Active Learning–Based Estimation of Extreme Roll Motion in Beam Seas with Supervised Dimensionality Reduction and Gradient Approximation

An efficient method is presented for estimating extreme nonlinear roll motion of a ship in beam seas in a given short-term sea state. With the aim of reducing the number of numerical simulations for roll motion simulation, this study employs an active learning framework, in which the active subspace is used for supervised dimensionality reduction. Heteroscedastic Gaussian process regression (hGPR) is used for meta-modeling the limit state function for a given extreme value, addressing incomplete stochastic spaces due to the use of active subspace. Beyond these implementations, a surrogate model of the roll motion, based on the Duffing oscillator with cubic damping and a parametrized roll excitation moment, is incorporated, by which one can further mitigate the computation time required to estimate gradient vectors of the limit state function. Numerical demonstrations are made using one-degree-of-freedom (1-DOF) and 2-DOF roll motion models, introducing a three-dimensional hull form for wave-induced roll excitation force. The latter model involves an implicit coupling with heave motion. Through a comprehensive comparison with crude Monte Carlo simulation (MCS) and first order reliability method, it turns out that the present approach enables efficient and accurate estimation of the exceedance probability for an excessive roll angle.