The Finite Element Method – Linear Static and Dynamic Finite Element Analysis

The Finite Element Method – Linear Static and Dynamic Finite Element Analysis
اسم المؤلف
Thomas J. R. Hughes
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19 فبراير 2024
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The Finite Element Method – Linear Static and Dynamic Finite Element Analysis
Thomas J. R. Hughes
Professor of Mechanical Engineering
Chairman of the Division of Applied Mechanics
Stanford UniversityContents
preface xv
A BRIEF GLOSSARY OF NOTATIONS XXII
Part One Linear Static Analysis
1 FUNDAMENTAL CONCEPTS; A SIMPLE ONE-DIMENSIONAL
BOUNDARY-VALUE PROBLEM 1
Introductory Remarks and Preliminaries 1
Strong, or Classical, Form of the Problem 2
Weak, or Variational, Form of the Problem 3
Eqivalence of Strong and Weak Forms; Natural Boundary
Conditions 4
Galerkin’s Approximation Method 7
Matrix Equations; Stiffness Matrix K 9
Examples: 1 and 2 Degrees of Freedom 13
Piecewise Linear Finite Element Space 20
Properties of K 22
Mathematical Analysis 24
Interlude: Gauss Elimination; Hand-calculation
Version 31
The Element Point of View 37
Element Stiffness Matrix and Force Vector 40
Assembly of Global Stiffness Matrix and Force Vector;
LM Array 42viii Contents
1.15 Explicit Computation of Element Stiffness Matrix
and Force Vector 44
1.16 Exercise: Bernoulli-Euler Beam Theory and Hermite
Cubics 48
Appendix 1.1 An Elementary Discussion of Continuity, Differentiability,
and Smoothness 52
References 55
2 FORMULATION OF TWO- AND THREE-DIMENSIONAL
BOUNDARY-VALUE PROBLEMS 57
Introductory Remarks 57
Preliminaries 57
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
Preliminary Concepts 109
Bilinear Quadrilateral Element 112
Isoparametric Elements 118
Linear Triangular Element; An Example
of “Degeneration” 120
Trilinear Hexahedral Element 123
Higher-order Elements; Lagrange Polynomials 126
Elements with Variable Numbers of Nodes 132Contents
Appendix 3.1
Appendix 3.II
Numerical Integration; Gaussian Quadrature 137
Derivatives of Shape Functions and Shape Function
Subroutines 146
Element Stiffness Formulation 151
Additional Exercises 156
Triangular and Tetrahedral Elements 164
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 stan¬
dard 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 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 4.1 Mathematical Preliminaries 263
4.1.1 Basic Properties of Linear Spaces 263
4.1.2 Sobolev Norms 266
4.1.3 Approximation Properties of Finite Element Spaces
in Sobolev Norms 268X Contents
4.1.4 Hypotheses on a(- , •) 273
Appendix 4.II Advanced Topics in the Theory of Mixed and Penalty
Methods: Pressure Modes and Error Estimates 276
by David S. Malkus
4.II.1
4.II.2
Pressure Modes, Spurious and Otherwise 276
Existence and Uniqueness of Solutions in the Pres¬
ence of Modes 278
4.II.3
4.II.4
4.II.5
4.H.6
Two Sides of Pressure Modes 281
Pressure Modes in the Penalty Formulation 289
The Big Picture 292
Error Estimates and Pressure Smoothing 297
References 303
5 THE C°-APPROACH TO PLATES AND BEAMS 310
5.1 Introduction 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 71: A Correct-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 366contents
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°-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 415Part Two Linear Dynamic Analysis
7 FORMULATION OF PARABOLIC, HYPERBOLIC, AND ELLIPTICEIGENVALUE 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 7.1 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 PARABOLICHYPERBOLIC 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
ContentsContents
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 582
by Bahram Nour-Omid
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 631
by Thomas J. R. Hughes, Robert M. Ferencz,
and Arthur M. Raefskyxiv Contents
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 797Index
    Absolute stability, 525
    Accuracy, 462
    Active column equation solver,
    554, 633
