An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems

Abstract

Purpose

The purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems.

Design/methodology/approach

The algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order accurate and conditionally stable. Its numerical stability, dissipation and dispersion are analyzed for the dynamic single-degree-of-freedom equation. The stability of the multi-degrees-of-freedom non-proportional damping system can be evaluated directly by the stability theory on ordinary differential equation.

Findings

The performance of the proposed algorithm is demonstrated by several numerical examples including the linear single-degree-of-freedom problem, non-linear two-degree-of-freedom problem, wave propagation problem in two-dimensional layer and seismic elastoplastic analysis of high-rise structure.

Originality/value

A new single-step second-order accurate explicit time integration algorithm is proposed to solve the linear and non-linear dynamic finite element equations. The algorithm has advantages on the numerical stability and accuracy over the existing modified central difference method and Chung-Lee method though the theory and numerical analyses.

Citation

Zhao, M., Li, H., Cao, S. and Du, X. (2019), "An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems", Engineering Computations, Vol. 36 No. 1, pp. 161-177. https://doi.org/10.1108/EC-07-2018-0312