AN INTRODUCTION TO THE FINITE ELEMENT METHOD Second Edition J. N. Reddv Oscar S. Wyatt Chair in Mechanical Engineering Texas A&M University College Station, Texas 77843 McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto
CONTENTS Preface to the Second Edition Preface to the First Edition Part 1 Preliminaries Introduction 3 1.1 General Comments 3 1.2 Historical Background 5 1.3 The Basic Concept of the Finite Element Method 5 1.3.1 General Comments 5 1.3.2 Approximation of the Circumference of a Circle 6 1.3.3 Approximate Determination of the Center of Mass 8 1.3.4 Solution of Differential Equation 10 1.3.5 Some Remarks 13 1.4 The Present Study 15 1.5 Summary 15 References for Additional Reading 16 Integral Formulations and Variational Methods 18 2.1 Need for Weighted-Integral Forms 18 2.2 Some Mathematical Concepts and Formulae 20 2.2.1 Boundary, Initial, and Eigenvalue Problems 20 2.2.2 Integral Relations 22 2.2.3 Functionals 26 2.2.4 The Variational Symbol 27 2.3 Weak Formulation of Boundary Value Problems 28 2.3.1 Introduction 28 2.3.2 Weighted-Integral and Weak Formulations 28 2.3.3 Linear and Bilinear Forms and Quadratic Functionals 33 2.3.4 Examples 35 ix
X CONTENTS 2.4 Variational Methods of Approximation 40 2.4.1 Introduction 40 2.4.2 The Rayleigh-Ritz Method 40 2.4.3 The Method of Weighted Residuais 51 2.5 Summary 57 Problems 59 References for Additional Reading 63 Part 2 Finite Element Analysis of One-Dimensional Problems 3 Second-Order Boundary Value Problems 67 3.1 Introduction 67 3.2 Basic Steps of Finite Element Analysis 70 3.2.1 Model Boundary Value Problem 70 3.2.2 Discretization of the Domain 72 3.2.3 Derivation of Element Equations 72 3.2.4 Connectivity of Elements 89 3.2.5 Imposition of Boundary Conditions 95 3.2.6 Solution of Equations 95 3.2.7 Postprocessing of the Solution 96 3.2.8 Radially Symmetrie Problems 103 3.3 Applications 105 3.3.1 keat Transfer 105 3.3.2 Fluid Mechanics 117 3.3.3 Solid Mechanics 123 3.4 Summary 127 Problems 128 References for Additional Reading 141 Bending of Beams 4.1 4.2 Introduction The Euler-Bernoulli Beam Element 4.2.1 Governing Equation 4.2.2 Discretization of the Domain 4.2.3 Derivation of Element Equations 4.2.4 Assembly of Element Equations 4.2.5 Imposition of Boundary Conditions 4.2.6 Solution 4.2.7 Postprocessing of the Solution 4.2.8 Examples 4.3 4.4 4.5 4.6 Plane Truss and Euler-Bernoulli Frame Elements The Timoshenko Beam and Frame Elements 4.4.1 Governing Equations 4.4.2 Weak Form 4.4.3 Finite Element Model Inclusion of Constraint Equations Summary 144 144 151 154 156 158 160 167 177 177 177 178 187 191
CONTENTS xi Problems 192 References for Additional Reading 198 5 Finite Element Error Analysis 199 5.1 Approximation Errors 199 5.2 Various Measures of Errors 200 5.3 Convergence of Solution 201 5.4 Accuracy of the Solution 202 5.5 Summary 207 Problems 207 References for Additional Reading 208 6 Eigenvalue and Time-Dependent Problems 209 6.1 Eigenvalue Problems 209 6.1.1 Introduction 209 6.1.2 Formulation of Eigenvalue Problems 210 6.1.3 Finite Element Models 213 6.1.4 Applications 216 6.2 Time-Dependent Problems 224 6.2.1 Introduction 224 6.2.2 Semidiscrete Finite Element Models 225 6.2.3 Time Approximations 227 6.2.4 Mass Lumping 232 6.2.5 Applications 233 6.3 Summary 241 Problems 241 References for Additional Reading 245 7 Numerical Integration and Computer Implementation 246 7.1 Isoparametric Formulations and Numerical Integration 246 7.1.1 Background 246 7.1.2 Natural Coordinates 248 7.1.3 Approximation of Geometry 249 7.1.4 Isoparametric Formulations 251 7.1.5 Numerical Integration 251 7.2 Computer Implementation 258 7.2.1 Introductory Comments 258 7.2.2 General Outline 259 7.2.3 Preprocessor 260 7.2.4 Calculation of Element Matrices (Processor) 262 7.2.5 Assembly of Element Equations (Processor) 265 7.2.6 Imposition of Boundary Conditions (Processor) 267 7.2.7 Solution of Equations and Postprocessing 269 7.3 Applications of the Computer Program FEM1DV2 270 7.3.1 General Comments 270 7.3.2 Illustrative Examples 271 7.4 Summary 286 Problems 287 References for Additional Reading 291
