Page 61 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

53
THE FINITE ELEMENT METHOD
Thus, the local coordinate L i varies from 0 on the side jk to 1 at the node i.From
Figure 3.9 it is obvious that
A i + A j + A k = A (3.53)
or
A i A j A k
+ + = 1 (3.54)
A A A
therefore
L i + L j + L k = 1 (3.55)
The relationship between the (x, y) coordinates and the natural, or area, coordinates are
given by
x = L i x i + L j x j + L k x k (3.56)
and
y = L i y i + L j y j + L k y k (3.57)
From Equations 3.55, 3.56 and 3.57, the following relations for the local coordinates
can be derived:
1
L i = (a i + b i x + c i y)
2A
1
L j = (a j + b j x + c j y)
2A
1
L k = (a k + b k x + c k y) (3.58)
2A
where the constants a, b and c are defined in Equation 3.40. Comparing with Equation 3.39,
it is clear that
L i = N i

L j = N j
L k = N k (3.59)
Thus, the local or area coordinates in a triangle are the same as the shape functions for
a linear triangular element. In general, the local coordinates and shape functions are the
same for linear elements irrespective of whether they are of one, two or three dimensions.
For a two-dimensional linear triangular element, with local coordinates L i , L j and L k ,
we have a simple formula for integration over the triangle, that is,

a b c a b c a!b!c!
L L L dA = N N N dA = 2A (3.60)
k
j
i
i
j
k
A A (a + b + c + 2)!
where ‘A’ is the area of a triangle. Note that L i ,L j and L k happen to be the shape functions
for a linear triangular element. Example 3.2.2 can also be solved using the local coordinates
via Equations 3.53 and 3.56, that is, on substituting the x and y coordinates of the three
points (Table 3.2) of the triangle into Equation 3.56, we obtain
x
L j = (3.61)
4