Program Realization And Application Of Nonlinear Natural Element Method

The Natural Element Method (NEM) is a recently developed meshless numerical method for solving partial differential equations, which is essentially the Galerkin Method based on the Natural Neighbor Interpolation (NNI). Its basic process is constructing the field variable interpolation of given point first, based on its natural neighbor nodes and second-order Voronoi cell. Then, the standard Galerkin Method is used to obtain the discrete system of control equations over the local Delaunay triangular sub-domain. As the variational principles of the partial differential equations in this work are already known, the control equations are discretized equivalently in variational method.The facts that NEM uses triangle or tetrahedron as background integration grid make mesh generation and adaptive refinement much easier. Compared with the Finite Element Method (FEM), the numerical results of NEM are better than those of triangular or tetrahedral FEM, almost equal to the results of quadrangular or hexahedral FEM. The NNI is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates the imposition of essential boundary conditions.The fundamental theory of Natural Element Method has been summarized, such as the concepts of Delaunay triangulation, Voronoi diagram, the computation of NNI and its derivatives, and so on. The algorithms for large deformation problem, coupled geometry and material nonlinearity problem, and Biot consolidation problem are presented. Based on these algorithms, computing programs have been built by MATLAB. Each of them has an example to check the veracity and advantage of NEM.