Biomedical Engineering Reference
In-Depth Information
Fig. 5.47 Two-dimensional
nodal distributions examples
and a three-dimensional
nodal distribution example
In Table
5.8
it is shown the number of integration points obtained for each
integration scheme considered in the present two-dimensional analysis. The
integration schemes respect the same nomenclature used in
Sect. 5.2.2
. The simple
nodal based integration scheme described in Section ''
Basic Integration Scheme
''
is the first integration scheme considered. In Table
5.8
this simple integration
scheme is called ''basic''. The other k 9 k integrations schemes presented in
Table
5.8
are
obtained
following
the
procedure
indicated
in
Section
The first three vibration modes are obtained with the NNRPIM, for the distinct
integration schemes, and compared with the first three vibration modes obtained
with FEM software, ABAQUS, for a regular nodal distribution of 4,000 nodes,
f
FEM
1
¼ 830 Hz, f
FEM
2
¼ 4,979 Hz and f
FEM
3
¼ 12,826 Hz. The difference between
the two methods is calculated by,
q
(f
i
)
meshless
(f
i
)
FEM
ð
ABAQUS
Þ
ffi
2
Error
ð
f
i
Þ
¼
q
(f
i
)
FEM
ð
ABAQUS
Þ
ð
5
:
15
Þ
ffi
2
The results for the distinct NNRPIM formulations are presented in Fig.
5.56
.As
it is visible for all formulations the integration scheme 1 9 1 seems to be suffi-
cient. There is no significant improvement in the accuracy of the solution when a
high order integration scheme is used. Therefore, in further examples the inte-
gration scheme used in the three considered NNRPIM formulations is the 1 9 1
integration scheme.
