Preface to the Third Edition ................................. xvii
Preface to the Second Edition ................................. xix
Preface to the First Edition .................................. xxi
About the Authors ........................................... xxiii
1 Equations of Heat Transfer and Fluid Mechanics ............... 1
1.1 Introduction ............................................ 1
1.1.1 Heat Transfer .................................... 1
1.1.2 Fluid Mechanics .................................. 2
1.2 Present Study ........................................... 3
1.3 Mathematical Preliminaries .............................. 3
1.3.1 Vectors and Tensors .............................. 3
1.3.2 Index Notation and Summation Convention .......... 5
1.3.3 The Del Operator and Calculus of Vectors and
Tensors .......................................... 7
1.4 Governing Equations of a Continuum ..................... 10
1.4.1 Introduction .................................... 10
1.4.2 Conservation of Mass; the Continuity Equation ... 10
1.4.3 Conservation of Momenta ......................... 11
1.4.4 Conservation of Energy .......................... 12
1.4.5 Equation of State ............................... 13
1.4.6 Constitutive Equations .......................... 14
1.4.7 Divergence and Advection Forms .................. 15
1.5 Governing Equations in Terms of Primitive Variables .... 16
1.5.1 Vector Form ..................................... 16
1.5.2 Cartesian Component Form ........................ 17
1.5.3 Cylindrical Component Form ...................... 17
1.5.4 Closure ......................................... 19
1.6 Porous Flow Equations .................................. 19
1.7 Low-Speed Compressible Flow Equations .................. 20
1.8 Auxiliary Transport Equations .......................... 22
1.9 Chemically Reacting Systems ............................ 23
1.10 Boundary Conditions .................................... 26
1.10.1 Viscous Flow Boundary Conditions ................ 26
1.10.2 Porous Flow Boundary Conditions ................. 29
1.10.3 Thermal and Transport Boundary Conditions ....... 30
1.10.4 Initial Conditions .............................. 31
1.11 Change of Phase ........................................ 32
1.12 Enclosure Radiation .................................... 34
1.13 Summary of Equations ................................... 36
Problems .................................................... 37
References for Additional Reading ........................... 40
2 The Finite Element Method ................................... 43
2.1 Introduction ........................................... 43
2.2 Model Differential Equation ............................ 44
2.3 Finite Element Approximation ........................... 45
2.4 Weighted-Integral Statements and Weak Forms ............ 47
2.4.1 Preliminary Comments ............................ 47
2.4.2 Weak Form Development ........................... 47
2.5 Finite Element Model ................................... 50
2.6 Interpolation Functions ................................ 52
2.6.1 Properties of Approximation Functions ........... 52
2.6.2 Linear Triangular Element ....................... 52
2.6.3 Linear Rectangular Element ...................... 54
2.6.4 Evaluation of Boundary Integrals ................ 55
2.7 Assembly of Elements ................................... 55
2.8 Time-Dependent Problems ................................ 58
2.8.1 Introduction .................................... 58
2.8.2 Semidiscretization .............................. 58
2.8.3 Temporal Approximation .......................... 60
2.9 Axisymmetric Problems .................................. 61
2.10 Library of Finite Elements ............................. 63
2.10.1 Introduction .................................... 63
2.10.2 Triangular Elements ............................. 63
2.10.3 Rectangular Elements ............................ 65
2.11 Numerical Integration .................................. 66
2.11.1 Preliminary Comments ............................ 66
2.11.2 Coordinate Transformations ...................... 68
2.11.3 Integration over a Master Rectangular Element ... 70
2.11.4 Integration over a Master Triangular Element .... 71
2.12 Modeling Considerations ................................ 72
2.12.1 Mesh Generation ................................. 72
2.12.2 Representation of Boundary Flux ................. 74
2.12.3 Imposition of Boundary Conditions ............... 74
2.13 Illustrative Examples .................................. 75
2.13.1 Example 1 ....................................... 76
2.13.1.1 Problem description .................... 76
2.13.1.2 Solution by linear triangular
elements ............................... 77
2.13.1.3 Solution by linear rectangular
elements ............................... 79
2.13.1.4 Discussion of the results .............. 79
2.13.1.1 Example 2 .............................. 81
2.13.3 Example 3 ....................................... 82
Problems .................................................... 83
References for Additional Reading ........................... 86
