Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory

Abstract

Random set theory is a general framework which comprises uncertainty in the form of probability boxes, possibility distributions, cumulative distribution functions, Dempster-Shafer structures or intervals; in addition, the dependence between the input variables can be expressed using copulas. In this paper, the lower and upper bounds on the probability of failure are calculated by means of random set theory. In order to accelerate the calculation, a well-known and efficient probability-based reliability method known as subset simulation is employed. This method is especially useful for finding small failure probabilities in both low-and highdimensional spaces, disjoint failure domains and nonlinear limit state functions. The proposed methodology represents a drastic reduction of the computational labor implied by plain Monte Carlo simulation for problems defined with a mixture of representations for the input variables, while delivering similar results. Numerical examples illustrate the efficiency of the proposed approach.

A new hybrid reliability‐based design optimization method under random and interval uncertainties

“…where P{⋅} represents the probability of an event, P L f and P U f are the lower and upper bounds of failure probability. The upper bound of failure probability is associated with the worst-case reliability and commonly considered in HRBDO-RI, 18,30 which is also applied in this work.…”

“…In Equations (2) and 7, the interval-constraint optimization problem can be handled by some alternative methods, such as the interior-point algorithm and the sampling method. 18 The interior-point algorithm based on the gradient information of the constraint function may easily trap into local solutions. Meanwhile, it may be time-consuming when an interval-constraint optimization is performed at each random variable sample.…”

“…In hybrid RBDO under random and interval uncertainties (HRBDO-RI), the reliability of constraints is often estimated by hybrid reliability analysis methods. Up to now, some methods for hybrid reliability analysis under random and interval uncertainties (HRA-RI) have been developed, which can be categorized into gradient-based methods, 17 sampling-based methods, 18 and metamodel-assisted methods. 19,20 As a gradient-based method, unified uncertainty analysis (UUA) method combines the first-order reliability method (FORM) and the interval-constrained optimization to assess the lower and upper bounds of failure probability, which is termed as FORM-UUA.…”

“…Nevertheless, the number of evaluations of constraints in the sampling-based methods is large, which limits their applications in the problems with time-consuming constraints. 18,22 Recently, metamodel techniques are popularly introduced in HRA-RI to improve the efficiency, where metamodels are employed to substitute the true constraints and sequentially updated to improve the estimation accuracy of failure probability bounds. Yang et al 19 propose an expected risk function to update the Kriging metamodel with refining the limit-state surface for HRA-RI.…”

Probability and interval hybrid reliability analysis based on adaptive local approximation of projection outlines using support vector machine

“…Aleatory and epistemic uncertainties exist in most engineering problems simultaneously (Alvarez, Uribe, & Hurtado, 2018;Balu & Rao, 2012;Der Kiureghian & Ditlevsen, 2009;Liu, Kuang, Yin, & Hu, 2017;Wang, Gao, Zhou, & Zhang, 2018;Zhang & Huang, 2010). Aleatory uncertainty is objective and irreducible, which is often described by the classical probability theory.…”