1. Introduction to Finite Element Methods
UNIT I
Introduction to Finite Element Methods

2. Numerical Methods – Definition and Advantages
Definition: Methods that seek quantitative approximations to the solutions of mathematical problems
Advantages:

3. What is a Numerical Method – An Example

4. What is a Numerical Method – An Example

5. What is a Numerical Method – An Example

6. What is a Numerical Method – An Example

7. What is a Numerical Method – An Example

8. What is a Finite Element Method

9. Discretization
1-D
2-D
?-D
3-D
Hybrid

10. Numerical Interpolation Non-exact Boundary Conditions
Approximation
Numerical Interpolation
Non-exact Boundary Conditions

11. Applications of Finite Element Methods
Structural & Stress Analysis
Thermal Analysis
Dynamic Analysis
Acoustic Analysis
Electro-Magnetic Analysis
Manufacturing Processes
Fluid Dynamics

12. Lecture 2
Review

13. Matrix Algebra Row and column vectors
Addition and Subtraction – must have the same dimensions
Multiplication – with scalar, with vector, with matrix
Transposition –
Differentiation and Integration

14. Matrix Algebra Determinant of a Matrix: Matrix inversion -
Important Matrices
diagonal matrix
identity matrix
zero matrix
eye matrix

15. Numerical Integration
Calculate:
Newton – Cotes integration
Trapezoidal rule – 1st order Newton-Cotes integration
Trapezoidal rule – multiple application

16. Numerical Integration
Calculate:
Newton – Cotes integration
Simpson 1/3 rule – 2nd order Newton-Cotes integration

17. Numerical Integration
Calculate:
Gaussian Quadrature
Trapezoidal Rule:
Gaussian Quadrature:
Choose
according to certain criteria

18. Numerical Integration
Calculate:
Gaussian Quadrature
2pt Gaussian Quadrature
3pt Gaussian Quadrature
Let:

19. Numerical Integration - Example
Calculate:
Trapezoidal rule
Simpson 1/3 rule
2pt Gaussian quadrature
Exact solution

20. Linear System Solver
Solve:
Gaussian Elimination: forward elimination + back substitution
Example:

21. Linear System Solver
Solve:
Gaussian Elimination: forward elimination + back substitution
Pseudo code:
Forward elimination:
Back substitution:
Do k = 1, n-1
Do i = k+1,n
Do j = k+1, n
Do ii = 1, n-1
i = n – ii
sum = 0
Do j = i+1, n
sum = sum +

22. Finite Element Analysis (F.E.A.) of 1-D Problems
UNIT II
Finite Element Analysis (F.E.A.) of 1-D Problems

23. Historical Background
Hrenikoff, 1941 – “frame work method”
Courant, 1943 – “piecewise polynomial interpolation”
Turner, 1956 – derived stiffness matrice for truss, beam, etc
Clough, 1960 – coined the term “finite element”
Key Ideas: - frame work method
piecewise polynomial approximation

24. Axially Loaded Bar Review: Stress: Stress: Strain: Strain:
Deformation:
Deformation:

25. Axially Loaded Bar
Review:
Stress:
Strain:
Deformation:

26. Axially Loaded Bar – Governing Equations and Boundary Conditions
Differential Equation
Boundary Condition Types
prescribed displacement (essential BC)
prescribed force/derivative of displacement (natural BC)

27. Axially Loaded Bar –Boundary Conditions
Examples
fixed end
simple support
free end

28. Potential Energy Elastic Potential Energy (PE) - Spring case
Unstretched spring
Stretched bar x
- Axially loaded bar
undeformed:
deformed:
- Elastic body

29. Potential Energy Work Potential (WE) Total Potential Energyf P
f: distributed force over a line
P: point force
u: displacement A B
Total Potential Energy
Principle of Minimum Potential Energy
For conservative systems, of all the kinematically admissible displacement fields,
those corresponding to equilibrium extremize the total potential energy. If the
extremum condition is a minimum, the equilibrium state is stable.

30. Potential Energy + Rayleigh-Ritz Approach
Example: f P A B
Step 1: assume a displacement field
f is shape function / basis function
n is the order of approximation
Step 2: calculate total potential energy

31. Potential Energy + Rayleigh-Ritz Approach
Example: f P A B
Step 3:select ai so that the total potential energy is minimum

32. Galerkin’s Method Example: f P Seek an approximation soB
Seek an approximation so
In the Galerkin’s method, the weight function is chosen to be the same as the shape
function.

