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Self-adaptive sampling for sequential surrogate modeling of time-consuming finite element analysis

  • Jin, Seung-Seop (Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology) ;
  • Jung, Hyung-Jo (Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology)
  • Received : 2015.12.28
  • Accepted : 2016.02.23
  • Published : 2016.04.25

Abstract

This study presents a new approach of surrogate modeling for time-consuming finite element analysis. A surrogate model is widely used to reduce the computational cost under an iterative computational analysis. Although a variety of the methods have been widely investigated, there are still difficulties in surrogate modeling from a practical point of view: (1) How to derive optimal design of experiments (i.e., the number of training samples and their locations); and (2) diagnostics of the surrogate model. To overcome these difficulties, we propose a sequential surrogate modeling based on Gaussian process model (GPM) with self-adaptive sampling. The proposed approach not only enables further sampling to make GPM more accurate, but also evaluates the model adequacy within a sequential framework. The applicability of the proposed approach is first demonstrated by using mathematical test functions. Then, it is applied as a substitute of the iterative finite element analysis to Monte Carlo simulation for a response uncertainty analysis under correlated input uncertainties. In all numerical studies, it is successful to build GPM automatically with the minimal user intervention. The proposed approach can be customized for the various response surfaces and help a less experienced user save his/her efforts.

Keywords

References

  1. Bastos, L.S. and O'Hagan, A. (2009), "Diagnostics for Gaussian process emulators", Technometrics, 51(4), 425-438. https://doi.org/10.1198/TECH.2009.08019
  2. Bellman, R. (2003), Dynamic Programming, Dover Publications, Mineola, N.Y.
  3. Box, G.E.P. and Behnken, D.W. (1960), "Some new three level designs for the study of quantitative variables", Technometrics, 2(4), 455-475. https://doi.org/10.1080/00401706.1960.10489912
  4. Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978), Statistics for Experimenters : An Introduction to Design, Data Analysis, and Model Building, Wiley, New York.
  5. Bucher, C. and Most, T. (2008), "A comparison of approximate response functions in structural reliability analysis", Probabilist. Eng. Mech., 23(2-3), 154-163. https://doi.org/10.1016/j.probengmech.2007.12.022
  6. Bucher, C.G. and Bourgund, U. (1990), "A fast and efficient response-surface approach for structural reliability problems", Struct. Safety, 7(1), 57-66. https://doi.org/10.1016/0167-4730(90)90012-E
  7. Dette, H. and Pepelyshev, A. (2010), "Generalized latin hypercube design for computer experiments", Technometrics, 52(4), 421-429. https://doi.org/10.1198/TECH.2010.09157
  8. DiazDelaO, F.A. and Adhikari, S. (2011), "Gaussian process emulators for the stochastic finite element method", Int. J. Numerical Meth. Eng., 87(6), 521-540. https://doi.org/10.1002/nme.3116
  9. Dubourg, V., Sudret, B. and Deheeger, F. (2013), "Metamodel-based importance sampling for structural reliability analysis", Probabilist. Eng. Mech., 33, 47-57. https://doi.org/10.1016/j.probengmech.2013.02.002
  10. Fang, K.T., Lin, D.K.J., Winker, P. and Zhang, Y. (2000), "Uniform design: Theory and application", Technometrics, 42(3), 237-248. https://doi.org/10.1080/00401706.2000.10486045
  11. Forrester, A.I.J. and Keane, A.J. (2009), "Recent advances in surrogate-based optimization", Prog. Aerospace Sci., 45(1-3), 50-79. https://doi.org/10.1016/j.paerosci.2008.11.001
  12. Forrester, A.I.J., Sobester, A.S. and Keane, A.J. (2008), Engineering Design Via Surrogate Modelling : A Practical Guide, J. Wiley, Chichester, West Sussex, England ; Hoboken, NJ.
