Research Papers: Design Automation

Heuristics-Enhanced Model Fusion Considering Incomplete Data Using Kriging Models

[+] Author and Article Information
Anton v. Beek

University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mr.v.beek@sjtu.edu.cn

Mian Li

University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mianli@sjtu.edu.cn

Chao Ren

Corporate Technology of Siemens Ltd.,
Shanghai 200240, China
e-mail: chao.ren@siemens.com

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 22, 2017; final manuscript received October 22, 2017; published online December 13, 2017. Assoc. Editor: Samy Missoum.

J. Mech. Des 140(2), 021403 (Dec 13, 2017) (11 pages) Paper No: MD-17-1357; doi: 10.1115/1.4038596 History: Received May 22, 2017; Revised October 22, 2017

Simulation models are widely used to describe processes that would otherwise be arduous to analyze. However, many of these models merely provide an estimated response of the real systems, as their input parameters are exposed to uncertainty, or partially excluded from the model due to the complexity, or lack of understanding of the problem's physics. Accordingly, the prediction accuracy can be improved by integrating physical observations into low fidelity models, a process known as model calibration or model fusion. Typical model fusion techniques are essentially concerned with how to allocate information-rich data points to improve the model accuracy. However, methods on subtracting more information from already available data points have been starving attention. Subsequently, in this paper we acknowledge the dependence between the prior estimation of input parameters and the actual input parameters. Accordingly, the proposed framework subtracts the information contained in this relation to update the estimated input parameters and utilizes it in a model updating scheme to accurately approximate the real system outputs that are affected by all real input parameters (RIPs) of the problem. The proposed approach can effectively use limited experimental samples while maintaining prediction accuracy. It basically tweaks model parameters to update the computer simulation model so that it can match a specific set of experimental results. The significance and applicability of the proposed method is illustrated through comparison with a conventional model calibration scheme using two engineering examples.

