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

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The Finite Element Method in Heat Transfer and Fluid Dynamics Third Edition The Finite Element Method in Heat Transfer and Fluid Dynamics Third Edition J. N. Reddy Department of Mechanical Engineering Texas A&M University College Station, Texas, USA 77843—3123 D. K. Gartling Engineering Sciences Center Sandia National Laboratories Albuquerque, New Mexico, USA 87185 CRC Press Boca Raton London New York Washington, D.C. • • • Library of Congress Cataloging-in-Publication Data To our wives Aruna and Laura vii Contents 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 viii CONTENTS 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 ix 2.13.2 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 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 ............................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 x CONTENTS 3.11.6 Material Motion ...............................................138
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