3 Conduction Heat Transfer .................................... 87
3.1 Introduction ........................................... 87
3.2 Semidiscrete Finite Element Model ...................... 88
3.3 Interpolation Functions ................................ 90
3.3.1 Preliminary Comments ............................ 90
3.3.2 B[exahedral (Brick) Elements .................... 91
3.3.3 Tetrahedral Elements ............................ 92
3.3.4 Prism Elements .................................. 93
3.3.5 Pyramid Elements ................................ 94
3.4 Numerical Integration .................................. 95
3.5 Computation of Surface Flux ............................ 96
3.6 Semidiscrete Finite Element Model ...................... 99
3.7 Solution of Nonlinear Equations ....................... 100
3.7.1 Preliminary Comments ........................... 100
3.7.2 Steady-State Problems .......................... 100
3.7.3 Transient Problems ............................. 102
3.7.3.1 General formulation ................... 102
3.7.3.2 Predictor-corrector methods ........... 105
3.7.3.3 Time step control ..................... 106
3.7.3.4 Initialization ........................ 107
3.7.3.5 Linear multi-step methods ............. 107
3.7.3.6 Convergence and stability ............. 108
3.7.3.7 Mode superposition methods ............ 111
3.8 Radiation Solution Algorithms ......................... 113
3.9 Variable Properties ................................... 118
3.9.1 Temperature-Dependent Properties ............... 118
3.9.2 Phase Change Properties ........................ 119
3.9.3 Anisotropic Properties ......................... 121
3.10 Post-Processing Operations ............................ 122
3.10.1 Heat Flux ...................................... 122
3.10.2 Heat Flow Function ............................. 124
3.11 Advanced Topics in Conduction ......................... 125
3.11.1 Introduction ................................... 125
3.11.2 Specialty Elements ............................. 126
3.11.3 Computational Boundary Conditions .............. 129
3.11.3.1 Contact boundary conditions ........... 129
3.11.3.2 Multipoint constraints ................ 132
3.11.3.3 Partially covered surfaces ............ 133
3.11.4 Bulk Nodes ..................................... 134
3.11.5 Reactive Materials ............................. 136
3.11.6 Material Motion ................................ 138
3.12 Example Problems ...................................... 139
3.12.1 Introduction ................................... 139
3.12.2 Element Convergence ............................ 139
3.12.3 Conduction/Radiation Solution .................. 141
3.12.4 Temperature-Dependent Conductivity ............. 143
3.12.5 Anisotropic Conductivity ....................... 143
3.12.6 One-Dimensional Stefan Problem ................. 145
3.12.7 Drag Bit Analysis .............................. 147
3.12.8 Brazing and Welding Analysis ................... 149
3.12.9 Investment Casting ............................. 152
Problems .............................................. 154
References for Additional Reading ..................... 155
4 Flows of Viscous Incompressible Fluids ..................... 161
4.1 Introduction .......................................... 161
4.1.1 Background ..................................... 161
4.1.2 Governing Equations ............................ 161
4.2 Mixed Finite Element Model ............................ 164
4.2.1 Weak Form ...................................... 164
4.2.2 Finite Element Model ........................... 165
4.3 Penalty Finite Element Models ......................... 167
4.3.1 Introduction ................................... 167
4.3.2 Penalty Function Method ........................ 168
4.3.3 Reduced Integration Penalty Model .............. 171
4.3.4 Consistent Penalty Model ....................... 171
4.4 Finite Element Models of Porous Flow .................. 172
4.5 Computational Considerations .......................... 174
4.5.1 Properties of the Matrix Equations ............. 174
4.5.2 Choice of Interpolation Functions .............. 175
4.5.2.1 Quadrilateral elements (2-D) .......... 176
4.5.2.2 Triangular elements (2-D) ............. 179
4.5.2.3 Hexahedral elements (3-D) ............. 179
4.5.2.4 Tetrahedral elements (3-D) ............ 180
4.5.3 Evaluation of Element Matrices in Penalty
Models ......................................... 180
4.5.4 Pressure Calculation ........................... 181
4.5.5 Traction Boundary Conditions ................... 184
4.6 Solution of Nonlinear Equations ....................... 186
4.6.1 General Discussion ............................. 186
4.6.2 Fully Coupled Solution Methods ................. 189
4.6.2.1 Picard method ......................... 189
4.6.2.2 Newton's method ....................... 190
4.6.2.3 Modified and quasi-Newton methods ..... 192
4.6.2.4 Continuation methods .................. 192
4.6.3 Pressure Correction/Projection Methods ......... 194
4.7 Time-Approximation Schemes ............................ 196
4.7.1 Preliminary Comments ........................... 196
4.7.2 Forward/Backward Euler Schemes ................. 197
