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Gao Xiaowei, Liu Huayu, Cui Miao, Yang Kai, Lyu Jun, Peng Haifeng, Ruan Bo. Generalized weak-form free element method. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-24-160
Citation: Gao Xiaowei, Liu Huayu, Cui Miao, Yang Kai, Lyu Jun, Peng Haifeng, Ruan Bo. Generalized weak-form free element method. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-24-160

GENERALIZED WEAK-FORM FREE ELEMENT METHOD

  • 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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