Variable Fidelity Surrogate Assisted Optimization Using A Suite of Low Fidelity Solvers

ABSTRACT

Variable-fidelity optimization (VFO) has emerged as an attractive method of performing, both, high-speed and high-fidelity optimization. VFO uses computationally inexpensive low-fidelity models, complemented by a surrogate to account for the difference between the high-and low-fidelity models, to obtain the optimum of the function efficiently and accurately. To be effective, however, it is of prime importance that the low fidelity model be selected prudently. This paper outlines the requirements for selecting the low fidelity model and shows pitfalls in case the wrong model is chosen. It then presents an efficient VFO framework and demonstrates it by performing transonic airfoil drag optimization at constant lift, subject to thickness constraints, using several low fidelity solvers. The method is found to be efficient and capable of finding the optimum that closely agrees with the results of high-fidelity optimization alone.

Variable-fidelity optimization (VFO) has emerged as an attractive method of performing, both, high-speed and high-fidelity optimization. VFO uses computationally inexpensive low-fidelity models, complemented by a surrogate to account for the difference between the high-and low-fidelity models, to obtain the optimum of the function efficiently and accurately. To be effective, however, it is of prime importance that the low fidelity model be selected prudently. This paper outlines the requirements for selecting the low fidelity model and shows pitfalls in case the wrong model is chosen. It then presents an efficient VFO framework and demonstrates it by performing transonic airfoil drag optimization at constant lift, subject to thickness constraints, using several low fidelity solvers. The method is found to be efficient and capable of finding the optimum that closely agrees with the results of high-fidelity optimization alone.

Cite this paper

M. Kashif Zahir and Z. Gao, "Variable Fidelity Surrogate Assisted Optimization Using A Suite of Low Fidelity Solvers,"*Open Journal of Optimization*, Vol. 1 No. 1, 2012, pp. 8-14. doi: 10.4236/ojop.2012.11002.

M. Kashif Zahir and Z. Gao, "Variable Fidelity Surrogate Assisted Optimization Using A Suite of Low Fidelity Solvers,"

References

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[22] R. M. Hicks and P. A. Henne, “Wing Design by Numerical Optimization,” Journal of Aircraft, Vol. 15, No. 7, 1978, pp. 407-412. doi:10.2514/3.58379

[23] P. Castonguay and S. Nadarajah, “Effect of Shape Parameterization on Aerodynamic Shape Optimization,” 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 8-11 January 2007.

[24] Y. Jin, “A Comprehensive Survey of Fitness Approximation in Evolutionary Computation,” Soft Computing, Vol. 9, No. 1, 2005, pp. 3-12. doi:10.1007/s00500-003-0328-5

[1] N. M. Alexandrov, C. R. Gumbert, L. L. Green, P. A. Newman and R. M. Lewis, “Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models,” Journal of Aircraft, Vol. 38, No. 6, 2001, pp. 1093-1101. doi:10.2514/2.2877

[2] A. I. J. Forrester, N. W. Bressloff and A. J. Keane, “Optimization Using Surrogate Models and Partially Converged Computational Fluid Dynamics Simulations,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, Vol. 462, No. 2071, 2006, pp. 2177-2204.

[3] A. I. J. Forrester, A. Sóbester and A. J. Keane, “MultiFidelity Optimization via Surrogate Modelling,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, Vol. 463, No. 2088, 2007, pp. 32513269.

[4] S. Gano, J. Renaud and B. Sanders, “Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function,” AIAA Journal, Vol. 43, No. 11, 2005, pp. 2422-2433. doi:10.2514/1.12466

[5] A. J. Keane, “Wing Optimization Using Design of Experiment, Response Surface, and Data Fusion Methods,” Journal of Aircraft, Vol. 40, No. 4, 2003, pp. 741-750. doi:10.2514/2.3153

[6] A. Nelson, J. Alonso and T. Pulliam, “Multi-Fidelity Aerodynamic Optimization Using Treed Meta-Models,” 25th AIAA Applied Aerodynamics Conference, Miami, 25-28 June 2007.

