Biomedical Engineering Reference
In-Depth Information
Fig. 4.1
Schematic representation of a generic influence-domain
2
represented in Fig.
4.1
. As indicated in
Fig.
4.1
the influence-domain of a generic interest point x
I
contains several field
nodes. It is possible to observe that compared with the influence-domain of x
I
, the
support-domain of the shape functions is much smaller. This size discrepancy
reduces the computational efficiency of the shape function construction procedure.
Notice that the contribution of node x
4
for the shape functions of interest point x
I
is zero, u
4
ð
x
I
Þ
¼0. In opposition the other nodes in the vicinity of x
I
, such are
nodes x
1
, x
2
and x
3
, contribute with non-zero values, u
1
ð
x
I
Þ 6
¼ 0, u
2
ð
x
I
Þ 6
¼ 0 and
u
3
ð
x
I
Þ 6
¼ 0. If the size of the influence-domain was coincident with the size of the
support-domain, all nodes belonging to the influence-domain of x
I
would actively
contribute for the construction of the shape function.
Similar to the influence-domain, usually the support-domain is centred in an
interest point, which can be a node or an integration point, and it can assume
several distinct geometric shapes and sizes. The most usual shapes are the circular
(as in Fig.
4.1
) and the rectangular shape.
It is possible to defined the support-domain size of an interest point x
I
with the
following expression,
X ¼ x
1
;
x
2
;
...
;
x
N
f
g 2
X
^
x
i
2
R
d
s
¼ b
s
d
a
ð
4
:
1
Þ
Being b
s
a dimensionless parameter governing the size of the support-domain
and d
a
is the average nodal spacing inside the support-domain. The parameter b
s
has
to be pre-established by the program user, before the analysis. The value adopted for
b
s
depends on the used meshless shape function and it should be determined by
preliminary numerical studies. Commonly benchmark examples, for which the
exact solutions are known, are used to determine the optimal value of b
s
. Previous
works on the MLS shape functions and RPI shape functions indicate that b
s
parameters ranging between b
s
¼½2
:
0
;
3
:
0
lead to stable and accurate solutions.
