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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