Rare event probability evaluation for static and dynamic structures based on direct probability integral method

Abstract

Efficient and accurate evaluation of the rare event probability is a crucial yet challenging task for the design of static and dynamic structures with uncertainties. This study establishes a novel level-wise representative points increment strategy for direct probability integral method (DPIM), which calculates accurately rare event probabilities (less than 10⁻³). Firstly, the two advantages of partitioning the probability space in DPIM, namely, enhanced accuracy in reliability estimation and flexibility in selecting representative points, are elaborated. Secondly, it is clarified that the error in reliability assessment using DPIM is caused by the imprecise simulation of important subdomains. The idea of increasing the number of important points is advanced to improve the precision of reliability assessment. Subsequently, inspired by subset simulation, the level-wise representative points increment strategy is proposed. This strategy effectively adds representative points within important subdomains by selecting new points from the low-level points. Embedding the points increment strategy into DPIM forms a unified and efficient method for rare event estimations of static and dynamic structures. Finally, the accuracy and efficiency of the proposed method are demonstrated in three typical examples by comparing with Monte Carlo simulation (MCS), Quasi-MCS and subset simulation. The results indicate that the proposed strategy significantly improves the accuracy of reliability assessment employing DPIM, and fulfils a versatile and precise analysis of rare event probabilities in both static and dynamic systems.