Predicting Uncertainty Propagation in a Highly Nonlinear System with a Stochastic Projection Method

Uncertainty quantification of limit-cycle oscillations

“…Clearly, the expansions fail in reproducing the baseline result (for n = 0). Pettit and Beran [30] and Millman et al [26] describe the computational problems associated with the WHe expansions in greater detail, but point to the failure of fixed-order WHe expansions at large-time to capture an increasing level of nonlinearity in the dependence of x i (t, n) on n.…”

“…In the B-spline results reported below, for which the standard deviations of the physical parameters are assumed to be fairly large, the corresponding rvs are confined to values within [À4, 4]. This sub-domain captures 99.99% of the possible responses [26].…”

“…Following Millman et al [26], splines can be used to approximate these kinds of projections. An important property of the multivariate B-splines they choose is that the splines are a compact support basis; i.e., the influence of any particular B-spline coefficient extends over a few intervals [7].…”

“…Attar and Dowell [1] employ MCS and a response surface method to map the output characteristics of a delta wing (as measured by the root-mean square of the wing tip displacement) to two input parameters considered variable: thickness and modulus of elasticity. As treated by Millman et al [26], Attar and Dowell gave special attention in the response surface mapping to the modeling of the bifurcation leading to LCO. Pettit and Beran [32] continued study of the airfoil, and related oscillatory phenomena, using two-dimensional Wiener-Haar expansions.…”

“…In their work, one structural parameter, the cubic stiffness coefficient, was considered random. Millman et al [26] alternatively proposed a stochastic projection method derived from B-splines and apply the technique to representing the probabilistic response of the airfoil in terms of a single output variable, the peak pitch angle. Two input variables were considered random, the cubic stiffness coefficient and the initial angle of attack.…”