Skip to main content
Skip to article control options
No AccessFluid Dynamics

Adaptive Hybrid Surrogate Modeling for Complex Systems

Published Online:

This paper explores the effectiveness of the recently developed surrogate modeling method, the Adaptive Hybrid Functions, when applied in conjunction with different sampling techniques and different sample sizes. The Adaptive Hybrid Functions is a hybrid surrogate modeling method that seeks to exploit the advantages of each component surrogate. To explore its effectiveness, the Adaptive Hybrid Functions is applied to model four disparate complex engineering systems, namely: 1) wind farm power generation, 2) product platform planning (for universal electric motors), 3) three-pane window heat transfer, and 4) onshore wind farm cost estimation. The effectiveness of the following three distinct sampling techniques is investigated: 1) Latin hypercube sampling, 2) Sobol’s quasi-random sequence, and 3) the Hammersley sequence sampling. Cross-validation is used to evaluate the accuracy of the resulting surrogate models. It was observed that the Sobol’s sequence and the Latin hypercube sampling techniques provided better accuracy in the case of high-dimensional problems, whereas the Hammersley sequence sampling technique performed better in the case of low-dimensional problems. It was further observed that a monotonic increase in modeling accuracy is not necessarily accomplished by simply increasing the sample size for different problems. Overall, the Adaptive Hybrid Functions provided acceptable to high accuracy in representing complex system behavior.


  • [1] Braha D., Minai A. and Bar-Yam Y., Complex Engineered Systems, Springer–Verlag, New York, 2006, pp. 1–21. CrossrefGoogle Scholar

  • [2] Queipo N., Haftka R., Shyy W., Goel T., Vaidyanathan R. and Tucker P., “Surrogate-Based Analysis and Optimization,” Progress in Aerospace Sciences, Vol. 41, No. 1, 2005, pp. 1–28. doi: PAESD6 0376-0421 CrossrefGoogle Scholar

  • [3] Wang G. and Shan S., “Review of Metamodeling Techniques in Support of Engineering Design Optimization,” Journal of Mechanical Design, Vol. 129, No. 4, 2007, pp. 370–380.doi: 1050-0472 CrossrefGoogle Scholar

  • [4] Jouhaud J. C., Sagaut P., Montagnac M. and Laurenceau J., “A Surrogate-Model Based Multidisciplinary Shape Optimization Method with Application to A 2D Subsonic Airfoil,” Journal of Computers and Fluids, Vol. 36, No. 3, 2007, pp. 520–529.doi: 0045-7930 CrossrefGoogle Scholar

  • [5] Neufeld D. and Chung K. B., “Aircraft Wing Box Optimization Considering Uncertainty in Surrogate Models,” Structural and Multidisciplinary Optimization, Vol. 42, No. 5, 2010, pp. 745–753.doi: 1615-1488 CrossrefGoogle Scholar

  • [6] Madsen J. I., Shyy W. and Haftka R. T., “Response Surface Techniques for Diffuser Shape Optimization,” AIAA Journal, Vol. 38, No. 9, 2000, pp. 1512–1518. doi: AIAJAH 0001-1452 LinkGoogle Scholar

  • [7] Myers R. and Montgomery D., Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 2nd ed., Wiley-Interscience, New York, 2002, pp. 13–72. Google Scholar

  • [8] Forrester A. and Keane A., “Recent Advances in Surrogate-Based Optimization,” Progress in Aerospace Sciences, Vol. 45, Nos. 1–3, 2009, pp. 50–79. doi: PAESD6 0376-0421 CrossrefGoogle Scholar

  • [9] Wilson B., Cappelleri D., Simpson T. and Frecker M., “Efficient Pareto Frontier Exploration Using Surrogate Approximations,” Optimization and Engineering, Vol. 2, No. 1, 2001, pp. 31–50. doi: OEPNBR 1389-4420 CrossrefGoogle Scholar

  • [10] Hardy R. L., “Multiquadric Equations of Topography and Other Irregular Surfaces,” Journal of Geophysical Research, Vol. 76, No. 8, 1971, pp. 1905–1915.doi: JGREA2 0148-0227 CrossrefGoogle Scholar