33. Galerkin’s Method
Example: f P A B 1 2 3 1 2 3

34. Finite Element Method – Piecewise Approximationu x u x

35. FEM Formulation of Axially Loaded Bar – Governing Equations
Differential Equation
Weighted-Integral Formulation
Weak Form

36. Approximation Methods – Finite Element Method
Example:
Step 1: Discretization
Step 2: Weak form of one element P1 P2 x1 x2

37. Approximation Methods – Finite Element Method
Example (cont):
Step 3: Choosing shape functions
- linear shape functions x x
x=-1
x=0
x=1 x1 l x2

38. Approximation Methods – Finite Element Method
Example (cont):
Step 4: Forming element equation
E,A are constant
Let , weak form becomes
Let , weak form becomes

39. Approximation Methods – Finite Element Method
Example (cont):
Step 5: Assembling to form system equation
Approach 1:
Element 1:
Element 2:
Element 3:

40. Approximation Methods – Finite Element Method
Example (cont):
Step 5: Assembling to form system equation
Assembled System:

41. Approximation Methods – Finite Element Method
Example (cont):
Step 5: Assembling to form system equation
Element 1
Element 2
Element 3 1 2 3 4
Approach 2: Element connectivity table
local node
(i,j)
global node index
(I,J)

42. Approximation Methods – Finite Element Method
Example (cont):
Step 6: Imposing boundary conditions and forming condense system
Condensed system:

43. Approximation Methods – Finite Element Method
Example (cont):
Step 7: solution
Step 8: post calculation

44. Summary - Major Steps in FEM
Discretization
Derivation of element equation
weak form
construct form of approximation solution over one element
derive finite element model
Assembling – putting elements together
Imposing boundary conditions
Solving equations
Postcomputation

45. Exercises – Linear Element
Example 1:
E = 100 GPa, A = 1 cm2

46. Linear Formulation for Bar Element
x=x1
x= x2 u1 u2
f(x)
L = x2-x1 u x
x=x2 1 f2 f1
x=x1

47. Higher Order Formulation for Bar Element1 3 u1 u3 u x u2 2 1 4 u1 u4 2 u x u2 u3 3 1 n u1 un 2 u x u2 u3 3 u4
…………… 4

48. Natural Coordinates and Interpolation Functionsx
x=-1
x=1 x
x=x1
x= x2
Natural (or Normal) Coordinate: 1 2 x
x=-1
x=1 1 3 2 x
x=-1
x=1 1 4 2 x
x=-1
x=1 3

49. Quadratic Formulation for Bar Element
x=-1
x=0
x=1 f3 f1 f2

50. Quadratic Formulation for Bar Element
f(x) P3 P1 P2
x=-1
x=0
x=1

51. Exercises – Quadratic Element
Example 2:
E = 100 GPa, A1 = 1 cm2; A1 = 2 cm2

52. Some Issues Non-constant cross section: Interior load point:
Mixed boundary condition: k

53. Finite Element Analysis (F.E.A.) of I-D Problems – Applications

54. Plane Truss Problems
Example 1: Find forces inside each member. All members have
the same length. F

55. UNIT II

56. Arbitrarily Oriented 1-D Bar Element on 2-D Plane
Q2 , v2 q
P2 , u2
Q1 , v1
P1 , u1

57. Relationship Between Local Coordinates and Global Coordinates

58. Relationship Between Local Coordinates and Global Coordinates

59. Stiffness Matrix of 1-D Bar Element on 2-D Plane
Q2 , v2 q
P2 , u2
Q1 , v1
P1 , u1

60. Arbitrarily Oriented 1-D Bar Element in 3-D Spaceax x gx bx y z 2 1 -
ax, bx, gx are the Direction Cosines of the bar in the x-y-z coordinate system -

61. Stiffness Matrix of 1-D Bar Element in 3-D Spaceax x gx bx y z 2 1 -

62. Matrix Assembly of Multiple Bar Elements
Element I
Element I I
Element I
I I

63. Matrix Assembly of Multiple Bar Elements
Element I
Element I I
Element I
I I

64. Matrix Assembly of Multiple Bar Elements
Apply known boundary conditions

65. Solution Procedures
u2= 4FL/5AE, v1= 0

66. Recovery of Axial Forces
Element I
Element I I
Element I
I I

67. Stresses inside members
Element I
Element I I
Element I
I I

68. FEM of 1-D Problems: Applications
Lecture 5
FEM of 1-D Problems: Applications

69. Torsional Shaft Review Assumption: Circular cross section
Shear stress:
Shear strain:
Deformation:

70. Finite Element Equation for Torsional Shaft

71. Bending Beam y Review x Pure bending problems: Normal strain:
Normal stress:
Normal stress with bending moment:
Moment-curvature relationship:
Flexure formula:

72. Bending Beam y Review q(x) x
Relationship between shear force, bending moment and
transverse load:
Deflection:
Sign convention: M + M - M V + - V V

73. Governing Equation and Boundary Condition
0<x<L
Boundary Conditions ----- {
Essential BCs – if v or is specified at the boundary.
Natural BCs – if or is specified at the boundary.

