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The Finite Element Methodin Heat Transfer and

Fluid Dynamics

Third Edition

J. N. ReddyDepartment of Mechanical Engineering

Texas A&M University

College Station, Texas, USA 778433123

D. K. GartlingEngineering Sciences Center

Sandia National Laboratories

Albuquerque, New Mexico, USA 87185

CRC Press

Boca Raton London New York Washington, D.C.

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Library of Congress Cataloging-in-Publication Data

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To our wives

Aruna and Laura

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1.11 Change of Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.12 Enclosure Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.13 Summary of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37References 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 . . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.2 Weak Form Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.1 Properties of Approximation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.2 Linear Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.3 Linear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.6.4 Evaluation of Boundary Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

2.7 Assembly of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.8 Time-Dependent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.8.2 Semidiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.8.3 Temporal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.9 Axisymmetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.10 Library of Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.10.2 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.10.3 Rectangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.11 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.11.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.11.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.11.3 Integration over a Master Rectangular Element . . . . . . . . . . . . . . . . . . 702.11.4 Integration over a Master Triangular Element . . . . . . . . . . . . . . . . . . . 71

2.12 Modeling Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.12.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.12.2 Representation of Boundary Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.12.3 Imposition of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.13 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.13.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.13.1.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.13.1.2 Solution by linear triangular elements . . . . . . . . . . . . . . . . . . . 772.13.1.3 Solution by linear rectangular elements . . . . . . . . . . . . . . . . . 792.13.1.4 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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2.13.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.13.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

863. Conduction Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.2 Semidiscrete Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.2 Hexahedral (Brick) Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.3 Tetrahedral Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.4 Prism Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.3.5 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.7.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.7.2 Steady-State Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.7.3 Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.7.3.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.7.3.2 Predictor-corrector methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.7.3.3 Time step control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.7.3.4 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.7.3.5 Linear multi-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.7.3.6 Convergence and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.7.3.7 Mode superposition methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.8 Radiation Solution Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.9 Variable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.9.1 Temperature-Dependent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.9.2 Phase Change Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.9.3 Anisotropic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.10 Post-Processing Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.10.1 Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.10.2 Heat Flow Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.11 Advanced Topics in Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.11.2 Specialty Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.11.3 Computational Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.11.3.1 Contact boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.11.3.2 Multipoint constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.11.3.3 Partially covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.11.4 Bulk Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.11.5 Reactive Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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3.11.6 Material Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.12 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.12.2 Element Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1393.12.3 Conduction/Radiation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.12.4 Temperature-Dependent Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.12.5 Anisotropic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.12.6 One-Dimensional Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.12.7 Drag Bit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.12.8 Brazing and Welding Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.12.9 Investment Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

References for Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4. Flows of Viscous Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.2 Mixed Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.2.1 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.2.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.3 Penalty Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.3.2 Penalty Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.3.3 Reduced Integration Penalty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.3.4 Consistent Penalty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.4 Finite Element Models of Porous Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1724.5 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.5.1 Properties of the Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.5.2 Choice of Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.5.2.1 Quadrilateral elements (2-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.5.2.2 Triangular elements (2-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.5.2.3 Hexahedral elements (3-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.5.2.4 Tetrahedral elements (3-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.5.3 Evaluation of Element Matrices in Penalty Models . . . . . . . . . . . . . . 1804.5.4 Pressure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.5.5 Traction Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.6 Solution of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.6.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.6.2 Fully Coupled Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.6.2.1 Picard method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.6.2.2 Newtons method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.6.2.3 Modied and quasiNewton methods . . . . . . . . . . . . . . . . . . . . 1924.6.2.4 Continuation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.6.3 Pressure Correction/Projection Methods . . . . . . . . . . . . . . . . . . . . . . . . 194

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4.7 Time-Approximation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.7.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.7.2 Forward/Backward Euler Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.7.3 AdamsBashforth/Trapezoid Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.7.4 Implicit Integration and Time Step Control . . . . . . . . . . . . . . . . . . . . . 1984.7.5 Explicit Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

4.8 Stabilized Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.8.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.8.2 Galerkin/Least-Squares Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.8.3 Polynomial Pressure Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.8.4 Variational Multiscale Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

4.9 Least-Squares Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.9.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.9.3 Least-Squares Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

4.9.4 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.9.5 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.10 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.10.1 Stress Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.10.2 Stream Function Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.10.3 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.11 Free Surface Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244.11.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244.11.2 Time-Independent Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244.11.3 Time-Dependent Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

4.12 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.12.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354.12.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.12.3 General Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

