Natural neighbour method based on the algorithm of local search

The natural neighbour method (or natural element method),which is based on the natural neighbour interpolation, is a method betweenmeshless and mesh. The discrete model of the domain \it\Omega in naturalneighbourmethod(NNM) consists of a set of distinct nodes, and a polygonal descriptionof the boundary. The whole displacement interpolations are constructed withrespect to the nature neighbour nodes and Voronoi tessellation of the givedpoint. The natural neighbours of the gived point have been definitelydefined. The properties of the natural neigbour interpolation are excellent.For instance, the conditions of linear consistency, partition of unitity,positivity, and delta properties are all satisfied in natural neigbourinterpolation. The disadvantages in element-free Galerkin method(EFG), suchas, the difficulties of imposition of essential boundary and treatment ofmaterial discontinuity, the complex algorithm of matrix inverse in thecomputation of Moving Least Squares(MLS) shape function, the uncertainchoice of the weight functions can be avoided in NNM. But, NNM is usuallyregarded as a mesh-based method beacause the delaunay triangulations fromthe whole solution domain are still needed for neighbour-search. In stead ofsearching for the natural neighbors from delauny triangulation of the wholedomain, an algorithm quantifies the natural neighbour nodes of the givenpoint based on the locally delaunay triangles is proposed for theimprovement of the NNM. Similar to the EFG method, the procedure ofinterpolation and construction in the improved NNM is meshless. As a result,the improved NNM can possesses both the excellent properties of the naturalneigbour interpolation and advantages of the EFG method. Numerical resultsshow that the excellent agreement with exact solution is obtained in thismethod. Convergence studies in the numerical examples also show that thepresent method possesses an excellent rate of convergence for both thedisplacement and strain energy.