A GFEM With Local Gradient Smoothed Approximation for 2D Solid Mechanics Problems

In this paper, a generalized finite element method (GFEM) with local gradient smoothed approximation (LGS-GFEM) using triangular meshes is proposed. The LGS-GFEM combines the idea of partition of unity (PU) and gradient smoothed technique to inherit their characteristics and avoid their respective shortcomings such as linear dependent problem, temporal instability problem , etc. Different from the classical finite element, the displacement field function is regarded as composed of the shape function and the node displacement , while LGS-GFEM assumes that the displacement field function consists of the finite element shape function and the node displacement function. In order to obtain the nodal displacement function, the second order Taylor expansion is considered. The derivative term in Taylor expansion is obtained by using gradient smoothed technique in a smoothed domain also known as the smoothed derivative term in this paper. The displacement in smoothed operation is interpolated by polynomial basis function and radial basis function in meshfree theory. The LGS-GFEM shape function obtained by the above method is a smoothed composite shape function, that is, synthesized by the shape function of finite element for PU and the smoothed nodal shape function for local gradient smoothed approximation. The smoothed composite shape function retains the ideal Kronecker property of the finite element shape function. In the process of smoothed integration, two kinds of integration schemes are proposed, which are polygon integration and equivalent circle domain integration, which correspond to LGS-GFEM-I and LGS-GFEM-II respectively. In addition, both the proposed LGS-GFEM-I and LGS-GFEM-II have some other important properties such as no extra DOFs, linear independent and temporal stability. In the analysis of two representative numerical examples of static and free vibration, the LGS-GFEM method is compared with the classical finite element of triangular (FEM-T3) and quadrilateral (FEM-Q4) elements. The results show that: (1) In the static analysis, LGS-GFEM shows better accuracy and insensitivity to mesh distortion than both FEM-T3 and FEM-Q4. In particular, in the static analysis, LGS-GFEM-II shows a significantly better convergence rate and computational efficiency than the other three methods. (2) Through the results of free vibration, it is found that there are no spurious non-zero energy modes in the modal frequencies of LGS-GFEM, which indicates that LGS-GFEM does not have temporal instability problem, and this problem often appears in the node-based smoothed finite element method (NS-FEM). Similarly, the modal frequencies calculated by LGS-GFEM are closer to the reference values. This indicate that the proposed method has great advantages in dynamic analysis.