The finite element method

GS1059_FM.indd Process CyanProcess CyanProcess MagentaProcess MagentaProcess Yellow YellowProcess Process Black 9/23/05 3:40:50 PM GS1059_Series.fm Page Monday, September 26, 2005 11:13 AM Series in Computational and Physical Processes in Mechanics and Thermal Sciences (Formerly the Series in Computational Methods in Mechanics and Thermal Sciences) W J Minkowycz and E M Sparrow, Editors Anderson, Tannehill and Pletcher, Computational Fluid Mechanics and Heat Transfer Aziz and Na, Perturbation Methods in Heat Transfer Baker, Finite Element Computational Fluid Mechanics Baker, Finite Element Computational Fluid Mechanics, Second Edition Beck, Cole, Haji-Sheikh and Litkouhi, Heat Conduction Using Green’s Functions Carey, Computational Grids Comini, del Giudice and Nonino, Finite Element Analysis in Heat Transfer Heinrich and Pepper, Intermediate Finite Element Method: Fluid Flow and Heat Transfer Application Jaluria, Computer Methods for Engineering Koenig, Modern Computational Methods Patankar, Numerical Heat Transfer and Fluid Flow Pepper and Heinrich, The Finite Element Method Shih, Numerical Heat Transfer Shyy, Udaykumar, Rao, and Smith, Computational Fluid Dynamics With Moving Boundaries Tannehill, Anderson and Pletcher, Computational Fluid Mechanics and Heat Transfer, Second Edition PROCEEDINGS Chung, Editor, Finite Elements in Fluids Chung, Editor, Numerical Modeling in Combustion Haji-Sheikh, Editor, Integral Methods in Science and Engineering - 90 Shih, Editor, Numerical Properties and Methodologies in Heat Transfer The Finite Element Method Basic Concepts and Applications S E C O N D E D I T I O N Darrell W Pepper University of Nevada Las Vegas, Nevada Juan C Heinrich University of New Mexico Albuquerque, New Mexico GS1059_FM.indd Process CyanProcess CyanProcess MagentaProcess MagentaProcess Yellow YellowProcess Process Black 9/23/05 3:40:51 PM GS1059_Discl.fm Page Thursday, August 18, 2005 12:10 PM Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-10: 1-59169-027-7 (Hardcover) International Standard Book Number-13: 978-1-59169-027-6 (Hardcover) Library of Congress Card Number 2005002971 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Pepper, D W (Darrell W.) The finite element method : basic concepts and applications / by Darrell W Pepper and Juan C Heinrich. 2nd ed p cm (Series in computational and physical processes in mechanics and thermal sciences) Includes bibliographical references and index ISBN 1-59169-027-7 Finite element method I Heinrich, Juan C II Title III Series TA347.F5P46 2005 620'.001'51825 dc22 2005002971 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of T&F Informa plc and the CRC Press Web site at http://www.crcpress.com GS1059_C000.fm Page v Friday, September 9, 2005 10:57 AM To our parents Weldon and Marjorie Pepper Carlos and Ruby Heinrich GS1059_C000.fm Page vi Friday, September 9, 2005 10:57 AM GS1059_C000.fm Page vii Friday, September 9, 2005 10:57 AM Preface It has been more than 13 years since the first edition of this book was published Many changes in the art of finite element methodology have occurred since then FORTRAN was the most prevalent programming language for scientific computing, and still exists in enhanced forms today C/C++ has become one of the preferred choices for much of the computing performed on PCs using WINDOWS As we progress into the 21st century, a new language is beginning to appear—JAVA—a platform-independent language that also runs on the Web The early finite element codes in the first edition were written in FORTRAN 77 and QuickBasic for graphical display under WINDOWS FORTRAN still appears to be the preferred scientific language for the scientist–engineer, and so we felt that it was necessary to include source listings in FORTRAN For this second edition, FORTRAN 95 versions of 1-D, 2-D, and 3-D codes are maintained on a Web site, located at femcodes.nscee.edu, along with several MATHCAD, MATLAB, and MAPLE algorithms Interactive C/C++ and JAVA versions of the 2-D codes are also available from the Web While the fundamental principles of the finite element method remain unchanged, applications of the method have continued to advance into new areas, including such fields as nanotechnology and biomedical This book is based on our experience in teaching the finite element method to both engineering students and experienced, practicing engineers in industry Much of the material stems from the AIAA home study course and ASME short courses that we have given over the last 17 years and from the suggestions and recommendations of the many participants There are many finite element books available in the literature today, and this book is among the multitude that continues to appear When teaching finite element methodology to students, we always found that some alteration or simplification of much of the material was required before the concepts were grasped Some of the confusion resulted from the mathematical “jargon” and