Finite Elements

Approaches computational engineering sciences from the
perspective of engineering applications

Uniting theory with hands-on computer practice, this book gives
readers a firm appreciation of the error mechanisms and control
that underlie discrete approximation implementations in the
engineering sciences.

Key features:

• Illustrative examples include heat conduction, structural
  mechanics, mechanical vibrations, heat transfer with convection and
  radiation, fluid mechanics and heat and mass transport

• Takes a cross-discipline continuum mechanics viewpoint

• Includes Matlab toolbox and .m data files on a companion
  website, immediately enabling hands-on computing in all covered
  disciplines

• Website also features eight topical lectures from the
  author's own academic courses

It provides a holistic view of the topic from covering the
different engineering problems that can be solved using finite
element to how each particular method can be implemented on a
computer. Computational aspects of the method are provided on a
companion website facilitating engineering implementation in an
easy way.

Autorentext

A. J. Baker is Professor Emeritus, Engineering Science and Computational Engineering, The University of Tennessee, USA. He is an elected Fellow of the International Association for Computational Mechanics (IACM) and the US Association for Computational Mechanics (USACM) and an Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA).

Zusammenfassung

Approaches computational engineering sciences from the perspective of engineering applications

Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences.

Key features:

• Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport
• Takes a cross-discipline continuum mechanics viewpoint
• Includes Matlab toolbox and .m data files on a companion website, immediately enabling hands-on computing in all covered disciplines
• Website also features eight topical lectures from the author's own academic courses
  It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each particular method can be implemented on a computer. Computational aspects of the method are provided on a companion website facilitating engineering implementation in an easy way.

Inhalt
Preface viii Notation xi

1 COMPUTATIONAL ENGINEERING SCIENCE 1

1.1 Engineering simulation 1

1.2 A problem solving environment 2

1.3 Problem statements in engineering 4

1.4 Decisions on forming WSN 6

1.5 Discrete approximate WSh implementation 8

1.6 Chapter summary 9

1.7 Chapter references 10

2 PROBLEM STATEMENTS 11

2.1 Engineering simulation 11

2.2 Continuum mechanics viewpoint 12

2.3 Continuum conservation law forms 12

2.4 Constitutive closure for conservation law PDEs 14

2.5 Engineering science continuum mechanics 18

2.6 Chapter references 20

3 SOME INTRODUCTORY MATERIAL 21

3.1 Introduction 21

3.2 Multi-dimensional PDEs, separation of variables 22

3.3 Theoretical foundations, GWSh 27

3.4 A legacy FD construction 28

3.5 An FD approximate solution 30

3.6 Lagrange interpolation polynomials 31

3.7 Chapter summary 32

3.8 Exercises 34

3.9 Chapter references 34

4 HEAT CONDUCTION35

4.1 A steady heat conduction example 35

4.2 Weak form approximation, error minimization 37

4.3 GWSN discrete implementation, FE basis38

4.4 Finite element matrix statement 41

4.5 Assembly of {WS}e to form algebraic GWSh 43

4.6 Solution accuracy, error distribution 45

4.7 Convergence, boundary heat flux 47

4.8 Chapter summary 47

4.9 Exercises 48

4.10 Chapter reference 48

5 STEADY HEAT TRANSFER, n =149

5.1 Introduction 49

5.2 Steady heat transfer, n = 1 50

5.3 FE k = 1 trial space basis matrix library 52

5.4 Object-oriented GWSh programming 55

5.5 Higher completeness degree trial space bases58

5.6 Global theory, asymptotic error estimate 62

5.7 Non-smooth data, theory generalization 66

5.8 Temperature dependent conductivity, non-linearity 69

5.9 Static condensation, p-elements 72

5.10 Chapter summary 75

5.11 Exercises 76

5.12 Computer labs 77

5.13 Chapter references 78

6 ENGINEERING SCIENCES, n =1 79

6.1 Introduction 79

6.2 The Euler-Bernoulli beam equation 80

6.3 Euler-Bernoulli beam theory GWSh reformulation 85

6.4 The Timoshenko beam theory 92

6.5 Mechanical vibrations of a beam 99

6.6 Fluid mechanics, potential flow 106

6.7 Electromagnetic plane wave propagation110

6.8 Convective-radiative finned cylinder heat transfer 112

6.9 Chapter summary 120

6.10 Exercises122

6.10 Computer labs 123

6.11 Chapter references 124

7 STEADY HEAT TRANSFER, n > 1 125

7.1 Introduction 125

7.2 Multi-dimensional FE bases and DOF 126

7.3 Multi-dimensional FE operations 129

7.4 The NC k = 1,2 basis FE matrix library 132

7.5 NC basis {WS}e template, accuracy, convergence 136

7.6 The tensor product basis element family 139

7.7 Gauss numerical quadrature, k = 1 TP basis library 141

7.8 Convection-radiation BC GWSh implementation 146

7.9 Linear basis GWSh template unification 150

7.10 Accuracy, convergence revisited 152

7.11 Chapter summary 153

7.12 Exercises155 7.13 Computer lab...