Spectral and multiresolution Wiener expansions of oscillatory stochastic processes

Abstract

Wiener chaos expansions are being evaluated for the representation of stochastic variability in the response of nonlinear aeroelastic systems, which often exhibit limit cycles. Preliminary studies with a simple nonlinear aeroelastic computational model have shown that the standard non-intrusive Wiener–Hermite expansion fails to maintain time accuracy as the simulation evolves. Wiener–Hermite expansions faithfully reproduce the short-term characteristics of the process but consistently lose energy after several mean periods of oscillation. This energy loss remains even for very high-order expansions. To uncover the cause of this energy loss and to explore potential remedies, the more elementary problem of a sinusoid with random frequency is used herein to simulate the periodic response of an uncertain system. As time progresses, coefficients of the higher order terms in both the Wiener–Hermite and Wiener–Legendre expansions successively gain and lose dominance over the lower-order coefficients in a manner that causes any fixed-order expansion in terms of global basis functions to fail over a simulation time of sufficient duration. This characteristic behavior is attributed to the continually increasing frequency of the process in the random dimension. The recently developed Wiener–Haar expansion is found to almost entirely eliminate the loss of energy at large times, both for the sinusoidal process and for the response of a two degree-of-freedom nonlinear system, which is examined as a prelude to the stochastic simulation of aeroelastic limit cycles. It is also found that Mallat's pyramid algorithm is more efficient and accurate for evaluating Wiener–Haar expansion coefficients than Monte Carlo simulation or numerical quadrature.