deep theoretical aspects of the technique We found that a much simpler approach was required before one can truly “appreciate” the detailed mathematical derivations and theory The primary intent of this book is to introduce the basic fundamentals of the finite element method in a clear, concise manner using simple examples Much attention is paid to the development of the discrete set of algebraic equations, beginning with simple one-dimensional problems that can be solved by inspection, continuing to two- and three-dimensional elements, and ending with three chapters dealing with applications The example problems are straightforward and can be worked out manually The computer codes that accompany this text will be helpful for many of the exercises, especially multidimensional homework problems; however, almost any one of the commercially available finite element codes available today can be used for the problems The FORTRAN source listings on the Web include simple 1-D and 2-D mesh generators and example data files Descriptions of the codes are included in the help file on the Web (see Appendix E) GS1059_C000.fm Page viii Friday, September 9, 2005 10:57 AM More exercises have been added to this revised edition, some of which involve working out the developments presented in the text in more detail In addition, we have added executable files on the Web that can be run using COMSOL, a commercial finite element code that has been developed by COMSOL, Inc (originally written to run under the MATLAB operating system) COMSOL is a very versatile finite element code that handles a wide variety of applications, including fluid flow, heat transfer, solid mechanics, and electrodynamics This package runs on PCs and is easy to use Because many finite element books are slanted toward the structurally oriented engineer, the nonstructural engineer must sift through a considerable amount of uninteresting concepts and applications before finding a relevant problem area We have found that most engineers are knowledgeable of the basic precepts of heat transfer and have, accordingly, directed the book toward heat flow and one degree of freedom (temperature) Many of the individuals whom we have instructed over the years have come from diverse backgrounds ranging from biology to nuclear physics; in nearly all cases, a simple generic approach focused on the transport and diffusion of heat (scalar transport) has allowed relatively easy mastering of finite elements We thank our colleagues and former short-course and home-study participants who greatly contributed to the presentation of the material in this second edition of our introductory text We especially thank Taylor & Francis for its helpful comments and editorial assistance, and Professor W.J Minkowycz and Professor E.M Sparrow for their continued patience in reading and for their suggestions for revising the manuscript We also express our thanks to Mrs Jeannie Pepper for her Herculean efforts in preparing the manuscript and the graphical images in this second edition Darrell W Pepper Juan C Heinrich GS1059_bookTOC.fm Page ix Saturday, September 10, 2005 11:08 AM Table of Contents Chapter Introduction 1.1 Background .1 1.2 Short History 1.3 Orientation .3 1.4 Closure .5 References Chapter The Method of Weighted Residuals and Galerkin Approximations 2.1 Background .7 2.2 Classical Solutions 2.3 The “Weak” Statement 2.4 Closure 17 Exercises 17 References .19 Chapter The Finite Element Method in One Dimension .21 3.1 3.2 Background 21 Shape Functions 21 3.2.1 Linear Elements 22 3.2.2 Quadratic Elements 25 3.2.3 Cubic Elements 27 3.3 Steady Conduction Equation 29 3.3.1 Galerkin Formulation 29 3.3.2 Variable Conduction and Boundary Convection 34 3.4 Axisymmetric Heat Conduction 39 3.5 Natural Coordinate System 41 3.6 Time Dependence 49 3.6.1 Spatial Discretization 49 3.6.2 Time Discretization 51 3.7 Matrix Formulation .54 3.8 Solution Methods 57 3.9 Closure 64 Exercises 64 References .69 Chapter 4.1 4.2 The Two-Dimensional Triangular Element 71 Background 71 The Mesh .72 GS1059_A001.fm Page 299 Friday, September 9, 2005 11:47 AM Thermophysical Properties of Some Common Materials | 299 Quantity Symbol SI English Conversion Heat flux (unit per length) q/L W /m Btu / hr ft W / m = 1.0403 Btu / hr ft Heat generation (per unit volume) Q W / m3 Btu / hr ft W / m = 0.096623 Btu / hr ft Length L m ft m = 3.2808 ft Mass m kg lbm kg = 2.20462 lbm Pressure p N / m2 lb f / in N / m = 1.45038 x 10 −4 lb f / in Specific heat cp kJ / kgC = 0.23884 Btu / lbm F k kJ / kgC W / mC Btu / lbm F Thermal conductivity Btu / hr ftF W / mC = 0.5778 Btu / hr ft F Thermal diffusivity α(k/ρcp) m2 / s ft /s ft/s m / s = 10.7639 ft / s m / s = 3.2808 ft / s lbm / ft s kg / m = 0.672 lbm / ft / s ft / s m / s = 10.7639 ft / s Velocity v m/s Viscosity: Dynamic m Viscosity: Kinematic ν (µ / ρ) V kg / m s m2 / s Volume m ft m = 35.3134 ft 3 C THERMOPHYSICAL PROPERTIES OF SOME COMMON MATERIALS Material Aluminum: Pure Aluminum: 