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

Maria Teresa Panaro (), Aldo Frediani (), Franco Giannessi () and Emanuele Rizzo ()
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Maria Teresa Panaro: Università di Pisa
Aldo Frediani: Università di Pisa
Franco Giannessi: Università di Pisa
Emanuele Rizzo: Università di Pisa

Chapter Chapter 17 in Variational Analysis and Aerospace Engineering, 2009, pp 313-342 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/978-0-387-95857-6_17

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