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 引用本文: 高效伟, 刘华雩, 崔苗, 杨恺, 吕军, 彭海峰, 阮波. 广义弱形式自由单元法. 力学学报, 待出版.
Gao Xiaowei, Liu Huayu, Cui Miao, Yang Kai, Lv Jun, Peng Haifeng, Ruan Bo. Generalized weak-form free element method. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Gao Xiaowei, Liu Huayu, Cui Miao, Yang Kai, Lv Jun, Peng Haifeng, Ruan Bo. Generalized weak-form free element method. Chinese Journal of Theoretical and Applied Mechanics, in press.

## GENERALIZED WEAK-FORM FREE ELEMENT METHOD

• 摘要: 自由单元法(FrEM)是一种单元配点法, 吸收了有限元法等参单元的插值稳定性与无网格法使用灵活的优点, 具有稳定性好和灵活性高的性能. FrEM在每个配置点只需要一个通过自由选择周围节点而形成的独立的等参单元, 不需要考虑单元之间节点的相互连接关系, 这种自由性使得人们可以按照学科性质形成所需要的单元, 因而FrEM既适用于固体力学, 也适用于流体力学问题的求解. 然而, FrEM所用单元需要至少有一个内部节点(用于方程的配点), 以致有限元法中单元节点相连的网格不能在FrEM中直接使用. 文章基于对应于配置点的等参单元形函数在与其不相连的边界上为0的特性和配置点等效节点力平衡关系, 提出了一种广义弱形式自由单元法(GFrEM). 该方法是一种使用单元逐点建立配置方程的单元配点法, 具有如下特点: (1)突破了现有自由单元法中单元需要内部节点的限制, 因而可以使用线性单元和高阶单元; (2)突破了现有自由单元法中一个配置点使用一个单元的限制, 可以使用与配置点相连的任意多个单元; (3)可以使用任意形式的规则单元和多边形/多面体单元; (4)如果使用单元节点相连的传统有限元法网格, 则GFrEM自然演变为传统的有限元法. 论文将给出几个固体力学方面的应用算例, 展示所提方法的有效性.

Abstract: Free element method (FrEM) is a type of element collocation method, which absorbs the advantages of the interpolation stability of isoparametric elements used in the finite element method and the flexibility of the mesh free method, and thus FrEM has properties of good stability and high flexibility. FrEM requires only one independent parametric element formed by freely selecting around nodes, without need to consider the connectivity between adjacent elements’ nodes. This freedom enables us to form elements according to the specific requirements of different disciplines, and thus FrEM is suitable for solving both solid and fluid mechanics problems. However, FrEM requires elements having at least one internal node using for collocating equations. Consequently, the meshes used in FEM, which have the node-to-node connection between elements, cannot be directly used in FrEM. This paper proposes a generalized weak–form free element method (GFrEM) based on the characteristic of the isoparametric shape function corresponding to the collocation point that is zero on boundaries not connected to the collocation point, and the equilibrium relationship of all equivalent nodal forces at the collocation point. This method builds equations point-by-point using around elements, which has the following features: (1) it breaks through the limitation on the existing free element method where the used elements require internal nodes, and thus allowing for the use of linear and higher-order elements; (2) it breaks through the restriction on the existing free element method where one collocation point can only be associated with one element, enabling multiple elements to be connected to a collocation point; (3) it allows for the use of arbitrary shaped regular elements as well as polygon/polyhedron elements; (4) if the traditional finite element meshes, with the feature of nodes connected to nodes between neighbor elements, are used, GFrEM naturally evolves into the conventional finite element method. Several application examples in solid mechanics will be presented in this paper to demonstrate the effectiveness of the proposed method.

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