Natural Element Method And Application

Numerical methods for solving partial diferential equations(PDEs) need someform of spatial discretization,or mesh of nodes,at which the solution is specified.Thecharacter of the mesh is an important factor in determining the stability and accu-racy of the method as well as the type of PDE that can be solved.Straightforwardmethods,such as the finite-diference method,are based on a regular spatial dis-cretization,whereas more flexible techniques such as the finite-element method di-vide the plane into triangles or rectangles,often with an irregular distribution.Inmany cases,the spatial mesh is constantly changing during the course of a calcula-tion.An typical example is a ‘self-adapting mesh’where nodes are added duringthe calculation to improve accuracy in a particular region,the position of which isdetermined by the solution itself.The most difcult situation to manage when themesh is required to evolve with the solution,namely,mesh nodes move during the cal-culation.Although an evolving mesh allows for material properties to be accuratelytransported at the nodes,large displacements quickly result in severe mesh distortionwhich in turn increases numerical instability and restricts accuracy.Here we showhow the fundamental geometrical concept of ‘natural neighbour’may be used toovercome these problems.Then,we present the C~1formulation and propose a com-putational methodology for its numerical implementation(Natural Element Method)for the solution of PDEs.The C~1shape functions interpolate to nodal function andnodal gradient values,which renders the interpolant amenable to application in aGalerkin scheme for the solution of fourth-order elliptic PDEs.