Quantifying initial condition and parametric uncertainties in a nonlinear aeroelastic system with an efficient stochastic algorithm

Computational fluid dynamics (CFD) methods have been coupled with structural solvers to provide accurate predictions of limit cycle oscillations (LCO). There is, however, a growing interest in understanding how uncertainties in flight conditions and structural parameters affect the character of an LCO response, leading to failure of an aeroelastic system. Uncertainty quantification of a stochastic system (parametric uncertainty) with stochastic inputs (initial condition uncertainty) has traditionally been analyzed with Monte Carlo simulations (MCS). Probability density functions (PDF) of the LCO response are obtained from the MCS to estimate the probability of failure. A CFD solution, however, can take days to weeks to obtain a single response, making the MCS method intractable for large problems. A candidate approach to efficiently estimate the PDF of an LCO response is the stochastic projection method. The classical stochastic projection method is a polynomial chaos expansion (PCE). The PCE approximates the response in the stochastic domain through a Fourier type expansion of the Wiener-Hermite polynomials. An LCO response can be characterized as a subcritical or supercritical bifurcation, and bifurcations are shown to be discontinuities in the stochastic domain. The PCE method, then, would be too inefficient for estimating the LCO response surface. 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. An MCS is performed on this response surface to estimate the PDF of the LCO response. The probability of failure is then computed from the PDF. 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. 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.