74. Weak Formulation for Beam Element
Governing Equation
Weighted-Integral Formulation for one element
Weak Form from Integration-by-Parts (1st time)

75. Weak Form from Integration-by-Parts ----- (2nd time)
Weak Formulation
Weak Form from Integration-by-Parts (2nd time)
V(x2)
x = x1
M(x2)
q(x) y x
x = x2
V(x1)
M(x1)
L = x2-x1

76. Weak Formulation Weak Form y(v) x Q1 q(x) Q3 Q2 Q4 x = x1 L = x2-x1

77. Ritz Method for Approximation
q(x)
y(v) Q1 Q3 Q2 Q4 x
x = x1
L = x2-x1
x = x2
where
Let w(x)= fi (x), i = 1, 2, 3, 4

78. Ritz Method for ApproximationQ3
x = x1
y(v) x
x = x2 Q1 Q2
L = x2-x1 Q4

79. Ritz Method for ApproximationQ3
x = x1
y(v) x
x = x2 Q1 Q2
L = x2-x1 Q4

80. Selection of Shape Function
The best situation is -----
Interpolation
Properties

81. Derivation of Shape Function for Beam Element – Local Coordinates
How to select fi???
and
where
Let
Find coefficients to satisfy the interpolation properties.

82. Derivation of Shape Function for Beam Element
How to select fi???
e.g.
Let
Similarly

83. Derivation of Shape Function for Beam Element
In the global coordinates:

84. Element Equations of 4th Order 1-D Model
y(v) u1
q(x) u3 u2 u4 x
x = x1
L = x2-x1
x = x2 f4 1 f1 1 f2 f3
x=x2
x=x1

85. Element Equations of 4th Order 1-D Model
y(v) u1
q(x) u3 u2 u4 x
x = x1
L = x2-x1
x = x2

86. Finite Element Analysis of 1-D Problems - Applications
Example 1. F L
Governing equation:
Weak form for one element
where

87. Finite Element Analysis of 1-D Problems
Example 1.
Approximation function: f1 f4
x=x1 f2 f3
x=x2

88. Finite Element Analysis of 1-D Problems
Example 1.
Finite element model:
Discretization:
P2 , v2
P3 , v3
P1 , v1 II
P4 , v4 I
III
M1 , q1
M2 , q2
M3 , q3
M4 , q4

89. Matrix Assembly of Multiple Beam Elements
Element I
Element I I

90. Matrix Assembly of Multiple Beam Elements
Element I
I I

91. Apply known boundary conditions
Solution Procedures
Apply known boundary conditions

92. Solution Procedures

93. Shear Resultant & Bending Moment Diagram

94. Plane Flame Frame: combination of bar and beam Q1 , v1 E, A, I, L
P1 , u1
P2 , u2
Q2 , q1
Q4 , q2

95. Finite Element Model of an Arbitrarily Oriented Frameq x y q x

96. Finite Element Model of an Arbitrarily Oriented Frame
local
global

97. Plane Frame Analysis - Example
Rigid Joint
Hinge Joint F F F F
Beam II
Bar
Beam I
Beam

98. Plane Frame Analysis
Q3 , v2
Q4 , q2
P2 , u2
P1 , u1
Q2 , q1
Q1 , v1

99. Plane Frame Analysis
Q1 , v2
Q3 , v3
P1 , u2
P2 , u3
Q2 , q2
Q4 , q3

100. Plane Frame Analysis

101. Plane Frame Analysis

102. Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction
UNIT IV
Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction

103. Heat Transfer Mechanisms
Conduction – heat transfer by molecular agitation within a material without any motion of the material as a whole.
Convection – heat transfer by motion of a fluid.
Radiation – the exchange of thermal radiation between two or more bodies. Thermal radiation is the energy emitted from hot surfaces as electromagnetic waves.

