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 |  موضوع: كتاب Finite Element Analysis Applications - A Systematic and Practical Approach الأربعاء 14 أغسطس 2024, 1:46 am | |
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أخواني في الله
أحضرت لكم كتاب
Finite Element Analysis Applications - A Systematic and Practical Approach
Zhuming Bi
Purdue University Fort Wayne,
Indiana, United States
و المحتوى كما يلي :
Table of Contents
1. Overview of Finite Element Analysis Methods
2. Decomposition
3. Description of Elements
4. Elemental Modelling
5. System Analysis and Modelling
6. Solutions to System Models
7. Integrated Computer Design Environments
8. Solid Mechanics Problems
9. Heat Transfer Problem
10. Flow Dynamics
11. Multi-Physics Simulations
12. Verification and Validation
Index
‘Note: Page numbers followed by “f” indicate figures, “t” indicate tables and “b” indicate boxes.’
A
Abaqus software, 229t
Active interactions, 31e32
p-Adaptive meshing, 270
Addition, 196e198
ADT. See Axiomatic design theory
(ADT)
Agros2D, 230te231t
Air-quenching
process, 425e430
for aluminum parts, 425f
challenges of air-quenching
simulation, 425e426
exemplifying outputs, 428
exemplifying results from parametric
study, 428e430, 431fe432f
simulation model, 426e428, 427f
Airflow characterization, 426
ALE mesh. See Arbitrary Lagrangian
Eulerian mesh (ALE mesh)
Aleatory uncertainty, 485
Algebra function, 77e78
Allow penetration contact condition,
157, 158t
Amount of computation, 49, 150, 211,
426
Analysis models
dimensions of analysis model,
150e153
models with linear or nonlinear
elements, 154
semisteady models, 154
single-or multiphysics, 153
steady models, 154
transient models, 154
types of, 149e155
what-if models and design studies,
154e155
Analytical method, 13e14
Analytical solution, 113e114
ANSYS software, 229t, 472be474b
ADPL, 480
exemplifying analysis types in, 412t
workbench, 233e234, 233f
Antisymmetric matrix, 192
apply_BCs () in C1DSpringModel, 246,
251t
Approximation approach, 22, 72e74
1-point approximation, 100
2-point approximation, 100e104
Arbitrary Lagrangian Eulerian mesh
(ALE mesh), 436, 437f
Architecture, 237
Aspect ratio, 52, 472be473b, 474f,
474b
Assembling process, 35e36, 37t
Attributes, 234e235, 238e239
Automatic meshing module, 268
Axiomatic design theory (ADT), 40e45
designer’s role in formulating FEA
model, 45f
example of coupled problem, 44f
example of decoupled engineering
problem, 43f
for system complexity decomposition,
41f
Axiomatic Design Theory, 4e5
Axisymmetric elements, models with,
153
B
Back substitution, 205e206
Backward substitution, 207e208
Banded matrix, 192
BCs. See Boundary conditions (BCs)
Beam elements, equivalent loads of,
167e171
Beam members, 47e48, 152
Bearings, 281
Benchmarking, 475e477
application, 488e489
for validation, 488e489, 489f
Bernoulli’s equation, 385
Biot number, 366
“Bonded” contact condition, 158t,
269e270
Bottom-up approaches, 15e16, 16f
Boundary conditions (BCs), 113e114,
122, 155e166, 426
applying, 233e234
of biomedical materials, 158be159b
constraints in assembly, 157e159
contact conditions and examples, 158t
on displacements of single object,
155e157
equivalent nodal loads, 160e166
and loads, 290e291, 291f
on node of beam member, 156t
on symmetric plane, 156be157b
tools for, 270e273
in 2D domain, 160f
2D solid domain, 163be164b
validation of, 480e483
Boundary element methods, 14e15
Boundary value problem, 122, 148
Bulky objects, 268, 268f
C
C1DPartDomain class, 247te249t
C1DSpringModel class, 247te249t
apply_BCs () in, 251t
create_sys_model () in, 250t
C3DPartDomain class, 259te260t
C3DTrussElement class, 259te260t
C3DTrussModel class, 259te260t
C3DTrussNode class, 259te260t
CAD. See Computer aided design
(CAD)
CAE. See Computer aided engineering
(CAE)
Calculation verification, 468e471, 469f,
469be471b
CalculiX, 230te231t
CALFEM. See Computer aided learning
of finite element method
(CALFEM)
Calibration of FE model, 478
Cartesian CS, 62
CBoundaryDoF class, 239, 242, 242f
CBoundaryLoad class, 239, 242, 242f
CDomain class, 239, 241
subclasses, 241f
CEigenSolver, 244
CElement class, 239, 243e244,
243f
Central processing units (CPUs), 237
Centrifugal loads, 272t
CEProperties class, 239, 244, 244f,
259te260t
CEquivalenceSolver, 244
CEuivalenceSolver, 245
