Mustafa A. Mohamad

Probabilistic Response and Rare Events in Mathieu's Equation under Correlated Parametric Excitation

Abstract We derive an analytical approximation to the probability distribution function (pdf) for the response of Mathieu׳s equation under parametric excitation by a random process with a spectrum peaked at the main resonant frequency, motivated by the problem of large amplitude ship roll resonance in random seas. The inclusion of random stochastic excitation renders the otherwise straightforward response to a system undergoing intermittent resonances : randomly occurring large amplitude bursts. Intermittent resonance occurs precisely when the random parametric excitation pushes the system into the instability region, causing an extreme magnitude response. As a result, the statistics are characterized by heavy-tails. We apply a recently developed mathematical technique, the probabilistic decomposition-synthesis method, to derive an analytical approximation to the non-Gaussian pdf of the response. We illustrate the validity of this analytical approximation through comparisons with Monte-Carlo simulations that demonstrate our result accurately captures the strong non-Gaussianinty of the response.

Sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

Significance Understanding the statistics of extreme events in dynamical systems of high complexity is of vital importance for reliability assessment and design. We formulate a method to pick samples optimally so that we have rapid convergence of the full statistics of a quantity of interest, including the tails that describe extreme events. This is important for large-scale problems in science and engineering, where we desire to predict the statistics of relevant quantities but can only afford a limited number of simulations or experiments due to their very expensive cost. We demonstrate our approach in a hydromechanical system with millions of degrees of freedom, where only 10–20 carefully selected samples can lead to accurate approximation of the extreme event statistics. We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems from a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the “next-best” data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach uses Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds using the posterior distribution of the inferred map. The next-best design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process uses only information from the inferred map, it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size and for problems where each sample is very expensive to obtain. We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to 3D irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using a limited number of samples.

Heavy-tailed response of structural systems subjected to stochastic excitation containing extreme forcing events

We characterize the complex, heavy-tailed probability distribution functions (pdf) describing the response and its local extrema for structural systems subjected to random forcing that includes extreme events. Our approach is based on the recent probabilistic decomposition-synthesis technique, where we decouple rare events regimes from the background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems) our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom structural system undergoing rare events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte-Carlo simulations.

Analytical Approximation of the Heavy-Tail Structure for Intermittently Unstable Complex Modes

In this work, we consider systems that are subjected to intermittent instabilities due to external, correlated stochastic excitation. These intermittent instabilities, though rare, give rise to heavy-tailed probability distribution functions (pdf). By making appropriate assumptions on the form of these instabilities, we formulate a method for the analytical approximation of the pdf of the system response. This method relies on conditioning the pdf of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. We illustrate our method to a prototype intermittent system, a complex mode in a turbulent signal, and show that the analytic results compare favorably with direct Monte Carlo simulations for a broad range of parameters.