Low-Cost Robust Airfoil Optimization by Variable-Fidelity Models and Stochastic Expansions

In this paper, we present a robust optimization algorithm for low computational cost air-foil design under aleatory uncertainty. Our approach exploits stochastic expansions derived from the Non-Intrusive Polynomial Chaos (NIPC) technique to create response surface approximation (RSA) models utilized in the optimization process. In this work, we employ a combined NIPC expansion approach, where both the design and the uncertain parameters are the input arguments of the RSA model. In order to reduce the computational complexity of the design process, the high-fidelity computational fluid dynamic (CFD) model is replaced by a suitably corrected low-fidelity one, the latter being evaluated using the same CFD solver but with a coarser mesh and relaxed convergence criteria. The model correction is realized at the response level using multi-point output space mapping (OSM).The OSM correction can be obtained without costly parameter extraction procedure and ensures that the low-fidelity model represents the high-fidelity one with sufficient accuracy. The proposed robust optimization algorithm is applied to the design of transonic airfoils with four deterministic design variables (the airfoil shape parameters and the angle of at-tack) and one aleatory uncertain variable (the Mach number). In terms of computational cost, the proposed surrogate-based technique outperforms the conventional approach that exclusively uses the high-fidelity model to create the RSA models: the design cost corresponds to only 12 equivalent high-fidelity model evaluations versus 42 for the conventional method