An improved higher-order time integration algorithm for structural dynamics

Based on the weighted residual method, a single-step time integration algorithm with higher-order accuracy and unconditional stability has been proposed, which is superior to the second-order accurate algorithms in tracking long-term dynamics. For improving such a higher-order accurate algorithm, this paper proposes a two sub-step higher-order algorithm with unconditional stability and controllable dissipation. In the proposed algorithm, a time step interval [t_k, t_k + h] where h stands for the size of a time step is divided into two sub-steps [t_k, t_k + γ h] and [t_k + γ h, t_k + h]. A non-dissipative fourth-order algorithm is used in the first sub-step to ensure low-frequency accuracy and a dissipative third-order algorithm is employed in the second sub-step to filter out the contribution of high-frequency modes. Besides, two approaches are used to design the algorithm parameter γ . The first approach determines γ by maximizing low-frequency accuracy and the other determines γ for quickly damping out high-frequency modes. The present algorithm uses ρ_∞ to exactly control the degree of numerical dissipation, and it is third-order accurate when 0 ≤ ρ_∞ < 1 and fourth-order accurate when ρ_∞ = 1. Furthermore, the proposed algorithm is self-starting and easy to implement. Some illustrative linear and nonlinear examples are solved to check the performances of the proposed two sub-step higher-order algorithm.