104. Heat Conduction in 1-D Governing equation: Steady state equation:
Heat flux q: heat transferred per unit area per unit time (W/m2)
Governing equation:
Q: heat generated per unit volume per unit time
C: mass heat capacity
k: thermal conductivity
Steady state equation:

105. Thermal Convection
Newton’s Law of Cooling

106. Thermal Conduction in 1-D
Boundary conditions:
Dirichlet BC:
Natural BC:
Mixed BC:

107. Weak Formulation of 1-D Heat Conduction (Steady State Analysis)
Governing Equation of 1-D Heat Conduction -----
0<x<L
Weighted Integral Formulation -----
Weak Form from Integration-by-Parts -----

108. Formulation for 1-D Linear Elementx1 x2 1 2 T1 x T2 f1
Let x2 x1
f1T1
f2T2

109. Formulation for 1-D Linear Element
Let w(x)= fi (x), i = 1, 2

110. Element Equations of 1-D Linear Elementx1 x2 1 2 T1 x T2 f1

111. 1-D Heat Conduction - Example
A composite wall consists of three materials, as shown in the figure below.
The inside wall temperature is 200oC and the outside air temperature is 50oC
with a convection coefficient of h = 10 W(m2.K). Find the temperature along
the composite wall. t1 t2 t3 x

112. Thermal Conduction and Convection- Fin
Objective: to enhance heat transfer
Governing equation for 1-D heat transfer in thin fin w t x dx
where

113. Fin - Weak Formulation (Steady State Analysis)
Governing Equation of 1-D Heat Conduction -----
0<x<L
Weighted Integral Formulation -----
Weak Form from Integration-by-Parts -----

114. Formulation for 1-D Linear Element
Let w(x)= fi (x), i = 1, 2

115. Element Equations of 1-D Linear Element
x=0
x=L 1 2 T1 x T2 f1

116. Finite Element Analysis of 2-D Problems
Lecture 7
Finite Element Analysis of 2-D Problems

117. 2-D Discretization
Common 2-D elements:

118. 2-D Model Problem with Scalar Function - Heat Conduction
Governing Equation
in W
Boundary Conditions
Dirichlet BC:
Natural BC:
Mixed BC:

119. Weak Formulation of 2-D Model Problem
Weighted - Integral of 2-D Problem -----
Weak Form from Integration-by-Parts -----

120. Weak Formulation of 2-D Model Problem
Green-Gauss Theorem -----
where nx and ny are the components of a unit vector, which is normal to the boundary G.

121. Weak Formulation of 2-D Model Problem
Weak Form of 2-D Model Problem -----
EBC: Specify T(x,y) on G
NBC: Specify on G
where is the normal
outward flux on the boundary G at the segment ds.

122. FEM Implementation of 2-D Heat Conduction – Shape Functions
Step 1: Discretization – linear triangular element T1
Derivation of linear triangular shape functions: T3
Let T2
Interpolation properties
Same

123. FEM Implementation of 2-D Heat Conduction – Shape Functions
linear triangular element – area coordinates T1 A2 A3 A1 T3 T2 f1 f2 f3

124. Interpolation Function - Requirements
Interpolation condition
Take a unit value at node i, and is zero at all other nodes
Local support condition
fi is zero at an edge that doesn’t contain node i.
Interelement compatibility condition
Satisfies continuity condition between adjacent elements over any element boundary that includes node i
Completeness condition
The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

125. Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element
Let
Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area
Of the element would be negative and the stiffness matrix can not be formed. f2 f1 f4 f3

126. FEM Implementation of 2-D Heat Conduction – Element Equation
Weak Form of 2-D Model Problem -----
Assume approximation:
and let w(x,y)=fi(x,y) as before, then
where

127. FEM Implementation of 2-D Heat Conduction – Element Equation

128. Assembly of Stiffness Matrices

129. Imposing Boundary Conditions
The meaning of qi: 3 3 1 1 1 2 2 3 3 1 1 1 2 2

130. Imposing Boundary Conditions
Consider
Equilibrium of flux:
FEM implementation:

131. Calculating the q Vector
Example:

132. 2-D Steady-State Heat Conduction - Example
AB and BC:
CD: convection
DA:
0.6 m C B
0.4 m y x

133. Finite Element Analysis of Plane Elasticity

134. Review of Linear Elasticity
Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition.
Stress: measurement of force intensity
with
2-D

135. Review of Linear Elasticity
Traction (surface force) :
Equilibrium – Newton’s Law

136. Review of Linear Elasticity
Strain: measurement of intensity of deformation
Generalized Hooke’s Law

137. Plane Stress and Plane Strain
Plane Stress - Thin Plate:

138. Plane Stress and Plane Strain
Plane Strain - Thick Plate:
Plane Stress:
Plane Strain:
Replace E by and by

139. Equations of Plane Elasticity
Governing Equations
(Static Equilibrium)
Strain-Deformation
(Small Deformation)
Constitutive Relation
(Linear Elasticity)

140. Specification of Boundary Conditions
EBC: Specify u(x,y) and/or v(x,y) on G
NBC: Specify tx and/or ty on G
where
is the traction on the boundary G at the segment ds.

