1. THE FINITE ELEMENT METHOD
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 3:
THE FINITE ELEMENT
METHOD

2. CONTENTS STRONG AND WEAK FORMS OF GOVERNING EQUATIONS
HAMILTON’S PRINCIPLE
FEM PROCEDURE
Domain discretization
Displacement interpolation
Formation of FE equation in local coordinate system
Coordinate transformation
Assembly of FE equations
Imposition of displacement constraints
Solving the FE equations
STATIC ANALYSIS
EIGENVALUE ANALYSIS
TRANSIENT ANALYSIS
REMARKS

3. STRONG AND WEAK FORMS OF GOVERNING EQUATIONS
System equations: strong form, difficult to solve.
Weak form: requires weaker continuity on the dependent variables (u, v, w in this case).
Weak form is often preferred for obtaining an approximated solution.
Formulation based on a weak form leads to a set of algebraic system equations – FEM.
FEM can be applied for practical problems with complex geometry and boundary conditions.

4. HAMILTON’S PRINCIPLE
“Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.”
An admissible displacement must satisfy:
The compatibility equations
The essential or the kinematic boundary conditions
The conditions at initial (t1) and final time (t2)

5. HAMILTON’S PRINCIPLE Mathematically where L=T-P+Wf (Kinetic energy)
(Potential energy)
(Work done by external forces)

6. FEM PROCEDURE Step 1: Domain discretization
Step 2: Displacement interpolation
Step 3: Formation of FE equation in local coordinate system
Step 4: Coordinate transformation
Step 5: Assembly of FE equations
Step 6: Imposition of displacement constraints
Step 7: Solving the FE equations

7. Step 1: Domain discretization
The solid body is divided into Ne elements with proper connectivity – compatibility.
All the elements form the entire domain of the problem without any overlapping – compatibility.
There can be different types of element with different number of nodes.
The density of the mesh depends upon the accuracy requirement of the analysis.
The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.

8. Step 2: Displacement interpolation
Bases on local coordinate system, the displacement within element is interpolated using nodal displacements.

9. Step 2: Displacement interpolation
N is a matrix of shape functions
Shape function for each displacement component at a node
where

10. Displacement interpolation
Constructing shape functions
Consider constructing shape function for
a single displacement component
Approximate in the form
pT(x)={1, x, x2, x3, x4,..., xp}
(1D)

11. Pascal triangle of monomials: 2D

12. Pascal pyramid of monomials : 3D

13. Displacement interpolation
Enforce approximation to be equal to the nodal displacements at the nodes
di = pT(xi) i = 1, 2, 3, …,nd or
de=P 
where ,

14. Displacement interpolation
The coefficients in  can be found by
Therefore, uh(x) = N( x) de

15. Displacement interpolation
Sufficient requirements for FEM shape functions
(Delta function property) 1.
(Partition of unity property – rigid body movement) 2. 3.
(Linear field reproduction property)

16. Step 3: Formation of FE equations in local coordinates
Since U= Nde
Strain matrix
e = LU
Therefore, 
e = L N de= B de e T Ve V c Π d B Bd ) ( 2 1 ε ò = V c T Ve e d B k ò = or
where
(Stiffness matrix)

17. Step 3: Formation of FE equations in local coordinates
Since U= Nde  or
where
(Mass matrix)

18. Step 3: Formation of FE equations in local coordinates
(Force vector)

19. Step 3: Formation of FE equations in local coordinates
(Hamilton’s principle) 
FE Equation

20. Step 4: Coordinate transformationx y x' y'
Local coordinate systems
Global coordinate systems
(Local)
(Global)
where , ,

21. Step 5: Assembly of FE equations
Direct assembly method
Adding up contributions made by elements sharing the node
(Static)

22. Step 6: Impose displacement constraints
No constraints  rigid body movement (meaningless for static analysis)
Remove rows and columns corresponding to the degrees of freedom being constrained
K is semi-positive definite

23. Step 7: Solve the FE equations
for the displacement at the nodes, D
The strain and stress can be retrieved by using e = LU and s = c e with the interpolation, U=Nd

24. STATIC ANALYSIS Solve KD=F for D Gauss elmination LU decomposition
Etc.

25. EIGENVALUE ANALYSIS [ K - li M ] fi = 0 (Homogeneous equation, F = 0)
Assume
Let 
(Roots of equation are the eigenvalues)
[ K - li M ] fi = 0
(Eigenvector)

26. EIGENVALUE ANALYSIS Methods of solving eigenvalue equation
Jacobi’s method
Given’s method and Householder’s method
The bisection method (Sturm sequences)
Inverse iteration
QR method
Subspace iteration
Lanczos’ method

27. TRANSIENT ANALYSIS
Structure systems are very often subjected to transient excitation.
A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks.
The discrete governing equation system usually requires a different solver from that of eigenvalue analysis.
The widely used method is the so-called direct integration method.

28. TRANSIENT ANALYSIS
The direct integration method is basically using the finite difference method for time stepping.
There are mainly two types of direct integration method; one is implicit and the other is explicit.
Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena
Explicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.

29. Newmark’s method (Implicit)
Assume that
Substitute into

30. Newmark’s method (Implicit)
where
Therefore,

31. Newmark’s method (Implicit)
Start with D0 and
Obtain
using
March forward in time
Obtain
using
Obtain Dt and
using

32. Central difference method (explicit)
(Lumped mass – no need to solve matrix equation)

33. Central difference method (explicit)
Find average velocity at time t = -t/2 using
Find using the average acceleration at time t = 0.
Find Dt using the average velocity at time t =t/2
Obtain D-t using
D0 and are prescribed and can be obtained from
Use to obtain assuming Obtain using
Time marching in half the time step
Central difference method (explicit)

34. REMARKS
In FEM, the displacement field U is expressed by displacements at nodes using shape functions N defined over elements.
The strain matrix B is the key in developing the stiffness matrix.
To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B.
The rest of the procedure is very much the same for all types of elements.