The Finite Element Method – Linear Static and Dynamic Finite Element Analysis
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
- Planar Truss 691
- Static Analysis of a Plane Strain Cantilever
Beam 705 - Dynamic Analysis of a Plane Strain Cantilever
Beam 705 - 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 - 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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