Biomedical Engineering Reference
In-Depth Information
Fig. 3.12
Triangular shape and quadrilateral shape and the respective integration points x
I
and for the quadrilateral shape the area is,
þ
det
A
I
¼
1
2
x
2
x
1
y
2
y
1
x
4
x
1
y
4
y
1
det
ð
3
:
12
Þ
x
3
x
1
y
3
y
1
x
3
x
1
y
3
y
1
This process is equivalent with the single integration point of the Gauss-
Legendre quadrature scheme for triangle and quadrilateral shapes, respectively.
Gauss-Legendre Quadrature Integration Scheme
From the previous basic integration scheme it is possible to obtain a more general
integration scheme. The sub-cell basic geometric forms in Fig.
3.12
are sub-
divided again, however in this case only as quadrilaterals. Using the triangle and
the quadrilateral sub-cells obtained in Fig.
3.12
, firstly it is determined the centre
of the geometric shape, x
C
, then middle points on the quadrilateral edges are
determined, x
ij
¼
ð
x
i
þ
x
j
Þ=
2, and thus new sub-quadrilaterals are defined,
Fig.
3.13
. It is possible to apply the Gauss-Legendre quadrature to the obtained
sub-quadrilaterals in order to obtain the integration points [
32
,
34
]. The process is
briefly described in
Sect. 3.3.1
. This process permits to fill each sub-quadrilateral
with k
k integration points. In Fig.
3.13
are shown distinct integrations schemes
for the triangular and the quadrilateral sub-cell simultaneously: a 1
1 quadrature
per sub-quadrilateral; and a 3
3 quadrature per sub-quadrilateral.
The integration weight of each integration point x
I
is obtained using the fol-
lowing expression,
A
h
4
_
I
¼ w
g
w
n
ð
3
:
13
Þ
being A
h
the area of the respective sub-quadrilateral, which can be obtained using
Eq. (
3.12
), and w
g
and w
n
are the Gauss-Legendre quadrature weights for an
isoparametric quadrilateral cell, Fig.
3.9
.
