Four-dimensional path integration method for estimating the stochastic roll response

Abstract

In this paper, the Markov theory is introduced in order to model the roll motion excited by random wave excitation. Specifically, a four-dimensional (4D) Markov dynamic system is established by combing the single-degree-of-freedom (SDOF) model used to represent the ship rolling behavior in random beam seas with a second-order linear filter used to approximate the stationary wave excitation moment. Based on the Markov property of the 4D coupled dynamic system, a 4D path integration (PI) method is applied in order to solve the Fokker-Planck equation, which governs the probabilistic properties of the coupled dynamic system. The theoretical principle and numerical implementation of the current state of the art 4D PI method are presented. Furthermore, the numerical robustness and accuracy of the 4D PI method are evaluated by means of the versatile Monte Carlo simulation (MCS) technique.