This article presents a proper generalized decomposition (PGD) based Padé 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 Padé 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.

中文翻译：

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用于随机频率响应分析的基于适当广义分解的 Padé 近似

本文提出了一种基于适当广义分解 (PGD) 的 Padé 近似，用于有效分析随机频率响应。由于随机响应相对于输入不确定性的高度非线性，利用多项式混沌的经典随机伽辽金 (SG) 方法在共振附近表现出缓慢收敛。此外，SG 方法的维数是确定性和随机近似空间的乘积，因此在带状频率范围内的分辨率在计算上是昂贵的，甚至是令人望而却步的。在这项研究中，为了解决这些问题，PGD 首先生成随机频率方程的解，作为确定性和随机分量的分离表示。对于带状频率范围计算，确定性向量被用作结合奇异值分解的简化基础。随后，基于 PGD 解应用 Padé 近似，随机频率响应由有理函数表示。通过各种数值研究，表明与 SG 方法相比，所提出的框架不仅提高了共振附近的精度，还提高了计算效率。