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.
Author(s): Jean-Michel Bergheau, Roland Fortunier
Publisher: Wiley-ISTE
Year: 2008
Language: English
Pages: 282
Finite Element Simulation of Heat Transfer......Page 6
Table of Contents......Page 8
Introduction......Page 14
PART 1. Steady State Conduction......Page 20
1.1.1. Thermal equilibrium equation......Page 24
1.1.2. Fourier law......Page 25
1.1.3. Boundary conditions......Page 26
1.2. Mathematical analysis......Page 27
1.2.1. Weighted residual method......Page 28
1.2.2.Weak integral formulation......Page 30
1.3.1. Physical modeling......Page 33
1.3.2.1. Analytical integration......Page 35
1.3.2.2. The finite difference method......Page 36
1.3.3. Collocation methods......Page 37
1.3.3.1. Point collocation......Page 38
1.3.3.2. Sub-domain collocation......Page 39
1.3.4.1. Polynomial functions......Page 40
1.3.4.2. Piecewise linear functions......Page 42
2.1.1.Mesh......Page 46
2.1.2. Nodal approximation......Page 49
2.2.Discrete problem formulation......Page 51
2.2.1. Element quantities......Page 52
2.2.2. Assembly......Page 54
2.3.1. Application of temperature boundary conditions......Page 56
2.3.2. Linear system solution......Page 59
2.3.2.1. Direct methods......Page 61
2.3.2.2. Iterative methods......Page 63
2.3.3. Storing the linear system matrix......Page 65
2.3.4. Analysis of results......Page 66
2.3.4.1. Smoothing the heat flux density......Page 67
2.3.4.2. Result accuracy......Page 69
2.4. Working example......Page 71
2.4.1.1.Mesh......Page 73
2.4.1.2. Nodal approximation......Page 74
2.4.2.1. Element quantities......Page 75
2.4.2.2. Assembly......Page 77
2.4.3.1. Application of boundary conditions......Page 78
2.4.3.2. Solution......Page 80
3.1.1. Reference element......Page 82
3.1.1.1. Triangular element with linear transformation functions......Page 84
3.1.1.2. Quadrangle element with linear transformation functions......Page 85
3.1.1.3. Quadrangle element with quadratic transformation functions......Page 87
3.1.2. Isoparametric elements......Page 88
3.1.3. Interpolation function properties......Page 92
3.2. Calculation of element quantities......Page 93
3.2.1. Expression in the reference frame......Page 94
3.2.2. Gaussian quadrature......Page 96
3.2.2.1. 1D numerical integration......Page 97
3.2.2.2. 2D and 3D numerical integration......Page 100
3.3. Some finite elements......Page 102
PART 2. Transient State, Non-linearities, Transport Phenomena......Page 104
4.1.1. The continuous problem......Page 108
4.1.2. Finite element approximation......Page 110
4.1.3. Linear case......Page 112
4.2.1. Modal method......Page 114
4.2.1.1. Determining the modal basis......Page 115
4.2.1.2. Projection on the modal basis......Page 117
4.2.2.Direct time integration......Page 118
4.2.3. Accuracy and stability of a direct integration algorithm......Page 122
4.2.3.1. Accuracy......Page 123
4.2.3.2. Stability......Page 124
4.2.3.3. Simplified analysis of the stability condition......Page 125
4.2.4.1. Space oscillations during thermal shock simulation......Page 127
4.2.4.2. Discrete maximum principle......Page 131
4.2.4.3. Initial temperatures during thermal contact simulation......Page 133
4.3.1. Physical modeling and approximation......Page 138
4.3.2. Numerical applications......Page 142
5.1.1. Formulation......Page 146
5.1.2. Non-linear equation system solution methods......Page 147
5.1.2.1. Newton-Raphson method......Page 150
5.1.2.2. Substitution method......Page 152
5.1.2.3. Quasi-Newton methods......Page 153
5.1.3.Line search method......Page 155
5.2.1. Physical properties......Page 156
5.2.2. Flux or volumetric heat source boundary conditions......Page 158
5.2.3. Modeling state changes......Page 160
5.2.3.1. Equivalent specific heat method......Page 161
5.2.3.2.Enthalpy solution method......Page 163
5.3. A temperature-enthalpy formulation......Page 165
5.3.1. Mathematical formulation......Page 166
5.3.2. Example......Page 169
6.1.1. Thermal balance......Page 172
6.1.2.Treating a simple case......Page 174
6.2. Resolution techniques......Page 177
6.2.1. Upwind technique......Page 178
6.2.2. SUPG method......Page 180
6.2.3. 2Dand 3DPetrov-Galerkin formulation......Page 183
PART 3. Coupled Phenomena......Page 186
7.1. Modeling radiative heat exchanges in a cavity......Page 192
7.1.1. Posing the problem......Page 193
7.1.2.Calculation of view factors......Page 197
7.1.3. Diffusion-radiation coupling......Page 200
7.1.3.1. Tangent matrix......Page 201
7.1.3.2. Substitution matrix......Page 202
7.2.1. Radiation between two walls......Page 203
7.2.2. Cylinder quenching......Page 206
8.1.1. Physical model and mathematical formulation......Page 210
8.1.2. Modeling the coupling......Page 213
8.2.1. Physical and geometric modeling......Page 215
8.2.2.Results......Page 216
9.1.1.1.Avrami kinetics......Page 218
9.1.2. Numerical integration......Page 220
9.1.3. The case of several phase changes......Page 223
9.1.4. Modeling the coupling......Page 224
9.2. Examples......Page 225
9.2.1. Phase transformation diagrams......Page 226
9.2.2. Steel quenching......Page 230
10.1. Finite element simulation of simultaneous diffusion and precipitation......Page 234
10.1.1. Governing equations......Page 235
10.1.2. Finite element formulation......Page 237
10.2.1. Mathematical formulation......Page 239
10.2.2. Numerical scheme......Page 241
10.3.1. Calculation of a phase diagram......Page 242
10.3.2.Carbon diffusion in a titanium steel......Page 243
11.1.1.Weak formulation......Page 246
11.1.2. Modeling the coupling......Page 247
11.1.3. Solving the coupled problem......Page 249
11.2. Resistance welding......Page 251
11.2.1. Implementing the model......Page 252
11.2.2.Results......Page 254
12.1. Introduction......Page 256
12.2. Magnetic vector potential formulation for magnetodynamics......Page 257
12.3. Coupled finite element-boundary element method......Page 260
12.3.1. Finite element formulation......Page 262
12.3.2. Boundary element formulation......Page 263
12.4. A harmonic balance method for the magnetodynamic problem......Page 264
12.5.1. Iterative coupling......Page 266
12.5.2. A direct method for magnetothermal coupling......Page 268
12.6. Application: induction hardening of a steel cylinder......Page 269
Bibliography......Page 272
Index......Page 280
