A benchmark study on the efficiency of unconstrained optimization algorithms in 2D-aerodynamic shape design

Optimization algorithms are used in various engineering applications to identify optimal shapes. We benchmark several unconstrained optimization algorithms (Nelder-Mead, Quasi-Newton, steepest descent) under variation of gradient estimation schemes (adjoint approach, finite differences). Flow fields are computed by solving the Reynolds-Averaged Navier-Stokes equations using the open source computational fluid dynamics code OpenFOAM. Design variables vary from N = 2 to N = 364. The efficiency of the optimization algorithms are benchmarked in terms of computation time, applicability and ease of use. Results for lift optimizations are presented for airfoils at a Reynolds number of 50,000. As a result, we find for a small number of design variables N_5 or less, the computational efficiency of all optimization algorithms to be similar, while the ease of use of the Nelder-Mead algorithm makes it a perfect choice. For intermediate and large number of design variables, gradient-based algorithms with gradient estimation through the solution of adjoint equations are unbeaten.