Graphics Programs Reference
In-Depth Information
The M-file that returns the function to be integratedis
_
functionz=fex6
15(x,y)
% Function used in Example 6.15.
z=((x-2)*(y-2))ˆ2;
Quadrature over a Triangular Element
3
4
Figure 6.8.
Quadrilateral with twocoincidentcorners.
1
2
A triangle may be viewedas adegenerate quadrilateral with twoofits corners
occupying the same location, as illustrated in Fig. 6.8. Therefore, the integration for-
mulasoveraquadrilateral region can also be used foratriangular element. However,
it iscomputationally advantageoustouse integration formulasspeciallydeveloped
for triangles, which we present without derivation.
15
3
y
x
A
Figure 6.9.
Triangular element.
A
P
1
2
A
3
2
1
Consider the triangular element in Fig. 6.9. Drawing straight lines from the point
P
in the triangle to each of the corners divides the triangle into three parts with areas
A
1
,
A
2
and
A
3
. The so-called
area coordinates
of
P
are definedas
A
i
A
,
α
i
=
i
=
1
,
2
,
3
(6.47)
where
A
is the area of the element.Since
A
1
+
A
2
+
A
3
=
A
, the area coordinates are
relatedby
α
1
+
α
2
+
α
3
=
1
(6.48)
Note that
α
i
ranges from 0(when
P
lies on the sideopposite to corner
i
)to1 (when
P
is atcorner
i
).
15
The triangle formulas areextensivelyusedinthe finite method analysis.See, for example, O.C.
Zienkiewicz and R.L Taylor,
The Finite Element Method
,Vol.1, 4th ed.,McGraw-Hill (1989).
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