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Finite Element Analysis - Method, Verification and Validation
Second Edition
Barna Szabó
Washington University in St. Louis
Ivo Babuška
The University of Texas at Austin
و المحتوى كما يلي :
Contents
Preface to the second edition xi
Preface to the first edition xiii
Preface xv
About the companion website xvii
1 Introduction to the finite element method 1
1.1 An introductory problem 3
1.2 Generalized formulation 6
1.2.1 The exact solution 6
1.2.2 The principle of minimum potential energy 11
1.3 Approximate solutions 12
1.3.1 The standard polynomial space 13
1.3.2 Finite element spaces in one dimension 16
1.3.3 Computation of the coefficient matrices 17
1.3.4 Computation of the right hand side vector 20
1.3.5 Assembly 21
1.3.6 Condensation 24
1.3.7 Enforcement of Dirichlet boundary conditions 24
1.4 Post-solution operations 26
1.4.1 Computation of the quantities of interest 26
1.5 Estimation of error in energy norm 30
1.5.1 Regularity 30
1.5.2 A priori estimation of the rate of convergence 31
1.5.3 A posteriori estimation of error 32
1.5.4 Error in the extracted QoI 36
1.6 The choice of discretization in 1D 38
1.6.1 The exact solution lies in Hk(I), k − 1 > p 38
1.6.2 The exact solution lies in Hk(I), k − 1 ≤ p 39
1.7 Eigenvalue problems 42
1.8 Other finite element methods 46
1.8.1 The mixed method 47
1.8.2 Nitsche’s method 48
2 Boundary value problems 51
2.1 Notation 51
2.2 The scalar elliptic boundary value problem 53vi Contents
2.2.1 Generalized formulation 53
2.2.2 Continuity 55
2.3 Heat conduction 55
2.3.1 The differential equation 57
2.3.2 Boundary and initial conditions 58
2.3.3 Boundary conditions of convenience 59
2.3.4 Dimensional reduction 61
2.4 Equations of linear elasticity – strong form 67
2.4.1 The Navier equations 70
2.4.2 Boundary and initial conditions 71
2.4.3 Symmetry, antisymmetry and periodicity 72
2.4.4 Dimensional reduction in linear elasticity 73
2.4.5 Incompressible elastic materials 76
2.5 Stokes flow 78
2.6 Generalized formulation of problems of linear elasticity 78
2.6.1 The principle of minimum potential energy 80
2.6.2 The RMS measure of stress 82
2.6.3 The principle of virtual work 83
2.6.4 Uniqueness 84
2.7 Residual stresses 87
2.8 Chapter summary 89
3 Implementation 91
3.1 Standard elements in two dimensions 91
3.2 Standard polynomial spaces 91
3.2.1 Trunk spaces 91
3.2.2 Product spaces 92
3.3 Shape functions 93
3.3.1 Lagrange shape functions 93
3.3.2 Hierarchic shape functions 95
3.4 Mapping functions in two dimensions 97
3.4.1 Isoparametric mapping 97
3.4.2 Mapping by the blending function method 99
3.4.3 Mapping algorithms for high order elements 101
3.5 Finite element spaces in two dimensions 102
3.6 Essential boundary conditions 103
3.7 Elements in three dimensions 103
3.7.1 Mapping functions in three dimensions 105
3.8 Integration and differentiation 106
3.8.1 Volume and area integrals 106
3.8.2 Surface and contour integrals 107
3.8.3 Differentiation 108
3.9 Stiffness matrices and load vectors 109
3.9.1 Stiffness matrices 109
3.9.2 Load vectors 110
3.10 Post-solution operations 111Contents vii
3.11 Computation of the solution and its first derivatives 111
3.12 Nodal forces 113
3.12.1 Nodal forces in the h-version 113
3.12.2 Nodal forces in the p-version 115