CFD. See Computational fluid dynamics
(CFD)
CFFixedDoF class, 247te249t
495Characteristic equation of system
model, 212
Child class, 238e239
Circular patterns, models for, 153
Classes, 234e235, 238e239
for one-dimensional spring systems,
247te249t
3D truss systems, 256, 259te260t
Clean-up process, 48e49, 49f
Clutches, 281
CMaterial class, 239, 241
subclasses, 241f
CModel class, 239e241
subclasses of root, 240f
CMultiPointDOF, 242
CNode class, 239, 243, 243f
Code Aster, 230te231t
Code verification, 465e468, 466f, 467b,
468f, 468b
Coefficient of friction (COF), 437e438,
448
Coherent multiphysics system, 407
Collocation methods, 124, 126
Commercial FEA packages, 228
Compatible mesh, 269e270, 269f
Completeness condition, 475
Composite beam, 417f, 417b, 418f
Computational methods, 13e14
Computational model, 463e464
Computational fluid dynamics (CFD),
379, 424
Computer aided design (CAD), 7,
13e14, 262, 478
commercial CAD/CAE software
tooleSolidWorks, 262e275
model, 46
tool, 10e11
Computer aided engineering (CAE), 7,
11e14, 262,
479
commercial CAD/CAE software
tooleSolidWorks, 262e275
tool to overcoming limitations of
human designers, 12f
Computer aided learning of finite
element method (CALFEM),
230te231t
Computer implementation
commercial CAD/CAE software
tooleSolidWorks, 262e275
computer programming, 227e237
of 1D spring systems, 245e246
applications of FEA program for,
246e256
static structure of FEA computer
implementation, 239e245
of 3D truss systems, 256e262
UML for OOP, 238e239
Computer programming, 227e237
analysis types in ANSYSs workbench,
233f
commercial and open access FEA
packages, 228
computer programs, 228e234, 231f
Matlab as programming language,
235e236
programming techniques, 234e235
system architecture, 237
Computers in engineering design, 7e11
characteristics of solution spaces and
DV, 9t
comparison of human designers and
computers, 10t
design activities types, 9t
example of computer aided synthesis,
11f
human designers and computers in
engineering design, 8f
Computing programs, 227
COMSOL Multiphysics software, 229t
Concentrated load, 283
Conceptual model, 462
Conduction, 342e343
Conjugate heat transfer, 19, 421e430
air-quenching process, 425e430
numerical solution to, 422e424
Constraints in assembly, 157e159
Contact conditions and examples, 158t
Contact model, 445
Continual models, 274
Continuity
condition, 475
equations, 384, 421
Convection, 342
Convergence
of energy, 475
study, 475
completeness condition, 475
continuity condition, 475
Coordinate systems (CS), 24e25, 61
GCS, 61e63, 63f
LCS, 61e63, 63f
natural CS, 69e72
transformation
in one-diemsnional space, 64e65
in three-dimensional space, 68e69
in two-dimensional space, 65e67
types, 61e62, 62f
Coordinate transformation, 63e69
coordinate transformation of GCS,
63e64
from LCS to GCS, 64e69
Coupled decomposition, 42
CPointLoad class, 247te249t
CPostPro class, 239
CPostProc class, 245, 245f
CPUs. See Central processing units
(CPUs)
Crack growth, 320e321
Crack initiation, 320e321
create_sys_model () in C1DSpring
Model, 246, 250t
CRs. See Customers requirements (CRs)
CS. See Coordinate systems (CS)
CSinglePointDoF, 242
CSolver class, 239, 244, 244f
CSpringElement class, 245e246,
247te249t
CSpringMaterial class, 245e246,
247te249t
CSpringNode class, 245e246,
247te249t
CSpringPostProc, 245
CTransSolver, 244
CTrussMaterial class, 259te260t
Customers requirements (CRs), 1e3
Cylindrical CS, 62e63
D
DAEs. See Differential algebraic
equations (DAEs)
Darcy’ law, 398e399
Data and data processing, 234e235
Deal. II, 230te231t
Decomposition, 21, 23e24, 31, 46e56,
149
ADT, 41e45
FEA as problem-solving tool, 39e41,
40f
mesh/meshing
control, 55e56, 56fe57f
and element types, 49e50
methods, 52e55
quality, 50e52
model preparation, 47e49, 48f
496 Indexmodular architecture of FEA code,
45e46
system
complexity, 33e39
description, 31e32, 32f
Decoupled decomposition, 42
Defeature, 47
Degrees of freedom (DoF), 268
Delaunay
property, 54e55
triangulation, 54e55
Dependent variables, 32
Design analysis, 6e7, 11e13, 154e155
Design optimization, 275
Design parameters (DP), 41
Design problem, 3e4, 4t
Design studies, 154e155
Design study in SolidWorks Simulation,
275
Design synthesis, 5e6, 6f, 11e13,
154e155
Design variable (DV), 4e5, 7, 32
Determinant of matrix, 193e195
