Proper generalized decomposition for multi-dimensional frequency response analysis

Abstract

In structural dynamics, frequency response is a steady-state solution of systems under harmonic force, providing crucial information about vibration characteristics. Traditionally, numerical simulation of the frequency response has been mainly conducted based on deterministic and linearity assumptions, and the finite element method (FEM) has been widely applied for spatial discretization. However, despite the remarkable advances in computer and simulation technologies, still difficult challenges in predicting actual responses are that many structural systems are subjected to various uncertainty and nonlinearity. When integrating these two properties into the frequency response analysis, multi-dimensional problems allowing additional dimensions as well as the existing spatial degrees of freedom (DOFs) and frequency are formulated. In these cases, the total dimension is a tensor product of each dimension, and computational cost can easily become intractable for high-dimensional problems due to the curse of dimensionality. The objective of this study is to develop an accurate and computationally efficient algorithm for multi-dimensional frequency response problems by utilizing the proper generalized decomposition (PGD). PGD relies on the concept of separation of variables, and the solutions of the governing equations are approximated as a low-rank separated representation. The progressive Galerkin approach is adopted to formulate subproblems defined in each dimension. Fixed-point iteration is then applied by solving subproblems in which other variables are fixed, and the update problem is additionally considered to improve the accuracy of the PGD approximation. In the case of stochastic FEM, the numerical model is characterized by random variables, and two strategies are proposed depending on the approximation of the stochastic space. The first one is based on the Padé approximant. Although the polynomial chaos (PC) basis, which utilizes orthogonal polynomials for random variables, is most widely used in the field of uncertainty quantification, it cannot reflect the non-smooth behavior of the frequency response around the resonance region and yields inaccurate results. To tackle this issue, this study first constructs the solution as a separated representation of spatial modes and stochastic coefficients expanded by PC basis. The Padé approximant is applied based on the PGD solution to reflect the non-smooth behavior, and the frequency response is finally represented as a surrogate model in the form of a rational function. Second, the collocation technique is used to discretize the dimensions of uncertainty and frequency. Unlike the PC basis, which is a global function for random variables, the collocation technique defines values only at collocation points due to the Dirac delta property. In the offline stage, PGD represents the solution as a low-rank sum of spatial modes and parametric functions. The spatial dimension is related to the DOFs discretized by the finite elements, and parametric functions for frequency and random variables are defined at the collocation points. The online stage utilizes the spatial modes computed at the offline stage as a reduced-order basis and generates a reduced-order model through Galerkin projection into a low-dimensional subspace. For both two approaches, the solutions for any random variables are then easily evaluated by solving the surrogate or reduced-order models, and the response statistics are estimated based on the Monte-Carlo simulation. In the case of nonlinearity, the harmonic balance method (HBM) approximates nonlinear frequency response utilizing a truncated Fourier series, resulting in a set of nonlinear algebraic equations formulated in the frequency domain. In this case, dimensions for the Fourier series are considered in addition to spatial degrees of freedom and frequency. PGD exploits the low-dimensional subspace of the HBM and constructs the solution by a separated representation of the spatial and harmonic components. During the continuation, the spatial modes acquired at the previous computation are utilized as a reduced-order basis. The numerical studies demonstrate that the proposed frameworks not only allow significant computational savings compared to the conventional methods, but also accurately reflect the complex stochastic and nonlinear behavior without prior information on the response characteristics.