3.12.3 Nodal forces and stress resultants 117
3.13 Chapter summary 117
4 Pre- and postprocessing procedures and verification 119
4.1 Regularity in two and three dimensions 119
4.2 The Laplace equation in two dimensions 120
4.2.1 2D model problem, uEX ∈ Hk(Ω), k − 1 > p 121
4.2.2 2D model problem, uEX ∈ Hk(Ω), k − 1 ≤ p 123
4.2.3 Computation of the flux vector in a given point 126
4.2.4 Computation of the flux intensity factors 128
4.2.5 Material interfaces 131
4.3 The Laplace equation in three dimensions 133
4.4 Planar elasticity 137
4.4.1 Problems of elasticity on an L-shaped domain 137
4.4.2 Crack tip singularities in 2D 139
4.4.3 Forcing functions acting on boundaries 142
4.5 Robustness 143
4.6 Solution verification 148
5 Simulation 155
5.1 Development of a very useful mathematical model 156
5.1.1 The Bernoulli-Euler beam model 156
5.1.2 Historical notes on the Bernoulli-Euler beam model 158
5.2 Finite element modeling and numerical simulation 159
5.2.1 Numerical simulation 159
5.2.2 Finite element modeling 160
5.2.3 Calibration versus tuning 163
5.2.4 Simulation governance 164
5.2.5 Milestones in numerical simulation 165
5.2.6 Example: The Girkmann problem 167
5.2.7 Example: Fastened structural connection 170
5.2.8 Finite element model 176
5.2.9 Example: Coil spring with displacement boundary conditions 180
5.2.10 Example: Coil spring segment 184
6 Calibration, validation and ranking 187
6.1 Fatigue data 187
6.1.1 Equivalent stress 188
6.1.2 Statistical models 189
6.1.3 The effect of notches 190
6.1.4 Formulation of predictors of fatigue life 190
6.2 The predictors of Peterson and Neuber 191viii Contents
6.2.1 The effect of notches – calibration 193
6.2.2 The effect of notches – validation 195
6.2.3 Updated calibration 197
6.2.4 The fatigue limit 199
6.2.5 Discussion 201
6.3 The predictor Gα 202
6.3.1 Calibration of ????(V, ????) 203
6.3.2 Ranking 204
6.3.3 Comparison of Gα with Peterson′s revised predictor 205
6.4 Biaxial test data 205
6.4.1 Axial, torsional and combined in-phase loading 206
6.4.2 The domain of calibration 208
6.4.3 Out-of-phase biaxial loading 210
6.5 Management of model development 218
6.5.1 Obstacles to progress 220
7 Beams, plates and shells 223
7.1 Beams 223
7.1.1 The Timoshenko beam 225
7.1.2 The Bernoulli-Euler beam 229
7.2 Plates 234
7.2.1 The Reissner-Mindlin plate 236
7.2.2 The Kirchhoff plate 240
7.2.3 The transverse variation of displacements 243
7.3 Shells 247
7.3.1 Hierarchic thin solid models 249
7.4 Chapter summary 254
8 Aspects of multiscale models 255
8.1 Unidirectional fiber-reinforced laminae 255
8.1.1 Determination of material constants 257
8.1.2 The coefficients of thermal expansion 258
8.1.3 Examples 258
8.1.4 Localization 261
8.1.5 Prediction of failure in composite materials 262
8.1.6 Uncertainties 263
8.2 Discussion 264
9 Non-linear models 265
9.1 Heat conduction 265
9.1.1 Radiation 265
9.1.2 Nonlinear material properties 266
9.2 Solid mechanics 266
9.2.1 Large strain and rotation 266
9.2.2 Structural stability and stress stiffening 270Contents ix
9.2.3 Plasticity 275
9.2.4 Mechanical contact 281
9.3 Chapter summary 287
Appendix A Definitions 289
A.1 Normed linear spaces, linear functionals and bilinear forms 289
A.1.1 Normed linear spaces 290
A.1.2 Linear forms 290
A.1.3 Bilinear forms 290