4.12.3.1 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.12.3.2 Integral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.12.3.3 One-point closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.12.3.4 Two-point closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.12.3.5 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.12.3.6 Direct numerical simulations (DNS) . . . . . . . . . . . . . . . . . . . 239

4.12.4 One-Point Closure Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . 2394.12.4.1 Zero-equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2404.12.4.2 One-equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

4.12.4.3 Two-equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.12.5 Finite Element Modeling of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 242

4.12.5.1 Zero-equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.12.5.2 Two-equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4.12.6 Variational Multiscale (VMS) Turbulence Modeling . . . . . . . . . . . . 244

4.13 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.13.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.13.2 Fluid Squeezed between Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . 248

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4.13.3 Flow of a Viscous Lubricant in a Slider Bearing . . . . . . . . . . . . . . . . 2504.13.4 Wall-Driven 2-D Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2524.13.5 Wall-Driven 3-D Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544.13.6 Evaluation of the EBE Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . 2564.13.7 Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2584.13.8 Flow Past a Submarine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2604.13.9 Crystal Growth from the Melt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.13.10 Mold Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.13.11 Examples Using Least-Squares Finite Element Models . . . . . . . . . 267

4.13.11.1 Kovasznay ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2674.13.11.2 Flow over a backward-facing step . . . . . . . . . . . . . . . . . . . . 2694.13.11.3 Flow past a cylinder at low Reynolds number . . . . . . . . 270

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273


5. Coupled Fluid Flow and Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 285

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

5.2 Nonisothermal Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2865.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2865.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

5.3 Mixed Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

5.4 Penalty Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.4.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.4.2 Reduced Integration Penalty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2945.4.3 Consistent Penalty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

5.5 Finite Element Models of Porous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

5.6 Nonisothermal, Low-Speed, Compressible Flows . . . . . . . . . . . . . . . . . . . . . . 2975.6.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2975.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.6.3 Mixed Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

5.7 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3025.7.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3025.7.2 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3035.7.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.12.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.12.2 Finite Element Modeling of Chemical Reactions . . . . . . . . . . . . . . . . 311

5.13 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125.13.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125.13.2 Concentric Tube Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

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5.13.3 Tube Flow with Change of Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135.13.4 Heated Cavity Boussinesq Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3145.13.5 Heated Cavity Acoustically Filtered Model . . . . . . . . . . . . . . . . . . 3165.13.6 Solar Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3175.13.7 Tube Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3205.13.8 Volumetrically Heated Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3225.13.9 Porous/Fluid Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3225.13.10 Curing of an Epoxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3265.13.11 Heated Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3295.13.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3366.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.2.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

6.2.3.1 Power-law model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.2.3.2 Carreau model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3406.2.3.3 Bingham model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

6.3 Finite Element Models of Inelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.3.2 Mixed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.3.3 Penalty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3436.3.4 Matrix Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

6.4 Solution Methods for Inelastic Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3466.5 Governing Equations of Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

6.5.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3506.5.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

6.5.2.1 Differential models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3526.5.2.2 Integral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

6.5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

6.6 Finite Element Model of Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.6.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.6.2 Summary of Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.6.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3586.6.4 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

6.7 Additional Models of Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3636.7.1 Explicitly Elliptic Momentum Equation Method . . . . . . . . . . . . . . . . . 3646.7.2 Elastic Viscous Stress Splitting Method . . . . . . . . . . . . . . . . . . . . . . . . . 365

6.8 Finite Element Model of Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

6.9 Unresolved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3686.9.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3686.9.2 Choice of Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

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7.5.6 Static Field Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4087.5.6.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4087.5.6.2 Steady current ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4097.5.6.3 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

7.5.7 Finite Element Models for EM Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 4097.5.7.1 Quasi-static potential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 4097.5.7.2 Gauge condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4127.5.7.3 Static eld equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

7.5.8 Solution Methods EM Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

7.6 Coupled Problems in Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157.6.2 Heat Conduction Viscous Fluid Interactions 1&2 . . . . . . . . . . . . . . 4167.6.3 Heat Conduction Quasi-Static Solid Interactions 1&3 . . . . . . . . . 4167.6.4 Heat Conduction Electric Field Interactions 1&4 . . . . . . . . . . . . . . 4187.6.5 Heat Conduction Electromagnetic Field Interactions 1&4&5 . . . 418

7.6.6 Viscous Flow

Quasi-Static Solid Interactions 2&3. . . . . . . . . . . . .

