Scholar 

  • Gu J, Li GY, Dong Z (2009) Hybrid and adaptive metamodel based global optimization. Paper presented at the Proceedings of the ASME 2009 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference, IDETC/CIE 2009, DETC2009–87121, august 30–September 2, 2009, San Diego, California, USA

  • Gu J, Li GY, Dong Z (2012) Hybrid and adaptive meta-model-based global optimization. Eng Optim 44(1):87–104

    Article  Google Scholar 

  • Gu X, Lu J, Wang H (2015) Reliability-based design optimization for vehicle occupant protection system based on ensemble of metamodels. Struct Multidiscip Optim 51(2):533–546

    Article  Google Scholar 

  • Hardy RL (1971) Multiquadratic equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915

    Article  Google Scholar 

  • Hedar A-R (2005) Test functions for unconstrained global optimization. http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO_files/Page2904.htm

  • Jie H, Wu Y, Ding J (2015) An adaptive metamodel-based global optimization algorithm for black-box type problems. Eng Optim 47(11):1459–1480

    Article  MathSciNet  Google Scholar 

  • Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodelling techniques under multiple modelling criteria. Struct Multidiscip Optim 23(1):1–13

    Article  Google Scholar 

  • Jones DR, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

    Article  MathSciNet  MATH  Google Scholar 

  • Krige DG (1953) A statistical approach to some mine valuation and allied problems on the Witwatersrand. Master's thesis, University of the Witwatersrand

  • Lee J (2007) A novel three-phase trajectory informed search methodology for global optimization. J Glob Optim 2007(38):61–77

    Article  MathSciNet  MATH  Google Scholar 

  • Lee Y, Choi D-H (2014) Pointwise ensemble of meta-models using v nearest points cross-validation. Struct Multidiscip Optim 50(3):383–394

    Article  Google Scholar 

  • Lophaven SN, Nielsen HB, Søndergaard J (2002) DACE - A MATLAB Kriging Toolbox-Version 2.0. In: Informatics and mathematical Modelling,Technical University of Denmark, Kgs. Lyngby, Denmark, rep. No. IMMREP-2002-12

  • Mckay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

    Article  MathSciNet  MATH  Google Scholar 

  • Myers RH, Montgomery DC (2002) Response surface methodology: process and product optimization using designed experiments. Wiley, Toronto

    MATH  Google Scholar 

  • Rodriguez JF, Renaud JE, Watson LT (1998) Trust Region Augmented Lagrangian Methods for Sequential Response Surface Approximation and Optimization. J Mech Des 120(1):58–66

    Article  Google Scholar 

  • Sacks J, Schiller SB, Welch W (1989a) Designs for computer experiments. Technometrics 31(1):41–47

    Article  MathSciNet  Google Scholar 

  • Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989b) Design and analysis of computer experiments. Stat Sci 4(4):409–423

    Article  MathSciNet  MATH  Google Scholar 

  • Sharif B, Wang GG, ElMekkawy TY (2008) Mode pursuing sampling method for discrete variable optimization on expensive black-box functions. J Mech Des 130(2):021402-1-11

  • Shi R, Liu L, Long T, Liu J (2016) An efficient ensemble of radial basis functions method based on quadratic programming. Eng Optim 48(7):1202–1225

    Article  MathSciNet  Google Scholar 

  • Shin YS, Grandhi RV (2001) A global structural optimization technique using an interval method. Struct Multidiscip Optim 22(5):351–363

    Article  Google Scholar 

  • Simpson TW, Peplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: Survey and recommendations. Eng Comput 17(2):129–150

    Article  MATH  Google Scholar 

  • Viana FAC, Gogu C, Haftka RT (2010) Making the most out of surrogate models: tricks of the trade. In: Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference IDETC/CIE 2010, August 15–18, 2010, Montreal, Quebec, Canada

  • Viana FAC, Haftka RT, Watson LT (2013) Efficient global optimization algorithm assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689

    Article  MATH  Google Scholar 

  • Volpi S et al (2015) Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification. Struct Multidiscip Optim 51(2):347–368

    Article  Google Scholar 

  • Wang GG (2003) Adaptive response surface method using inherited Latin hypercube design points transactions of the ASME. J Mech Des 125(2):210–220

    Article  Google Scholar 

  • Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. ASME J Mech Des 129(4):370–380. https://doi.org/10.1115/1.2429697

  • Wang GG, Dong Z, Aitchisonc P (2001) Adaptive response surface method - a global optimization scheme for approximation-based design problems. Eng Optim 33(6):707–733

    Article  Google Scholar 

  • Wang LQ, Shan S, Wang GG (2004) Mode-pursuing sampling method for global optimization on expensive black-box functions. Eng Optim 36(4):419–438

    Article  Google Scholar 

  • Wilde D (1978) Globally optimal design. Wiley, New York

    Google Scholar 

  • Wujek BA, Renaud JE (1998a) New adaptive move-limit management strategy for approximate optimization, part1. AIAA J 36(10):1911–1921

    Article  Google Scholar 

  • Wujek BA, Renaud JE (1998b) New adaptive move-limit management strategy for approximate optimization, part2. AIAA J 36(10):1922–1934

    Article  Google Scholar 

  • Ye P, Pan G (2017) Global optimization method using ensemble of metamodels based on fuzzy clustering for design space reduction. Eng Comput 33(3):573–585

    Article  Google Scholar 

  • Yin H, Fang H, Wen G, Xiao Y (2018) On the ensemble of metamodels with multiple regional optimized weight factors. Struct Multidiscip Optim 2018(6):1–19

    Google Scholar 

  • Zhao L, Choi KK, Lee I (2011) Metamodeling method using dynamic kriging for design optimization. AIAA J 49(9):2034–2046

    Article  Google Scholar 

Download references

Acknowledgements

National Natural Science Foundation of China under grant number 51505138 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, People’s Republic of China

    Nianfei Gan

  2. Crash & Safety Department, Guangzhou Automobile Group Co., Ltd., Automotive Engineering Institute, Guangzhou, Guangdong Province, People’s Republic of China

    Jichao Gu

Authors
  1. Nianfei Gan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jichao Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jichao Gu.

Additional information

Responsible Editor: Somanath Nagendra

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gan, N., Gu, J. Hybrid meta-model-based design space exploration method for expensive problems. Struct Multidisc Optim 59, 907–917 (2019). https://doi.org/10.1007/s00158-018-2109-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-018-2109-x

Keywords

Search

Navigation