A.2 Convergence in the space X 291
A.2.1 The space of continuous functions 291
A.2.2 The space Lp(Ω) 291
A.2.3 Sobolev space of order 1 291
A.2.4 Sobolev spaces of fractional index 292
A.3 The Schwarz inequality for integrals 293
Appendix B Proof of h-convergence 295
Appendix C Convergence in 3D: Empirical results 297
Appendix D Legendre polynomials 301
D.1 Shape functions based on Legendre polynomials 302
Appendix E Numerical quadrature 303
E.1 Gaussian quadrature 303
E.2 Gauss-Lobatto quadrature 304
Appendix F Polynomial mapping functions 307
F.1 Interpolation on surfaces 308
F.1.1 Interpolation on the standard quadrilateral element 309
F.1.2 Interpolation on the standard triangle 309
Appendix G Corner singularities in two-dimensional elasticity 311
G.1 The Airy stress function 311
G.2 Stress-free edges 312
G.2.1 Symmetric eigenfunctions 313
G.2.2 Antisymmetric eigenfunctions 315
G.2.3 The L-shaped domain 315
G.2.4 Corner points 317
Appendix H Computation of stress intensity factors 319
H.1 Singularities at crack tips 319
H.2 The contour integral method 320
H.3 The energy release rate 321
H.3.1 Symmetric (Mode I) loading 322x Contents
H.3.2 Antisymmetric (Mode II) loading. 323
H.3.3 Combined (Mode I and Mode II) loading. 323
H.3.4 Computation by the stiffness derivative method. 323
Appendix I Fundamentals of data analysis 325
I.1 Statistical foundations 325
I.2 Test data 326
I.3 Statistical models 328
I.4 Ranking 335
I.5 Confidence intervals 335
Appendix J Estimation of fastener forces in structural connections 337
Appendix K Useful algorithms in solid mechanics 341
K.1 The traction vector 341
K.2 Transformation of vectors 342
K.3 Transformation of stresses 343
K.4 Principal stresses 344
K.5 The von Mises stress 344
K.6 Statically equivalent forces and moments 345
K.6.1 Technical formulas for stress 348
Bibliography 351
Index 357xi
Index
a
adaptive methods 38
ADINA 252
admissible functions 80, 84
Airy stress function 311
Almansi strain 181
analytic function 31, 289
angular velocity 43
antisymmetry 121
assembly 21
asymptotic consistency 237, 250
b
Babuska-Brezzi condition 47 ̆
basis function 3
basis vectors
curvilinear 247
Bayes factor 190, 335
Bayes’ theorem 325, 326
beam models 223
Bernoulli-Euler beam 229
Timoshenko beam 226
bending moment 235
Bernoulli-Euler beam model 156, 268
biharmonic equation 241, 311
bilinear form 7, 290
binormal 347
boundary condition 58
antisymmetric 9
antisymmetry 59, 72
convection 58
Dirichlet 8, 53, 54
displacement 71
essential 8, 53, 103
flux 58
homogeneous 53
kinematic 71
natural 9, 10, 53
Neumann 8, 53, 54
of convenience 53
periodic 9, 60, 72
radiation 265
Robin 9, 53, 54
symmetric 9
symmetry 59, 72
temperature 58
traction 71
Winkler spring 71
boundary layer 253
buckling load factor 273
c
calculus of variations 12
calibration 160, 187, 330
calibration set 193
Cauchy stress 181
censored data 328
coefficient
of convective heat transfer 58
of dynamic viscosity 78
of thermal conduction 57, 61, 64
of thermal conductivity 112
of thermal expansion 68
spring 71
coefficient of determination 194
coefficient of variation 194
coil spring 346
complex potentials 311
Finite Element Analysis: Method, Verification and Validation, Second Edition. Barna Szabó and Ivo Babuška.
 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.