4207.6.7 Viscous Flow Electric Field Interactions 2&4 . . . . . . . . . . . . . . . . . . 4217.6.8 Viscous Flow Electromagnetic Field Interactions 2&4&5 . . . . . . 4227.6.9 Quasi-Static Solid Electromagnetic Field Interactions 3&4&5 . . 423

7.7 Implementation of Coupled Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

7.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4267.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4267.8.2 Thermal-Stress Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4267.8.3 Thermal-Electromagnetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4287.8.4 Fluid-Solid Interaction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.8.5 Fluid-Electromagnetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4368. Parallel Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.2 Parallel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4408.2.1 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

8.2.1.1 Granularity of the processing elements . . . . . . . . . . . . . . . . . . . 4408.2.1.2 Topology of interconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4408.2.1.3 Distribution of control across the processing elements . . . . 4418.2.1.4 Memory access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

8.2.2 Languages and Communication Utilities . . . . . . . . . . . . . . . . . . . . . . . . . 4428.2.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

8.2.3.1 Algorithmic effi

ciency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4438.2.3.2 Actual/Benecial efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4438.2.3.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

8.3 FEM and Parallel Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4448.3.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4448.3.2 Generic FEM Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4458.3.3 External Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4458.3.4 Internal Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

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8.3.5 Solution Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4478.3.5.1 Element matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4478.3.5.2 Matrix solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4488.3.5.3 Solution control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

8.3.6 Internal Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3.7 External Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3.8 Other Parallel Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

8.3.8.1 Nonlocal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4528.3.8.2 Multiphysics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454


Appendix A: Computer Program 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

Appendix B: Solution of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 477

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

B.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

B.3 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479B.3.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

B.3.2 Solution Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

479References for Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Appendix C: Fixed Point Methods and Contraction Mappings 485

C.1 Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

C.2 Chord Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

C.3 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

C.4 The NewtonRaphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

C.5 Descent Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488


Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

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Preface to the Third Edition

Computational uid mechanics (CFD) and computational heat transfer (CHT)continue to evolve at a signicant pace and have become an ever increasing presencein standard engineering design and analysis practice. The seemingly endless increasein computing power in both single processor and parallel environments has allowedrealistic problems of signicant complexity and delity to be routinely solved and

utilized in technological advances. Commercial software has made rapid progress inproviding a broad spectrum of analysis capabilities to a variety of industries. Thoughsoftware is increasingly robust, accurate CFD and CHT simulations still require aknowledgeable user, with a background in both mechanics and numerical methods.The present edition of this book remains focused on providing the informationrequired by an individual who is interested in good numerical methods for the studyand understanding ofuid mechanics and heat transfer phenomena.

This book remains practical in scope and content with an emphasis oncomputational procedures that we have found effective on a wide spectrum ofapplications. Little, if any, material has been deleted from the second edition. Newmaterial has been added primarily in the rst ve chapters and reects the researchthrusts over the last eight years. Chapter 1 contains the general description of

the boundary value problems of interest and has been expanded with a section onmathematical preliminaries and a section on low-speed compressible ows. Chapter2 continues with the introduction of the nite element method and is essentiallyunchanged. The thermal conduction and radiation problem is discussed in greatdetail in Chapter 3, and has a new discussion of mode superposition methods anda more detailed account of radiation solution methods. The isothermal, viscousow problem is the topic of Chapter 4. The section on stabilized methods has beenexpanded with more discussion of variational multiscale methods (VMM), and a newsection on least-squares nite element models (LSFEM) has been added. Chapter5 extends the nite element method to non-isothermal ows and now includes asection on the formulation of low-speed, compressible ows. Non-Newtonian owproblems, both inelastic and viscoelastic, are included in Chapter 6, which is largely

unchanged. Chapter 7 remains focused on the formulations and algorithms formultidisciplinary problems involving uid mechanics, heat transfer, solid mechanics,and electromagnetics; the last chapter is on parallel computing, including a generaldiscussion of the parallel architecture and the implementation of nite elementmodels. Organization of the text, equation numbering, references, and symbolsretain the same style as used in the previous editions. References have been addedas needed without the removal of older citations. We believe it is important to retainthe historical record.