    Algorithm for constructing inter¬
    polation functions, 176-77
    Algorithmic damping ratio, 505
    a-method, 532
    Amplification factor, 466
    Amplification matrix, 492
    Amplitude decay, 505
    Assembly algorithm, 43
    Assembly operator, 44
    A-stable, 525
    Aubin-Nitsche method, 190
    Augmented matrix, 32
    Average acceleration method,
    494-95
    Axial force, 367
    Axial strain, 367
    Axisymmetric shells (see Shells
    of revolution, rings and
    tubes)
    Axisymmetry, 101-3
    Babuska-Brezzi condition, 208, 292
    Back substitution, 33, 642
    Backward difference method, 460
    Backward Euler method, 460
    Banded matrix, 23
    Bandwidth, 23
    B-approach, 232
    Barlow curvature points, 50
    Barlow stress points, 31
    Basis, 463
    Basis functions, 9
    Bazzi-Anderheggen p-method,
    551
    Beams (see also Bernoulli-Euler
    beam theory):
    assumptions, 363-64
    cross-section properties, 367
    element stiffness matrix and
    load vector, 375
    local-global transformations,
    367
    matrix formulation, 373
    strain-displacement equations,
    366
    strong form, 371
    variational equation, 369
    weak form, 372
    Bending moments, 367
    Bemoulli-Euler beam theory,
    48-51
    Best approximation property, 186
    Bilinear quadrilateral element,
    112
    Biquadratic Lagrange element,
    129
    Blank common, 633
    Block power method, 577
    Body force, 76
    Bossak’s method, 550
    Boundary, 59
    Boundary conditions, 2
    Boundary heat flux calculations
    107
    Boundary traction calculations,
    107
    Brick elements, 123, 136
    BTCS method, 480
    Bubble function, 130, 134
    Bubnov-Galerkin method, 8
    Bulk modulus, 192
    C*(ft), 52
    CtW, 52
    Capacity, 419
    Capacity matrix, 422
    Cauchy stress tensor, 76
    C°-elements, 110
    C’-elements, 110
    Central difference method,
    494-95
    Chain rule, 44
    Change of variables formula:
    Dirac delta function, 158
    one dimension, 44
    three dimension, 140
    two dimensions, 138
    7798 Index
    ’aracteristic velocity, 510
    rolesky decomposition, 644
    rcular plates, 328-32, 339-42,
    346, 347, 350, 358
    osed unit interval, 2
    •efficient of heat transfer, 71
    ‘factors, 149-50
    ‘[location schemes, 530
    impacted column equation
    solver, 554
    •mpacted column storage, 633
    <mpatible elements, 110
    ‘mpleteness of finite element
    functions, 110-11
    ‘mpleteness of function spaces,
    265
    ‘nditional consistency, 481
    ‘nditional stability, 466
    60
    •nductivity matrix, 60
    elements, 110
    ‘nservation of total energy, 457
    insistency, 462
    insistent mass, 436, 507
    institutive equation:
    clastostatics, 76
    jeat conduction, 60
    nstrained media problems, 192
    ‘retrained variational problem,
    194
    •retraint ratio, 209, 223, 289,
    324
    <ntinuously differentiable func¬
    tions, 52
    ntinuous pressure elements,
    215-16
    nvection-diffusion equation,
    161
    •nvergence, 462
    criterion for eigen¬
    values, 592
    rrectors, 553, 562, 657
    uples, 367
    ack elements, 159 (see also
    Singular elements)
    ank-Nicolson method, 460, 480
    eeping flow, 193
    itically damped, 524
    ‘deal sampling frequency, 493
    ossed triangles, 224, 286
    out factorization, 485, 636
    ibic beam element, 520
    ibic four-node element in one
    dimension, 128
    tbic Hermite shape functions,49
    triangular element, 169
    Curvature, 50, 314, 367
    Dahlquist’s theorem, 525
    Data structure, 633
    Deflation, 626
    Degeneration, 120, 125, 126,
    180-81
    Density, 419, 423
    Derivatives of shape functions,
    146-50, 174
    Destination array (see ID array)
    Deviatoric components, 233
    Diagonal scaling, 642
    Dilatational components, 233
    Dipole, 50
    Dirac delta function, 24, 158
    Direct stiffness method, 41
    Discontinuous pressure elements,
    210-14
    Discrete Kirchhoff approach, 359
