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 引用本文: 高山, 史东华, 郭永新. Hamel 框架下几何精确梁的离散动量守恒律[J]. 力学学报, 2021, 53(6): 1712-1719.
Gao Shan, Shi Donghua, Guo Yongxin. DISCRETE MOMENTUM CONSERVATION LAW OF GEOMETRICALLY EXACT BEAM IN HAMEL'S FRAMEWORK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1712-1719.
 Citation: Gao Shan, Shi Donghua, Guo Yongxin. DISCRETE MOMENTUM CONSERVATION LAW OF GEOMETRICALLY EXACT BEAM IN HAMEL'S FRAMEWORK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1712-1719.

## DISCRETE MOMENTUM CONSERVATION LAW OF GEOMETRICALLY EXACT BEAM IN HAMEL'S FRAMEWORK

• 摘要: Hamel场变分积分子是一种研究场论的数值方法, 可以通过使用活动标架规避几何非线性带来的计算复杂度, 同时数值上具有良好的长时间数值表现和保能动量性质. 本文在一维场论框架下, 以几何精确梁为例, 从理论上探究Hamel场变分积分子的保动量性质. 具体内容包括: 利用活动标架法对几何精确梁建立动力学模型, 通过变分原理得到其动力学方程, 利用其动力学方程及Noether定理得到系统动量守恒律; 将几何精确梁模型离散化, 通过变分原理得到其Hamel场变分积分子, 利用Hamel场变分积分子和离散Noether定理得到离散动量守恒律, 并给出离散动量的一阶近似表达式; Hamel场变分积分子可在计算中利用系统对称性消除系统运动带来的非线性问题, 但此框架中离散对流速度、离散对流 应变及位形均不共点, 而这种错位导致离散动量中出现级数项, 本文对几何精确梁的离散动量与连续形式的关系及其应 用进行了讨论, 并通过算例验证了结论. 上述证明方法也同样适用一般经典场论场景下的Hamel场变分积分子. Hamel场变分积分子的动量守恒为进一步研究其保结构性质提供了参考依据.

Abstract: Hamel's field variational integrators are numerical schemes for classical field theory. It reduces computational cost caused by geometrical nonlinearity and exhibits a long-term energy stability and momentum-preserving property numerically. In the framework of one-dimensional field theory, taking geometrically exact beam as an example, this paper investigates theoretically discrete momentum conservation law of Hamel's field variational integrator. The major studies of this paper include the following aspects: The dynamical model of geometrically exact beam is established by using moving frame methods, dynamical equations of geometrically exact beam are obtained by variational principle, a momentum conservation law is then obtained through its dynamical equations and Noether theorem; For discrete model of geometrically exact beam, a discrete momentum conservation law is given by utilizing Hamel's field variational integrator of geometrically exact beam and discrete Noether theorem, and then the first order approximation of discrete momentum is proposed. Hamel's field variational integrators use system's symmetry to simplify the geometrical nonlinearity. It locates discrete convective velocities, discrete convective strain and configurations at different nodes on the spatial-temporal grid, thus leading to a series term in the expression of discrete momentum. This paper discusses the relation between the expression of continuous and the corresponding discrete one. Analytical and numerical examples are proposed to verify the conclusion. The proposed proof above is also applicable to the case in classical field theory and motivates further investigation of structure-preserving properties of Hamel's field variational integrator.

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