1. MANE 4240 & CIVL 4240 Introduction to Finite Elements
Prof. Suvranu De
Convergence of analysis results

2. Reading assignment:
Lecture notes
Summary:
Concept of convergence
Criteria for monotonic convergence :
completeness (rigid body modes + constant strain) +
compatibility
Incompatible elements and the patch test
Rate of convergence

3. Errors that affect finite element solution results
Type of error
Source
1. Discretization error
Use of FE interpolations for geometry and solution variables
2. Numerical integration
Evaluation of FE element matrices and vectors using numerical integration
3. Round off
This error is due to the finite precision arithmetic used in digital computers

4. What is “convergence”?
“Convergence” of FE solution results to the exact solution of the mathematical model
FE scheme exhibits convergence if the
Discretization error →0 as the mesh is made infinitely fine (i.e., element size →0)
Physical system
Mathematical model
FE model

5. Mesh refinement
h-refinement
p-refinement
h=element size
p=polynomial order

6. Convergence in energy and displacement
u : exact displacement solution to a problem that makes the potential energy of the system a minimum
corresponding stress
and strain
Exact strain energy of the body
uh : FE solution (‘h’ refers to the element size)
Approximate strain energy of the body

7. Calculation of strain energies
Example:
Consider a linear elastic bar with varying cross section 1 2 x
The governing differential (equilibrium) equation
P=3E/80
80cm
Eq(1)
E: Young’s modulus
Boundary conditions
Analytical solution

8. Note The exact strain energy of the system is
If we discretize the problem using a single linear finite element, the stiffness matrix is
The strain energy of the FE system is
Note

9. Convergence in strain energy
Monotonic convergence
Nonmonotonic convergence

10. Convergence in displacement
Monotonic convergence
Nonmonotonic convergence

11. Criteria for monotonic convergence 1. COMPLETENESS 2. COMPATIBILITY
© 2002 Brooks/Cole Publishing / Thomson Learning™

12. CONDITION 1. COMPLETENESS
This requires that the displacement interpolation functions must be chosen so that the elements can represent
1. Rigid body modes
2. Constant strain states

13. Rigid body modes

14. The # of rigid body modes of an element = # of zero eigenvalues of the element stiffness matrix

15. Constant strain states
Strain computed using linear finite elements
Actual variation of strain e x

16. Mathematical implication of the two conditions (rigid body modes + constant strain state)
Inside a finite element (of any order) in 1D
but this is just a polynomial…
Hence
The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1), ie any 2 node element and higher is complete

17. Mathematical implication of the two conditions (rigid body modes + constant strain state)
Inside a finite element (of any order) in 2D
but this is just a polynomial…
Hence
The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1).

18. Mathematical implication of the two conditions (rigid body modes + constant strain state)
The element displacement approximation must be at least a COMPLETE polynomial of degree one
k=1 1D 2D

19. In 2D, the minimum displacement assumption needs to be
Translation along x
Translation along y
Rigid body rotation about z-axis

20. CONDITION 2. COMPATIBILITY
The assumed displacement variations are continuous within elements and across inter-element boundaries
Ensures that strains are bounded within elements and across element boundaries.
If ‘u’ is discontinuous across element boundaries then the strains blow up in-between elements and this leads to erroneous contributions to the potential energy of the structure
Physical meaning: no gaps/cracks open up when the finite element assemblage is loaded

21. Nonconforming elements and the patch test
Conforming = compatible
Nonconforming = incompatible
Ideal: Conforming elements
Observation: Certain nonconforming elements also give good results, at the expense of nonmonotonic convergence
Nonconforming elements:
satisfy completeness
do not satisfy compatibility
result in at least nonmonotonic convergence if the element assemblage as a whole is complete, i.e., they satisfy the PATCH TEST

22. PATCH TEST:
1. A patch of elements is subjected to the minimum displacement boundary conditions to eliminate all rigid body motions
2. Apply to boundary nodal points forces or displacements which should result in a state of constant stress within the assemblage
3. Nodes not on the boundary are neither loaded nor restrained.
4. Compute the displacements of nodes which do not have a prescribed value
5. Compute the stresses and strains
The patch test is passed if the computed stresses and strains match the expected values to the limit of computer precision.

24. NOTES:
1. This is a great way to debug a computer code
2. Conforming elements ALWAYS pass the patch test
3. Nodes not on the boundary are neither loaded nor restrained.
4. Since a patch may also consist of a single element, this test may be used to check the completeness of a single element
5. The number of constant stress states in a patch test depends on the actual number of constant stress states in the mathematical model (3 for plane stress analysis. 6 for a full 3D analysis)

25. CONVERGENCE RATE
This is a measure of how fast the discretization error goes to zero a the mesh is refined
Convergence rate depends on the order of the complete polynomial (k) used in the displacement approximation
k=1
k=2
k=3

26. It can be shown that for (1) a sufficiently refined mesh and (2) for problems whose analytical solution does not contain singularities
Convergence in strain energy : order 2k
Convergence in displacements : order p=k+1
C and C1 are constants independent of ‘h’ but dependent on
1. the analytical solution
2. material properties
3. type of element used

27. Ex: for a domain discretized using 4 node plane stress/strain elements (k=1)
Large C shifts curve up
slope = 2

28. Important property of finite element solution:
When the conditions of monotonic convergence are satisfied (compatibility and completeness) the finite element strain energy always underestimates the strain energy of the actual structure
Strain energy of mathematical model
Strain energy of FE model