An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach

460 Pages·2004·26.21 MB·English

by  Prem K. Kythe

Prem K. Kythe
Dongming Wei
An Introduction
to Linear and No nlinear
Finite Element Analysis
A Computational Approach
With 152 illustrations
Springer Science+Business Media, LLC
Prem K. Kythe Dongming Wei
Department of Mathematics Department of Mathematics
University of New Orleans University of New Orleans
New Orleans, LA 70148-0001 New Orleans, LA 70148-0001
U.S.A. U.S.A.
Library of Congress Cataloging-in-Publication Data
Kythe, Prem K.
An introduction to linear and nonlinear finite element analysis : a computational approach
Iby Prem K. Kythe, Dongming Wei.
p. cm.
Inc\udes bibliographical references and index.
ISBN 978-1-4612-6466-8 ISBN 978-0-8176-8160-9 (eBook)
DOI 10.1007/978-0-8176-8160-9
1. Structural analysis (Engineering) 2. Finite element method. 1. Wei, Dongming, 1958-
II. Title
TA 646.K98 2003
620'.0042-<ic22 2003057713
CIP
AMS Subject Classifications: 65N30,65-01
$®
Printed on acid-free paper.
©2004 Springer Science+Business Media New York
Originally published by Birkhauser Boston in 2004
Softcover reprint of the hardcover 1s t edition 2004
AII rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC),
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with
any form of information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-6466-8 SPIN 10888337
Typeset by TechType Works, Inc., Gretna, LA.
Mathematica® is a registered Trademark ofWolfram Research, Inc.
MA TLA~ is a registered trademark of Math Works, Inc.
9 8 7 6 543 2 1
Contents
Preface ix
Notation xvii
1 Introduction 1
1.1 Historical Sketch 1
1.2 Euler-LagrangeEquations 2
1.2.1 Functionals 2
1.2.2 Boundary Conditions 3
1.3 WeakVariationalForm 3
1.4 Galerkin Method 9
1.5 Rayleigh-RitzMethod 11
1.6 Exercises 18
2 One-Dimensional Shape Functions 25
2.1 Local andGlobalLinear ShapeFunctions 25
2.2 Local andGlobal Quadratic ShapeFunctions 28
2.3 Parametric Coordinates 31
2.4 Hermite Shape Functions 31
2.5 Exercises 34
3 One-Dimensional Second-Order Equation 37
3.1 Galerkin Finite Element Method 37
3.1.1 Linear Elements forSecond-OrderEquation " 38
3.1.2 Local andGlobalMatrices 42
vi CONTENTS
3.1.3 Quadratic ElementsforSecond-Order Linear Problems 45
3.1.4 MixingTwoDifferentElements 47
3.2 TwoDependentVariables 63
3.5 Exercises 66
4 One-Dimensional Fourth-Order Equation 75
4.1 Euler-Bernoulli BeamEquation 75
4.1.1 FiniteElementEquation 75
4.1.2 BoundaryConditions 77
4.2 Exercises 86
5 Two-Dimensional Elements 89
5.1 LinearThree-Node TriangularElements 89
5.2 Bilinear Four-NodeRectangularElements 91
5.3 GlobalShapeFunctions 94
5.4 Triangular Coordinates ......................................... 96
5.5 Shape Functions ontheSidesofaTriangle 99
5.5.1 Linear ShapeFunctions ontheSidesofaTriangle 99
5.5.2 QuadraticShapeFunctionsontheSidesofaTriangle 101
5.6 Exercises 105
6 Two-Dimensional Problems 107
6.1 SingleDependent VariableProblems 107
6.1.1 LocalWeakFormulation 108
6.1.2 Finite ElementEquation 109
6.1.3 Evaluation ofStiffnessMatrix andLoadVector ............. 110
6.1.4 Evaluation ofBoundaryIntegrals.......................... III
6.1.5 Assembly ofElementMatrices 112
6.2 Exercises ..................... 129
7 More Two-Dimensional Problems 137
7.1 HeatTransfer ........................................... 137
7.2 Torsion 144
7.3 Seepage 146
7.4 Fluid Flows 152
7.5 Exercises .................................................... 164
8 Axisymmetric Heat Transfer 175
8.1 Radial Symmetry 175
8.2 Linear Elements 182
8.3 Linear Elements forHeatTransferinFluids 194
8.4 Nonlinear HeatTransfer 196
8.4.1. Gauss-Legendre IntegrationMethod 198
8.5 Exercises 202
CONTENTS vii
9 Transient Problems 207
9.1 Classical Methods 207
9.2 One-DimensionalTransientProblems 212
9.3 Time-DependentHeatConduction 217
9.3.1 Derivation ofFinite Element Equations 218
9.3.2 Numerical Time Integration 221
9.4 Two-Dimensional Transient Problems 228
9.5 Exercises 233
