Page 79 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

71
THE FINITE ELEMENT METHOD
where
1
N i =
6V (a i + b i x + c i y + d i z) with i = 1, 2, 3, 4 (3.148)
The volume of the tetrahedron is expressed as
 
1 x 1 y 1 z 1
 1 x 2 y 2 z 2 
6V = det   (3.149)
1 x 3 y 3 z 3
 
1 x 4 y 4 z 4
Also note that
∂N 1 b 1
=
∂x 6V
∂N 1 c 1
=
∂y 6V
∂N 1 d 1
= (3.150)
∂z 6V
Therefore, the gradient matrix of the shape functions can be written as
 
1 b 1 b 2 b 3 b 4
[B] =  c 1 c 2 c 3 c 4  (3.151)
6V
d 1 d 2 d 3 d 4
where
 
1 y 2 z 2
b 1 =−det  1 y 3 z 3  (3.152)
1 y 4 z 4
 
x 2 1 z 2
c 1 =−det  x 3 1 z 3  (3.153)
x 4 1 z 4
 
x 2 y 2 1
d 1 =−det  x 3 y 3 1  (3.154)
x 4 y 4 1

Similarly, the other terms in Equation 3.151 can also be determined. We therefore
summarize all the terms as follows:
b-terms
b 1 = (y 2 − y 4 )(z 3 − z 4 ) − (y 3 − y 4 )(z 2 − z 4 )
b 2 = (y 3 − y 4 )(z 1 − z 4 ) − (y 1 − y 4 )(z 3 − z 4 )
b 3 = (y 1 − y 4 )(z 2 − z 4 ) − (y 2 − y 4 )(z 1 − z 4 )

b 4 = b 1 + b 2 + b 3 (3.155)