Neumann accelerations of generalized polynomial chaos expansions for larger dynamic systems

Abstract

This work proposes a method for accelerating generalized polynomial chaos (gPC) expansions, using Neumann series, for uncertainty quantification of complex structures with uncertain damping and spring parameters. Often, classical Monte Carlo simulations are used for uncertainty quantification, however, for larger systems and frequency sweeps this method becomes very computationally expensive. Neumann series have been used with Monte Carlo simulations to decrease computation times, however, the Neumann series does not always converge. More recently gPC expansions have been used for uncertainty quantification. gPC is a stochastic spectral method that develops a surrogate model, whose coefficients can be used to calculate statistical moments. gPC requires sampling the response at predetermined quadrature points, which may become expensive for large systems. This study proposes a new approach where Neumann series are used to calculate the response at gPC quadrature points, in order to accelerate gPC computation times. The validity of this method is shown using large, viscously damped systems. Damping and stiffening elements fully connect degrees of freedom to each other and to ground. First the damping elements are taken as the uncertain parameter, then stiffening elements are taken as the uncertain parameter, and frequency dependent responses such as power dissipated are calculated.