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

Abstract

Optimization algorithms are used in various engineering applications to identify optimal shapes. In this work, we benchmark several unconstrained optimization algorithms (Nelder–Mead, Quasi-Newton, steepest descent) under variation of gradient estimation schemes (adjoint equations, 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: (a) computation time, and (b) applicability and ease of use. Results for lift optimizations are presented for airfoils at a Reynolds number of $ \text{ Re}=50,000 $. As a result, we find for a small number of design variables $ N \approx 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 a low number of design variables. For intermediate and large number of design variables, gradient-based algorithms with gradient estimation through the solution of adjoint equations are unbeaten.

Keywords

• airfoil optimization
• airfoil parametrization
• computational fluid dynamics
• openfoam
• gradient-based
• gradient-free
• adjoint approach
• optimization
• automatic design