The Finite Element Method in Heat Transfer and Fluid Dynamics Third Edition

J. N. Reddy D. K. Gartling

@ CRC Press Taylor & Francis Group

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

Contents

Preface t o t h e Third Edi t ion xvii

Preface t o t h e Second Edi t ion xix

Preface t o t h e First Ed i t ion xxi

A b o u t t h e A u t h o r s xxiii

1. Equat ions of H e a t Transfer and Fluid Mechan ic s 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

V l l l C O N T E N T S

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

CONTENTS i x

2.13.2 Example 2 81 2.13.3 Example 3 82

Problems 83

References for Additional Reading 86

3. Conduc t ion H e a t 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 Hexahedral (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 I l l

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

X C O N T E N T S

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

CONTENTS XI

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

Х П C O N T E N T S

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 F low and H e a t 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

C O N T E N T S X l l l

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 s 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

Xiv CONTENTS

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

CONTENTS XV

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 2&4 421 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 Proces s ing 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

Xvi CONTENTS

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

A p p e n d i x A: C o m p u t e r P r o g r a m FEM2DHT 457

A. 1 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

A p p e n d i x B: So lut ion of Linear Equat ions 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

A p p e n d i x C: F i x e d Point M e t h o d s and Contract ion M a p p i n g s 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 I n d e x 491