finite elements

2. • (FEM hereafter) is a numerical technique
finding approximate solutions for differential
equations.
• FEM solves the boundary or initial value
problem by dividing the complex geometry
into simple small elements.
• Approximated solutions for problems under
certain boundary condition can be obtained
by solving a system of equations.

3. Description
• FEM cuts a structure into several elements (pieces of the structure).
• Then reconnects elements at “nodes” as if nodes were pins or drops
of glue that hold elements together.
• This process results in a set of simultaneous algebraic equations.

4. • Governing Equations & Boundary Conditions
Elastic Problems
Thermal Problems
Fluid Flow
Electrostatics
etc

5. • Governing L(phi)+f=0
Equations:
• Boundary B(phi)+g=0
conditions:
Set of Simultaneous
algebraic equations
[K]{u}={F}
FEM
Approximate

6. [K]{u}={F} {u}=[K] {F}
Property
Behaviour
Action
Property [K] Behaviour {u} Action {F}
Elastic Stiffness Displacement Force
Thermal Conductivity Temperature Heat Source
Fluid Viscosity Velocity Body Force
Unknown

7. It is very difficult to make the algebraic equations for the entire domain
• Divide the domain into a number of small, simple elements
• A field quantity is interpolated by a polynomial over an element
• Adjacent elements share the DOF at connecting nodes.
• Put all the element equations together
Finite Element: Small piece of structure

8. Finite Element
• Arbitrary Domain

9. Galerkin’s Approach
• Governing Equation
( )
( ) 0
L u P
L u P

 
( )
L u P e
 
0 1
u a x
a
 
Residual Error

10. Galerkin’s Approach
0
V
WedV 

Weighted
Function
‘W’ any arbitrary
value
If above integral is zero then
We will say that estimate of ubar
is an accurate approximation
0 1
u a x
a
 

11. Galerkin’s Method
1. Select a set of basis functions
2. Weight Field
'
i
G s
i i
u QG
  0 1
u a x
a
 
i i
W G
  0 1
W b x
b
 
W should satisfy the BC of U
At X=0 u=0
At X=L u=0
At X=0 W=0
At X=L W=0

12. Galerkin’s Method
3. Find such that
i
Q
0
V
WedV 

( ( ) ) 0
V
W L P
w dV
 


13. Governing Equations
/ / / 0
/ / / 0
/ / / 0
x yx z
y
x x
y xy zy
z yz xz z
y y y f
y y y f
y y y f
  
  
  
      
      
      




14. Glaerkin Method
• The approximate solution of the balance
equation
/ / / 0
/ / / 0
/ / / 0
x yx zx x
y xy zy
z yz xz z
y
R
Ry
Rz
x y y y f
y y y f
y y y f
  
  
  
        
        
       

 
 



15. Galerkin’s
• Reducing residual over the domain
i=‘1’ to ‘n’ (n=no of equations)
N=Shape Function
W=Weighted Function
d ....... d
i n
R
w w R
       
i i
w N


16. Stiffness Matrix
Integral Form
[ ] ([ ][ ]) [ ][ ][ ]
T
K D N C D N dxdydz

[ ]{ } { }
K u F

Compliance Matric

17. Elements Type Table

18. Simple Beam