Stability of the Bathe implicit time integration methods in the presence of physical damping

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 ρ_∞-Bathe and β₁/β₂- 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 β₁/β₂-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.