Efficient Aerodynamic Shape Optimization

Since the present author first became involved in computational fluid dynamics, around 1970, the landscape has changed dramatically. At that time, panel methods had just come into use, and the world’s fastest super computer, the Control data 6600, had only 131000 words of memory (about 1 megabyte). Prior to the break-through of Murman and Cole [1970], no viable algorithms for computing transonic flow with shock waves had been discovered. By 1980 the standard for super-computing was represented by the Cray 1, which achieved a performance of about 100 megaflops, but at least initially it was hard to obtain a Cray with more than 128 megabytes of memory. At the present time numerous laptops are available with processing speeds of 2-3 gigaherz, and a gigabyte of memory, well beyond the power of the Cray XMP of the mid-eighties. In fact the speed of the Intel microprocessors has increased more than one thousand fold in 17 years, between the 80386 of 1986 and the current Pentium 4. These developments were unimaginable in 1970. There have been almost equally dramatic advances in algorithms, at least for some aerodynamic problems. Stemming in part from the pioneering work of Godunov, many effective shock capturing algorithms have been developed. Moreover, whereas the available methods for solving the steady state Euler equations in 1980 required 5000-10000 iterations to reach a reasonable level of convergence, and none would converge completely to machine zero, solutions of the Euler equations for flows around airfoils can now be obtained in 3-5 steps. These developments are reviewed by the author in an article for the Encyclopedia of Computational Mechanics. Some problems such as the prediction of transition and separation, or the formulation of universal turbulence model, remain recalcitrant. Nevertheless the combined advances in software and hardware have made it feasible to tackle problems of many orders of magnitude greater complexity than could contemplated 30 years ago. Even at the outset, intelligent use of computational fluid dynamics (CFD) could have an important impact on design, and the present author has always recognized that the real challenge was not just to predict the flow over a give shape, but to find a superior shape, optimal accordingly to some useful criteria. In fact the author’s first CFD program, Syn1 (July 1970) provided a complete solution to the inverse problem of designing an airfoil in ideal (irrotational and incompressible) flow which would produce a specified target pressure distribution. Stemming from discussions with Malcolm James at Douglas Aircraft, the method finds the conformal mapping which transforms a circle to the required airfoil. It is on extension of Lighthill’s method, which is described by Thwaites, as an incomplete solution because it requires the target velocity to be specified in the circle plane. The input to Syn1 is the target pressure as a function of the arc length s. Then since the potential along the profile is