Structural and Multidisciplinary Optimization

, Volume 47, Issue 1, pp 63–76 | Cite as

Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design

Research Paper

Abstract

Robust design is an effective approach to design under uncertainty. Many works exist on mitigating the influence of parametric uncertainty associated with design or noise variables. However, simulation models are often computationally expensive and need to be replaced by metamodels created using limited samples. This introduces the so-called metamodeling uncertainty. Previous metamodel-based robust designs often treat a metamodel as the real model and ignore the influence of metamodeling uncertainty. In this study, we introduce a new uncertainty quantification method to evaluate the compound effect of both parametric uncertainty and metamodeling uncertainty. Then the new uncertainty quantification method is used for robust design. Simplified expressions of the response mean and variance is derived for a Kriging metamodel. Furthermore, the concept of robust design is extended for metamodel-based robust design accounting for both sources of uncertainty. To validate the benefits of our method, two mathematical examples without constraints are first illustrated. Results show that a robust design solution can be misleading without considering the metamodeling uncertainty. The proposed uncertainty quantification method for robust design is shown to be effective in mitigating the effect of metamodeling uncertainty, and the obtained solution is found to be more “robust” compared to the conventional approach. An automotive crashworthiness example, a highly expensive and non-linear problem, is used to illustrate the benefits of considering both sources of uncertainty in robust design with constraints. Results indicate that the proposed method can reduce the risk of constraint violation due to metamodel uncertainty and results in a “safer” robust solution.

Keywords

Parametric uncertainty Metamodeling uncertainty Uncertainty quantification Kriging  Robust design 

