Page 80 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
72
c-terms
c 1 = (x 3 − x 4 )(z 2 − z 4 ) − (x 2 − x 4 )(z 3 − z 4 )
c 2 = (x 1 − x 4 )(z 3 − z 4 ) − (x 3 − x 4 )(z 1 − z 4 )
c 3 = (x 2 − x 4 )(z 1 − z 4 ) − (x 1 − x 4 )(z 2 − z 4 )
c 4 =−(c 1 + c 2 + c 3 ) (3.156)
d-terms
d 1 = (x 2 − x 4 )(y 3 − y 4 ) − (x 3 − x 4 )(y 3 − y 4 )
d 2 = (x 3 − x 4 )(y 1 − y 4 ) − (x 1 − x 4 )(y 3 − y 4 )
d 3 = (x 1 − x 4 )(y 2 − y 4 ) − (x 2 − x 4 )(y 1 − y 4 )

d 4 =−(d 1 + d 2 + d 3 ) (3.157)

A volume coordinate system for the tetrahedron can be established in a similar manner
as were the area coordinates for a triangle. In the tetrahedron, four distance ratios are used,
each normal to sides L 1 ,L 2 ,L 3 and L 4 .
Note that L 1 + L 2 + L 3 + L 4 = 1.
The linear shape functions are related to the volume coordinate as follows:

N 1 = L 1 ; N 2 = L 2 ; N 3 = L 3 and N 4 = L 4 (3.158)
The volume integrals can easily be evaluated from the relationship,

a b c d a!b!c!d!
L L L L dV = 6V (3.159)
1
2 3 4
V (a + b + c + d + 3)!
For a quadratic tetrahedron,
2
2
2
T = α 1 + α 2 x + α 3 y + α 4 z + α 5 x + α 6 y + α 7 z + α 8 xy + α 9 yz + α 10 zx (3.160)
Therefore, ten nodes will exist in a quadratic tetrahedron as shown in Figure 3.22.
The element may also have curved surfaces on the boundaries. As before, the temperature
distribution can be rewritten in terms of the shape functions as

T = N 1 T 1 + N 2 T 2 + N 3 T 3 + N 4 T 4 + N 5 T 5
+N 6 T 6 + N 7 T 7 + N 8 T 8 + N 9 T 9 + N 10 T 10 (3.161)
The shape functions can be expressed in terms of local coordinates as

N 1 = L 1 (2L 1 − 1)

N 2 = L 2 (2L 2 − 1)
N 3 = L 3 (2L 3 − 1)
N 4 = L 4 (2L 4 − 1)