Development Of The Natural Element Method And Its Application

The Natural Element Method (NEM) is a newly coming numerical method to solve the Partial Differential Equations. NEM shape function on the one hand assumes the characteristics of meshless method and on the other hand possesses the advantages of traditional finite element method, so it's a most promising numerical method for geotechnical engineering problems. The paper studies on this method and its application.Firstly, the paper reviews the history and development trend of the Meshless methods in detail, together with some comments on the advantages and also some existing problems to be further studied. Secondly, the current status of the Natural Element Method(NEM),the construction of the NEM shape function and its properties and advantages and the discrete formulation are discussed in detail.In chapter 3, issues regarding the numerical integration of the weak form are addressed. The partition of unity quadrature method and a single point quadrature rule based on the average strain of Delaunay triangle are proposed respectively. Considering both the accuracy and the efficiency, the point quadrature rule based on the average strain of Delaunay triangle is a better selection, especially in the adaptive analysis, though its convergence and accuracy is a little lower than the partition of unit quadrature method. In chapter 4, preliminary study is performed on the adaptive analysis of NEM. The solution error is estimated using the ZZ error estimator which is based on the recovery stress filed acquired through the MLS/NNI stress recovery scheme according to the character of the NEM. Then new nodes are inserted automatically according to the error distribution information acquired, which just involves in the local adjustment of the Delaunay Tessellation. The implementation of adaptive process is effect and efficient. In chapter 5, issues regarding the modeling the strong/weak discontinuities are addressed. NEM is extended in the framework of the Partition of Unity. Crack and interface between different materials are considered respectively for illustration the idea. In chapter 6, The Weak-Strong Form(WSF) which combines the advantages of the Local Petrov-Galerkin method and the Collocation method is discussed. NEM based WSF is implemented and applied to 3D analysis.In chapter 7, The Goodman element is introduced to NEM in order to model the rock discontinuities. The scale effect of jointed rock mass is studied in the view of fractal...