Research Papers

An Adaptive Response Surface Method Using Bayesian Metric and Model Bias Correction Function

[+] Author and Article Information
Lei Shi

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: lshi1340@gmail.com

Ren-Jye Yang

Research and Advanced Engineering,
Ford Motor Company,
2101 Village Rd,
Dearborn, MI 48121
e-mail: ryang@ford.com

Ping Zhu

State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: pzhu@sjtu.edu.cn

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 31, 2013; final manuscript received November 12, 2013; published online January 10, 2014. Assoc. Editor: Wei Chen.

J. Mech. Des 136(3), 031005 (Jan 10, 2014) (8 pages) Paper No: MD-13-1043; doi: 10.1115/1.4026095 History: Received January 31, 2013; Revised November 12, 2013

The Bayesian metric was used to select the best available response surface in the literature. One of the major drawbacks of this method is the lack of a rigorous method to quantify data uncertainty, which is required as an input. In addition, the accuracy of any response surface is inherently unpredictable. This paper employs the Gaussian process based model bias correction method to quantify the data uncertainty and subsequently improve the accuracy of a response surface model. An adaptive response surface updating algorithm is then proposed for a large-scale problem to select the best response surface. The proposed methodology is demonstrated by a mathematical example and then applied to a vehicle design problem.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Oberkampf, W. L., Diegert, K. V., Alvin, K. F., and Rutherford, B. M., 1998, “Variability, Uncertainty, and Error in Computational Simulation,” ASME Proceedings of the 7th AIAA/ASME Joint thermophysics and heat transfer conference, ASME-HTD-Vol.357-2.
Apley, D., Liu, J., and Chen, W., 2006, “Understanding the Effects of Model Uncertainty in Robust Design With Computer Experiments,”ASME J. Mech. Des., 128(4), pp. 945–958. [CrossRef]
Zhan, Z., Fu, Y., Yang, R. J., and Peng, Y. H., 2012, “Bayesian Based Multivariate Model Validation Method Under Uncertainty for Dynamic Systems,” ASME J. Mech. Des., 134(3), p. 0345022. [CrossRef]
Kim, C., and Choi, K. K., 2008, " Reliability-Based Design Optimization Using Response Surface Method With Prediction Interval Estimation,” ASME J. Mech. Des., 130(12), p. 121401. [CrossRef]
Yang, R. J., Wang, N., Tho, C. H., Bobineau, J. P., and Wang, B. P., 2005, “Metamodeling Development for Vehicle Frontal Impact Simulation,” ASME J. Mech. Des, 127(5), pp. 1014–1020. [CrossRef]
Gu, L., 2001, “A comparison of Polynomial Based Regression Models in Vehicle Safety Analysis,” ASME Design Engineering Technical Conferences, ASME Paper No.: DETC/DAC-21083.
Kurtaran, H., Eskandarian, A., Marzougui, D., and Bedewi, N. E., 2002, “Crashworthiness Design Optimization Using Successive Response Surface Approximations,” Comput. Mech.29(4–5), pp. 409–421. [CrossRef]
Fang, H., Rais-Rohani, M., Liu, Z., and Horstemeyer, M. F., 2005, “A Comparative Study of Metamodeling Methods for Multiobjective Crashworthiness Optimization,” Comput. Struct.83(25–26), pp. 2121–2136. [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, pp. 439–457. [CrossRef]
Shi, L., Yang, R. J., and Zhu, P., 2012, “A Method for Selecting Surrogate Models in Crashworthiness Optimization,” Struct. Multidiscip. Optim., 46(2), pp. 159–170. [CrossRef]
Xiong, Y., Chen, W., Tsui, K., and Apley, D. W., 2009, “A Better Understanding of Model Updating Strategies in Validating Engineering Models,” Comput. Methods Appl. Mech. Eng., 198, pp. 1327–1337. [CrossRef]
Kennedy, M. C., and O'Hagan, A., 2001, “Bayesian Calibration of Computer Models,” J. R. Stat. Soc. Ser. B (Methodol.), 63(3), pp. 425–464. [CrossRef]
Wang, S. C., Chen, W., and Tsui, K. L., 2009, “Bayesian Validation of Computer Models,” Technometrics, 51(4), pp. 439–451. [CrossRef]
Arendt, P., Apley, D., and Chen, W., 2012, “Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability,” ASME J. Mech. Des., 134(10), p. 100908. [CrossRef]
Gearhart, C., and Wang, B. P., 2001, “Bayesian Metrics for Comparing Response Surface Models of Data With Uncertainty,” Struct. Multidiscip. Optim., 22, pp. 198–207. [CrossRef]
Sivia, D. S., and Skilling, J., 2006, Data Analysis: A Bayesian Tutorial, 2nd ed., Oxford University Press, New York.
Park, J., and Jeon, J., 2002, “Estimation of Input Parameters in Complex Simulation Using a Gaussian Process Metamodel,” Probab. Eng. Mech., 17, pp. 219–225. [CrossRef]
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P., 1989, “Design and Analysis of Computer Experiments,” Stat. Sci., 4(4), pp. 409–435. [CrossRef]
Shi, L., Yang, R. J., and Zhu, P., 2012, “An Adaptive Response Surface Method for Crashworthiness Optimization,” Eng. Optim., 2012, p. 734815. [CrossRef]
National Crash Analysis Center (NCAC), 2001, "Public Finite Element Model Archive.” Available at: http://www.ncac.gwu.edu/archives/model/index.html


Grahic Jump Location
Fig. 1

Response surfaces with and without model bias correction updating

Grahic Jump Location
Fig. 2

Algorithm of response surface updating methodology

Grahic Jump Location
Fig. 3

Comparison of structure deformation between finite element simulation and test: (a) Full frontal impact and (b) 40% offset impact

Grahic Jump Location
Fig. 4

Acceleration history comparison of rocker at B-pillar: (a) Full frontal impact, and (b) 40% offset impact (Note: the physical test data for full frontal impact and 40% offset impact can be referred to the link.3

Grahic Jump Location
Fig. 5

Design variable selection for main front-end structure



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In