Efficient Quantification of Aerodynamic Uncertainties Using Gradient-Employing Surrogate Methods

Dishi Liu

Abstract. Uncertainty quantification (UQ) in aerodynamic simulations is hindered by the high computational cost of CFD models. With gradient information obtained efficiently by using an adjoint solver, gradient-employing surrogate methods are promising in speeding up the UQ process. To investigate the efficiency of UQ meth­ods we apply gradient-enhanced radial basis functions, gradient-enhanced point – collocation polynomial chaos, gradient-enhanced Kriging and quasi-Monte Carlo (QMC) quadrature to a test case where the geometry of an RAE2822 airfoil is per­turbed by a Gaussian random field parameterized by 10 independent variables. The four methods are compared in their efficiency in estimating some statistics and the probability distribution of the uncertain lift and drag coefficients. The results show that with the same computational effort the gradient-employing surrogate methods achieve better accuracy than the QMC does.

1 Introduction

In aerodynamic simulations it is beneficial to consider uncertainties in the inputs, the formulation and the numerical error of the CFD model. In this work our concern is confined to the uncertainties in the model’s input and probabilistic approaches for uncertainty quantification (UQ) for CFD models. The uncertainties in the input propagates to the system response quantities (SRQ) through the model. Minor un­certainties can have an amplified impact in some instances and lead to occurrences of rare catastrophic events. Quantifying the uncertainties associated with the SRQ enhances the reliability of the simulations and enables robust design optimization. Most often this UQ process is done in a probabilistic framework in which the input

Dishi Liu

German Aerospace Center (DLR), Institute fur Aerodynamik und Stromungstechnik, Lilienthalplatz 7, 38108 Braunschweig, Germany e-mail: dishi. liu@dlr. de

B. Eisfeld et al. (Eds.): Management & Minimisation of Uncert. & Errors, NNFM 122, pp. 283-296. DOI: 10.1007/978-3-642-36185-2_12 © Springer-Verlag Berlin Heidelberg 2013

uncertainties are represented by random variables, and the consequent uncertainties in the SRQ are quantified by determining its probability distribution or statistical moments.

However, uncertainties in the input, especially those spatially or temporally dis­tributed, like geometric uncertainties, often generate a large number of variables. The “curse of dimensionality” prohibits the use of tensor-product quadratures. In [16] and [21] sparse grid quadratures were employed in aerodynamic UQ problems due to uncertain airfoil geometry. Nevertheless, if the number of variables is lar­ger than 10 even sparse grid methods suffer limitations in applicability [17]. The high computational cost of CFD models also makes the traditional sampling meth­ods such as Monte-Carlo and its variance-reduced variants (e. g. Latin Hypercube method) not efficient due to their slower error convergence rate.

Surrogate methods are gaining more attention in UQ as they provide approxim­ations of the CFD model which are much cheaper to evaluate while maintaining a reasonable accuracy so that the UQ can be performed on the basis of a large number of samples evaluated on the surrogate model. E. g. [12] shows a Kriging surrogate method better than plain Monte Carlo and Latin Hypercube methods in estimating the mean value of a bivariate Rosenbrock function. A comparative study of surrog­ate methods that are not employing gradients [23] shows Kriging is more accurate than radial basis functions and multivariate polynomial in approximating some 10- variate test functions.

Gradient-employing also give an edge to surrogate methods if the gradients are obtained at a relatively lower cost than that of the SRQ, which is the case when an adjoint CFD solver [5] is used and the number of SRQ is less than the number of variables. It should be noted that the gradient information cannot be effectively utilized by the UQ methods based on direct sampling of the CFD model. A naive augmentation of samples by finite difference brings no benefit because the augment­ing samples are not statistically independent of the original ones.

Different sampling schemes are adopted by surrogate methods, majorly of two groups: “on-grid” sampling and scattered sampling. The former is used in some methods based on polynomial approximations, e. g. stochastic collocation methods

[2] , and affected by the “curse of dimensionality” if the number of variables is large. The latter is more robust since it admits an arbitrary number of samples and arbitrary sample sites. This flexibility not only makes it tolerate sample failures (due to, e. g. poor convergence, as often observed in CFD models), but also makes an incorpor­ation of pre-existing or additional samples possible and enables run-time adaptive sampling.

In this work we apply three gradient-employing surrogate methods, i. e. gradient – enhanced radial basis functions (GERBF), gradient-enhanced point-collocation polynomial chaos (GEPC) and gradient-enhanced Kriging (GEK) [13], and for the purpose of comparison, also the quasi-Monte Carlo quadrature, to a UQ test case where an RAE2822 airfoil is subject to random geometric perturbations, and we compare their efficiency in estimating some statistics and probability distribution of the resulting uncertain lift and drag coefficients. The number of CFD model evaluations is kept small (< 200) in this numerical comparison to make it relev­ant to large-scale industrial applications.