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Research Papers

Conservative Surrogate Model Using Weighted Kriging Variance for Sampling-Based RBDO

[+] Author and Article Information
Liang Zhao

Schlumberger,
1310 Rankin Road,
Houston, TX 77073
e-mail: lzhao01@slb.com

K. K. Choi

Department of Mechanical and
Industrial Engineering,
College of Engineering,
The University of Iowa,
Iowa City, IA 52242
e-mail: kkchoi@engineering.uiowa.edu

Ikjin Lee

Department of Mechanical Engineering,
University of Connecticut,
Storrs, CT 06269
e-mail: ilee@engr.uconn.edu

David Gorsich

US Army RDECOM/TARDEC,
Warren, MI 48397-5000
e-mail: gorsichd@tacom.army.mil

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received July 12, 2012; final manuscript received February 12, 2013; published online July 2, 2013. Assoc. Editor: Irem Y. Tumer.

J. Mech. Des 135(9), 091003 (Jul 02, 2013) (10 pages) Paper No: MD-12-1356; doi: 10.1115/1.4024731 History: Received July 12, 2012; Revised February 12, 2013

In sampling-based reliability-based design optimization (RBDO) of large-scale engineering applications, the Monte Carlo simulation (MCS) is often used for the probability of failure calculation and probabilistic sensitivity analysis using the prediction from the surrogate model for the performance function evaluations. When the number of samples used to construct the surrogate model is not enough, the prediction from the surrogate model becomes inaccurate and thus the Monte Carlo simulation results as well. Therefore, to count in the prediction error from the surrogate model and assure the obtained optimum design from sampling-based RBDO satisfies the probabilistic constraints, a conservative surrogate model, which is not overly conservative, needs to be developed. In this paper, a conservative surrogate model is constructed using the weighted Kriging variance where the weight is determined by the relative change in the corrected Akaike Information Criterion (AICc) of the dynamic Kriging model. The proposed conservative surrogate model performs better than the traditional Kriging prediction interval approach because it reduces fluctuation in the Kriging prediction bound and it performs better than the constant safety margin approach because it adaptively accounts large uncertainty of the surrogate model in the region where samples are sparse. Numerical examples show that using the proposed conservative surrogate model for sampling-based RBDO is necessary to have confidence that the optimum design satisfies the probabilistic constraints when the number of samples is limited, while it does not lead to overly conservative designs like the constant safety margin approach.

