EXTREME-VALUE PREDICTION FOR NON-LINEAR STOCHASTIC OSCILLATORS VIA NUMERICAL SOLUTIONS OF THE STATIONARY FPK EQUATION

Abstract

The accuracy of two well-established numerical methods is demonstrated, and the importance of “bandwidth” examined, for computationally efficient Markov based extreme-value predictions associated with finite duration stationary sample paths of a non-linear oscillator driven by Gaussian white noise. By making the Poisson assumption of independent upcrossings, extreme exceedance probabilities are predicted via the mean threshold crossing rate, using numerical solutions of the stationary Fokker–Planck (FPK) equation. With bandwidth initially ignored, predicted exceedances using the Weighted Residual methods of Bhandari and Sherrer, Soize and Kunert, and the Finite Element method of Langley, are compared with nominally “exact” predictions for a heavily damped Duffing-type model obtained by using an explicit FPK solution—the FE method being established as superior. Predictions via FE solutions are then compared with very long Monte Carlo simulations, in which bandwidth effects are included. Two lightly damped non-linear oscillator models are examined, both with cubic stiffness, but different damping mechanisms—one model again being of simple Duffing-type with linear damping, the other being appropriate to single co-ordinate random vibration of a clamped–clamped beam, with wholly non-linear damping. The realistic damping parameter values assigned to the beam model are statistically equivalent to the linear damping level chosen for the simple model, at just above 1%. At this overall damping level, results clearly demonstrate that, for the probability levels and durations considered, bandwidth is only important for the linearly damped model—for the beam model with non-linear damping, bandwidth can be ignored, allowing accurate extreme exceedance predictions by using only the stationary FPK equation. The paper also demonstrates that the “limiting decay rate of the first-passage probability”—advocated by Crandall, Roberts and others, as a criterion for deciding the barrier height above which first-passage times can be assumed independent—proves grossly over-conservative as a corresponding criterion for deciding the independence level for use in stationary extreme-value prediction.