A time-domain POD approach based on numerical implicit and explicit Green’s functions for 3D elastodynamic analysis

Abstract

This paper is concerned with the development of time integration methods based on numerical Green’s functions applied in conjunction with the proper orthogonal decomposition (POD) technique to solve three-dimensional elastodynamic problems discretized by the FEM. The convolution integral in the velocity expression of these methods is firstly carried out through integration by parts, and then approximated numerically, resulting in a simpler and more accurate convolution than that reported in the literature. To compute the solution samples, required by the POD technique, the Implicit Green’s functions Approach (ImGA)-Newmark method rewritten in terms of the ultimate spectral radius is employed. The proposed ImGA scheme is truly self-starting and easy to implement with just one free parameter. To solve the reduced-order model, a new highly accurate time integration scheme based on the Explicit Green’s functions Approach (ExGA)-Runge–Kutta method is proposed, considering the Gauss–Lobatto-Legendre (GLL) points to perform both the time integration of the Green’s functions and the approximation of the convolution integrals. To verify the accuracy and efficiency of the proposed formulation, the numerical results are compared with those from the full-order model. Moreover, undamped solution samples are utilized to build a reduced-order model that can take into account a Rayleigh damping matrix.