The Research On The Natural Element Method In Solid Mechanics

The Natural Element Method (NEM) is a recently proposed novel numerical, which has been applied for the solution of partial differential equations, boundary-value problems in different fields of engineering and the applied sciences. In this work, the application of the Natural Element Method (NEM) to the problems in two-dimensional elastostatics is presented.We assume a discrete model of the domain Ω consists of a set of distinct nodes N, and a polygonal description of the boundary Ω. In the Natural Element Method, the whole interpolants are constructed with respect to the natural neighbour nodes and Voronoi tessellation of the given point, and a standard displacement-based Galerkin procedure over the Delaunay triangular sub-domain is used to obtain the discrete system of linear equations. As a result, the numerical integration of the equilibrium equations can be analytically calculated at the triangular quadrature meshes. The solution of the discrete system of linear equations (Kd=f) is carried out to obtain the nodal displacement vector d.Natural neighbour interpolants is a multivariate data interpolation scheme, which has primarily been used in data interpolation and modeling of geophysical phenomena. The 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 c°. 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 non-convex bodies (cracks) using NEM is also described. The NEM interpolant is unique for a given set of distinct points (nodes) in the plane. The Delaunay triangles are used in the numerical computation of the NEM interpolant. However, unlike the finite element method where angle restrictions are imposed on the triangles for the convergence of the method, there are no such constraints on the shape, size, and angles of the triangles in NEM. This facilitates random configuration of nodes in space. Recent work on Naturalneighbour interpolants and its application to the modeling of complex fluid-structure interaction phenomena does indicate the merits of the method for the solution of PDEs and suggests that it could be a promising numerical tool in the realm of solid mechanics.Application of NEM to various problems in solid mechanics, which include, a static crack problem, the displacement patch test, the equilibrium patch test, the cantilever beam, and plate with a hole problem are presented and comparisons made to results obtained using finite elements as well as to reference solutions to validate the accuracy and convergence of NEM. Excellent agreement with exact (analytical) solutions is obtained, which exemplifies the accuracy and robustness of NEM and suggests its potential application, provides impetus for its application to other classes of problems such as crack growth, plates, and large deformations in solid as well as fluid mechanics.