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

TL;DR

This paper introduces a high-order implicit time integration algorithm for linear and nonlinear dynamics that improves computational efficiency and accuracy, especially in nonlinear problems, while controlling numerical dissipation.

Contribution

The authors present a new algorithm that eliminates mass matrix factorization, directly computes acceleration with high order accuracy, and handles nonlinearity via numerical integration within each time step.

Findings

The algorithm achieves high accuracy for both linear and nonlinear problems.

It effectively suppresses high-frequency oscillations in wave propagation simulations.

Numerical results confirm the method's superior performance over second-order schemes.

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. numerical dissipation in the algorithm is controlled by a user-specified parameter, 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. The proposed algorithm 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. 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 noticably better at suppressing spurious high-frequency oscillations and producing reliable and useable acceleration responses.