Deterministic model, 273
Deviatoric stresses, 381,
386
Diagonal matrix, 190
Differential algebraic equations (DAEs),
407
Differential equations types, 132e135
Differentiation of matrix, 195
Dimensionless natural CS, 61, 94,
97e98
Dimensions of analysis model,
150e153
with 1D truss/beam elements, 152
with axisymmetric elements, 153
exemplified models by 1D or 2D
elements, 151te152t
for objects with symmetric plane(s) or
circular patterns, 153
with plane stress/strain elements, 153
with shell elements, 152
Direct formulation, 116e118, 119t
Direct methods, 22, 204
Direct Sparse, 274
Dirichlet BC, 160
Discrete models, 274
Discretization, 73, 462e463, 462f
Distributed and unified numeric
environment (DUNE),
230te231t
Divide and conquer strategy, 14e15,
20e21, 25e26
DoF. See Degrees of freedom (DoF)
DP. See Design parameters (DP)
Drop test, 271t
DUNE. See Distributed and unified
numeric environment (DUNE)
DV. See Design variable (DV)
Dynamic analysis, 266, 271t
Dynamic loads, characteristics of, 327t
Dynamic models, 273e274
Dynamics behaviors, 37, 38f
E
EHL. See Elastohydrodynamic
lubrication (EHL)
Eigenvalue problems, 211e221
element modeling of, 142e145
rectangle element, 143e144
triangle element, 144e145
natural frequencies from subspace
iteration, 216e219
PDEs for, 134e135
range of natural frequencies, 213e214
transformation method, 220e221, 222t
transient problems, 222
Eigenvalue system, 311
Elastohydrodynamic lubrication (EHL),
437e449
case study, 441e449
friction characteristics and design
factors, 438e440
mathematic models, 440e441
Electric circuit
calculating voltage, 254t
example, 253e255, 253f
Elements, 20e21, 24e25
description, 61
coordinate transformation, 63e69
CS types, 61e62
GCS, 62e63
interpolation, 72e77
LCS, 62e63
natural CS and shape functions,
69e72
nodes and, 61
numerical integration, 97e106
one-dimensional quadratic and cubic
elements, 77e84
in two-dimensional elements,
84e97
modeling process, 113, 233e234
of 2D equilibrium problems,
135e139
of 2D propagate problems, 139e141
differential equations types,
132e135
direct formulation, 116e118, 119t
of eigenvalue problems, 142e145
governing equations of engineering
problems, 113e116, 114te115t,
116f
minimum potential energy principle,
118e122
weighted residual methods, 122e130
types, 49e50
validation, 480
Elliptic PDEs, 132e134
Elmer, 230te231t
Endurance limit, 323
Energy
conservation model, 386e388,
421e423
convergence of, 475
equation, 421
Engineering, 13
design process, 1e7
computers in, 7e11
design analysis, 6e7
design problem, 3e4, 4t
design synthesis, 5e6, 6f
implementation of solution, 7
solution space, 4e5, 5f
steps and activities in design
process, 2f
governing equations of engineering
problems, 113e116
judgment, 488
process, 1e3
Enthalpy, 387e388
Epistemic uncertainty, 485e488
Equilibrium problems, PDEs for,
133e134
Equivalent loads, 166e176
of beam elements, 167e171
determining vector of equivalent nodal
loads, 173be174b
loading conditions on cantilever beam,
170be171b
reactional forces of beam under
pressure, 167f
of rectangle elements, 174e176
for transverse loads on beam, 170t
of triangle elements, 171e174, 172f
Index 497Equivalent nodal loads, 160e166
rectangle elements, 161e164, 162f
triangle elements, 165e166
Error(s)
quantification, 459e460
sources, 459e464
error quantification, 459e460
errors of idealization, 462e463
errors of mathematic models,
463e464, 464f
errors of model or analysis types,
464
system inputs, 461
Essential boundary conditions (Essential
BC), 122, 160, 481e483
Euler equations, 407
Euler parameter, values of, 367
Eulerian mesh, 436
Experimental methods, 13e14
Explicit models, 274
External loads, 154e155, 156be157b,
166. See also Equivalent loads
F
Failure diagnose, 20
Fast Finite Elements (FFEPlus), 274
Fasteners, 35e36, 47, 281
Fatigue analysis, 19, 37, 266, 271t,
320e334. See also Modal
analysis
case study of, 323e334
characteristics of dynamic loads,
327t
dimensions of rod, 328t
failure modes, stresses, and safety
factors, 332t
FEA simulation on new design, 329,
334t
FEA simulation on original design,
327e329, 333t
preparation of model data, 325e327
tie rod as two-force member, 331f
verification of FEA simulation,
330e334
linear elastic fracture mechanics
method, 321e322, 322f
