An Inverse Design Method for Wings Using Integral Equations and Its Recent Progress

  • Kisa Matsushima
  • Susumu Takanashi
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 65)


An inverse design method using integral equations is explicated. Two other recently developed design methods are introduced. They are the extension of the first method. The formulation of an inverse problem used in each design method is also discussed for three major design categories. Each method designs a wing section shape which realizes the prescribed target pressure distribution by iterating a residual-correction loop which consists of a flow simulation and an inverse problem. It starts with the initial guess of current wing shape. The residual, defined as the difference between the target and simulated current pressure distributions, is compensated by solving the inverse problem. The inverse problem determines the section geometry of the wing. Each of three inverse problems here are formulated to be an integral equation system by mathematically converting the partial differential equations which govern the flowfield. The first inverse problem is for transonic wing design. The second one is for supersonic and the third one is for design of multiple wing systems. Emphasis is put on the discussion of the formulation. Works on wing design using the method with the first inverse problem are cited. Design problems by the second and third ones are also presented.


Inverse Problem Geometrical Correction Wing Surface Inverse Design Wing Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Laburujere, Th. E. and Slooff, J. W.: Computational Methods for the Aerodynamic Design of Aircraft Components, Annu. Rev. Fluid Mech., 25 (1993), pp. 183–214.CrossRefGoogle Scholar
  2. [2]
    Dulikravich, G. S.: Shape Inverse Design and Optimization for Three-Dimensional Aerodynamics, AIAA-95–0695, 1995.Google Scholar
  3. [3]
    Takanashi, S.: Iterative Three-Dimensional Transonic Wing Design Using Integral Equations, J. Aircraft, Vol. 22, No. 8, pp. 655–660, 1985.CrossRefGoogle Scholar
  4. [4]
    Hirose, N., Takanashi, S. and Kawai, N.: Transonic Airfoil Design Based on Navier-Stokes Equation to Attain Arbitrarily Specified Pressure Distribution — an Iterative Procedure, AIAA-85–1592, 1985.Google Scholar
  5. [5]
    Xia, Z. X., Zhu, Z. Q. and Vu, L. Y.: A Computational Method for Inverse Design of Transonic Airfoil and Wing, AIAA-93–3482-CP, 1993.Google Scholar
  6. [6]
    Lorentzen, L.; Development of Inverse Airfoil Design for Transonic Applications, FFA TN 1993–37, 1993.Google Scholar
  7. [7]
    Hua, J., Yang, Q. Z., Xi, D. K., Zhang Z. Y., Pu, D. W., Zhang Z. L. and Wang L.: Design and Experimental Investigation of Transonic Natural Laminar Flow Wings, ICAS-94–4,7,3, 1994.Google Scholar
  8. [9]
    Obayashi, S. and Takanashi, S.: Genetic Optimization of Target Pressure Distributions for Inverse Design Methods, AIAA-95–1649-CP, 1995.Google Scholar
  9. [8]
    Bartelheimer, W.: An Improved Integral Equation Method for the Design of Transonic Airfoils and Wings, AIAA-95–1688-CP, 1995.Google Scholar
  10. [10]
    Fujii, K. and Takanashi, S.: Aerodynamic Aircraft Design Methods and Their Notable Applications, ICIDES-III, pp.31–45, 1991, and References Therein.Google Scholar
  11. [11]
    Heaslet, M. A. and Spreiter, J. R.: Three Dimensional Transonic Flow Theory Applied to Slender Wings and Bodies, NACA Report 1318, 1957.Google Scholar
  12. [12]
    Nørstrud, H.,: High Speed Flow Past Wings, NASA CR-2246, 1973.Google Scholar
  13. [13]
    Sinbo, Y., Yoshida, K., Iwamiya T., Takaki R., and Matsushima, K., Aerodynamic Design of Scaled Supersonic Experimental Airplane, NAL International CFD Workshop on SST Desgn, March, 1998.Google Scholar
  14. [14]
    Jeong, S., Matsushima, K., Iwamiya, T., Obayashi, S., and Nakahashi, K., Inverse Design Method for Wings of Supersonic Transport. AIAA-98–0602, Jan., 1998.Google Scholar
  15. [15]
    Matsushima, K., Iwamiya, T., Jeong, S. and Obayshi, S.: Aerodynamic Wing Design for NAL’s SST Using an Iterative Inverse Approach, NAL International CFD Workshop on SST Design, March, 1998.Google Scholar
  16. [16]
    Lomax, H., Heaslet, M. A. and Fuller, F. B.: Integrals and Integral equations in Linearized Wing Theory, NACA Rep. 1054, 1951.Google Scholar
  17. [17]
    Takaki, R., Iwamiya, T., Aoki, A.: CFD Analysis Applied to the Supersonic Research Airplane, NAL International CFD Workshop on SST Design, March, 1998.Google Scholar
  18. [18]
    Narramore, J. C., and Beaty, T. D.: An Inverse Method for Multielement High-Lift Systems, AIAA-75–879, 1975.Google Scholar
  19. [19]
    Shigemi, M.: A Solution of Inverse Problems for Multi-Element Aerofoils though Application of Panel Method, Trans. Japan Soc. Aero. Space Sciences, Vol. 28, No. 80, 1985.Google Scholar
  20. [20]
    Matsushima, K. and Takanashi, S.: An Inverse Design Method for Transonic Multiple Wing Systems on Integral Equations, AIAA-96–2465, June, 1996.Google Scholar
  21. [21]
    Matsushima, K., Takanashi, S. and Iwamiya, T.: An Inverse Design Method for Transonic Multiple Wing Systems using Integral Equations, J. Aircraft, Vol. 34, No. 3, 1997.Google Scholar
  22. [22]
    Fujii, K. and Obayashi, S.: High Resolution Upwind Scheme for Vortical Flow Simulations, J. Aircraft, vol. 26, No. 12, pp. 1123–1129, 1989.CrossRefGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1999

Authors and Affiliations

  • Kisa Matsushima
    • 1
  • Susumu Takanashi
    • 2
  1. 1.FUJITSU Ltd.Mihama-Ku, ChibaJapan
  2. 2.National Aerospace LaboratoryChofu City, TokyoJapan

Personalised recommendations