Study On Natural Element Method And Its Applications

Natural Element Method (NEM) is one of the new meshless methods. It has attractedthe attention of many researchers because of good properties of its shape functions. Thispaper introduces the basic concepts and the realizing process of NEM. NEM is applied tosolve a free interface problem and the membrane deformation problem described by anelliptic variational inequality and a parabolic variational inequality respectively. Numericalresults show the accuracy and efficiency of the present method.The main contents in this paper are as follows:1. First we introduce some basic concepts in NEM, some useful natural neighbourinterpolation methods. By patch experiments, it shows the efficiency of naturalneighbour interpolation methods. Next we present the realizing process of NaturalElement Method. Finally many numerical examples are given to check thecomputational effect of NEM.2. A kind of free interface problem is introduced. Applying duality method to deal withthe constraint condition of the elliptic variational inequality we proposed the NEM fora kind of interface problem’s NEM method. Numerical example is given. We discussedthe choice of the optimal parameter and the effect of the iterative times and number ofcentre points. We compare the results of NEM and over-relaxation method, it shows thevalidation and efficiency of NEM.3. Consider the membrane deformation problem. Discreting time with difference methodand spatial variable with natural element method, we proposed NEM for the membranedeformation problem. Numerical example shows the accuracy and effectiveness of thismethod.