[7] S. G. Lehner, L. B. Lurati, S. C. Smith, G. C. Bower, E. J. Cramer, W. A. Crossley, F. Engelson, I. Kroo, S. C. Smith and K. E. Willcox, “Advanced Multidisciplinary Optimization Techniques for Efficient Subsonic Aircraft Design,” 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, 4-7 January 2010.

[8] D. Huang, T. Allen, W. Notz and R. Miller, “Sequential Kriging Optimization Using Multiple-Fidelity Evaluations,” Structural and Multidisciplinary Optimization, Vol. 32, No. 5, 2006, pp. 369-382. doi:10.1007/s00158-005-0587-0

[9] S. J. Leary, A. Bhaskar and A. J. Keane, “A KnowledgeBased Approach to Response Surface Modelling in Multifidelity Optimization,” Journal of Global Optimization, Vol. 26, No. 3, 2003, pp. 297-319,. doi:10.1023/A:1023283917997

[10] T. W. Simpson, V. Toropov, V. Balabanov and F. A. C. Viana, “Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come or Not,” 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, 10-12 September 2008.

[11] A. J. Booker, J. E. Dennis, P. D. Frank, D. B. Serafini, V. Torczon and M. W. Trosset, “A Rigorous Framework for Optimization of Expensive Functions by Surrogates,” Structural and Multidisciplinary Optimization, Vol. 17, No. 1, 1999, pp. 1-13.

[12] M. Drela and H. Youngren, “Xfoil 6.94 User Guide,” 2001.

[13] W. Su, Z. Gao and Y. Zuo, “Application of RBF Neural Network Ensemble to Aerodynamic Optimization,” 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 7-10 January 2008.

[14] J. W. Slater, “RAE 2822 Transonic Airfoil: Study #4.” http://www.grc.nasa.gov/WWW/wind/valid/raetaf/raetaf04/raetaf04.html

[15] P. H. Cook, M. A. McDonald and M. C. P. Firmin, “Aerofoil Rae 2822: Pressure Distributions, and Boundary Layer and Wake Measurements,” Experimental Data Base for Computer Program Assessment, AGARD Report ar 138, 1979.

[16] A. Oyama, S. Obayashi, K. Nakahashi and T. Nakamura, “Aerodynamic Optimization of Transonic Wing Design Based on Evolutionary Algorithm,” 3rd International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, 10-12 May 2000.

[17] A. I. J. Forrester, A. Sóbester and A. J. Keane, “Engineering Design via Surrogate Modelling: A Practical Guide,” Wiley, New York, 2008. doi:10.1002/9780470770801

[18] I. M. Sobol, “Distribution of Points in a Cube and Approximate Evaluation of Integrals,” Zh. Vych. Mat. Mat. Fiz., Vol. 7, 1967, pp. 784-802.

[19] D. R. Jones, M. Schonlau and W. J. Welch, “Efficient Global Optimization of Expensive Black-Box Functions,” Journal of Global Optimization, Vol. 13, No. 4, 1998, pp. 455-492. doi:10.1023/A:1008306431147

[20] J. Sacks, W. J. Welch, T. J. Mitchell and H. P. Wynn, “Design and Analysis of Computer Experiments,” Statistical Science, Vol. 4, No. 4, 1989, pp. 409-423. doi:10.1214/ss/1177012413

[21] F. A. C. Viana, “Surrogates Toolbox User’s Guide,” 2010. http://fchegury.googlepages.com

[22] R. M. Hicks and P. A. Henne, “Wing Design by Numerical Optimization,” Journal of Aircraft, Vol. 15, No. 7, 1978, pp. 407-412. doi:10.2514/3.58379

[23] P. Castonguay and S. Nadarajah, “Effect of Shape Parameterization on Aerodynamic Shape Optimization,” 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 8-11 January 2007.

[24] Y. Jin, “A Comprehensive Survey of Fitness Approximation in Evolutionary Computation,” Soft Computing, Vol. 9, No. 1, 2005, pp. 3-12. doi:10.1007/s00500-003-0328-5