Biomedical Engineering Reference
In-Depth Information
nodal mesh, which can follow a regular or irregular distribution. In meshless
methods the absence of elements, which pre-establish the nodal connectivity in the
FEM, requires the application of an interpolation or approximation technique,
based on a moving local nodal domain, to permit the construction of the meshless
shape function for the approximation of the field variable.
In this chapter two distinct meshless shape function construction techniques are
studied in detail: an approximation shape function approach and an interpolation
shape function approach. Both presented shape functions techniques are desig-
nated as locally supported, since the shape functions are constructed for an arbi-
trary interest point x
I
using only a small set of field nodes spatially localized in the
vicinity of x
I
. This small set of field nodes in the vicinity of x
I
is called the
support-domain of the shape function. Generally, the support-domain of the shape
function constructed for an arbitrary interest point x
I
is coincident with the
influence-domain of x
I
. The constructed shape function assumes non-zero values
inside the support-domain and is null outside the support-domain.
The construction and development of shape functions assume great importance
in meshless methods [
3
], since the shape functions construction methodology
should be able to use only the nodes discretizing the domain without the need of
any pre-established mesh providing the nodal connectivity. The shape functions
construction methodology should also be computationally efficient, with the pur-
pose of being a proficient FEM alternative, and capable to deal effortlessly with
random nodal distributions, in order to solve practical engineering problems.
Additionally the methodology should be: numerically stable; present a certain
order of consistency; compactly supported; compatible; and, if possible, satisfy the
Kronecker delta property [
4
,
5
].
In the following sections the moving least squares approximation (MLS)
methodology and the radial point interpolation (RPI) technique for the construc-
tion of meshless shape functions are described in detail. Presently, the MLS and
the RPI are among the most popular techniques used to construct meshless shape
functions. Nonetheless, other appropriate methodologies to construct meshless
shape functions are available in the literature and extensively described in [
4
,
5
].
4.2 Support-Domain
Most of the procedures used to construct the meshless shape function apply the
support-domain concept. The shape function support-domain can be defined as the
set of field nodes that directly contribute to the construction of the shape function.
Usually the shape function support-domain of an interest point is coincident with
the interest point influence-domain.
The size and shape of the support-domain should be carefully selected, since the
accuracy of the approximation and the computational efficiency of the shape
function construction strongly depends on this parameter. To illustrate this last
remark, consider the two-dimensional domain X
2
R
discretized with a nodal set