  • [11] Mullur A. and Messac A., “Extended Radial Basis Functions: More Flexible and Effective Metamodeling,” AIAA Journal, Vol. 43, No. 6, 2005, pp. 1306–1315. doi: AIAJAH 0001-1452 LinkGoogle Scholar

  • [12] Messac A. and Mullur A., “A Computationally Efficient Metamodeling Approach for Expensive Multiobjective Optimization,” Optimization and Engineering, Vol. 9, No. 1, 2008, pp. 37–67. doi: OEPNBR 1389-4420 CrossrefGoogle Scholar

  • [13] Simpson T., Peplinski J., Koch P. and Allen J., “Metamodels for Computer-Based Engineering Design: Survey and Recommendations,” Engineering with Computers, Vol. 17, No. 2, 2001, pp. 129–150. doi: ENGCE7 0177-0667 CrossrefGoogle Scholar

  • [14] Basudhar A. and Missoum S., “Adaptive Explicit Decision Functions for Probabilistic Design and Optimization Using Support Vector Machines,” Computers and Structures, Vol. 86, Nos. 19–20, 2008, pp. 1904–1917. doi: CMSTCJ 0045-7949 CrossrefGoogle Scholar

  • [15] Yun Y., Yoon M. and Nakayama H., “Multi-Objective Optimization Based on Meta-Modeling by Using Support Vector Regression,” Optimization and Engineering, Vol. 10, No. 2, 2009, pp. 167–181. doi: OEPNBR 1389-4420 CrossrefGoogle Scholar

  • [16] Zerpa L., Queipo N., Pintos S. and Salager J., “An Optimization Methodology of Alkaline-Urfactant-Polymer Flooding Processes Using Field Scale Numerical Simulation and Multiple Surrogates,” Journal of Petroleum Science and Engineering, Vol. 47, Nos. 3–4, 2005, pp. 197–208. doi: JPSEE6 0920-4105 CrossrefGoogle Scholar

  • [17] Goel T., Haftka R., Shyy W. and Queipo N., “Ensemble of Surrogates,” Structural and Multidisciplinary Optimization, Vol. 33, No. 3, 2007, pp. 199–216. doi: 1615-1488 CrossrefGoogle Scholar

  • [18] Zhang J., Chowdhury S. and Messac A., “An Adaptive Hybrid Surrogate Model,” Structural and Multidisciplinary Optimization, Vol. 46, No. 2, 2012, pp. 223–238. doi: 1615-1488 CrossrefGoogle Scholar

  • [19] Zhang J., Chowdhury S., Messac A., Zhang J. and Castillo L., “Surrogate Modeling of Complex Systems Using Adaptive Hybrid Functions,” Proceedings of the ASME 2011 International Design Engineering Technical Conferences (IDETC), Vol. 5: 37th Design Automation Conference, Parts A and B, Washington, DC, Aug. 2011, pp. 775–790. Google Scholar

  • [20] Sanchez E., Pintos S. and Queipo N., “Toward an Optimal Ensemble of Kernel-Based Approximations with Engineering Applications,” Structural and Multidisciplinary Optimization, Vol. 36, No. 3, 2008, pp. 247–261.doi: 1615-1488 CrossrefGoogle Scholar

  • [21] Acar E. and Rais-Rohani M., “Ensemble of Metamodels with Optimized Weight Factors,” Structural and Multidisciplinary Optimization, Vol. 37, No. 3, 2009, pp. 279–294.doi: 1615-1488 CrossrefGoogle Scholar

  • [22] Viana F., Haftka R. and Steffen V., “Multiple Surrogates: How Cross-Validation Errors Can Help Us to Obtain the Best Predictor,” Structural and Multidisciplinary Optimization, Vol. 39, No. 4, 2009, pp. 439–457.doi: 1615-1488 CrossrefGoogle Scholar

  • [23] Acar E., “Various Approaches for Constructing an Ensemble of Metamodels Using Local Measures,” Structural and Multidisciplinary Optimization, Vol. 42, No. 6, 2010, pp. 879–896.doi: 1615-1488 CrossrefGoogle Scholar

  • [24] Zhou X., Ma Y. and Li X., “Ensemble of Surrogates with Recursive Arithmetic Average,” Structural and Multidisciplinary Optimization, Vol. 44, No. 5, 2011, pp. 651–671.doi: 1615-1488 CrossrefGoogle Scholar

