Page 153 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN MULTI-DIMENSIONS
5.6.1 Galerkin’s method for linear triangular axisymmetric elements
The Galerkin method for axisymmetric equations results in the following integral form 145
k r ∂ ∂T ∂ T
2
N i r + k z 2 + G d
= 0 (5.65)

r ∂r ∂r ∂z
The spatial approximation of temperature is given by Equation 5.61. As in the previous
sections, the substitution of the spatial approximation will result in the familiar final form
of the matrix equation as
[K]{T}={f} (5.66)
where
T T
[K] = [B] [D][B]d
+ h[N] [N]d (5.67)

Here,
   
∂T ∂N i ∂N j ∂N k
 
 
∂x  ∂r ∂r ∂r  1 b i b j b k
 
[B] = =   = (5.68)
 ∂N i ∂N j ∂N k  2A c i c j c k
 ∂T 
 
 
∂y ∂z ∂z ∂z
and

k r 0
[D] = (5.69)
0 k z
In Equation 5.67, the volume
is defined as
dV = 2πr dA (5.70)
where r is the radius, which varies and can be approximated using linear shape functions as
r = N i r i + N j r j + N k r k (5.71)
Substituting into Equation 5.67 and integrating, we obtain
 2   2 
b b i b j b i b k c c i c j c i c k
2πrk r i 2 2πrk z i 2
[K] =  b i b j b j b j b k  +  c i c j c j c j c k 
4A 2 4A 2
b i b k b j b k b k c i c k c j c k c k
 
3r i + r j r i + r j 0.0
2πhl ij
+  r i + r j r i + 3r j 0.0  (5.72)
12
0.0 0.0 0.0
where
r i + r j + r k
r = (5.73)
3
Similarly,

T T T
{f}= G[N] d
− q[N] d + hT a [N] d

      

0
2πGA 211 r i  2πql jk   2πhT a l ij 2r i + r j
=  121  r j − 2r j + r k + r i + 2r j (5.74)
12   6   6  
112 r k r j + 2r k 0