2024-T6 Boron Carbon: Diamond Carbon: Amorphous Copper: Pure Copper: Bronze Gold Iron: Pure Iron: Carbon Steel (AISI 1010) Iron: Stainless Steel (AISI 304) Lead Nickel: Pure Nickel: Inconel (X-750) Platinum Silver Tin Tungsten Zinc Zirconium * At 300 K Melting Point (K) ρ (kg/m3) 933 775 2573 — 1500 1358 1293 1336 1810 — 1670 601 1728 1665 2045 1235 505 3660 693 2125 2701 2770 2500 3500 1950 8933 8800 19300 7870 7832 7900 113450 8900 8510 21450 10500 7310 19300 7140 6570 cp (J/kg K) K (W/m K) α × 106 (m2/s)* 903 875 1107 509 — 385 420 129 447 434 477 129 944 439 133 235 228 132 389 278 237 177 27.0 300 1.6 401 52 317 80.2 63.9 14.9 35.3 90.7 11.7 71.6 429 66.6 174 116 22.7 97.1 73.0 9.76 — — 117 14 127 23.1 18.8 3.95 24.1 23.0 3.1 25.1 174 40.1 68.3 41.8 12.4 GS1059_A001.fm Page 300 Friday, September 9, 2005 11:47 AM 300 | Appendices D DIMENSIONLESS GROUPSa Biot number Bi = hL / k Eckert number Fourier number Ec = u / cρ ∆T Fo = αt / L2 Friction factor f = τ w /(1 / 2ρu ) Froude number Fr = u / gl Grashof number Gr = β gL3 ∆T / ν2 Mach number (for perfect gas) M = u / usound = u /( γ RT / M )1 / Nu = hL / κ Nusselt number Péclet number Pe = (Re)(Pr) Prandtl number Pr = µcρ / κ Rayleigh number Ra = (Gr )(Pr) Reynolds number  /µ Re = uL / ν = ρuL / µ = mL Schmidt number Sc = ν / D = µ / ρD Stanton number St = ( Nu)(Re)(Pr) = h / ρcρu Ste or Ja = cρ ∆T / ∆h Sr = ν L / u Stefan or Jakob number Strouhal number a The symbol L in the dimensionless groups stands for generic length and is defined according to the particular geometry being described; i.e., it may be diameter, hydraulic diameter, plate length, etc Physical Constants Stefan–Boltzmann constant Universal gas constant σ = 5.67051 x 10 −8 W /(m K ) R = 8.31441 x 103 J / kmol K E COMPUTER PROGRAMS A set of computer codes for solving one-, two-, and three-dimensional (1-D, 2-D, and 3-D) problems is available on the Web (located at: femcodes.nscee.edu), along with source listings for several MAPLE, MATHCAD, and MATLAB files In addition, COMSOL files are included that use COMSOL Viewer, which is a limited, free version of COMSOL 3.1 that lets users examine specific data sets The FORTRAN codes are written in FORTRAN 95 and consist of source and executable versions, including example data sets All of the codes and examples run on a PC Minimum PC systems requirements include the following: Intel Pentium or equivalent processor Microsoft Windows 95/98/2000/XP GS1059_A001.fm Page 301 Friday, September 9, 2005 11:47 AM Computer Programs | 301 256 MB RAM (for Windows and COMSOL) 40 MB of available disk space 16-bit graphics adaptor and display to support 256 simultaneous colors at 640 × 480 resolution; 24-bit color at 800 × 600 (or greater) resolution In addition, there are C++ and JAVA versions of the 2-D codes available on the Web The C++ version runs under Windows and allows the user to directly input boundary conditions using mouse drag-and-drop routines; results are instantly produced as display contours along with an adapted (refined) mesh The JAVA version permits the user to input boundary conditions directly into the code on the Web site and instantly displays contours along with an adapted mesh Both codes use bilinear quadrilateral elements along with local mesh refinement If users require assistance or have questions regarding access to the software, they should contact personnel identified in the Web site It is recommended that a subdirectory be created on the hard drive of the PC for storing and modifying the codes Copying the files from the Web into the subdirectory can be accomplished using Windows drag and drop Once the files are copied, the computer codes can be edited and run on the PC Most of the codes require data files that are read to establish a mesh (preprocessing) and boundary conditions, then create an output file for displaying the results (post-processing) E.1 Source Codes The source codes permit examination of the programming structure, and allow changes to be made to the codes as desired Any suitable FORTRAN compiler contains an editor that allows the user to examine the source listing and change lines as desired This can also be accomplished using a word processor, but be sure to save the code with a FOR extension when finished Once the codes are changed and updated, they will need to be recompiled to produce an executable version The code listings in MAPLE, MATHCAD, and MATLAB will all execute within their program kernels; i.e., the user must have working versions of these software packages to run these files Similarly, the user must have COMSOL Viewer (free) to run the COMSOL files A review of the FORTRAN source programs will reveal a great deal of similarity among the 1-D, 2-D, and 3-D code listings The 2-D and 3-D FEM versions were built on the 1-D FEM code structure These FORTRAN codes all assume only one degree of freedom per node (1 DOF/node) It is left to the user to alter the codes accordingly to accommodate more than DOF per node (as in