4.7.3 Adams-Bashforth/Trapezoid Rule ................. 198
4.7.4 Implicit Integration and Time Step Control ..... 198
4.7.5 Explicit Integration ........................... 199
4.8 Stabilized Methods .................................... 200
4.8.1 Preliminary Comments ........................... 200
4.8.2 Galerkin/Least-Squares Formulation ............. 202
4.8.3 Polynomial Pressure Projection ................. 204
4.8.4 Variational Multiscale Methods ................. 205
4.9 Least-Squares Finite Element Models ................... 212
4.9.1 Introduction ................................... 212
4.9.2 Governing Equations ............................ 215
4.9.3 Least-Squares Formulation ...................... 216
4.9.4 Finite Element Model ........................... 217
4.9.5 Computational Aspects .......................... 218
4.10 Post-Processing ....................................... 219
4.10.1 Stress Computation ............................. 219
4.10.2 Stream Function Computation .................... 221
4.10.3 Particle Tracking .............................. 223
4.11 Free Surface Flows .................................... 224
4.11.1 Preliminary Comments ........................... 224
4.11.2 Time-Independent Free Surfaces ................. 224
4.11.3 Time-Dependent Free Surfaces ................... 229
4.12 Turbulence ............................................ 235
4.12.1 Preliminary Comments ........................... 235
4.12.2 Governing Equations ............................ 236
4.12.3 General Turbulence Models ...................... 237
4.12.3.1 Correlations .......................... 238
4.12.3.2 Integral methods ...................... 238
4.12.3.3 One-point closure ..................... 238
4.12.3.4 Two-point closure ..................... 238
4.12.3.5 Large eddy simulation ................. 238
4.12.3.6 Direct numerical simulations (DNS) .... 239
4.12.4 One-Point Closure Turbulence Models ............ 239
4.12.4.1 Zero-equation model ................... 240
4.12.4.2 One-equation model .................... 240
4.12.4.3 Two-equation model .................... 241
4.12.5 Finite Element Modeling of Turbulence .......... 242
4.12.5.1 Zero-equation model ................... 242
4.12.5.2 Two-equation model .................... 243
4.12.6 Variational Multiscale (VMS) Turbulence
Modeling ....................................... 244
4.13 Numerical Examples .................................... 247
4.13.1 Preliminary Comments ........................... 247
4.13.2 Fluid Squeezed between Parallel Plates ......... 248
4.13.3 Flow of a Viscous Lubricant in a Slider
Bearing ........................................ 250
4.13.4 Wall-Driven 2-D Cavity Flow .................... 252
4.13.5 Wall-Driven 3-D Cavity Flow .................... 254
4.13.6 Evaluation of the EBE Iterative Solvers ........ 256
4.13.7 Backward Facing Step ........................... 258
4.13.8 Flow Past a Submarine .......................... 260
4.13.9 Crystal Growth from the Melt ................... 262
4.13.10 Mold Filling .................................. 263
4.13.11 Examples Using Least-Squares Finite Element
Models ........................................ 267
4.13.11.1 Kovasznay flow ....................... 267
4.13.11.2 Flow over a backward-facing step ..... 269
4.13.11.3 Flow past a cylinder at low
Reynolds number ...................... 270
Problems ................................................ 273
References for Additional Reading ....................... 275
5 Coupled Fluid Flow and Heat Transfer ....................... 285
5.1 Introduction .......................................... 285
5.2 Nonisothermal Incompressible Flows .................... 286
5.2.1 Governing Equations ............................ 286
5.2.2 Boundary Conditions ............................ 288
5.3 Mixed Finite Element Model ............................ 289
5.4 Penalty Finite Element Model .......................... 293
5.4.1 Preliminary Comments ........................... 293
5.4.2 Reduced Integration Penalty Model .............. 294
5.4.3 Consistent Penalty Model ....................... 295
5.5 Finite Element Models of Porous Flow .................. 295
5.6 Nonisothermal, Low-Speed, Compressible Flows .......... 297
5.6.1 Governing Equations ............................ 297
5.6.2 Boundary Conditions ............................ 299
5.6.3 Mixed Finite Element Model ..................... 299
5.7 Solution Methods ...................................... 302
5.7.1 General Discussion ............................. 302
5.7.2 Newton's Method ................................ 303
5.7.3 Segregated Equation Methods .................... 304
5.8 Convection with Change of Phase ....................... 306
5.9 Convection with Enclosure Radiation ................... 308
5.10 Post-Computation of Heat Flux ......................... 308
5.11 Turbulent Heat Transfer ............................... 310
5.12 Chemically Reacting Systems ........................... 311
5.12.1 Preliminary Comments ........................... 311
5.12.2 Finite Element Modeling of Chemical
Reactions ...................................... 311
5.13 Numerical Examples .................................... 312
