Biomedical Engineering Reference
In-Depth Information
Fig. 5.21 a Generic solid
mechanics problem.
b Quarter of the square plate
under parabolic stress
c ¼ 0
:
0001 and p = 1.0001. The nodal connectivity is enforced using the second
degree influence-cells and the integration points are obtained using the ''basic''
nodal based integration scheme.
Besides the RPIM and the NNRPIM, the presented benchmark examples were
analysed with the finite element method (FEM) [
1
,
14
]. For the two-dimensional
analyses, it were considered discretizations with quadrilateral elements (FEM-4n)
and triangular elements (FEM-3n). For the three-dimensional analyses are con-
sidered hexahedral elements (FEM-8n) and tetrahedral elements (FEM-4n).
5.3.1 Square Plate Under Parabolic Stress
The proposed example combines several objectives: to analyse the behaviour of
the NNRPIM when random irregular nodal distributions are used in the analysis; to
compare the computational cost of the NNRPIM with other numerical methods;
and to study the convergence and accuracy level of the NNRPIM.
In this example it is considered the solid domain X
2
represented in
Fig.
5.21
a, with the following material properties: E = 1 kPa and t = 0.3. In the
natural boundary, C
t
2
X, of the solid domain the following stress field is applied,
R
x
2
L
2
y
2
r
xx
ð
x
Þ
¼r
0
D
2
L
2
2 D
2
Þ
x
2
þ
y
2
L
2
ð
ð
5
:
7
Þ
r
yy
ð
x
Þ
¼r
0
L
2
D
2
2 x
y
L
2
r
xy
ð
x
Þ
¼
r
0
being r
0
= 100 Pa. Due the problem symmetry the study of the complete problem
can be reduced to the analysis of only one quarter of the solid domain, Fig.
5.21
b.
Therefore, the displacement constrains on the essential boundary, C
u
2
X, must be
considered
as
represented
in
Fig.
5.21
b:
u ¼ 0:
8
y
2
R
^
x ¼ 0
and