  • [25] Zhang J., Chowdhury S. and Messac A., “A New Robust Surrogate Model: Reliability Based Hybrid Functions,” 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2011. Google Scholar

  • [26] Cressie N., Statistics for Spatial Data, Wiley, New York, 1993, pp. 105–210. CrossrefGoogle Scholar

  • [27] Lophaven S., Nielsen H. and Sondergaard J., “DACE—A Matlab Kriging Toolbox, Version 2.0,” Technical Univ, of Denmark TR-IMM-REP-2002-12, 2002. Google Scholar

  • [28] Cherrie J. B., Beatson R. K. and Newsam G. N., “Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in Rn,” SIAM Journal on Scientific Computing, Vol. 23, No. 5, 2002, pp. 1549–1571.doi: 1095-7197 CrossrefGoogle Scholar

  • [29] Chowdhury S., Zhang J., Messac A. and Castillo L., “Unrestricted Wind Farm Layout Optimization (UWFLO): Investigating Key Factors Influencing the Maximum Power Generation,” Renewable Energy, Vol. 38, No. 1, 2012, pp. 16–30.doi: RNENE3 0960-1481 CrossrefGoogle Scholar

  • [30] GE Energy 1.5MW Wind Turbine Brochure, General Electric,, 2010. Google Scholar

  • [31] Simpson T., Siddique Z. and Jiao R., Product Platform and Product Family Design: Methods and Applications, Springer-Verlag, New York, 2006. CrossrefGoogle Scholar

  • [32] Chowdhury S., Messac A. and Khire R. A., “Comprehensive Product Platform Planning (CP3) Framework,” Journal of Mechanical Design, Vol. 133, No. 10, 2011, p. 101004.doi: 1050-0472 CrossrefGoogle Scholar

  • [33] Chowdhury S., Messac A. and Khire R. A., “Comprehensive Product Platform Planning (CP3) Using Mixed-Discrete Particle Swarm Optimization and a New Commonality Index,” ASME 2012 International Design Engineering Technical Conferences (IDETC), Chicago, IL, 12–15 Aug. 2012. Google Scholar

  • [34] Zhang J., Messac A., Chowdhury S. and Zhang J., “Adaptive Optimal Design of Active Thermally Insulated Windows Using Surrogate Modeling,” AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Orlando, FL, 12–15 April 2010. Google Scholar

  • [35] Zhang J., Chowdhury S., Messac A. and Castillo L., “A Response Surface-Based Cost Model for Wind Farm Design,” Energy Policy, Vol. 42, 2012, pp. 538–550.doi: ENPYAC CrossrefGoogle Scholar

  • [36] McKay M., Conover W. and Beckman R., “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code,” Technometrics, Vol. 21, No. 2, 1979, pp. 239–245. TCMTA2 0040-1706 CrossrefGoogle Scholar

  • [37] Sobol I., “Uniformly Distributed Sequences with an Additional Uniform Property,” USSR Computational Mathematics and Mathematical Physics, Vol. 16, No. 5, 1976, pp. 236–242. doi: CMMPA9 0041-5553 CrossrefGoogle Scholar

  • [38] Bratley P. and Fox B., “Algorithm 659: Implementing Sobol’s Quasirandom Sequence Generator,” ACM Transactions on Mathematical Software, Vol. 14, No. 1, 1988, pp. 88–100. doi: ACMSCU 0098-3500 CrossrefGoogle Scholar

  • [39] Kalagnanam J. and Diwekar U., “An Efficient Sampling Technique for Off-Line Quality Control,” Technometrics, Vol. 39, No. 3, 1997, pp. 308–319. doi: TCMTA2 0040-1706 CrossrefGoogle Scholar

  • [40] Goldberg M., Jobs and Economic Development Impact (JEDI) Model, National Renewable Energy Lab., Golden, CO, Oct. 2009. Google Scholar

  • [41] Jin R., Chen W. and Simpson T., “Comparative Studies of Metamodelling Techniques Under Multiple Modelling Criteria,” Structural and Multidisciplinary Optimization, Vol. 23, No. 1, 2001, pp. 1–13.doi: 1615-1488 CrossrefGoogle Scholar

  • [42] Hastie T., Tibshirani R. and Friedman J., The Elements of Statistical Learning, Springer-Verlag, New York, 2001. CrossrefGoogle Scholar