Archives of Computational Methods in Engineering

, Volume 26, Issue 1, pp 119–141 | Cite as

Solving the Two Objective Evolutionary Shape Optimization of a Natural Laminar Airfoil and Shock Control Bump with Game Strategies

  • Z. TangEmail author
  • Y. Chen
  • L. Zhang
  • J. Périaux
Original Paper


In order to improve the performances of a civil aircraft at transonic regimes, it is critical to develop new computational optimization methods reducing friction drag. Natural laminar flow (NLF) airfoil/wing design remain efficient methods to reduce the turbulence skin friction. However, the existence of wide range of favorable pressure gradient on a laminar flow airfoil/wing surface leads to strong shock waves occurring at the neighborhood of the trailing edge of the airfoil/wing. Consequently, the reduction of the friction drag due to the extension of the laminar flow surface of the airfoil is compensated with an increase of the shock wave induced drag. In this paper, an evolutionary algorithm (EAs) hybridized with different games (cooperative Pareto game, competitive Nash game and hierarchical Stackelberg game) for comparison is implemented to optimize the airfoil shape with a larger laminar flow range and a weaker shock wave drag simultaneously due to a shock control bump (SCB) active device. Numerical experiments demonstrate that each game coupled to the EAs optimizer can easily capture either a Pareto front, a Nash equilibrium or a Stackelberg equilibrium of this two-objective shape optimization problem. From the analysis/synthesis of 2D results it is concluded that a variety of laminar flow airfoils with greener aerodynamic performances can be significantly improved due to optimal SCB shape and position when compared to the baseline airfoil geometry. This methodology illustrate the potentiality of such an approach to solve the challenging shape optimization of the NLF wings in industrial design environments.


Natural laminar airfoil Shock control bump Pareto game Nash game Stackelberg game Evolutionary optimization 



This work has also benefited partially from the support of EU-China international cooperation projects from EC and MIIT. Acknowledgements also dedicate to NUAA and CIMNE colleagues for fruitful discussions on game theory and for institutions’ partial support provided during crossed visits of the authors to CIMNE and NUAA.


