A critical review/look at “Optimal implicit single-step time integration methods with equivalence to the second-order-type linear multistep methods for structural dynamics: Accuracy analysis based on an analytical framework”

Abstract

A critical look and review of the so-called generalized single-step time integration method by Zhang (CMAME, 418(2024), 116503) is proved and demonstrated to be not new, but identical to and within the existing GS4-II computational framework. The following are addressed: (1) Firstly, it is claimed that 16 parameters were introduced (somewhat misleading as evident in what follows) to obtain a more generalized single-step mathematical formulation. We show that 4 conditions are made redundant with minimum consistency requirements, and thus, the framework is not new and is identical to the original version of the GS4-II computational framework with 12 parameters. (2) Then, the overshooting behavior is revisited, and the analysis, missteps, and information are clarified and corrected in this paper, which is significant. (3) Next, the time shift phenomenon is also revisited to show the recovery of the order of time accuracy in the acceleration, which is misunderstood in much of the existing literature. (4) Lastly, each design in the so-called newly proposed schemes already exists and is found in the GS4-II computational framework. In particular, via GS4-II we additionally prove and demonstrate that the so-called “Optimal Equivalent Single-step with Single parameter (OESS)” scheme by Zhang (CMAME, 418(2024), 116503) is nothing but identical to the existing Three-Parameters Optimal/Generalized-$\alpha$ method within the GS4-II framework for physically undamped problems. Furthermore, it is noteworthy to point out that also within the GS4-II framework, for physically damped problems, U0/U0${}^{\ast }$, TPO/G-$\alpha$, and OESS all share the same undesired overshooting deficiency in comparison to V0/V0${}^{\ast }$. Numerical examples validate the issues identified about the accuracy and overshooting analysis.