Biomedical Engineering Reference
In-Depth Information
being h in the present example a constant value: h ¼ h
x
¼ h
y
¼ 0
:
2 m, Fig.
5.2
.
Consequently the shape parameter c suggested in the literature is c ¼ 7
:
15. It is
visible in Fig.
5.3
that such values for the parameters conducts to a significantly
low medium displacement error: E
med
10
5
. However the same c
¼
7
:
15 leads
to a very high value of the sum of the interior equivalent forces: f
tot
10
9
, which
is unacceptable, it should be zero.
This anomaly is explained with the effective lack of the Kronecker delta
property on the RPIM shape functions for high values of c.In
Sect. 4.4.4
it was
possible to observe in Fig.
4.23
that the value of the shape parameter c regulates
the silhouette of the shape function. Considering small values for c the constructed
shape function becomes cone-shaped. If c is increased then the peak of the shape
function becomes flat. As a consequence, the value of the shape parameter c affects
the accuracy of the solution. However the shape parameter c cannot be null,
because c ¼ 0 leads ill-conditioned or singular moment matrices [
11
,
12
].
It is now understandable that for high values of c the shape function does not
pass exactly on the nodes. To prove the statement, during the previous analyses the
shape function vector constructed for node n
15
, u
ð
x
15
Þ
, was saved. The u
ð
x
15
Þ
values on nodes n
9
, n
14
, n
16
and n
21
(all inside the support-domain of n
15
) were
used to calculate the following medium value,
u
med
¼
1
4
u
9
ð
x
15
Þþ
u
14
ð
x
15
Þþ
u
16
ð
x
15
Þþ
u
21
ð
x
15
Þ
ð
Þ
ð
5
:
5
Þ
Note that these four nodes are the closest nodes to node n
15
and if u
ð
x
15
Þ
possesses the Kronecker delta property, u
i
ð
x
15
Þ
should be zero for all i
6
¼ 15, and
consequently u
med
¼ 0. However, it is visible in Fig.
5.3
a, b that u
med
only sta-
bilizes for c
0
:
1 and even for those values of c, u
med
is different of zero,
u
med
10
7
, indicating an effective lack of the Kronecker delta property.
As in the RPIM early works [
9
,
10
], in this example the essential boundary
conditions were directly imposed, as in FEM, since it was initially assumed that
the RPI shape functions possess the Kronecker delta property. This is the reason
why f
tot
is different from zero. However, it is acceptable to consider, for c
0
:
1,
that the RPIM shape functions possesses the Kronecker delta property.
Next, the optimal value of the shape parameter p is pursue. The procedure to
find p
opt
is similar with the previous shown example. Considering c
opt
¼ 0
:
0001,
the shape parameter p is changed between p ¼½10
4
;
5
until an optimal p is
achieved. All other considerations regarding the material properties, the geometric
and the boundary conditions remain the same as in previous analyses. Also the two
nodal distributions previously considered and the respective background integra-
tion mesh Fig.
5.1
, are once more assumed.
The obtained results are shown in Fig.
5.4
. As it is possible to visualize, p
1
is an optimal value since E
med
show a minimum for this value for both the regular
and the irregular nodal distribution. Additionally, when p % 1 is considered the
sum of the interior equivalent forces f
tot
present a very low value, as it should.
