Uncertainty Quantification with a B-Spline Stochastic Projection

Abstract

Presented is a new stochastic algorithm for computing the probability density functions and estimating bifurcations of nonlinear equations of motion whose system parameters are characterized by a Gaussian distribution. Polynomial and Fourier chaos expansions, which are spectral methods, have been used successfully to propagate parametric uncertainties in nonlinear systems. However, it is shown that bifurcations in the time domain are manifested as discontinuities in the stochastic domain, which are problematic for solution with these spectral approaches. Because of this, a new algorithm is introduced based on the stochastic projection method but employing a multivariate B spline. Samples are obtained by choosing nodes on the stochastic axes. These samples are used to build an interpolating function in the stochastic domain. Monte Carlo simulations are then very efficiently performed on this interpolating function to estimate probability density functions of a response. The results from this nonintrusive and non-Galerkin approach are in excellent agreement with Monte Carlo simulations of the governing equations, but at a computational cost 2 orders of magnitude less than a traditional Monte Carlo approach. The probability density functions obtained from the stochastic algorithm provide a rapid estimate of the probability of failure for a nonlinear pitch and plunge airfoiL.