The document introduces finite element methods (FEM) as computational techniques for solving complex engineering problems that cannot be addressed by traditional analytical methods. It explains the fundamental process of breaking down a physical body into smaller elements, employing interpolation functions to estimate unknown variables, and assembling these into a system of equations. Key advantages of FEM include the ability to handle complex geometries, varying material properties, and nonlinear effects, making it a versatile tool in fields such as structural analysis, fluid mechanics, and heat transfer.

1. Introduction to Finite Element Methods

2. Need for Computational Methods
• Solutions Using Either Strength of Materials or Theory of
Elasticity Are Normally Accomplished for Regions and
Loadings With Relatively Simple Geometry
• Many Applicaitons Involve Cases with Complex Shape,
Boundary Conditions and Material Behavior
• Therefore a Gap Exists Between What Is Needed in
Applications and What Can Be Solved by Analytical Closed-
form Methods
• This Has Lead to the Development of Several
Numerical/Computational Schemes Including: Finite
Difference, Finite Element and Boundary Element Methods

3. Introduction to Finite Element Analysis
The finite element method is a computational scheme to solve field problems in
engineering and science. The technique has very wide application, and has been used on
problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations,
electrical and magnetic fields, etc. The fundamental concept involves dividing the body
under study into a finite number of pieces (subdomains) called elements (see Figure).
Particular assumptions are then made on the variation of the unknown dependent
variable(s) across each element using so-called interpolation or approximation functions.
This approximated variation is quantified in terms of solution values at special element
locations called nodes. Through this discretization process, the method sets up an
algebraic system of equations for unknown nodal values which approximate the
continuous solution. Because element size, shape and approximating scheme can be
varied to suit the problem, the method can accurately simulate solutions to problems of
complex geometry and loading and thus this technique has become a very useful and
practical tool.

4. Advantages of Finite Element Analysis
- Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying
Element Size and Type (Approximation Scheme)
- Time Dependent and Dynamic Effects Can Be Included
- Can Handle a Variety Nonlinear Effects Including Material
Behavior, Large Deformations, Boundary Conditions, Etc.

5. Basic Concept of the Finite Element Method
Any continuous solution field such as stress, displacement,
temperature, pressure, etc. can be approximated by a
discrete model composed of a set of piecewise continuous
functions defined over a finite number of subdomains.
Exact Analytical Solution
x
T
Approximate Piecewise
Linear Solution
x
T
One-Dimensional Temperature Distribution

6. Two-Dimensional Discretization
-1
-0.5
0
0.5
1
1.5
2
2.5
3
1
1.5
2
2.5
3
3.5
4
-3
-2
-1
0
1
2
x
y
u(x,y)
Approximate Piecewise
Linear Representation

7. Discretization Concepts
x
T
Exact Temperature Distribution, T(x)
Finite Element Discretization
Linear Interpolation Model
(Four Elements)
Quadratic Interpolation Model
(Two Elements)
T1
T2
T2
T3 T3
T4 T4
T5
T1
T2
T3
T4 T5
Piecewise Linear Approximation
T
x
T1
T2
T3 T3
T4 T5
T
T1
T2
T3
T4 T5
Piecewise Quadratic Approximation
x
Temperature Continuous but with
Discontinuous Temperature Gradients
Temperature and Temperature Gradients
Continuous

8. Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism (Brick)
3-D Continua

9. Discretization Examples
One-Dimensional
Frame Elements
Two-Dimensional
Triangular Elements
Three-Dimensional
Brick Elements

10. Basic Steps in the Finite Element Method
Time Independent Problems
- Domain Discretization
- Select Element Type (Shape and Approximation)
- Derive Element Equations (Variational and Energy Methods)
- Assemble Element Equations to Form Global System
[K]{U} = {F}
[K] = Stiffness or Property Matrix
{U} = Nodal Displacement Vector
{F} = Nodal Force Vector
- Incorporate Boundary and Initial Conditions
- Solve Assembled System of Equations for Unknown Nodal
Displacements and Secondary Unknowns of Stress and Strain Values

