• We will look at manipulation of the mechanics quantities
(displacement, strain, stress) using shape functions
• The approach is quite general, and is used to formulate a number of different elements
• We will use a specific example to make the development more concrete (Q4)
• We will start from the nodal displacement representation, work toward strain and stress, and finally element stiffness
• There is a lot going on here, pay attention to both the overall themes and the detailed steps …

• Note: we will use a for x and b for h because I can’t remember, pronounce, or legibly write “xi” and “eta” actual element master element


• Interpolation involves the summation of nodal values multiplied by corresponding shapes functions
 x
 y
 


N i x i

    c
N i y i
 u
 v
 


N i u i
N i v i

    d geometry interpolation
 
 



 x u
1
1 y
1 v
1 x
2
- where y

2 x
3 u
2 v
2 u
3 field variable interpolation y
3 v
3 u
4 x
4 y
4
 T v
4
 T nodal coordinates nodal displacements
  


N
1
0
0 N
1
N
0
2
0
N
2
N
0
3
0
N
3
N
0
4
0
N
4

 shape functions


• Start with the element displacement field
• We have to give it some functional form in order to work with it
• Let it be defined over the element by our interpolation scheme
     d

N
1

1
4

1
 a
 
1
 b

N
2

1
4

1
 a
 
1
 b


N
3

1
4

1
 a
 
1
 b

N
4

1
4

1
 a
 
1
 b


[N] = the element shape functions in master element coordinates
{d} = the nodal (discrete) displacement values


• Now calculate element strains from the displacement field
     u
  







 x
0

 y
0

 y

 x













 x y xy


 










0
 x
 y
0

 y








 u
 v



 x
• This is just the usual strain-displacement relationship written



• Let’s now work toward an expression for element strain
    
N d
- or -
     d







 x y xy


 










0
 x
 y
0

 y

 x



 this operation cannot be done directly





N
1
0
0 N
1
N
0
2
0
N
2
N
0
3
0
N
3
N
0
4
0
N
4









 u
1 v u
1
 v u
2
2
3



 v u
4
3










• We have a bit of a difficulty here with direct substitution  v
4
– The shape functions (N
1
, N
2
, N
3
, N element coordinates (a,b)
4
) are defined in terms of the master
– But we need to differentiate in terms of the global coordinates (x,y)

• Given any function of the master element coordinates (a,b): f
  
• We can find derivatives with respect to global (x,y) by using the chain rule:
 f
 a

 f
 x
 x
 a

 f
 y
 y
 a
 f
 b

 f
 x
 x
 b

 f
 y
 y
 b
• We can combine and rearrange these relationships to get our derivatives:


 f , a
 f , b



  
 f , f , y x


 f , f , y x

    
1

 f , a
 f , b






• The Jacobian matrix is an important part of element formulation:
  


 x , a x , b y , a y , b







N i , a
N i , b x i x i
• For the Q4 element this becomes:


N i , a
N i , b y y i i


 note the Jacobian matrix is a function of location within the master element
  
1
4








1
1

 b a





1
1

 b a

 

1
1

 b a

 

1

1

 a b






 x x x x
1
2
3
4 y
1 y y
2
3




 y
4 local coordinate derivatives of the shape functions global coordinate locations of the element nodes

• The Jacobian contains information about element size and shape
  


J
11

J
21
J
J
12


22

• The Jacobian determinant (j) is a scaling factor that relates the differential j
 det
  
J
11
J
22

J
21
J
22
• The Jacobian inverse (
G
) relates global coordinate system (x,y) function
  
J

1 
1 j

J

 
J
22
21

J
12
J
11






G
11
G
21
G
12
G
22





• Here is a single Q4 element (highly-distorted, not recommended)
• Notice how sub-region size and distortion varies within the element
• The Jacobian captures local area and distortion differences large area high distortion small area low distortion

• This is one measure of element quality (which affects element accuracy)
• Ratio of the highest to lowest quadrature point Jacobian determinant
• It is 1.0 for any square or rectangular element (same j throughout element)
• It increases as element distortion increases

• Start with the usual strain-displacement relationship in a slightly different form:
  






 x
 y xy


 



