Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Maruyama, D., Liu, D., Görtz, S.: Comparing surrogates for estimating aerodynamic uncertainties of airfoils. In: Hirsch, C. et al. (eds.) Uncertainty Management for Robust Industrial Design in Aeronautics, Chap. 13, pp. xx–xx. (2017)
Galle, M., Gerhold, T., Evans, J.: Parallel computation of turbulent flows around complex geometries on hybrid grids with the DLR-TAU code. In: Ecer, A., Emerson, D.R. (eds.) In: Proceedings 11th Parallel CFD Conference, Williamsburg, VA, North-Holland, 23–26 May 1999
Gerhold, T., Hannemann, V., Schwamborn, D.: On the validation of the DLR-TAU code. In: Nitsche, W., Heinemann, H.J., Hilbig, R. (eds.) New Results in Numerical and Experimental Fluid Mechanics, Notes on Numerical Fluid Mechanics, vol. 72, pp. 426–433. Vieweg (1999). ISBN 3-528-03122-0
Schwamborn, D., Gerhold, T., Heinrich, R.: The DLR TAU-code: recent applications in research and industry, invited lecture. In: Wesseling, P., Oate, E., Priaux, J. (eds.) Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006), The Netherlands (2006)
Allmaras, S.R., Johnson, F.T., Spalart, P.R.: Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model. In: Seventh International Conference on Computational Fluid Dynamics (ICCFD7), ICCFD7-1902, Hawaii, July 2012
Sobol, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Zh. Vychisl. Mat. Mat. Fiz. 7(4), 784–802 (1967)
Joe, S., Kuo, F.Y.: Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 29, 49–57 (2003)
Joe, S., Kuo, F.Y.: Constructing Sobol sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30, 2635–2654 (2008)
Liu, D., Litvinenko, A., Schillings, C., Schulz, V.: Quantification of airfoil geometry-induced aerodynamic uncertainties—comparison of approaches. SIAM/ASA J. Uncertainty Quant. (2016)
Liu, D., Maruyama, D., Görtz, S.: Geometrical uncertainties—accuracy of parametrization and its influence on UQ and RDO. In: Hirsch, C. et al. (eds.) Uncertainty Management for Robust Industrial Design in Aeronautics, Chap. 51, pp. xx–xx. (2017)
Maruyama, D., Görtz, S., Liu, D.: Robust design measures for airfoil shape optimization. In: Hirsch, C. et al. (eds.) Uncertainty Management for Robust Industrial Design in Aeronautics, Chap. 32, pp. xx–xx. (2017)
Dwight, R., Han, Z.H.: Efficient uncertainty quantification using gradient-enhanced Kriging. AIAA Paper 2009–2276 (2009)
Shimoyama, K., Kawai, S., Alonso, J.J.: Dynamic adaptive sampling based on Kriging surrogate models for efficient uncertainty quantification. In: 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. AIAA paper 2013-1470 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Maruyama, D., Liu, D., Görtz, S. (2019). Surrogate Model-Based Approaches to UQ and Their Range of Applicability. In: Hirsch, C., Wunsch, D., Szumbarski, J., Łaniewski-Wołłk, Ł., Pons-Prats, J. (eds) Uncertainty Management for Robust Industrial Design in Aeronautics . Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-319-77767-2_43
Download citation
DOI: https://doi.org/10.1007/978-3-319-77767-2_43
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77766-5
Online ISBN: 978-3-319-77767-2
eBook Packages: EngineeringEngineering (R0)