11. Common Sources of Error in FEA
• Domain Approximation
• Element Interpolation/Approximation
• Numerical Integration Errors
(Including Spatial and Time Integration)
• Computer Errors (Round-Off, Etc., )

12. Measures of Accuracy in FEA
Accuracy
Error = |(Exact Solution)-(FEM Solution)|
Convergence
Limit of Error as:
Number of Elements (h-convergence)
or
Approximation Order (p-convergence)
Increases
Ideally, Error  0 as Number of Elements or
Approximation Order  

13. Two-Dimensional Discretization Refinement
(Discretization with 228 Elements)
(Discretization with 912 Elements)
(Triangular Element)
(Node)




14. One Dimensional Examples
Static Case
1 2
u1 u2
Bar Element
Uniaxial Deformation of Bars
Using Strength of Materials Theory
Beam Element
Deflection of Elastic Beams
Using Euler-Bernouli Theory
1 2
w1
w2
2
1
dx
du
a
u
q
cu
au
dx
d
,
:
ion
Specificat
Condtions
Boundary
0
)
(
:
Equation
al
Differenti




)
(
,
,
,
:
ion
Specificat
Condtions
Boundary
)
(
)
(
:
Equation
al
Differenti
2
2
2
2
2
2
2
2
dx
w
d
b
dx
d
dx
w
d
b
dx
dw
w
x
f
dx
w
d
b
dx
d



15. Two Dimensional Examples
u1
u2
1
2
3 u3
v1
v2
v3
1
2
3



Triangular Element
Scalar-Valued, Two-Dimensional
Field Problems
Triangular Element
Vector/Tensor-Valued, Two-
Dimensional Field Problems
y
x n
y
n
x
dn
d
y
x
f
y
x


















,
:
ion
Specificat
Condtions
Boundary
)
,
(
:
Equation
ial
Different
Example
2
2
2
2
y
x
y
y
x
x
y
x
n
y
v
C
x
u
C
n
x
v
y
u
C
T
n
x
v
y
u
C
n
y
v
C
x
u
C
T
F
y
v
x
u
y
E
v
F
y
v
x
u
x
E
u




































































































22
12
66
66
12
11
2
2
Conditons
Boundary
0
)
1
(
2
0
)
1
(
2
ents
Displacem
of
Terms
in
Equations
Field
Elasticity

16. Development of Finite Element Equation
• The Finite Element Equation Must Incorporate the Appropriate Physics
of the Problem
• For Problems in Structural Solid Mechanics, the Appropriate Physics
Comes from Either Strength of Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using Direct, Variational-
Virtual Work or Weighted Residual Methods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and
external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations
develop the “weak form” that can be incorporated into a FEM equation. This
method is particularly suited for problems that have no variational statement. For
stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that
found by variational methods.
Direct Method
Based on physical reasoning and limited to simple cases, this method is
worth studying because it enhances physical understanding of the process

17. Simple Element Equation Example
Direct Stiffness Derivation
1 2
k
u1 u2
F1 F2
}
{
}
]{
[
rm
Matrix Fo
in
or
2
Node
at
m
Equilibriu
1
Node
at
m
Equilibriu
2
1
2
1
2
1
2
2
1
1
F
u
K
F
F
u
u
k
k
k
k
ku
ku
F
ku
ku
F





























Stiffness Matrix Nodal Force Vector

18. Common Approximation Schemes
One-Dimensional Examples
Linear Quadratic Cubic
Polynomial Approximation
Most often polynomials are used to construct approximation
functions for each element. Depending on the order of
approximation, different numbers of element parameters are
needed to construct the appropriate function.
Special Approximation
For some cases (e.g. infinite elements, crack or other singular
elements) the approximation function is chosen to have special
properties as determined from theoretical considerations

