Meshless Finite Element Ideas

It is widely acknowledged that 3-D mesh generation remains one of the most man-hours consuming techniques within computational mechanics. The problem of mesh generation is that the time remains unbounded, even using the most sophisticated mesh-generator. For a given distribution of points, it is possible to obtain a mesh very quickly, but it may require several iterations, including manual interaction, to achieve an acceptable mesh. On the other hand, standard meshless methods rely on the node connectivity to define the interpolations. Unfortunately, the correct choice of the connectivities may also be an unbounded problem. In that case, the use of a meshless method may be superfluous. A meshless method is presented in this paper called Meshless Finite Element Method (MFEM). The method has the advantages of a good meshless method concerning the ease of introduction of connectivities in a bounded time of order n, and the condition that the shape functions depends only on the node positions. Furthermore, the method proposed also shares several of the advantages of the Finite Element Method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C continuity between elements, allowing the treatment of material discontinuities, and (c) an easy introduction of the boundary conditions. The MFEM can be seen either as a finite element method using elements with different geometric shapes, or as a meshless method with clouds of nodes formed by all the nodes that are in the same empty sphere. In either case, whether as a meshless method or as a standard FEM, the method satisfies the raison d’etre of the meshless procedures: it gives the node connectivities in a bounded time of order n. Sergio R. Idelsohn, Eugenio Oñate, Nestor Calvo, Facundo Del Pin