The natural element method in solid mechanics

The application of the Natural Element Method (NEM)^1,2 to boundary value problems in two-dimensional small displacement elastostatics is presented. The discrete model of the domain fl consists of a set of distinct nodes N, and a polygonal description of the boundary ∂Ω. In the Natural Element Method, the trial and test functions are constructed using natural neighbour interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N. The interpolants are smooth (C°°) everywhere, except at the nodes where they are C⁰. In one-dimension, NEM is identical to linear finite elements. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and non-convex bodies (cracks) using NEM is also described. A standard displacement-based Galerkin procedure is used to obtain the discrete system of linear equations. Application of NEM to various problems in solid mechanics, which include, the patch test, gradient problems, bimaterial interface, and a static crack problem are presented. Excellent agreement with exact (analytical) solutions is obtained, which exemplifies the accuracy and robustness of NEM and suggests its potential application in the context of other classes of problems-crack growth, plates, and large deformations to name a few.