Orthonormal Polynomial Bases for Airfoil Design

Shape parameterization plays an important role in the process of Aerodynamic Shape Optimization (ASO). A good parameterization allows the optimizer to easily explore the design space at multiple scales, while not adversely affecting the convergence or accuracy of the solution. Over the last several decades, a wide variety of methods have been proposed and characterized for representing aerodynamic shapes, including various polynomial representations, classes of splines, and radial basis functions. Our research into the application of convex optimization techniques to aerospace design has motivated us to revisit and expand these techniques, particularly for those with convex representations. For such parameterizations, when coupled with convex objective functions and constraints, convergence is guaranteed to globally optimal solutions in polynomial time. This guarantee allows for comparison of the results obtained with different shape representations without potentially adverse complications related to selection of initial conditions, local optima, or noisy gradients. In the course of this work we make several contributions. First, we provide orthonormal extensions to several common polynomial bases. Second, we demonstrate efficient objective function representation for supersonic drag minimization problems when employing an integral form of such orthonormal bases. Finally, we develop an analytic solution to a sample benchmark problem and compare the solutions offered by a finite linear combination of design variables with basis functions, to the underlying continuous result.