Researches On Complex Variable Meshless Manifold Method

Meshless method is a class of new numerical methods developed in recent years. The meshless methods only need the information at nodes, and don't discretize the domain into a mesh. The advantages of meshless methods are the simpler pre- processing and higher precision. Meshless method is a hot topic of researches on science and engineering computation.The meshless manifold method, in which the idea of numerical manifold method is introduced, is one of the meshless methods. It has great computational cost because of a large number of nodes selected in the domain of problem.In this dissertation, for the disadvantage of great computational cost of the meshless manifold method, the complex variable meshless manifold method (CVMMM) is developed,and is applied to solve elasticity, fracture, transient heat conduction, and elastodynamics problems respectively.On the basis of the complex variable theory, the approximation function of a 2D problem is formed with 1D basis function when the shape function is formed. The complex variable numerical manifold method (CVNMM) for elasticity is presented in this paper. The corresponding formulae of the CVNMM for elasticity are obtained in detail.The meshless manifold method (MMM) has a great number of nodes, which results smaller computational efficiency. And the MMM can form an ill-conditioned or singular equations sometimes. For the disadvantages of the MMM, combining the complex variables moving least-square (CVMLS) approximation with the MMM, the complex variable meshless manifold method (CVMMM) for 2D elasticity is presented in this paper, and the corresponding formulae of the CVMMM are obtained.In CVMMM, the analytical solutions of displacements at the tip of a crack are used to enrich the approximation function of the CVMLS approximation, and the formulae of the approximation function for fracture problems are obtained. On the basis of least potential energy theory, the CVMMM for fracture problems is presented, and the corresponding formulae of the enriched CVMMM are obtained. Finally, the enriched CVMNMM is compared with the traditional meshless manifold method.For steady heat conduction problems, the CVMMM is applied to form approximation function of transient heat conduction problems. Time discretization is given by the traditional two-point differential method, and space domain discretization is given by the CVMLS approximation. The penalty method is employed to apply the essential boundary conditions. On basis of the Galerkin weak form of transient heat conduction problems, the CVMMM for transient heat conduction problems is presented, and the corresponding formulae are obtained.The CVMMM is also used to analyze transient elastodynamics problems. The Newmark time integration method is used for time history analysis, and the CVMLS approximation is used to discretize the space domain. Then the CVMMM for elastodynamics is presented in the paper.In order to show the efficiency of the CVMMM in the dissertation, the MATLAB codes of the methods above are written. Some numerical examples are given, and the validity and efficiency of these methods are demonstrated.