A comparative study of the stochastic averaging method and the path integration method for nonlinear ship roll motion in random beam seas

Abstract

In this paper, the energy-based stochastic averaging method and the path integration (PI) method are applied to study the stochastic response of the nonlinear roll motion in random beam seas. Specifically, the Markov diffusion theory is applied to describe the random roll motion such that the probabilistic properties of the ship roll motion are governed by the Fokker-Planck (FP) equation. The stochastic averaging method focuses on the roll energy envelope process and reduces the difficulty in calculating the stochastic response via a dimensional reduction of the original system. In contrast, the path integration method is based on the Markov property of the original dynamical system and provides approximate solutions to the FP equation by linking the explicitly known local solutions. Although the stochastic averaging technique is well established in the study of the stochastic responses of various dynamical systems, its accuracy for determining the high-level roll response has not yet been a focus of study. This paper aims to provide a comparative study for the performance of the two abovementioned methods, additionally, the advantages and shortcomings of using each method for studying the nonlinear roll motion in beam seas are demonstrated by practical calculations.

Reduction of nonlinear dynamical systems excited by random loads

“…Due to reduction of random nonlinear problems of higher order to lower order systems, the proposed theory is applicable for important problems, such as the stability of ships in large amplitude ocean waves, cf. [1,3,4]. The necessary computation time is very low, since the proposed analytical expressions involve integrals, which can be evaluated using standard quadrature formulas.…”

Systematic matrix formulation for efficient computational path integration

“…Most formulations of the numerical evaluation of PI solutions to the Chapman-Kolmogorov equations lead to a computationally expensive iterative method [2,5,6,[19][20][21], where the integral in the CK is evaluated directly while using an interpolation for the spatial discretisation of the PDF. Recently there were two major efforts in order to reduce the computation time of the iterations within a PI formulation.…”