    Discrete Poisson equation for
    pressure, 203
    Discrete Rayleigh-Ritz approxi¬
    mation, 585
    Discrete Rayleigh-Ritz reduction,
    574
    Discretization, 7, 65
    Displacement, 366
    Displacement difference equation
    form, 527
    Displacement vector, 11, 76
    Distributions, 25
    Divergence theorem, 60
    DKQ, 359, 361, 362
    DKT, 361
    DLEARN coding techniques, 632
    DLEARN examples:
    dynamic analysis of a plane
    strain cantilever beam:
    description, 705-6
    input file, 706
    output, 717-29
    implicit-explicit dynamic analy¬
    sis of a rod:
    description, 715-16
    input file, 717
    output file, 717-29
    planar truss:
    description, 691-92
    input file, 692
    output, 693-704
    static analysis of a plane strain
    cantilever beam:
    description, 705
    input file, 705
    DLEARN input instructions:
    boundary bonditions data, 672
    coordinate data, 667
    element data:
    three-dimensional, elastic
    truss element, 687
    two-dimensional, isotropic
    elasticity element, 680
    execution control, 664
    input data echo, 663
    kinematic initial condition data,
    678
    load-time functions, 677
    nodal history data, 667
    prescribed nodal forces and
    kinematic boundary condi¬
    tions, 673
    time sequence data, 666
    DLEARN program listing,
    734-96
    DLEARN program structure:
    global control, 651
    initialization phase, 651-53
    solution phase, 653-59
    DLEARN storage in blank com¬
    mon:
    dynamic analysis data, 646
    equation system data, 650
    static analysis data, 647
    time sequence and time history
    data, 645-46
    DLEARN storage requirements
    for four-node element,
    647-50
    DLEARN subroutine index,
    729-34
    Domain, 1
    Doubly curved shells:
    element force vector, 396
    element stiffness matrix, 396
    fiber coordinate systems, 387
    geometry, 384
    kinematics, 388
    lamina coordinate systems, 385
    reduced constitutive equation,
    389
    strain-displacement matrix, 392
    stress resultants, 399
    Douglas, 27
    Drilling degrees of freedom, 404
    Driven cavity flow, 230-31,
    282-85
    DuFort-Frankel method, 481
    Dupont, 27
    Dynamic storage allocation,
    633-44Index
    Eigenvalue problems:
    buckling of a thin beam, 431
    free vibration of an elastic rod,
    430
    free vibration of a thin beam,
    433
    generalized, 570
    standard, 571
    standard error estimates, 433
    Elastic coefficients, 76
    Elastic membrane, 428
    Elastodynamics (see Hyperbolic
    problems)
    Elastostatics:
    axisymmetric formulation,
    101-3
    element displacement vector,
    91-92
    element force vector, 90
    element stiffness matrix, 90
    element strain-displacement ma¬
    trix:
    axisymmetric case, 102
    three-dimensional case, 90
    two-dimensional case, 90
    Galerkin formulation, 84
    matrix formulation, 87
    strong form, 77
    summary of important equa¬
    tions, 98-99
    weak form, 78
    Element body forces, 162-63
    Element boundary forces, 161-63
    Element-by-element (EBE) im¬
    plicit methods, 483
    Element force vector, 41
    Element groups, 633
    Element nodes array (see IEN ar¬
    ray)
    Elements, 20
    Element stiffness implementation,
    151-56
    Element stiffness matrix, 41
    Elements with variable numbers
    of nodes, 132-35
    Empty set, 58
    Energy inner product, 186, 273
    Energy method (see Stability via
    the energy method)
    Energy norm, 186, 273
    Energy stability, 472
    Enriched bilinear displacementsconstant pressure quadrilat¬
    eral, 259
    Equation of motion, 423
    Equilibrium equations, 77
    Equivalence theorem, 221, 330
    Error, 186
    Error equation, 470
    Error estimates:
    elliptic boundary-value prob¬
    lems, 189
    elliptic eigenvalue problems,
    433
    semidiscrete Galerkin approxi¬
    mations, 456
    Error in the derivative, 29
    Essential boundary conditions, 6
