Introduction to Finite element method

1. Introduction to
Finite Element Method
Dr. Santosh Chavan
Assistant Professor
Department of Mechanical Engineering

2. Scope, Purpose, and Scale
Small-scale, single part
optimization
Large-scale simulation, e.g.
vehicle crash-worthiness
These may both employ the same software,
but with significantly different models!

3. Large-Scale Simulation, Full Vehicle
Models
Part of the CAE Domain
Such analyses and simulations
likely include multi-physics
and media-structure
interaction

4. Actual Geometry
Derived
Idealized
Geometry
Eng. Analysis.
Model (FEM)
Simplify
Idealize
De-Feature
Pave
Mesh
Discretize
Analysis Model Creation (FEA)
A mechanical engineer, a structural engineer, and a piping
engineer may each require different forms of geometry
capture.

5. Geometry Representation
Same Object ...
Multiple/Different Forms of
Geometry Capture
1D
Line
(Curv
e)
3D Solid
(Volume)
2D Surface
(Shell)

6. Exploded View - Same Geometry,
Different Data Capture
1D Line
(Curve)
3D Solid
(Volume)
2D Surface
(Shell)

7. INTRODUCTION
 The finite element method (FEM), sometimes referred to as finite element analysis (FEA),
is a computational technique used to obtain approximate solutions of boundary value
problems in engineering.
 Simply stated, a boundary value problem is a mathematical problem in which one or more
dependent variables must satisfy a differential equation everywhere within a known domain
of independent variables and satisfy specific conditions on the boundary of the domain.
 Boundary value problems are also sometimes called field problems. The field is the domain
of interest and most often represents a physical structure.
 The field variables are the dependent variables of interest governed by the differential
equation. The boundary conditions are the specified values of the field variables (or related
variables such as derivatives) on the boundaries of the field.
 Depending on the type of physical problem being analyzed, the field variables may include
physical displacement, temperature, heat flux, and fluid velocity to name only a few.

8. A GENERAL PROCEDURE FOR FINITE
ELEMENT ANALYSIS
• Certain steps in formulating a finite element analysis of a physical problem are common
to all such analyses, whether structural, heat transfer, fluid flow, or some other problem.
• These steps are embodied in commercial finite element software packages. The steps are
described as follows.
PREPROCESSING
The preprocessing step is, quite generally, described as defining the model and
includes
 Define the geometric domain of the problem.
 Define the element type(s).
 Define the material properties of the elements.
 Define the geometric properties of the elements (length, area, and the like).
 Define the element connectivity's (mesh the model).
 Define the physical constraints (boundary conditions).
 Define the loadings.

9. SOLUTION / PROCESSING
• During the solution phase, finite element software assembles the governing algebraic
equations in matrix form and computes the unknown values of the primary field
variable(s).
• The computed values are then used by back substitution to compute additional, derived
variables, such as reaction forces, element stresses, and heat flow.
POSTPROCESSING
Analysis and evaluation of the solution results is referred to as postprocessing.
Check equilibrium.
Calculate factors of safety.
Plot deformed structural shape.
Animate dynamic model behavior.
Produce color-coded temperature plots.
While solution data can be manipulated many ways in postprocessing, the most
important objective is to apply sound engineering judgment in determining whether
the solution results are physically reasonable.

10. EXAMPLES OF FINITE ELEMENT ANALYSIS

11. 1. Mathematical Model
Physical
Problems
Mathematical
Model
Solution
Identify control variables
Assumptions (empirical law)
(1) Modeling
(2) Types of solution
Exact Eq. Approx. Eq.
Exact Sol. ◎ ◎
Approx. Sol. ◎ ◎
Eq.
Sol.

12. (3) Methods of Solution
method

13. (3) Method of Solution
A. Classical methods
They offer a high degree of insight, but the problems are difficult or
impossible to solve for anything but simple geometries and loadings.
B. Numerical methods
(I) Energy: Minimize an expression for the potential energy of the
structure over the whole domain.
(II) Boundary element: Approximates functions satisfying the
governing differential equations not the boundary conditions.
(III) Finite difference: Replaces governing differential equations
and boundary conditions with algebraic finite difference
equations.
(IV) Finite element: Approximates the behavior of an irregular,
continuous structure under general loadings and constraints
with an assembly of discrete elements.

