Finite Element Simulation of Heat Transfer

✍ Scribed by Jean-Michel Bergheau, Roland Fortunier

• Publisher: Wiley-ISTE
• Year: 2008
• Tongue: English
• Leaves: 281
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Finite Element Simulation of Heat Transfer......Page 5
Table of Contents......Page 7
Introduction......Page 13
PART 1. Steady State Conduction......Page 19
1.1.1. Thermal equilibrium equation......Page 23
1.1.2. Fourier law......Page 24
1.1.3. Boundary conditions......Page 25
1.2. Mathematical analysis......Page 26
1.2.1. Weighted residual method......Page 27
1.2.2.Weak integral formulation......Page 29
1.3.1. Physical modeling......Page 32
1.3.2.1. Analytical integration......Page 34
1.3.2.2. The finite difference method......Page 35
1.3.3. Collocation methods......Page 36
1.3.3.1. Point collocation......Page 37
1.3.3.2. Sub-domain collocation......Page 38
1.3.4.1. Polynomial functions......Page 39
1.3.4.2. Piecewise linear functions......Page 41
2.1.1.Mesh......Page 45
2.1.2. Nodal approximation......Page 48
2.2.Discrete problem formulation......Page 50
2.2.1. Element quantities......Page 51
2.2.2. Assembly......Page 53
2.3.1. Application of temperature boundary conditions......Page 55
2.3.2. Linear system solution......Page 58
2.3.2.1. Direct methods......Page 60
2.3.2.2. Iterative methods......Page 62
2.3.3. Storing the linear system matrix......Page 64
2.3.4. Analysis of results......Page 65
2.3.4.1. Smoothing the heat flux density......Page 66
2.3.4.2. Result accuracy......Page 68
2.4. Working example......Page 70
2.4.1.1.Mesh......Page 72
2.4.1.2. Nodal approximation......Page 73
2.4.2.1. Element quantities......Page 74
2.4.2.2. Assembly......Page 76
2.4.3.1. Application of boundary conditions......Page 77
2.4.3.2. Solution......Page 79
3.1.1. Reference element......Page 81
3.1.1.1. Triangular element with linear transformation functions......Page 83
3.1.1.2. Quadrangle element with linear transformation functions......Page 84
3.1.1.3. Quadrangle element with quadratic transformation functions......Page 86
3.1.2. Isoparametric elements......Page 87
3.1.3. Interpolation function properties......Page 91
3.2. Calculation of element quantities......Page 92
3.2.1. Expression in the reference frame......Page 93
3.2.2. Gaussian quadrature......Page 95
3.2.2.1. 1D numerical integration......Page 96
3.2.2.2. 2D and 3D numerical integration......Page 99
3.3. Some finite elements......Page 101
PART 2. Transient State, Non-linearities, Transport Phenomena......Page 103
4.1.1. The continuous problem......Page 107
4.1.2. Finite element approximation......Page 109
4.1.3. Linear case......Page 111
4.2.1. Modal method......Page 113
4.2.1.1. Determining the modal basis......Page 114
4.2.1.2. Projection on the modal basis......Page 116
4.2.2.Direct time integration......Page 117
4.2.3. Accuracy and stability of a direct integration algorithm......Page 121
4.2.3.1. Accuracy......Page 122
4.2.3.2. Stability......Page 123
4.2.3.3. Simplified analysis of the stability condition......Page 124
4.2.4.1. Space oscillations during thermal shock simulation......Page 126
4.2.4.2. Discrete maximum principle......Page 130
4.2.4.3. Initial temperatures during thermal contact simulation......Page 132
4.3.1. Physical modeling and approximation......Page 137
4.3.2. Numerical applications......Page 141
5.1.1. Formulation......Page 145
5.1.2. Non-linear equation system solution methods......Page 146
5.1.2.1. Newton-Raphson method......Page 149
5.1.2.2. Substitution method......Page 151
5.1.2.3. Quasi-Newton methods......Page 152
5.1.3.Line search method......Page 154
5.2.1. Physical properties......Page 155
5.2.2. Flux or volumetric heat source boundary conditions......Page 157
5.2.3. Modeling state changes......Page 159
5.2.3.1. Equivalent specific heat method......Page 160
5.2.3.2.Enthalpy solution method......Page 162
5.3. A temperature-enthalpy formulation......Page 164
5.3.1. Mathematical formulation......Page 165
5.3.2. Example......Page 168
6.1.1. Thermal balance......Page 171
6.1.2.Treating a simple case......Page 173
6.2. Resolution techniques......Page 176
6.2.1. Upwind technique......Page 177
6.2.2. SUPG method......Page 179
6.2.3. 2Dand 3DPetrov-Galerkin formulation......Page 182
PART 3. Coupled Phenomena......Page 185
7.1. Modeling radiative heat exchanges in a cavity......Page 191
7.1.1. Posing the problem......Page 192
7.1.2.Calculation of view factors......Page 196
7.1.3. Diffusion-radiation coupling......Page 199
7.1.3.1. Tangent matrix......Page 200
7.1.3.2. Substitution matrix......Page 201
7.2.1. Radiation between two walls......Page 202
7.2.2. Cylinder quenching......Page 205
8.1.1. Physical model and mathematical formulation......Page 209
8.1.2. Modeling the coupling......Page 212
8.2.1. Physical and geometric modeling......Page 214
8.2.2.Results......Page 215
9.1.1.1.Avrami kinetics......Page 217
9.1.2. Numerical integration......Page 219
9.1.3. The case of several phase changes......Page 222
9.1.4. Modeling the coupling......Page 223
9.2. Examples......Page 224
9.2.1. Phase transformation diagrams......Page 225
9.2.2. Steel quenching......Page 229
10.1. Finite element simulation of simultaneous diffusion and precipitation......Page 233
10.1.1. Governing equations......Page 234
10.1.2. Finite element formulation......Page 236
10.2.1. Mathematical formulation......Page 238
10.2.2. Numerical scheme......Page 240
10.3.1. Calculation of a phase diagram......Page 241
10.3.2.Carbon diffusion in a titanium steel......Page 242
11.1.1.Weak formulation......Page 245
11.1.2. Modeling the coupling......Page 246
11.1.3. Solving the coupled problem......Page 248
11.2. Resistance welding......Page 250
11.2.1. Implementing the model......Page 251
11.2.2.Results......Page 253
12.1. Introduction......Page 255
12.2. Magnetic vector potential formulation for magnetodynamics......Page 256
12.3. Coupled finite element-boundary element method......Page 259
12.3.1. Finite element formulation......Page 261
12.3.2. Boundary element formulation......Page 262
12.4. A harmonic balance method for the magnetodynamic problem......Page 263
12.5.1. Iterative coupling......Page 265
12.5.2. A direct method for magnetothermal coupling......Page 267
12.6. Application: induction hardening of a steel cylinder......Page 268
Bibliography......Page 271
Index......Page 279

✦ Subjects

Топливно-энергетический комплекс;Тепло- и массообмен;