Abstract
Response surface methods for the approximation of outputs of computer experiments such as the Kriging method often suffer from a lack of accuracy or efficiency. Many computationally expensive samples are needed for the globally correct reproduction of an unknown response. We investigate adaptive sampling strategies, which can automatically identify critical regions of an input-parameter domain and require less samples than traditional one-stage approaches like Latin hypercube designs. Furthermore, we propose a new method which makes use of the assumption that the aerodynamic responses are not of arbitrary structure, but rather related to other instances of a mutual problem class. Both approaches are validated with numerical test cases, showing that they produce more accurate surrogate models using less samples than traditional approaches.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bui-Thanh, T., Damodaran, M., Willcox, K.: Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA Journal 42(8), 1505–1516 (2004)
Busby, D., Farmer, C.L., Iske, A.: Hierarchical nonlinear approximation for experimental design and statistical data fitting. SIAM Journal on Scientific Computing 29(1), 49–69 (2007)
Chang, K.J., Haftka, R.T., Giles, G.L., Kao, P.-J.: Sensitivity-based scaling for approximating structural response. Journal of Aircraft 30, 283–288 (1993)
Chung, H.-S., Alonso, J.J.: Using gradients to construct cokriging approximation models for high-dimensional design optimization problems. In: 40th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2002–2317 (2002)
Crombecq, K., De Tommasi, L., Gorissen, D., Dhaene, T.: A novel sequential design strategy for global surrogate modeling. In: Proceedings of the 2009 Winter Simulation Conference (WSC), pp. 731–742 (2009)
Davies, R.H., Twining, C.J., Taylor, C.J.: Statistical Models of Shape—Optimisation and Evaluation. Springer (2008)
Dwight, R., Han, Z.-H.: Efficient uncertainty quantification using gradient-enhanced Kriging. In: Proceedings of 11th AIAA Conference on Non-Deterministic Approaches, Palm Springs CA. AIAA-2009-2276. AIAA (2009)
Everson, R., Sirovich, L.: Karhunen–Loève procedure for gappy data. Journal of the Optical Society of America A 12(8), 1657–1664 (1995)
Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Progress in Aerospace Sciences 45(1-3), 50–79 (2009)
Forrester, A.I.J., Sóbester, A., Keane, A.J.: Multi-fidelity optimization via surrogate modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 463(2088), 3251–3269 (2007)
Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling—A Practical Guide. Wiley (2008)
Gramacy, R.B., Lee, H.K.H.: Adaptive design and analysis of supercomputer experiments. Technometrics 51(2), 130–145 (2009)
Han, Z.-H., Görtz, S.: Hierarchical kriging model for variable-fidelity surrogate modeling. AIAA Journal 50(9), 1885–1896 (2012)
Han, Z.-H., Görtz, S., Hain, R.: A variable-fidelity modeling method for aero-loads prediction. In: Dillmann, A., Heller, G., Klaas, M., Kreplin, H.-P., Nitsche, W., Schröder, W. (eds.) New Results in Numerical and Experimental Fluid Mechanics VII. NNFM, vol. 112, pp. 17–25. Springer, Heidelberg (2010)
Han, Z.-H., Zimmermann, R., Görtz, S.: A new cokriging method for variable-fidelity surrogate modeling of aerodynamic data. In: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, pp. 4–7 (January 2010)
Jin, R., Chen, W., Sudjianto, A.: On sequential sampling for global metamodeling in engineering design. In: Proceedings of DETC 2002 ASME 2002 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Montreal, Canada (2002)
Koehler, J., Owen, A.: Computer experiments. In: Ghosh, S., Rao, C. (eds.) Handbook of Statistics, 13: Design and Analysis of Experiments, pp. 261–308. North-Holland (1996)
Krige, D.G.: A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa 52(6), 119–139 (1951)
Kroll, N., Fassbender, J. (eds.): MEGAFLOW—Numerical Flow Simulation for Aircraft Design. Springer (2005)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis 40(2), 492–515 (2003)