Companion Website: www.wiley.com/go/szabo/finite_element_analysis358 Index
compliance matrix
elastic-plastic 279
condensation 24
condition number 13, 41
confidence interval 335
conservation law 56
consistency 160
consistency condition 277
constant
Stefan-Boltzmann 265
constraint
rigid body 84
contact 281
contact zone 281
continuity 45, 55
C1 243
contour integral 108
contour integral method 130
contour plot 112
convergence 291
monotonic 81
radius of 121
coordinate space 289
coordinates
Eulerian 266
Lagrangian 266
triangular 94
cross product 347
cumulative distribution function
329
marginal 207
cycle ratio 188
d
d’Alembert’s principle 70
data analysis
parametric 187, 325
degrees of freedom 17, 25
derivative
fractional 30
design rules 2
diameter of an element 16
differential volume 106
differentiation 108
dimension 51
dimensional reduction 61
dipole problem 172
director functions 223, 247
Dirichlet, see boundary condition 53
discretization 2, 39
display grid 112
divergence theorem 52, 54, 79
domain
of calibration 194
e
effectivity index 125
eigenfunction
Mode I, 313
Mode II, 313
eigenpair 43
eigenvalue 313
element
eight-node quadrilateral 94
four-node quadrilateral 93
nine-node quadrilateral 94
six-node triangle 95
three-node triangle 95
element stiffness matrix 161
emissivity 265
endurance limit 189, 331
energy norm 7, 54, 292
energy release rate 140, 321
energy space 7
equations of motion 70
equations of static equilibrium 70
equilibrium
dynamic 69
equivalence of norms 292
equivalent stress 188, 276
error
a priori estimate 296
model form 3
numerical 2
of approximation 3
pollution 151
error estimation 33
asymptotic 31
essential infimum 289
essential supremum 289
Euler-Lagrange equation 12
evidence 326
extension
p-version 151, 174Index 359
extraction 2
extraction function 27, 29, 117, 129, 179
f
fastener 337
fatigue limit 189, 191, 199, 330,
331
fatigue strength 200, 331
field functions 223
finite element 16
hexahedral 104
pentahedral 104
standard quadrilateral 91
standard triangular 91
tetrahedral 104
finite element mesh 16
finite element method 3
h-version 166
p-version 166
finite element model 2
finite element space
anisotropic 252
hierarchic 33
flank angle 192
flux
heat 56, 59
vector 53
flux intensity factor 128
forcing function 3, 142
Fourier’s law 56
free body 179
frequency 43
function
even 60
odd 60
functional
quadratic 11
g
Galerkin orthogonality 13
gap elements 283
generalized form 1
generalized solution 10, 55
uniqueness of 84
Girkmann problem 167
global number 23
grading factor 16
h
h-extension 17
h-version 17
hierarchic spaces 150
highly stressed volume 203
Hilbert space 292
Hooke’s law 69, 158
generalized 69
hourglassing 162
hp-extension 17
hp-version 17
i
incompressibility
condition of 77
index notation 51
infimum 289
initial condition 58, 72
integration 106
interpolation 307
isoparametric mapping, see mapping 97
j
Jacobian
determinant 106
matrix 106, 112
inverse of 108
k
Kronecker delta 51
l
L-shaped domain 315
Lagrange polynomial 13
Lamé constants 69
Laplace equation 120
law of parsimony 175
Lebesque constant 105, 307
legacy codes 168
Legendre polynomial 13, 97, 301, 303
likelihood function 190, 330
likelihood ratio 198
linear form 7, 290
linear independence 5
linear space 290
Lobatto point 20360 Index
locking 143, 253
shear 229
log likelihood 330
m
mapping
by blending functions 99, 307
high order elements 101
improper 107
inverse 100, 112
isoparametric 97, 101
linear isoparametric 97
proper 106
quadratic isoparametric 97
subparametric 97
superparametric 97
mapping function 17
marginal likelihood 326
marginalization 326, 332
material stiffness matrix 81
mathematical model 2, 155
hierarchic models 223
range of validity 159
Matlab 330
matrix
Gram 19
mass 19
stiffness 18
maximum likelihood 189
maximum norm 307
mean optimal set 252, 308
median 196
membrane analogy 1
membrane force 234
mesh
geometric 16
quasiuniform 16, 280
radical 16
uniform 16
Mode I eigenfunctions 313
model form error 171
model hierarchy 265
modulus of elasticity 68, 157
modulus of rigidity, see shear modulus
69
Mohr circle 236
moment of inertia 349
n
NASTRAN 165
Navier equations 70
Neumann, see boundary condition 53
neutral axis 157
neutral plane 157
Newton-Raphson method 100, 112
Nitsche’s method 48
nodal displacement 161
nodal force 29, 113, 161
nodal set 307
node point 16
