An Improved Sub-Step Composite Time Integration Formulation With Enhanced Performance on Linear and Nonlinear Dynamics

Abstract

An improved time integration scheme is proposed for linear and nonlinear dynamics. The proposed scheme has two free parameters which control numerical dissipation and accuracy effectively. Basic properties including spectral stability, algorithmic accuracy, algorithmic damping, period elongation and overshooting behavior are investigated. The influences of algorithmic parameters on these properties are quantified. The effectiveness of the proposed scheme for linear and nonlinear dynamics is evaluated through some numerical examples. Analytical and numerical results demonstrate that the proposed scheme has the following significant characteristics: (1) desirable accuracy can be obtained for various linear and nonlinear problems, when compared with other effective schemes; (2) for nonlinear problems, new scheme also shows good performance; (3) the proposed scheme has simple formulation and good compatibility for various dynamic problems, and thus, is a promising candidate for practical analysis.

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

“…It is difficult in Equation ( 26) to obtain the analytical expression of 𝛾 1 in terms of 𝜌 ∞ 3 . Figure 1A depicts the unconditionally stable region of TSSC, fulfilling Equation (26). For each given 𝜌 ∞ 3 ∈ (−1, 1), Equation ( 26) provides the upper and lower…”

“…As reviewed in Section 1, a noticeable number of studies 11,16,19,[23][24][25][26][27][28][29][30][31]33 have reported some composite three-sub-step implicit methods. However, there are some identities among them.…”

“…𝜌 ∞ -TTBDF method. 28 Therefore, all composite three-sub-step implicit methods compared in this study are TTBDF, 24 TRNewton3, 16 two improved Wen schemes, 26,27 𝜌 ∞ -TTBDF, 28 the Yuan scheme, 30 MSSTH3, 33 and TR-TS. 34 Apart from these, the third-order accurate 𝜌 ∞ -Bathe method 43 is also compared in this paper, although it only uses two sub-steps.…”

“…(a) The algorithmic parameters used in these plots are highly recommended in their original studies. For example, 𝜃 = 3∕4, (p = 0.33, q = 0.28), and (𝛼 = 0.5, 𝛽 = 0.28) are suggested for TTBDF, 24 the Wen scheme, 26 and the Yuan scheme, 30 respectively. (b) The Yuan scheme 30 is conditionally stable but the minimum period errors are provided.…”

“…In detail, the study 27 further imposes an additional condition, namely q = p∕ √ 2, to make the effective stiffness matrices within the first two sub-steps identical, while the general cases of p and q are analyzed in the literature. 26 Note that these composite three-sub-step implicit methods reviewed above are L-stable, namely achieving the most level of numerical dissipation in the high-frequency range. Li et al 28 reported at the first time two composite three-sub-step implicit methods with controllable numerical dissipation, which are named as 𝜌 ∞ -TTBDF in this paper for convenience.…”