Meshless Natural Element Method And Its Application

The reserech work for meshless methods has been lasted for twenty and more years. The meshless method possesses many outstanding advantages compared with element-based methods such as the finite element method, the boundary element method et al. In recent years, a lot of sorts of meshless methods were presented by native or foreign scholars and applied to the engineering practice. Many theoretic and applied achievements were acquired.The meshless natural element method is a new numerical method based on the Voronoi diagram and dual Delaunay triangularization structure for a set of randomly distributed nodes. In the meshless natural element method, a local Petrov-Galerkin formulation is employed to abtain the discrete system equations,where trial functions are constructed using natural neighbour interpolants, and test functions are constructed using the finite element shape functions for a triangle with three nodes or its Bernstein-Bezier basis functions. The meshless natural element method has some advantages when it is employed to analyze material discontinuous problems, such as composite materials, phase transformantions, inclusions in a matrix with different material properties et al., and singularity problems such as a crack propagation, large deformation problems in solid and fluid mechanics, and plate bending problems governing by the fourth-order partial differential equations. This numerical method has broad prospects for applications.Firstly, in the thesis the meshless natural element method is applied to slove elasticity ploblems with heterogeneous materials and crack propagation ploblems. The, treatment of material discontinuities using the meshless natural element method is presented. The influence of node numbers in a non-convex boundary on the solution accuracy is investigated.Then the meshless natural element method is applied to slove bending problems for the Kirchhoff plate. Shape functions for plate deflection function, which possess C1 continuous requirement are constructed using natural neighbour interpolants. Test functions are constructed by using finite element shape functions for a triangle with three nodes as variables of Bernstein-Bezier basis functions. The discrete system equations are formulated by the local Petrov-Galerkin method.Lastly, the meshless natural element method is employed to slove bending...