141. UNIT V

142. Weak Formulation for Plane Elasticity
are components of traction on the boundary G
where

143. Finite Element Formulation for Plane Elasticity
Let
where
and

144. Constant-Strain Triangular (CST) Element for Plane Stress Analysis
Let

145. Constant-Strain Triangular (CST) Element for Plane Stress Analysis

146. 4-Node Rectangular Element for Plane Stress Analysis
Let

147. 4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis:
and

148. Loading Conditions for Plane Stress Analysis

149. Evaluation of Applied Nodal Forces

150. Evaluation of Applied Nodal Forces

151. Element Assembly for Plane Elasticity5 6 B 3 4 3 4 A 1 2

152. Element Assembly for Plane Elasticity1 2 3 4 6 5 A B

153. Comparison of Applied Nodal Forces

154. Discussion on Boundary Conditions
Must have sufficient EBCs to suppress rigid body
translation and rotation
For higher order elements, the mid side nodes cannot be
skipped while applying EBCs/NBCs

155. Plane Stress – Example 2

156. Plane Stress – Example 3

157. Evaluation of Strains

158. Evaluation of Stresses
Plane Stress Analysis
Plane Strain Analysis

159. Finite Element Analysis of 2-D Problems – Axi-symmetric Problems

160. Axi-symmetric Problems
Definition:
A problem in which geometry, loadings, boundary
conditions and materials are symmetric about one axis.
Examples:

161. Axi-symmetric Analysis
Cylindrical coordinates:
quantities depend on r and z only
3-D problem D problem

162. Axi-symmetric Analysis

163. Axi-symmetric Analysis – Single-Variable Problem
Weak form:
where

164. Finite Element Model – Single-Variable Problem
where
Ritz method:
Weak form
where

165. Single-Variable Problem – Heat Transfer
Weak form
where

166. 3-Node Axi-symmetric Element1 2

167. 4-Node Axi-symmetric Elementh 4 3 b 1 2 x a z r

168. Single-Variable Problem – Example
Step 1: Discretization
Step 2: Element equation

169. Time-Dependent Problems

170. Time-Dependent Problems
In general,
Key question: How to choose approximate functions?
Two approaches:

171. Model Problem I – Transient Heat Conduction
Weak form:

172. Transient Heat Conduction
and
let:
ODE!

173. Time Approximation – First Order ODE
Forward difference approximation - explicit
Backward difference approximation - implicit

174. Time Approximation – First Order ODE
a - family formula:
Equation

175. Time Approximation – First Order ODE
Finite Element Approximation

176. Stability of – Family Approximation
Example
Stability

177. FEA of Transient Heat Conduction
a - family formula for vector:

178. Stability Requirment where
Note: One must use the same discretization for solving
the eigenvalue problem.

179. Transient Heat Conduction - Example

180. Transient Heat Conduction - Example

181. Transient Heat Conduction - Example

182. Transient Heat Conduction - Example

183. Transient Heat Conduction - Example

184. Transient Heat Conduction - Example

185. Transient Heat Conduction - Example

186. Model Problem II – Transverse Motion of Euler-Bernoulli Beam
Weak form:
Where:

187. Transverse Motion of Euler-Bernoulli Beam
and
let:

188. Transverse Motion of Euler-Bernoulli Beam

189. ODE Solver – Newmark’s Scheme
where
Stability requirement:
where

190. ODE Solver – Newmark’s Scheme
Constant-average acceleration method (stable)
Linear acceleration method (conditional stable)
Central difference method (conditional stable)
Galerkin method (stable)
Backward difference method (stable)

191. Fully Discretized Finite Element Equations

192. Transverse Motion of Euler-Bernoulli Beam

193. Transverse Motion of Euler-Bernoulli Beam

194. Transverse Motion of Euler-Bernoulli Beam

195. Transverse Motion of Euler-Bernoulli Beam