1. Isoparametric Elements Element Stiffness Matrices
Structural Mechanics
Displacement-based Formulations

2. General Approach – Specific Example
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 …

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

4. Bilinear Quadrilateral (Q4)
Interpolation involves the summation of nodal values multiplied by corresponding shapes functions
geometry interpolation
field variable interpolation
- where -
nodal coordinates
nodal displacements
shape functions

5. Q4 - Displacements 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
{u} = displacements continuously defined (all components) over an element
[N] = the element shape functions in master element coordinates
{d} = the nodal (discrete) displacement values

6. Strain from {u}
Now calculate element strains from the displacement field
This is just the usual strain-displacement relationship written in compact form with an operator matrix

7. Q4 – Strain from {d}
Let’s now work toward an expression for element strain
We have a bit of a difficulty here with direct substitution
The shape functions (N1, N2, N3, N4) are defined in terms of the master element coordinates (a,b)
But we need to differentiate in terms of the global coordinates (x,y)
this operation cannot be done directly
- or -

8. Coordinate Transformation
Given any function of the master element coordinates (a,b):
We can find derivatives with respect to global (x,y) by using the chain rule:
We can combine and rearrange these relationships to get our derivatives:

9. The Jacobian
The Jacobian matrix is an important part of element formulation:
For the Q4 element this becomes:
note the Jacobian matrix is a function of location within the master element
local coordinate derivatives of the shape functions
global coordinate locations of the element nodes

10. Jacobian Interpretation
The Jacobian contains information about element size and shape
The Jacobian determinant (j) is a scaling factor that relates the differential area of the actual element to the differential area of the master element
The Jacobian inverse (G) relates global coordinate system (x,y) function derivatives to master element coordinate system (a,b) function derivatives

11. Jacobian (determinant) Ratio
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

12. Strain/Displacement for Q4
Start with the usual strain-displacement relationship in a slightly different form:
Now add the Jacobian approach to master/global coordinate derivative transformation:

13. Strain/Displacement cont.
Now represent the displacement field master element derivatives in terms of the shape functions:

14. All Together Now … - or - [B] organization
shape function derivatives, master coordinates
- or -
Jacobian inverse terms, master to global coordinate transformation
nodal displacements, global coordinates

15. Stress
If we have strain, we can get to stress by bringing in material properties
We have to be a little careful here, this simple expression assumes:
No initial (residual, assembly) stresses present
Linear elastic behavior
The general form above does accommodate anisotropic behavior
If we further limit ourselves to 2D, isotropic, plane stress, we can write:

16. Element Stiffness Matrix
Recall where the element stiffness matrix fits into the finite element formulation:
Take it as a given for the present that the element stiffness matrix [k] is:
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

17. Integration in Master Coordinates
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):
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

18. Quadrature Read “quadrature” as “numerical integration”
Why do we want to numerically integrate to establish [k]?
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
this varies point-by-point too …
these contain Jacobian inverse terms which vary point-by-point within the element

19. Gauss Points
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 2nd order and 3rd 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 = (total = 4.0)
Note: the quadrature rule is independent of element order (Q4, Q8, Q9)

20. Computational Procedure
Clear the array that will contain [k]
Loop over integration points in the a direction (i=1 to ni)
Set sampling point location ai and the weight factor Wi (i.e. +/ , 1.0)
Loop over integration points in the b direction (j=1 to nk)
Set sampling point location bk and the weight factor Wk (i.e. +/ , 1.0)
Call shape function subroutine, return [B] and j=det[J] at the point (ai, bk)
Calculate [B]T[E][B]tjWiWk, and add into [k]
End loop k
End loop i
Shape function subroutine
Input is the local Gauss point coordinates (a,b)
Access database for element global node coordinates (i.e. x1,y1,x2,y2,x3,y3,x4,y4)
Calculate [J], j = det[J], [g] = [J]-1 and [B]

21. Efficiency One-time element calculations File vs. DRAM storage
The shape function local derivatives are the same for a given element type and integration scheme
They are only calculated once and stored each time the shape function subroutine is called
File vs. DRAM storage
If you had to access a hard drive file each time a node location was required, the shape function subroutine calculations could be slow
But you need enough DRAM to store the locations, along with all the other information required during the analysis

22. Element Distortion
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

23. Intra-Element Jacobian Variation
Here is a single Q4 element (highly-distorted, not recommended)
For integration purposes, element characteristics are sampled at the discrete Gauss points shown
How well does the point represent the region around it?
Gauss point
approx area represented by the point

24. Geometric Nonlinearity
The configuration of the element changes as soon as it deforms.
The global locations of the nodes is different at each load increment.
Since global node locations enters into the element stiffness calculation …
The element stiffness evolves during a high-deformation sequence
A single load step is often broken up into a series of increments …
And the element stiffness is updated at each increment.
original element
load increment 1
load increment 2 …