selection of fatigue analysis methods,
323
strain-life method, 321
stress-life method, 323
Fatigue failure, 271t, 321e322, 325f
Fatigue strength, 323, 338e339
FBDs. See Free body diagrams (FBDs)
FEA. See Finite element analysis (FEA)
FEAP, 230te231t
FEATool, 230te231t
FEBio. See Finite elements for
biomechanics (FEBio)
FFEPlus. See Fast Finite Elements
(FFEPlus)
Finite difference methods, 14e15
Finite element analysis (FEA), 1,
13e20, 31, 97e98, 114e116,
187, 227, 262, 281e282, 410,
455, 460
advantages and disadvantages, 20, 21t
applications, 17e20, 18f
of FEA program for 1D spring
systems, 246e256
approach, 61
CAE, 11e13
classification of mathematic models
and application of modeling, 273f
classification of problem solving tools,
12f, 13e15
commercial FEA packages, 228
computers in engineering design, 7e11
data and methods in procedure, 232f
engineering design process, 1e7
general modeling procedure, 20e22,
21f, 23f
hardware and software architecture,
237f
methodology, 235
model, 33, 33f, 149
of heat transfer problems, 347e349
nodes and elements in, 354t
preparing inputs for, 444
for structural analysis, 283, 283f
modular architecture of FEA code,
45e46
open access FEA packages, 228,
230te231t
organization of book, 23e26, 24f
planning of V&V in modeling,
458e459, 458f
as problem-solving tool, 39e41, 40f
simulation
on new design, 329, 334t
on original design, 327e329, 333t
simulation results of rod, 330f
verification of, 330e334
software tools, 229t
solutions, 455e456
static structure of FEA computer
implementation, 239e245, 240f
top-down and bottom-up approaches,
15e16
Finite elements, 15, 49
Finite elements for biomechanics
(FEBio), 230te231t
First law of thermodynamics, 341
Fixed geometry restraints, 158be159b,
272t
Fixed Hinge restraints, 272t
Flip algorithm, 55
Flow pressure (FP), 385
Flow simulation, 19, 271t
Fluid flow, 380
Fluid mechanics, 26, 379, 435. See also
Solid mechanics problems
as branch of continuous mechanics,
380f
energy conservation model, 386e388
groundwater problems, 398e401
mathematical models, 380e386
pipe network, 388e392, 389fe390f
2D incompressible and irrotational
flow, 392e398, 394fe395f
Fluid movement, 379
Fluid structural interaction (FSI),
19e20, 431e449
EHL, 437e449
governing equations of FSI, 434e436
fluid mechanics, 435
fluidestructure interactions,
435e436
structural dynamics, 434e435
mesh adaptation, 436e437
problem and discretized domains,
433f
Force loads, 272t
Forced airflow, 425
Forward iteration, 214
Forward reduction, 205
Forward substitution, 207
Fourier number, 366
Fourier’s law, 343
FP. See Flow pressure (FP)
FR. See Functional requirements (FR)
Frame members, 152
Free body diagrams (FBDs), 253, 459
Free-stream velocity, 394e398
FreeFem++, 230te231t
Frequency analysis, 271t
Friction
498 Indexcharacteristics and design factors,
438e440
normal loads, 439
shapes, 440
surface finish, 440
temperature, 440
viscosity, 439e440
from contact and fluid flow, 448
FSI. See Fluid structural interaction
(FSI)
Functional requirements (FR), 4e5, 31
G
Galerkin methods, 26, 124, 129e130,
138
Gap-load curve determination, 445e446
Gaussian elimination, 204e206,
205be206b
GausseLegendre integration, 101, 102t,
103be104b
Generalization, 238e239
Geometric models, 478e479
Geometric shapes, 39
GetFEM++, 230te231t
Global coordinate system (GCS),
61e63, 63f, 233e234
coordinate transformation from LCS to
GCS, 64e69
coordinate transformation of, 63e64
Goodman’s equation, 332
Graphic methods, 13e14
Graphic user interfaces (GUIs), 262
Gravity, 272t
Groundwater
flow, 398e399
problems, 398e401
GUIs. See Graphic user interfaces
(GUIs)
H
H-adaptive meshing, 270
h-refinement, 77e78
Heat
conduction, 343
convection, 343
radiation, 344
transfer analysis, 19
Heat transfer, 341e342
boundary conditions in heat transfer
problems, 346f
examples in applications, 342f
FEA modeling of heat transfer
problems, 347e349
governing equations of, 342e346
conduction, 343
heat convection, 343
heat radiation, 344
mathematic models, 344e346
1D steady heat transfer problems,
349e354, 350f, 352f
problems, 26
transient heat transfer problems,
366e374, 368f, 370f
2D steady heat transfer problems,
355e365
Heat transfer coefficient (HTC), 426
Helmholtz equation, 142
Hermes Project, 230te231t
HTC. See Heat transfer coefficient
