Abstract
Second-order reliability methods are commonly used for the computation of reliability, defined as the probability of satisfying an intended function in the presence of uncertainties. These methods can achieve highly accurate reliability predictions owing to a second-order approximation of the limit-state function around the Most Probable Point of failure. Although numerous formulations have been developed, the lack of full-scale comparative studies has led to a dubiety regarding the selection of a suitable method for a specific reliability analysis problem. In this study, the performance of commonly used second-order reliability methods is assessed based on the problem scale, curvatures at the Most Probable Point of failure, first-order reliability index, and limit-state contour. The assessment is based on three performance metrics: capability, accuracy, and robustness. The capability is a measure of the ability of a method to compute feasible probabilities, i.e., probabilities between 0 and 1. The accuracy and robustness are quantified based on the mean and standard deviation of relative errors with respect to exact reliabilities, respectively. This study not only provides a review of classical and novel second-order reliability methods, but also gives an insight on the selection of an appropriate reliability method for a given engineering application.
Highlights
Reliability analysis is widely used in industrial applications, ranging from structural (Chojaczyk et al 2015; Mansour et al 2019a; Hultgren et al 2021a) and mechanical engineering (Papadimitriou et al 2020; Sandberg et al 2017) to materials design (Liu et al 2018; Mansour et al 2019b; Alzweighi et al 2020)
Except Zhao’s, Mansour’s, Park’s and second-order saddlepoint approximation (SOSPA), see Fig. 3, a variable transformation U → Z aiming at rotating the coordinate system so that the last coordinate coincides with the vector u∗ from the origin to the Most Probable Point of failure (MPP) is performed
The results show that the accuracy of all second-order reliability methods (SORM) except Zhao’s, Mansour’s, and SOSPA methods is slightly higher for small-scale problems
Summary
Reliability analysis is widely used in industrial applications, ranging from structural (Chojaczyk et al 2015; Mansour et al 2019a; Hultgren et al 2021a) and mechanical engineering (Papadimitriou et al 2020; Sandberg et al 2017) to materials design (Liu et al 2018; Mansour et al 2019b; Alzweighi et al 2020). Each category is further divided into the following classes: small and large for the problem scale; small, large, positive, negative, and mixed for the curvatures; small and large for the first-order reliability index and parabolic or full quadratic for the limit-state contour Using this detailed testing scheme, the applicability of the different SORMs is evaluated based on the characteristics of the second-order Taylor approximation of the limit-state function. This includes three relatively new methods, Mansour and Olsson’s method, the second-order saddlepoint approximation (SOSPA), and Park and Lee’s method, as well as a number of traditional methods, i.e., Breitung’s method, Tvedt’s method, Hohenbichler and Rackwitz’s method, Koyluoglu and Nielsen’s method, Cai and Elishakoff’s method, and Zhao and Ono’s method
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