Aiaa 2003–0043 Airfoil Design Using a Genetic Algorithm and an Inverse Method

In this paper, optimal airfoil shapes are found through manipulation of the velocity distribution by a genetic algorithm. The airfoil geometries are generated by an inverse method from velocity distribution parameters, and a viscous-flow analysis code is used to determine proper fitness values for candidate airfoils based on preset performance criteria. The method is compared with the more traditional approach of direct geometry manipulation for a simple single-objective aerodynamic optimization problem for a symmetric airfoil. The inverse and direct approaches are compared using a simple genetic algorithm and a hybrid genetic algorithm, where the hybrid method is formed by combining a simple genetic algorithm and a specialized local search method. Finally, the method is used to design a cambered airfoil that outperforms the existing state-of-the-art. Results indicate that using the design variables defining the velocity distribution in the inverse method has great potential for increasing the efficiency of airfoil shape optimization using genetic algorithms. Introduction In the past, several researchers have developed optimization methods that directly adjust airfoil shapes by way of spline supports, orthogonal shape functions, linear combinations of known airfoils, or geometry perturbations of an airfoil that is known to be close to an optimum. These direct-design approaches have been used within a wide range of optimization algorithms including classical methods and genetic algorithms (GAs). In their GA-based optimization method, Holst and Pulliam used the PARSEC method to parameterize airfoil geometries using 10 control variables that represent typical geometry characteristics. Using this method, broad geometry constraints can be met by ∗Graduate Research Assistant, 306 Talbot Laboratory. Student Member AIAA. [email protected] †Associate Professor, 306 Talbot Laboratory. Senior Member AIAA. [email protected] Copyright (c) 2003 by B. Allen Gardner and Michael S. Selig. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. simply fixing a single control variable. By employing only 10 variables, however, the method searches a small design space compared with most other directdesign approaches. Thus, these 10 geometry controls are not sufficient to describe all possible airfoil shapes including unconventional designs. Fanjoy and Crossley developed a method to optimize airfoil shapes by using 21 design variables representing the control points of a B-spline. They found that this method could be used to represent nearly any arbitrary shape. The B-spline, however, produced airfoils with small surface “waves” between control points. Unfortunately, this “waviness” is reflected in the velocity distribution. Viccini and Quagliarella also investigated a GA-based airfoil optimization scheme using B-splines, and concluded that unlike a gradient-based approach increasing the number of control variables used in their GA did not impose a proportional computational cost. They also found that a classical, conjugate gradient method was able to solve their airfoil optimization problem with about 1/5 of number of flow-solver solutions required by their best GA so long as a suitable starting point was chosen. A problem with these direct-design approaches is that the relationship between geometry and performance is highly nonlinear. That is, a small change in the surface shape of an airfoil can cause ripples in the velocity distribution, greatly degrading performance. Because of this phenomenon, there has been interest in perturbing the geometry in such away as to produce reasonable velocity distributions with each perturbation. A way to bypass this is by using an inverse method to parameterize the airfoil. An inverse method allows the velocity distribution to be directly controlled rather than anticipated from geometry perturbations. Concerns about using the inverse method have been expressed in the past because of the risk of defining a velocity distribution that cannot be physically achievable. This situation, however, can be partially avoided by using iteration schemes in the inverse method. The inherent robustness of GAs has made them increasingly popular in engineering applications. GAs are capable of searching for optimal solutions within a