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# An Improved Higher-Order Time Integration Algorithm for Structural Dynamics

Yi Ji1,2, Yufeng Xing1,*
1 Institute of Solid Mechanics, Beihang University, Beijing, 100083, China
2 Shen Yuan Honors College, Beihang University, Beijing, 100083, China
* Corresponding Author: Yufeng Xing. Email:

Computer Modeling in Engineering & Sciences 2021, 126(2), 549-575. https://doi.org/10.32604/cmes.2021.014244

Received 13 September 2020; Accepted 14 October 2020; Issue published 21 January 2021

### Abstract

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 [tk, tk + h] where h stands for the size of a time step is divided into two sub-steps [tk, tk + γh] and [tk + γh, tk + h]. A non-dissipative fourth-order algorithm is used in the rst sub-step to ensure low-frequency accuracy and a dissipative third-order algorithm is employed in the second sub-step to lter out the contribution of high-frequency modes. Besides, two approaches are used to design the algorithm parameter γ. The rst approach determines γ by maximizing low-frequency accuracy and the other determines γ for quickly damping out highfrequency 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.

### Keywords

Time integration algorithm; two-sub-step; higher-order accuracy; controllable dissipation; unconditional stability

Ji, Y., Xing, Y. (2021). An Improved Higher-Order Time Integration Algorithm for Structural Dynamics. CMES-Computer Modeling in Engineering & Sciences, 126(2), 549–575.

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