A high-order implicit time integration method for linear and nonlinear dynamics with efficient computation of accelerations

Abstract

An algorithm for a family of self-starting high-order implicit time integration schemes with controllable numerical dissipation is proposed for both linear and nonlinear transient problems. This work builds on the previous works of the authors on elastodynamics by presenting a new algorithm that eliminates the need for factorization of the mass matrix providing benefit for the solution of nonlinear problems. The improved algorithm directly obtains the acceleration at the same order of accuracy of the displacement and velocity using vector operations (without additional equation solutions). The nonlinearity is handled by numerical integration within a time step to achieve the desired order of accuracy. The new algorithm fully retains the desirable features of the previous works: 1. The order of accuracy is not affected by the presence of external forces and physical damping; 2. The amount of numerical dissipation in the algorithm is controlled by a user-specified parameter ${\rho }_{\infty }$, leading to schemes ranging from perfectly nondissipative $A$-stable to $L$-stable; 3. The effective stiffness matrix is a linear combination of the mass, damping, and stiffness matrices as in the trapezoidal rule, leading to high efficiency for large-scale problems. The proposed algorithm, with its elegance and computational advantages, is shown to replicate the numerical results demonstrated on linear problems in previous works. Additional numerical examples of linear and nonlinear vibration and wave propagation are presented herein. Notably, the proposed algorithms show the same convergence rates for nonlinear problems as linear problems, and very high accuracy. It was found that second-order time integration methods commonly used in commercial software produce significantly polluted acceleration responses for a common class of wave propagation problems. The high-order time integration schemes presented here perform noticeably better at suppressing spurious high-frequency oscillations and producing reliable and usable acceleration responses. The source code written in MATLAB is available for download at: https://github.com/ChongminSong/HighOrderTimeIngt_PartialFraction.git.