14. 2. Finite Element Method
(1) Definition
FEM is a numerical method for solving a system of
governing equations over the domain of a continuous
physical system, which is discretized into simple geometric
shapes called finite element.
Continuous system
Time-independent PDE
Time-dependent PDE
Discrete system
Linear algebraic eq.
ODE

15. (2) Discretization
Modeling a body by dividing it into an equivalent system of
finite elements interconnected at a finite number of points on
each element called nodes.
Continuous system Discretization system
System
Discretization
Nodes
Finite Elements

16. 3. Historical Background

17. 4. Analytical Processes of Finite Element Method
(1) Structural stress analysis problem
A. Conditions that solution must satisfy
a. Equilibrium
b. Compatibility
c. Constitutive law
d. Boundary conditions
Above conditions are used to generate a system of equations
representing system behavior.
B. Approach
a. Force (flexibility) method: internal forces as unknowns.
b. Displacement (stiffness) method: nodal disp. As unknowns.
For computational purpose, the displacement method is more
desirable because its formulation is simple. A vast majority of
general purpose FE software's have incorporated the displacement
method for solving structural problems.

18. (2) Analysis procedures of linear static structural analysis
A. Build up geometric model
a. 1D problem
line
b. 2D problem
surface
c. 3D problem
solid

19. B. Construct the finite element model
a. Discretize and select the element types
(a) element type
1D line element
2D element
3D brick element
(b) total number of element (mesh)
1D:
2D:
3D:

20. b. Select a shape function
1D line element: u=ax+b
c. Define the compatibility and constitutive law
d. Form the element stiffness matrix and equations
(a) Direct equilibrium method
(b) Work or energy method
(c) Method of weight Residuals
e. Form the system equation
Assemble the element equations to obtain global system equation
and introduce boundary conditions

21. C. Solve the system equations
a. elimination method
Gauss’s method (Nastran)
b. iteration method
Gauss Seidel’s method
Displacement field strain field stress field
D. Interpret the results (postprocessing)
a. deformation plot b. stress contour

22. 5. Applications of Finite Element Method
Structural Problem Non-structural Problem
Stress Analysis
- truss & frame analysis
- stress concentrated problem
Buckling problem
Vibration Analysis
Impact Problem
Heat Transfer
Fluid Mechanics
Electric or Magnetic Potential

23. Basic Steps of Solving FEM
1. Derive an equilibrium equation from the potential energy
equation in terms of material displacement.
2. Select the appropriate finite elements and corresponding
interpolation functions. Subdivide the object into
elements.
3. For each element, re-express the components of the
equilibrium equation in terms of interpolation functions
and the element’s node displacements.
4. Combine the set of equilibrium equations for all the
elements into a single system and solve the system for the
node displacements for the whole object.
5. Use the node displacements and the interpolation
functions of a particular element to calculate
displacements (or other quantities) for points within the
element.

24. Steps in FEM
1. Discretize and Select Element Type
2. Select a Displacement Function
3. Define Strain/Displacement and Stress/Strain
Relationships
4. Derive Element Stiffness Matrix & Eqs.
5. Assemble Equations and Introduce B.C’s
6. Solve for the Unknown Displacements
7. Solve for Element Stresses and Strains
8. Interpret the Results 24

25. Typical Application of FEM
• Structural/Stress Analysis
• Fluid Flow
• Heat Transfer
• Electro-Magnetic Fields
• Soil Mechanics
• Acoustics
25

26. Advantages
• Irregular Boundaries
• General Loads
• Different Materials
• Boundary Conditions
• Variable Element Size
• Easy Modification
• Dynamics
• Nonlinear Problems (Geometric or Material)
26

27. End of Chapter One

28. Definitions for this section
For an element, a stiffness matrix
is a matrix such that
where relates local coordinates
and nodal displacements
to local forces of a single element.
Bold denotes vector/matrices.
k̂
f̂ = k̂ d̂,
k̂ x̂,ŷ,ẑ
( )
d̂
f̂
28

29. Spring Element
k
1 2
L
x̂
f̂1x̂ ,d̂1x̂ f̂2x̂ ,d̂2x̂
29

30. Definitions
f̂1x̂ local nodal force
d̂1x̂ displacement
node
k - spring constant
x̂ local
coordinate
direction
f̂2x̂ local nodal force
d̂2x̂ displacement
node
These are scalar values
30

