Robust Optimization for Aerodynamic Problems Using Polynomial Chaos and Adjoints

Abstract

The central theme of this paper is to show how one can combine Polynomial Chaos Expansions (PCE) and adjoint theory to efficiently obtain sensitivities for robust optimal control and optimization. Two different formulations to obtain the sensitivities are presented: 1) Semi-intrusive: builds an adjoint equation for each PCE mode and and uses the combination of the gradient for each basis-adjoint pair to form the overall gradient, 2) Non-intrusive: solves an adjoint solution for each sample point to form a PCE for the gradient. The latter is used to obtain all the results in the paper. However, the two methods are compared to show that the latter involves an approximation which can lead to different solutions depending on the magnitude of the higher-order modes. The resulting gradient is then used in an iterative optimization procedure. Using an analytical problem to determine the trade-off between cost and accuracy of some PCE methods, the optimization algorithm is applied to an airfoil and turbine vane optimization problem. The robust optimal solutions for the airfoil optimization problem are compared against a multi-point design approach and shown to result in better designs in the constrained (not included here) and unconstrained case. In the turbine vane case, the mean loss is reduced by close to 4 tenths of a point.