    Estimation of eigenvalues, 452
    Euclidean basis vectors, 85-86
    Euclidean decomposition of a sec¬
    ond-rank tensor, 78
    Euler-Lagrange equations, 5
    Explicit methods, 461
    Explicit predictor-corrector meth¬
    ods. 553
    Exponential shape functions, 47
    Factorization, 637
    Fiber, 384
    Fiber numerical integration, 398
    Finite difference equations, 479
    Finite difference stencil, 31
    Finite element, 20
    Finite element domain, 20
    Finite Taylor expansion, 28
    Flop, 642
    Force vector, 11
    Forward difference method, 460
    Forward Euler method, 460
    Forward reduction, 32, 639
    Fourier coefficients, 463
    Fourier law, 60
    Fox-Goodwin method, 493
    Fractional-step algorithm, 474
    Frames (see Beams)
    FTCS method, 479
    Function spaces, 8
    Fundamental lemma of the cal¬
    culus of variations, 6
    Galerkin equation, 9
    Galerkin method, 8
    Gauss elimination:
    example, 35
    hand-calculation algorithm, 33
    Gaussian quadrature (see Numeri¬
    cal integration)
    Gear’s methods. 526
    Generalized derivative, 17
    Generalized displacements, 243
    Generalized Fourier law, 60
    Generalized functions, 25
    Generalized Hooke’s law, 76
    Generalized Jacobi method, 578
    Generalized solution, 4
    Generalized step function, 21
    Generalized trapezoidal methods
    commutative diagram, 465
    convergence, 468
    equations, 460
    implementations, 460-61
    modal reduction to SDOF fom
    462
    SDOF model problem, 464
    stability, 465-67
    Geometric stiffness, 432
    Ghost eigenvalues, 594
    Givens method, 572, 619
    Green’s function, 25
    Growth/decay estimates, 457
    //‘(D). 54
    Half-bandwidth, 23
    Heat conduction:
    axisymmetric formulation, 101
    element force vector, 69
    element stiffness matrix, 69
    element temperature vector, 71
    Galerkin formulation, 64
    matrix formulation, 67
    strong form, 61
    summary of important equa¬
    tions, 99-100
    weak form, 61
    Heat equation, 61, 419, 422
    Heat flux, 107
    Heat flux vector, 60
    Heat supply, 60
    Heaviside function, 25
    Hermite shape functions, 49
    Hermitian matrix, 564
    Heterosis plate element, 335
    Heterosis shell element, 401
    Higher-order elements, 126
    Higher-order mass, 446, 507
    Hilber-Hughes-Taylor method
    (see a -method)
    Hilbert projection theorem, 280
    Hilbert space, 266
    Homogeneity:
    elastic coefficients, 155
    elastostatics, 76
    heat conduction, 60
    Hooke’s law. 76800 Index
    ubolt’s method, 529
    urglass modes, 239, 254
    urglass stabilization operator,
    254
    ,urglass stiffness, 251
    method, 572,
    582
    drostatic pressure, 193
    problems:
    •natrix formulation, 424
    .emidisercte Galerkin formula¬
    tion, 424
    strong form, 423
    weak form, 423
    array:
    iefinrtion:
    elastostatics, 85
    heat conduction, 66
    sxample:
    elastostatics, 94
    heat conduction, 72-73
    N array:
    definition, 71
    example:
    elastostatics, 94
    heat conduction, 72-73
    plicit-explicit element mesh
    partitions, 461
    plicit-explicit methods, 553
    plicit methods, 461
    .ompatible elements, 110, 243
    :ompatible modes, 243
    impressible elasticity, 192-93
    lex-free notation, 63
    :rtial inner product, 584
    initesimal rigid-body motions,
    88
    initesimal strain tensor, 76
    tial condition, 418
    tial strain, 105
    tial stress, 104-5
    tial-stress stiffness matrix, 104,
    432
    ter product, 264
    ‘egration by parts, 60
    terior element boundaries, 68
    terpolation estimate, 189
    terpolation functions, 9
    terpolation property, 114
    verse function theorem, 119
    verse iteration, 579
    ms-Guyan reduction, 576
    OFLEX, 361
    Isoparametric elements, 118, 271
    Isotropy:
    elastic coefficients, 155
    elastostatics, 83
    heat conduction, 60
    Jacobian determinant, 119
    Jacobi method, 572
