The Natural Element Method And Its Implementation

The natural element method uses the Voronoi tessellation,the natural neighbournode and the natural neighbour interpolation to construct the approximatefunction,and the solution of equation is based on the Delaunay triangulation bythe Galerkin method. We can calculate the integral on the back-ground trianglemesh. The natural neighbour interpolation is a multi-variables interpolation,whichis used in the natural neighbour interpolation.Based on the definition of the Laplace interpolation and the angle relationof a tiangle, I get the explicit formula about the neighbour coordinates. Usingthe NEM in one-dimension numerical experiment,I prove the NEM is equivalentto the FEM in one dimension,theoretically. Considered the number of the naturalneighbour nodes in the NEM for the two-dimensional questions, I find two caseswhich confirm the equivalence between the NEM and the FEM: the number ofnatural neighbour nodes is 3 and 4. There are experiments for the two-dimensionalline elasticity question in the numerical experiment part.The essential difference between the natural element method and the finiteelement method is the constructure of the trial and test function.There are twonatural neighbor interpolants to construct the base-functions of the NEM: naturalneighbor-based Sibson interpolation and Laplace interpolation(non-Sibsonianinterpolation). Laplace interpolation is easier than Sibson interpolation in application.