31. Stiffness Relationship for a Spring
ˆ
f1x
f̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k11 k12
k21 k22
é
ë
ê
ê
ù
û
ú
ú
d̂1x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
31

32. Steps in Process
1) Discretize and Select Element Type
2) Select a Displacement Function
3) Define Strain/Displacement and Stress/Strain
Relationships
4) Derive Element Stiffness Matrix & Eqs.
5) Assemble Equations and Introduce B.C.’s
6) Solve for the Unknowns (Displacements)
7) Solve for Element Stresses and Strains
8) Interpret the Results
32

33. Step 1 - Select the Element Type
L
T
k
1 2
x̂
T
d̂1x̂ d̂2x̂
33

34. Step 2 - Select a Displacement Function
Assume a displacement function
Assume a linear function.
Number of coefficients = number of d-o-f
Write in matrix form.
û
û = a1 + a2x̂
û = 1 x̂
[ ]
a1
a2
ì
í
î
ü
ý
þ
34

35. û(0) = a1 + a2 (0) = d̂1x = a1
û(L) = a1 + a2 (L) = d̂2 = a2L + d̂1x
û
Express as function of and
d̂1x d̂2x
a2 =
d̂2x - d̂1x
L
Solve for a2 :
35

36. û = a1 + a2x̂
Substituting back into:
Yields: û =
d̂2x - d̂1x
L
æ
è
ç
ç
ö
ø
÷
÷x̂ + d̂1x
36

37. In matrix form:
û = 1-
x̂
L
x̂
L
é
ë
ê
ù
û
ú
d̂1x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
û = N1 N2
[ ]
d̂1x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
Where :
N1 =1-
x̂
L
and N2 =
x̂
L
37

38. Shape Functions
N1 and N2 are called Shape Functions or
Interpolation Functions. They express the
shape of the assumed displacements.
N1 =1 N2 =0 at node 1
N1 =0 N2 =1 at node 2
N1 + N2 =1
Recall Fourier Transform, in which the basis
functions are sinusoidal functions.
Here, the bases are linear functions! 38

39. 1 2
N1
L
39

40. 1 2
N2
L
40

41. 2
1
N1 N2
L
41

42. Step 3 - Define Strain/Displacement and
Stress/Strain Relationships
T = kd
d = û(L)- û(0)
d = d̂2x - d̂1x
T - tensile force  - total elongation
Here is where
physics comes into
play!
42

43. Step 4 - Derive the Element
Stiffness Matrix and Equations
f̂1x = -T
f̂2x = T
T = - f̂1x = k d̂2x - d̂1x
( )
T = f̂2x = k d̂2x - d̂1x
( )
f̂1x = k d̂1x - d̂2x
( )
f̂2x = k d̂2x - d̂1x
( )
43

44. ˆ
f1x
ˆ
f2 x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k -k
-k k
é
ë
ê
ù
û
ú
d̂1x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
k̂
é
ë
ù
ûº
k -k
-k k
é
ë
ê
ù
û
ú Stiffness Matrix
44

45. Step 5 - Assemble the Element Equations to Obtain the
Global Equations and Introduce the B.C.
K
[ ] = k̂(e)
é
ë
ù
û
e=1
N
å
F
{ } = f̂(e)
{ }
e=1
N
å
Note: not simple addition!
An example later.
(e) indicates
“element” index
45

46. Step 6 - Solve for Nodal
Displacements
Solve :
K
[ ] d
{ } = F
{ }
46

47. Step 7 - Solve for Element Forces
Once displacements at each
node are known, then substitute
back into element stiffness equations
to obtain element nodal forces.
47

48. Two Spring Assembly
k1
1 2
k2
1 2
3
x
F3x F2x
48

49. For element 1:
f̂1x
f̂3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k1 -k1
-k1 k1
é
ë
ê
ù
û
ú
d̂1x
d̂3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
For element 2 :
ˆ
f3x
f̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k2 -k2
-k2 k2
é
ë
ê
ù
û
ú
d̂3x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
49

50. Elements 1 and 2 remain connected
at node 3. This is called the continuity
or compatibility requirement.
d3x
(1)
= d3x
(2)
= d3x
Continuity/Compatibility Condition
50

