Vol.:(0123456789)1 3Engineering with Computers https://doi.org/10.1007/s00366-022-01743-1 ORIGINAL ARTICLE A unified scheme for nonlinear dynamic direct time integration methods: a comparative study on the application of multi‑point methods Mojtaba Shahraki1 · Farzad Shahabian1 · Ali Maghami1 Received: 31 January 2022 / Accepted: 14 September 2022 © The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2022 Abstract In this article, we first present a unified scheme to apply nonlinear dynamic time integration methods. The unified scheme covers many existing time integration methods as exceptional cases. This paper has investigated time integration methods, including the Newmark, Wilson, Houbolt, and ????∞-Bathe method. We then implement the multi-point methods as the nonlinear solution schemes along with the direct time integration methods in nonlinear dynamic analysis. Also, a unified scheme for applying single-point and multi-point methods is presented. Finally, we demonstrate with numerical examples that the unified scheme provides a framework for comparing direct time integration methods. We also investigate the performance of multi-point methods as nonlinear solution methods in detail.

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