(HTC)
Human designers, 11
Hybrid mesh, 49e50
Hyperbolic PDEs, 132
I
Idealization, 47e48
errors of, 462e463
of solid model, 232e233
Identity matrix, 190
IGES. See Initial Graphics Exchange
Specification (IGES)
Immovable restraints, 272t
Implicit models, 274
Incompatible mesh, 269e270, 269f
Incremental algorithm, 55
Independent variables, 22, 32
Inertia loads, 327t
Information subsystem, 4e5
Information technology (IT), 12e13,
234e235
Initial Graphics Exchange Specification
(IGES), 264t
Insulated boundary, 346
Integral operation
over one-dimensional domain, 98e104
integral using 1-point approximation,
100
integral using 2-point approximation,
100e104
over two-dimensional domain,
104e106
Integration
integrated design, 7, 9t
integrated engineering analysis
environment, 25e26
of matrix, 195e196
Interfaces at contacts, 34e35
Interference detection tool, 268e269
Interpolation, 72e77
example of state variables, 73t
in one-dimensional linear element,
74e77
Taylor expansion, 73e74
in two-dimensional elements,
84e97
Intrinsic energy, 387
Inverse iteration, 214, 215t
Irreducible uncertainty, 485
IT. See Information technology (IT)
Iterative methods, 204, 208e211,
210be211b
J
Jacobi’ method, 210
Jacobian check, 52, 54f
Jacobian ratio, 52
Kk-ε
model, 423e424
Kinetic energy (KE), 385
Kronecker delta, 381
L
Lagrange interpolation functions, 84
Lagrange’s equation, 313
Lagrangian mesh, 436
Large Problem Direct Sparse (LPDS),
274
Lattice analogy, 17
Law of heat conduction, 343
Laws of thermodynamics, 341
LCS. See Local coordinate system
(LCS)
Least square methods, 124, 128e129
Linear elastic fracture mechanics
method, 321e322, 322f
Linear elements, models with, 154
Linear model(s), 273
analysis, 18
Linear system model, 203e204
Linear truss member, 287
Load-sharing principle, 447
Local coordinate system (LCS), 61e63,
63f, 120
Index 499Local coordinate system (LCS)
(Continued)
coordinate transformation from LCS to
GCS, 64e69
2D beam member in, 314e315, 314f
2D truss member in, 312e314, 313f
Local mesh control, 268
LPDS. See Large Problem Direct Sparse
(LPDS)
LS-DYNA software, 229t
LU decomposition, 206e208, 208b
M
Maclaurin series. See also Taylor
expansion
Manufacturing process, analysis of, 19
MasoneCoffin relation of fatigue life,
321
Mass conservations, 421
equation, 382e384
in internal fluid flow, 383f
Materials, 478
library, 264e266, 266f
Web Portal, 266
Mathematic/mathematical models, 25,
344e346, 380e386, 440e441
errors of, 463e464, 464f
mass conservation equation, 382e384,
383f
momentum conservation equation,
384e386
Matlab, 245
as programming language, 235e236
Matrices, 187e203
computation in matrix manipulation,
211, 212t
operations, 196e203
addition, 196e198
multiplication by scalar, 200
partition of matrix, 202e203
product of matrices, 201
subtraction, 196e198
transpose of combined matrices,
198e200
transpose of matrix with product of
scalar, 201
transpose of product of two matrices,
202
properties, 192e196
determinant of matrix, 193e195
differentiation of matrix, 195
integration of matrix, 195e196
trace of matrix, 193
transpose of matrix, 192e193
types, 190e192, 191f
Mechanical design, 26, 281
Mechanical subsystem, 4e5
Mesh
adaptation, 436e437
control, 55e56, 56fe57f
quality, 50e52
refinement, 270
types, 49e50, 50f, 51t
Meshing, 232e233
methods, 52e55, 53t
process, 47, 49
verification, 471e474, 472f,
472be474b
Minimum potential energy
methods, 22
principle, 118e122
Minor’s Rule, 323, 332e333
Modal analysis, 19, 311e319. See also
Fatigue analysis
examples of products, 312f, 320f
modeling of 2D frame element,
315e316
2D beam member in LCS, 314e315,
314f
of 2D structure with frame members,
317e319
2D truss member in LCS, 312e314,
313f
Modal validation, 478e484
assembly of parts or components, 480
assembly verification, 481fe482f
of boundary conditions, 480e483
detailed features, 479
of element types, 480
geometric models, 478e479
materials, 478
singularity of fixed supports, 484, 484f
symmetry, 483, 483f
Model data preparation, 325e327
Model template, 149
Model tuning, 478
Model updating, 478
Modular programming, 234e235
Module-based design, 15
Momentum conservations, 421
equation, 384e386
Momentum equation, 421
Monolithic approach, 410
Monolithic method, 433
Multiphysics, 153
problems, 37e38, 38f