19. One-Dimensional Bar Element






 
 

udV
f
u
P
u
P
edV j
j
i
i
}
]{
[
:
Law
Strain
-
Stress
}
]{
[
}
{
]
[
)
(
:
Strain
}
{
]
[
)
(
:
ion
Approximat
d
B
d
B
d
N
d
N
E
Ee
dx
d
u
x
dx
d
dx
du
e
u
x
u
k
k
k
k
k
k





















 

L
T
T
j
i
T
L
T
T
fdx
A
P
P
dx
E
A
0
0
]
[
}
{
}
{
}
{
]
[
]
[
}
{ N
δd
δd
d
B
B
δd

 

L
T
L
T
fdx
A
dx
E
A
0
0
]
[
}
{
}
{
]
[
]
[ N
P
d
B
B
Vector
ent
Displacem
Nodal
}
{
Vector
Loading
]
[
}
{
Matrix
Stiffness
]
[
]
[
]
[
0
0





















j
i
L
T
j
i
L
T
u
u
fdx
A
P
P
dx
E
A
K
d
N
F
B
B
}
{
}
]{
[ F
d
K 

20. One-Dimensional Bar Element
A = Cross-sectional Area
E = Elastic Modulus
f(x) = Distributed Loading
dV
u
F
dS
u
T
dV
e i
V
i
S
i
n
i
ij
V
ij
t





 


Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
For One-Dimensional Case

 






 udV
f
u
P
u
P
edV j
j
i
i

(i) (j)
Axial Deformation of an Elastic Bar
Typical Bar Element
dx
du
AE
P i
i 

dx
du
AE
P
j
j 

i
u j
u
L
x
(Two Degrees of Freedom)

21. Linear Approximation Scheme
 
Vector
ent
Displacem
Nodal
}
{
Matrix
Function
ion
Approximat
]
[
}
]{
[
1
)
(
)
(
1
2
1
2
1
2
1
2
2
1
1
2
1
1
2
1
2
1
2
1
1
2
1























































d
N
d
N
nt
Displaceme
Elastic
e
Approximat
u
u
L
x
L
x
u
u
u
u
x
u
x
u
L
x
u
L
x
x
L
u
u
u
u
L
a
a
u
a
u
x
a
a
u
x (local coordinate system)
(1) (2)
i
u j
u
L
x
(1) (2)
u(x)
x
(1) (2)
1(x) 2(x)
1
k(x) – Lagrange Interpolation Functions

22. Element Equation
Linear Approximation Scheme, Constant Properties
Vector
ent
Displacem
Nodal
}
{
1
1
2
]
[
}
{
1
1
1
1
1
1
1
1
]
[
]
[
]
[
]
[
]
[
2
1
2
1
0
2
1
0
2
1
0
0



















































































u
u
L
Af
P
P
dx
L
x
L
x
Af
P
P
fdx
A
P
P
L
AE
L
L
L
L
L
AE
dx
AE
dx
E
A
K
o
L
o
L
T
L
T
L
T
d
N
F
B
B
B
B






























1
1
2
1
1
1
1
}
{
}
]{
[
2
1
2
1 L
Af
P
P
u
u
L
AE o
F
d
K

23. Quadratic Approximation Scheme
  }
]{
[
)
(
)
(
)
(
4
2
3
2
1
3
2
1
3
3
2
2
1
1
2
3
2
1
3
2
3
2
1
2
1
1
2
3
2
1
d
N
nt
Displaceme
Elastic
e
Approximat
































u
u
u
u
u
x
u
x
u
x
u
L
a
L
a
a
u
L
a
L
a
a
u
a
u
x
a
x
a
a
u
x
(1) (3)
1
u 3
u
(2)
2
u
L
u(x)
x
(1) (3)
(2)
x
(1) (3)
(2)
1
1(x) 3(x)
2(x)



































3
2
1
3
2
1
7
8
1
8
16
8
1
8
7
3
F
F
F
u
u
u
L
AE
Equation
Element

24. Lagrange Interpolation Functions
Using Natural or Normalized Coordinates
1
1 




(1) (2)
)
1
(
2
1
)
1
(
2
1
2
1








)
1
(
2
1
)
1
)(
1
(
)
1
(
2
1
3
2
1

















)
1
)(
3
1
)(
3
1
(
16
9
)
3
1
)(
1
)(
1
(
16
27
)
3
1
)(
1
)(
1
(
16
27
)
3
1
)(
3
1
)(
1
(
16
9
4
3
2
1



































