Dishi Liu

Surrogate Model-Based Approaches to UQ and Their Range of Applicability

Efficient surrogate modeling approaches are presented in the context of robust design. The type of surrogate model, and the number and distribution of the sample points are discussed. The test case is the UMRIDA BC-02 airfoil with two uncertain operational and 10 uncertain geometrical parameters. Statistics of the quantity of interest (QoI) are evaluated based on surrogate models of the QoI. Here, the QoI is lift coefficient or drag coefficient. Both Kriging and gradient-enhanced Kriging (GEK) surrogate models are considered. The surrogate models are generated based on scattered samples of QoI. A Sobol sequence is used to generate samples with a low-discrepancy distribution, for which the QoI and its gradients with respect to the uncertain parameters are evaluated with a Computational Fluid Dynamics (CFD) solver and its adjoint counterpart. The mean and standard deviation of the QoI are efficiently evaluated by using GEK with more than 12 samples for large numbers of uncertainty parameters more than 10. The accuracy of the surrogate models is also investigated in terms of the derived robust design solutions. The error dispersion of the stochastic objective function due to the sample distribution affects the optimal solution. Thirty sample points are necessary to reduce the error dispersion to within one drag count, which is considered to be on the same order of magnitude as the epistemic uncertainty due to CFD errors.

Comparing Surrogates for Estimating Aerodynamic Uncertainties of Airfoils

Different surrogate models are compared in terms of their efficiency in estimating statistics of aerodynamic coefficients of the RAE2822 airfoil due to geometric input uncertainties. A comparison with direct integration and polynomial chaos methods is also performed. The aerodynamic coefficients and their partial gradients with respect to the uncertain input parameters are computed with a CFD solver and its adjoint counterpart. Reference statistics are computed in order to quantify the error of the different methods. The efficiency of the different methods is discussed in terms of the error in estimating a statistical quantity as a function of the number of CFD (including adjoint) computations used to construct the surrogate model. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods for the same computational cost. Sampling techniques are discussed in the context of estimating stochastic quantities used for risk management. While the mean and standard Deviation (used for mean-risk approach) can be efficiently computed by distributing the samples in the input parameter space with its probability density function, the maximum or minimum value (used for worst-case scenario) can be led more accurately by an expected improvement based adaptive sampling technique. This fact indicates that advanced sampling techniques are required for evaluating both the mean risk and worst-case risk at the same time.

AN EFFICIENT AERODYNAMIC SHAPE OPTIMIZATION FRAMEWORK FOR ROBUST DESIGN OF AIRFOILS USING SURROGATE MODELS

This paper deals with developing an efficient Robust Design Optimization (RDO) framework. The goal is to obtain an aerodynamic shape that is less sensitive to small random geometry perturbations and to uncertain operational conditions. The initial shape is the RAE2822 airfoil which is parameterized with 10 design variables. The robust design formulation used is based on an expectation measure. The goal was to minimize the sum of the mean and standard deviation of the drag coefficient of the RAE 2822 airfoil for a given nominal lift coefficient. Here, we focus on improving the methods used for computing the statistics of the aerodynamic performance of the airfoil in every optimization cycle. A relatively small number of samples is evaluated with CFD and used to construct surrogate models based on Kriging and gradient-enhanced Kriging. The aerodynamic performance statistics, which are used to evaluate the robust objective function, are estimated by using quasi Monte Carlo (QMC) sampling with many samples evaluated on the surrogate models. A large number of geometrical uncertainties is parameterized by using a truncated Karhunen-Loeve expansion, which enables a significant reduction of the dimensionality of the problem and thus of the surrogate models. By varying the number of samples used to build the surrogate model and by comparing the two types of surrogate modeling methods, it is confirmed that the robust objective function can be evaluated accurately with at most 30 CFD computations and corresponding adjoint computations.

Efficient Quantification of Aerodynamic Uncertainty due to Random Geometry Perturbations

The effort of quantifying the aerodynamic uncertainties caused by uncertainties in the airfoil geometry is hindered by the large number of the variables and the high computational cost of the CFD model. To identify efficient methods addressing this challenge, four promising methods, gradient-enhanced Kriging (GEK), gradient-assisted polynomial chaos (GAPC), maximum entropy method and quasi-Monte Carlo quadrature are applied to a test case where the geometry of an RAE2822 airfoil is perturbed by a Gaussian random field parameterized by nine independent variables. The four methods are compared in their efficiency of estimating some statistics and probability distribution of the uncertain lift and drag coefficients. The results show that the two surrogate method, GEK and GAPC, both utilizing gradient information obtained by an adjoint CFD solver, are more efficient in this situation. Their advantage is expected to increase as the number of variables increases.

Robust Design Measures for Airfoil Shape Optimization

Two kinds of robustness measures are introduced and applied to design optimization of the UMRIDA BC-02 transonic airfoil test case under uncertainty. Robust design optimization (RDO) aims at minimizing the mean and standard deviation of the drag coefficient. Reliability-based design optimization (RBDO) targets minimizing the maximum drag coefficient. Both robustness measures are efficiently evaluated by using efficient sampling techniques assisted by a gradient-enhanced Kriging model. The airfoil is parameterized with 10 deterministic design variables, which are optimized by a gradient-free Subplex algorithm. The nominal airfoil geometry is assumed to be perturbed by a Gaussian random field which is parameterized by 10 independent variables through a truncated Karhunen–Loeve expansion. Two operational parameters are also considered uncertain. The airfoil obtained by optimizing the two robustness measures has similar geometrical features and shows better performance in terms of the robustness measures than the initial and the deterministically designed airfoils. The strength and location of the shock wave of the robustly designed airfoils are shown to be less sensitive to random geometrical perturbations than the initial and deterministically designed airfoils.

General Introduction to Surrogate Model-Based Approaches to UQ

This chapter introduces two popular surrogate modeling methods which can be used to quantify uncertainties such as statistics of the aerodynamic coefficients from scattered data obtained by computational fluid dynamics (CFD) simulations. One is Kriging, which is able not only to interpolate predicted data but also to provide statistical information at unsampled locations in the parameter space based on Bayesian statistics. The other one is the radial basis function (RBF) method. The RBF method is also a powerful nonlinear interpolation method which exactly interpolates the samples, and its various radial basis function types support the interpolated values locally or globally when appropriately selected. Both methods can make use of gradient information, if available, to improve the model accuracy.

Geometrical Uncertainties—Accuracy of Parametrization and Its Influence on UQ and RDO Results

A large number of geometrical uncertainties of a transonic RAE2822 airfoil are parameterized by a truncated Karhunen–Loeve expansion (KLE), and the influence of the truncation on the statistics of aerodynamic quantities is investigated both in terms of efficiency and accuracy. Direct integration of a very large number of quasi-Monte Carlo samples computed with CFD is used to compute the mean and standard deviation for different levels of truncation, i.e., for different numbers of uncertain parameters. We show that a parameterization based on a well-truncated KLE can efficiently reduce the number of geometrical uncertainties while maintaining accuracy. Excessive truncation will not improve the efficiency of surrogate-based statistics integration and will inevitably lead to a loss of accuracy of the estimated statistics. This is attributed to the use of a gradient-enhanced surrogate model that employs an adjoint flow solver to compute the gradient of the aerodynamic coefficients with respect to the uncertain parameters. All partial gradients can be computed at the cost of one adjoint solution; i.e., the cost of computing all partial gradients is independent of the number of uncertain parameters. It is also shown that a loss of accuracy due to an improper truncation may influence the results of robust design optimization.