Graphics Programs Reference
In-Depth Information
weobtain
A j n
ξ i i )
1
n
n
I
=
A i f (
ξ i
) d
η =
A i f (
1
i
=
1
j
=
1
i
=
1
or
n
n
I
=
A i A j f (
ξ i j )
(6.40)
i
=
1
j
=
1
The number of integrationpoints n in each coordinate directioniscalled the in-
tegration order . Figure 6.7(a) shows the locations of the integrationpoints usedin
third-orderintegration ( n
=
3).Because the integration limits were the “standard”
limits (
1)ofGauss-Legendrequadrature, the weights and the coordinates of the
integrationpoints are aslisted Table 6.3.
In order to apply quadrature to the quadrilateral element in Fig. 6.7(b), we must
first map the quadrilateral into the “standard” rectangle in Fig. 6.7(a). By mapping
we mean a coordinate transformation x
1
,
)that results in one-to-
onecorrespondence betweenpoints in the quadrilateral and in the rectangle. The
transformation that does the jobis
=
x (
ξ,η
), y
=
y (
ξ,η
4
4
x (
ξ,η
)
=
N k (
ξ,η
) x k
y (
ξ,η
)
=
N k (
ξ,η
) y k
(6.41)
k
=
1
k
=
1
where( x k ,
y k ) are the coordinates of corner k of the quadrilateral and
1
4 (1
N 1 (
ξ,η
)
=
ξ
)(1
η
)
1
4 (1
N 2 (
ξ,η
)
=
+ ξ
)(1
η
)
(6.42)
1
4 (1
N 3 (
ξ,η
)
=
+ ξ
)(1
+ η
)
1
4 (1
N 4 (
ξ,η
)
=
ξ
)(1
+ η
)
The functions N k (
), known as the shape functions , are bilinear (linear in each
coordinate).Consequently,straight lines remain straight uponmapping. Inparticular,
note that the sides of the quadrilateral are mappedinto the lines
ξ,η
ξ
1 and
η
1.
Because mapping distorts areas, an infinitesimal area element dA
=
dx dy of the
quadrilateral is not equaltoits counterpart d
ξ
d
η
of the rectangle. It can be shown
that the relationship between the areas is
dx dy
= |
J (
ξ,η
)
|
d
ξ
d
η
(6.43)
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