References

  1. Allen JK, Seepersad C, Choi HJ, Mistree F (2006) Robust design for multiscale and multidisciplinary applications. J Mech Design 128(4):832–843CrossRefGoogle Scholar
  2. Apley DW, Liu J, Chen W (2006) Understanding the effects of model uncertainty in robust design with computer experiments. J Mech Design 128:946–958CrossRefGoogle Scholar
  3. Chen W, Allen JK, Tsui KL, Mistree F (1996) A procedure for robust design: minimizing variations caused by noise factors and control factors. J Mech Design 118(4):478–485CrossRefGoogle Scholar
  4. Chen W, Wiecek M, Zhang J (1999) Quality utility: a compromise programming approach to robust design. J Mech Design 121(2):179–187CrossRefGoogle Scholar
  5. Choi JH, Lee WH, Park JJ, Youn BD (2008) A study on robust design optimization of layered plates bonding process considering process uncertainties. Struct Multidisc Optim 35(6):531–540CrossRefGoogle Scholar
  6. Du XP, Chen W (2000) Towards a better understanding of modeling feasibility robustness in engineering design. J Mech Design 122(4):385–394CrossRefGoogle Scholar
  7. Fonseca JR, Friswell MI, Lees AW (2007) Efficient robust design via Monte Carlo sample reweighting. Int J Numer Meth Eng 69:2279–2301MATHCrossRefGoogle Scholar
  8. Forrester AIJ, Sobester A, Keane AJ (2007) Multi-fidelity optimization via surrogate modelling. P Roy Soc A-Math Phy 463:3251–3269MathSciNetMATHCrossRefGoogle Scholar
  9. Ghosh D, Farhat C (2007) Strain and stress computations in stochastic finite element methods. Int J Numer Meth Eng 74:1219–1239MathSciNetCrossRefGoogle Scholar
  10. Gu L, Yang RJ, Tho CH, Makowskit M, Faruquet Q, Li Y (2001) Optimization and robustness for crashworthiness of side impact. Int J Veh Des 26:348–360CrossRefGoogle Scholar
  11. Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modelling criteria. Struct Multidisc Optim 23(1):1–13CrossRefGoogle Scholar
  12. Jin R, Du XP, Chen W (2003) The use of metamodeling techniques for optimization under uncertainty. Struct Multidisc Optim 25(2):99–116CrossRefGoogle Scholar
  13. Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plan Infer 134(1):268–287MathSciNetMATHCrossRefGoogle Scholar
  14. Jung DH, Lee BC (2002) Development of a simple and efficient method for robust optimization. Int J Numer Meth Eng 53:2201–2215MathSciNetMATHCrossRefGoogle Scholar
  15. Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc B 63(3):325–364MathSciNetCrossRefGoogle Scholar
  16. Kim C, Choi KK (2008) Reliability-based design optimization using response surface method with prediction interval estimation. J Mech Design 130(12):121–401CrossRefGoogle Scholar
  17. Kleijnen JPC (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716MathSciNetMATHCrossRefGoogle Scholar
  18. Lee KH, Kang DH (2006) A robust optimization using the statistics based on Kriging metamodel. J Mech Sci Technol 20(8):1169–1182CrossRefGoogle Scholar
  19. Lin Y, Luo D, Bailey T, Khire R, Wang JC, Simpson TW (2008) Model validation and error modeling to support sequential sampling. In: Proceeding of the ASME 2008 international design engineering technical conferences & computers and information in engineering conference, Brooklyn, New YorkGoogle Scholar
  20. Mera NS (2007) Efficient optimization processes using Kriging approximation models in electrical impedance tomography. Int J Numer Meth Eng 69:202–220MathSciNetMATHCrossRefGoogle Scholar
  21. Nechval NA, Nechval KN, Purgailis M, Berzins G, Rozevskis U (2011) Improvement of statistical decisions under parametric uncertainty. AIP Conf Proc 1394(47). doi:10.1063/1.3649935
  22. Paciorek CJ (2003) Nonstationary Gaussian processes for regression and spatial modelling. Dissertation, Carnegie Mellon UniversityGoogle Scholar
  23. Pan F, Zhu P (2011) Design optimization of vehicle roof structures: benefits of using multiple surrogates. Int J Crashworthiness 16(1):85–95MathSciNetCrossRefGoogle Scholar
  24. Park IP, Grandhi RV (2011) Quantifying multiple types of uncertainty in physics-based simulation using Bayesian model averaging. AIAA J 49(5):1037–1045CrossRefGoogle Scholar
  25. Park IP, Amarchinta HK, Grandhi RV (2010) A Bayesian approach for quantification of model uncertainty. Reliab Eng Syst Safe 95:777–785CrossRefGoogle Scholar
  26. Picheny V, Ginsbourger D, Roustant O, Haftka RT, Kim NH (2010) Adaptive designs of experiments for accurate approximation of a target region. J Mech Design 132(7):071008CrossRefGoogle Scholar
  27. Reinert JM, Apostolakis GE (2006) Including model uncertainty in risk-informed decision making. Ann Nucl Energy 33:354–369CrossRefGoogle Scholar
  28. Riley ME, Grandhi RV (2011) Quantification of model-form and predictive uncertainty for multi-physics simulation. Comput Struct 89(11):1206–1213CrossRefGoogle Scholar
  29. Simpson TW, Peplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2):129–150MATHCrossRefGoogle Scholar
  30. Taguchi G, Chowdhury S, Taguchi S (2000) Robust engineering. McGraw Hill Education Pvt. Ltd., New YorkGoogle Scholar
  31. Turner CJ, Campbell MI, Crawford RH (2003) Generic sequential sampling for metamodel approximations. In: Proceedings of ASME 2003 design engineering technical conferences and computers and information in engineering conference, Chicago, IllinoisGoogle Scholar
  32. Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Design 129(4):370–380MathSciNetCrossRefGoogle Scholar
  33. Wang S, Chen W, Tsui KL (2009) Bayesian validation of computer models. Technometrics 51(4):439–451MathSciNetCrossRefGoogle Scholar
  34. Xiong Y, Chen W, Apley D, Ding XR (2007) A non-stationary covariance-based kriging method for metamodeling in engineering design. Int J Numer Meth Eng 71(6):733–756MATHCrossRefGoogle Scholar
  35. Xiong Y, Chen W, Tsui KL, Apley D (2009) A better understanding of model updating strategies in validating engineering models. Comput Method Appl M 198(15–16):1327–1337MATHCrossRefGoogle Scholar
  36. Xiu DB (2006) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2):293–309MathSciNetGoogle Scholar
  37. Youn BD, Choi KK, Yang RJ, Gu L (2004) Reliability-based design optimization of crashworthiness of vehicle side impact. Struct Multidisc Optim 26:272–283CrossRefGoogle Scholar
  38. Zhang Y, Zhu P, Chen GL (2007) Lightweight design of automotive front side rail based on robust design. Thin Wall Struct 45:670–676CrossRefGoogle Scholar
  39. Zhu P, Zhang Y, Chen GL (2009) Metamodel-based lightweight design of an automotive front-body structure using robust optimization. P I Mech Eng D- J Aut 223(9):1133–1147CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Siliang Zhang
    • 1
  • Ping Zhu
    • 1
  • Wei Chen
    • 1
    • 2
  • Paul Arendt
    • 2
  1. 1.The State Key Laboratory of Mechanical System and Vibration, Shanghai Key Laboratory of Digital Autobody EngineeringShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

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