A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics

Abstract

Summary A comprehensive study of A‐stable linear two‐step time integration methods for structural dynamics analysis is presented in this paper. An optimal A‐stable linear two‐step (OALTS) time integration method is revealed with desirable performance on low‐frequency accuracy and high‐frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second‐order equations of structural dynamics; is implicit, A‐stable, and second‐order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.

On the optimization of n-sub-step composite time integration methods

“…In this section, the representative methods in the literature, including the single-step generalized-α method [9] (G-α) and the linear two-step method [33] (LTS) are also considered for comparison. As the employed methods are all implicit, their computational cost is mainly spent on the iterative calculation when used for nonlinear problems, or the matrix factorization for linear problems.…”

“…In the multi-step class, the Dahlquist's barrier certainly works, but in terms of accuracy, the linear twostep method [24,33] is superior to most existing singlestep methods under the same degree of algorithmic dissipation. In this class, BDFs (backward differentiation formulas) [11,16] also represent a widely-used branch, particularly useful for stiff problems owing to the strong algorithmic dissipation.…”

Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods

“…After determining the parameters of the linear multi-step methods, the parameters of the single-step methods are directly obtained by comparing the respective characteristic polynomials. To our knowledge, apart from LMS2 [31,44], the second-order unconditionally stable schemes of the higher-step methods have not been proposed yet.…”

“…These single-step methods can display the identical spectral characteristics of the corresponding linear multi-step methods [45]. However, the linear two-step method with optimal parameters [31,44] can provide better accuracy than most existing single-step methods under the same degree of algorithmic dissipation.…”

“…This paper focuses on second-order accurate and unconditionally stable schemes of the linear multi-step methods. To acquire these accuracy and stability properties, the linear two-step method has been originally studied in [31,32,44]. Seeking higher accuracy, the linear three-, and four-step methods are discussed in this paper.…”