Local Kronecker delta property of the MLS approximation and feasibility of directly imposing the essential boundary conditions for the EFG method

The element-free Galerkin (EFG) method is a promising method for solving many engineering problems. Because the shape functions of the EFG method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property, special techniques are required to impose the essential boundary conditions. In this paper, it is proved that the MLS shape functions satisfy the Kronecker delta property when the number of nodes in the support domain is equal to the number of the basis functions. According to this, a local Kronecker delta property, which is satisfying the Kronecker delta property only at boundary nodes, can be obtained in one- and two-dimension. This local Kronecker delta property is an inherent property of the one-dimensional MLS shape functions and can be obtained for the two-dimensional MLS shape functions by reducing the influence domain of each boundary node to weaken the influence between them. The local Kronecker delta property provides the feasibility of directly imposing the essential boundary conditions for the EFG method. Four numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the direct method and Lagrange multiplier method shows the feasibility of the direct method.