Stochastic Dynamic Analysis and Reliability of a Vessel Rolling in Random Beam Seas

Abstract

This article presents a four-dimensional (4D) path integration (PI) approach to study the stochastic roll response and reliability of a vessel in random beam seas. Specifically, a 4D Markov dynamic system is established by combing the single-degree-offreedom model used to represent the ship rolling behavior in random beam seas with a second-order linear filter used to approximate the stationary roll excitation moment. On the basis of the Markov property of the coupled 4D dynamic system, the response statistics of roll motion can be obtained by solving the Fokker-Planck equation of the dynamic system via the 4D PI method. The theoretical principle and numerical implementation of the current state of the art 4D PI method are presented. Moreover, the numerical robustness and accuracy of the 4D PI method are evaluated by comparing with the results obtained by the application of Monte Carlo simulation (MCS). The influence of the restoring terms and the damping terms on the stochastic roll response are investigated. Furthermore, based on the well-known Poisson assumption and the response statistics yielded by the 4D PI technique, evaluation of the reliability associated with high-level response is performed. The performance of the Poisson estimate for different levels of external excitations is evaluated by the versatile MCS technique.

Filter models for prediction of stochastic ship roll response

“…where b 44 , b 44q , c 1 , c 3 are relative roll parameters [24]. In this case, the spectrum of the relative wave excitation moment, S mm (ω), can be expressed as:…”

“…In Figure 9, the mean upcrossing rate obtained by the 4D PI approach and the Rice formula (24) and the empirical estimation of the upcrossing rateˆ() v as well as the 95% confidence interval provided by the Monte Carlo simulation are plotted.…”

“…It can be observed in Figures 6 and 7 that the PDF of the roll response is symmetric. Further, the mean upcrossing rate of the roll response can be determined by the Rice formula (24), benefiting from the joint PDF of the roll response obtained by the 4D PI approach:…”

“…In their work, the PI method has been shown to be capable of providing satisfactory estimation of the stochastic response, even in the tail region for low probability levels. Recently, this algorithm was successfully extended to 4D for studying the stochastic roll response of a ship in random beam seas [23,24].…”

GPU computing for accelerating the numerical Path Integration approach

“…To the best knowledge of the authors, the numerical solution of a 4D system calculated by the method described in section 2 has not been reported before. We should mention though that the PDF of 4D systems has been computed in [11,30], applying a variation of the PI method where the Fourier transform is utilized to convert the integration in Equation 2.2 to a convolution. This approach has been found to increase the efficiency of the PI code with respect to the execution time.…”