structural problems involving u,v deflections per node, etc.) The compiled versions of the computer codes (extension EXE) were made using Compaq Visual FORTRAN (version 6.5), developed by Compaq Computer Corporation This compiler works easily and provides good diagnostics—and permits C++ coding to be embedded within the FORTRAN The FORTRAN codes were also compiled using Lahey and Ryan–McFarland FORTRAN compilers, and produced equally good executable codes Any changes to the FORTRAN codes contained on GS1059_A001.fm Page 302 Friday, September 9, 2005 11:47 AM 302 | Appendices the Web will require both compiling and linking—the user must have a FORTRAN compiler resident on the PC The FORTRAN codes can also be uploaded onto larger class machines, if desired (e.g., workstations, mainframes, etc.) The programs and data files are listed in seven subdirectories and grouped for convenience to permit association of files with one another The principal subdirectories are as follows: Subdirectory 1-D: MESH-1D.FOR*; MESH-1D.EXE FEM-1D.FOR; FEM-1D.EXE PLOTFEM.EXE (graphics display) EX301.DAT; EX302.DAT; EX305.DAT; EX306.DAT LIN-1D.DAT; QUAD-1D.DAT; CUBIC-1D.DAT Subdirectory 2-D: MESH-2D.FOR and MESH-2D.EXE FEM-2D.FOR and FEM-2D.EXE PLOTFEM.EXE EX405.DAT; EX503.DAT; EX504.DAT EX901.DAT; EX902.DAT; EX903.DAT EXS1.DAT; EXS2.DAT; EXS3.DAT; EXS4.DAT Subdirectory 3-D: FEM-3D.FOR; FEM-3D.EXE HEX-3D.DAT; TET-3D.DAT EX705.DAT Subdirectory MAPLE: EXMPL301; EXMPL302; EXMPL305; EXMPL306 EXMPL405; EXMPL503; EXMPL504 EXMPL705 Subdirectory MATHCAD: EX301.MCD; EX302.MCD; EX305.MCD; EX306.MCD EX405.MCD; EX503.MCD; EX504.MCD EX705.MCD * The FOR extension denotes source code listing in FORTRAN; the DAT extensions are data sets; the EXE files are executable versions of the codes GS1059_A001.fm Page 303 Friday, September 9, 2005 11:47 AM Computer Programs | 303 Subdirectory MATLAB: M-files: Linear1D; Quadratic1D; Lintriangle2D; Quadtriangle2D; Linquadrilateral2D; Quadquadrilateral2D; Lintetra3D EXMTL301; EXMTL302; EXMTL305; EXMTL306 EXMTL405; EXMTL503; EXMTL504 EXMTL801; EXMTL901; EXMTL902; EXMTL903 EXMTL705 Subdirectory COMSOL: EX301.FL; EX302.FL; EX305.FL; EX306.FL EX405.FL; EX503.FL; EX504.FL; EX705.FL EXS1.FL; EXS2.FL; EXS3.FL; EXS4.FL EX801.FL; EX901.FL; EX902.FL; EX903.FL EX1001.FL; EX1002.FL PROB1001.FL; PROB1002.FL; PROB1003.FL PLOTFEM.EXE is problem dependent with the total number of nodes limited to 101 In addition, the FORTRAN codes produce output files that can be read by TECPLOT, which is a well-known post-processing graphics program used by many engineers and scientists Information regarding flowcharts and subroutine descriptions of the FORTRAN 1-D and 2-D mesh and FEM codes can be obtained from the Web site: femcodes nscee.edu GS1059_A001.fm Page 304 Friday, September 9, 2005 11:47 AM GS1059_Index.fm Page 305 Monday, September 12, 2005 9:32 AM Index A Abiabatic right-end boundary condition, 37 Addition of matrices, 294 Analytical solution, nonconvective transport, 269 Area coordinates, 79–82, 80–86, 84 Argyris studies, Armaly studies, 290 Assembly process, 16 Atkinson studies, 61–62 Axisymmetric conduction one-dimensional elements, 39–41 two-dimensional triangular element, 97–100 B Baker, Pepper and, studies, 51, 263 Baker studies, Bandwidth, 32, 112–113, 112–116, 115–117 Becker studies, 2–3 Belegundu, Chandrupatla and, studies, 3, 235 Bickford studies, 2, 235 Bilinear quadrilateral, 175, 274 Bilinear quadrilateral shape functions, 174–175 Bilinear rectangular element, 131–133, 133 Biot number, 300 Blackwell and Pepper studies, 290 Bohn and Garboczi studies, Boundary conditions, see also Dirichlet condition axisymmetric conduction equation, 99 conduction, triangular elements, 93 convective transport, 257, 260 Galerkin approximation, 233 natural convection, 286, 286 nonconvective transport, 267–268 potential flow, 252, 252–253 quadratic triangular elements, 106 thermal stresses, 244 time-dependent diffusion equation, 110 two-dimensional elasticity, 230 Boundary convection one-dimensional elements, 34–39 two-dimensional quadrilateral element, 149–155, 151–155 two-dimensional triangular element, 92, 94, 94–97 Boussinesq approximation, 282 Brooks, Hughes and, studies, 260 Bulk modulus, groundwater flow, 270 Burgers equation and studies, 265 C Canale, Charpra and, studies, 59–62 Carey, Oden and, studies, 239 Carey and Oden studies, 131, 165 Carey studies, 116, 165 Cartesian cases and properties axisymmetric conduction equation, 98 element matrices, 181, 183 inviscid flow, 184 natural coordinate system, 136, 172 quadratic quadrilateral element, 157 C/C++ solution methods, 60 source codes, 4, 301 Cell Péclet number, 258, 264 Centered implicit method, 51 Chandrupatla and Belegundu studies, 3, 235 Chapra and Canale studies, 59–62 Charafi, Portela and, studies, Cheung, Zienkiewicz and, studies, Cholesky’s methods banded matrices, 63 LU decomposition, 61 two-dimensional triangular elements, 72 Classical solutions, 8, 8–9 Clough studies, Coarse