Random dynamical system in time domain: A POD-PC model

Abstract

Propagating uncertainties through mechanical systems has been widely studied for the last thirty years. In particular metamodels based on polynomial chaos expansion (PCE) have been successfully developed in the context of both intrusive and non-intrusive methods. However, modelling random dynamical systems is much more challenging and requires increasing computational resources when the time integration becomes longer. Therefore separating the time aspect of the dynamical response and the random contributions is an appealing approach, which has been used in this paper. Thus, a non-intrusive method is proposed by associating a proper orthogonal decomposition (POD) and a PCE. The POD-PC model was applied on three examples. On two examples, the method was very efficient not only in calculating the first two statistical moments, but also in estimating the responses corresponding to several samples of the random parameters. As to the last example, the Kraichnan-Orszag three-mode problem, the model was able to estimate well the evolution of the first two statistical moments for the first part of the time duration, but a noticeable discrepancy occurred for the remaining part. It seems that the model is more appropriate to estimate the transient response of a random dynamical system than the steady-state response.