Biomedical Engineering Reference
In-Depth Information
Table 17.1 Gaussian quadrature up to
n
int
=3.
n
int
ξ
i
W
i
1
ξ
1
= 0
W
1
= 2
−
1
√
3
,
1
√
3
2
ξ
1
=
ξ
2
=
W
1
=
W
2
=
1
5
,
ξ
2
= 0,
ξ
3
=
5
5
9
,
W
2
=
8
9
3
ξ
1
=−
W
1
=
W
3
=
g
(
ξ
)
W
i
ξ
i
-1
1
ξ
Figure 17.9
Numerical integration of a function
φ
(
ξ
).
to obtain optimal accuracy for polynomial expressions of
g
(
ξ
). In Table
17.1
the
location of the integration points and the associated weight factors are given up to
n
int
=
3. For two-dimensional problems the above generalizes to
1
1
f
(
x
,
y
)
d
=
f
(
x
(
ξ
,
η
),
y
(
ξ
,
η
))
j
(
ξ
,
η
)
d
ξ
d
η
e
−
1
−
1
1
1
=
g
(
ξ
,
η
)
d
ξ
d
η
,
(17.39)
−
1
−
1
with
j
(
ξ
,
η
) according to Eq. (
16.50
) and
1
1
n
int
n
int
g
(
ξ
,
η
)
d
ξ
d
η
≈
g
(
ξ
i
,
η
j
)
W
i
W
j
.
(17.40)
−
1
−
1
i
=
1
j
=
1
The above integration scheme can be elaborated for the 9-node rectangular
Lagrangian element in Fig.
17.10
using Table
17.2
.
As was already remarked in Section
17.3
integration over a triangular domain is
not trivial. For a triangular element that is formed by degeneration from a quadri-
lateral element the integration can be performed in the same way, with the same
integration points, as given above.
In the case that triangular coordinates are used, the evaluation of the integrals is
far from trivial. If the triangle coordinates
λ
1
and
λ
2
are maintained by eliminating
