Page 96 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
88
∂T
∂T
∂T
(3.242)
˜
k x
l + k y
˜ m + k z
˜ n + h(T − T a ) = 0 on surface S 3
∂z
∂y
∂x
where l, ˜m and ˜n are surface normals, h is the heat transfer coefficient, k is the thermal
˜
conductivity and q is the heat flux.
3.4.1 Variational approach
The variational integral, I, corresponding to the above differential equation with its bound-
ary conditions is given by
2 2 2
1 ∂T ∂T ∂T
I(T ) = k x + k y + k z − 2GT d
2
∂x ∂y ∂z
1 2

+ qT ds + h(T − T a ) ds (3.243)
2
S 2 S 3
The given domain
is divided into ‘n’ number of finite elements with each element
having ‘r’ nodes. The temperature is expressed in each element by
r
e
T = N i T i = [N]{T} (3.244)
i=1
where [N] = [N i ,N j ,... ,N r ] = shape function matrix and
 
 T i 
 
 
T j
{T}= (3.245)
...
 
 
T r
is the vector of nodal temperatures.
The finite element solution to the problem involves selecting the nodal values of T so
as to make the function I(T ) stationery. In order to make I(T ) stationery, with respect to
the nodal values of T , we require that
n
∂I

δI (T ) = = 0 (3.246)
∂T i
i=1
where n is the total number of discrete values of T assigned to the solution domain. Since
T i are arbitrary, Equation 3.246 holds good only if
∂I
= 0for i = 1, 2,... ,n (3.247)
∂T i
The functional I(T ) can be written as a sum of individual functions, defined for the
assembly of elements, only if the shape functions giving piece-wise representation of T