    Joints, 20
    Kinematic boundary conditions,
    563, 655
    Kinematic condition of incom¬
    pressibility, 193
    Kinetic energy, 512
    Kirchhoff mode concept, 324, 353
    k,m-regular, 189, 269
    Knots, 20
    Kronecker delta, 21
    Krylov sequence, 586
    LAO), 54
    Lagrange elements, 130, 138, 139
    Lagrange-multiplier method, 195
    Lagrange plate elements, 327
    Lagrange polynomials, 127, 176
    Lagrange shell elements, 400
    Lagrange-type interpolation over
    tetrahedra, 171
    Lagrange-type interpolation over
    triangles, 166-69
    Lame parameters, 83, 192
    Lamina, 384
    Lanczos algorithm, 582-90
    example, 590
    summary (table), 588
    Lanczos vectors, 586
    LANSEL eigenvalue package,
    600-29
    Lax equivalence theorem, 470
    LBB condition, 208
    Leap frog method, 480
    Least squares, 227
    Limitation principle, 226
    Linear acceleration method, 493
    Linear multistep (LMS) methods
    for first-order equations,
    523
    Linear multistep (LMS) methods
    for second-order equations,
    526
    Linear nonconforming triangle,
    250
    Linear one-dimensional finite ele¬
    ment, 37
    Linear spaces, 263
    Linear tetrahedral element, 126,
    170
    Linear triangular element, 120,
    167
    Linear triangular plate element,
    355
    LM array:
    definition:
    elastostatics, 92
    heat conduction, 72
    one-dimensional model prob¬
    lem, 42
    example:
    elastostatics, 94
    heat conduction, 72-73
    Lobatto element, 440
    Lobatto quadrature, (see Numeri¬
    cal integration)
    Local spurious modes, 287
    Local truncation error, 468, 529
    Location matrix (see LM array)
    Locking, 323
    Locking element, 293
    LORA, 345, 351
    Loss of orthogonality, 595
    Lumped mass, 436-45, 507
    nodal quadrature, 436
    row-sum, 444
    special lumping, 445
    Macaulay bracket, 25
    Macroelement, 224, 259
    Mass, consistent (see Consistent
    mass)
    Mass, higher-order (see Higherorder mass)
    Mass, lumped, 436—45, 507 (see
    also Lumped mass)
    Mass matrices for shell elements,
    564
    Mass matrix, 426
    Matched methods, 505
    Matrix equations, 11
    Mean-dilatation approach, 232
    Mean incompressibility, 161
    Mean-value theorem, 28
    Mechanisms, 240
    Memory manager, 644Index
    Memory pointer dictionary, 631,
    644
    Mesh, 7
    Mesh locking, 208
    Mesh parameter, 189
    Mesh partitions, 552
    Midpoint rule, 460
    Minimum potential energy princi¬
    ple, 188
    Misconvergence, 594
    Mixed boundary-value problem of
    linear elastostatics, 77
    Mixed formulation of elasticity:
    element arrays, 204-6
    Galerkin formulation, 200
    matrix formulation, 200-204
    strong form, 198
    weak form, 199
    Mixed method, 195, 197
    Modal analysis, 487, 540
    Moment tensor, 314
    Multicorrector iteration, 656
    Multiple eigenvalues, 595
    Natural boundary conditions, 6
    Natural coordinates, 112
    Natural norm, 265
    Nearly incompressible case, 217
    Newmark method:
    commutative diagram, 494
    displacement-difference equa¬
    tion form, 527
    equations, 490-91
    error equation, 496
    high-frequency behavior,
    498-500
    implementation, 491
    predictors, 491
    stability conditions, 492-93
    truncation error, 496
    viscous damping, 500
    Newton’s law of heat transfer, 71
    Nodal points, 20
    Nodes, 20
    Nonconforming elements, 110,
    242
    Nonlocking element, 293
    Norm, 265
    Numerical dispersion, 504
    Numerical dissipation, 504
    Numerical integration:
    Gaussian quadrature, 141-45
    Lobatto quadrature. 440
    rules for tetrahedra, 172, 174
    rules for triangles, 172-74
    Simpson’s rule, 141
    trapezoidal rule, 140
    One-dimensional model problem:
    element force vector, 41
    element stiffness matrix, 41
    Galerkin formulation, 9
    matrix formulation, 11
    strong form, 3
    summary of important equa¬
    tions, 100
