A proper generalized decomposition based Pade approximant for stochastic frequency response analysis

Abstract

This article presents a proper generalized decomposition (PGD) based Pade approximant for efficient analysis of the stochastic frequency response. Due to the high nonlinearity of the stochastic response with respect to the input uncertainties, the classical stochastic Galerkin (SG) method utilizing polynomial chaos exhibits slow convergence near the resonance. Furthermore, the dimension of the SG method is the product of deterministic and stochastic approximation spaces, and hence resolution over a banded frequency range is computationally expensive or even prohibitive. In this study, to tackle these problems, the PGD first generates the solution of stochastic frequency equations as a separated representation of deterministic and stochastic components. For the banded frequency range computations, the deterministic vectors are exploited as a reduced basis in conjunction with singular value decomposition. Subsequently, the Pade approximant is applied based on the PGD solution, and the stochastic frequency response is represented by a rational function. Through various numerical studies, it is demonstrated that the proposed framework improves not only the accuracy in the vicinity of resonance but also the computational efficiency, compared with the SG method.