Reddy, J. N.

An Introduction to the Finite Element Method

Preface xiv Introduction 1(26) General Comments 1(1) Mathematical Models 2(7) Numerical Simulations 9(4) The Finite Element Method 13(11) The Basic Idea 13(1) The Basic Features 13(8) Some Remarks 21(2) A Brief Review of History and Recent Developments 23(1) The Present Study 24(1) Summary 24(3) Problems 25(1) References for Additional Reading 26(1) Mathematical Preliminaries, Integral Formulations, and Variational Methods 27(76) General Introduction 27(4) Variational Principles and Methods 27(1) Variational Formulations 28(1) Need for Weighted-Integral Statements 28(3) Some Mathematical Concepts and Formulae 31(10) Coordinate Systems and the Del Operator 31(2) Boundary Value, Initial Value, and Eigenvalue Problems 33(3) Integral Identities 36(3) Linear and Bilinear Functionals 39(2) Elements of Calculus of Variations 41(17) Introduction 41(1) Variational Operator and First Variation 41(3) Fundamental Lemma of Variational Calculus 44(1) The Euler Equations 44(3) Natural and Essential Boundary Conditions 47(7) Hamilton's Principle 54(4) Integral Formulations 58(16) Introduction 58(1) Weighted-Integral and Weak Formulations 58(6) Linear and Bilinear Forms and Quadratic Functionals 64(2) Examples 66(8) Variational Methods 74(23) Introduction 74(1) The Ritz Method 74(2) Approximation Functions 76(1) Examples 77(14) The Method of Weighted Residuals 91(6) Summary 97(6) Problems 98(4) References for Additional Reading 102(1) Second-Order Differential Equations in One Dimension: Finite Element Models 103(52) Background 103(2) Basic Steps of Finite Element Analysis 105(36) Model Boundary Value Problem 105(1) Discretization of the Domain 106(2) Derivation of Element Equations 108(17) Connectivity of Elements 125(7) Imposition of Boundary Conditions 132(1) Solution of Equations 132(2) Postcomputation of the Solution 134(7) Some Remarks 141(5) Axisymmetric Problems 146(4) Model Equation 146(1) Weak Form 147(1) Finite Element Model 148(2) Summary 150(5) Problems 151(3) References for Additional Reading 154(1) Second-Order Differential Equations in One Dimension: Applications 155(78) Preliminary Comments 155(1) Discrete Systems 156(6) Linear Elastic Spring 156(2) Torsion of Circular Shafts 158(1) Electrical Resistor Circuits 159(2) Fluid Flow through Pipes 161(1) Heat Transfer 162(19) Governing Equations 162(4) Finite Element Models 166(1) Numerical Examples 166(15) Fluid Mechanics 181(2) Governing Equations 181(1) Finite Element Model 181(2) Solid and Structural Mechanics 183(11) Preliminary Comments 183(1) Finite Element Model of Bars and Cables 184(1) Numerical Examples 185(9) Plane Trusses 194(20) Introduction 194(1) Basic Truss Element 194(1) General Truss Element 195(7) Constraint Equations: Penalty Approach 202(9) Constraint Equations: A Direct Approach 211(3) Summary 214(19) Problems 215(16) References for Additional Reading 231(2) Beams and Frames 233(58) Introduction 233(1) Euler--Bernoulli Beam Element 233(28) Governing Equation 233(1) Discretization of the Domain 234(1) Derivation of Element Equations 234(9) Assembly of Element Equations 243(2) Imposition of Boundary Conditions 245(2) Postprocessing of the Solution 247(1) Numerical Examples 248(13) Timoshenko Beam Elements 261(13) Governing Equations 261(1) Weak Form 262(2) General Finite Element Model 264(2) Consistent Interpolation Elements 266(4) Reduced Integration Element 270(1) Numerical Examples 271(3) Plane Frame Elements 274(7) Introductory Comments 274(1) Frame Element 274(7) Summary 281(10) Problems 282(8) References for Additional Reading 290(1) Eigenvalue and Time-Dependent Problems 291(52) Eigenvalue Problems 291(23) Introduction 291(1) Formulation of Eigenvalue Problems 292(3) Finite Element Formulation 295(19) Time-Dependent Problems 314(23) Introduction 314(2) Semidiscrete Finite Element Models 316(2) Parabolic Equations 318(6) Hyperbolic Equations 324(2) Mass Lumping 326(2) Applications 328(9) Summary 337(6) Problems 337(5) References for Additional Reading 342(1) Computer Implementation 343(66) Numerical Integration 343(13) Background 343(2) Natural Coordinates 345(1) Approximation of Geometry 346(1) Isoparametric Formulations 347(1) Numerical Integration 348(8) Computer Implementation 356(14) Introductory Comments 356(1) General Outline 357(2) Preprocessor 359(1) Calculation of Element Matrices (Processor) 360(3) Assembly of Element Equations (Processor) 363(2) Imposition of Boundary Conditions (Processor) 365(2) Solving Equations and Postprocessing 367(3) Applications of Program FEMID 370(31) General Comments 370(1) Illustrative