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In recent years, methods were proposed so as to efficiently perform time-variant reliability analysis. However, importance sampling (IS) for time-variant reliability analysis is barely studied in the literature. In this paper, an IS framework is proposed. A multi-dimensional integral is first derived to define the time-variant cumulative probability of failure, which has the similar expression to the classical definition of time-invariant failure probability. An IS framework is then developed according to the fact that time-invariant random variables are commonly involved in time-variant reliability analysis. The basic idea of the proposed framework is to simultaneously apply time-invariant IS and crude Monte Carlo simulation on time-invariant random variables and stochastic processes, respectively. Thus, the probability of acquiring failure trajectories of time-variant performance function is increased. Two auxiliary probability density functions are proposed to implement the IS framework. However, auxiliary PDFs available for the framework are not limited to the proposed two. Three examples are studied in order to validate the effectiveness of the proposed IS framework.
Jian Wang; Xiang Gao; Zhili Sun. An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes. Sustainability 2021, 13, 7776 .
AMA StyleJian Wang, Xiang Gao, Zhili Sun. An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes. Sustainability. 2021; 13 (14):7776.
Chicago/Turabian StyleJian Wang; Xiang Gao; Zhili Sun. 2021. "An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes." Sustainability 13, no. 14: 7776.
Crude Monte Carlo simulation (MCS) is the most robust and easily implemented method for performing time-variant reliability analysis (TRA). However, it is inefficient, especially for high reliability problems. This paper aims to present a random simulation method called the multilevel Monte Carlo (MLMC) method for TRA to enhance the computational efficiency of crude MCS while maintaining its accuracy and robustness. The proposed method first discretizes the time interval of interest using a geometric sequence of different timesteps. The cumulative probability of failure associated with the finest level can then be estimated by computing corrections using all levels. To assess the cumulative probability of failure in a way that minimizes the overall computational complexity, the number of random samples at each level is optimized. Moreover, the correction associated with each level is independently computed using crude MCS. Thereby, the proposed method can achieve the accuracy associated with the finest level at a much lower computational cost than that of crude MCS, and retains the robustness of crude MCS with respect to nonlinearity and dimensions. The effectiveness of the proposed method is validated by numerical examples.
Jian Wang; Xiang Gao; Zhili Sun. A Multilevel Simulation Method for Time-Variant Reliability Analysis. Sustainability 2021, 13, 3646 .
AMA StyleJian Wang, Xiang Gao, Zhili Sun. A Multilevel Simulation Method for Time-Variant Reliability Analysis. Sustainability. 2021; 13 (7):3646.
Chicago/Turabian StyleJian Wang; Xiang Gao; Zhili Sun. 2021. "A Multilevel Simulation Method for Time-Variant Reliability Analysis." Sustainability 13, no. 7: 3646.
Even though a great number of methods have been developed for time-variant reliability analysis (TRA), crude Monte Carlo simulation (MCS) is still widely used alone or combined with other methods to enhance the efficiency and accuracy of TRA. Multilevel Monte Carlo (MLMC) is introduced to TRA herein in order to reduce the computational complexity of MCS. MLMC first discretizes the time interval of interest by a geometric sequence of different timesteps. The finer grid is associated with smaller discretization error but larger computational cost. Accounting for the computational complexity of each level and the contribution that each level makes to the time-variant failure probability, the allocation of computational resource is then optimized in order to achieve a satisfying estimation of failure probability at the minimal computational cost. MLMC retains the robustness of crude MCS with respect to nonlinearity and dimensions of the time-variant performance function. Two benchmark examples are adopted to validate the effectiveness of the proposed MLMC. The results show that achieving the same accuracy level of results, the proposed method saves considerably computational cost relative to crude MCS.
Jian Wang; Xiang Gao; Runan Cao; Zhili Sun. A Multilevel Monte Carlo Method for Performing Time-Variant Reliability Analysis. IEEE Access 2021, 9, 31773 -31781.
AMA StyleJian Wang, Xiang Gao, Runan Cao, Zhili Sun. A Multilevel Monte Carlo Method for Performing Time-Variant Reliability Analysis. IEEE Access. 2021; 9 (99):31773-31781.
Chicago/Turabian StyleJian Wang; Xiang Gao; Runan Cao; Zhili Sun. 2021. "A Multilevel Monte Carlo Method for Performing Time-Variant Reliability Analysis." IEEE Access 9, no. 99: 31773-31781.
Importance sampling methods are extensively used in time-independent reliability analysis. However, the kind of methods is barely studied in the field of time-variant reliability analysis. This article presents an importance sampling method for time-variant reliability analysis. It increases the probability of sampling failure trajectories of a time-variant performance function. To develop the method, the instantaneous performance function at a predefined time instant is regarded as a time-independent one. A time-independent importance sampling is first implemented on the instantaneous performance function in order to obtain instantaneous samples of stochastic processes and random variables. Then, conditional trajectories of stochastic processes are generated on the condition of instantaneous samples achieved above, which utilizes the correlationship among instantaneous uncertainties at different time instants associated with stochastic processes. Subsequently, trajectories of the time-variant performance function are obtained. Validation results show that comparing with crude Monte Carlo simulation, the proposed method remarkably increases the probability of sampling failure trajectories. The efficiency and accuracy of the proposed method are demonstrated.
Jian Wang; Runan Cao; Zhili Sun. Importance Sampling for Time-Variant Reliability Analysis. IEEE Access 2021, 9, 20933 -20941.
AMA StyleJian Wang, Runan Cao, Zhili Sun. Importance Sampling for Time-Variant Reliability Analysis. IEEE Access. 2021; 9 ():20933-20941.
Chicago/Turabian StyleJian Wang; Runan Cao; Zhili Sun. 2021. "Importance Sampling for Time-Variant Reliability Analysis." IEEE Access 9, no. : 20933-20941.