by Dishi Liu, Daigo Maruyama and Stefan Görtz (German Aerospace Center)
Within the framework of the project “Uncertainty Management for Robust Industrial Design in Aeronautics” (UMRIDA), funded by the European Union, several machine learning-based predictive models were compared in terms of their efficiency in estimating statistics of aerodynamic performance of aerofoils. The results show that the models based on both samples and gradients achieve better accuracy than those based solely on samples at the same computational costs.
The UMRIDA project (Uncertainty Management for Robust Industrial Design in Aeronautics) is a collaborative project funded by the European Union that aims at a technology readiness level of robust design under a large number of simultaneous uncertainties. This work on machine learning-based predictive models was done in the UMRIDA project for a more efficient quantification of uncertainties in aerodynamic performance, by the Institute of Aerodynamics and Flow Technology in German Aerospace Center at Braunschweig, Germany.
In the context of uncertainty quantification and robust design, the typical quantities of interest are statistics of drag coefficient (Cd) and lift coefficient (Cl) of aerofoils. Uncertain input parameters are operational parameters such as the angle of attack, the Mach number, the Reynolds number, and an inherently large number of geometric parameters.
We made two comparisons of various methods in their efficiency of quantifying aerodynamic performance uncertainties caused by operational and/or geometric uncertainties. First, we compare two surrogate integration methods based on machine learning processes, Gaussian process model (Kriging) and gradient-enhanced Kriging (GEK), with a direct integration method based on quasi-Monte Carlo (QMC) quadrature on a viscous test case. Second, we compare the QMC quadrature and four machine learning-based integration methods, namely GEK, polynomial chaos combined with a sparse Gauss-Hermite quadrature, gradient-enhanced radial basis functions (GERBF) and a gradient-enhanced polynomial chaos collocating method on an inviscid test case.
Viscous test case: Comparison of Kriging and GEK
This comparison is based on a CFD model of the viscous flow around the RAE2822 aerofoil. We opt for DLR’s unstructured RANS solver TAU , the Spalart-Allmaras turbulence model, and a central flux discretisation scheme. The domain is discretised by a hybrid unstructured grid in which the aerofoil has 380 surface nodes. The uncertainties come from a random Mach number and angle of attack, together with a random perturbation to the original aerofoil geometry at every surface grid point. The two operational variables are assumed to be beta-distributed around M=0.729 and α=2.31°, respectively. The perturbations in Mach number and angle of attack are with a support within ±2% of the nominal values. The geometry perturbation is modelled by a random field parameterised into 24 independent Gaussian variables through a truncated KLE .
Quasi-Monte Carlo quadrature and two machine learning-based UQ methods, Kriging and gradient-enhanced Kriging, are applied to the test case and their efficiency is compared in estimating two statistics (mean, standard deviation) of the coefficient of lift (Cl). The accuracy of the estimates is judged by comparing to reference statistics which are based on 10,000 QMC samples. The computational cost is measured in terms of “compensated evaluation number” Nc to take the cost of gradient evaluation into account.
Figure 1 shows the error convergence in the two statistics of Cl by using various methods. GEK is seen to converge faster than Kriging in estimating both statistics. This can obviously be attributed to the greater amount of information utilised by the former at the same computational cost with the help of the adjoint TAU solver. It can also be observed that if the sample number is small Kriging may perform worse than QMC.
Figure 1: Convergence of estimates of Cl statistics to the reference statistics, reproduced from  with permission.
Inviscid test case: Comparison of four machine learning-based integration methods and direct integration
This comparison is based on a CFD model of the inviscid flow around the RAE2822 aerofoil at a Mach number of 0.73 and a 2.0° degree angle of attack. We use the TAU flow solver, opting for a central flux discretisation, scalar dissipation, and a backward Euler solver. The domain is discretised by a 193-by-33 structured grid in which the aerofoil has 128 surface nodes.
The source of uncertainty is a random perturbation to the original aerofoil geometry, which is modelled by a random field parameterised into nine independent Gaussian variables through a KLE . The five methods in the comparison are applied to the test case and compared in terms of their efficiency in estimating statistics of Cl and Cd, as well as the probability distribution functions (pdf). The reference values of these statistics are obtained from an integration of four million quasi-Monte Carlo (QMC) samples of the CFD model.
The results of this comparison are detailed in . It is observed that generally the gradient-employing machine learning-based methods perform better than direct integration methods. This can be ascribed to the fact that the former utilises more information at the same computational cost. This advantage comes from the cheaper cost of the gradients computed by an adjoint solver in the case that the number of response quantities of interest is smaller than the number of variables (in our case, two versus nine), and the advantage would increase for a larger number of variables or fewer response quantities. The results indicate that GEK and GERBF are the most efficient methods. Besides the cheaper gradients, this could be attributed to properties of the kernel functions they use and the effort of tuning model parameters.
 T. Gerhold, V. Hannemann, D. Schwamborn: “On the validation of the DLR-TAU code”, in New Results in Numerical and Experimental Fluid Mechanics, in: Notes on Numerical Fluid Mechanics, vol. 72, Vieweg, pp. 426–433, 1999.
 D. Liu, A. Litvinenko, C. Schillings, V. Schulz: “Quantification of airfoil geometry induced aerodynamic uncertainties - comparison of approaches”, SIAM-ASA journal on Uncertainty Quantification, vol. 5, no. 1, pp. 334–352, 2017.
 D. Maruyama, D. Liu, and S. Görtz: “Comparing surrogates for estimating aerodynamic uncertainties of airfoils”, in Uncertainty Management for Robust Industrial Design in Aeronautics. Springer, 2019, pp. 213–228.
Dishi Liu, TU Braunschweig, Germany