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The authors thank the publisher for the opportunity to prepare this third edition.The rst author thanks colleagues Drs. Vinu Unnikrishnan and Ginu Unnikrishnanfor their help with the scanning ofgures from the rst two editions and preparingthe subject index, and Feifei Cheng for proofreading of the manuscript. Thesecond author again thanks his numerous present and former colleagues at SandiaNational Laboratories who continue to present challenging engineering applicationsin applied and computational mechanics. Specic acknowledgments must go to Drs.Mike Glass, Rick Givler, Charles Hickox, Roy Hogan, Mario Martinez and PhilSackinger for collaboration and assistance in much of the algorithm developmentand demonstration simulations cited in this work. Portions of this book are takenfrom research work performed at Sandia National Laboratories under Contract No.DEAC0494AL85000 awarded by the U.S. Department of Energy and are usedwith permission. The authors dedicate this book to their wives, Aruna and Laura,who have graciously tolerated the authors preoccupation with the writing of thebook.

J. N. Reddy

College Station, [email protected]

D. K. Gartling

Albuquerque, New [email protected]

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xix

Preface to the Second Edition

In the six years since the rst edition of this book appeared some signicantchanges have occurred in the area of computational mechanics in general, and incomputational uid mechanics and heat transfer in particular. Foremost amongthese changes has been the extraordinary increase in performance in desktopcomputing platforms and the arrival in signicant numbers of parallel computers.This widespread availability of capable computing hardware has predictably lead tothe increased demand for computer simulation of products and processes during theengineering design and manufacturing process. Our original thesis that the niteelement method was very well suited to general purpose and commercial softwarecontinues to hold true, as numerous programs are now available for the simulation ofall types of applied mechanics problems. The range of applications ofnite elementanalysis in uid mechanics and heat transfer has become quite remarkable withcomplex, realistic simulations being carried out on a routine basis. The combinationof hardware performance and reliable nite element algorithms has made theseadvances possible. Another signicant change in computational mechanics is theincrease in multidisciplinary (multiphysics) problems and their solution via niteelement methods. Again, the increase in hardware performance has contributed tothese types of computationally intensive problems. However, the inroads made bythe nite element method in all areas of mechanics have also had a positive inuenceon coupled analysis. The commonality ofnite element formulation, approximationand solution among the various boundary value problems in mechanics eases

considerably the contemplation of multiphysics solutions and software. A

nalchange in the numerical simulation arena comes from the implementation side ofthe nite element method. The demand for software capability and reliability hasincreased in step with the hardware performance. The use of parallel computershas added to the complexity of the implementation. All of these attributes lead tothe conclusion that nite element implementation, if done well, will require somesignicant knowledge from areas in computer science. The time of general purposecodes being developed and maintained by one or two individuals is past and multi-talented teams now provide the most modern software.

Our focus for the present edition of this book remains the same the educationof the individual who is interested in good numerical methods for the study of uidmechanics and heat transfer phenomena. The text remains practical in scope and

content with an emphasis on computational procedures that we have found effectiveon a wide spectrum of applications. Little, if any, material has been deleted fromthe rst edition. New material has been added in almost all chapters along withsome rearrangement of topics to improve overall clarity and maintain the step-wise addition of increasingly complex material. Chapter 1 contains the generaldescription of the boundary value problems of interest and has been augmentedwith a section on chemically reactive systems and additional discussion of changeof phase. Chapter 2 continues with the introduction of the nite element method

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xx PREFACE

and is essentially unchanged. The thermal conduction and radiation problem isdiscussed in great detail in Chapter 3, and it has new sections covering specializednite elements and advanced topics in thermal analysis. The advanced topics sectionincludes descriptions of difficult boundary conditions, such as multipoint constraints,contact and bulk nodes, material motion and kinematics, and methods for chemicallyreactive solids. The isothermal, viscous ow problem is the topic of Chapter 4.New sections in this chapter cover stabilized nite element methods and a generaldiscussion of methods for free surface problems; the section on turbulence modelinghas also been moved to this chapter. Chapter 5 extends the nite element methodto non-isothermal ows and is largely unchanged. Non-Newtonian ow problems,both inelastic and viscoelastic, are now included in a revised and updated Chapter6. A completely new Chapter 7 is focused on formulations and algorithms formultidisciplinary problems involving uid mechanics, heat transfer, solid mechanics,and electromagnetics. This chapter outlines many of the possible types of coupling,describes the nite element equations for each mechanics area, and presents a

number of realistic numerical examples. The last chapter on advanced topics is nowdevoted exclusively to a discussion of parallel computing including some generaldiscussion of the parallel architecture and sections on parallel implementation ofnite element models. Organization of the text, equation numbering, references,and symbols retain the same style as used in the rst edition.

