Structural and Multidisciplinary Optimization

, Volume 59, Issue 2, pp 421–438 | Cite as

Constraint aggregation for large number of constraints in wing surrogate-based optimization

  • Ke-Shi Zhang
  • Zhong-Hua HanEmail author
  • Zhong-Jian Gao
  • Yuan Wang
Research Paper


The method of aggregating a large number of constraints into one or few constraints has been successfully applied to wing structural design using gradient-based local optimization. However, numerical difficulties may occur in the case that the local curvatures of the aggregated constraint become extremely large and then ill-conditioned Hessian matrix may be yielded. This paper aims to test different methods of constraint aggregation within the framework of a gradient-free optimization, which makes use of cheap-to-evaluate surrogate models to find the global optimum. Three constraint aggregation approaches are investigated: the maximum constraint approach, the constant parameter Kreisselmeier-Steinhauser (KS) function, and the adaptive KS function. We also explore methods of aggregating constraints over the entire structure and within sub-domains. Examples of structural optimization and aero-structural optimization for a transport aircraft wing are employed and the results show that (1) the KS function with a larger constant parameter ρ can lead to better optimization results than the adaptive method, as the active constraints are approximated more accurately; (2) lumping the constraints within sub-domains instead of all together can improve the accuracy of the aggregated constraint and therefore helps find a better design. Finally, it is concluded from current test cases that the most efficient way of handling large-scale constraints for wing surrogate-based optimization is to aggregate constraints within sub-domains and with a relatively large constant parameter.


Wing design Surrogate-based optimization Constraint aggregation Kreisselmeier-Steinhauser function Aero-structural optimization 



Computational fluid dynamics


Computational structural dynamics


Expected improvement


Finite element method


Genetic algorithms




Lower confidence bounding


Latin hypercube sampling


Multidisciplinary analysis


Mean squared error


Minimizing surrogate prediction


Probability of improvement


Surrogate-based optimization


Sequential quadratic programming



The authors are grateful for the thoughtful comments and valuable suggestions given by the anonymous reviewers.

Funding information

This research was sponsored by the National Natural Science Foundation of China (NSFC) under grant no. 11772261 and Aeronautical Science Foundation of China under grant no. 2016ZA53011.

Supplementary material

158_2018_2074_MOESM1_ESM.rar (8.5 mb)
ESM 1 (RAR 8715 kb)


