An Example of a Pitfall in Aerodynamic Shape Optimization

The Applied Aerodynamics technical committee of the American Institute of Aeronautics and Astronautics (AIAA) launched an optimization discussion group, ADO-DG, in 2013. One of the four benchmark test cases is based on the NACA0012 airfoil and solutions of the Euler equations with prescribed objective function and geometric constaints. Volunteer participants were invited by ONERA to submit their optimized shapes for uniform pressure drag assessment through grid convergence. The assessment required three phases. Initial investigations with a simple protocol proved insufficient and raised some questions. This led to further investigations which showed that for most optimized airfoils there exists a lower drag branch and a higher drag branch in the drag (Mach number) diagram. Hysteresis was observed when performing a downward Mach number sweep and an upward Mach number sweep. For several airfoils, the design Mach number falls within the overlap range of the two branches, which indicates that two solutions may exist for the same airfoil. Two kinds of supersonic flow structure can be distinguished. One kind produces a lower drag branch, the other a higher drag branch. The jump from one branch to the other involves non symmetrical converged solutions. These further investigations required a final assessment, where a protocol was instituted to ensure that the lower drag branch was captured by taking advantage of the hysteresis. However, nonuniqueness may confound an optimization algorithm, as the same set of design variables can give two different objective function values.