Page 78 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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which gives
1
L 1 =
4 THE FINITE ELEMENT METHOD
1
L 2 =
8
5
L 3 = (3.144)
8
Substituting into Equation 3.142 gives
 
8.5
 
 ∂N 4 
   
∂x 8
   
= (3.145)
 ∂N 4  −1.5
   
 
∂y 16
Similarly, other derivatives can also be calculated.
3.2.8 Three-dimensional elements
The amount of data required to establish the computational domain and boundary conditions
become significantly greater in three dimensions than for two-dimensional problems. It is
therefore obvious that the amount of computational work/cost increases by a considerable
extent. Therefore, appropriate three-dimensional elements need to be used. The tetrahedron
and brick-shaped hexahedron elements are developed (Figure 3.21) in this section, which
are extensions of the linear triangle and quadrilateral elements in two dimensions.
The linear temperature representation for a tetrahedron element (three-dimensional lin-
ear element) is given by
T = α 1 + α 2 x + α 3 y + α 4 z (3.146)
As discussed previously for 2D elements, the constants of Equation 3.146 can be deter-
mined and may be written in the following form:
(3.147)
T = N 1 T 1 + N 2 T 2 + N 3 T 3 + N 4 T 4

4
8 (−1,1,1) 7(1,1,1) 6
4
4 (−1,1,−1)
5
3 (1,1,−1)
2
5 (−1,−1,1)
6 (1,−1,1)
1 3
1
3 1 (−1,−1,−1) 2 (1,−1,−1) 2
(c)
(a) (b)
Figure 3.21 Three-dimensional elements, (a) tetrahedron, (b) hexahedron and (c) prism