Lam, X., Kim, Y., Hoang, A., Park, C.: Coupled aerostructural design optimization using the Kriging model and integrated multiobjective optimization algorithm. Journal of Optimization Theory and Applications 142, 533–556 (2009)
Laurenceau, J., Meaux, M.: Comparison of gradient and response surface based optimization frameworks using adjoint method. AIAA Paper, 2008-1889 (2008)
Laurenceau, J., Sagaut, P.: Building efficient response surfaces of aerodynamic functions with kriging and cokriging. AIAA Journal 46(2), 498–507 (2008)
Leifsson, L., Koziel, S.: Variable-fidelity aerodynamic shape optimization. In: Koziel, S., Yang, X.-S. (eds.) Computational Optimization, Methods and Algorithms. SCI, vol. 356, pp. 179–210. Springer, Heidelberg (2011)
Liu, W.: Development of Gradient-Enhanced Kriging Approximations for Multidisciplinary Design Optimization. PhD thesis, University of Notre Dame (2003)
Lockwood, B., Anitescu, M.: Gradient-enhanced universal kriging for uncertainty propagation. Nuclear Science and Engineering 170(2), 168–195 (2012)
Lophaven, S.N., Nielsen, H.B., Søndergaard, J.: DACE—A MATLAB kriging toolbox. Technical Report IMM-REP-2002-12, Technical University of Denmark, Copenhagen (2002)
Martin, J., Simpson, T.: Use of adaptive metamodeling for design optimization. In: AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. AIAA (September 2002)
Matheron, G.: Principles of geostatistics. Economic geology 58(8), 1246–1266 (1963)
Morris, M.D., Mitchell, T.J., Ylvisaker, D.: Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction. Technometrics 35(3), 243–255 (1993)
Robinson, T., Eldred, M., Willcox, K., Haimes, R.: Strategies for multifidelity optimization with variable dimensional hierarchical models. In: Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2nd AIAA Multidisciplinary Design Optimization Specialist Conference), Newport, RI (2006)
Rosenbaum, B., Schulz, V.: Comparing sampling strategies for aerodynamic kriging surrogate models. ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik (2012), doi:10.1002/zamm.201100112
Rosenbaum, B., Schulz, V.: Efficient response surface methods based on generic surrogate models. SIAM Journal on Scientific Computing (2012) arXiv:1206.4172 (submitted to)
Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Statistical Science 4(4), 409–423 (1989)
Santner, T.J., Williams, B.J., Notz, W.: The Design and Analysis of Computer Experiments. Springer (2003)
Schwamborn, D., Gerhold, T., Heinrich, R.: The DLR TAU-code: Recent applications in research and industry. In: European conference on computational fluid dynamics, ECCOMAS CFD (2006)
Shan, S., Wang, G.: Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Structural and Multidisciplinary Optimization 41, 219–241 (2010)
Tang, C., Gee, K., Lawrence, S.: Generation of aerodynamic data using a design of experiment and data fusion approach. In: 43rd AIAA Aerospace Sciences meeting, Reno, Nevada (2005)
Viana, F.A.C., Venter, G., Balabanov, V.: An algorithm for fast optimal Latin hypercube design of experiments. International Journal for Numerical Methods in Engineering 82(2), 135–156 (2010)
Volkwein, S.: Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 81(2), 83–97 (2001)
Wackernagel, H.: Multivariate Geostatistics: An Introduction with Applications. Springer (2003)
Yamazaki, W., Rumpfkeil, M., Mavriplis, D.: Design optimization utilizing gradient/hessian enhanced surrogate model. AIAA Paper, 2010-4363 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rosenbaum, B., Schulz, V. (2013). Response Surface Methods for Efficient Aerodynamic Surrogate Models. In: Kroll, N., Radespiel, R., Burg, J., Sørensen, K. (eds) Computational Flight Testing. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38877-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-38877-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38876-7
Online ISBN: 978-3-642-38877-4
eBook Packages: EngineeringEngineering (R0)