Faes, M. G. R., Valdebenito, M. A., Yuan, X. K., Wei, P. F., & Beer, M. (2021). Augmented reliability analysis for estimating imprecise first excursion probabilities in stochastic linear dynamics. Adv. Eng. Softw., 155, 102993.
Abstract: Imprecise probability allows quantifying the level of safety of a system taking into account the effect of both aleatory and epistemic uncertainty. The practical estimation of an imprecise probability is usually quite demanding from a numerical viewpoint, as it is necessary to propagate separately both types of uncertainty, leading in practical cases to a nested implementation in the so-called double loop approach. In view of this issue, this contribution presents an alternative approach that avoids the double loop by replacing the imprecise probability problem by an augmented, purely aleatory reliability analysis. Then, with the help of Bayes' theorem, it is possible to recover an expression for the failure probability as an explicit function of the imprecise parameters from the augmented reliability problem, which ultimately allows calculating the imprecise probability. The implementation of the proposed framework is investigated within the context of imprecise first excursion probability estimation of uncertain linear structures subject to imprecisely defined stochastic quantities and crisp stochastic loads. The associated augmented reliability problem is solved within the context of Directional Importance Sampling, leading to an improved accuracy at reduced numerical costs. The application of the proposed approach is investigated by means of two examples. The results obtained indicate that the proposed approach can be highly efficient and accurate.
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Yuan, X. K., Faes, M. G. R., Liu, S. L., Valdebenito, M. A., & Beer, M. (2021). Efficient imprecise reliability analysis using the Augmented Space Integral. Reliab. Eng. Syst. Saf., 210, 107477.
Abstract: This paper presents an efficient approach to compute the bounds on the reliability of a structure subjected to uncertain parameters described by means of imprecise probabilities. These imprecise probabilities arise from epistemic uncertainty in the definition of the hyper-parameters of a set of random variables that describe aleatory uncertainty in some of the structure's properties. Typically, such calculation involves the solution of a so-called double-loop problem, where a crisp reliability problem is repeatedly solved to determine which realization of the epistemic uncertainties yields the worst or best case with respect to structural safety. The approach in this paper aims at decoupling this double loop by virtue of the Augmented Space Integral. The core idea of the method is to infer a functional relationship between the epistemically uncertain hyper-parameters and the probability of failure. Then, this functional relationship can be used to determine the best and worst case behavior with respect to the probability of failure. Three case studies are included to illustrate the effectiveness and efficiency of the developed methods.
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Yuan, X. K., Liu, S. L., Faes, M., Valdebenito, M. A., & Beer, M. (2021). An efficient importance sampling approach for reliability analysis of time-variant structures subject to time-dependent stochastic load. Mech. Syst. Sig. Process., 159, 107699.
Abstract: Structural performance is affected by deterioration processes and external loads. Both effects may change over time, posing a challenge for conducting reliability analysis. In such context, this contribution aims at assessing the reliability of structures where some of its parameters are modeled as random variables, possibly including deterioration processes, and which are subjected to stochastic load processes. The approach is developed within the framework of importance sampling and it is based on the concept of composite limit states, where the time-dependent reliability problem is transformed into a series system with multiple performance functions. Then, an efficient two-step importance sampling density function is proposed, which splits time-invariant parameters (random variables) from the time-variant ones (stochastic processes). This importance sampling scheme is geared towards a particular class of problems, where the performance of the structural system exhibits a linear dependency with respect to the stochastic load for fixed time. This allows calculating the reliability associated with the series system most efficiently. Practical examples illustrate the performance of the proposed approach.
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Yuan, X. K., Liu, S. L., Valdebenito, M. A., Faes, M. G. R., Jerez, D. J., Jensen, H. A., et al. (2021). Decoupled reliability-based optimization using Markov chain Monte Carlo in augmented space. Adv. Eng. Softw., 157, 103020.
Abstract: An efficient framework is proposed for reliability-based design optimization (RBDO) of structural systems. The RBDO problem is expressed in terms of the minimization of the failure probability with respect to design variables which correspond to distribution parameters of random variables, e.g. mean or standard deviation. Generally, this problem is quite demanding from a computational viewpoint, as repeated reliability analyses are involved. Hence, in this contribution, an efficient framework for solving a class of RBDO problems without even a single reliability analysis is proposed. It makes full use of an established functional relationship between the probability of failure and the distribution design parameters, which is termed as the failure probability function (FPF). By introducing an instrumental variability associated with the distribution design parameters, the target FPF is found to be proportional to a posterior distribution of the design parameters conditional on the occurrence of failure in an augmented space. This posterior distribution is derived and expressed as an integral, which can be estimated through simulation. An advanced Markov chain algorithm is adopted to efficiently generate samples that follow the aforementioned posterior distribution. Also, an algorithm that re-uses information is proposed in combination with sequential approximate optimization to improve the efficiency. Numeric examples illustrate the performance of the proposed framework.
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Yuan, X. K., Liu, S. L., Valdebenito, M. A., Gu, J., & Beer, M. (2021). Efficient procedure for failure probability function estimation in augmented space. Struct. Saf., 92, 102104.
Abstract: An efficient procedure is proposed to estimate the failure probability function (FPF) with respect to design variables, which correspond to distribution parameters of basic structural random variables. The proposed procedure is based on the concept of an augmented reliability problem, which assumes the design variables as uncertain by assigning a prior distribution, transforming the FPF into an expression that includes the posterior distribution of those design variables. The novel contribution of this work consists of expressing this target posterior distribution as an integral, allowing it to be estimated by means of sampling, and no distribution fitting is needed, leading to an efficient estimation of FPF. The proposed procedure is implemented within three different simulation strategies: Monte Carlo simulation, importance sampling and subset simulation; for each of these cases, expressions for the coefficient of variation of the FPF estimate are derived. Numerical examples illustrate performance of the proposed approaches.
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