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 引用本文: 高效伟, 徐兵兵, 吕军, 彭海峰. 自由单元法及其在结构分析中的应用[J]. 力学学报, 2019, 51(3): 703-713.
Xiaowei Gao, Bingbing Xu, Jun Lü, Haifeng Peng. FREE ELEMENT METHOD AND ITS APPLICATION IN STRUCTURAL ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 703-713.
 Citation: Xiaowei Gao, Bingbing Xu, Jun Lü, Haifeng Peng. FREE ELEMENT METHOD AND ITS APPLICATION IN STRUCTURAL ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 703-713.

## FREE ELEMENT METHOD AND ITS APPLICATION IN STRUCTURAL ANALYSIS

• 摘要: 通过吸收有限元与无网格法的优点,提出了一种新的数值方法------自由单元法.此方法在离散方面,采用有限元法中的等参单元,表征几何形状和进行物理量的插值;在算法方面,采用单元配点技术,逐点产生系统方程.主要特点是,在每个配置点只需要一个和周围自由选择的节点而形成的一个独立的等参单元,因而不需要考虑物理量在单元之间的相互连接关系与导数连续性问题. 本文介绍强形式与弱形式两种自由单元法,前者直接由控制方程和边界条件直接产生系统方程,后者通过在自由单元上建立控制方程的加权余量式产生弱形式积分式,并通过像传统有限元法中的积分过程建立系统方程组.本文提出的方法是一种单元配点法,对于域内点为了获得较高的导数精度,需要采用至少具有一个内部点的等参单元,为此除了可使用各阶次的拉格朗日四边形单元外, 还 给出了七节点三角形等参单元,用于模拟较为复杂的几何形状问题.

Abstract: By absorbing advantages of the finite element and meshless methods, a new numerical method, free element method, is proposed in the paper. In the discretization, the isoparametric elements as used in FEM are employed to represent the geometry and interpolate physical variables; and in the algorithm, the point collocation technique using elements is employed to generate the system of equations point by point. The main feature of the method is that only one independent element formed by freely selecting surrounding points is required for each collocation point, without need to consider the connective relationship between adjacent elements and the continuity of physical variables and their spatial derivatives at interfaces of the connected elements. Two types of free element methods, the strong-form method and weak form method, will be described in the paper. The former directly generates the system of equations from the governing equations and the Neumann boundary conditions, while the latter establishes the weak-form integral expression of the governing equations by the weighted residual technique over the free element first and then generates the system of equations through an integration process similar to that employed in the standard FEM. The method proposed in the paper is an element collocation method. To achieve highly accurate spatial derivatives for internal collocation points of the computational domain, isoparametric elements with at least one internal node are required. For this purpose, apart from the arbitrary order quadrilateral Lagrange elements, a new seven-node triangle element is constructed in the paper, which can be used to model problems with complex geometries.

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