A new concept for initialization of single-step time integration algorithms: Eliminating pesky high-frequency overshooting behavior

In this paper, we propose a new concept of a modified initialization for three-value single-step time integration algorithms in structural dynamics problems. High-frequency spurious modes have long been recognized as a primary cause of undesirable overshooting in numerical time integration. This undesired overshooting behavior has attracted sustained research interest over the past several decades, particularly in the context of linear multi-step and equivalent single-step methods, with the aim of mitigating such adverse effects. Unfortunately, these deleterious aspects have been an issue for the research community at large. We present a novel and simple methodology to completely circumvent and resolve the issue of overshooting that can be straightforwardly implemented in research and commercial software. The primary contribution is a modified initialization procedure that is necessary and completely eliminates overshooting under arbitrary conditions, and preserves the global second-order time accuracy. The theoretical framework and design principles are demonstrated under the umbrella of the Generalized Single-Step Single-Solve computational framework for second-order time-dependent problems (GS4-II), which encompasses several new and improved designs and includes subsets such as HHT-α, TPO/G-α, WBZ, etc. We introduce a novel conceptual framework and a straightforward implementation architecture for the new concept of modified initialization data with clear mathematical and physical interpretations, which fully eliminates overshooting in these schemes for all application scenarios. Numerical analyses and various illustrative examples confirm that the proposed approach completely eliminates overshooting across a broad spectrum of three-value single-step methods and preserves second-order time accuracy, marking a significant advance in single-step time integration algorithms.