51. Assemble Global force matrix
F3x = f̂3x
(1)
+ ˆ
f3x
(2)
F2x = f̂2x
(2)
F1x = f̂1x
(1)
This is just adding.
51

52. F3x = -k1d1x + k1d3x + k2d3x - k2d2x
F2x = -k2d3x + k2d2x
F1x = k1d1x - k1d3x
In the matrix form:
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k1 0 -k1
0 k2 -k2
-k1 -k2 k1 + k2
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d1x
d2x
d3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
or
F
[ ]= K
[ ] d
{ }
Substitution will give
you these equations
52

53. Global Force Matrix: Global Displacement Matrix:
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
d1x
d2x
d3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
Global Stiffness Matrix:
k1 0 -k1
0 k2 -k2
-k1 -k2 k1 + k2
é
ë
ê
ê
ê
ù
û
ú
ú
ú
But, this way is really cumbersome!
Computers can’t do this either!
53

54. Assembly of [K] -
An Alternative Method
k1
1 2
k2
1
2
3 x
F3x F2x
54

55. For element 1:
f̂1x
f̂3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k1 -k1
-k1 k1
é
ë
ê
ù
û
ú
d̂1x
d̂3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
For element 2 :
ˆ
f3x
f̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
k2 -k2
-k2 k2
é
ë
ê
ù
û
ú
d̂3x
d̂2x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
Recall that
55

56. Assembly of [K] - An
Alternative Method
node 1 3
[k(1)
] =
k1 -k1
-k1 k1
é
ë
ê
ê
ù
û
ú
ú
node 3 2
[k(2)
] =
k2 -k2
-k2 k2
é
ë
ê
ê
ù
û
ú
ú
Insert row and
column 2 with zeros
Flip row, flip
columns, and insert
row 1 with zeros
1
3
3
2
56

57. Expand Local [k] matrices to
Global Size
k1
1 0 -1
0 0 0
-1 0 1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d̂1x
(1)
d̂2x
(1)
d̂3x
(1)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
f1x
(1)
f2x
(1)
f3x
(1)
ì
í
ï
î
ï
ü
ý
ï
þ
ï
k2
0 0 0
0 1 -1
0 -1 1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d̂1x
(2)
d̂2x
(2)
d̂3x
(2)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
f1x
(2)
f2x
(2)
f3x
(2)
ì
í
ï
î
ï
ü
ý
ï
þ
ï Computers
can do this!
57

58. Net Force
f1x
(1)
0
f3x
(1)
ì
í
ï
î
ï
ü
ý
ï
þ
ï
+
0
f2x
(2)
f3x
(2)
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
58

59. k1
1 0 -1
0 0 0
-1 0 1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d̂1x
(1)
d̂2x
(1)
d̂3x
(1)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
+k2
0 0 0
0 1 -1
0 -1 1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d̂1x
(2)
d̂2x
(2)
d̂3x
(2)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
59

60. d̂1x
(1)
d̂2x
(1)
d̂3x
(1)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
d̂1x
(2)
d̂2x
(2)
d̂3x
(2)
ì
í
ï
ï
î
ï
ï
ü
ý
ï
ï
þ
ï
ï
=
d1x
d2x
d3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
Compatibility
Displacements of the shared nodes are equal.
60

61. k1 0 -k1
0 k2 -k2
-k1 -k2 k1 + k2
é
ë
ê
ê
ê
ù
û
ú
ú
ú
d1x
d2x
d3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
61
Assembling the Components

62. Boundary Conditions
• Must Specify B.C’s to prohibit rigid body
motion.
• Two type of B.C’s
– Homogeneous - displacements = 0
– Nonhomogeneous - displacements = nonzero
value
62

63. Homogeneous B.C’s
• Delete row and column corresponding to
B.C.
• Solve for unknown displacements.
• Compute unknown forces (reactions)
from original (unmodified) stiffness
matrix.
63

64. k1
1 2
k2
1
2
3 x
F3x F2x
64

65. k1 0 -k1
0 k2 -k2
-k1 -k2 k1 + k2
é
ë
ê
ê
ê
ù
û
ú
ú
ú
0
d2x
d3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
=
F1x
F2x
F3x
ì
í
ï
î
ï
ü
ý
ï
þ
ï
k2 -k2
-k2 k1 + k2
é
ë
ê
ù
û
ú
d2x
d3x
ì
í
î
ü
ý
þ
=
F2x
F3x
ì
í
î
ü
ý
þ
F1x = -k1 d3x
Example: Homogeneous BC, d1x=0
65

66. 66
Reference books

67. Thank You