norm 290
L
2 166
energy 166
notch sensitivity 192, 194
notch sensitivity factor 202
nuisance parameter 326
numerical error 2
numerical simulation 2
p
p-convergence 31
p-extension 17
p-version 17
path-independent integral 129
permutation symbol 52
physical reality 155
physics-based model 156
piecewise polynomials 243
plane strain 74
plane stress 237
plasticity 275
plates
Kirchhoff plate 240
Reissner-Mindlin plate 236
Poisson’s equation 349
Poisson’s ratio 68
polar moment of inertia 350
pollution error, see error 151
posterior probability 326
potential energy 55, 80
sign of 12
potential flow 55
predicted median 333
predictor of fatigue life 187
pressure 78Index 361
principal direction 59
principal moments 236
principal stress 344
principal structural element 164
principle of minimum potential
energy
application to beams 224
application to plates 238
principle of virtual work 83
prior probability 326
probability
Bayesian 325
frequentist 325
product rule 325, 332
product space 92, 95
anisotropic 250
profile likelihood 335
propping effect 172
pull-back polynomials 98
q
QoI, see quantity of interest 2
quadrature 107, 303
Gauss-Lobatto 303, 304
Gaussian 303
qualified test record 193
quantile 333
quantity of interest 2, 26, 155, 170, 171, 187,
265
quarter point element 99
quasiuniform, see mesh 16
r
radiation 58
random fatigue limit 189, 330
ranking 187
rate of convergence 148
algebraic 31
exponential 32
Rayleigh quotient 44
reduced integration 162
reduction factor 274
region of primary interest 150, 239
region of secondary interest 150
regularity 16, 30
representative volume element 191, 255
residual stress 87
resolvent set 271
Reynolds number 78
rigid body
displacement 84
rotation 84, 102, 269
Robin, see boundary condition 53
robustness 143
rod 348
runout 188, 328, 330
s
S-N curve 189, 330
S-N data 189, 328
Saint-Venants’s principle 185
scalar product 52
Schwarz inequality 37, 293
scope
of calibration 202
of validation 205
secant modulus 278
seminorm 290
separation of variables 42
shadow function 137, 141
shape function 13, 161
hierarchic 15
shape function vector 109
shape functions
3-dimensional 104
Lagrange 93
shear correction factor 226, 227, 237, 238
shear force 234
shear modulus 69
shearing tractions 349
shell
hierarchic models 247
hyperboloidal 252
structural 247
shell model
Naghdi 248
Novozhilov-Koiter 249
shift theorem 32
simulation 1, 155
simulation governance 164, 167, 193, 205,
219
smart application 186
Sobolev space 291
solution verification 168362 Index
solver
direct 26
iterative 26
space
energy 79
Euclidean 51
finite element 17, 102
infinite-dimensional 7
span 5, 12
spectrum 43, 271
continuous 271
point 271
spring rate 177
square integrable function 291
stability 47, 160
stability parameter 49
standard element 103
statical equivalence 345
stationary problems 57
statistical inference 325
steady state problems 57
Steklov method 133
stiffness matrix 18
elastic-plastic 279
Stokes equations 78
strain 157
Almansi 267
enginering shear 68
equivalent elastic 276
equivalent plastic 276
Green 268
infinitesimal 67
mechanical 69
normal 67
shear 67
thermal 69
total 69
volumetric 76
stress
transformation of 344
resultant 226
stress concentration factor 190
fatigue 191
stress deviator tensor 344
stress intensity factor 139, 321
generalized 153
stress invariant 202
stress stiffening 181, 271
stress vector 341
StressCheck 51, 280
superconvergence 130, 182
supremum 289
surface finish 326
surface integral 108
survival function 207
symmetry 121
axial 65
t
T-stress 139, 319
Taylor series 35
technical formulas 185
temperature
absolute 58, 265
test function 80, 290
test section 188
test space 8, 9, 290
test stress 188
thin solid model 249, 268, 273
traction 71
traction vector 341
trial function 290
trial space 7, 9, 290
trunk space 91, 95
anisotropic 249
truss 160
tuning 164
twisting moment 235
u
uncertainty 78
aleatory 163
uniqueness 9
v
validation 166, 187, 195
validation metric 205
validation scenario 196Index 363
validation set 193, 196
variational crime 162
vector
transformation of 342
verification 148, 166
data 119
solution 152
virtual displacement 83
virtual experimentation 265
virtual work 83
viscosity 78
von Mises stress 184, 189, 202, 276
von Mises yield criterion 345
w
Winkler spring, see boundary condition 71
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