POD-RBF hyper-reduction method for fast finite element analysis of nonlinear dynamic problems

Abstract

This paper proposes a new hyper-reduction method for fast finite element analysis of nonlinear dynamic problems using proper orthogonal decomposition (POD) and radial basis function (RBF) interpolation. In the offline stage, displacement and internal force snapshots are collected from full-order FE simulations of nonlinear dynamic problems with training load cases. POD basis vectors are extracted from the displacement snapshots using the singular value decomposition (SVD). RBF coefficients for the internal force snapshots are also computed in the offline stage. The proposed POD-RBF hyper-reduction method efficiently estimates the reduced internal force vectors and the reduced tangent stiffness matrices using RBF interpolation with respect to reduced generalized coordinates. A snapshot selection strategy combining K-means clustering and greedy sampling algorithms is used to reduce the size of solution snapshots, which further enhances the efficiency of the present method. Numerical results show that the POD-RBF hyper-reduction method can be efficiently and effectively used to quickly solve nonlinear dynamic problems in a reduced-order space.