5.13.1 Preliminary Comments ........................... 312
5.13.2 Concentric Tube Flow ........................... 312
5.13.3 Tube Flow with Change of Phase ................. 313
5.13.4 Heated Cavity - Boussinesq Model ............... 314
5.13.5 Heated Cavity - Acoustically Filtered Model .... 316
5.13.6 Solar Receiver ................................. 317
5.13.7 Tube Bundle .................................... 320
5.13.8 Volumetrically Heated Fluid .................... 322
5.13.9 Porous/Fluid Layer ............................. 322
5.13.10 Curing of an Epoxy ............................ 326
5.13.11 Heated Channel ................................ 329
5.13.12 Closure ....................................... 331
References for Additional Reading ..................... 331
6 Flows of Non-Newtonian Fluids .............................. 335
6.1 Introduction .......................................... 335
6.2 Governing Equations of Inelastic Fluids ............... 336
6.2.1 Conservation Equations ......................... 336
6.2.2 Boundary Conditions ............................ 337
6.2.3 Constitutive Equations ......................... 338
6.2.3.1 Power-law model ....................... 339
6.2.3.2 Carreau model ......................... 340
6.2.3.3 Bingham model ......................... 340
6.3 Finite Element Models of Inelastic Fluids ............. 341
6.3.1 Introduction ................................... 341
6.3.2 Mixed Model .................................... 341
6.3.3 Penalty Model .................................. 343
6.3.4 Matrix Evaluations ............................. 344
6.4 Solution Methods for Inelastic Fluids ................. 346
6.5 Governing Equations of Viscoelastic Fluids ............ 350
6.5.1 Conservation Equations ......................... 350
6.5.2 Constitutive Equations ......................... 351
6.5.2.1 Differential models ................... 352
6.5.2.2 Integral models ....................... 355
6.5.3 Boundary Conditions ............................ 356
6.6 Finite Element Model of Differential Form ............. 357
6.6.1 Preliminary Comments ........................... 357
6.6.2 Summary of Governing Equations ................. 357
6.6.3 Finite Element Model ........................... 358
6.6.4 Solution Methods ............................... 362
6.7 Additional Models of Differential Form ................ 363
6.7.1 Explicitly Elliptic Momentum Equation Method ... 364
6.7.2 Elastic Viscous Stress Splitting Method ........ 365
6.8 Finite Element Model of Integral Form ................. 367
6.9 Unresolved Problems ................................... 368
6.9.1 General Comments ............................... 368
6.9.2 Choice of Constitutive Equation ................ 369
6.9.3 Uniqueness and Existence of Solutions .......... 370
6.9.4 Numerical Algorithm Problems ................... 370
6.9.5 Equation Change of Type ........................ 371
6.9.6 Closure ........................................ 372
6.10 Numerical Examples .................................... 372
6.10.1 Preliminary Comments ........................... 372
6.10.2 Buoyancy Driven Flow in a Cavity ............... 372
6.10.3 Driven Cavity Flow ............................. 374
6.10.4 Squeeze Film Flow .............................. 374
6.10.5 Time-Dependent Poiseuille Flow ................. 377
6.10.6 Four-to-One Contraction Problem ................ 380
Problems .............................................. 381
References for Additional Reading ..................... 382
7 Multiphysics Problems ...................................... 387
7.1 Introduction .......................................... 387
7.2 Coupled Boundary Value Problems ....................... 387
7.3 Fluid Mechanics and Heat Transfer ..................... 388
7.3.1 Introduction ................................... 388
7.3.2 Continuum Equations ............................ 388
7.3.3 Finite Element Models .......................... 390
7.4 Solid Mechanics ....................................... 390
7.4.1 Introduction ................................... 390
7.4.2 Kinematics of Deformation ...................... 391
7.4.2.1 Descriptions of motion ................ 391
7.4.2.2 Displacement vector ................... 393
7.4.2.3 Deformation gradient tensor ........... 393
7.4.2.4 Green strain tensor ................... 394
7.4.3 Kinetics ....................................... 395
7.4.3.1 Stress measures ....................... 395
7.4.3.2 Equilibrium statements ................ 395
7.4.4 Constitutive Relations ......................... 396
7.4.5 Boundary Conditions ............................ 397
7.4.6 Finite Element Models .......................... 397
7.4.7 Solution Methods ............................... 399
7.5 Electromagnetics ...................................... 399
7.5.1 Introduction ................................... 399
7.5.2 Maxwell's Equations ............................ 400
7.5.2.1 Constitutive relations ................ 400
7.5.2.2 Electromagnetic forces and volume
heating ............................... 402
7.5.2.3 Quasi-static approximation ............ 402
7.5.3 Electromagnetic Potentials ..................... 403