discretization, 238 Codes, see Source codes Computer program exercises, 161–164, 162–164, see also the exercises at end of each chapter Computer programs, 300–303 COMSOL bandwidth, 116 basics, 300 source codes, 4, 301, 303 two-dimensional triangular elements, 73–74 Conductance/conduction matrix basics, 56 quadratic triangular elements, 100, 104 steady-state, boundary convection, 94 triangular elements, 92 Conduction, two-dimensional triangular element, 90, 90–93, 92 Connectivity, 52, 113 Conte studies, 60, 62 305 GS1059_Index.fm Page 306 Monday, September 12, 2005 9:32 AM 306 | Index Convective boundary condition one-dimensional elements, 34–39 two-dimensional quadrilateral element, 149–155, 151–155 two-dimensional triangular element, 92, 94, 94–97 Convective transport, applications basics, 4, 251, 277 convective transport, 255–265, 259, 261–264 exercises, 277–278 groundwater flow, 269–274, 273–274 lubrication, 274–277, 275–276 nonlinear convective transport, 265–269, 266, 269 potential flow, 251–255, 252, 254–256 Corner nodes, 176–177 Cosines, directional, 178–179, 178–180 Cowper studies, 87–88 Crank-Nicolson-Galerkin method, 51 Cubic elements, 21, 27–28 Cuthill-McGee studies, 116 D Dawe studies, 10, 239 Degrees of freedom Galerkin approximation, 31, 234–235 three-dimensional solid elements, 246 Derivative of matrix, 297–298 Desai studies, Determinant of matrix, 295–296 Diagonally dominant matrices, 59 Diagonal matrices, 60 Diffusion equation, time-dependent, 90, 107–112, 110 Diffusion integral, quadratic triangular elements, 102–104 Dimensionless groups, 300 Dirac delta function, 12 Directional cosines, 178–179, 178–180 Dirichlet condition, see also Boundary conditions computer program exercises, 163 convective boundary conditions, 37–39 one-element heat conduction, 205 potential energy, 241 “Do loops,” 55 Domain, natural convection, 286, 286 E EAGLE code, 74 Eckert number, 300 Eight-noded quadratic quadrilateral, 175, 175–176 Elder studies, 287 Element diffusion matrices, 96 Element mass matrix, 108, see also Mass matrix Element matrix, 147, 179–183 Element mesh, see also Mesh basics, 21 three-dimensional elements, 191–194, 192–193 two-dimensional quadrilateral element, 129–131, 130 Ern and Guermond studies, Exercises, see the end of each chapter F FEMAP, 74 FEM-1D source codes, 301, 303 two-dimensional triangular elements, 74 FEM-2D, 75 Finite element mesh, 21 Finite element method, 1–5 Finlayson studies, 2–3 First derivatives, limitations, 11–12 Fix, Strang and, studies, 239 Fletcher studies, 2, 9, 184, 187 Fluxes, 31 Force vector, 100, 258 FORTRAN/FORTRAN 95 basics, 300 source codes, 4, 301–302 symmetric banded matrices, 62 upper/lower triangular matrices, 60 Fourier number, 300 Fourier’s law, Friction number, 300 Froude number dimensionless groups, 300 natural convection, 287 penalty function algorithm, 286 viscous fluid flow, 282 G Galerkin approximation and formulation, see also Weighted residuals and Galerkin approximations axisymmetric heat conduction, 40 convective transport, 257, 259, 261, 264 groundwater flow, 273 historical developments, lubrication, 276 natural coordinate system, 47 nonconvective transport, 269, 269 GS1059_Index.fm Page 307 Monday, September 12, 2005 9:32 AM Index one-dimensional elements, 29–31, 32, 33–34 one-element heat conduction, 205–206 potential energy, 239–240 potential flow, 254 solid mechanics, 92, 231, 231–238, 236 spatial discretization, 50 steady conduction equation, 145, 150–151 time-dependent heat transfer, 213 Galerkin studies, GAMBIT, 74 Garboczi, Bohn and, studies, Gartling studies, 290 Gaussian quadratures matrix algebra, 293 numerical integration, 204, 204 quadratic quadrilateral element, 158 two-dimensional quadrilateral element, 141–144, 142–144 Gauss-Seidel method convective transport, 258 solution methods, 59 two-dimensional triangular elements, 72 Gauss’s theorem, 205 Gewali, Pepper and, studies, 193 Global system Galerkin formulation, 29–31, 33 linear elements, 22 matrix formulation, 55 natural coordinate system, 45 quadratic elements, 26 Grashof number, 300 Green-Gauss theorem, 285 Green’s theorem conduction, triangular elements, 91 Galerkin approximation, 232 inviscid flow, 185 potential flow, 254 steady conduction equation, 145 GRIDGEN, 74 Gross studies, 275 Groundwater flow, 269–274, 273–274 Guermond, Ern and, studies, H Hageman and Young studies, 57, 59 Half-bandwith, 62, 114 Hammer studies, 87 Heat flux, 34 Heat transfer, 281–283 Heinrich, Yu and, studies, 260 Heinrich and Pepper studies calculating flux, 39 fluid flow, | forced convection, 287 Gaussian quadratures, 142 linear triangle, 176 mesh generation, 116 momentum equations, 265 penalty function algorithm, 285 penalty method, 283 Petrov-Galerkin algorithm, 260, 263 quadrilateral mesh, 131 shape functions, 135 three-dimensional problems, 281 Heinrich and Vionnet studies, 286 Heinrich and Wadhwa studies, 276 Heinrich and Yu studies, 263 Heinrich and Zienkiewicz studies, 259 Hermites, cubic, 28 Heuser