Biomedical Engineering Reference
In-Depth Information
Fig. 5.22
a
5 9 5
regular
nodal
distribution.
b
Irregular
nodal
distribution
with k = 5.
c Irregular nodal distribution with k = 2
v ¼ 0:
8
x
2
^
y ¼ 0, being u ¼
f
u
;
v
g
. The analytical displacement field [
15
],
which can be obtained from the analytical stress field of Eq. (
5.7
), is described as,
R
3 L
2
x
y
2
x
3
x
3
L
2
2 D
2
þ
x
y
2
L
2
u
ð
x
Þ
¼
r
0
E
ð
Þ
D
2
t
3 L
2
D
2
ffi
ð
5
:
8
Þ
x
2
L
2
2D
2
v
ð
x
Þ
¼
r
0
E
y
þ
y
3
x
2
y
L
2
y
3
3 D
2
3 L
2
t
L
2
D
2
Firstly it is studied the behaviour of the NNRPIM when random irregular nodal
distributions are used in the analysis. To create random irregular nodal distribu-
tions the following procedure is used. A uniform nodal distribution is constructed,
with all nodes equally spaced and aligned, as in Fig.
5.22
a, then all the nodes
x
2
X
n
C are affected with,
¼ x
i
þ
r
1
h
k
x
new
i
cos 2 r
2
p
ð
Þ
ð
5
:
9
Þ
¼ y
i
þ
r
1
h
k
y
new
i
sin 2 r
2
p
ð
Þ
Being x
i
the initial node n
i
coordinates, x
ne
i
the new obtained node n
i
coor-
dinates and h is the distance shown in Fig.
5.22
a. The random parameter is defined
by r N
ð
0
;
1
Þ
and k is a parameter that controls the irregularity level of the nodal
distribution. The three nodal distributions presented in Fig.
5.22
show the effect of
the irregularity parameter k on the nodal distribution; notice that if k = ? the
nodal distribution is perfectly regular, Fig.
5.22
a, and with the decrease of k the
nodal distribution becomes more and more irregular, Fig.
5.22
b, c.
The proposed mechanical problem was analysed considering several irregular
nodal distributions with 21
21 ¼ 441 nodes, varying the irregularity parameter
from k ¼ 100 (practically a regular mesh) to k ¼ 2 (extremely irregular mesh).
Each irregular nodal distribution was used to analyse the problem considering the
NNRPIM, the FEM-4n, the FEM-3n and the RPIM. The results of the medium
displacement error, Eq. (
5.1
), are presented in Fig.
5.23
a. The results regarding the
