The Hybrid Natural Element Method

As a new numerical method, Natural Element Method (NEM) has beenproposed at the end of last century. Sibson and non-Sibonian interpolation which arecalled Natural Neighbor Interpolation (NNI) are employed to constitute trial functionin NEM. Because of meshless and easy application, the NEM is one of numericalmethod which has a bright prospect using to solution the partial differentialequations.Because of the inaccurate stress solution and poor ability to obtain the stress ofnode, the tradition Natural Element Method(NEM) are rarely used to solve complexquestion. In this paper, the hybrid finite element method was introduced into NEMand the hybrid natural element method (HNEM) has been provided. And the HNEMis applied to solve elasticity problems, elastoplasticity problems, larger deflectionproblems and viscoelasticity problems respectively. The main researches of thisthesis are as follows.Based on the complex variables moving least-square, a new stress recoveryalgorithm for NEM is presented. Compared with the traditional stress recoveryalgorithm, the new method can improve calculation efficiency.Using NNI to be displacement and stress interpolation function respectively，the HNEM and Hellinger-Reissner variation principle are introduced into analysiselastic problems and the HNEM for elasticity is provided. Comparing with NEM, theHNEM has more precision solution of stress and can obtain the stress of nodesdirectly. In the other hand, the HNEM has higher computational efficiency thanthestress recovery algorithm.On the basis of HNEM, the incremental form HNEM for nonlinear analysis ispresented. Combined with the incremental Hellinger-Reissner variation principle, theformula of HENM for the elastic-plastic problem is deduced, and is used to analyszethe elastic-plastic mechanics problem. Its numerical precision is better than NEMand the calculation efficiency is higherThe creep properties of linear viscoelastic material are analyzed by usingHNEM. The material property of viscoelastic material has close relations with time,so it is difficult to analyze this problem. Based on the detailed analysis of the linearviscoelastic material characteristics, the creep process of material can be expressed by deformation changes on some time points. The HNEM for linear viscoelasticproblem is presented based on Hellinger-Reissner variation principle. The numericalexamples show that the method in this paper can depict the creep properties ofmaterials accurately, and can obtain the stress of nodes directly and quickly.The HNEM is applied to elastic large deformation problem in this dissertation.Combining with the incremental Hellinger-Reissner variation principle which accordwith lager deformation analysis the HENM for two-dimensional elastic largedeformation problems is presented based on the total Lagrangian formulation, andthe corresponding formulae are obtained. The Newton-Raphson iteration isemployed in the mumerical implementation. The advantages of HNEM are higherprecision and faster calculation.Based on the incremental Hellinger-Reissner variation principle and thecharacters of geotechnical engineering，the HNEM has been used to analyze theproblems of geotechnical engineering，such as tunnel excavation. In order tosimulate the process of tunnel support, the method which can be increase nodes hasbeen proposed.In order to show the efficiency of the HNME in the dissertation, the MATLABcodes of the methods above have been written. The numerical examples are provided,and the validity and efficiency of these methods are demonstrated.