studies, Hexahedron, 199–201, 199–203, 209–213 Hinton, Owen and, studies, Historical background, 1–3 Hollig studies, Hooke’s law, 229, 245 Howell studies, 215, 218 Huebner and Thornton studies, 81 Huebner studies, 2, 10, 162, 276 Hughes and Brooks studies, 260 Hughes studies, 118 Hutton studies, I Ill-conditioned matrices, 59n Incompressible flow, 281–283 Interpolation function bilinear rectangular elements, 131 hexahedron, 199 quadratic rectangular elements, 133 Inverse Jacobian, 147–148 Inverse of matrix, 296–297 Inviscid flow, 184–187, 185–186 Isaacson and Keller studies, 57, 59, 62 Isoparametric elements, 186 Isoparametric two-dimensional elements basics, 4, 171, 187 bilinear quadrilateral shape functions, 174–175, 175 directional cosines, 178–179, 178–180 eight-noded quadratic quadrilateral, 175, 175–176 element matrices, 147, 179–183 exercises, 187–189 inviscid flow example, 184–187, 185–186 linear triangle, 176–177, 177 307 GS1059_Index.fm Page 308 Monday, September 12, 2005 9:32 AM 308 | Index natural coordinate system, 172–174 quadratic triangle, 177, 177–178 shape functions, 174–179 J Jacobian matrix area coordinates, 83–85 determinants, matrix, 295–296 element matrices, 181 inverse, steady conduction equation, 147–148 inviscid flow, 186 natural coordinate system, 45–47, 137, 140, 172–173 numerical integration, 87, 202–203 quadratic quadrilateral element, 157 quadratic triangular elements, 101 Jakob number, 300 JAVA basics, 301 solution methods, 60 source codes, Johnson studies, Journal of Computation Physics, 63 Linear triangular two-dimensional elements, 110 Liu and Quek studies, Load vector axisymmetric conduction equation, 99 basics, 56 Galerkin approximation, 234 hexahedron, 212 one-element heat conduction, 206 steady-state, boundary convection, 96 tetrahedron, 209 time-dependent heat transfer, 214 triangular elements, 92 Local coordinate system Galerkin formulation, 33 hexahedron, 202 linear elements, 24 matrix formulation, 55 quadratic elements, 26 Long and Pepper studies, 263 Lower triangular matrices, 59 Lubrication, 274–277, 275–276 LU decomposition, 60–61, 63 Lumped mass matrix, see Mass lumping M K Kattan studies, Keller, Isaacson and, studies, 57, 59, 62 Kelly studies, 260 Kronecker delta function, 28, 40 L Lagrangian interpolation, 28 Laplace equations groundwater flow, 272 historical background, potential flow, 252, 254 Linear elements conduction, triangular elements, 91–92, 92 natural coordinate system, 47–48 one-dimensional elements, 22–24, 23–24 time discretization, 52 Linear shape functions, 75–78, 78 Linear system of equations, 268 Linear triangle area coordinates, 80–81 isoparametric two-dimensional elements, 176–177, 177 numerical integration, 86 Linear triangular elements, 108 Mach number, 300 MAPLE, 4, 300–302 Mass lumping, 51n, 117–118 Mass matrix, see also Matrices element matrices, 183 hexahedron, 209–210 one-element heat conduction, 206 spatial discretization, 51 tetrahedron, 207 time-dependent diffusion, 108, 160 time discretization, 51n, 54 Materials, thermophysical properties, 299 Material stiffness matrix, 229, see also Stiffness matrix MATHCAD, 4, 300–302 MATLAB basics, 300 source codes, 4, 301, 303 two-dimensional triangular elements, 73, 75 Matrices, see also Mass matrix addition and subtraction, 294 algebra, 197, 293–298 basics, 293–294 derivative of matrix, 297–298 determinant of matrix, 295–296 ill-conditioning, 59n inverse of matrix, 296–297 GS1059_Index.fm Page 309 Monday, September 12, 2005 9:32 AM Index inversion limitations, 59 multiplication, 295 one-dimensional elements, 54–56 sparse matrix, 56 transpose of matrix, 298 Mesh, 1, 72–75, 73–74, see also Element mesh MESH-1D, 74 MESH-2D, 73–75, 116 Method of square roots, 61 Mid-side nodes eight-noded quadratic quadrilateral, 176 hexahedron, 202 quadratic triangles, 178 Mixed boundary conditions, 38 Morton, Richtmeyer and, studies, 51 Multi-dimensional elements, 45, 54 Multiplication, matrix algebra, 295 N Natural convection, 286, 286–287, 288–290 Natural coordinate system hexahedron, 200 isoparametric two-dimensional elements, 172–174 one-dimensional elements, 41–48, 42 quadratic quadrilateral element, 157 two-dimensional quadrilateral element, 133–134, 136–141, 137–139 Navier-Stokes equations, 281–282 Neumann conditions computer program exercises, 163 convective boundary conditions, 38 one-element heat conduction, 205 Newton iteration, 275 Newton’s method, 265, 266, 268–269 Nodes banded matrices, 63 bandwidth, 112–113 displacements, 231 eight-noded quadratic quadrilateral, 176 Galerkin approximation, 231 numbering, 63, 112–113 quadratic triangles, 177–178 quadratic triangular elements, 100 temperature, equilibrium state, 96 tetrahedron, 194 Nonlinear convective transport, 265–269, 266, 269 Numerical integration natural coordinate system, 42 three-dimensional elements, 202–204, 203–204 | 309 two-dimensional quadrilateral element, 141–144, 142–144 two-dimensional triangular element, 84, 86–89, 88 Numerical linear algebra, 