Advertisement

Variational Approach to the Problem of the Minimum Induced Drag of Wings

  • Maria Teresa Panaro
  • Aldo Frediani
  • Franco Giannessi
  • Emanuele Rizzo
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)

Abstract

A closed form solution of the problem of the minimum induced drag of a finite span straight wing was given by Prandtl. In this chapter, a mathematical theory, based on a variational approach, is proposed in order to revise such a problem and provide one with a support for optimizing more complex wing configurations, which are becoming of interest for future aircraft. The first step of the theory consists in finding a class of functions (lift distributions) for which the induced drag functional is well defined. Then, in this class, the functional to be minimized is proved to be strictly convex; taking into account this result, it is proved that the global minimum solution exists and is unique. Subsequently, we introduce the image space analysis associated with a constrained extremum problem; this allows us to define the Lagrangian dual of the problem of the minimum induced drag and show how such a dual problem can supply a new approach to the design. After having obtained the Prandtl exact solution in the context of a variational formulation, a numerical algorithm, based on the Ritz method, is presented, and its convergence is proved.

Keywords

Dual Problem Variational Approach Ritz Method Isoperimetric Problem Global Minimum Point 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Boltyanski, H. Martini and V. Soltarr: Geometric Methods and Optimization Problems. Kluwer Academic Publishers, Dordrecht, 1999.Google Scholar
  2. 2.
    F. Giannessi: Constrained optimization and Image Space Analysis. Vol. 1: Separation of Sets and Optimality Conditions. Springer, New York, 2005.Google Scholar
  3. 3.
    F. Giannessi: On the theory of Lagrangian duality. Optimization Letters, Vol. 1, pp 9–20, Springer, New York, 2005.Google Scholar
  4. 4.
    Munk M.: The minimum induced drag in airfoils, NACA 121(1924).Google Scholar
  5. 5.
    Munk M.: Isoperimetrische Aufgaben aus der Theorie des Fluges, Inaugural Dissertation 1919, Gottinga (1919).Google Scholar
  6. 6.
    Prandtl L.: Induced Drag of Multiplanes, NACA TN 182 (1924).Google Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  • Maria Teresa Panaro
    • 1
  • Aldo Frediani
    • 2
  • Franco Giannessi
    • 3
  • Emanuele Rizzo
    • 4
  1. 1.Dipartimento di Matematica, “L. Tonelli,”Università di PisaPisaItaly
  2. 2.Dipartimento di Ingegneria Aerospaziale, “L. Lazzarino,”Università di PisaPisaItaly
  3. 3.Dipartimento di Matematica, “L. Tonelli,”Università di PisaPisaItaly
  4. 4.Dipartimento di Ingegneria Aerospaziale, “L. Lazzarino,”Università di PisaPisaItaly

Personalised recommendations