The finite element method : linear static and dynamic finite element analysis

목차 (Table of Contents)

• CONTENTS
• PREFACE = XV
• A BRIEF GLOSSARY OF NOTATIONS = XXII
• Part One Linear Static Analysis
• 1 FUNDAMENTAL CONCEPTS ; A SIMPLE ONE-DIMENSIONAL BOUNDARY-VALUE PROBLEM = 1
• 1.1 Introductory Remarks and Preliminaries = 1
• 1.2 Strong, or Classical, Form of the Problem = 2
• 1.3 Weak, or Variational, Form of the Problem = 3
• 1.4 Eqivalence of Strong and Weak Forms : Natural Boundary Conditions = 4
• 1.5 Galerkin's Approximation Method = 7
• 1.6 Matrix Equations ; Stiffness Matrix K = 9
• 1.7 Examples : 1 and 2 Degrees of Freedom = 13
• 1.8 Piecewise Linear Finite Element Space = 20
• 1.9 Properties of K = 22
• 1.10 Mathematical Analysis = 24
• 1.11 Interlude : Gauss Elimination ; Hand-calculation Version = 31
• 1.12 The Element Point of View = 37
• 1.13 Element Stiffness Matrix and Force Vector = 40
• 1.14 Assembly of Global Stiffness Matrix and Force Vector ; LM Array = 42
• 1.15 Explicit Computation of Element Stiffness Matrix and Force Vector = 44
• 1.16 Exercise : Bernoulli-Euler Beam Theory and Hermite Cubics = 48
• Appendix 1.Ⅰ An Elementary Discussion of Continuity, Differentiability, and Smoothness = 52
• References = 55
• 2 FORMULATION OF TWO-AND THREE-DIMENSIONAL BOUNDARY-VALUE PROBLEMS = 57
• 2.1 Introductory Remarks = 57
• 2.2 Preliminaries = 57
• 2.3 Classical Linear Heat Conduction : Strong and Weak Forms ; Equivalence = 60
• 2.4 Heat Conduction : Galerkin Formulation ; Symmetry and Positive-definiteness of K = 64
• 2.5 Heat Conduction : Element Stiffness Matrix and Force Vector = 69
• 2.6 Heat Conduction : Data Processing Arrays ID, IEN, and LM = 71
• 2.7 Classical Linear Elastostatics: Strong and Weak Forms ; Equivalence = 75
• 2.8 Elastostatics : Galerkin Formulation, Symmetry, and Positive-definiteness of K = 84
• 2.9 Elastostatics : Element Stiffness Matrix and Force Vector = 90
• 2.10 Elastostatics : Data Processing Arrays ID, IEN, and LM = 92
• 2.11 Summary of Important Equations for Problems Considered in Chapters 1 and 2 = 98
• 2.12 Axisymmetric Formulations and Additional Exercises = 101
• References = 107
• 3 ISOPARAMETRIC ELEMENTS AND ELEMENTARY PROGRAMMING CONCEPTS = 109
• 3.1 Preliminary Concepts = 109
• 3.2 Bilinear Quadrilateral Element = 112
• 3.3 Isoparametric Elements = 118
• 3.4 Linear Triangular Element ; An Example of "Degeneration" = 120
• 3.5 Trilinear Hexahedral Element = 123
• 3.6 Higher-order Elements ; Lagrange Polynomials = 126
• 3.7 Elements with Variable Numbers of Nodes = 132
• 3.8 Numerical Integration ; Gaussian Quadrature = 137
• 3.9 Derivatives of Shape Functions and Shape Function Subroutines = 146
• 3.10 Element Stiffness Formulation = 151
• 3.11 Additional Exercises = 156
• Appendix 3.Ⅰ Triangular and Tetrahedral Elements = 164
• Appendix 3.Ⅱ Methodology for Developing Special Shape Functions with Application to Singularities = 175
• References = 182
• 4 MIXED AND PENALTY METHODS, REDUCED AND SELECTIVE INTEGRATION, AND SUNDRY VARIATIONAL CRIMES = 185
• 4.1 "Best Approximation" and Error Estimates : Why the standard FEM usually works and why sometimes it does not = 185
• 4.2 Incompressible Elasticity and Stokes Flow = 192
• 4.2.1 Prelude to Mixed and Penalty Methods = 194
• 4.3 A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit = 197
• 4.3.1 Strong Form = 198
• 4.3.2 Weak Form = 198
• 4.3.3 Galerkin Formulation = 200
• 4.3.4 Matrix Problem = 200
• 4.3.5 Definition of Element Arrays = 204
• 4.3.6 Illustration of a Fundamental Difficulty = 207
• 4.3.7 Constraint Counts = 209
• 4.3.8 Discontinuous Pressure Elements = 210
• 4.3.9 Continuous Pressure Elements = 215
• 4.4 Penalty Formulation : Reduced and Selective Integration Techniques ; Equivalence with Mixed Methods = 217
• 4.4.1 Pressure Smoothing = 226
