An efficient approach to optimal aerodynamic design. II - Implementation and evaluation

Abstract

An efficient method for optimal aerodynamic design of airfoils is being developed. It is based on the integration of three component elements: 1) analytical representation of the geometry, 2) analytical determination of aerodynamic sensitivities based on the Euler equations, and 3) a computational package for optimal design subject to arbitrary constraints. Analytical determination of aerodynamic sensitivities coupled with the analytical geometry representation leads to more than two orders of magnitude cost reduction with respect to the conventional cornput+ tional fluid dynamics (CFD) approach. Results are presented for a two-dimensional airfoil design to a specified pressure distribution or lift coefficient with design constraints. Comparisons are made with a conventional CFD Euler design method illustrating the efficiency improvement of the analytic method. -2 Introduction With the advent of high speed computers, design methods using CFD flow analysis tools coupled with optimization techniques have increasingly become the convention. The overall problem of effective optimal aerodynamic design requires an integrated procedure. First, a method for representing geometry to an arbitrary accuracy with minimaldegrees of f r e e dom (e.g., number of design variables) is needed. Second, the aerodynamic sensitivity ( i e . , change in surface pressure with respect to design variable changes) must be determined accurately. Finally, an optimization procedure is needed which determines the changes in the design variables which optimize a given payoff function (e.g., minimum drag at constant lift). Many ideas have been proposed for functionally representing airfoil geometry.'-' These often make use of a finite number of arbitrary shape functions to describe the geometry or perturbations to a baseline geometry. Because the number of shape functions is * Lead Engineer Aerodynamics, Member AlAA t M C D O ~ ~ I D O U ~ I ~ ~ Corporation F~IIOW, ASSOC. F ~ I I O W AIAA t Senior Tech. Specialist -Aerodynamics, Assoe. Fellow AIAA Copyright 01993 by McDonnell Douglas Corporation Published by the Amepican Imtitute of Aeronautics and Astronautics. Ine. with permission. finite and their selection is somewhat arbitrary, direct control of surface smoothness and accuracy is compromised; after-the-fact smoothing is often required. Another deficiency of this approach is the inability to represent all possible shapes (ie., lack of completeness) due to non-orthogonal construction. This limitation may prevent the design from achieving the 'true optimum shape. Optimiiation methods depend on accurate aerodynamic sensitivity derivatives to drive the geometry toward an optimum design. TypicalIy, the method used to calculate aerodynamic sensitivity derivatives is finite geometric pertGbations coupled with a CFD flow analysis ~ o d e . l ~ This procedure consists of computing a baseline flow solution followed by solutions for each (finitely) perturbed airfoil shape produced by small, arbitrary and finite perturbations of each design variable. The distribution of aerodynamic sensitivity over the airfoil surface corresponding to each design variable perturbation is calculated by finite differences. The main disadvantage of the diect CFD approach is the computational cost for each flow solution. The number of flow solutions required to calculate the sensitivities is proportional to the number of design variables and can become very large as the number of design variables increases. Fnrthermore, to obtain accurate derivatives each flow analysis must be well converged which increases computational time and cost. The choice of perturbation magnitude may also cause inaccuracies in the derivative calculation and, therefore, must be addressed. If the geometric perturbation is too large, an inaccurate sensitivity is obtained. If too small a perturbation is used, accuracy significance is lost due to CFD truncation error. There is also ambiguity in pairing surface points on the finitely perturbed shape with corresponding points on the baseline airfoil. Aerodynamic sensitivity derivatives may be obtained without finite perturbations by analyticallydifferentiating the system of nonlinear governing equations and solving the resulting system of equations This approach involves fewer calculations than the finite difference method, thereby reducing the cost. The system of equations, however, is very large requiring enormous computer storage for even two-dimensional problems.