simulation, 26
systems, 407
classification of, 410f
conjugate heat transfer, 421e430
FSI, 431e449
mathematic models, 407e409
simulation models of multiphysics
systems, 410e411
structural-thermal analysis, 412e420
Multiplication by scalar, 200
N
NASA Structural Analysis
(NASTRAN), 17
Nastran software, 229t
Natural BC, 122, 160
Natural CS, 69e72
two-dimensional linear triangle element
under, 94e95
Natural frequencies
range, 213e214
from subspace iteration, 216e219
NaviereStokes equations, 379, 407,
422e423
Neumann boundary conditions, 160,
480e481
Newton’s law, 343
Newtonian fluid flow, 382, 439e440
“No penetration” contact condition,
157, 158t, 158be159b,
269e270
Nodes, 20e21, 61
Nominal load, 327t
Non-Newtonian fluid flow, 382,
439e440
Nonlinear analysis, 271t
Nonlinear elements, models with, 154
Nonlinear fluid flow, 382
Nonlinear model, 18, 273
Nonlinear system model, 203e204
Normal loads, 439
Null matrix, 190
Numerical algorithm
adequacy verification, 467
property verification, 467
Numerical integration, 97e106
integral operation over two-dimensional
domain, 104e106
integral over one-dimensional domain,
98e104
500 IndexNumerical methods, 13e15, 236
Numerical simulation, 113e114, 467
Numerical solution to conjugate heat
transfers, 422e424
O
Object Oriented Finite EleMent solver
(OOFEM), 230te231t
Object-oriented approach
(OO approach), 25e26
Object-oriented programming (OOP),
227, 234e235, 241
procedural programming vs., 236t
UML for, 238e239
ODE. See Ordinary differential equation
(ODE)
One-dimension (1D)
CS transformation in one-dimensional
space, 64e65
cubic elements, 77e78, 83e84
integral over one-dimensional domain,
98e104
interpolation in one-dimensional linear
element, 74e77
models with 1D truss/beam elements,
152
quadratic elements, 77e83
spring systems, 245
applications of FEA program for,
246e256
apply_BCs () in C1DSpringModel,
251t
computer implementation of,
245e246
create_sys_model () in C1DSpringModel, 250t
definition of classes for,
247te249t
displacement of spring assemblage,
252t
electric circuit example, 253e255,
253f
equivalent stiffness coefficient (k) of
physical systems, 251t
example, 246e253
pipe flow network example,
255e256, 255f
spring assemblage example, 252f
steady heat transfer problems,
349e354, 350f, 352f
system models from 1D elements,
176e179
OO approach. See Object-oriented
approach (OO approach)
OOFEM. See Object Oriented Finite
EleMent solver (OOFEM)
OOP. See Object-oriented programming
(OOP)
Open access FEA packages, 228
OpenFOAM, 230te231t
Operation forces, 327t
Optimization, 275
Order of PDE, 132
Ordinary differential equation (ODE),
475e476
Original design, 7, 9t
Overload failure, 320e321
Pp
-refinement, 77e78
Parabolic PDEs, 132
Parameter calibration, 478
Parametric design, 7, 9t
Parent class, 238e239
Partial differential equations (PDEs),
15, 39e40, 113, 122, 132, 132t,
475e476, 485e488
exacting and weak solutions to,
122e123
types, 132e135
for eigenvalue problems, 134e135
for equilibrium problems, 133e134
for propagation problems, 134
Partition of matrix, 202e203
Partitioned approach, 410
Partitioned matrix, 192
Partitioned method, 433
Passive interactions, 31
PE. See Potential energy (PE)
Phreatic zone, 398e399
Pipe flow network example, 255e256,
255f
Pipe network, 388e392, 389fe390f
Plane strain, 299e300
dam as plane strain model, 309f
examples of parts, 308f
problems, 306e311
stress state in plane strain model, 308f
Plane stress, 299e306, 300fe301f,
303f, 306f, 311b
models with plane stress/strain
elements, 153
Poisson effect, 484
Poisson equations, 114e116, 116f
Postprocessing phase, 22, 228e232,
275
Potential energy (PE), 384e385
Potential function, 393
Power method, 214, 216, 220
Preprocessing phase, 22, 228e234
Pressure, 272t
Probabilistic model, 273
Probability Bound, 486b
Problem solving tools
classification of, 12f, 13e15
FEA as, 39e41, 40f
Procedural programming, 234e235
OOP vs., 236t
Processing phase, 22, 228e232
Product of matrices, 201
Programming techniques, 234e235
evolution, 234f
Propagation problems, PDEs for, 134
Q
Quenching processes, 19
R
Radiation, 342
Random access memory (RAM), 274
RANS equations. See Reynolds
Averaged NaviereStokes
equations (RANS equations)
Rayleigh quotient, 213e214
Rectangle element(s), 161e164, 162f.
