Rapid Aerodynamic Shape Optimization Under Parametric and Turbulence Model Uncertainty: A Stochastic Gradient Approach

Abstract

Aerodynamic optimization is ubiquitous in the design of most engineering systems interacting with fluids. A common approach is to optimize a performance function -subject to some constraints -defined by a choice of an aerodynamic model, e.g., turbulence RANS model, and at nominal operating conditions. Practical experience indicates that such a deterministic, i.e., single-point, approach may result in considerably sub-optimal designs when the adopted aerodynamic model does not lead to accurate flow 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 forward and/or adjoint solves, at each iteration of a gradient-based scheme. This, in turn, renders the design of complex aerodynamic systems computationally expensive, if not infeasible. To tackle this difficulty, we consider a variant of the stochastic gradient descent method where, in each optimization iteration, a stochastic approximation of the objective, constraints, and their gradients are generated. This is done via a small number of forward/adjoint solves corresponding to random selections of the uncertain parameters and aerodynamic model. The methodology is applied to the robust optimization of the standard NACA-0012 airfoil in a low-Mach-number turbulent flow regime and subject to parametric and turbulence model uncertainty. With a cost that is a small factor larger than that of the deterministic approach, the stochastic gradient approach significantly improves the performance (mean and variance) of the aerodynamic design for a wide range of operating conditions and turbulence models.