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Enhanced studies on the composite sub-step algorithm for structural dynamics: The Bathe-like algorithm

机译：结构动力学复合子步算法的增强研究：类Bathe算法

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摘要

The Bathe algorithm is superior to the trapezoidal rule in solving nonlinear problems involving large deformations and long-time durations. Generally, the parameter γ = 2 - √2 is highly recommended due to its optimal numerical properties. This paper further studies this implicit composite sub-step algorithm and thus presents a class of the Bathe-like algorithm. It not only gives a novel family of composite algorithms whose numerical properties are the exactly same as the original Bathe algorithm with γ = 2 - √2, but also provides the generalized alternative to the original Bathe algorithm with any γ. In this study, it has been shown that the Bathe-like algorithm, including the original Bathe algorithm, can reduce to two common single-step algorithms: the trapezoidal rule and the backward Euler formula. Besides, a new parameter called the algorithmic mode truncation factor is firstly defined to describe the numerical property of the Bathe-like algorithm and it can estimate which modes to be damped out. Finally, numerical experiments are provided to show the superiority of the Bathe-like algorithm over some existing methods. For example, the novel Bathe-like algorithms are superior to the original Bathe algorithm when solving the highly nonlinear pendulum.
机译：Bathe算法在解决涉及大变形和长时间的非线性问题方面优于梯形法则。通常，由于其最佳的数值特性，强烈建议使用参数γ= 2-√2。本文进一步研究了这种隐式复合子步算法，从而提出了一类类似于Bathe的算法。它不仅提供了一个新颖的复合算法系列，其数值属性与原始的带有γ= 2-√2的Bathe算法完全相同，而且还为具有任何γ的原始Bathe算法提供了广义的替代方案。在这项研究中，已经证明，包括原始Bathe算法在内的Bathe-like算法可以简化为两个常见的单步算法：梯形规则和后向Euler公式。此外，首先定义了一个称为算法模式截断因子的新参数，以描述类Bathe算法的数值特性，并且可以估计要衰减的模式。最后，通过数值实验证明了类似Bathe的算法在某些现有方法上的优越性。例如，当求解高度非线性摆时，新颖的类Bathe算法优于原始的Bathe算法。

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