Page 145 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN MULTI-DIMENSIONS
l h, T a k 137
T 1
q
2a

i j
2b

Figure 5.11 Rectangular element with different boundary conditions

From Equation 3.89, Chapter 3 (with origin at k), the shape functions for a rectangular
element are given as
x y

N i = 1 − 1 −
2b 2a
x y
N j = 1 −
2b 2a
xy
N k =
4ab
y x
N l = 1 − (5.35)
2a 2b
The gradient matrix of the shape functions is
 
∂N i ∂N j ∂N k ∂N l
 ∂x ∂x ∂x ∂x  1 −(2a − y) (2a − y) y −y
[B] =   = (5.36)
∂N i ∂N j ∂N k ∂N l
  4ab −(2b − x) −x x (2b − x)
∂y ∂y ∂y ∂y
The stiffness matrix is given by

T T
[K] = [B] [D][B]dV + h[N] [N]d (5.37)

where

k x 0
[D] = (5.38)
0 k y
Substituting, the [B] and [D] matrices into the above equation, results in a 4 × 4matrix.
We leave the algebra to the readers to work out. A typical term in the matrix is
2b 2a 2b 2a
k x 2 k y 2
(2a − y) dx dy + (2b − x) dx dy
2 2
2 2
16a b 16a b
0 0 0 0
2b 2a xy
+ dx dy (5.39)
0 0 4ab