    weak form, 4
    One-step multivalue methods, 492
    One-to-one, 118
    Onto, 118
    Open set, 57-59
    Open unit interval, 2
    Operator splitting, 552
    Optimal collocation methods, 531
    Optimally constrained, 300
    Order of accuracy, 30, 468
    Order of convergence, 30
    Orthogonality, 264, 571
    Orthonormality, 571
    Overconstrained, 300
    Overdamped, 524
    Overshoot, 537 _
    Parabolic problems:
    matrix formulation, 421
    semidiscrete Galerkin formula¬
    tion, 420
    strong form, 419
    weak form, 419
    Parallel processing, 486
    Parent domain, 112
    Parent tetrahedron, 170
    Parent triangle, 165
    Park’s method, 526
    Partitioned form, 573
    Pascal triangles, 139
    Patch test, 238, 248, 256, 259-61
    Penalty formulation of incom¬
    pressible elasticity, 217,
    289
    Penalty method, 196
    Pergola roof, 334
    Perpendicular, 264
    Petrov-Galerkin method, 9
    Pinched cylinder, 401
    Plane strain, 83, 103, 237
    Plane stress, 83, 103
    Bl
    Plate theory (see Reissner-Mind
    plate theory)
    Poisson-Kirchhoff theory, 310,;
    Poisson’s ratio, 83
    Positive-definite matrix, 23
    Post processing, 107
    Potential energy, 188
    Preconditioned conjugate gradi¬
    ents (table), 485
    Predictor-corrector algorithms,
    473, 476
    Predictor-multicorrector al¬
    gorithms, 562
    Predictors, 553, 562, 654-55
    Prescribed boundary displace¬
    ments, 77
    Prescribed boundary heat flux, 6
    Prescribed boundary temperature
    61
    Prescribed boundary tractions, 75
    Pressure modes, 207, 277
    Pressure smoothing, 227
    Principal invariants, 498, 528
    Principal roots, 529
    Profile, 554, 633
    Projects, 261-62
    Pseudonormal, 384
    QUAD4, 345, 351, 362, 395
    Quadratic tetrahedral element,
    171
    Quadratic three-node element in
    one dimension, 128
    Quadratic triangular element, 136
    168
    Quasi-uniform, 269
    Range, 1
    Rank check, 257, 637
    Rank deficiency, 191, 239, 278,
    332-34
    Rate of convergence, 468
    Rayleigh damping, 426, 492
    Rayleigh quotient, 435, 452
    Rectangular plates, 338-40,
    346-49, 363
    Reduced integration, 221, 327
    Reduced integration beam ele¬
    ments, 376
    Reduced system, 571
    Region of absolute stability, 525
    Regular, 269
    Regularized element array, 486802 Index
    jsner-Mindlin plate theory:
    ssumptions, 310
    oundary conditions, 324-26
    onstitutive equation, 313
    onvergence criteria, 322
    lenient stiffness matrix and
    load vector, 321
    jnetnatics, 311-12
    oatrix formulation, 319
    rain-displacement equations,
    313
    Tong form, 317
    ariational equation, 315
    zeak form, 318
    ative error, 36
    ative period error, 505
    irdered Crout EBE precondi¬
    tioner, 486
    idual, 585
    idual bending flexibility, 378
    -idual forces, 656
    toring orthogonality, 598
    urn of banished Ritz vectors,
    597
    imbic plates, 346-47, 351,
    352
    gs (see Shells of revolution,
    rings and tubes)
    tz value, 585
    z vector, 585
    tation, 314, 366
    uth-Hurwitz criterion, 530
    yal road method (see FoxGoodwin method)
    tl’yev’s method, 481
    iwarz inequality, 264
    OF model problem, 464
    nor element, 159-60
    ective orthogonalization meth¬
    ods, 598
    ective reduced integration,
    221, 327
    MILOOF, 361
    condition, 598
    elements, 135, 138,
    139
    it closure, 58
    4 intersection, 58
    4 union, 58
    ape functions, 9, 112-37,
    165-71
    quadrilaterals and bricks,
    112-37
    tetrahedra, 170-71
    triangles, 165-70
    Shape function subroutines,
    146-50
    Shear constraints, 323
    Shear correction factors, 391
    Shear force, 314, 367
    Shear modulus, 192
    Shear strain, 312, 323, 367
    Shells (see Doubly curved shells)
    Shells as an assembly of flat ele¬
    ments, 404
    Shells of revolution, rings and
    tubes:
    boundary conditions, 414
