Quantifying Uncertainty
Pettit, C.L. and Beran, P.S.. "Spectral and multiresolution Wiener expansions of oscillatory stochastic processes" Journal of Sound and Vibration. 294
(4-5).
JUL 25 2006.
pp. 752--779.
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.
Witteveen, J.A.S., Loeven, A., Sarkar, S., and Bijl, H.. "Probabilistic collocation for period-1 limit cycle oscillations" Journal of Sound and Vibration. 311
(1-2).
MAR 18 2008.
pp. 421--439.
In this paper probabilistic collocation for limit cycle oscillations (PCLCO) is proposed. Probabilistic collocation (PC) is a non-intrusive approach to compute the polynomial chaos description of uncertainty numerically. Polynomial chaos can require impractical high orders to approximate long-term time integration problems, due to the fast increase of required polynomial chaos order with time. PCLCO is a PC formulation for modeling the long-term stochastic behavior of dynamical systems exhibiting a periodic response, i.e. a limit cycle oscillation (LCO). In the PC method deterministic time series are computed at collocation points in probability space. In PCLCO, PC is applied to a time-independent parametrization of the periodic response of the deterministic solves instead of to the time-dependent functions themselves. Due to the time-independent parametrization the accuracy of PCLCO is independent of time. The approach is applied to period-1 oscillations with one main frequency subject to a random parameter. Numerical results are presented for the harmonic oscillator, a two-dof airfoil flutter model and the fluid-structure interaction of an elastically mounted cylinder. (C) 2007 Elsevier Ltd. All rights reserved.