10 Single Nonlinear One-Dimensional Problems 241
10.1 Newton's method 241
10.2 Radiation HeatTransfer 243
10.3 StressAnalysis ofPlasticRods 252
10.4 Power-Law Pressure DrivenFlowbetweenTwoPlates 256
10.5 Mixing-LengthEquation forTurbulentFlowinPipes 259
10.6 Rayleigh-RitzandNonlinear GradientMethods 261
10.7 Exercises 266
11 Plane Elasticity 269
11.1 Stress-StrainRelations 269
11.2 Constant-StrainTriangularElement 271
11.3 VirtualDisplacementFinite Element Model ......... 272
11.4 WeakFormFinite Element Model 273
11.5 Stiffness Matrix andLoadVector 275
11.5.1 LinearCST Element 275
11.5.2 Assembly ofCSTElements 276
11.5.3 BilinearRectangular Element 284
11.5.4 TwoorMore Bilinear Rectangular Elements 288
11.6 Exercises 291
12 Stokes Equations and Penalty Method 297
12.1 Equality-ConstrainedPrograms andLagrange Multipliers 297
12.2 Penalty Formulation forLinear StokesEquation 299
12.3 Penalty LinearTriangular StokesElement 301
12.4 Penalty BilinearRectangular StokesElement 309
12.5 Penalty LinearTriangular Power-lawStokesElement 311
12.6 Solutions byConjugate Gradient Methods 313
12.7 Exercises 321
13 Vibration Analysis 323
13.1 Hamiltonian Principle 323
13.2 Free Axial VibrationsofanElastic Rod " 327
13.3 FreeVibrations ofaEulerElastic Beam 330
13.4 Free In-Plane Vibrations ofanElastic Plate 334
viii CONTENTS
13.5 AxialVibrationsofaPlasticRod 342
13.6 EigenvalueProblems 344
13.6.1 Hyperbolic Equations 344
13.6.2 ParabolicEquations 346
13.6.3 HelmholtzEquation 347
13.7 Exercises 347
14 Computer Codes 351
14.1 Mathematica Codes 351
14.2 AnsysCodes 363
14.3 MatlabCodes 368
14.4 FortranCodes 374
A Integration Formulas 393
B Special Cases 397
C Temporal Approximations 405
C.l First-Order Derivative ....... 405
C.2 Second-OrderDerivative 406
D Isoparametric Elements 409
D.l Introduction 409
D.Ll Local~-Syatem 409
D.l.2 TriangularElements 411
D.1.3 Rectangular Elements 412
D.2 CurvilinearCoordinates 414
D.2.1 One-Dimensional Integrals 416
D.2.2 Two-DimensionalIntegrals 417
D.3 PascalTriangle 418
E Green's Identities 421
F Gaussian Quadrature 423
G Gradient-Based Methods 429
G.l Method ofSteepestDescent 429
G.2 Conjugate GradientMethod 431
Bibliography 433
Index 439
Preface
This textbook on linear and nonlinear finiteelement methods has evolved out of
teaching ofasequence oftwocoursesattheUniversity ofNewOrleans forthepast
12years to a variety of students from the engineering,physics, geophysics, and
mathematicsdepartments. Thefirstofthesecoursesisanintroductiondesigned for
undergraduate and beginning graduate students, whereas the second is a follow
up mainly for graduate students. In the past we used a number of textbooks and
selected material at different times on a trial basis, and we developed our own
notes and a repertoire of representative engineering and technological problems
thatcater tothedisciplinesofourstudents.
As everyone knows the market is replete with textbooks on finite element
analysis. Then why do weneed another textbook onthis subject? Toanswer this
question, we next describe how we have used existing books and why we felt a
need towriteourowntextbook.
(i) Although there are many excellent books on the market, very few are de
signed for students from different disciplines and use widely available modern
software such as Mathematica, Matlab, Fortran, and commercial finite element
codes such asAnsys.
(ii)Inourview,thefiniteelementtechnologyoftodayhasgrownintoabasictool
foralmost every fieldinengineeringandapplied sciences. Finite elementcourses
havebecome moreandmorepopular inbothundergraduateandgraduatecurricula
incolleges ofengineeringandsciences throughouttheworld. The newgeneration
of undergraduate and graduate students is more accustomed to using computers
for everyday homework assignments and research than ever before. There are
a few introductory finiteelement textbooks that provide a good combination of
x PREFACE
methodology,applications,computerimplementation, and hands-on projects.
