Researches On Meshless Method With Complex Variables And Its Applications

The numerical methods in engineering such as FEM (finite element method) and BEM (boundary element method) have been successfully used. But the existence and formation of mesh in these methods also cause some difficulties when they are used. The meshless method being presently developed will wholly or partially eliminate the mesh. This method is the hot point and recent trend in the study of engineering calculation.With the problems like ill-conditioning, precision and efficiency involved in moving least-square approximation, the moving least-square approximation with complex variables is presented in this paper. The advantage of the moving least-square approximation with complex variables shows that the number of the undetermined constants in the trial function is decreased, therefore the solving efficiency is increased. Meanwhile, The basis function of the moving least-square approximation with complex variables is discussed in this paper, and the formula based on the orthogonal basis function for the moving least-square approximation with complex variables is obtained. And the advantage of the method reveals that the ill-conditioning equation system won't be formed when the shape functions are obtained. As a result, the solving precision gets much better. With this basis, the meshless method with complex variables is given in this paper, and this method has the following advantages such as less distribution nodes, higher precision as well as faster calculation. To the meshless method with complex variables, the influences of the weight function, basic function, the influence domain of the node as well as the distribution density of nodes to thesolving precision are studied in this paper, and some new conclusions are obtained. To the problem of crack, the enriched meshless method with complex variables is presented in this paper. By using the analysis solution of displacements at the tip of a crack, the basis function of meshless method with complex variables will be enriched, and the precision that the meshless method with complex variables used to solve the crack problems will be increased. Since the meshless method can't deal with the boundary conditions well, so the coupling of meshless method with complex variables and finite element method is presented in this paper. And the new coupling approximation function is discussed. The problems existed in the old coupling methods are solved, and the solving precision also will be increased. Moreover, The nonlinear analysis to the concrete member allowed to crack is carried with the meshless method with complex variables proposed in this paper. The method of nodal diffusion domain which is suitable for the treatment of cracks in the meshless method is shown. And the calculations prove that it is very well to deal with the nonlinear concrete problems by using meshless method with complex variables.A series of creative studies for the moving least-square approximation with complex variables as well as the meshless method with complex variables are carried out in this paper. These studies will promote the development of meshless method, and at the same time the study on the meshless method is available for the application in engineering.