See also Triangle element
eigenvalue problems, 143e144
equivalent loads of, 174e176
modeling of, 357e360
2D equilibrium problems, 136e138
Redesign, 7, 9t
Reducible uncertainty, 485e488
Remote loads, 272t
Remote restraints, 272t
Residual, 123
Residual function (R(x)), 122e123
Reynold’s equation, 441
Reynolds Averaged NaviereStokes
equations (RANS equations),
423
Reynolds number, 388e389
RFEM software, 229t
Roller/Sliding, 272t
“Roller” condition, 158be159b
Routine design, 7, 9t
Index 501S
Saturated zone, 398e399
Scalar, 188
multiplication by, 200
transpose of matrix with product of,
201
Second law of thermodynamics, 341
Selection
design, 7, 9t
of fatigue analysis methods, 323, 324t
Semisteady models, 154
Sensitivity
analysis, 488
study, 488
Shafts, 281
Shape(s), 440
functions, 69e72, 80e83, 81be83b,
86, 143, 161, 165, 475
utilities, 233e234
Shear modulus, 285
Shear stress, 379
Shear thinning, 440
Shell elements, models with, 152
Shrink fit contact condition, 157, 158t
SimScale, 230te231t
Simulation models of multiphysics
systems, 410e411
Single-physics, 153, 408
Singularity of fixed supports, 484,
484f
Sleipner-A oil platform, 455, 456f
Small-or medium sized enterprises
(SMEs), 20
SeN curve. See Strengthenumber of
cycles (SeN curve)
Solid mechanical problems, 26. See also
Fluid mechanics
fatigue analysis, 320e334
modal analysis, 311e319
structural analysis, 282e311,
282fe283f
plane strain problems, 306e311,
308fe309f
plane stress, 299e306, 300fe301f,
303f, 306f, 311b
truss structures, 286e299
SolidWork(s), 472be474b
analysis types, 270, 271t
CAD/CAE interface, 262e264
commercial CAD/CAE software tool,
262e275
design optimization, 275
displacement boundary conditions for
structural analysis, 272t
file formats compatible to, 265f
formats of solid models, 264t
loads boundary conditions for structural
analysis, 272t
materials library, 264e266, 266f
meshing tool, 268e270
postprocessing, 275
Simulation software, 229t
SolidWorks/simulation for FEA, 263f
solvers to FEA models, 273e274
tools for boundary conditions,
270e273
Solution
control, 233e234
space, 4e5, 5f, 7
Solvers to FEA models, 273e274
Spatial discretization, 410
Spherical CS, 62
“Split line”, 479, 479f
Spring assemblage
displacement of, 252t, 257t, 261t
example, 252f
Spring system example, 246e253
State variables, 407
Static analysis, 45e46, 203, 271t,
292be294b, 295f, 311b,
327e328, 330
Static models, 273e274
Static structure, 238
of FEA computer implementation,
239e245, 240f
CBoundaryDoF class, 242, 242f
CBoundaryLoad class, 242
CDomain class, 241
CElement class, 243e244, 243f
CEProperties class, 244,
244f
CMaterial class, 241
CModel class, 239e241
CNode class, 243, 243f
CPostProc class, 245, 245f
CSovler class, 244, 244f
Statistical turbulence models, 423
Steady models, 154
Steady-state model. See Static models
Steady-state transfer, 19
STEP files, 264t
STereoLithography (STL), 264t
Strain-life method, 321
Stream function, 392
Strengthenumber of cycles
(SeN curve), 266, 267f, 323,
326
in finite element analysis, 326f
Stress-life method, 323
Stribeck curve, 438e439, 439f
Strong-coupling, 153, 411
Strongly coupled partitioned system,
410
Structural analysis, 18, 282e311,
282fe283f
displacement boundary conditions for,
272t
loads boundary conditions for,
272t
plane strain problems, 306e311,
308fe309f
plane stress, 299e306, 300fe301f,
303f, 306f, 311b
truss structures, 286e299
Structural design, 282
Structural dynamics, 434e435
Structural-thermal analysis, 412e420
modeling of 2D frame members,
415e420, 415f
structural analysis under thermal load,
412e415
thermal-stress analysis applications,
413f
Structured mesh, 49e50
Structures, 234e235
Subdomain methods, 124, 127
