Following the introduction of the standard Bathe implicit time integration scheme, there have been many research efforts in the development of time integration methods employing multi-step composite strategies. However, the stability analysis of these methods has been primarily conducted in the absence of physical damping, under the assumption that physical damping would render the analysis results more stable. This study demonstrates that physical damping can, in fact, impair the stability of a time integration. We consider the -Bathe and - Bathe methods, and confirm that the use of the previously reported parameters is within the stable range in the presence of physical damping. For the -Bathe method, a semi-analytic stability analysis provides an expanded range of applicable parameters. The results from the theoretical analysis are illustrated through simple numerical examples that could be used as benchmark problems to check the stability of a time integration method in the presence of physical damping.

中文翻译：

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存在物理阻尼时 Bathe 隐式时间积分方法的稳定性

随着标准 Bathe 隐式时间积分方案的推出，人们在开发采用多步复合策略的时间积分方法方面做出了许多研究工作。然而，这些方法的稳定性分析主要是在没有物理阻尼的情况下进行的，假设物理阻尼会使分析结果更加稳定。这项研究表明，物理阻尼实际上会损害时间积分的稳定性。我们考虑 -Bathe 和 -Bathe 方法，并确认先前报告的参数的使用在存在物理阻尼的情况下处于稳定范围内。对于 -Bathe 方法，半解析稳定性分析提供了更广泛的适用参数范围。理论分析的结果通过简单的数值例子进行说明，这些例子可以用作基准问题来检查存在物理阻尼的情况下时间积分方法的稳定性。