Elastoplastic Mechanical Problem In The Element-free Galerkin Method

Element-free Galerkin method(EFGM) , similar to Finite element method, is a new numerical method developed recently. In EFGM, in order to get a numerical solution for a partial differential equation, the shape function is constructed by moving least squares(MLS), the control equation is derived from the weak form of variational equation and essential boundary conditions are imposed by penalty function method. The advantages of EFGM are : (1) only nodal datas are necessary, i.e. there is no need to join nodes into elements; (2) high accuracy can be achieved; (3) postprocess is easy, etc.The mathematical basis of EFGM is moving least squares method. To use MLS, it is only necessary to construct an array of nodes in the domain under consideration. Just because of this, EFGM is completely free. Moving least squares interpolants do not pass through the data because the interpolation functions are not equal to unity at the nodes unless the weight functions are singular. This is of disadvantage in EFGM for it complicates the imposition of essential boundary conditions and the application of point loads. However, these disadvantages are heavily outweighted by its advantages.EFGM has been successfully used to solve elastic problems. In this paper, EFGM is applied to solve elasto-plastic problems. Firstly, the discrete equations in differential form, which is used to solve elasto-plastic problems , is derived using variational principle. In addition, it is discussed how to solve problems involving point loads with EFGM. During the solving process, the increment form can approximate the differential form and Newton-Raphson iteration techniques are introduced into the computation. Then the corresponding computer program is developed and several examples are given. The results of all examples are compared with those of ANSYS and they show good agreement. The closeness of the results obtained by these two methods verifies the reliability of the theory in the present paper and also shows that in solving elasto-plastic problems, EFGM still possesses some advantages such as good stability and high rate of convergence. At last, the advantages of EFGM and some key issues are also discussed in this paper.