The Finite Element Method (FEM)

These notes were developed from a set originally devised for Part IV Engineering undergraduate students. Sticking to one dimension allows one to treat some fairly “advanced” topics without too much math and algebra, e.g. time-dependent problems and non-linear problems, and are particularly useful for developing some first  computer code.

 

Please send any errors, typos, comments, etc. to pa.kelly@auckland.ac.nz (I am always interested to hear who is reading these notes, and where you are from).

 

To reference these notes, see here.

 

Return to Mechanics Lecture Notes

 

 

Table of Contents -OK!

 

*

  = a topic which may be skipped (certainly in a first reading/course)

 

 

PART 1: ONE-DIMENSIONAL FEM

 

 

 

 

 

 

 

Last updated

1.

Introduction and Background to the FEM

1

 

24-Jan-20

 

1.1

Weighted Residual Methods

1

 

 

*

1.2  Galerkin’s Method: Further Applications    

15

 

 

*

1.3  The Variational Approach

22

 

 

 

1.4

Summary

26

 

 

 

1.5

Problems

27

 

 

 

 

 

 

 

 

2.

The (Galerkin) Finite Element Method

31

 

24-Jan-20

 

2.1

Approximate Solution and Nodal Values

31

 

 

2.2

Trial Functions

32

 

 

2.3

The Standard Galerkin FEM

34

 

 

*

2.4  Adaptive Meshing

43

 

 

 

2.5

Local Coordinate Systems

44

 

 

 

*

2.6  Approximate Error in the Finite Element Method

49

 

 

 

*

2.7  Quadratic C0 Elements

52

 

 

 

*

2.8  Cubic Hermite Finite Elements

58

 

 

 

*

2.9  Finite Differences

61

 

 

 

2.10

Application: Static Elasticity (Structural Mechanics)

63

 

 

 

2.11

Problems

73

 

 

 

2.12

Appendix to Chapter 2

75

 

 

 

 

 

3.

NonStandard Galerkin FEM

81

 

24-Jan-20

 

*

3.1  The Penalty Method

81

 

 

*

3.2  The Mixed Method

85

 

 

*

3.3  Mixed Finite Volume / Covolume Methods

90

 

 

*

3.4  Non-mixed Method for Conservative Elements

93

 

 

*

3.5  Problems

95

 

 

 

 

 

 

4.

Finite Element Methods for Partial Differential Equations

97

 

24-Jan-20

 

4.1

First Order Systems

98

 

 

4.2

Second Order Systems

120

 

 

4.3

Application: Elastodynamics

135

 

 

4.4

Problems

147

 

 

 

4.5

Appendix to Chapter 4

148

 

 

 

 

 

 

 

5.

Non-Linear Differential Equations

151

 

27-Mar-23

 

5.1

Methods for the solution of Non-Linear Equations

151

 

 

5.2

The FEM for the Solution of Non-Linear ODEs

161

 

 

5.3

The FEM for the Solution of Non-Linear PDEs

163

 

 

5.4

Problems

167

 

 

5.5

Appendix to Chapter 5

169

 

 

 

 

 

 

 

 

PART 2: TWO-DIMENSIONAL FEM

 

 

 

 

 

 

 

 

6.

Quadrilateral Elements

171

 

27-Mar-23

 

6.1

The (Q4) Bilinear Element

171

 

 

6.2

Numerical Integration

179

 

 

6.3

The Q6 Bilinear Element

181

 

 

6.4

Higher Order Quadrilateral Elements

182

 

 

 

6.5

Problems

183

 

 

 

 

 

 

 

 

7.

Triangular Elements

185

 

27-Mar-23

 

7.1

The Linear Triangular Element

185

 

 

7.2

Numerical Integration

191

 

 

7.3

The Nonconforming Linear Triangle

192

 

 

7.4

The Quadratic Triangular Element

197

 

 

7.5

Problems

198

 

 

 

 

8.

2-D FEM: Poisson’s Equation

199

 

27-Mar-23

 

8.1

Solution using Q4 elements

199

 

 

8.2

Solution using Rotated Nonconforming Q1 Element

210

 

 

8.3

Solution using Triangular Elements

214

 

 

8.4

Solution using Nonconforming Triangular Element

220

 

 

8.5

Appendix to Chapter 8

225

 

 

 

 

9.

2D (Plane) Linear Elasticity

229

 

27-Mar-23

 

9.1

Governing Equations

229

 

 

9.2

Derivation of the FEM Equations

231

 

 

 

9.3

Example Problems

234

 

 

 

9.4

Problems

245

 

 

 

 

 

 

 

 

Index

 

 

 

Answers to Selected Problems: Chapter 2                                                                

Answers to Selected Problems: Chapter 3                                                                

Answers to Selected Problems: Chapter 4                                                                

Answers to Selected Problems: Chapter 5                                                                

Answers to Selected Problems: Chapter 6                                                                

Answers to Selected Problems: Chapter 7

Answers to Selected Problems: Chapter 8

Answers to Selected Problems: Chapter 9

 

 

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.