Robust optimization of aerodynamic loadings for airfoil inverse designs

  • C. J. B. Reis
  • N. Manzanares-Filho
  • A. M. G. de LimaEmail author
Technical Paper


For the modern design of more realistic aerodynamic shapes, disturbances caused by uncertain operating conditions must be conveniently considered, since they can affect significantly the performance of the designed systems. In this situation, the concept of robust design in conjunction with optimization tools is strongly recommended, since the interest is to maximize the performance and its robustness, simultaneously. Clearly, the great number of exact evaluations normally required to compute the robustness makes the robust optimization in aerodynamics computationally prohibitive, particularly when direct methods are chosen. To overcome this drawback, inverse methods appear as an interesting option, provided a robust aerodynamic loading can be furnished previously at low computational costs. However, few works have been dedicated to this subject in the open literature, which motivates the present study. The focus is to apply the robustness concept to optimize the velocity (or pressure) distributions for airfoil inverse designs, using a boundary layer method to predict the aerodynamic coefficients prior to the knowledge of the final airfoil shape. Here, the velocity distribution is parameterized using B-spline polygons with a set of control points, where the design variables are the ordinates of these points in the parameterization. The resulting robust multiobjective optimization problem involves the performance of the airfoil as a first objective function and its robustness introduced as additional objective to be optimized simultaneously. To illustrate the usefulness of the proposed robust design method, an example of drag minimization for an isolated airfoil is addressed and the aerodynamic coefficients for the optimal airfoils are compared with the corresponding obtained by experiments from the open literature.


Aerodynamics Airfoils Inverse methods Robust design Multiobjective optimization 



The authors are grateful to the FAPEMIG for the financial support to their research projects APQ-01865-18 (N. Manzanares-Filho) and PPM-00548-18 (A.M.G. de Lima) and the Brazilian Research Council—CNPq for the continued support to their research work, especially through research projects 302026/2016-9 (A.M.G. de Lima). The authors would also like to thank Mr. L.L.F. Soares (a doctorate student at UNIFEI) for his invaluable help with CFD calculations using ANSYS Fluent.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Kontoleontos EA, Giannakoglou KC, Koubogiannis DG (2005) Robust design of compressor cascade airfoils using evolutionary algorithms and surrogate models. In: Proceedings of the 1st international conference on experiments/process/simulation/optimization (IC-EpsMsO), AthensGoogle Scholar
  2. 2.
    Reis CJB, Manzanares-Filho N, de Lima AMG (2016) Robust optimization of turbomachinery cascades using inverse methods. J Braz Soc Mech Sci Eng 38:297–305CrossRefGoogle Scholar
  3. 3.
    Huyse L, Lewis M (2001) Aerodynamic shape optimization of two dimensional airfoils under uncertain operating conditions. ICASE Report No. 2001-1, ICASE, NASA Langley Research Center, Hampton, Virginia, USAGoogle Scholar
  4. 4.
    Sampaio R, Soize C (2007) On measures of nonlinearity effects for uncertain dynamical systems applications to a vibro-impact system. J Sound Vib 303:659–674CrossRefGoogle Scholar
  5. 5.
    Taguchi G, Elsayed EA, Hsiang TC (1989) Quality engineering in production systems. McGraw-Hill, New YorkGoogle Scholar
  6. 6.
    Florian A (1992) An efficient sampling scheme: updates Latin-hyper-cube sampling. Probab Eng Mech 7:123–130CrossRefGoogle Scholar
  7. 7.
    Beyer H, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196:3190–3218MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eschenauer J, Koski J, Osyczka A (1990) Multicriteria design optimization. Springer, BerlinCrossRefGoogle Scholar
  9. 9.
    Lee KH, Park GJ (1996) Robust optimization considering tolerance of design variables. J Comput Struct 79:77–86CrossRefGoogle Scholar
  10. 10.
    Srinivas N, Deb K (1993) Multiobjective using non-dominated sorting in genetic algorithms. Technical Report, Department of Mechanical Engineering, Institute of Technology, IndiaGoogle Scholar
  11. 11.
    The Mathworks Inc (2017) Genetic algorithm and direct search toolbox guide. Accessed 04 Nov 2017
  12. 12.
    Obayashi S, Takanashi S (1996) Genetic optimization of target pressure distributions for inverse design methods. AIAA J 34:881–886CrossRefGoogle Scholar
  13. 13.
    Rogers D (2001) An introduction to NURBS: with historical perspective. Academic Press, San DiegoGoogle Scholar
  14. 14.
    Kumar A, Nair PB, Keane AJ, Shahpar S (2008) Robust design using bayesian Monte Carlo. Int J Numer Methods Eng 73:1497–1517CrossRefGoogle Scholar
  15. 15.
    Drela M (1998) Pros and cons of airfoil optimization. Front Comput Fluid Dyn. CrossRefzbMATHGoogle Scholar
  16. 16.
    Squire HB, Young AD (1938) The calculation of the profile drag of airfoils. Aeronautical Research Committee, Repts. and Memoranda No. 1838Google Scholar
  17. 17.
    Young AD (1989) Boundary layers. AIAA Education Series. AIAA, Washington, DCGoogle Scholar
  18. 18.
    Manzanares-Filho N, Albuquerque RBF, Sousa BS, Santos LGC (2018) A comparative study of controlled random search algorithms with application to inverse airfoil design. Eng Optim 50:996–1015MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lewis RI (1991) Vortex element methods for fluid dynamic analysis of engineering systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  20. 20.
    Gostelow JP (1984) Cascade aerodynamics. J Fluid Mech 153:503Google Scholar
  21. 21.
    Manzanares-Filho N (1994) Axial flow turbomachinery analysis” (in portuguese). Thesis, Instituto Tecnológico de Aeronáutica-ITA, BrazilGoogle Scholar
  22. 22.
    Moran J (1984) An introduction to theoretical and computational aerodynamics. Wiley, New YorkGoogle Scholar
  23. 23.
    The Mathworks Inc (2019) Multiobjective genetic algorithm options Accessed 01 Mar 2019
  24. 24.
    Abbott IH, von Doenhoff AE (1959) Theory of wing sections. Dover Publications Inc, New YorkGoogle Scholar
  25. 25.
    Drela M (2007) A user’s guide to MSES 3.05. MIT Department of Aeronautics and AstronauticsGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Mechanical Engineering InstituteFederal University of ItajubáItajubáBrazil
  2. 2.School of Mechanical EngineeringFederal University of UberlândiaUberlândiaBrazil

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