Introduction - Meshless Methods in Biomechanics - page 3

opinion that such integration schemes are less accurate, which is not true. Meshless
methods using these integrations schemes are considered 'truly' meshless methods.
In this work a hybrid integration scheme, totally nodal dependent, is presented and
applied, ensuring that the proposed meshless method is 'truly' meshless.
1.1.4 Relevant Meshless Methods
The initially created meshless methods used approximation functions, since it
produces smoother solutions, the implementation of the influence-domain concept
was easier and the background integration scheme was nodal independent. The
first meshless method using the Moving Least Square approximants (MLS) in the
construction of the approximation function was the Diffuse Element Method
(DEM) [ 6 ]. The MLS was proposed by Lancaster and Salkauskas [ 7 ] for surface
fitting. Belytschko evolved the DEM and developed one of the most popular
meshless methods, the Element Free Galerkin Method (EFGM) [ 8 ], which uses a
nodal independent background integration mesh. One of the oldest meshless
methods is the Smooth Particle Hydrodynamics Method (SPH) [ 9 ], which is in the
origin of the Reproducing Kernel Particle Method (RKPM) [ 10 ].
Another very popular approximant meshless method is the meshless local Pet-
rov-Galerkin method (MLPG) [ 11 ], initially created to solve linear and nonlinear
potential problems, which later evolved towards the Method of the Finite Spheres
(MFS) [ 12 ]. The Finite Point Method (FPM) [ 13 - 15 ] uses for integration proposes a
stabilization technique in the collocation point method. Another approximation
method, distinct from the previous, is the Radial Basis Function Method (RBFM)
[ 16 , 17 ]. It uses the radial basis functions, respecting a Euclidean norm, to
approximate the variable fields within the entire domain or in small domains. It
does not require an integration mesh and, in opposition to the previous referred
meshless methods, uses the strong form formulation. Initially used to approximate
multidimensional data [ 18 ] only latter it was applied by others [ 19 , 20 ] to the
analysis of solid mechanics differential equations.
Although approximants meshless methods have been successfully applied in
computational mechanics there were several problems not completely solved. One
of these problems, and perhaps the most important unsolved issue, was the lack of
the Kronecker delta property on the approximation functions, which difficult the
imposition of essential and natural boundary conditions.
To address the above problem, several interpolant meshless methods were
developed in the last few years. The most relevant are the Point Interpolation
Method (PIM) [ 21 ], the Point Assembly Method [ 22 ], the Radial Point Interpo-
lation Method (RPIM) [ 23 , 24 ], Meshless Finite Element Method (MFEM) [ 25 ].
The Natural Neighbour Finite Element Method (NNFEM) [ 26 , 27 ] or the Natural
Element Method (NEM) [ 28 - 30 ] are meshless methods that use the Sibson
interpolation functions. The combination between the NEM and the RPIM origi-
nated the Natural Neighbour Radial Point Interpolation Method (NNRPIM) [ 31 ],