In the last decade a considerable effort of research has been paid in the development of a series of novel numerical tools that have been referred to as meshless or meshfree methods. These methods share the characteristics of no need of explicit connectivity information, as in the standard form of the FEM. This connectivity is generated in a process transparent to the user, thus alleviating the preprocessing stage of the method.

In this paper a numerical study of the behaviour of the alpha-shapes-based Natural Element Method (alpha-NEM) in the field of solid mechanics is presented. The NEM, that can be considered as a meshless method. uses natural neighbour interpolation to build the discrete system of equations of the Galerkin method, thus imposing no constrains in the relative position of nodes. It also differs to other methods of this kind in its capability to accurately reproduce essential boundary conditlons along convex boundaries. The alpha-shape-based approach of this method (alpha-NEM) generalizes this behaviour to nonconvex domains, enables us to construct models entirely in terms of nodes while keeping Galerkin formulation and allows to simulate material discontinuities in a straightforward manner with a full compatible formulation.

Numerical proofs of this behaviour are presented in two-dimensional cases and in the context of linear elastostatics. First, some examples are analyzed and the differences between the alpha-NEM, the standard formulation of the NEM and FEM are pointed out. The paper is completed with some examples on piece-wise homogeneous domains.

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