Optimum aerodynamic shape design under uncertainty by utility theory and metamodeling - Du, Xiaosong; Leifsson, Leifur (Aerospace science and technology, 2019)
In this work, utility theory is introduced to optimum aerodynamic shape design under uncertainty (also called robust aerodynamic design). Specifically, utility theory is used to formulate the objective function of the optimization problem. The advantage of the proposed approach over the commonly used weighted sum method is that it does not require the use of weighting factors or the addition of constraints on the statistical moments. The polynomial chaos expansion metamodel with the least-angle regression and the hyperbolic truncation scheme are used to accelerate the uncertainty propagation and the optimization process. The proposed approach is demonstrated on the optimum airfoil shape design under uncertainty at steady transonic conditions. Two design cases are considered. In both cases the drag coefficient is to be minimized subject to constraints on the lift coefficient and the airfoil thickness at two chordwise locations. The first case treats the Mach number as uncertainty parameter and the second case treats the Mach number and the lift coefficient at target as uncertain. The proposed method is compared with deterministic single-point optimization and multi-point optimization, and the standard robust design formulations. Results show that the proposed approach is capable of obtaining airfoil design of the lowest mean drag coefficient with the smallest standard deviation as compared to the other approaches. In the first case, the proposed approach yields a design with a mean drag coefficient comparable to the standard robust design (around 97.6 drag counts, cts), but with a significantly lower variance (0.6 cts versus 2.5 cts). In the second case, the proposed approach yielded a design with a higher mean drag coefficient than the standard approach (98.4 cts versus 97.6 cts), but with a lower standard deviation (4.1 cts versus 4.8 cts). In both cases, the deterministic approaches yield designs with comparable or lower drag coefficients but with significantly larger standard deviations.