An improved stochastic method for conducting system probabilistic analysis by solving FPK equations

Abstract

This paper proposes an improved stochastic approach to solve the Fokker–Planck–Kolmogorov (FPK) equation for conducting system probabilistic analysis. The FPK equations, associated with the system under random excitation, are solved to provide the probabilistic solutions, which are essential for system reliability analysis. The proposed method is developed based on the optimization-oriented exponential-polynomial-closure (OEPC) method, and enhances OEPC by ensuring the limitation property of the probabilistic solution, named ’improved OEPC’ method. The key innovation of this study lies in introducing constraints for undetermined parameters to ensure the limitation property of the probabilistic solution. The limitation property means that the solution of FPK approaches zero as the state variables in FPK approach infinity. This improvement guarantees the solution accuracy across a wider spatial range when dealing with rare events, such as system failures of mechanical structures. The methodology is verified by investigating three different types of system: the Duffing oscillator, the ship rolling system, and the wind-excited frame tower system. The results show that the improved OEPC method significantly enhances the tail behavior of the probabilistic solution for complex nonlinear stochastic systems compared to the conventional OEPC method. Additionally, the improved OEPC method outperforms the Gaussian closure method in terms of solution accuracy and demonstrates considerably higher efficiency compared to Monte Carlo simulation.