57 Numerical Recipes, 60 Nusselt number, 300 O Oden, Carey and, studies, 131, 165 Oden and Carey studies, 239 Oden and Reddy studies, 165, 239 Oden studies, 2, 75 One-dimensional elements axisymmetric heat conduction, 39–41 basics, 4, 21, 64 boundary convection, 34–39 cubic elements, 27–28 element mesh, 192 exercises, 64–68 Galerkin formulation, 29–31, 32, 33–34 linear elements, 22–24, 23–24 matrix formulation, 54–56 natural coordinate system, 41–48, 42 node numbering, 63, 112 quadratic elements, 25–26, 25–27 shape function, 21–28 solution methods, 57–63, 62 spatial discretization, 49–51 steady conduction equation, 29–39 time dependence, 49–54 time-dependent diffusion equation, 109–110 time discretization, 51–54 variable conduction, 34–39 One-element heat conduction, 205–213 Orientation, 3–5 Ortega and Rheinboldt studies, 269 Owen and Hinton studies, P Pascal’s triangle, 195 PATRAN, 74 pcGRIDGEN, 74 Péclet number, see also Cell Péclet number dimensionless groups, 300 natural convection, 287 viscous fluid flow, 282 Penalty function algorithm, 283–286 Pepper, Blackwell and, studies, 290 Pepper, Heinrich and, studies calculating flux, 39 fluid flow, GS1059_Index.fm Page 310 Monday, September 12, 2005 9:32 AM 310 | Index forced convection, 287 Gaussian quadratures, 142 linear triangle, 176 mesh generation, 116 momentum equations, 265 penalty function algorithm, 285 penalty method, 283 Petrov-Galerkin algorithm, 260, 263 quadrilateral mesh, 131 shape functions, 135 three-dimensional problems, 281 Pepper, Long and, studies, 263 Pepper and Baker studies, 51, 263 Pepper and Gewali studies, 193 Persson and Strang studies, 73, 75 Petrov-Galerkin formulation convective transport, 259, 261, 262 nonconvective transport, 269, 269 penalty function algorithm, 284 Pintur studies, Plane stress and strain Galerkin approximation, 235, 237 two-dimensional elasticity, 229–230 PLOTFEM.EXE, 303 Poisson equation and ratio groundwater flow, 270 historical background, two-dimensional elasticity, 230 Polynomial order, shape functions, 75 Portela and Charafi studies, Positive definite matrices, 63n Potential energy, solid mechanics, 238–241, 241 Potential flow, convective transport, 251–255, 252, 254–256 Prandtl number, 287, 300 Q Quadratic elements basics, 21 Galerkin formulation, 33 natural coordinate system, 44–45, 47–48 one-dimensional elements, 25–26, 25–27 Quadratic quadrilateral element, 139, 156–158, 156–160 Quadratic rectangular element, 133–135, 134 Quadratic shape functions, 78–79, 79 Quadratic triangle area coordinates, 82 isoparametric two-dimensional elements, 177, 177–178 Quadratic triangular element conduction, triangular elements, 91 two-dimensional triangular element, 79, 100, 100–106 Quadrilateral elements, see also Two-dimensional quadrilateral elements conduction, triangular elements, 91 convective transport, 261 Quek, Liu and, studies, R Radiation, 214–217, 216–217 Rayleigh number, 287, 300 Rayleigh-Ritz approximation application limitations, 17 historical developments, 2–3 potential energy, 239 “weak” statement, 9–10 Rayleigh studies, Reciprocity theorem, 218 Reddy, Oden and, studies, 165, 239 Reddy studies, 2–3, 9, 228, 233, 235 Reynolds equation and number dimensionless groups, 300 lubrication, 274–275 natural convection, 287 penalty function algorithm, 286 viscous fluid flow, 282 Rheinboldt, Ortega and, studies, 269 Richtmeyer and Morton studies, 51 Ritz studies, Roache studies, S Schmidt number, 300 Second-order, centered implicit method, 51 Segerlind studies, 2, 81, 118 Semi-bandwith symmetric matrices, 62 two-dimensional mesh, 114, 116 Semi-discrete Galerkin formulation, 51, 108 Shape factors, three-dimensional elements, 217–219 Shape functions hexahedron, 199–201 isoparametric two-dimensional elements, 174–179 one-dimensional elements, 21–28 tetrahedron, 197–198, 206 three-dimensional elements, 194–202 two-dimensional quadrilateral element, 131–135 GS1059_Index.fm Page 311 Monday, September 12, 2005 9:32 AM Index two-dimensional triangular element, 75–79 “weak” statement, Shear modulus, groundwater flow, 270 SIAM, see Society of Industrial and Applied Mathematics (SIAM) Smith studies, Society of Industrial and Applied Mathematics (SIAM), 63 Solid mechanics basics, 4, 227, 247–248 exercises, 248–250 Galerkin approximation, 92, 231, 231–238, 236 potential energy, 238–241, 241 thermal stresses, 242–244, 244 three-dimensional solid elements, 245–247, 247 two-dimensional elasticity, 227–230, 228 Solin studies, Solution methods, one-dimensional elements, 57–63, 62 SOR, see Successive overrelaxation (SOR) Sorenson studies, 74 Source codes, 3–5, 301–303 Southwell studies, Sparse matrix, 56 Spatial discretization, 49–51 Square matrices, 59 Stanton number, 300 Stasa studies, 235 Steady conduction, 29–39 Steady-state conduction two-dimensional quadrilateral element, 137, 144–155, 147, 151–155 two-dimensional triangular element, 92, 94, 94–97 Stefan-Boltzmann constant, 215 Stefan number, 300 Stiffness matrix basics, 56 convective