• 4.5 An Extension of Reduced and Selective Integration Techniques = 232
• 4.5.1 Axisymmetry and Anisotropy : Prelude to Nonlinear Analysis = 232
• 4.5.2 Strain Projection : The $$\bar B$$-approach = 232
• 4.6 The Patch Test ; Rank Deficiency = 237
• 4.7 Nonconforming Elements = 242
• 4.8 Hourglass Stiffness = 251
• 4.9 Additional Exercises and Projects = 254
• Appendix 4.Ⅰ Mathematical Preliminaries = 263
• 4.Ⅰ.1 Basic Properties of Linear Spaces = 263
• 4.Ⅰ.2 Sobolev Norms = 266
• 4.Ⅰ.3 Approximation Properties of Finite Element Spaces in Sobolev Norms = 268
• 4.Ⅰ.4 Hypotheses on a(ㆍ,ㆍ) = 273
• Appendix 4.Ⅱ Advanced Topics in the Theory of Mixed and Penalty Methods : Pressure Modes and Error Estimates / by David S. Malkus = 276
• 4.Ⅱ.1 Pressure Modes, Spurious and Otherwise = 276
• 4.Ⅱ.2 Existence and Uniqueness of Solutions in the Presence of Modes = 278
• 4.Ⅱ.3 Two Sides of Pressure Modes = 281
• 4.Ⅱ.4 Pressure Modes in the Penalty Formulation = 289
• 4.Ⅱ.5 The Big Picture = 292
• 4.Ⅱ.6 Error Estimates and Pressure Smoothing = 297
• References = 303
• 5 THE $$C^0$$-APPROACH TO PLATES AND BEAMS = 310
• 5.1 lntroduction = 310
• 5.2 Reissner-Mindlin Plate Theory = 310
• 5.2.1 Main Assumptions = 310
• 5.2.2 Constitutive Equation = 313
• 5.2.3 Strain-displacement Equations = 313
• 5.2.4 Summary of Plate Theory Notations = 314
• 5.2.5 Variational Equation = 314
• 5.2.6 Strong Form = 317
• 5.2.7 Weak Form = 317
• 5.2.8 Matrix Formulation = 319
• 5.2.9 Finite Element Stiffness Matrix and Load Vector = 320
• 5.3 Plate-bending Elements = 322
• 5.3.1 Some Convergence Criteria = 322
• 5.3.2 Shear Constraints and Locking = 323
• 5.3.3 Boundary Conditions = 324
• 5.3.4 Reduced and Selective Integration Lagrange Plate Elements = 327
• 5.3.5 Equivalence with Mixed Methods = 330
• 5.3.6 Rank Deficiency = 332
• 5.3.7 The Heterosis Element = 335
• 5.3.8 T1 : A Coffect-rank, Four-node Bilinear Element = 342
• 5.3.9 The Linear Triangle = 355
• 5.3.10 The Discrete Kirchhoff Approach = 359
• 5.3.11 Discussion of Some Quadrilateral Bending Elements = 362
• 5.4 Beams and Frames = 363
• 5.4.1 Main Assumptions = 363
• 5.4.2 Constitutive Equation = 365
• 5.4.3 Strain-displacement Equations = 366
• 5.4.4 Definitions of Quantities Appearing in the Theory = 366
• 5.4.5 Variational Equation = 368
• 5.4.6 Strong Form = 371
• 5.4.7 Weak Form = 372
• 5.4.8 Matrix Formulation of the Variational Equation = 373
• 5.4.9 Finite Element Stiffness Matrix and Load Vector = 374
• 5.4.10 Representation of Stiffness and Load in Global Coordinates = 376
• 5.5 Reduced Integration Beam Elements = 376
• References = 379
• 6 THE $$C^0$$-APPROACH TO CURVED STRUCTURAL ELEMENTS = 383
• 6.1 Introduction = 383
• 6.2 Doubly Curved Shells in Three Dimensions = 384
• 6.2.1 Geometry = 384
• 6.2.2 Lamina Coordinate Systems = 385
• 6.2.3 Fiber Coordinate Systems = 387
• 6.2.4 Kinematics = 388
• 6.2.5 Reduced Constitutive Equation = 389
• 6.2.6 Strain-displacement Matrix = 392
• 6.2.7 Stiffness Matrix = 396
• 6.2.8 External Force Vector = 396
• 6.2.9 Fiber Numerical Integration = 398
• 6.2.10 Stress Resultants = 399
• 6.2.11 Shell Elements = 399
• 6.2.12 Some References to the Recent Literature = 403
• 6.2.13 Simplifications : Shells as an Assembly of Flat Elements = 404
• 6.3 Shells of Revolution ; Rings and Tubes in Two Dimensions = 405
• 6.3.1 Geometric and Kinematic Descriptions = 405
• 6.3.2 Reduced Constitutive Equations = 407
• 6.3.3 Strain-displacement Matrix = 409
• 6.3.4 Stiffness Matrix = 412
• 6.3.5 External Force Vector = 412
• 6.3.6 Stress Resultants = 413
• 6.3.7 Boundary Conditions = 414
• 6.3.8 Shell Elements = 414
• References = 415
• Part Two Linear Dynamic Analysis
• 7 FORMULATION OF PARABOLIC, HYPERBOLIC, AND ELLIPTIC- EIGENVALUE PROBLEMS = 418
• 7.1 Parabolic Case : Heat Equation = 418