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Le Gratiet, L. , 2013, “ Bayesian Analysis of Hierarchical Multifidelity Codes,” SIAM/ASA J. Uncertainty Quantif., 1(1), pp. 244–269. [CrossRef]
Goh, J. , Bingham, D. , Holloway, J. P. , Grosskopf, M. J. , Kuranz, C. C. , and Rutter, E. , 2013, “ Prediction and Computer Model Calibration Using Outputs From Multifidelity Simulators,” Technometrics, 55(4), pp. 501–512. [CrossRef]
Akil, N. , Claude, G. , and Dongbin, X. , 2014, “ A Stochastic Collocation Algorithm With Multifidelity Models,” SIAM J. Sci. Comput., 36(2), pp. A495–A521. [CrossRef]
Shishi, C. , Zhen, J. , Shuxing, Y. , Faniel, W. , and Wei, C. , 2016, “ Nonhierarchical Multi-Model Fusion Using Spatial Random Processes,” Int. J. Numer. Methods Eng., 106(7), pp. 503–528. [CrossRef]
Chen, S. , Jiang, Z. , Yang, S. , and Chen, W. , 2016, “ Multimodel Fusion Based Sequential Optimization,” AIAA J., 55(1), pp. 241–254. [CrossRef]
Kennedy, M. C. , and Hagan, A. O. , 2001, “ Bayesian Calibration of Computer Models,” R. Statisitcal Soc., 63(3), pp. 425–465. [CrossRef]
Wang, S. , Chen, W. , and Tsui, K.-L. , 2009, “ Bayesian Validation of Computer Models,” Technometrics, 51(4), pp. 439–451. [CrossRef]
Sacks, J. , Welch, W. J. , Mitchell, J. S. B. , and Henry, P. W. , 1989, “ Design and Analysis of Computer Experiments,” Statisitcal Sci., 4(4), pp. 409–423. [CrossRef]
Arendt, P. D. , and Apley, D. W. , 2015, “ Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability,” ASME J. Mech. Des., 134(10), p. 101402.
Jiang, Z. , Li, W. , Apley, D. W. , and Chen, W. , 2015, “ A Spatial-Random-Process Based Multidisciplinary System Uncertainty Propagation Approach With Model Uncertainty,” ASME J. Mech. Des., 137(10), p. 101402.
Zhu, Z. , Hu, Z. , and Du, X. , 2015, “ Reliability Analysis for Multidisciplinary Systems Involving Stationary Stochastic Processes,” ASME Paper No. DETC2015-46168.
Jiang, Z. , Chen, W. , and German, B. J. , 2016, “ Multidiciplinary Statistical Sensitivity Analysis Considering Both Aleatory and Epistemic Uncertainties,” AIAA, 54(4), pp. 1326–1338. [CrossRef]
Jiang, Z. , Chen, S. , Apley, D. W. , and Chen, W. , 2016, “ Reduction of Epistemic Model Uncertainty in Simulation-Based Multidisciplinary Design,” ASME J. Mech. Des., 138(8), p. 081403.
Rasmussen, C. E. , and Williams, C. K. I. , 2004, Gaussian Processes for Machine Learning, Vol. 1, MIT Press, Cambridge, MA.
Do, C. B. , 2007, “ Gaussian Processes,” Stanford University, Stanford, CA, accessed Dec. 5, 2017, https://see.stanford.edu/materials/aimlcs229/cs229-gp.pdf
Xi, Z. , and Yang, R.-J. , 2015, “ Stochastic Model Bias Correction of Dynamic System Responses for Simulation-Based Reliability Analysis,” ASME Paper No. DETC2015-46938.
Gorguluarslan, R. M. , and Choi, S.-K. , 2014, “ An Improved Stochastic Upscaling Method for Multiscale Engineering Systems,” ASME Paper No. DETC2014-34418.
Rosen, D. W. , and Park, S.-I. , 2015, “ Material Characterization of Additively Manufactured Part Via Multi-Level Stochastic Upscaling Method,” ASME Paper No. DETC2015-46822.
Mlakar, M. , Petelin, D. , Tušar, T. , and Filipič, B. , 2015, “ GP-DEMO: Differential Evolution for Multiobjective Optimization Based on Gaussian Process Models,” Eur. J. Oper. Res., 243(2), pp. 347–361. [CrossRef]
Wu, C.-C. , Chen, Z. , and Tang, G.-R. , 1998, “ Component Tolerance Design for Minimum Quality Loss and Manufacturing Cost,” Comput. Ind., 35(3), pp. 223–232. [CrossRef]
Yan, H. , Wu, X. , and Yang, J. , 2015, “ Application of Monte Carlo Method in Tolerance Analysis,” Procedia CIRP, 27(13), pp. 281–285. [CrossRef]
Besag, J. , Green, P. , Higdon, D. , and Mengersen, K. , 1989, “ Design and Analysis of Computer Experiments,” Stat. Sci., 4(4), pp. 409–435. http://www.stat.ohio-state.edu/~comp_exp/jour.club/Sacks89.pdf
Song, W. , Lee, E. , Gea, H. C. , and Xu, L. , 2015, “ Topology Optimization With Load Uncertainty as an Inhomogeneous Eigenvalue Problem,” ASME Paper No. DETC2015-46912.
Zhu, Z. , and Du, X. , 2015, “ Extreme Value Metamodeling for System Reliability With Time-Dependent Functions,” ASME Paper No. DETC2015-46162.
Cho, H. , Choi, K. , and Lamb, D. , 2014, “ Confidence-Based Method for Reliability-Based Design Optimization,” ASME Paper No. DETC2014-34644.
Moon, M.-Y. , Choi, K. , Cho, H. , Gaul, N. , Lamb, D. , and Gorsich, D. , 2015, “ Development of a Conservative Model Validation Approach for Reliable Analysis,” ASME Paper No. DETC2015-46982.
Gaul, N. J. , Cowles, M. K. , Cho, H. , Choi, K. K. , and Lamb, D. , 2015, “ Modified Bayesian Kriging for Noisy Response Problems for Reliability Analysis,” ASME Paper No. DETC2015-47370.
Lichtenstern, A. , 2013, “ Kriging Methods in Spatial Statistics,” Bachelor's thesis, Technische Universitat München, Munich, Germany, pp. 1–97. http://mediatum.ub.tum.de/doc/1173364/file.pdf
Wackernagel, H. , 2013, “ Geostatistical Structure Analysis: The Variogram,” MINES ParisTech, Paris, France, accessed Dec. 5, 2017, http://hans.wackernagel.free.fr
Tipping, M. E. , 2006, “ Bayesian Inference: An Introduction to Principles and Practice in Machine Learning From Least-Squares to Bayesian Inference,” Miketipping, 1(1), pp. 41–62. http://www.miketipping.com/papers/met-mlbayes.pdf
Gallager, R. G. , 2013, Theory for Applications, Vol. 3, Springer-Verlag, Berlin.
Bayram, U. , and Acar, E. , 2015, “ Tolerance Analysis With Multiple Surrogate Models,” Acta Phys. Polonica A, 128(2), pp. 447–449. [CrossRef]
Wang, P. , and Cui, X. , 2015, “ Reliability Analysis and Design Considering Disjointed Active Failure Regions,” ASME Paper No. IMECE2015-52985.
Drignei, D. , Mourelatos, Z. , Kosova, E. , and Baseski, I. , 2015, “ Time-Dependent Reliability Using Metamodels With Transformed Random Inputs,” ASME Paper No. DETC2015-46823.
Hamel, J. , 2015, “ Sequential Cooperative Robust Optimization (Scro) for Multi-Objective Design Under Uncertainty,” ASME Paper No. DETC2015-47885.
Hans, W. , 2003, An Introduction With Applications, Vol. 3, Springer-Verlag, Berlin.
Richard, W. , and Margaret, A. O. , 2007, Statistics in Practice, Vol. 2, Wiley, Chichester, UK.
Wang, C. , Qiu, J. , Liu, G. , and Zhang, Y. , 2014, “ Testability Evaluation Using Prior Information of Multiple Sources,” Chin. J. Aeronaut., 27(4), pp. 867–874. [CrossRef]
Yu, K. , Chen, C. W. , Reed, C. , and Dunson, D. B. , 2013, “ Bayesian Variable Selection in Quantile Regression,” Stat. Its Interface, 6(2), pp. 261–274. [CrossRef]
Moala, F. , Ramos, P. , and Achcar, J. , 2013, “ Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors,” Rev. Colomb. Estadística, 36(2), pp. 321–338. http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0120-17512013000200009
Fink, D. , 1997, “ A Compendium of Conjugate Priors,” Mag. Western History, 1(1), pp. 1–47. https://www.johndcook.com/CompendiumOfConjugatePriors.pdf
Raiffa, H. , and Schlaifer, R. , 1961, Applied Statistical Decision Theory, 5 ed., Harvard University, Boston, MA.
Pandey, B. N. , and Bandyopadhyay, P. , 2012, “ Bayesian Estimation of Inverse Gaussian Distribution,” Stat. Comput. Simul., 1(1), pp. 1–17. https://arxiv.org/abs/1210.4524
Banerjee, A. K. , and Bhattacharyya, G. K. , 1979, “ Bayesian Results for the Inverse Gaussian Distribution With an Application,” Technometrics, 21(2), pp. 247–251. [CrossRef]
Tsionas, E. G. , 2004, “ Bayesian Inference for Multivariate Gamma Distributions,” Stat. Comput., 14(3), pp. 223–233. [CrossRef]
Beek, A. , and Li, M. , 2016, “ Tolerance Allocation and Calibration With Limited Emphirical Data,” ASME Paper No. DETC2016-59328.
Ferson, S. , Oberkampf, W. L. , and Ginzburg, L. , 2008, “ Model Validation and Predictive Capability for the Thermal Challenge Problem,” Comput. Methods Appl. Mech. Eng., 197(29–32), pp. 2408–2430. [CrossRef]
Ferson, S. , Oberkamp, F. W. , and Ginzburg, L. , 2009, “ Validation of Imprecise Probability Models,” Int. J. Reliab. Saf., 3(1), pp. 3–22. [CrossRef]
Murphy, K. P. , 2007, “ Conjugate Bayesian Analysis of the Gaussian Distribution,” University of British Columbia, Vancouver, BC, Canada, accessed Dec. 5, 2017, https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf


