Abstract
Simulation-based methods can be used for accurate uncertainty quantification and prediction of the reliability of a physical system under the following assumptions: (1) accurate input distribution models and (2) accurate simulation models (including accurate surrogate models if utilized). However, in practical engineering applications, often only limited numbers of input test data are available for modeling input distribution models. Thus, estimated input distribution models are uncertain. In addition, the simulation model could be biased due to assumptions and idealizations used in the modeling process. Furthermore, only a limited number of physical output test data is available in the practical engineering applications. As a result, target output distributions, against which the simulation model can be validated, are uncertain and the corresponding reliabilities become uncertain as well. To assess the conservative reliability of the product properly under the uncertainties due to limited numbers of both input and output test data and a biased simulation model, a confidence-based reliability assessment method is developed in this paper. In the developed method, a hierarchical Bayesian model is formulated to obtain the uncertainty distribution of reliability. Then, we can specify a target confidence level. The reliability value at the target confidence level using the uncertainty distribution of reliability is the confidence-based reliability, which is the confidence-based estimation of the true reliability. It has been numerically demonstrated that the proposed method can predict the reliability of a physical system that satisfies the user-specified target confidence level, using limited numbers of input and output test data.
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Abbreviations
- \( {\mathbf{x}}_i^e,{\mathbf{x}}^e \) :
-
Input test data for X i and input test data vector
- \( {\mathbf{y}}_i^e,{\mathbf{y}}^e \) :
-
ith physical output test data and physical output test data vector
- B i (x):
-
Unknown model bias for ith performance measure, G i (x)
- F Re (Re| y e , x e):
-
CDF of reliability given input/output test data
- \( {G}_i\left(\mathbf{x}\right),{G}_i^{true}\left(\mathbf{x}\right) \) :
-
Biased constraint and true constraint of ith performance measure
- f X (x; ζ, ψ):
-
Joint PDF of x
- AKDE:
-
Adaptive kernel density estimation
- CAE:
-
Computer-aided engineering
- CCDF:
-
Complementary CDF
- CDF:
-
Cumulative distribution function
- DKG:
-
Dynamic Kriging
- DOD:
-
Department of Defense
- DOE:
-
Design of experiment
- h 0 , h 0 (i) :
-
Bandwidth and ith realization of h0 in MCMC sampling
- HPC:
-
High-performance computing
- K :
-
Kernel
- M :
-
Number of MCS samples used in sampling-based Bayesian analysis
- MCMC:
-
Markov Chain Monte Carlo
- MCS:
-
Monte Carlo simulation
- nMCS :
-
Number of MCS samples used in sampling-based reliability analysis
- n p :
-
Prior sample size
- PDF:
-
Probability density function
- RBDO:
-
Reliability-based design optimization
- Re :
-
Reliability
- X i , x :
-
ith input random variable, input random variable vector
- P(h 0, ζ, ψ|y e, x e):
-
Joint PDF of bandwidth, input distribution type, and input distribution parameters given input/output test data
- P(h 0|ζ, ψ, x e):
-
Prior distribution of bandwidth given ζ and ψ
- P(ζ |ψ, x e):
-
Probability of input distribution type given ψ and xe
- P(ψ |x e):
-
Probability of input distribution parameter given xe
- f(ρ|h 0, ζ, ψ, y e, x e):
-
Conditional PDF of reliability
- ζ :
-
Input distribution type
- ψ :
-
Input distribution parameter
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Acknowledgments
The authors sincerely appreciate the technical and financial support of the Automotive Research Center (ARC) in accordance with Cooperative Agreement W56HZV-04-2-0001 U.S. Army Tank Automotive Research, Development and Engineering Center (TARDEC); and HPC support from the DOD High Performance Computing Modernization Program.
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Moon, MY., Cho, H., Choi, K.K. et al. Confidence-based reliability assessment considering limited numbers of both input and output test data. Struct Multidisc Optim 57, 2027–2043 (2018). https://doi.org/10.1007/s00158-018-1900-z
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DOI: https://doi.org/10.1007/s00158-018-1900-z