(1) (2) (3)

(1) (2) (3) (4)








j
i
j
i
j
i
,
0
,
1
)
(

25. Simple Example
P
A1,E1,L1 A2,E2,L2
(1) (3)
(2)
1 2

































0
0
0
0
0
1
1
0
1
1
1
Element
Equation
Global
)
1
(
2
)
1
(
1
3
2
1
1
1
1
P
P
U
U
U
L
E
A

































)
2
(
2
)
2
(
1
3
2
1
2
2
2
0
1
1
0
1
1
0
0
0
0
2
Element
Equation
Global
P
P
U
U
U
L
E
A
























































3
2
1
)
2
(
2
)
2
(
1
)
1
(
2
)
1
(
1
3
2
1
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
0
0
Equation
System
Global
Assembled
P
P
P
P
P
P
P
U
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
Loading
ed
Distribut
Zero
Take

f

26. Simple Example Continued
P
A1,E1,L1 A2,E2,L2
(1) (3)
(2)
1 2
0
0
Conditions
Boundary
)
2
(
1
)
1
(
2
)
2
(
2
1




P
P
P
P
U












































P
P
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
0
0
0
Equation
System
Global
Reduced
)
1
(
1
3
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1




























P
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
3
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
L
E
A ,
,
Properties
m
For Unifor





















P
U
U
L
AE 0
1
1
1
2
3
2
P
P
AE
PL
U
AE
PL
U 



 )
1
(
1
3
2 ,
2
,
Solving

27. One-Dimensional Beam Element
Deflection of an Elastic Beam
2
2
4
2
3
1
1
2
1
1
2
2
2
4
2
2
2
3
1
2
2
2
1
2
2
1
,
,
,
,
,
dx
dw
u
w
u
dx
dw
u
w
u
dx
w
d
EI
Q
dx
w
d
EI
dx
d
Q
dx
w
d
EI
Q
dx
w
d
EI
dx
d
Q
















































I = Section Moment of Inertia
E = Elastic Modulus
f(x) = Distributed Loading

(1) (2)
Typical Beam Element
1
w
L
2
w
1
 2

1
M 2
M
1
V 2
V
x
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces








 
 

wdV
f
w
Q
u
Q
u
Q
u
Q
edV 4
4
3
3
2
2
1
1

 




L
T
L
dV
f
w
Q
u
Q
u
Q
u
Q
dx
EI
0
4
4
3
3
2
2
1
1
0
]
[
}
{
]
[
]
[ N
d
B
B T
(Four Degrees of Freedom)

28. Beam Approximation Functions
To approximate deflection and slope at each
node requires approximation of the form
3
4
2
3
2
1
)
( x
c
x
c
x
c
c
x
w 



Evaluating deflection and slope at each node
allows the determination of ci thus leading to
Functions
ion
Approximat
Cubic
Hermite
the
are
where
,
)
(
)
(
)
(
)
(
)
( 4
4
3
3
2
2
1
1
i
u
x
u
x
u
x
u
x
x
w










29. Beam Element Equation

 




L
T
L
dV
f
w
Q
u
Q
u
Q
u
Q
dx
EI
0
4
4
3
3
2
2
1
1
0
]
[
}
{
]
[
]
[ N
d
B
B T















4
3
2
1
}
{
u
u
u
u
d ]
[
]
[
]
[ 4
3
2
1
dx
d
dx
d
dx
d
dx
d
dx
d 





N
B



















 
2
2
2
2
3
0
2
3
3
3
6
3
6
3
2
3
3
6
3
6
2
]
[
]
[
]
[
L
L
L
L
L
L
L
L
L
L
L
L
L
EI
dx
EI
L
B
B
K T































































L
L
fL
Q
Q
Q
Q
u
u
u
u
L
L
L
L
L
L
L
L
L
L
L
L
L
EI
6
6
12
2
3
3
3
6
3
6
3
2
3
3
6
3
6
2
4
3
2
1
4
3
2
1
2
2
2
2
3


































 