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References

Haldar, A., and Mahadevan, S., 2000, Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, New York.
Yi, K., Choi, K. K., Kim, N. H., and Botkin, M. E., 2007, “Design Sensitivity Analysis and Optimization for Minimizing Springback of Sheet-Formed Part,” Int. J. Numer. Methods Eng., 71, pp. 1483–1511. [CrossRef]
Yu, X., and Du, X., 2006, “Reliability-Based Multidisciplinary Optimization for Aircraft Wing Design,” Struct. Infrastruct. Eng., 2(3–4), pp. 277–289. [CrossRef]
Lee, I., Choi, K. K., Noh., Y., Zhao., L., and Gorsich., D., 2011, “Sampling-Based Stochastic Sensitivity Analysis Using Score Functions for RBDO Problems With Correlated Random Variables,” ASME J. Mech. Des., 133(2), p. 021003. [CrossRef]
Lee, I., Choi, K. K., and Zhao, L., 2011, “Sampling-Based RBDO Using the Dynamic Kriging (D-Kriging) Method and Stochastic Sensitivity Analysis,” J. Struct. Multidiscip. Optim., 44(3), pp. 299–317. [CrossRef]
Rubinstein, R. Y., and Kroese, D. P., 2008, Simulation and the Monte Carlo Method, Wiley-Interscience, New York.
Krige, D. G., 1951, “A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand,” J. Chem., Metall. Min. Eng. Soc. S Afr., 52(6), pp. 119–139.
Martin, J. D., and Simpson, T. W., 2005, “Use of Kriging Models to Approximate Deterministic Computer Models,” AIAA J., 43(4), pp. 853–863. [CrossRef]
Forrester, A., Sobester, A., and Keane, A. J., 2008, Engineering Design Via Surrogate Modeling: A Practical Guide, Wiley, Hoboken, NJ.
Chiles, J. P., and Delfiner, P., 1999, Geostatistics: Modeling Spatial Uncertainty, Wiley, New York.
Suykens, J. A. K., and Vandewalle, J., 1999, “Least Square Support Vector Machine Classifiers,” Neural Process. Lett., 9(3), pp. 293–300. [CrossRef]
Smola, A. J., and Scholkopf, B., 2004, “A Tutorial on Support Vector Regression,” Stat. Comput., 14(3), pp. 199–222. [CrossRef]
Xie, L., Gu, N., Li, D., Cao, Z., Tan, M., and Nahavandi, S., 2013, “Concurrent Control Chart Patterns Recognition With Singular Spectrum Analysis and Support Vector Machine,” Comput. Ind. Eng., 64(1), pp. 280–289. [CrossRef]
Song, H., Choi, K. K., Lee, I., Zhao, L., and Lamb, D., 2011, “Adaptive Virtual Support Vector Machine for the Reliability Analysis of High-Dimensional Problems,” 37th ASME Design Automation Conference, Aug. 28–31, Washington, DC.
Zhao, L., Choi, K. K., and Lee, I., 2011, “Metamodeling Method Using Dynamic Kriging for Design Optimization,” AIAA J., 49(9), pp. 2034–2046. [CrossRef]
Zhao, L., Choi, K. K., and Lee, I., 2009, “Response Surface Method Using Sequential Sampling for Reliability-Based Design Optimization,” Proceedings of the ASME IDETC/CIE 2009, Aug. 30th–Sept 2nd, San Diego, Paper No. DETC2009-87084.
Chan, K. Y., Skerlos, S. J., Papalambros, P., 2007, “An Adaptive Sequential Linear Programming Algorithm for Optimal Design Problems With Probabilistic Constraints,” ASME J. Mech. Des., 129(2), pp. 140–149. [CrossRef]
Elishakoff, I., 2004, Safety Factors and Reliability: Friends or Foes? Kluwer Academic, Norwell, MA.
Acar, E., Kale, A., and Haftka, R. T., 2007, “Comparing Effectiveness of Measures That Improve Aircraft Structural Safety,” J. Aerosp. Eng., 20(3), pp. 186–199. [CrossRef]
Picheny, V., Kim, N. H., and Haftka, R. T., 2008, “Conservative Predictions Using Surrogate Modeling,” 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials, April 7–10, Schaumburg, IL.
Picheny, V., 2009, “Improving Accuracy and Compensating for Uncertainty in Surrogate Modeling,” Ph.D. dissertation, University of Florida, Gainesville, FL.
Picheny, V., Ginsbourger, D., Roustant, O., Haftka, R. T., and Kim, N. H., 2010, “Adaptive Designs of Experiments for Accurate Approximation of a Target Region,” ASME J. Mech. Des., 132(7), p. 071008. [CrossRef]
Hertog, D., Kleijnen, J., and Siem, A., 2006, “The Correct Kriging Variance Estimated by Bootstrapping,” J. Oper. Res. Soc., 57, pp. 400–409. [CrossRef]
Luna, S. D., and Young, A., 2003, “The Bootstrap and Kriging Prediction Intervals,” Scand. J. Stat., 30, pp. 175–192. [CrossRef]
Picheny, V., Kim, N. H., and Haftka, R. T., 2010, “Application of Bootstrap Method in Conservative Estimation of Reliability With Limited Samples,” Struct. Multidiscip. Optim., 41(2), pp. 205–217. [CrossRef]
Viana, F. A. C., and Haftka, R. T., 2009, “Cross Validation Can Estimate How Well Prediction Variance Correlates With Error,” AIAA J., 47(9), pp. 2266–2270. [CrossRef]
Viana, F. A. C., Haftka, R. T., and Steffen, V., 2009, “Multiple Surrogates: How Cross-Validation Errors Can Help Us to Obtain the Best Predictor,” Struct. Multidiscip. Optim., 39(4), pp. 439–457. [CrossRef]
Viana, F. A. C., Picheny, V., and Haftka, R. T., 2010, “Using Cross Validation to Design Conservative Surrogates,” AIAA J., 48(10), pp. 2286–2298. [CrossRef]
Xiong, F., Chen, W., Xiong, Y., and Yang, S. X., 2011, “Weighted Stochastic Response Surface Method Considering Sample Weights,” J. Struct. Multidiscip. Optim., 44, pp. 837–849. [CrossRef]
Akaike, H., 1974, “A New Look at the Statistical Model Identification,” IEEE Trans. Autom. Control, 19(6), pp. 716–723. [CrossRef]
Hoeting, J. A., Davis, R. A., Merton, A. A., and Thompson, S. E., 2006, “Model Selection for Geostatistical Models,” Ecol. Appl., 16(1), pp. 87–98. [CrossRef] [PubMed]
Hurvich, C. M., and Tsai, C. L., 1989, “Regression and Time Series Model Selection in Small Samples,” Biometrika, 76, pp. 297–307. [CrossRef]
McDonald, M., and Mahadevan, S., 2008, “Design Optimization With System-Level Reliability Constraints,” ASME J. Mech. Des., 130(2), p. 021403. [CrossRef]
Lee, I., Choi, K. K., and Gorsich, D., 2010, “System Reliability-Based Design Optimization Using the MPP-Based Dimension Reduction Method,” Struct. Multidiscip. Optim., 41(6), pp. 823–839. [CrossRef]
Burnham, K. P., and Anderson, D. R., 2002, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Springer, New York.
Goel, T., Haftka, R., and Shyy, W., 2008, “Comparing Error Estimation Measures for Polynomial and Kriging Approximation of Noise-Free Functions,” Struct. Multidiscip. Optim., 38(5), pp. 429–442. [CrossRef]
Xiong, F., Xiong, Y., Chen, W., and Yang, S., 2009, “Optimizing Latin Hypercube Design for Sequential Sampling of Computer Experiments,” Eng. Optim., 41(8), pp. 793–810. [CrossRef]
Guo, J., and Du, X., 2009, “Reliability Sensitivity Analysis With Random and Interval Variables,” Int. J. Numer. Methods Eng., 78, pp. 1585–1617. [CrossRef]

Figures

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Fig. 1

One-dimensional problem. (a) 7 samples and (b) 6 samples.

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Fig. 2

Leave-one-out Kriging prediction and importance values. (a) Kriging prediction without sample #2, (b) Kriging prediction without sample #3, (c) Kriging prediction without sample #4, (d) Kriging prediction without sample #5, and (e) Kriging prediction without sample #6.

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Fig. 3

Conservative surrogate model using weighted Kriging variance

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Fig. 4

Cost and constraint functions plot

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Fig. 5

Different conservative surrogate models and optimum designs

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Fig. 6

Cantilever tube design

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