Graphics Programs Reference
In-Depth Information
where
⎡
⎣
⎤
⎦
∂
x
∂ξ
∂
y
∂ξ
J
(
ξ,η
)
=
(6.44a)
∂
x
∂η
∂
y
∂η
isknown as the
Jacobian matrix
of the mapping. Substituting fromEqs. (6.41)
and (6.42) and differentiating, we find that the components of the Jacobian matrix
are
1
4
J
11
=
[
−
(1
−
η
)
x
1
+
(1
−
η
)
x
2
+
(1
+
η
)
x
3
−
(1
+
η
)
x
4
]
1
4
J
12
=
−
(1
−
η
)
y
1
+
(1
−
η
)
y
2
+
(1
+
η
)
y
3
−
(1
+
η
)
y
4
]
(6.44b)
[
1
4
J
21
=
[
−
(1
−
ξ
)
x
1
−
(1
+
ξ
)
x
2
+
(1
+
ξ
)
x
3
+
(1
−
ξ
)
x
4
]
1
4
J
22
=
[
−
(1
−
ξ
)
y
1
−
(1
+
ξ
)
y
2
+
(1
+
ξ
)
y
3
+
(1
−
ξ
)
y
4
]
Wecan nowwrite
1
1
,
=
ξ,η
,
ξ,η
|
ξ,η
|
ξ
η
f
(
x
y
)
dx dy
f
[
x
(
)
y
(
)]
J
(
)
d
d
(6.45)
A
−
1
−
1
Since the right-hand side integral istaken over the “standard” rectangle, itcan be
evaluatedusing Eq. (6.40).Replacing
f
(
) in Eq. (6.40) by the integrand in Eq. (6.45),
we get the following formula for Gauss-Legendrequadratureoveraquadrilateral
region:
ξ,η
n
n
A
i
A
j
f
x
(
ξ
i
,η
j
)
J
(
ξ
i
,η
j
)
I
=
ξ
i
,η
j
)
,
y
(
(6.46)
i
=
1
j
=
1
The
coordinates of the integrationpoints and theweights canagainbeobtained
from Table 6.3.
ξ
and
η
gaussQuad2
The function
gaussQuad2
computes
A
y
)
dx dy
overaquadrilateral element
with Gauss-Legendrequadratureofintegration order
n
. The quadrilateral is de-
finedbythe arrays
x
and
y
, which contain the coordinates of the four corners
f
(
x
,
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