Introduction - Meshless Methods in Biomechanics - page 4

which is the object of this work. More recently, the Natural Radial Element
Method (NREM) [ 32 - 34 ] was developed. The NREM is an efficient and accurate
truly meshless method, which presents a low order nodal connectivity.
1.2 Natural Neighbour Radial Point Interpolation Method
All the numerical examples presented in this topic are analysed considering the
Natural Neighbour Radial Point Interpolation Method (NNRPIM). The NNRPIM
is the product of the combination of the Radial Point Interpolators (RPI) with the
Natural Neighbours geometric concept. The RPI started with the Point Interpo-
lation Method (PIM) [ 21 ]. The technique consisted in constructing polynomial
interpolants, possessing the Kronecker delta property, based only on a group of
arbitrarily distributed points. However this technique has too many numerical
problems, for instance the perfect alignment of the nodes produces singular
solutions in the interpolation function construction process. As so, this technique
evolved and the Radial Point Interpolation Method (RPIM) [ 23 ] was created.
Within this meshless method the Radial Basis Function (RBF) was added in the
construction process of the interpolation function, stabilizing the procedure. The
RBF used in these early works were the Gaussian and the multiquadric RBF.
Initially the RBF was developed for data surface fitting, and later, with the work
developed by Kansa [ 16 , 17 ], the RBF was used for solving partial differential
equations. However the RPIM uses, unlike Kansa's algorithm, the concept of
''influence-domain'' instead of ''global-domain'', generating sparse and banded
stiffness matrices, more adequate to complex geometry problems.
The NNRPIM is the next step in the RPI. In order to impose the nodal con-
nectivity, the 'influence-domain' is substituted by the 'influence-cell' concept. In
order to obtain the influence-cells the NNRPIM relies on geometrical and math-
ematical constructions such as the Voronoï diagrams [ 35 ] and the Delaunay tes-
sellation [ 36 ]. Thus, resorting to Voronoï cells, a set of influence-cells are created
departing from an unstructured set of nodes. The Delaunay triangles, which are the
dual of the Voronoï cells, are applied to create a node-depending background mesh
used in the numerical integration of the NNRPIM interpolation functions. Due to
the integration mesh total dependency on the nodal distribution, the NNRPIM can
be considered a truly meshless method. Unlike the FEM, where geometrical
restrictions on elements are imposed for the convergence of the method, in the
NNRPIM there are no such restrictions, which permits a total random node dis-
tribution for the discretized problem. The NNRPIM interpolation functions, used
in the Galerkin weak form, are constructed in a similar process to the RPIM, with
some differences that modify the method performance.
Although the NNRPIM is a recent developed meshless method [ 31 ] it has been
extended to many fields of the computational mechanics, such as the static analysis
of isotropic and orthotropic plates [ 37 ] and the functionally graded material plate
analysis [ 38 ], the 3D shell-like approach [ 39 ] for laminated plates and shells [ 40 ].