Estimating extreme characteristics of stochastic non-linear systems

Abstract

The nature of the ocean is random and, as a result, marine structures face extreme, non-linear load effects. Estimating extreme responses of said structures often reduces to purely probabilistic approaches involve a considerable amount of conjecture. A method is developed in this paper to conserve original system information without the need for high fidelity Monte Carlo Simulations. Here, an iterative linearization of a non-linear system is developed to estimate extremes for the non-linear system. The Matched Upcrossing Equivalent Linear System (MUELS) method finds linear systems with zero-upcrossing periods equivalent to that of the non-linear system under a specified forcing spectrum. These linear systems are used as input into the Design Loads Generator, where ensembles of extreme linear realizations at the exposure period of interest, and the input time series that lead to those extremes, are generated. These input time series are valid inputs into the non-linear system and create a set of non-linear realizations that approximate the set of non-linear extremes at the exposure period of interest. As an example, this paper introduces and applies this method to the Duffing Oscillator at various levels of increasing cubic stiffness, forced by an ITTC wave spectrum, for exposure periods as long as 108 hours representing O(1010) zero-crossing maxima. Quantitative comparisons are made with Monte Carlo simulations, GEVD estimates, and expected values of time series near extreme local maxima.