Skip to main content
SpringerLink
Log in

An Improved PC-Kriging Method for Efficient Robust Design Optimization

  • Conference paper
  • First Online:
Advances in Mechanical Design (ICMD 2019)

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 77))

Included in the following conference series:

Abstract

The polynomial-chaos-kriging (PC-Kriging) method has been derived as a new uncertainty propagation approach and widely used for robust design optimization in a straightforward manner, of which the statistical moments would be estimated through directly conducting Monte Carlo simulation (MCS) on the PC-Kriging model. However, the computational cost still cannot be negligible because thousands of statistical moment estimations might be performed during robust optimization, especially for highly nonlinear and complicated engineering problems. An analytical statistical moment estimation method is derived for PC-Kriging in this work to reduce the computational cost rather than referring to MCS. Meanwhile, a sequential sampling strategy is applied for PC-Kriging model construction, in which the sample points are not generated all at once, but sequentially allocated in the region with the largest prediction uncertainty to improve the accuracy of PC-Kriging model as much as possible. Through testing on three mathematical examples and an airfoil robust optimization design problem, it is noticed that the improved PC-Kriging method with analytical statistical moment estimation and sequential sampling strategy is more efficient than the traditional ones, demonstrating its effectiveness and advantage.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

D :

= dimension of random variables

\( \varvec{F} \) :

= information matrix in PC model

Ma :

= mach number of flight flow field

N :

= number of sample points

P :

= number of coefficients in PC model

PC:

= polynomial chaos

PCK:

= polynomial chaos kriging

p :

= order of PC model

\( R( \bullet ) \) :

= auto-correlation function

R :

= lift-to-drag ratio

α :

= flight angle of attack

\( \beta \) :

= coefficient of PC model

\( \sigma^{2} \) :

= prior variance of the gaussian random process

\( {\mathbf{\varphi }}\left( \bullet \right) \) :

= multi-dimensional orthogonal polynomial

References

  1. Cook, L.W., Jarrett, J.P.: Robust airfoil optimization and the importance of appropriately representing uncertainty. AIAA J. 55(11), 1–15 (2017). https://doi.org/10.2514/1.j055459

    Article  Google Scholar 

  2. Zhang, Y., Zhu, P., Chen, G.L.: Lightweight design of automotive front side rail based on robust optimization. Thin-Walled Struct. 45(7), 670–676 (2007). https://doi.org/10.1016/j.tws.2007.05.007

    Article  Google Scholar 

  3. Cheng, Q., Wang, S.W., Yan, C.C.: Robust optimal design of chilled water systems in buildings with quantified uncertainty and reliability for minimized life-cycle cost. Energy Build. 126(15), 159–169 (2016). https://doi.org/10.1016/j.enbuild.2016.05.032

    Article  Google Scholar 

  4. Ben-Tal, A., Nemirovski, A.: Robust optimization – methodology and applications. Math. Program. 92(3), 453–480 (2002). https://doi.org/10.1007/s101070100286

    Article  MathSciNet  MATH  Google Scholar 

  5. Xiu, D.B., Karniadakis, G.E.M.: The Wiener-askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). https://doi.org/10.1137/s1064827501387826

    Article  MathSciNet  MATH  Google Scholar 

  6. Dodson, M., Parks, G.T.: Robust aerodynamic design optimization using polynomial chaos. J. Aircr. 46(2), 635–646 (2015). https://doi.org/10.2514/1.39419

    Article  Google Scholar 

  7. Wei, X., Feng, B.W., Liu, Z.Y.: Ship uncertainty optimization design based on multidimensional polynomial chaos expansion method. Ship Eng. 1(40), 42–47 (2018)

    Google Scholar 

  8. Kim, N.H., Wang, H., Queipo, N.V.: Efficient shape optimization under uncertainty using polynomial chaos expansions and local sensitivities. AIAA J. 44(5), 1112–1116 (2006). https://doi.org/10.2514/1.13011

    Article  Google Scholar 

  9. Fisher, J., Bhattacharya, R.: Linear quadratic regulation of systems with stochastic parameter uncertainties. Automatica 45(12), 2831–2841 (2009). https://doi.org/10.1016/j.automatica.2009.10.001

    Article  MathSciNet  MATH  Google Scholar 

  10. Prabhakar, A., Fisher, J., Bhattacharya, R.: Polynomial chaos based analysis of probabilistic uncertainty in hypersonic flight dynamics. J. Guid. Control Dyn. 33(1), 222–234 (2010). https://doi.org/10.2514/1.41551

    Article  Google Scholar 

  11. Wang, F.G., Xiong, F.F., Jiang, H., Song, J.M.: An enhanced data-driven polynomial chaos method for uncertainty propagation. Eng. Optim. 50(2), 1–20 (2017). https://doi.org/10.1080/0305215x.2017.1323890

    Article  Google Scholar 

  12. Schobi, R., Sudret, B., Wiart, J.: Polynomial-chaos-based Kriging. Statistics 5(2), 55–63 (2015). https://doi.org/10.1615/int.j.uncertaintyquantification.2015012467

