This paper further analyzes three Bathe algorithms (γ-Bathe, β1∕β2-Bathe and ρ∞-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 β1∕β2-Bathe algorithm is algebraically identical to the ρ∞-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 ρ∞-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.

中文翻译：

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波传播问题的三种沐浴算法和实现的进一步评估

本文进一步分析了三种Bathe算法（γ-洗澡，β1∕β2-洗澡和ρ∞-Bathe) 揭示了它们的未知特性。分析首先表明三种Bathe算法可以覆盖梯形规则和后向欧拉公式两种常见的积分方案，并且二阶β1∕β2-Bathe 算法在代数上与ρ∞-沐浴算法。通过对广义两子步 Newmark 算法的表述，证明了普通 Newmark 方法不能被认为是ρ∞-沐浴算法。对于波传播问题，通过同时考虑空间和时间离散化，为三种Bathe算法找到了减少色散误差的最佳Courant-Friedrichs-Lewy（CFL）数，而改进的积分规则用于单元质量和刚度矩阵以减少波传播方向的各向异性。给出了三种Bathe算法的推荐最优算法参数，帮助用户有效解决各种动态和波传播问题。