    element force vector, 412
    element stiffness matrix, 412
    fiber coordinate systems, 407
    geometry, 405
    kinematics, 407
    lamina coordinate systems, 406
    reduced constitutive equations,
    407
    strain-displacement matrix, 409
    stress resultants, 412
    Shifting, 574
    Singular elements, 175-82
    Skew-symmetric second-rank ten¬
    sor, 79
    Skyline, 633
    Slope, 50
    Small displacements superposed
    upon large, 104
    Sobolev imbedding theorems, 268
    Sobolev norms, 186, 266-67
    Sobolev spaces, 54, 267
    Sobolev’s theorem, 54
    Space of pressures, 198
    Spectral radius, 497
    Spectral stability, 497
    Spectral transformations, 575,
    583
    Spectrum slicing, 578
    Speed of sound, 510
    Spurious roots, 529
    Spurious zero-energy modes (see
    Rank deficiency)
    Square plates, 328-29, 337-39,
    356, 357, 359
    Stability, 462
    Stability polynomial, 524, 527
    Stability via the energy method:
    generalized trapezoidal meth¬
    ods, 471
    implicit-explicit algorithms, 559
    Newmark methods, 556
    predictor-corrector methods,
    557
    Standard element families,
    137-38
    Standard error estimate, 190
    Statically condensed elastic
    coefficient matrix, 103
    Static condensation, 246, 573
    Static load patterns, 574
    Stiffly stable, 526
    Stiffness matrix, 11
    Stokes flow, 193
    Strain-displacement equations. 76
    Strain energy, 187, 512
    Strain projection, 232, 236
    Strain tensor, 76
    Stress tensor, 76
    String on an elastic foundation,
    46
    Structural dynamics (see Hyper¬
    bolic problems)
    Structural dynamics algorithms:
    comparison, 532
    discussion, 535
    Sturm sequence check, 578
    Subspace iteration method, 577
    Superconvergence, 27
    Sylvester’s inertia theorem, 578,
    627
    Symmetric bilinear forms, 7
    Symmetric matrix, 12
    Symmetric second-rank tensor,
    79
    Taylor’s formula with remainder,
    27
    Temperature, 60
    Tetrahedral coordinates, 170
    Tetrahedral elements, 126,
    170-71
    Thermal expansion coefficients,
    105
    Thin plate, 313
    Three-node quadratic element,
    157-58
    Time-step estimates:
    linear beam elements, 515-17
    quadrilateral and hexahedral el¬
    ements, 517
    three-node quadratic rod ele¬
    ment, 514
    two-node linear heat conduction
    element, 515Index
    two-node linear rod element,
    513-14
  5. 342, 362
    Torsionless axisymmetric analy¬
    sis, 236
    Torsionless axisymmetric case,
    101
    Total energy, 512
    Total potential energy function,
    194
    Tractions, 77
    Transition element, 159-60
    Transverse displacement, 314
    Trapezoidal rule (see Average ac¬
    celeration method)
    Trial solutions, 3
    Trial vectors, 574, 585
    Triangular coordinates, 166
    Triangular elements, 121, 136,
    138, 139, 167-69, 180-81
    Tridiagonal matrix, 589
    Trilinear hexahedral element, 123
    Truncation errors, 496
    Tubes (see Shells of revolution,
    rings and tubes)
    Twist, 367
    Twisted ribbon, 351, 353, 354
    Twisting moment, 367
    Two-point boundary-value prob¬
    lems, 2
    Unconditional stability, 466
    Underconstrained, 300
    Underdamped, 524
    Unified single-step methods, 552
    Uniform reduced integration, 221,
    327, 414
    Unit outward normal vector,
    57-58
    Unit roundoff error, 595
    Unit step function, 25
    Upper triangular matrices, 636
    Variational crimes, 8
    81
    Variational equation, 4
    Variations, 4
    Virtual displacement principle, «
    78
    Virtual work principle, 4, 78
    Viscous damping, 426
    Von Neumann method, 479, 52:
    Wave equation, 427, 506
    Weak solution, 4
    Wedge-shaped elements, 125,
    171-72
    Weighted residual methods, 9
    Weighting functions, 4
    Wilson-0 method, 530
    Winkler foundation, 428
    Young’s modulus, 83
    Zero-energy modes (see Rank
    deficiency)

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