Uncertainty quantification in stochastic models for extreme loads

Many response parameters for offshore structures such as wave energy converters (WECs), wind turbines, oil and gas platforms, etc. can be modeled as stochastic processes. The extreme of such a response process over any selected interval of time is a random variable. Having accurate estimates of such extremes during a structure's life is crucial in structural design, but there are challenges in their estimation due to various sources of uncertainty. These include uncertainty from environmental conditions or the climate/weather as well as from short-term simulations of these stochastic processes at appropriate time and space resolution. Together, these uncertainty sources make up a high-dimension vector of random variables (that can be on the order of thousands). Many offshore structures must withstand many years of exposure and use return periods for design that are on order of 50 to 100 years. The focus of this study is on rare events or response levels that are associated with very low probabilities of exceedance (e.g., on the order of $10⁻⁶ over a typical 1-hour duration). Time-domain simulations of dynamic offshore structures can be computationally expensive even for a single simulation. Various approaches can be adopted in practice to account for uncertainties in extreme response prediction. Monte Carlo Simulation (MCS) is the most common for exhaustive prediction of the response for all conditions. Since MCS can be computationally very demanding, the development of efficient surrogate models is presented to more efficiently deal with these uncertainties. A proposed method, in this study, is based on the use of an ensemble of multiple polynomial chaos expansion (M-PCE) surrogate models to propagate the uncertainty from the environment through the stochastic input simulation to eventual design load prediction. In particular, each PCE model in the ensemble provides an approximate relationship between the structural response and the underlying environmental variables, while variability in the short-term simulations is accounted for by the multiple surrogates. M-PCE helps overcome the curse of dimensionality since, instead of dealing with development of a high-dimensional surrogate model, the M-PCE ensemble includes multiple low-dimensional PCE models, each defined in terms of only the long-term environmental variables, which are of low dimension. It is found that the M-PCE ensemble can efficiently predict long-term extreme loads associated at exceedance probability levels (in 1 hour) of $10⁻⁵ or higher. Next, by considering MCS and M-PCE as high-fidelity and low-fidelity models, respectively, this study proposes a bi-fidelity approach that combines M-PCE and MCS outputs so as to control, or even eliminate bias introduced by the use of the M-PCE ensemble alone. The approach takes advantage of the robustness of MCS on the one hand and the efficiency of M-PCE model on the other. The key idea is that many of the model simulations are carried out using the inexpensive M-PCE ensemble while a very small number of simulations use the costly high-fidelity model. In this way, the new method significantly enhances the efficiency of MCS and improves the accuracy of the M-PCE ensemble. Finally, this dissertation explores the use of a combination of sliced inverse regression (SIR) and polynomial chaos expansion in uncertainty quantification of response extremes. The SIR procedure is adopted to reduce the original high-dimensional problem to a low-dimensional one; then, the PCE model is employed as a surrogate in the reduced-dimension space in this SIR-PCE scheme. All the proposed approaches including the M-PCE ensemble, the bi-fidelity MPCE-MCS and the SIR-PCE scheme can help mitigate the curse of dimensionality issue; thus, they are all viable approaches for probabilistic assessment of high-dimensional stochastic models, especially when predicting very rare long-term extreme response levels for offshore structures. The proposed methods are validated using examples ranging from benchmarking analytical functions to offshore structures that include studies on a maximum wave elevation, a linear single-degree-of-freedom system response, and a nonlinear wave energy converter. All the proposed methods are found to be efficient and need significantly less effort to achieve unbiased estimations of extreme response levels compared with MCS.