A two-step time integration method with desirable stability for nonlinear structural dynamics

Time integration methods are frequently used in the analysis of general dynamic systems. Numerical experiments have shown that for nonlinear dynamic systems, some time integration methods that are unconditionally stable for linear systems may fail. Inspired by this phenomenon, this work first proposes a parameter spectral analysis theory to examine the stability of time integration methods as applied to nonlinear systems. The trapezoidal rule is found to be unstable when stiffness softens, which validates the effectiveness of the present theory. Furthermore, it can be found that the present theory is consistent with the BN-stability theory. Next, according to this parameter spectral analysis theory, a two-step time integration method possessing desirable stability for both linear and nonlinear systems is designed, which has second-order accuracy for undamped systems, first-order accuracy for damped systems, and controllable dissipation. The superiority in accuracy and stability of the present two-step method is illustrated by comparing the present method with the trapezoidal rule, the OALTS method and the TGCD method.