XU CONTENTS Part 3 Finite Element Analysis of Two-Dimensional Problems 8 Single-Variable Problems 295 8.1 Introduction 295 8.2 Boundary Value Problems 297 8.2.1 The Model Equation 297 8.2.2 Finite Element Discretization 298 8.2.3 Weak Form 299 8.2.4 Finite Element Model 301 8.2.5 Interpolation Functions 303 8.2.6 Evaluation of Element Matrices and Vectors 311 8.2.7 Assembly of Element Equations 318 8.2.8 Postprocessing 322 8.2.9 Axisymmetric Problems 323 8.2.10 AnExample 324 8.3 Some Comments on Mesh Generation and Imposition of Boundary Conditions 334 8.3.1 Discretization of a Domain 334 8.3.2 Generation of Finite Element Data 336 8.3.3 Imposition of Boundary Conditions 339 8.4 Applications 340 8.4.1 Heat Transfer 340 8.4.2 Fluid Mechanics 353 8.4.3 Solid Mechanics 365 8.5 Eigenvalue and Time-Dependent Problems 370 8.5.1 Introduction 370 8.5.2 Parabolic Equations 372 8.5.3 Hyperbolic Equations 379 8.6 Summary 384 Problems 386 References for Additional Reading 402 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 404 9.1 Library of Elements and Interpolation Functions 404 9.1.1 Introduction 404 9.1.2 Triangulär Elements 405 9.1.3 Rectangular Elements 411 9.1.4 The Serendipity Elements 417 9.2 Numerical Integration 421 9.2.1 Preliminary Comments 421 9.2.2 Coordinate Transformations 423 9.2.3 Integration over a Master Rectangular Element 429 9.2.4 Integration over a Master Triangulär Element 433 9.3 Modeling Considerations 439 9.3.1 Preliminary Comments 439 9.3.2 Element Geometries 439 9.3.3 Mesh Generation 441 9.3.4 Load Representation 446 9.4 Summary 448
CONTENTS Xlll Problems 448 References for Additional Reading 453 10 Plane Elasticity 455 10.1 Introduction 455 10.2 Governing Equations 456 10.2.1 Assumptions of Plane Elasticity 456 10.2.2 Basic Equations 457 10.3 Weak Formulations 459 10.3.1 Preliminary Comments 459 10.3.2 Principle of Virtual Displacements in Matrix Form 459 10.3.3 Weak Form of the Governing Differential Equations 461 10.4 Finite Element Model 461 10.4.1 Matrix Form of the Model 461 10.4.2 Weak Form Model 464 10.4.3 Eigenvalue and Transient Problems 465 10.5 Evaluation of Integrals 465 10.6 Assembly and Boundary and Initial Conditions 468 10.7 Examples 469 10.8 Summary 476 Problems 476 References for Additional Reading 480 11 Flows of Viscous Incompressible Fluids 482 11.1 Preliminary Comments 482 11.2 Governing Equations 483 11.3 Velocity-Pressure Finite Element Model 484 11.4 Penalty-Finite Element Model 488 11.4.1 Penalty Function Method 488 11.4.2 Formulation of the Flow Problem as a Constrained Problem. 489 11.4.3 Lagrange Multiplier Formulation 491 11.4.4 Penalty Function Formulation 491 11.4.5 Computational Aspects 492 11.5 Examples 494 11.6 Summary 502 Problems - 503 References for Additional Reading 506 12 Bending of Elastic Plates 508 12.1 Introduction 508 12.2 Classical Plate Model 510 12.2.1 Displacement Field 510 12.2.2 Virtual Work Statement 510 12.2.3 Finite Element Model 514 12.3 Shear Deformable Plate Model 516 12.3.1 Displacement Field 516 12.3.2 Virtual Work Statement 517 12.3.3 Finite Element Model 518 12.3.4 Shear Locking and Reduced Integration 519 12.4 Eigenvalue and Time-Dependent Problems 520
XIV CONTENTS 12.5 Examples 522 12.6 Summary 529 Problems 529 References for Additional Reading 531 13 Computer Implementation 533 13.1 Introduction 533 13.2 Preprocessor 534 13.3 Element Computations: Processor 535 13.4 Applications of the Computer Program FEM2DV2 540 13.4.1 Introduction 540 13.4.2 Description of Mesh Genrators 548 13.4.3 Applications (Illustrative Examples) 551 13.5 Summary 563 Problems 570 References for Additional Reading 575 Part 4 Advanced Topics 14 Weighted-Residual Finite Element Models, and Finite Element Models of Nonlinear and Three-Dimensional Problems 579 14.1 Introduction 579 14.2 Alternative Formulations 580 14.2.1 Introductory Comments 580 14.2.2 Weighted-Residual Finite Element Models 580 14.2.3 Mixed Formulations 590 14.3 Nonlinear Problems 594 14.3.1 General Comments 594 14.3.2 Large-Deflection Bending of (Euler-Bernoulli) Beams 595 14.3.3 Solution Methods for Nonlinear Algebraic Equations 597 14.3.4 The 2-D Navier-Stokes Equations 598 14.4 Three-Dimensional Problems 599 14.5 Summary 601 Problems 602 References for Additional Reading 606 Appendixes 1 Fortran Listing of FEM1DV2 609 2 Fortran Listing of FEM2DV2 640 Index 679