Rapid aerodynamic shape optimization under uncertainty using a stochastic gradient approach

A common approach in aerodynamic design is to optimize a performance function-provided some constraints-defined by a choice of an aerodynamic model at nominal operating conditions. Practical experience indicates that such a deterministic approach may result in considerably sub-optimal designs when the adopted aerodynamic model does not lead to accurate predictions, or when the actual operating conditions differ from those considered in the design. One approach to address this shortcoming is to consider an average or robust design, wherein the statistical moments of the performance function, given the uncertainty in the operating conditions and the aerodynamic model, is optimized. However, when the number of uncertain inputs is large or the performance function exhibits significant variability, an accurate evaluation of these moments may require a large number of function evaluations at each optimization iteration, rendering the problem significantly expensive. To tackle this difficulty, we consider a variant of the stochastic gradient descent method where in each iteration, a stochastic approximation of the objective, constraints, and their gradients is generated. This is done via a small number of forward/adjoint solutions corresponding to random selections of the uncertainties. The methodology is applied to the robust optimization of the NACA-0012 airfoil subject to operating condition and turbulence model uncertainty. With a cost that is only a small factor larger than that of the deterministic methodology, the stochastic gradient approach significantly improves the performance of the aerodynamic design for a wide range of operating conditions and turbulence models.