Optimization and Engineering

, Volume 12, Issue 3, pp 349–369 | Cite as

Airfoil design for compressible inviscid flow based on shape calculus

  • Stephan Schmidt
  • Caslav Ilic
  • Volker Schulz
  • Nicolas R. Gauger


Aerodynamic design based on the Hadamard representation of shape gradients is considered. Using this approach, the gradient of an objective function with respect to a deformation of the shape can be computed as a boundary integral without any additional “mesh sensitivities” or volume quantities. The resulting very fast gradient evaluation procedure greatly supports a one-shot optimization strategy and coupled with an appropriate shape Hessian approximation, a very efficient shape optimization procedure is created that does not deteriorate with an increase in the number of design parameters. As such, all surface mesh nodes are used as shape design parameters for optimizing a variety of lifting and non-lifting airfoil shapes using the compressible Euler equations to model the fluid.


Shape derivative Preconditioning One-shot Drag reduction Euler flow 


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  1. Arian E, Ta’asan S (1996) Analysis of the Hessian for aerodynamic optimization: Inviscid flow. Technical Report 96-28, Institute for Computer Applications in Science and Engineering (ICASE) Google Scholar
  2. Arian E, Vatsa VN (1998) A preconditioning method for shape optimization governed by the Euler equations. Technical Report 98-14, Institute for Computer Applications in Science and Engineering (ICASE) Google Scholar
  3. Delfour MC, Zolésio J-P (2001) Shapes and geometries: analysis, differential calculus, and optimization. In: Advances in design and control. SIAM, Philadelphia Google Scholar
  4. Eppler K, Schmidt S, Schulz V, Ilic C (2009) Preconditioning the pressure tracking in fluid dynamics by shape Hessian information. J Optim Theory Appl 141(3):513–531 MATHCrossRefMathSciNetGoogle Scholar
  5. Gauger NR (2003) Das Adjungiertenverfahren in der aerodynamischen Formoptimierung. Forschungsbericht DLR-FB–2003-05, Deutsches Zentrum für Luft- und Raumfahrt eV Google Scholar
  6. Gherman I (2008) Approximate partially reduced SQP approaches for aerodynamic shape optimization problems. PhD thesis, University of Trier, Trier, Germany Google Scholar
  7. Giles MB, Pierce NA (1997) Adjoint equations in CFD: duality, boundary conditions and solution behaviour. AIAA, 97-1850 Google Scholar
  8. Haack W (1941) Geschoßformen kleinsten Wellenwiderstandes. Ber Lilienthal-Gesellschaft 136(1):14–28 Google Scholar
  9. Hazra SB, Schulz V (2004) Simultaneous pseudo-timestepping for PDE-model based optimization problems. BIT Numer Math 44(3):457–472 MATHCrossRefMathSciNetGoogle Scholar
  10. Hazra SB, Schulz V (2005) How to profit from adjoints in one-shot pseudotime-stepping optimization. In: Evolutionary and deterministic methods for design, optimization and control with applications to industrial problems, EUROGEN Google Scholar
  11. Hazra SB, Schulz V (2006) Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with state constraints. SIAM J Sci Comput 28(3):1078–1099 MATHCrossRefMathSciNetGoogle Scholar
  12. Hicks RM, Henne PA (1978) Wing design by numerical optimization. J Aircr 15(7):407–412 CrossRefGoogle Scholar
  13. Jameson A (1988) Aerodynamic design via control theory. J Sci Comput 3(3):233–260 MATHCrossRefGoogle Scholar
  14. Jameson A (1990) Automatic design of transonic airfoils to reduce the shock induced pressure drag. In: Proceedings of the 31st Israel annual conference on aviation and aeronautics, Tel Aviv, pp 5–17 Google Scholar
  15. Jameson A (1994) Optimum aerodynamic design via boundary control. In: AGARD-VKI lecture series, optimum design methods in aerodynamics. von Karman Institute for Fluid Dynamics, Rhode Saint Genese Google Scholar
  16. Jameson A, Schmidt W, Turkel E (1981) Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. In: AIAA 14th fluid and plasma dynamic conference, 23–25 June 1981, pp 1981–1259 Google Scholar
  17. Kroll N, Schwamborn D, Becker K, Rieger H, Thiele F (eds) (2009) MEGADESIGN and MegaOpt—German initiatives for aerodynamic simulation and optimization in aircraft design. Springer, Berlin MATHGoogle Scholar
  18. Nemec M, Zingg DW (2002) A Newton–Krylov algorithm for aerodynamic design using the Navier–Stokes equations. AIAA J 40(6):1146–1154 CrossRefGoogle Scholar
  19. Nemec M, Zingg DW, Pulliam TH (2003) Multipoint and multi-objective aerodynamic shape optimization. AIAA J 42(6):1057–1065 CrossRefGoogle Scholar
  20. Schmidt S (2010) Efficient large scale aerodynamic design based on shape calculus. PhD thesis, University of Trier, Germany Google Scholar
  21. Schmidt S, Schulz V (2009) Impulse response approximations of discrete shape Hessians with application in CFD. SIAM J Control Optim 48(4):2562–2580 MATHCrossRefMathSciNetGoogle Scholar
  22. Schmidt S, Schulz V (2010) Shape derivatives for general objective functions and the incompressible Navier-Stokes equations. Control Cybern 39(3):677–713 MathSciNetGoogle Scholar
  23. Sokolowski J, Zolésio J-P (1992) Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin MATHGoogle Scholar
  24. Ta’asan S (1991) One-shot methods for optimal control of distributed parameter systems I: Finite dimensional control. ICASE, 91-2 Google Scholar
  25. Ta’asan S, Kuruvila G, Salas MD (1991) Aerodynamic design and optimization in one shot. AIAA, 92-0025 Google Scholar
  26. Widhalm M, Ronzheimer A, Hepperle M (2007) Comparison between gradient-free and adjoint based aerodynamic optimization of a flying wing transport aircraft in the preliminary design. In: AIAA 25th applied aerodynamics conference, 25–28 June 2007. American Institute of Aeronautics and Astronautics, New York Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Stephan Schmidt
    • 1
  • Caslav Ilic
    • 2
  • Volker Schulz
    • 1
  • Nicolas R. Gauger
    • 3
  1. 1.Department of MathematicsUniversity of TrierTrierGermany
  2. 2.German Aerospace Center (DLR)BraunschweigGermany
  3. 3.Computational Mathematics GroupRWTH Aachen UniversityAachenGermany

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