Biomedical Engineering Reference
In-Depth Information
Table 14.1 Gaussian quadrature points 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
,
5
5
9
,
W
2
=
8
9
3
ξ
1
=−
ξ
2
=
0,
ξ
3
=
W
1
=
W
3
=
ξ
)
g
(
ξ
)
g
(
ξ
ξ
-1
+1
+1
-1
1
1
-
√
3
√
3
(a)
(b)
Figure 14.7
(a) Trapezoidal integration (b) 2-point Gauss integration.
which corresponds to the shaded area in Fig.
14.7
(a). For trapezoidal integration
the integration point positions
ξ
i
are given by
ξ
1
=−
1,
ξ
2
=
1,
(14.76)
while the associated weighting factors are
W
1
=
1,
W
2
=
1.
(14.77)
The trapezoidal integration rule integrates a linear function exactly. A 2-point
Gaussian integration rule, as depicted in Fig.
14.7
(b) may yield a more accurate
result since this integration rule integrates up to a third order function exactly
using two integration points only. In this case the integral is approximated by
1
g
(
ξ
)
d
ξ
≈
g
+
g
.
−
1
√
3
1
√
3
ξ
=
ξ
=
(14.78)
−
1
The location of the Gaussian integration (quadrature) points and the associated
weighting factors are summarized in Table
14.1
.
Application to element coefficient matrix
Use of the local coordinate sys-
tem, with isoparametric formulation and numerical integration to the element
coefficient matrix yields
