Quantifying Uncertainty
Beran, P.S., Pettit, C.L., and Millman, D.R.. "Uncertainty quantification of limit-cycle oscillations" Jounral of Computational Physics. 217
(1).
2006.
pp. 217--247.
Different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit-cycle oscillation, subject to parametric variability. The aeroelastic system is that of a rigid airfoil, supported by pitch and plunge structural coupling, with nonlinearities in the component in pitch. The nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit-cycle oscillations. Uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil. Stochastic projections of the time-domain and cyclic equations governing system response are carried out, leading to both intrusive and non-intrusive computational formulations. Non-intrusive formulations are examined using stochastic projections derived from Wiener expansions involving Haar wavelet and B-spline bases, while Wiener-Hermite expansions of the cyclic equations are employed intrusively and non-intrusively. Application of the B-spline stochastic projection is extended to the treatment of aerodynamic nonlinearities, as modeled through the discrete Euler equations. The methodologies are compared in terms of computational cost, convergence properties, ease of implementation, and potential for application to complex aeroelastic systems.
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.
Millman, D.R., King, P.I., and Beran, P.S.. "Airfoil pitch-and-plunge bifurcation behavior with Fourier chaos expansions" Journal of Aircraft. 42
(2).
MAR-APR 2005.
pp. 376--384.
A stochastic projection method is employed to obtain the probability distribution of pitch angle of an airfoil in pitch and plunge subject to probabilistic uncertainty in both the initial pitch angle and the cubic spring coefficient of the restoring pitch force. Historically, the selected basis for the stochastic projection method has been orthogonal polynomials, referred to as the polynomial chaos. Such polynomials, however, result in unacceptable computational expense for applications involving oscillatory motion, and a new basis, the Fourier chaos, is introduced for computing limit-cycle oscillations. Unlike the polynomial chaos expansions, which cannot predict limit-cycle oscillations, the Fourier chaos expansions predict both subcritical and supercritical responses even with low-order expansions and high-order nonlinearities. Bifurcation diagrams generated with this new approximate method compare well to Monte Carlo simulations.
Millman, D.R., King, P.I., Maple, R.C., Beran, P.S., and Chilton, L.K.. "Estimating the probability of failure of a nonlinear aeroelastic system" Journal of Aircraft. 43
(2).
MAR-APR 2006.
pp. 504--516.
A limit-cycle oscillation (LCO) can be characterized by a subcritical or supercritical bifurcation, and bifurcations are shown to be discontinuities in the stochastic domain. The traditional polynomial-chaos-expansion method, which is a stochastic projection method, is too inefficient for estimating the LCO response surface because of the discontinuities associated with bifurcations. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. A Monte Carlo simulation (MCS) is performed on this response surface to estimate the probability density function (PDF) of the LCO response. The stochastic projection method via B-splines is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. A probability of failure based upon certain failure criteria can then be computed from the estimated PDF. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations.