Further assessment of three Bathe algorithms and implementations for wave propagation problems. (English) Zbl 1535.65102
Summary: This paper further analyzes three Bathe algorithms \(( \gamma \)-Bathe, \( \beta_1/ \beta_2\)-Bathe and \(\rho_\infty \)-Bathe) with their unknown properties revealed. The analysis shows firstly that three Bathe algorithms can cover two common integration schemes, trapezoidal rule and backward Euler formula, and that the second-order \(\beta_1/ \beta_2\)-Bathe algorithm is algebraically identical to the \(\rho_\infty \)-Bathe algorithm. Via formulation of the generalized two-sub-step Newmark algorithm, it is shown that the common Newmark method cannot be considered as a special case of the \(\rho_\infty \)-Bathe algorithm. For wave propagation problems, optimal Courant-Friedrichs-Lewy (CFL) numbers for reducing dispersion errors are found for the three Bathe algorithms by considering spatial and temporal discretizations simultaneously, while the modified integration rules are used for the element mass and stiffness matrices to reduce the anisotropy in wave propagating directions. The recommended optimal algorithmic parameters are given for the three Bathe algorithms to help users effectively solve various dynamic and wave propagation problems.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 70J35 | Forced motions in linear vibration theory |
Keywords:
dispersion error; Newmark method; time integration; trapezoidal rule; wave propagation; Bathe’s methodReferences:
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