An Improved Higher-Order Time Integration Algorithm for Structural Dynamics

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 [t k , t k + h] where h stands for the size of a time step is divided into two sub-stepsA 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 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.

Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

“…The burden due to the expansion of matrix dimension limits the use of this type of methods in structural dynamics. Also, the differential quadrature method (DQM) [6][7][8][9] and the weighted residual method (WRM) [10][11][12][13] were also frequently utilized in the construction of higher-order methods. In general, these methods can reach up to (2n-1)th-order accuracy for dissipative schemes and (2n)th-order accuracy for non-dissipative schemes, wherein n is the number of sampling grid points per step.…”