Optimization of a Class of n-Sub-Step Time Integration Methods for Structural Dynamics

Abstract

This paper develops a family of optimized [Formula: see text]-sub-step time integration methods for structural dynamics, in which the generalized trapezoidal rule is used in the first [Formula: see text] sub-steps, and the last sub-step employs [Formula: see text]-point backward difference formula. The proposed methods can achieve second-order accuracy and unconditional stability, and their degree of numerical dissipation can range from zero to one. Also, the proposed methods can achieve the identical effective stiffness matrices for all sub-steps, reducing computational costs in the analysis of linear systems. Using the spectral analysis, optimized algorithmic parameters are presented, ensuring that the proposed methods can accurately calculate different types of dynamic problems such as wave propagation, stiff and nonlinear systems. Besides, with the increase in the number of sub-steps, the accuracy of the proposed methods can be enhanced without extra workload compared with single-step methods. Numerical experiments show that the proposed methods perform better in different dynamic systems.

A three-sub-step composite method for the analysis of rigid body rotations with Euler parameters

“…To flexibly control the degree of high-frequency dissipation, adopting similar ideas, the TR and backward interpolation formula (BIF) were used together to develop composite methods. Among this type of controllably dissipative composite methods [37][38][39][40][41], the three-sub-step TR-TR-BIF [37] proposed by the present authors effectively fills the contradiction in accuracy, dissipation, stability, and efficiency. The TR-TR-BIF [37] was originally developed for solving structural dynamic problems, then it was applied in the analysis of seismic responses [42,43] and multibody dynamics without rotation [44].…”

“…In recent decades, the composite methods [32][33][34][35][36][37][38][39][40][41][42][43][44][45] based on the multi-sub-step concept have attracted wide attention in the field of structural dynamics. This type of method was birthed in the research of Bank et al [32], in which the TR and the three-point BDF were combined in a time element for analyzing the transient responses of silicon devices and circuits.…”

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

“…Another second-order three-sub-step implicit method with controllable numerical dissipation was also proposed in Reference 31, but it is algebraically identical to the earlier published 𝜌 ∞ -TTBDF 28 method, as shown in Appendix A. Recently, Ji and Xing 32 analyze an implicit family of composite n-sub-step algorithms with equal sub-step sizes, and the trapezoidal rule and (n + 1)-point backward difference scheme are used in the first (n − 1) sub-steps and last sub-step, respectively. However, the resulting second-order algorithms 32 have been proposed in other literature.…”

“…However, the resulting second-order algorithms 32 have been proposed in other literature. 21,28,33 For example, the second-order conservative sub-step algorithms 32 is essentially the trapezoidal rule; the two(three)-sub-step schemes for solving nonlinear and stiff problems have already been developed in the literature; 21,28 and entirely new contributions in Reference 32 are to develop some first-order accurate schemes for solving wave propagation problems.…”

“…Recently, Ji and Xing 32 analyze an implicit family of composite $$$ n $$$‐sub‐step algorithms with equal sub‐step sizes, and the trapezoidal rule and $$$ \left(n+1\right) $$$‐point backward difference scheme are used in the first $$$ \left(n-1\right) $$$ sub‐steps and last sub‐step, respectively. However, the resulting second‐order algorithms 32 have been proposed in other literature 21,28,33 . For example, the second‐order conservative sub‐step algorithms 32 is essentially the trapezoidal rule; the two(three)‐sub‐step schemes for solving nonlinear and stiff problems have already been developed in the literature; 21,28 and entirely new contributions in Reference 32 are to develop some first‐order accurate schemes for solving wave propagation problems.…”