The study and application of Element-free Galerkin Method (EFGM) to three-dimensional problems is presented. Based on moving least squares method (MLSM), EFGM formulates the discrete model by using only a set of nodes, so the pre-processing of the method is simplified and the approximating function is Continuous with higher-order differentials over the entire domain. The relevant theories of EFGM are introduced and the derivation of discrete equations using variational principle is presented. The visibility criterion is used to deal with the effect of discontinuities on the influence domain of a gauss point. Contribution to the system of equilibrium equations of the reciprocity between two surfaces of discontinuities is considered. To investigate the accuracy of the proposed method, a cantilever beam and Laxiwa arch dam-foundation system are analyzed In detail. The results show that the method is effective and accurate in 3D problems.
Thus far finite element method has reached a high degree of effectiveness in structure failure analysis. However some problems appeared in its application, among which the following are the most remarkable Ones. (1) Pre-processing of finite element methods is always a far more time-consuming and hard task, especially in three-dimensional problems, even with powerful mesh generators. (2) Finite element method always seem to exhibit volumetric locking when the Poisson ratio is close to 0.5. (3) Despite the continuity of results of displacement, they always appear discontinuous and a post-processing smoothing work of those variables is necessary. (4) Mesh refinement of finite element method in problems of crack growth and large deformation is accompanied by a heavy computational burden. As these problems are associated with the fundamental characters of finite element methods, meshless methods have been developed to deal with them. Meshless methods formulate the discrete model by using only a set of nodes, with a suitable weight function on compact support and a proper integral approach, and the equilibrium equations can be congregated to solve the problem. On eliminating parts of the mesh structure used in finite element methods, meshless methods can simplify the pre-processing of the numerical simulation and it becomes possible to solve large classe of problems which are very awkward with mesh-based methods.
Nayroles, et al (1992) were evidently the first to promote mesh less method in 1992. Moving least-square approximations were used in a Galerkin method called the diffuse element method (DEM). Later, lots of mesh less methods have been developed, such as the element-free Galerkin method (EFGM), meshless local Petrov-Galerkin method (MLPGM), wavelet- Galerkin method (WGM), reproducing kernel particle method (RKPM), partition of unity method (PUM), natural element method (NEM), manifold method (MM) and so on. Among all this methods, the EFGM, which is refined and modified from the diffuse element method by Belytschko et al (1994), is probably the most influential one.
The EFGM constructs the trial and test functions for variational principle (weak form) using moving leasts-quare interpolants and solves the partial differential equation using the Galerkin method.