transport, 258 element matrices, 182–183 hexahedron, 210–211 numerical integration, 204 one-element heat conduction, 206 potential energy, 240 quadratic quadrilateral element, 158 quadratic triangular elements, 104 tetrahedron, 207–208 three-dimensional solid elements, 247 triangular elements, 92 Strang, Persson and, studies, 73, 75 Strang and Fix studies, 239 Strouhal number, 300 Structured mesh, 74 Stuffing, matrix formulation, 55 | 311 Subtraction of matrices, 294 Successive overrelaxation (SOR), 60 Symmetric banded matrices, 62–63 T Tangent matrix, 268 Taylor, Zienkiewicz and, studies, 2, 10, 118, 165 Taylor series expansion, 266 Tetrahedron numerical integration, 203 three-dimensional elements, 194–198, 194–198, 206–209 three-dimensional solid elements, 246 Thermal stresses, 242–244, 244 Thermophysical material properties, 299 Thompson studies, 74 Thornton, Huebner and, studies, 81 Three-dimensional elements bandwidth, 112 basics, 4, 191, 219 element mesh, 191–194, 192–193 exercises, 220–224 hexahedron, 199–201, 199–202, 209–213 natural coordinate system, 42 numerical integration, 202–204, 203–204 one-element heat conduction, 205–213 radiation, 214–217, 216–217 shape factors, 217–219 shape functions, 194–202 tetrahedron, 194–198, 194–198, 206–209 time-dependent heat conduction, 213–219 Three-dimensional solid elements, 245–247, 247 Time dependence one-dimensional elements, 49–54 three-dimensional elements, 213–219 two-dimensional quadrilateral element, 143, 157–158, 160–161 two-dimensional triangular element, 90, 107–112, 110 Time derivatives, matrix formulation, 55–56 Time discretization, 51–54 Total force vector, 100 Transpose of matrix, 298 Triangular elements, see also Two-dimensional triangular elements conduction, 90, 90–93, 92 numerical integration, 87, 88 TrueGrid, 74 Turner studies, Two-dimensional elasticity, 227–230, 228 Two-dimensional elements, see also Isoparametric two-dimensional elements GS1059_Index.fm Page 312 Monday, September 12, 2005 9:32 AM 312 | Index bandwidth, 112 element mesh, 193 natural coordinate system, 42 Two-dimensional mesh, 114 Two-dimensional quadrilateral elements basics, 4, 129, 164–165 bilinear rectangular element, 131–133, 133 boundary convection, 149–155, 151–155 computer program exercises, 161–164, 162–164 element mesh, 129–131, 130 exercises, 165–168 Gaussian quadratures, 141–144, 142–144 natural coordinate system, 133–134, 136–141, 137–139 numerical integration, 141–144, 142–144 quadratic quadrilateral element, 139, 156–158, 156–160 quadratic rectangular element, 133–135, 134 shape functions, 131–135 steady-state conduction, 137, 144–155, 147, 151–155 time-dependent diffusion, 143, 157–158, 160–161 Two-dimensional triangular elements area coordinates, 79–82, 80–86, 84 axisymmetric conduction equation, 97–100 bandwidth, 32, 112–113, 112–116, 115–117 basics, 4, 71–72, 119 boundary convection, 92, 94, 94–97 conduction, 90, 90–93, 92 exercises, 119–126 linear shape functions, 75–78, 78 mass lumping, 117–118 mesh, 72–75, 73–74 numerical integration, 84, 86–89, 88, 202 quadratic shape functions, 78–79, 79 quadratic triangular element, 79, 100, 100–106 shape functions, 75–79 steady-state conduction, 92, 94, 94–97 time-dependent diffusion equation, 90, 107–112, 110 U Units, 298–299 Universal elements, 73 Upper triangular matrices, 60 V Varga studies, 57, 59 Variable conduction, 34–39 Verruijt studies, 269, 271 Vionnet, Heinrich and, studies, 286 Viscous fluid flow basics, 4, 281, 290 exercises, 290–292 heat transfer, 281–283 historical background, incompressible flow, 281–283 natural convection, 286, 286–287, 288–290 penalty function algorithm, 283–286 VisualFEA/CBT, W Wadhwa, Heinrich and, studies, 276 “Weak” statement bandwidth, 113 weighted residuals and Galerkin approximations, 7, 9, 9–17, 11, 13–14 Web sites, 5, 119, see also Source codes Weighted residuals and Galerkin approximations basics, 7, 17, 47 classical solutions, 8, 8–9 conduction, triangular elements, 90–91 exercises, 17–19 “weak” statement, 9, 9–17, 11, 13–14 X Young, Hageman and, studies, 57, 59 Young’s modulus groundwater flow, 269–270 two-dimensional elasticity, 230 Young studies, 63 Yu, Heinrich and, studies, 263 Yu and Heinrich studies, 260 Y Zienkiewicz, Heinrich and, studies, 259 Zienkiewicz and Cheung studies, Zienkiewicz and Taylor studies, 2, 10, 118, 165 Zienkiewicz studies, 43, 51, 81, 135, 176, 195 ... practitioners of the finite element method now employ Galerkin’s method to establish the approximations to the governing equations The underlying theme in this book likewise follows Galerkin’s method The. .. serves as the actual beginning of the finite element method, utilizing the one-dimensional element? ??in fact, the entire framework of the method is presented in this chapter Reinforcement of the basic... MAPLE, and MATLAB (and the supplementary software COMSOL), have helped to spread the training and use of the method The development of the finite element method using these mathematical symbolic