Robust Airfoil Optimization Under Inherent and Model-Form Uncertainties Using Stochastic Expansions

Abstract

The objective of this paper was to introduce a computationally efficient approach for robust aerodynamic optimization under aleatory (inherent) and epistemic (model-form) uncertainties using stochastic expansions that are based on Non-Intrusive Polynomial Chaos method. The stochastic surfaces were used as surrogates in the optimization process. To create the surrogates, a combined non-intrusive polynomial chaos expansion approach was utilized, which is a function of both the design and the uncertain variables. In this paper, two stochastic optimization formulations were given: (1) optimization under pure aleatory uncertainty and (2) optimization under mixed (aleatory and epistemic) uncertainty. The formulations were demonstrated for the drag minimization of NACA 4-digit airfoils described with three geometric design variables over the range of uncertainties at transonic flow conditions. The deterministic CFD simulations were performed to solve steady, 2-D, compressible, turbulent RANS equations. The pure aleatory uncertainty case included the Mach number as the uncertain variable. For the mixed uncertainty case, a k factor which is multiplied with the turbulent eddy-viscosity coefficient is introduced to the problem as the epistemic uncertain variable. The results of both optimization cases confirmed the effectiveness of the robust optimization approach with stochastic expansions by giving the optimum airfoil shape that has the minimum drag over the range of aleatory and epistemic uncertainties. The optimization under pure aleatory uncertainty case required 90 deterministic CFD evaluations, whereas the optimization under mixed uncertainty case required 126 CFD evaluations to create the stochastic response surfaces, which show the computational efficiency of the proposed stochastic optimization approach. The stochastic optimization methodology described in this paper is general in the sense that it can be applied to aerodynamic optimization problems that utilize different shape parameterization techniques.

Multi-fidelity robust aerodynamic design optimization under mixed uncertainty

“…The aerodynamic response (e.g., the drag coefficient) should be in the form of the combination of probability distribution due to the effect of aleatory input uncertainty and interval distribution which indicate the effect of epistemic uncertainty. This paper attempts to further reduce the computational cost of the robust design procedure introduced in Zhang et al [4] and builds upon the recent study by the authors [5], which focused on robust optimization under inherent uncertainties only. The proposed approach is based on replacing the computationally expensive High-Fidelity (HF) CFD model by its inexpensive representation referred to as the Corrected Low-Fidelity (CLF) model.…”

Multifidelity Physics-Based Method for Robust Optimization Applied to a Hovering Rotor Airfoil

“…Other research focused more on epistemic uncertainties, which represent a lack of knowledge associated with the modeling process, that are reducible through the introduction of additional information [22]. Some works, like the one presented in [23], combine aleatory uncertainty on the freestream Mach number with an epistemic uncertain variable, i.e., the kinematic eddy viscosity of the Spalart-Allmaras model [24].…”

Surrogate-based robust design optimization of airfoil using inexpensive Monte Carlo method

“…CFD model in aerodynamics) and investigate the probabilistic behavior of the output model. Different methods are available to propagate aleatory uncertainty into CFD model e.g., Monte Carlo simulation (MCS) [10][11][12][13][14], Method of moment (Taylor series expansion) [15,16] and NonIntrusive Polynomial chaos [17][18][19]. Among above methods, most straightforward approach is Monte Carlo simulation (MCS) but it requires large number of performance evaluations (e.g., CFD, FEA) for obtaining accurate results.…”