The second author thanks his numerous present and former colleagues atSandia National Laboratories who continue to provide a wealth of challengingproblems in applied and computational mechanics. Specic acknowledgmentsmust go to Drs. Mike Glass, Rick Givler, Charles Hickox, Roy Hogan and PhilSackinger for collaboration and assistance in much of the algorithm developmentand demonstration simulations cited in this work. Portions of this book are adaptedfrom work performed at Sandia National Laboratories under Contract No. DE

AC0494AL85000 awarded by the U.S. Department of Energy and are used withpermission. The authors dedicate this book to their wives, Aruna and Laura, whohave graciously tolerated the authors preoccupation with the writing of the book.

J. N. Reddy

College Station, Texas

D. K. Gartling

Albuquerque, New Mexico

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xxi

Preface to the First Edition

The numerical simulation ofuid mechanics and heat transfer problems has becomea routine part of engineering practice as well as a focus for fundamental andapplied research. Though there are still various topical areas where our physicalunderstanding and/or ineffective numerical algorithms limit the investigation, alarge number of complex phenomena can now be condently studied via numericalsimulation. Though nite difference methods have and will continue to playa major role in computational uid dynamics (CFD) and heat transfer, niteelement techniques have spurred the explosive development of general purposemethods and the growth of commercial software. The inherent strengths of thenite element method such as unstructured meshes, element-by-element formulationand processing, and the simplicity and rigor of boundary condition applicationare being coupled with modern developments in automatic mesh generation,adaptive meshing, and improved solution techniques to produce accurate and reliablesimulation packages that are widely accessible. Improvements in computer hardwareand system software (e.g., powerful workstations and window environments) havecontributed signicantly to streamlining the numerical simulation process. The niteelement method in uid mechanics and heat transfer has rapidly caught up with thewell-established solid mechanics community in simulation capabilities.

As in any rapidly developing eld, the education of the non-expert usercommunity is of primary importance. The present text is an attempt to ll a needfor those interested in using the nite element method in the study ofuid mechanics

and heat transfer. It is a pragmatic book that views numerical computation as ameans to an endwe do not dwell on theory or proof. Other fundamental andtheoretical textbooks that cover these aspects are available or anticipated. Theemphasis here is on presenting a useful methodology for a limited but signicantclass of problems dealing with heat conduction, incompressible viscous ows, andconvection heat transfer.

The text has been developed out of our experience and course notes used inteaching graduate courses and continuing education courses to a wide spectrum ofstudents. To gain the most from the book the student should have a reasonablebackground in uid mechanics and heat transfer as would normally be found inmost mechanical, aerospace, chemical or engineering mechanics curriculums. Anintroductory knowledge of nite element techniques would be very helpful but

not essential; some familiarity with basic numerical analysis, linear algebra, andnumerical integration would also be of assistance.

Our approach to the nite element method for uid mechanics and heat transferhas been designed as a series of incremental steps of increasing complexity. InChapter 1, the continuum boundary value problems that form the central focusof the book are described in some detail. We have tried to be as general aspossible in describing the varied physical phenomena that may be encounteredwithin the limits of non-isothermal, incompressible, viscous ows. Chapter 2

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introduces the nite element method by application to a simplied, two-dimensionalheat conduction problem. All of the necessary machinery for constructing weakforms of a partial differential equation and building a nite element model areintroduced here and demonstrated by application. Chapter 3 recaps parts of Chapter2 and extends the nite element method to three dimensions, time dependence,and practical applications in conduction heat transfer. Isothermal viscous uidmechanics formulations are described in Chapter 4 along with a signicant sectionon the solution of nonlinear equations developed from the ow problem. Chapter 5extends the viscous ow problem to consider convective heat transfer formulationsand applications. Inelastic non-Newtonian ows and free surface problems arecovered in Chapter 6. The complex topic of viscoelastic ow simulation is surveyedin Chapter 7. The last chapter concludes the text with a survey of several advancedtopics, including turbulence modeling. The coverage of each topic is sufficientto allow the reader to understand the basic methodology, use existing simulationsoftware with condence, and allow development of some simpler, special purpose

computer codes. Example problems ranging from simple benchmarks to practicalengineering solutions are included with each topical area. Adequate references tothe relevant literature have also been included for those desiring a more encyclopediccoverage of a specic topic.

