| Meshless method is a new numerical method on problem for determining solution of differential equation. It is an approximation method based on nodes, and doesn't need a mesh entirely or partly. Then the discontinuous problems and the extremely large deformations problems can be solved without the re-meshing techniques. In this paper, after introducing the meshless method simply, we derive the moving least squares approximation (MLS) in detail. MLS makes no require for the residual of derivative approximation. It only requires that the residual of the approximation functions on every node reaches the minimum and it may bring an evident error when the smooth of field variable is required. Then the generalized moving least squares approximation (GMLS) is derived under adding the residual of high orders derivative. For the sake of decrease of the computing time, modifying generalized moving least squares approximation is constructed under adding the residual of arbitrary high orders derivative only at the portion nodes. With the derived MGMLS approximation function, we derive the Galerkin meshless method for plane problems of elasticity mechanics and weighted least squares meshless method for plane piezoelectric problems. Finally numerical examples with three shape functions are analyzed in detail. The numerical examples show that the modifying generalized moving least squares approximation has high accuracy not only for function value but also one or higher orders derivative. |