(iii) Of the books we have used, some incorporate commercial codes into
the text. A few books introduce applications of the finite element method in
electromagnetics together with other traditional applications in solid and fluid
mechanics. There are a few other books that deal exclusively with the finite
elementanalysis of electromegnetic problems. Such books are aimed at special
readers inspecializedfieldsofinterestandarenotsuitableforageneralcurriculum.
(iv)Wehavefoundonlyonespecializedbookonfiniteelementanalysisdealing
with non-Newtonian fluidflows,but itisdesigned forchemical engineers. Other
books donotconsiderthistopicatall. Eveninbooks withtraditionalapplications,
onlyafewgivedetails about vibrationalfiniteelementanalysis, althoughthere are
other books that deal exclusively withthistopic.
(v)Webelieve that thebasic theory offiniteelementanalysis should be made
accessible to all undergraduate students inengineering and sciences with a min
imum background in mathematics, computer science, physics, and any specific
engineering field. Our experiencewith the books wehaveused isthat only a few
are designed to serve abroad range ofengineering and science majors and to ex
pose students to hands-on application projects using the computational software
tools that are nowwidely available oncampuses.
(vi) Wehave found that inour introductory course itisvery helpful todevote
sufficient time to cover equations in one space dimension and include problems
with applications in thefollowing areas: structural,thermal,Newtonian and non
Newtonian fluids, mechanical vibration,electromagnetics, variational principles,
and thermal elastic coupled systems. The underlying mathematics for the first
courseare kept at aminimum by using only basic techniques, such as the Gauss
Jordaneliminationmethodandthebackwardsubstitutionmethodforsolvinglinear
systemsofalgebraicequation, integration byparts formulasinone ortwo dimen
sions,chain rule fordifferentiation,andchange ofvariable formulaforintegration
and Newton's method forsolvingnonlinearsystems ofalgebraicequations.
(vii)Inthesecondcourseweintroduceadvancedtechniquesforsolvingdiscrete
algebraicsystems,viz.,linesearch,methodofsteepestdescent, linearandnonlinear
conjugategradientmethods, Newton's methods andpenaltymethod.
During thepastdecade wehaveaccumulatedvaluableexperienceandabsorbed
manyexcellentideasfromotherauthors inteachingthesetwoservicecourses. The
result isa general service applied finiteelement textbook that not only combines
alltheabove ideasandexperiencebutisdiverseenoughinapplicationsandsimple
enoughintheoryforstudentsinallengineeringandsciencefields. Wehavekeptthe
mathematicaltheory toaminimum,justsufficientforunderstandingthebeginner's
whys and hows offiniteelement methods.
PREFACE xi
Salient Features
This textbook presents basic theoretical material and practical applications for
solving problems of interest in engineering and technology. The book covers a
two-semestercourseandisdesigned forjunior/seniorundergraduateandfirst-year
graduatestudentsfromengineering,geophysics,physics, andappliedmathematics
who have had a course in calculus ofseveral variables, differential equations as
well aslinear algebraandhave some knowledge ofcomputers.
This book
• Fulfillstheneedforaservicecourse taughttostudentswithdiversebackground.
• Provides thorough instructions and training to these diverse students, unlike
most textbooks thataredesignedwith specificapplications,e.g.,forstructural
engineers,mechanicalengineers,electricalengineers,chemicalengineers,geo
physicists,orapplied mathematicians.
• Involvesstudents intheuseofwidelyavailable modern softwaresuchasMathe
matica, Matlab, andtheuseofcommercialfiniteelementcodes such asAnsys.
• Deals with problems of varied types, including linear and nonlinear problems
in radiation heat transfer, mechanics of elastic and plastic media, continuum
mechanics,non-Newtonianfluidflows,andelectromagnetics.
• Provides accessibilitytoallstudents withdiverse backgroundswithaminimum
ofmathematical analysis.
• Serves a broad range ofengineering and science students and exposesthem to
applications from different disciplines, unlike books that deal with only one
specific fields.
• Inthefirst-semestercourse, presents one-dimensionalequationsand theirappli
cations inthefollowing areas: structuralanalysis, heatconduction,fluidflows,
mechanical vibration, electromagnetics, variational principles, and thermoe
lastic coupledsystems.
• In the second-semester course, presents advanced techniques for solving dis
cretealgebraic systems,viz.,linesearch, methodofsteepestdescent,conjugate
gradientmethods, Newton's methods,and penalty method.
• Offers supporting andother interestingmaterial via theInternetatthe authors'
web sites.
• Provides service, accessibility, and applications toward the developmentof the
subject.