7.5.4 Boundary and Interface Conditions .............. 405
7.5.5 Gauge Conditions ............................... 407
7.5.6 Static Field Problems .......................... 408
7.5.6.1 Electrostatics ........................ 408
7.5.6.2 Steady current flow ................... 409
7.5.6.3 Magnetostatics ........................ 409
7.5.7 Finite Element Models for EM Fields ............ 409
7.5.7.1 Quasi-static potential equations ...... 409
7.5.7.2 Gauge condition ....................... 412
7.5.7.3 Static field equations ................ 413
7.5.8 Solution Methods - EM Fields ................... 414
7.6 Coupled Problems in Mechanics ......................... 415
7.6.1 Introduction 415
7.6.2 Heat Conduction - Viscous Fluid Interactions
1&2 ............................................ 416
7.6.3 Heat Conduction - Quasi-Static Solid
Interactions 1&3 ............................... 416
7.6.4 Heat Conduction - Electric Field Interactions
1&4 ............................................ 418
7.6.5 Heat Conduction - Electromagnetic Field
Interactions 1&4&5 ............................. 418
7.6.6 Viscous Flow — Quasi-Static Solid
Interactions 2&3 ............................... 420
7.6.7 Viscous Flow - Electric Field Interactions
2Ы ............................................. 421
7.6.8 Viscous Flow - Electromagnetic Field
Interactions 2&4&5 ............................. 422
7.6.9 Quasi-Static Solid - Electromagnetic Field
Interactions 3&4&5 ............................. 423
7.7 Implementation of Coupled Algorithms .................. 424
7.8 Numerical Examples .................................... 426
7.8.1 Introduction ................................... 426
7.8.2 Thermal-Stress Example ......................... 426
7.8.3 Thermal-Electromagnetic Example ................ 428
7.8.4 Fluid-Solid Interaction Example ................ 431
7.8.5 Fluid-Electromagnetic Example .................. 432
References for Additional Reading ..................... 436
8 Parallel Processing ........................................ 439
8.1 Introduction .......................................... 439
8.2 Parallel Systems ...................................... 440
8.2.1 Classification ................................. 440
8.2.1.1 Granularity of the processing
elements .............................. 440
8.2.1.2 Topology of interconnections .......... 440
8.2.1.3 Distribution of control across the
processing elements ................... 441
8.2.1.4 Memory access ......................... 442
8.2.2 Languages and Communication Utilities .......... 442
8.2.3 Performance .................................... 443
8.2.3.1 Algorithmic efficiency ................ 443
8.2.3.2 Actual/Beneficial efficiency .......... 443
8.2.3.3 Scalability ........................... 444
8.3 FEM and Parallel Processing ........................... 444
8.3.1 Preliminary Comments ........................... 444
8.3.2 Generic FEM Steps .............................. 445
8.3.3 External Preprocessing ......................... 445
8.3.4 Internal Preprocessing ......................... 447
8.3.5 Solution Processing ............................ 447
8.3.5.1 Element matrices ...................... 447
8.3.5.2 Matrix solvers ........................ 448
8.3.5.3 Solution control ...................... 450
8.3.6 Internal Postprocessing ........................ 451
8.3.7 External Postprocessing ........................ 451
8.3.8 Other Parallel Issues .......................... 451
8.3.8.1 Nonlocal data ......................... 452
8.3.8.2 Multiphysics simulations .............. 452
8.4 Summary ........................................... 454
References for Additional Reading ..................... 454
Appendix A: Computer Program FEM2DHT .......................... 457
A.l Introduction .......................................... 457
A.2 Heat Transfer and Related Problems .................... 457
A.3 Flows of Viscous Incompressible Fluids ................ 458
A.4 Description of the Input Data ......................... 458
A.5 Source Listings of Selective Subroutines .............. 469
Reference for Additional Reading ........................... 470
Appendix B: Solution of Linear Equations ...................... 477
B.l Introduction .......................................... 477
B.2 Direct Methods ........................................ 478
B.3 Iterative Methods ..................................... 479
B.3.1 General Comments ............................... 479
B.3.2 Solution Algorithms ............................ 479
References for Additional Reading .......................... 483
Appendix C: Fixed Point Methods and Contraction Mappings ...... 485
C.l Fixed Point Theorem ................................... 485
C.2 Chord Method .......................................... 486
C.3 Newton's Method ....................................... 487
C.4 The Newton-Raphson Method ............................. 488
C.5 Descent Methods ....................................... 488
References for Additional Reading .......................... 489
Subject Index ................................................. 491
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