# Variables

The variables of our problem describe the aerodynamic shape of the aircraft. Over the last two decades, many engineers have tried to describe the shape of a wing accurately with only a few parameters while maintaining flexibility. There arc three general classes of aerodynamic shape functions:

• Linear combination of existing wing sections. This geometric approach to the airfoil design problem is a derivative of the original NACA method {330}. Although this method is compu­tationally efficient, at DA this method has often given only trivial solutions (c. g. the old air­foil) due to the low flexibility inherent to this geometrical representation. In addition, there is little physical foundation for the implicit assumption that the drag (objective! of a linear combination of airfoil geometries is linearly or at all related to the drag of the individual air­foils.

• Analytical shape functions arc linearly superimposed to define a wing geometry. Examples Of these include Hieks-Hcnnc functions. Wagner functions and the patched polynomials dis­cussed in reference 1344j. as well as the Legendre polynomials used by Reneaux (346J and

the splines used by Coscntino (336]. Unfortunately, the number of parameters required for these shape functions arc usually too high to allow multipoint three-dimensional design. Eve­ry lime, a minimum of 30-40 points arc required to define each airfoil. Even if only five air­foils are sufficient for defining a wing, about 200 parameters will be necessary. This number of variables is simply too great for practical optimization at present. Coscntino optimizes a 3D wing by allowing variations over only a small portion of the wing. Lee has proposed patched polynomials as a way of flcxiblcly modeling an airfoil section with only sixteen pa­rameters. Although this approach does produce a flexible geometry, the performance of the sections designed with this method cannot compete with those produced by experienced en­gineers using inverse methods.

• Special aero-functions were proposed by Rcncaux [346] for designing airfoils with a mini­mum number of parameters. This approach automates the steps that an expenenced designer follows when using an inverse method. ‘Good* pressure distribution types are defined, along with the shape or special aero-functions that produce these pressure distributions. Unfortu­nately. for a given pressure distribution type, these aero-functions vary with Mach number, and this method is therefore impractical for multipoint designs. This method has all the dis­advantages of inverse methods discussed in the Introduction.

At DA (352) the author has introduced another type of function, a highly nonlinear spe­cial aero-function, which can optimize the shape of a 3D wing with adequate flexibility using current computer technology. It is now being used for the industrial design of transonic and su­personic wings. Its application will be shown in the next section. These shape functions produce the aircraft geometry in any resolution.