Estimating extreme responses using a non-uniform phase distribution

Random processes are often modeled as a summation of a finite number of sinusoidal components. Various individual time series are created through the randomly chosen phase angles associated with each component. A specific event of the random process is characterized by the time at which the event happens and the chosen set of phase angles. Together, the time and the phase angle constitute the phase of each component. If many samples of a given event are cataloged, a histogram of the phases can be generated to produce a phase probability density function (PDF) that relates the event to the spectrum of the random process and the number of components used in the simulation.; Simulation of moderately rare events showed the component phase PDFs to be non-uniform and non-identically distributed. These PDFs were modeled using a single parameter, modified Gaussian distribution and used to generate design time series with a specific event at a specific time. To eliminate the need for Monte Carlo simulation, the single parameter of the phase distribution of each component was determined by comparing the PDF of the rare event as calculated using the non-uniform phase distributions to the PDF of the rare event as calculated using Extreme Value Theory. This approach is convenient and efficient as the phase parameters do not have to be estimated via Monte Carlo simulation; it is useful as the parameters can be generated for extremely rare events as easily as moderately rare events. In addition, the comparison to Extreme Value Theory helps to quantify the risk associated with rare events. An example application involving the springing of a Great Lakes bulk carrier shows how the method of nonuniform phases correctly predicts the build up of waves over several periods that produces a large bending moment.