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Technical Briefs

Bayesian Based Multivariate Model Validation Method Under Uncertainty for Dynamic Systems

[+] Author and Article Information
Zhenfei Zhan, Yan Fu

Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48121

Ren-Jye Yang1

Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48121ryang@ford.com

Yinghong Peng

School of Mechanical Engineering,  Shanghai Jiao Tong University, Shanghai 200240, P.R. China

1

Corresponding author

J. Mech. Des 134(3), 034502 (Feb 29, 2012) (7 pages) doi:10.1115/1.4005863 History: Received May 16, 2011; Revised January 13, 2012; Accepted January 23, 2012; Published February 28, 2012; Online February 29, 2012

Validation of computational models with multiple, repeated, and correlated functional responses for a dynamic system requires the consideration of uncertainty quantification and propagation, multivariate data correlation, and objective robust metrics. This paper presents a new method of model validation under uncertainty to address these critical issues. Three key technologies of this new method are uncertainty quantification and propagation using statistical data analysis, probabilistic principal component analysis (PPCA), and interval-based Bayesian hypothesis testing. Statistical data analysis is used to quantify the variabilities of the repeated tests and computer-aided engineering (CAE) model results. The differences between the mean values of test and CAE data are extracted as validation features, and the PPCA is employed to handle multivariate correlation and to reduce the dimension of the multivariate difference curves. The variabilities of the repeated test and CAE data are propagated through the data transformation to the PPCA space. In addition, physics-based thresholds are defined and transformed to the PPCA space. Finally, interval-based Bayesian hypothesis testing is conducted on the reduced difference data to assess the model validity under uncertainty. A real-world dynamic system example which has one set of the repeated test data and two stochastic CAE models is used to demonstrate this new approach.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Bayesian based multivariate model validation method under uncertainty

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Figure 2

A driver side occupant restraint system model

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Figure 3

Time history plots for the test data and CAE model A results: (a) chest deflection, (b) chest acceleration in x-direction, (c) belt load at anchor, (d) belt load at retractor, (e) belt load at shoulder, (f) left femur load in z-direction, (g) right femur load in z-direction, (h) head acceleration in x-direction, (i) upper neck load, (j) upper neck moment, (k) pelvis acceleration in x-direction

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Figure 4

Time history plots for the test data and CAE model B results: (a) chest deflection, (b) chest acceleration in x-direction, (c) belt load at anchor, (d) belt load at retractor, (e) belt load at shoulder, (f) left femur load in z-direction, (g) right femur load in z-direction, (h) head acceleration in x-direction, (i) upper neck load, (j) upper neck moment, (k) pelvis acceleration in x-direction

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Figure 5

Normalized mean curves of test and models A and B

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Figure 6

PPCA dimension reduction of: (a) model A and (b) model B

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Figure 7

Bayesian confidence results and histogram charts at b = 15% of reference data: (a) model A and (b) model B

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