L
L
fL
dx
f
dx
f
L
L
T
6
6
12
]
[
0
4
3
2
1
0
N

30. FEA Beam Problem
f
a b
Uniform EI
































































































0
0
0
0
6
6
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
/
2
/
3
/
1
/
3
0
0
/
3
/
6
/
3
/
6
0
0
/
1
/
3
/
2
/
3
0
0
/
3
/
6
/
3
/
6
2 )
1
(
4
)
1
(
3
)
1
(
2
)
1
(
1
6
5
4
3
2
1
2
2
2
3
2
3
2
2
2
3
2
3
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
EI
1
Element







































































)
2
(
4
)
2
(
3
)
2
(
2
)
2
(
1
6
5
4
3
2
1
2
2
2
3
2
3
2
2
2
3
2
3
0
0
/
2
/
3
/
1
/
3
0
0
/
3
/
6
/
3
/
6
0
0
/
1
/
3
/
2
/
3
0
0
/
3
/
6
/
3
/
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
Q
Q
Q
Q
U
U
U
U
U
U
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
EI
2
Element
(1) (3)
(2)
1 2

31. FEA Beam Problem



















































































































)
2
(
4
)
2
(
3
)
2
(
2
)
1
(
4
)
2
(
1
)
1
(
3
)
1
(
2
)
1
(
1
6
5
4
3
2
1
2
3
2
2
3
2
2
3
3
2
2
3
2
3
0
0
6
6
12
/
2
/
3
/
6
/
1
/
3
/
2
/
2
/
3
/
6
/
3
/
3
/
6
/
6
0
0
/
1
/
3
/
2
0
0
/
3
/
6
/
3
/
6
2
Q
Q
Q
Q
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
a
a
a
a
a
b
a
a
a
b
a
b
a
a
a
a
a
a
a
a
EI
System
Assembled
Global
0
,
0
,
0 )
2
(
4
)
2
(
3
)
1
(
1
2
)
1
(
1
1 





 Q
Q
U
w
U
Conditions
Boundary
0
,
0 )
2
(
2
)
1
(
4
)
2
(
1
)
1
(
3 


 Q
Q
Q
Q
Conditions
Matching






































































0
0
0
0
0
0
6
12
/
2
/
3
/
6
/
1
/
3
/
2
/
2
/
3
/
6
/
3
/
3
/
6
/
6
2
4
3
2
1
2
3
2
3
3
2
2
3
3
a
fa
U
U
U
U
a
a
a
a
a
b
a
a
a
b
a
b
a
EI
System
Reduced
Solve System for Primary Unknowns U1 ,U2 ,U3 ,U4
Nodal Forces Q1 and Q2 Can Then Be Determined
(1) (3)
(2)
1 2

32. Special Features of Beam FEA
Analytical Solution Gives
Cubic Deflection Curve
Analytical Solution Gives
Quartic Deflection Curve
FEA Using Hermit Cubic Interpolation
Will Yield Results That Match Exactly
With Cubic Analytical Solutions

33. Truss Element
Generalization of Bar Element With Arbitrary Orientation
x
y
k=AE/L



 cos
,
sin c
s

34. Frame Element
Generalization of Bar and Beam Element with Arbitrary Orientation

(1) (2)
1
w
L
2
w
1
 2

1
M 2
M
1
V 2
V
2
P
1
P
1
u 2
u













































































4
3
2
2
1
1
2
2
2
1
1
1
2
2
2
3
2
3
2
2
2
3
2
3
4
6
0
2
6
0
6
12
0
6
12
0
0
0
0
0
2
6
0
4
6
0
6
12
0
6
12
0
0
0
0
0
Q
Q
P
Q
Q
P
w
u
w
u
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
AE
L
AE
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
AE
L
AE
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation