A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM)

In this paper a new extension of the mesh-free natural element method (NEM) is presented. In this approach, coined as constrained natural element method (C-NEM), a visibility criterion is introduced to select natural neighbours in the computation of the shape functions. The computation of these shape functions is based on a modified, constrained Voronoi diagram. With this technique, some difficulties inherent to this method in non-convex domains are avoided and the analysis of problems involving cracks or discontinuities are now easily performed. As the NEM satisfies the Kronecker delta property, the imposition of essential boundary conditions is trivial, unlike other mesh-free methods. The C-NEM technique provides a description of integration cells that allows the use of the stabilized conforming nodal integration (SCNI) scheme instead of Gauss integration to enhance computational efficiency and accuracy. Two numerical examples in elastostatics are reported to evaluate the potential of the proposed technique in highly non-convex geometries, like a crack, through which the solution becomes discontinuous