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

In: Variational Analysis and Aerospace Engineering

Author

Listed:
  • Maria Teresa Panaro

    (Università di Pisa)

  • Aldo Frediani

    (Università di Pisa)

  • Franco Giannessi

    (Università di Pisa)

  • Emanuele Rizzo

    (Università di Pisa)

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.

Suggested Citation

  • Maria Teresa Panaro & Aldo Frediani & Franco Giannessi & Emanuele Rizzo, 2009. "Variational Approach to the Problem of the Minimum Induced Drag of Wings," Springer Optimization and Its Applications, in: Variational Analysis and Aerospace Engineering, chapter 0, pages 313-342, Springer.
    • Handle: RePEc:spr:spochp:978-0-387-95857-6_17
    DOI: 10.1007/978-0-387-95857-6_17
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