Grahic Jump Location
Fig. 1

Flowchart of the proposed HEMF method

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Fig. 2

Relation between ato and ero

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Fig. 3

Configuration of friction ring assembly

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Fig. 4

Results of the HEMF method on a friction ring assembly compared to a conventional updating method: (a) discrepancy between computer simulation, and experimental response friction ring model; (b) verification of the HEMF model for friction ring assembly; (c) response distributions for slacked mean, and tightened standard deviation for friction ring assembly; (d) individual parameter distributions for slacked mean, and tightened standard deviation of the friction ring assembly in millimeter; (e) response distributions for tightened mean and slacked standard deviation of the friction ring assembly; and (f) individual parameter distributions for tightened mean and slacked standard deviation of the friction ring assembly in millimeter

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Fig. 5

The electrical amplifier circuit and its components

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Fig. 6

Discrepancy in computer simulation and experimental responses of the electrical amplifier

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Fig. 7

Results of the HEMF method on an electrical amplifier assembly compared to the conventional method: (a) probability density functions for calibrated electrical amplifier response for the same design as its training data in Volts; (b) u-pooling results for calibrated electrical amplifier response for the same design as its training data; (c) probability density functions for calibrated electrical amplifier response for tightened standard deviation and perturbed mean in Volts; (d) u-pooling results for calibrated electrical amplifier response for tightened standard deviation and perturbed mean; (e) probability density functions for calibrated electrical amplifier response for slacked standard deviation and perturbed mean in Volts; and (f) u-pooling results for calibrated electrical amplifier response for slacked standard deviation and perturbed mean




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