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