Page 72 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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The quantity (dx/dζ) is called the Jacobian of the coordinate transformation and is
denoted by [J]. For a one-dimensional coordinate, transformation [J] is calculated using
dx dN i dN j THE FINITE ELEMENT METHOD
dN k
[J] = = x i + x j + x k (3.108)
dζ dζ dζ dζ
Example 3.2.4 Derive the shape function derivatives for a one-dimensional quadratic ele-
ment that has nodal coordinates x i = 2, x j = 4 and x k = 6.
The Jacobian matrix is written as
dx
[J] =
dζ
dN i dN j dN k
= x i + x j + x k
dζ dζ dζ

1 1
= − + ζ 2 + (−2ζ)4 + + ζ 6
2 2
= 2 + 8ζ − 8ζ = 2 (3.109)
thus,
1
−1
[J] = (3.110)
2
The shape function derivatives are written as follows:
   
 dN i   dN i     
1 −1 ζ
   
   
   dζ     
+
 dx     − + ζ   
       
     2   4 2 
dN j −1 dN j 1
= [J] = −2ζ = −ζ (3.111)
 1

 dx   dζ  2    1 
     ζ 
+ ζ
       
       + 
 dN k 
   2 4 2
 dN k 
   
dx dζ
For two-dimensional cases, we may express x and y as functions of ζ and η,thatis,
x = x(ζ, η) and y = y(ζ, η) (3.112)
Since we deal with Cartesian derivatives for the calculation of the stiffness matrix, we
transform the derivatives of the shape functions using the chain rule as follows,
∂N i ∂N i ∂x ∂N i ∂y
(x, y) = +
∂ζ ∂x ∂ζ ∂y ∂ζ
∂N i ∂N i ∂x ∂N i ∂y
(x, y) = + (3.113)
∂η ∂x ∂η ∂y ∂η
which can be written as
      
∂x ∂y

 ∂N i   ∂N i   ∂N i 
     
∂ζ  ∂ζ ∂ζ  ∂x ∂x
     
=   = [J] (3.114)
 ∂x ∂y 
∂N i   ∂N i   ∂N i 
     
∂η ∂η ∂η ∂y ∂y
     