Examples 370(31) Summary 401(8) Problems 401(5) References for Additional Reading 406(3) Single-Variable Problems in Two Dimensions 409(116) Introduction 409(1) Boundary Value Problems 410(32) The Model Equation 410(1) Finite Element Discretization 411(1) Weak Form 412(3) Finite Element Model 415(2) Derivation of Interpolation Functions 417(8) Evaluation of Element Matrices and Vectors 425(11) Assembly of Element Equations 436(4) Postcomputations 440(1) Axisymmetric Problems 441(1) A Numerical Example 442(11) Some Comments on Mesh Generation and Imposition of Boundary Conditions 453(5) Discretization of a Domain 453(2) Generation of Finite Element Data 455(1) Imposition of Boundary Conditions 456(2) Applications 458(32) Conduction and Convection Heat Transfer 458(14) Fluid Mechanics 472(13) Solid Mechanics 485(5) Eigenvalue and Time-Dependent Problems 490(14) Introduction 490(1) Parabolic Equations 491(8) Hyperbolic Equations 499(5) Summary 504(21) Problems 504(18) References for Additional Reading 522(3) Interpolation Functions, Numerical Integration, and Modeling Considerations 525(52) Introduction 525(1) Element Library 525(15) Triangular Elements 525(7) Rectangular Elements 532(5) The Serendipity Elements 537(2) Hermite Cubic Interpolation Functions 539(1) Numerical Integration 540(21) Preliminary Comments 540(3) Coordinate Transformations 543(6) Integration over a Master Rectangular Element 549(8) Integration over a Master Triangular Element 557(4) Modeling Considerations 561(8) Preliminary Comments 561(1) Element Geometries 562(1) Mesh Generation 563(4) Load Representation 567(2) Summary 569(8) Problems 570(5) References for Additional Reading 575(2) Flows of Viscous Incompressible Fluids 577(30) Preliminary Comments 577(1) Governing Equations 577(2) Velocity-Pressure Formulation 579(4) Weak Formulation 579(2) Finite Element Model 581(2) Penalty Function Formulation 583(5) Preliminary Comments 583(1) Formulation of the Flow Problem as a Constrained Problem 583(1) Lagrange Multiplier Model 584(1) Penalty Model 585(3) Time Approximation 588(1) Computational Aspects 588(3) Properties of the Matrix Equations 588(1) Choice of Elements 589(1) Evaluation of Element Matrices in the Penalty Model 590(1) Postcomputation of Stresses 591(1) Numerical Examples 591(11) Summary 602(5) Problems 603(2) References for Additional Reading 605(2) Plane Elasticity 607(28) Introduction 607(1) Governing Equations 607(5) Plane Strain 607(1) Plane Stress 608(2) Summary of Equations 610(2) Weak Formulations 612(2) Preliminary Comments 612(1) Principle of Virtual Displacements in Vector Form 612(1) Weak Form of the Governing Differential Equations 613(1) Finite Element Model 614(3) General Model 614(3) Eigenvalue and Transient Problems 617(1) Evaluation of Integrals 617(3) Assembly of Finite Element Equations 620(2) Examples 622(7) Summary 629(6) Problems 629(4) References for Additional Reading 633(2) Bending of Elastic Plates 635(32) Introduction 635(2) Classical Plate Theory 637(9) Displacement Field 637(1) Virtual Work Statement 638(4) Finite Element Model 642(1) Plate Bending Elements 643(3) Shear Deformation Plate Theory 646(7) Displacement Field 646(2) Virtual Work Statement 648(2) Finite Element Model 650(2) Shear Locking and Reduced Integration 652(1) Eigenvalue and Time-Dependent Problems 653(2) Examples 655(8) Summary 663(4) Problems 663(2) References for Additional Reading 665(2) Computer Implementation of Two-Dimensional Problems 667(44) Introduction 667(2) Preprocessor 669(1) Element Computations (Processor) 669(6) Applications of the Computer Program FEM2D 675(28) Introduction 675(6) Description of Mesh Generators 681(5) Applications (Illustrative Examples) 686(17) Summary 703(8) Problems 705(4) References for Additional Reading 709(2) Prelude to Advanced Topics 711(46) Introduction 711(1) Alternative Finite Element Models 711(14) Introductory Comments 711(1) Weighted Residual Finite Element Models 712(10) Mixed Formulations 722(3) Three-Dimensional Problems 725(11) Heat Transfer 726(1) Flows of Viscous Incompressible Fluids 727(1) Elasticity 728(3) Three-Dimensional Finite Elements 731(4) A Numerical Example 735(1) Nonlinear Problems 736(7) General Comments 736(1) Bending of Euler--Bernoulli Beams 736(2) The Navier--Stokes Equations in Two Dimensions 738(1) Solution Methods for Nonlinear Algebraic Equations 739(1) Numerical Examples 740(3) Errors in Finite Element Analysis 743(7) Types of Errors 743(1) Measures of Errors 744(1) Convergence and Accuracy of Solutions 745(5) Summary 750(7) Problems 751(2) References for Additional Reading 753(4) Index 757