Finite Element Simulation of Heat Transfer

Jean-Michel Bergheau is presently Director of Research at Ecole Nationale d'ingenieurs de Saint-Etienne and head of the Computational Mechanics research group of the Laboratory of Tribology and Systems Dynamics. His research interests concern the modeling of coupled physical phenomena applied to the numerical simulation of manufacturing processes such as welding, heat treatment and machining, and the understanding of the mechanical phenomena involved in interfaces. Roland Fortunier conducted research abroad, in Germany and Canada, of the nuclear and the steel industry. He was appointed Professor of the Ecole des Mines in France in 1995.

Introduction 11

PART 1.  Steady State Conduction 17

Chapter 1.  Problem Formulation 21

1.1. Physical modeling 21

1.2. Mathematical analysis 24

1.3. Working example 30

Chapter 2.  The Finite Element Method 43

2.1. Finite element approximation 43

2.2. Discrete problem formulation 48

2.3. Solution 53

2.4. Working example 68

Chapter 3.  Isoparametric Finite Elements 79

3.1. Definitions 79

3.2. Calculation of element quantities 90

3.3. Some finite elements 99

PART 2.  Transient State, Non-linearities, Transport Phenomena 101

Chapter 4.  Transient Heat Conduction 105

4.1. Problem formulation . 105

4.2. Time integration 111

4.3. Working example 135

Chapter 5.  Non-linearities 143

5.1. Formulation and solution techniques 143

5.2. Traditional non-linearities 153

5.3. A temperature-enthalpy formulation 162

Chapter 6. Transport Phenomena 169

6.1. Highlighting instabilities 169

6.2. Resolution techniques 174

PART 3. Coupled Phenomena 183

Chapter 7.  Radiation Exchanges in a Chamber 189

7.1. Modeling radiative heat exchanges in a cavity 189

7.2. Examples 200

Chapter 8. Fluid-Structure Coupling in a Pipe 207

8.1 Modeling the fluid 207

8.2. Example 212

Chapter 9. Thermometallurgical Coupling 215

9.1. Modeling phase changes 215

9.2. Examples 222

Chapter 10. Thermochemical Coupling 231

10.1. Finite element simulation of simultaneous diffusion and precipitation 231

10.2. Calculation of precipitation 236

10.3. Examples 239

Chapter 11. Electrothermal Coupling 243

11.1. Electrokinetic modeling 243

11.2. Resistance welding 248

Chapter 12. Magnetothermal Coupling 253

12.1. Introduction 253

12.2. Magnetic vector potential formulation for magnetodynamics 254

12.3. Coupled finite element-boundary element method 257

12.4. A harmonic balance method for the magnetodynamic problem 261

12.5. Coupling magnetodynamics with heat transfer .263

12.6. Application: induction hardening of a steel cylinder 266

Bibliography 269

Index 277