  1. Akguen MA, Haftka RT, Wu KC, Walsh JL, Garcelon JH (2001) Efficient structural optimization for multiple load cases using adjoint sensitivities. AIAA J 39(3):511–516CrossRefGoogle Scholar
  2. Brooks TR, Kenway GK, Martins JRRA (2017) Undeflected common research model (uCRM): an aerostructural model for the study of high aspect ratio transport aircraft wings. In: 35th AIAA Applied Aerodynamics Conference, AIAA paper 2017-4456, Denver, Colorado, 5–9 JuneGoogle Scholar
  3. Chernukhin O, Zingg DW (2013) Multimodality and global optimization in aerodynamic design. AIAA J 51(6):1342–1354CrossRefGoogle Scholar
  4. Courrier N, Boucard PA, Soulier B (2016) Variable-fidelity modeling of structural analysis of assemblies. J Glob Optim 64(3):577–613MathSciNetCrossRefzbMATHGoogle Scholar
  5. Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186:311–338CrossRefzbMATHGoogle Scholar
  6. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79CrossRefGoogle Scholar
  7. Ha H, Oh S, Yee K (2014) Feasibility study of hierarchical Kriging model in the design optimization process. J Korean Soc Aero Spac Sci 42(2):108–118Google Scholar
  8. Haftka RT, Villanueva D, Chaudhuri A (2016) Parallel surrogate-assisted global optimization with expensive functions – a survey. Struct Multidiscip Optim 54(1):3–13MathSciNetCrossRefGoogle Scholar
  9. Haghighat S, Martins JRRA, Liu HHT (2013) Aeroservoelastic design optimization of a flexible wing. J Aircr 49(2):432–443CrossRefGoogle Scholar
  10. Han Z-H (2016) SurroOpt: a generic surrogate-based optimization code for aerodynamic and multidisciplinary design. In: 30th Congress of the International Council of the Aeronautical Sciences, ICAS paper 2016_0281, Daejeon, Korea, 25–30 SeptemberGoogle Scholar
  11. Han Z-H, Goertz S (2012) Hierarchical Kriging model for variable-fidelity surrogate modeling. AIAA J 50(5):1285–1296CrossRefGoogle Scholar
  12. Han Z-H, Goertz S, Zimmermann R (2013) Improving variable-fidelity surrogate modeling via gradient enhanced Kriging and a generalized hybrid bridge function. Aerosp Sci Technol 25(1):177–189CrossRefGoogle Scholar
  13. Han Z-H, Zhang Y, Song C-X, Zhang K-S (2017) Weighted gradient-enhanced Kriging for high-dimensional surrogate modeling and design optimization. AIAA J 55(12):4330–4346CrossRefGoogle Scholar
  14. Han Z-H, Chen J, Zhang K-S, Zhu Z, Song W-P (2018a) Aerodynamic shape optimization of natural-laminar-flow wing using surrogate-based approach. AIAA J 56(7):2579–2593CrossRefGoogle Scholar
  15. Han Z-H, Abu-Zurayk M, Görtz S, Ilic C (2018b) Surrogate-based aerodynamic shape optimization of a wing-body transport aircraft configuration. In: Heinrich R (ed) AeroStruct: enable and learn how to integrate flexibility in design. AeroStruct 2015. Notes on numerical fluid mechanics and multidisciplinary design, vol 138. Springer, Cham, pp 257–282Google Scholar
  16. Jo Y, Yi S, Choi S, Lee DJ, Choi DZ (2016) Adaptive variable-fidelity analysis and design using dynamic fidelity indicators. AIAA J 54(11):3565–3579CrossRefGoogle Scholar
  17. Kennedy GJ, Martins JRRA (2014) A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures. Finite Elem Anal Des 87(15):56–73CrossRefGoogle Scholar
  18. Kenway GKW, Martins JRRA (2014) Multi-point high fidelity aerostructural optimization of a transport aircraft configuration. J Aircr 51(1):144–160CrossRefGoogle Scholar
  19. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. In: International Federation of Active Controls Symposium on Computer-Aided Design of Control System, Zurich, SwitzerlandGoogle Scholar
  20. Laurenceau J, Meaux M, Montagnac M, Sagaut P (2012) Comparison of gradient-based and gradient-enhanced response-surface-based optimizers. AIAA J 48(5):981–994CrossRefGoogle Scholar
  21. Leifsson L, Koziel S (2015) Aerodynamic shape optimization by variable-fidelity computational fluid dynamics models: a review of recent progress. J Comput Sci 10:45–54MathSciNetCrossRefGoogle Scholar
  22. Leifsson L, Koziel S, Tesfahunegn YA (2016) Multiobjective aerodynamic optimization by variable-fidelity models and response surface surrogates. AIAA J 54(2):531–541CrossRefGoogle Scholar
  23. Liu J, Song WP, Han ZH, Zhang Y (2017) Efficient aerodynamic shape optimization of transonic wings using a parallel infilling strategy and surrogate models. Struct Multidiscip Optim 55(3):925–943CrossRefGoogle Scholar
  24. Mortished C, Ollar J, Jones R, Benzie P, Toropov V, Sienz J (2016) Aircraft wing optimization based on computationally efficient gradient-enhanced Kriging. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA paper 2016-0420, San Diego, California, US, 4–8 JanuaryGoogle Scholar
  25. Parr JM, Holden CME, Forrester AIJ, Keane AJ (2010) Review of efficient surrogate infill sampling criteria with constraint handling. In: 2nd international conference on engineering optimization, Lisbon, 6–9 SeptemberGoogle Scholar
  26. Poon NMK, Martins JRRA (2007) An adaptive approach to constraint aggregation using adjoint sensitivity analysis. Struct Multidiscip Optim 34(1):61–73CrossRefGoogle Scholar
  27. Queipo NV, Haftka RT, Shyy W, Goela T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 45(1):1–28CrossRefGoogle Scholar
  28. Raspanti C, Bandoni J, Biegler L (2000) New strategies for flexibility analysis and design under uncertainty. Comput Chem Eng 24(9–10):2193–2209CrossRefGoogle Scholar
  29. Sasena MJ, Papalambros P, Goovaerts P (2002) Exploration of meta modeling sampling criteria for constrained global optimization. Eng Optim 34(3):263–278CrossRefGoogle Scholar
  30. Schonlau M (1997) Computer experiments and global optimization. PhD thesis, University of Waterloo, 1997Google Scholar
  31. Viana FAC, Simpson TW, Balabanov V, Toropov V (2014) Metamodeling in multidisciplinary design optimization: how far have we really come? AIAA J 52(4):670–690CrossRefGoogle Scholar
  32. Wrenn GA (1989) An indirect method for numerical optimization using the Kreisselmeier-Steinhauser function. Technical report CR-4220, NASAGoogle Scholar
  33. Zhang KS, Han ZH, Li WJ, Song WP (2008) Coupled aerodynamic/structural optimization of subsonic transport wing using a surrogate model. J Aircr 45(6):2167–2170CrossRefGoogle Scholar
  34. Zhang Y, Han Z-H, Zhang K-S (2018) Variable-fidelity expected improvement for efficient global optimization of expensive functions. Struct Multidiscip Optim.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ke-Shi Zhang
    • 1
  • Zhong-Hua Han
    • 1
    Email author
  • Zhong-Jian Gao
    • 1
  • Yuan Wang
    • 1
  1. 1.School of AeronauticsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

Personalised recommendations