Abstract
Reliability analysis aims at quantitatively assessing the risks of structures and infrastructure systems considering various sources of aleatory or epistemic uncertainties. This concept can be interpreted as reliability updating (RU) to track the change of risks, fusing new data emerged in the operation period. Optimal allocation of resources can be therefore prepared for reasonable maintenance and rehabilitation. In the state-of-the-art method for RU with equality information, auxiliary random variables are introduced to transform the problem into an inequality one. However, the joint events derived in aforementioned approach are typically very rare, which can be computationally cumbersome via simulation techniques. To enhance the performance of RU with equality information, this paper proposes a new reliability updating approach that decomposes the joint event term into two separate multipliers and subsequently computes conditional probability through subset simulation (SS). Moreover, this work analyzes the statistical properties for RU through SS. An adaptive manner is proposed to dynamically adjust the batch sample size for each subset so that the consistency of simulation results can be ensured. Compared to past SS-based reliability updating approaches, the proposed method is computationally easier to handle and substantially more robust. Three numerical examples together with a practical application of structural health monitoring are investigated to demonstrate the computational superiority of the proposed method.
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Abbreviations
- \(c\) :
-
A constant required to formulate the acceptance domain in rejection sampling
- \(F\) :
-
The failure event
- \(F_{i}\) :
-
The subset failure event \(\left( {i = 1,2, \ldots ,m} \right)\)
- \(f_{{\varepsilon_{e} }}\) :
-
The probability density function of \(\varepsilon_{e}\)
- \(f_{{\varepsilon_{s} }}\) :
-
The probability density function of \(\varepsilon_{s}\)
- \(g( \cdot )\) :
-
The performance function
- \(h( \cdot )\) :
-
The limit state function after introducing \(U\) for the RU with equability information
- \(J( \cdot )\) :
-
The limit state function of \(F \cap Z\)
- \(L( \cdot )\) :
-
The likelihood function
- \(m_{s}\) :
-
The measurement of a property of the system
- \(m_{e}\) :
-
The measurement of a property of the external loading
- \(N_{F \cap Z}\) :
-
The number of simulations of performance function for estimating \({\text{ Pr}}\left( {F \cap Z} \right)\) with SS
- \(N_{F}\) :
-
The number of simulations of performance function for estimating \({\text{ Pr}}\left( F \right)\) with SS
- \(N_{Z}\) :
-
The number of simulations of performance function for estimating \({\text{ Pr}}\left( Z \right)\) with SS
- \(N_{ss}^{*}\) :
-
The number of samples in each subset in the last iteration
- \(\Delta N_{ss}\) :
-
The number of extra sampled added by MCMC
- \({\text{Pr}}\left( F \right)\) :
-
The prior probability of failure
- \({\text{Pr}}\left( {F{|}Z} \right)\) :
-
The updated probability of failure with information \(Z\)
- \({\text{Pr}}\left( {F \cap Z} \right)\) :
-
The probability of the joint even \(F \cap Z\)
- \({\text{Pr}}\left( Z \right)\) :
-
The probability of observing the information \(Z\)
- \({\text{Pr}}\left( {Z{|}F} \right)\) :
-
The probability of observing the information \(Z\) if the failure event occurs
- \(P\) :
-
Representing a standard uniform variable
- \(p\) :
-
Representing a realization of a uniform variable
- \(p_{i}\) :
-
The intermediate probability of failure for subset simulation
- \(p_{i}^{*}\) :
-
The updated intermediate probability of failure by adding external samples
- \(S^{i}\) :
-
The set of samples for each subfailure events \(\left( {i = 1,2, \ldots ,m} \right)\)
- \(U\) :
-
Representing a standard normal variable
- \(u\) :
-
Representing a realization of a Gaussian random variable
- \({\varvec{X}}\) :
-
The vector of random variables
- \(Z\) :
-
The observed information
- \(\varepsilon_{e}\) :
-
The measurement error for \(m_{s}\)
- \(\varepsilon_{s}\) :
-
The measurement error for \(m_{e}\)
- \({\Theta }\left( {\varvec{X}} \right)\) :
-
The function parameterized by \({\varvec{X}}\) with a realization notation \({\uptheta }\)
- \({\Theta }_{e} \left( {\varvec{X}} \right)\) :
-
Representing the uncertainty of the external loading
- \({\Theta }_{s} \left( {\varvec{X}} \right)\) :
-
Representing the uncertainty of the system characteristic
- \(\Omega_{f}\) :
-
The target domain for the subset simulation
- \(\Omega_{i}\) :
-
The subset domain \(\left( {i = 1,2, \ldots ,m} \right)\)
- \(N_{F|Z}^{{\text{R}}}\) :
-
The number of simulations of performance function for estimating \({\text{ Pr}}\left( {F|Z} \right)\) with RUSS
- \(N_{F|Z}^{{{\text{OR}}}}\) :
-
The number of simulations of performance function for estimating \({\text{ Pr}}\left( {F|Z} \right)\) with Opt-RUSS
- \({\text{COV(}} \cdot {)}\) :
-
The coefficient of variation for an estimator
- \({\text{E(}} \cdot {)}\) :
-
The mean of an estimator
- \({\text{Var(}} \cdot {)}\) :
-
The variance of an estimator
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 52308196) and New Faculty Basic Research Capability Enhancement Program (No. 590123060). It was also partially funded by China Postdoctoral Science Foundation (No. 2022M710886) and Guangdong Provincial Key Laboratory of Green Construction and Intelligent Operation & Maintenance for Offshore Infrastructure. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the support. The support is greatly appreciated.
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The algorithms and step-by-step implementation approach for the proposed Opt-RUSS are elaborated in the paper. Readers can use MATLAB and UQLab package to implement the algorithms and generate the results presented in the paper. The codes can be accessed upon reader’s requirement.
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Wang, Z., Zhao, Y., Song, C. et al. A new interpretation on structural reliability updating with adaptive batch sampling-based subset simulation. Struct Multidisc Optim 67, 7 (2024). https://doi.org/10.1007/s00158-023-03720-8
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DOI: https://doi.org/10.1007/s00158-023-03720-8