Study Of Theory And Algorithm For The Smoothed Finite Element Method

The finite element method (FEM) is an important numerical method which has been widely used in various disciplines and engineering practice. Finite element method based on displacement theory is used in most of the current commercial finite element software. However, theory based on displacement usually makes the system stiffness matrix of the results more rigid, in addition, the displacement smaller and the natural frequency larger. In recent years, researchers such as LIU G R proposed a series of smoothed finite element methods (S-FEM) with distinctive features by combining the FEM and smoothed strain technology of Mesh-less method. They can be classified into four categories, including cell-based smoothed finite element method (CS-FEM), node-based smoothed finite element method (NS-FEM), edge-based smoothed finite element method (ES-FEM) and face-based smooth finite element method (FS-FEM).The basic knowledge of FEM and key technique were elaborated in this paper. The solving process is summarized and discrete modes are also introduced using method of analogy. NS-FEM and ES-FEM are mainly suitable for2D problems. The related theory of NS-FEM and ES-FEM are studied in contrast to the FEM. Generation of smooth domain, smooth matrix computing methods, shape function construction and computing process are also investigated. The related theory of volumetric locking, shear locking and their mechanism are also studied.In this paper, the diffsets of fixed beam with uniform load and curved bar which is forced on one end are analyzed by using matlab programming. The results show that the stiffness matrix of NS-FEM is softer, and the displacement and strain energy are the upper limits of the solution, also the stress, strain, strain energy have good precision, and the NS-FEM does not produce volume locking phenomenon. The method to eliminate the volume locking is proposed by combining the NS-FEM and ES-FEM method. The shear locking can be solved by combining the ES-FEM method and discrete interval method. The methods can be verified by analyzing the diffsets of the bent lever with one end loaded and the four edges simply supported square plate with the centre loaded.FS-FEM is mainly used in3D problems. This paper details the discretization process of surface domain and the solution of strain matrixes. It focuses on the programming process of3D smoothed finite element method, especially on the generation of element-nodes matrix, element-faces matrix and nodes-faces matrix. Taking the discretization of cuboid for example, this paper deals with the process, such as element mesh, representation of related matrix, the solution of shape function, the generation of the smooth strain matrix and stiffness matrix. Finally, the solving procedure of FS-FEM is proposed using a solving procedure diagram. A3D cantilever beam is illustrated to validate the accuracy of the numerical results of stress and displacement by FS-FEM.