This study was funded by National natural Science Foundation of China (NSFC) under Grant Number 11272149.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Quadrio M, Ricco P (2011) The laminar generalized stokes layer and turbulent drag reduction. J Fluid Mech 667:135–157MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Du YQ, Karniadakis GE (2000) Suppressing wall turbulence by means of a transverse traveling wave. Science 288(5469):1230–1234CrossRefGoogle Scholar
  3. 3.
    Zhu ZQ, Wu ZC, Ding JC (2011) Laminar flow control technology and application. Chin J Aeronaut 32:765–784Google Scholar
  4. 4.
    Fujino M, Yoshizaki Y, Kawamura Y (2003) Natural-laminar-flow airfoil development for a lightweight business jet. J Aircr 40(4):609–615CrossRefGoogle Scholar
  5. 5.
    Chen YB, Tang ZL, Sheng JD (2016) Multi objective optimization for natural laminar flow airfoil in transonic flow. Chin J Comput Phys 33(3):283–296Google Scholar
  6. 6.
    Cebeci T, Cousteix J (2005) Modeling and computation of boundary-layer flows. Horizons Publishing Inc., Long BeachzbMATHGoogle Scholar
  7. 7.
    Stock HW, Haase W (1999) Feasibility study of \(e^N\) transition prediction in Navier-Stokes methods for airfoils. AIAA J 37(10):1187–1196CrossRefGoogle Scholar
  8. 8.
    Krumbein A, Krimmelbein N, Schrauf G (2009) Automatic transition prediction in hybrid flow solver, part 1: methodology and sensitivities. J Aircr 46(4):1176–1190CrossRefGoogle Scholar
  9. 9.
    Stock HW (2002) Airfoil validation using coupled Navier-Stokes and \(e^N\) transition prediction methods. J Aircr 39(1):51–58CrossRefGoogle Scholar
  10. 10.
    Crouch JD, Ng LL (2000) Variable N-factor method for transition prediction in three-dimensional boundary layers. AIAA J 38(2):211–216CrossRefGoogle Scholar
  11. 11.
    Windte J, Scholz U, Radespiel R (2006) Validation of the rans-simulation of laminar separation bubbles on airfoils. Aerosp Sci Technol 10(6):484–494CrossRefzbMATHGoogle Scholar
  12. 12.
    Cebeci T, Cousteix J (2003) Modeling and computation of boundary layer flow: laminar, turbulent and transitional boundary layers in compressible flows. Horizons Publishing, Long BeachzbMATHGoogle Scholar
  13. 13.
    Cebeci T, Keller HB (1971) Shooting and parallel shooting methods for solving the Falker-Skan boundary-layer equation. J Comput Phys 7:289–290CrossRefzbMATHGoogle Scholar
  14. 14.
    Cebeci T (2004) Stability and transition: theory and application. Springer, BerlinzbMATHGoogle Scholar
  15. 15.
    Langtry RB, Menter FR (2009) Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes. AIAA J 47(12):2894–2906CrossRefGoogle Scholar
  16. 16.
    Krumbein A (2005) An automatic transition prediction and application to high-lift multi-element configurations. J Aircr 42(5):1150–1164CrossRefGoogle Scholar
  17. 17.
    Lee JD, Jameson A (2009) Natural-laminar-flow airfoil and wing design by adjoint method and automatic transition prediction. In: 47th AIAA aerospace sciences meeting including the New Horizons forum and aerospace exposition, aerospace sciences meetings, Orlando, FloridaGoogle Scholar
  18. 18.
    Somers DM (1981) Design and experimental results for a natural laminar flow airfoil for general aviation application. NASA Technical PaperGoogle Scholar
  19. 19.
    Driver J, Zing DW (2002) Optimzed natural-laminar-flow airfoils. AIAA J 40(6):4–6Google Scholar
  20. 20.
    Giles MB, Drela M (1987) Two-dimensional transonic aerodynamic design method. AIAA J 25(9):1199–1206CrossRefGoogle Scholar
  21. 21.
    Amoignon O, Pralits J, Hanifi A, Berggren M, Henningson D (2006) Shape optimization for delay of laminar-turbulent transition. AIAA J 44(5):1009–1024CrossRefGoogle Scholar
  22. 22.
    Farin G (1990) Curves and surfaces for computer aided geometric design. Academic Press, New YorkzbMATHGoogle Scholar
  23. 23.
    Monterde J (2001) Singularities of rational bezier curves. Comput Aided Geom Des 18(8):805–816MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fawaz Z, Xu YG, Behdinan K (2005) Hybrid evolutionary algorithm and application to structural optimization. Struct Multidiscip Optim 30(3):219–226CrossRefGoogle Scholar
  25. 25.
    Periaux J, Gonzalez LF, Lee DS (2015) Evolutionary optimization and games strategies for advanced multi physics design in aerospace engineering. Springer, BerlinGoogle Scholar
  26. 26.
    Jameson A, Martinelli L, Pierce NA (1998) Optimum aerodynamic design using the Navier-Stokes equations. ThCFD 10(1–4):213–237zbMATHGoogle Scholar
  27. 27.
    Ashill PR, Fulker LJ, Shires A (1992) A novel technique for controlling shock strength of laminar flow aerofoil sections. In: Proceeding 1st European forum on laminar flow technology, 16–18 March, Hamburg, GermanyGoogle Scholar
  28. 28.
    Qin N, Zhu Y, Shaw ST (2004) Numerical study of active shock control for transonic aerodynamics. Int J Numer Method Heat Fluid Flow 14(4):444–466CrossRefzbMATHGoogle Scholar
  29. 29.
    Chowdhury S, Dulikravich GS (2010) Improvements to single-objective constrained predator-prey evolutionary optimization algorithm. Struct Multidiscip Optim 41(4):541–554MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wong WS, Qin N, Sellars N, Holden H, Babinsky H (2008) A combined experimental and numerical study of flow structures over three-dimensional shock control bumps. Aerosp Sci Technol 12(6):436–447CrossRefzbMATHGoogle Scholar
  31. 31.
    Tian Y, Liu P, Feng P (2011) Shock control bump parametric research on supercritical airfoil. Sci China Technol Sci 54(11):2935–2944CrossRefGoogle Scholar
  32. 32.
    Nash JF (1951) Non-cooperative games. Ann Math 54(2):286–295MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Aubin JP (1979) Mathematical methods of game and economic theory. North-Holland Publishing Co., AmsterdamzbMATHGoogle Scholar
  34. 34.
    Tang ZL (2006) Multi-objective optimization strategies using adjoint method and game theory in aerodynamics. Acta Mech Sin 22(4):307–314MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tang ZL, Dong J (2009) Couplings in multi-criterion aerodynamic optimization problems using adjoint methods and game strategies. Chin J Aeronaut 22(1):1–8MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tang Z, Zhang LH (2016) Nash equilibrium and multi criterion aerodynamic optimization. J Comput Phys 314:107–126MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sefrioui M (1998) Algorithmes evolutionnaires pour le calcul scientifique, Application à l’Electromagnetisme et à la Mécanique des Fluides Numérique, Ph.D. thesis, Universite Pierre et Marie Curie, ParisGoogle Scholar
  38. 38.
    Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  39. 39.
    Tang ZL, Chen YB, Zhang LH (2017) Natural laminar flow shape optimization in transonic regime with competitive Nash game strategy. Appl Math Model 48:534–547MathSciNetCrossRefGoogle Scholar
  40. 40.
    Aumann RJ, Dreze JH (1974) Cooperative games with coalition structures. Int J Game Theory 3(4):217–237MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ungureanu V (2008) Equilibria in Pareto-Nash-Stackelberg games. In: World Congress of the Game Theory Society Games 2008 Evanston, Illinois, July 13–17 2007, pp 1–13Google Scholar
  42. 42.
    Li MQ, Yang SX, Liu XH (2016) Pareto or non-Pareto: bi-criterion evolution in multiobjective optimization. IEEE Trans Evol Comput 20(5):645–665CrossRefGoogle Scholar
  43. 43.
    Greiner D, Periaux J, Emperador JM, Blas Galvàn B, Winter G (2016) Game theory based evolutionary algorithms: a review with nash applications in structural engineering optimization problems. Arch Comput Method Eng 23(3):1–48zbMATHGoogle Scholar
  44. 44.
    Wang L, Gao HW, Petrosyan L, Qiao H, Sedakov A (2016) Strategically supported cooperation in dynamic games with coalition structures. Sci China Math 59(5):1015–1028MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2017

Authors and Affiliations

  1. 1.College of Aerospace EngineeringNanjing University of Aeronautics and Astronautics (NUAA)NanjingChina
  2. 2.AVIC Aerodynamics Research InstituteHaerbinChina
  3. 3.International Center for Numerical Methods in Engineering (CIMNE)Universitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.Faculty of Information TechnologyUniversity of JyväskyläJyväskyläFinland

Personalised recommendations