Abstract
The research of military vehicle protection against the under-belly blast is a particular topic that enormously relies on computer-aided engineering (CAE) simulation technology because of the high cost of relevant real physical tests. CAE model with high predictive performance is required to obtain accurate results. An efficient model calibration process is necessary to improve CAE modeling. Traditional model calibration method uses visual comparison; this cannot provide any quantitative, and cannot consider uncertainty. This paper would study an inverse model calibration process that combines error assessment of response time histories (EARTH) and Bayesian method. Experimental data and computational data would be compared, and the discrepancies between them are quantified by EARTH metric. And Bayesian method is applied to address the uncertainties of discrepancies quantization caused by model parameters uncertainties. Comparison of model confidence level can be quantified by Bayes factor. Then, the calibration process can be considered as an optimization problem solved by optimization-based method. The optimization objectives are minimizing quantified discrepancies between experimental data and computational data and maximizing model’s Bayes factor. The model parameters which need to be calibrated would be improved during this process. An example case of an occupant loaded in vehicle under-belly blast scenario is used to describe the raised method.
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Abbreviations
- \(B\) :
-
Bayes factor
- \(C\) :
-
Computational data time history
- \({d}_{\mathrm{DTW}}\) :
-
DTW distance
- \({\varvec{d}}[i,j]\) :
-
Cost matrix
- \(\Delta \) :
-
Discrepancy between test and CAE data
- \(\delta \) :
-
Model inadequacy
- \(\boldsymbol{\Sigma }\) :
-
Covariance matrix of model parameters
- \({\boldsymbol{\Sigma }}_{\varepsilon }\) :
-
Covariance matrix of EARTH errors
- \({\varepsilon }_{\mathrm{CAE}}\) :
-
CAE uncertainty
- \({\varepsilon }_{m}\) :
-
Magnitude error
- \({\varepsilon }_{m,b}\) :
-
Magnitude error of baseline model
- \({\varepsilon }_{m,n}\) :
-
Dimensionless magnitude error
- \({\varepsilon }_{\mathrm{meas}}\) :
-
Measurement uncertainty
- \({\varepsilon }_{p}\) :
-
Phase error
- \({\varepsilon }_{p,b}\) :
-
Phase error of baseline model
- \({\varepsilon }_{p,n}\) :
-
Dimensionless phase error
- \({\varepsilon }_{\mathrm{param}}\) :
-
Parameters uncertainty
- \({\varepsilon }_{s}\) :
-
Slope error
- \({\varepsilon }_{s,b}\) :
-
Slope error of baseline model
- \({\varepsilon }_{s,n}\) :
-
Dimensionless slope error
- \({\varepsilon }_{\mathrm{overall}}\) :
-
Overall error
- \({H}_{0}\) :
-
Null hypotheses
- \({H}_{1}\) :
-
Alternative hypotheses
- \({\varvec{L}}\) :
-
Likelihood probability
- \({\varvec{\mu}}\) :
-
Mean matrix of model parameters
- \({{\varvec{\mu}}}_{\varepsilon }\) :
-
Mean matrix of EARTH errors
- \({\mu }_{\varepsilon m}\) :
-
Mean magnitude error
- \({\mu }_{\varepsilon p}\) :
-
Mean phase error
- \({\mu }_{\varepsilon s}\) :
-
Mean slope error
- \(r(n)\) :
-
Cross-correlation
- \({\sigma }_{\varepsilon m}\) :
-
Standard deviation of magnitude error
- \({\sigma }_{\varepsilon p}\) :
-
Standard deviation of phase error
- \({\sigma }_{\varepsilon s}\) :
-
Standard deviation of slope error
- \(T\) :
-
Experimental data time history
- \({\varvec{v}}\) :
-
Variance matrix of model parameters
- \({{\varvec{v}}}_{\varepsilon }\) :
-
Variance matrix of EARTH errors
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Acknowledgements
This work was supported by the National Natural Science Foundation of China [Grant Number 11802140]. The authors also would like to acknowledge Nanjing University of Science and Technology (NJUST) Vehicle Engineering institute for test and equipment support, and LetPub for its linguistic assistance during the preparation of this manuscript.
Funding
The National Natural Science Foundation of China [Grant Number 11802140].
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The authors cannot provide the keyword files for FEA and detail physical test data because of their confidentiality. So, the results presented in this paper cannot be reproduced perfectly. However, the readers can still replicate the calibration process with your own research object in similar area with codes for calculation available. For software using, FEM was conducted by Hypermesh 2017; FEA was conducted by LS-Dyna with solver version R11; multi-objective optimization solving was conducted by modeFRONIER 2019.
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Zheng, Q., Wu, M., Sun, X. et al. Combined Bayesian and error assessment-based model calibration method for vehicle under-belly blast with uncertainty. Struct Multidisc Optim 65, 136 (2022). https://doi.org/10.1007/s00158-022-03226-9
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DOI: https://doi.org/10.1007/s00158-022-03226-9