Polynomial chaos expansion in structural dynamics: Accelerating the convergence of the first two statistical moment sequences
Résumé
Polynomial chaos solution for the frequency response of linear non-proportionally damped dynamic systems has been considered. It has been observed that for lightly damped systems the convergence of the solution can be very poor in the vicinity of the deterministic resonance frequencies. To address this, Aitkenós transformation and its generalizations are suggested. The proposed approach is successfully applied to the sequences defined by the first two moments of the responses, and this process significantly accelerates the polynomial chaos convergence. In particular, a 2-dof system with respectively 1 and 2 parameter uncertainties has been studied. The first two moments of the frequency response were calculated by Monte Carlo simulation, polynomial chaos expansion and Aitkenós transformation of the polynomial chaos expansion. Whereas 200 polynomials are required to have a good agreement with Monte Carlo results around the deterministic eigenfrequencies, less than 50 polynomials transformed by the Aitkenós method are enough. This latter result is improved if a generalization of Aitkenós method (recursive Aitkenós transformation, Shankós transformation) is applied. With the proposed convergence acceleration, polynomial chaos may be reconsidered as an efficient method to estimate the first two moments of a random dynamic response.