• 7.2 Hyperbolic Case : Elastodynamics and Structural Dynamics = 423
• 7.3 Eigenvalue Problems : Frequency Analysis and Buckling = 429
• 7.3.1 Standard Error Estimates = 433
• 7.3.2 Alternative Definitions of the Mass Matrix ; Lumped and Higher-order Mass = 436
• 7.3.3 Estimation of Eigenvalues = 452
• Appendix 7.Ⅰ Error Estimates for Semidiscrete Galerkin Approximations = 456
• References = 457
• 8 ALGORITHMS FOR PARABOLIC PROBLEMS = 459
• 8.1 One-step Algorithms for the Semidiscrete Heat Equation : Generalized Trapezoidal Method = 459
• 8.2 Analysis of the Generalized Trapezoidal Method = 462
• 8.2.1 Modal Reduction to SDOF Form = 462
• 8.2.2 Stability = 465
• 8.2.3 Convergence = 468
• 8.2.4 An Alternative Approach to Stability : The Energy Method = 471
• 8.2.5 Additional Exercises = 473
• 8.3 Elementary Finite Difference Equations for the One-dimensional Heat Equation ; the von Neumann Method of Stability Analysis = 479
• 8.4 Element-by-element(EBE) Implicit Methods = 483
• 8.5 Modal Analysis = 487
• References = 488
• 9 ALGORITHMS FOR HYPERBOLIC AND PARABOLIC-HYPERBOLIC PROBLEMS = 490
• 9.1 One-step Algorithms for the Semidiscrete Equation of Motion = 490
• 9.1.1 The Newmark Method = 490
• 9.1.2 Analysis = 492
• 9.1.3 Measures of Accuracy : Numerical Dissipation and Dispersion = 504
• 9.1.4 Matched Methods = 505
• 9.1.5 Additional Exercises = 512
• 9.2 Summary of Time-step Estimates for Some Simple Finite Elements = 513
• 9.3 Linear Multistep(LMS) Methods = 523
• 9.3.1 LMS Methods for First-order Equations = 523
• 9.3.2 LMS Methods for Second-order Equations = 526
• 9.3.3 Survey of Some Commonly Used Algorithms in Structural Dynamics = 529
• 9.3.4 Some Recently Developed Algorithms for Structural Dynamics = 550
• 9.4 Algorithms Based upon Operator Splitting and Mesh Partitions = 552
• 9.4.1 Stability via the Energy Method = 556
• 9.4.2 Predictor／Multicorrector Algorithms = 562
• 9.5 Mass Matrices for Shell Elements = 564
• References = 567
• 10 SOLUTION TECHNIQUES FOR EIGENVALUE PROBLEMS = 570
• 10.1 The Generalized Eigenproblem = 570
• 10.2 Static Condensation = 573
• 10.3 Discrete Rayleigh-Ritz Reduction = 574
• 10.4 Irons-Guyan Reduction = 576
• 10.5 Subspace Iteration = 576
• 10.5.1 Spectrum Slicing = 578
• 10.5.2 Inverse Iteration = 579
• 10.6 The Lanczos Algorithm for Solution of Large Generalized Eigenproblems / by Bahram Nour-Omid = 582
• 10.6.1 Introduction = 582
• 10.6.2 Spectral Transformation = 583
• 10.6.3 Conditions for Real Eigenvalues = 584
• 10.6.4 The Rayleigh-Ritz Approximation = 585
• 10.6.5 Derivation of the Lanczos Algorithm = 586
• 10.6.6 Reduction to Tridiagonal Form = 589
• 10.6.7 Convergence Criterion for Eigenvalues = 592
• 10.6.8 Loss of Orthogonality = 595
• 10.6.9 Restoring Orthogonality = 598
• 10.6.10 LANSEL Package = 600
• References = 629
• 11 DLEARN - A LINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS PROGRAM / by Thomas J. R. Hughes ; Robert M. Ferenez ; Arthur M. Raefsky = 631
• 11.1 Introduction = 631
• 11.2 Description of Coding Techniques Used in DLEARN = 632
• 11.2.1 Compacted Column Storage Scheme = 633
• 11.2.2 Crout Elimination = 636
• 11.2.3 Dynamic Storage Allocation = 644
• 11.3 Program Structure = 650
• 11.3.1 Global Control = 651
• 11.3.2 Initialization Phase = 651
• 11.3.3 Solution Phase = 653
• 11.4 Adding an Element to DLEARN = 659
• 11.5 DLEARN User's Manual = 662
• 11.5.1 Remarks for the New User = 662
• 11.5.2 Input Instructions = 663
• 11.5.3 Examples = 691
• 1. Planar Truss = 691
• 2. Static Analysis of a Plane Strain Cantilever Beam = 705
• 3. Dynamic Analysis of a Plane Strain Cantilever Beam = 705
• 4. Implicit-explicit Dynamic Analysis of a Rod = 715
• 11.5.4 Subroutine Index for Program Listing = 729
• 11.5.5 Program Listing = 734
• References = 796
• INDEX = 797