    Article  MathSciNet  Google Scholar 

  13. Kersaudy, P., Sudret, B., Varsier, N., Wiart, J.: A new surrogate modeling technique combining Kriging and polynomial chaos expansions application to uncertainty analysis in computational dosimetry. J. Comput. Phys. 286(14), 103–117 (2015). https://doi.org/10.1016/j.jcp.2015.01.034

    Article  MathSciNet  MATH  Google Scholar 

  14. Schobi, R., Sudret, B.: Imprecise structural reliability analysis using PC-Kriging. In: 25th European Safety and Reliability Conference (2015). https://doi.org/10.1201/b19094-549

    Chapter  Google Scholar 

  15. Xiong, F.F., Yang, S.X., Liu, Y., Chen, S.S.: Analysis Method of Engineering Probability Uncertainty. Science Press, Beijing (2015)

    Google Scholar 

  16. Sepahvand, K., Marburg, S., Hardtke, H.J.: Uncertainty quantification in stochastic systems using polynomial chaos expansion. Int. J. Appl. Mech. 2(2), 305–353 (2010). https://doi.org/10.1142/s1758825110000524

    Article  MATH  Google Scholar 

  17. Xiong, F.F.: Weighted stochastic response surface method considering sample weights. Struct. Multidiscip. Optim. 43(6), 837–849 (2011). https://doi.org/10.1007/s00158-011-0621-3

    Article  Google Scholar 

  18. Hosder, S., Walters, R.W., Balch, M.: Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: AIAA Non-Deterministic Approaches Conference (2007). https://doi.org/10.2514/6.2007-1939

  19. Hampton, J., Doostan, A.: Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280(12), 363–386 (2015). https://doi.org/10.1016/j.jcp.2014.09.019

    Article  MathSciNet  MATH  Google Scholar 

  20. An, D., Choi, J.H.: Efficient reliability analysis based on Bayesian framework under input variable and metamodel uncertainties. Struct. Multidiscip. Optim. 46(4), 533–547 (2012). https://doi.org/10.1007/s00158-012-0776-6

    Article  MathSciNet  MATH  Google Scholar 

  21. Xiong, F.F.: Robust design optimization considering metamodel uncertainty. J. Mech. Eng. 50(19), 136–143 (2014, in Chinese). https://doi.org/10.3901/jme.2014.19.136

    Article  Google Scholar 

  22. Miller, F.P., Vandome, A.F., Mcbrewster, J.: Inverse Transform Sampling. Alphascript Publishing, German (2010)

    Google Scholar 

  23. Lockwood, B.A., Anitescu, M.: Gradient-enhanced universal Kriging for uncertainty propagation. Nucl. Sci. Eng. 170(2), 168–195 (2012). https://doi.org/10.13182/nse10-86

    Article  Google Scholar 

  24. Ganesh Ram, R.K., Cooper, Y.N., Bhatia, V.: Design optimization and analysis of NACA0012 airfoil using computational fluid dynamics and genetic algorithm. Appl. Mech. Mater. 664(22) 111–116 (2014). https://www.scientific.net/AMM.664.111

    Article  Google Scholar 

  25. Liang, Y., Cheng, X.Q., Li, Z.N., Xiang, J.W.: Multi-objective robust airfoil optimization based on non-uniform rational B-spline (NURBS) representation. Sci. China Ser. E: Technol. Sci. 53(10), 2708–2717 (2010). https://doi.org/10.1007/s11431-010-4075-4

    Article  MATH  Google Scholar 

  26. Chen, X., Agarwal, R.K.: Optimization of wind turbine blade airfoils using a multi-objective genetic algorithm. J. Aircr. 50(2), 519–527 (2013). https://doi.org/10.2514/1.c031910

    Article  Google Scholar 

  27. Cheng, F.Y., Li, D.: Multiobjective optimization design with Pareto genetic algorithm. J. Struct. Eng. 123(9), 1252–1261 (1997). https://doi.org/10.1061/(asce)0733-9445(1997)123:9(1252)

    Article  Google Scholar 

Download references

Acknowledgement

The grant support from Science Challenge Project (No. TZ2018001) and Hongjian Innovation Foundation (No. BQ203-HYJJ-Q2018002) is greatly acknowledged.

Author information

Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Qizhang Lin, Fenfen Xiong & Fenggang Wang

  2. Science and Technology on Complex Aviation Systems Simulation Laboratory, Beijing, 100076, China

    Chao Chen

  3. Beijing Electromechanical Engineering Institute, Beijing, 100074, China

    Shishi Chen

Authors
  1. Qizhang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fenfen Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shishi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fenggang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fenfen Xiong .

Editor information

Editors and Affiliations

  1. Zhejiang University, Hangzhou, China

    Jianrong Tan

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lin, Q., Chen, C., Xiong, F., Chen, S., Wang, F. (2020). An Improved PC-Kriging Method for Efficient Robust Design Optimization. In: Tan, J. (eds) Advances in Mechanical Design. ICMD 2019. Mechanisms and Machine Science, vol 77. Springer, Singapore. https://doi.org/10.1007/978-981-32-9941-2_33

Download citation

  • DOI: https://doi.org/10.1007/978-981-32-9941-2_33

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-32-9940-5

  • Online ISBN: 978-981-32-9941-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation