Efficient estimation of temporal exceeding probability for ship responses in broadband wave fields

In this paper, we develop an efficient method to evaluate the temporal exceeding probability of ship motion (percentage of time for the ship motion to be above a given high threshold) in an irregular wave field. Our method builds on our previous work Gong et al. (2022), which converts the calculation into a sampling problem in the space of wave group parameters, within which an acquisition-based sequential sampling method is developed to reduce the number of required samples. Two critical advancements are achieved in this paper relative to Gong et al. (2022). (1) We develop a new wave group parameterization method, which allows the framework to be applied to general broadband wave fields. (2) We incorporate the variability regarding each parameterized wave group (e.g., varying wave form and initial condition of the ship encountering the group) into the final estimation of a single value of the temporal exceeding probability. Our complete framework is tested for a ship subject to a wave field with a JONSWAP spectrum, for different ship motion dynamical models, spectral bandwidths, and exceeding thresholds. The results show that for most cases our method provides a result with O(15 %) error or below within 210 samples, with the ground truth obtained from a continuous simulation that is more than 2300 times more expensive than our method. We also demonstrate the benefits of sequential sampling (with an acquisition function updated due to (2)) compared to standard random or Latin hypercube (LH) samplings, in terms of the mean error of the results.