1 0 0 0



0 0 0 1
0 1 1 0






 u , u , y v , x x v , y




• Now add the Jacobian approach to master/global coordinate derivative transformation:


 u , x u , y 
 v , x




 v , y





G
11
G
21
G
G
12
22
0
0
0 0
G
11
G
0
0
12
0 0
G
21
G
22








 u , a u , b v , a v , b







• Now represent the displacement field master element derivatives in terms of the shape functions:

 u , a u , b 
 v , a




 v , b





N
1, a
N
1, b
0 N
0
0
1, a
N
N
2, a
2, b
0 N
0 N
0
2, a
N
3, a
3, b
0 N
0 N
0
3, a
N
4, a
4, b
0 N
0
0
4, a
0 N
1, b
0 N
2, b
0 N
3, b
0 N
4, b






 v u
4 v
3
4




 v u
2
3
2





 u
1 v u
1















 
- or -








 x y xy


 



1 0 0 0



0 0 0 1
0 1 1 0


G
11






G
0
0
21
G
G
0
0
12
22
G
G
0
0
11
21
G
G
0
0
12
22




 




N
1, a
N
1, b
0 N
0
0
1, a
N
N
2, a
2, b
0 N
0 N
0
2, a
N
0 N
1, b
0 N
2, b
0 N
0
3, a
3, b
N
0 N
0
3, a
3, b
N
4, a
4, b
0 N
0 N
0
0
4, a
4, b






 v u
4 v
3
4






 v u
3
2



 u
1 v u
2
1 organization shape function












  d
Jacobian inverse terms, master to global coordinate transformation derivatives, master coordinates nodal displacements, global coordinates


• If we have strain, we can get to stress by bringing in material properties
      
E B d
• We have to be a little careful here, this simple expression assumes:
– No initial (residual, assembly) stresses present
–
–

The general form above does accommodate anisotropic behavior
• If we further limit ourselves to 2D, isotropic, plane stress, we can write:
  
1

E

2

1




1

0 0
0
0

1
  
2




  
1  





1

0
E
E
 
1
E
E
0
0
0
1
G





G


E
  


• Recall where the element stiffness matrix fits into the finite element formulation:
  d
 r
• Take it as a given for the present that the element stiffness matrix [k] is:
   

B
T
E B t dA
• An integral over the element area in global coordinates (t = thickness)
• Why is integration required?
– Think about what [k] does
– For displacements applied to the element nodes, it determines the required force
– If one element is larger than another, the force required ought to be greater for the same nodal displacements
– If an element has a rotated orientation, a coordinate axis displacement can produce forces with multiple coordinate components

• It is not easy to integrate for the terms in [k] using the global coordinate system (elements are generally distorted and not aligned with global axes)
• But we can do this instead (matrix dimensions for a Q4 element):
8 x 8 symm

 1

1
 1

1
 
T
8 x 3
    t j da db
3 x 3 3x8
• Integrate over the master element
• It is undistorted and aligned with the coordinate system
• Adjust for the change in coordinates by bringing in the Jacobian determinant j

• Read “quadrature” as “numerical integration”
• Why do we want to numerically integrate to establish [k]?

8 x 8 symm

 1

1
 1

1
  T
8 x 3
    t j da db
3 x 3 3x8 these contain Jacobian inverse terms which vary point-by-point within the element this varies point-bypoint too …
• To integrate directly is still computationally expensive, even with the change to local coordinates
• Quadrature involves sampling at discrete points, multiplying by a weighting factor, and summing to get an estimate of the integral

• Gauss quadrature is a method of numerical integration that has optimal characteristics when the underlying functions have polynomial form
• The figure shows Gauss points for 2 nd order and 3 rd order quadrature
– For (a), all four points have a weight of 1.0 (total = 4.0)
– For (b): 1,3,7,9 weight = .3086; 2,4,6,8 weight = .4938; 5 weight = .7901 (total = 4.0)
• Note: the quadrature rule is independent of element order (Q4, Q8, Q9)

• One of the reasons a distorted element is less than ideal:
– The integral is estimated by discrete sampling at specific locations within the element
– If the element is not distorted, the sampled points are highly representative of the un-sampled near by regions of the element
– If the element is highly distorted, the sampled points are not representative of the un-sampled regions of the element

• If you get “inside out element” errors
• Verify-Element-Normals as a fringe or vector plot (rotate the model to see the vector orientation)

• Use Modify-Element-Reverse to get them all going in the same (positive Z, I think, check this) direction