  13. Friedman, J. H. (1991), "Multivariate adaptive regression splines", Annals of Statistics, 19(1), 1-67. https://doi.org/10.1214/aos/1176347963
  14. Goel, T., Haftka, R.T., Shyy, W. and Watson, L.T. (2008), "Pitfalls of using a single criterion for selecting experimental designs", Int. J. Numer. Meth. Eng., 75(2), 127-155. https://doi.org/10.1002/nme.2242
  15. Gramacy, R.B. and Lee, H.K.H. (2012), "Cases for the nugget in modeling computer experiments", Statist. Comput., 22(3), 713-722. https://doi.org/10.1007/s11222-010-9224-x
  16. Hyndman, R.J. and Koehler, A.B. (2006), "Another look at measures of forecast Accuracy", Int. J. Forecast., 22(4), 679-688. https://doi.org/10.1016/j.ijforecast.2006.03.001
  17. Jones, D.R. (2001), "A taxonomy of global optimization methods based on response surfaces", J. Global Optim., 21(4), 345-383. https://doi.org/10.1023/A:1012771025575
  18. Jones, D.R., Schonlau, M. and Welch, W.J. (1998), "Efficient global optimization of expensive black-box functions", J. Global Optim., 13(4), 455-492. https://doi.org/10.1023/A:1008306431147
  19. Kennedy, M. C. and O'Hagan, A. (2001), "Bayesian calibration of computer models", J. Roy. Statist. Soc. Series B-Statistical Methodology, 63, 425-450. https://doi.org/10.1111/1467-9868.00294
  20. Kiefer, J. and Wolfowitz, J. (1959), "Optimum designs in regression problems", Annals Mathematical Statist., 30(2), 271-294. https://doi.org/10.1214/aoms/1177706252
  21. Kleijnen, J.P.C. and van Beers, W.C.M. (2004), "Application-driven sequential designs for simulation experiments: Kriging metamodelling", J. Operational Res. Soc., 55(8), 876-883. https://doi.org/10.1057/palgrave.jors.2601747
  22. Krige, D.G. (1994), "A Statistical approach to some basic mine valuation problems on the witwatersrand", J. South African Institute of Mining and Metallurgy, 94(3), 95-111.
  23. Mckay, M.D., Beckman, R.J. and Conover, W.J. (1979), "A comparison of three methods for selecting values of input variables in the analysis of output from a computer code", Technometrics, 21(2), 239-245.
  24. Moriasi, D.N., Arnold, J.G., Van Liew, M.W., Bingner, R.L., Harmel, R.D. and Veith, T.L. (2007), "Model evaluation guidelines for systematic auantification of accuracy in watershed simulations", Transactions of the Asabe, 50(3), 885-900. https://doi.org/10.13031/2013.23153
  25. Queipo, N.V., Haftka, R.T., Shyy, W., Goel, T., Vaidyanathan, R. and Tucker, P.K. (2005), "Surrogate-based analysis and optimization", Prog. Aerospace Sci., 41(1), 1-28. https://doi.org/10.1016/j.paerosci.2005.02.001
  26. Rougier, J., Sexton, D.M.H., Murphy, J.M. and Stainforth, D. (2009), "Analyzing the climate sensitivity of the Hadsm3 climate model using ensembles from different but related experiments", J. Climate, 22(13), 3540-3557. https://doi.org/10.1175/2008JCLI2533.1
  27. Sacks, J., Welch, W.J., Mitchell, T.J. and Wynn, H.P. (1983), "Design and analysis of computer experiments", Statist. Sci., 4(4), 409-423. https://doi.org/10.1214/ss/1177012413
  28. Xiong, Y., Chen, W., Apley, D. and Ding, X.R. (2007), "A non-stationary covariance-based kriging method for metamodelling in engineering design", Int. J. Numer. Meth. Eng., 71(6), 733-756. https://doi.org/10.1002/nme.1969
  29. Ye, K.Q. (1998), "Orthogonal column latin hypercubes and their application in computer experiments", J. American Statistical Association, 93(444), 1430-1439. https://doi.org/10.1080/01621459.1998.10473803
  30. Zhang, Z., Jiang, C., Han, X., Hu, D. and Yu, S. (2014), "A response surface approach for structural reliability analysis using evidence theory", Adv. Eng. Software, 69, 37-45. https://doi.org/10.1016/j.advengsoft.2013.12.005

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