The text is organized into major sections within each chapter. Equations arenumbered consecutively within each ma jor section. Within a section, reference toan equation is by its sequential number; references to equations outside the currentsection have a full section, equation number citation. Vectors, tensors, and matricesare denoted by boldface letters. The vectors of interpolation (shape) functions inthis book are denoted by Greek symbols (,,). Bibliographic information forliterature cited in the text is numbered sequentially within each chapter and collectedat the end of the chapter.

The rst author would like to thank M. S. Ravisankar for his help with Chapter8 topics and Praveen Gramma for reading the manuscript. The second authorwould like to thank his numerous present and former colleagues at Sandia NationalLaboratories who have provided a seemingly endless stream of challenging problemsand taught him much about the practice of computational mechanics. Specicacknowledgments must go to Drs. Charles Hickox, Rick Givler, Roy Hogan, PhilSackinger, Randy Schunk, Rekha Rao, and Steve Rottler for their suggestionsand comments on early versions of the text. Portions of this book are adaptedfrom work performed at Sandia National Laboratories under Contract No. DEAC0476DP00789 awarded by the U.S. Department of Energy and are used withpermission.

J. N. ReddyCollege Station, Texas

D. K. Gartling

Albuquerque, New Mexico

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xxiii

About the Authors

J. N. Reddy earned a Ph.D. in Engineering Mechanics from the University of Alabamain Huntsville, worked as a Postdoctoral Fellow at the University of Texas at Austin, wasResearch Scientist for Lockheed Missiles and Space Company during l97475, and taught atthe University of Oklahoma from 1975 to 1980 and Virginia Polytechnic Institute & StateUniversity from 1980 to 1992. Currently, he is a Distinguished Professor and the inauguralholder of the Oscar S. Wyatt Endowed Chair at Texas A&M University, College Station. Dr.Reddy has published over 400 journal papers and 16 textbooks on theoretical formulationsand numerical simulations of problems in solid and structural mechanics, computationaluid dynamics, numerical heat transfer, computational biology, geology and geophysics,mechanics of nanosystems, and applied mathematics.

Dr. Reddy is the recipient of numerous honors and awards, including the 1998 NathanM. Newmark Medal from the American Society of Civil Engineers, the 2003 ComputationalSolid Mechanics award from the U.S. Association of Computational Mechanics, the 2004Distinguished Research Award from the American Society for Composites, and an honorarydegree (Honoris Causa) from the Technical University of Lisbon, Portugal (2009). Dr.Reddy is a fellow of the American Academy of Mechanics, the American Institute ofAeronautics and Astronautics, the American Society of Civil Engineers, the AmericanSociety of Mechanical Engineers, the American Society for Composites, InternationalAssociation of Computational Mechanics, U.S. Association of Computational Mechanics,the Aeronautical Society of India, and the Institution of Structural Engineers, U.K. Dr.Reddy serves on the editorial boards of two dozen journals, and as the Editor-in-Chief ofApplied Mechanics Reviews, Mechanics of Advanced Materials and Structures, InternationalJournal of Computational Methods in Engineering Science and Mechanics, and InternationalJournal of Structural Stability and Dynamics.

As a result of his extensive publications of archival journal papers and books, Dr. Reddyis recognized by ISI Highly Cited Researchers with over 10,000 citations and an H-index ofover 40 to his credit. A more complete resume with links to journal papers can be found athttp://authors.isihighlycited.com/ or http://www.tamu.edu/acml.

David K. Gartling is a Senior Scientist in the Engineering Sciences Center at SandiaNational Laboratories, Albuquerque, New Mexico. He earned his B.S. and M.S. in AerospaceEngineering at the University of Texas at Austin and completed the diploma course atthe von Karman Institute for Fluid Dynamics in Brussels, Belgium. After completion ofhis Ph.D. in Aerospace Engineering at the University of Texas at Austin, he joined thetechnical staff at Sandia National Laboratories. Dr. Gartling was a Visiting Associate

Professor in the Mechanical Engineering Department at the University of Sydney, Australia,under a Fulbright Fellowship, and later he was a Supervisor in the Fluid and ThermalSciences Department at Sandia National Laboratories. Dr. Gartling has published numerouspapers dealing with nite element model development and nite element analysis of heattransfer and uid dynamics problems of practical importance. He is the recipient of the2001 Computational Fluid Dynamics Award from the U.S. Association of ComputationalMechanics and is a fellow of the American Society of Mechanical Engineers. Dr. Gartling ispresently a member of several professional societies, serves on the editorial boards of several

journals, and is the Co-Editor of International Journal for Numerical Methods in Fluids.

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