Finite Element Simulation of Heat Transfer

This book introduces the finite element method applied to the resolution of industrial heat transfer problems. Starting from steady conduction, the method is gradually extended to transient regimes, to traditional non-linearities, and to convective phenomena. Coupled problems involving heat transfer are then presented. Three types of couplings are discussed: coupling through boundary conditions (such as radiative heat transfer in cavities), addition of state variables (such as metallurgical phase change), and coupling through partial differential equations (such as electrical phenomena).  A review of the various thermal phenomena is drawn up, which an engineer can simulate. The methods presented will enable the reader to achieve optimal use from finite element software and also to develop new applications.

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