Biomedical Engineering Reference
In-Depth Information
1
2
(1
−
ξ
2
)(1
−
η
)
N
5
=
1
2
(1
+
ξ
)(1
−
η
2
)
N
6
=
1
2
(1
2
)(1
N
7
=
−
ξ
+
η
)
1
2
(1
2
) .
N
8
=
−
ξ
)(1
−
η
(17.35)
Other examples may be found in Zienkiewicz [
18
] and Hughes [
10
].
17.5
Numerical integration
Let
f
:
e
→
IR be some function, and assume that the integral:
f
(
x
)
dx
,
(17.36)
e
over the domain
e
of an element is to be computed. In finite element computa-
tions there is a mapping from the
x
-space to the
ξ
-space, such that (see Section
16.7
, on isoparametric elements)
1
)
dx
d
ξ
f
(
x
)
dx
=
f
(
)
φ
(
ξ
)
ξ
(
ξ
d
ξ
.
(17.37)
e
−
1
This integral can be approximated with a numerical integration rule:
1
n
int
g
(
ξ
)
d
ξ
≈
g
(
ξ
i
)
W
i
,
(17.38)
−
1
i
=
1
where
ξ
i
denotes the location of an integration point and
W
i
the associated weight
factor.
In Fig.
17.9
an interpretation is given of the above numerical integration rule.
At a discrete number of points
ξ
i
within the interval
ξ
∈
[
−
1,
+
1] the function
value
g
(
ξ
i
) is evaluated. Related to each point
ξ
i
a rectangle is defined with height
g
(
ξ
i
) and width
W
i
. Note that it is not necessary that the point
ξ
i
is located on the
symmetry line of the rectangle. By adding up the surfaces
g
(
ξ
i
)
W
i
of all rectangles
an approximation is obtained of the total surface underneath the function, which
is the integral. It is clear that the weight factor
W
i
in Eq. (
17.38
) can be interpreted
as the width of the interval around
ξ
i
.
The integration rule that is mostly used is the
Gaussian quadrature
. In that
case the locations of the integration points and weight factors are chosen so as