Subspace method, 220
Subtraction, 196e198
Supermatrix, 192
Surface finish, 440
Surface force, 283
Symmetric matrix, 191
Symmetric plane(s)
BCs on, 156be157b
models for objects with, 153
Symmetry, 483, 483f
restraints, 272t
System analysis and modeling
assembling system models, 176e182
BCs, 155e166
equivalent loads, 166e176
FEA, 149
types of analysis models, 149e155
users’ inputs for model types, boundary
conditions and loads, 150f
System architecture, 237
502 IndexSystem boundaries, clarification of,
34e35
System complexity, 23e24, 33e39
assembling process, 35e36, 37t
dynamics, 37, 38f
geometric shapes, 39
multiphysics problems, 37e38, 38f
uncertainties, 34e35, 34fe36f
System description, 31e32, 32f
System inputs, 461
System model(s), 32, 216
assembling, 176e182
from 1D elements, 176e179
solutions to
eigenvalue problems, 211e221
FEA, 187
matrices, 187e203
solutions to system of linear
equations, 203e211
tensors, 187e203
vectors, 187e203
system parameters and load conditions
of elements, 178t
2D element, 179e182
System modeling, 25, 233e234
System of linear equations, solutions to,
203e211
computation in matrix manipulation,
211, 212t
Gaussian elimination, 204e206,
205be206b
iterative methods, 208e211,
210be211b
LU decomposition, 206e208, 208b
System parameters, 32, 34e35
T
Taylor expansion, 73e74, 123
Taylor exploration, 72e73
Taylor series, 73, 78, 100
Taylor’s theorem, 74
Temperature, 440
Tensors, 187e203, 188f
Thermal analysis, 271t
Thermal load, structural analysis under,
412e415
Thermodynamics, 341
Thin objects, 268
Third law of thermodynamics, 341
Three-dimension (3D)
CS transformation in three-dimensional
space, 68e69
model, 150
truss system
computer implementation of,
256e262
definition of classes, 256, 259te260t
displacement of spring assemblage,
261t
example of, 256e262, 261f
static structure of C3DTrussModel
for, 256
static structure of FEA computer
implementation, 258f
Tie rods, 325e326, 331
characteristics of dynamic load on,
327f
material properties of, 326t
parametric representation of, 328f
Time
increments, 410
integrators, 410
time-dependent problems, 211
Tonsorial definition of strain rates, 381
Top-down approaches, 15e16, 16f
Trace of matrix, 193
Transformation method, 220e221, 222t
Transient heat transfer, 19
problems, 366e374, 368f, 370f
Transient models, 154
Transient problems, 222
Transpose
of combined matrices, 198e200
of matrix, 192e193
with product of scalar, 201
of product of two matrices, 202
Triangle element(s), 165e166. See also
Rectangle element
eigenvalue problems, 144e145
equivalent loads of, 167e171,
172f
modeling of triangular elements,
360e362
2D equilibrium problems, 138e139
Triangular matrix, 191
Truss elements, 286e290
Truss members, 152, 290, 294f
assigning structural members as, 295f
Truss structures, 286e299, 286f, 293f,
297f
applying displacement boundary
conditions and load on, 298f
boundary conditions and loads,
290e291, 291f
deflection of three-dimensional truss
structure, 299f
examples, 292e299
joints in, 291f
meshing of three-dimensional truss
structure, 299f
from reference nodes and planes, 298f
truss elements, 286e290
Turbulent viscosity, 423e424
Two-dimension (2D)
CS transformation in two-dimensional
space, 65e67
heat transfer finite element analysis
model, 364t
integral operation over two-dimensional
domain, 104e106, 105be106b
incompressible and irrotational flow,
392e398, 394fe395f
interpolation in two-dimensional
elements, 84e97
two-dimensional linear triangle
element, 90e93, 91be93b
two-dimensional linear triangle
element under natural CS, 94e95
two-dimensional quadratic rectangle
element, 87e89
two-dimensional quadratic triangle
element, 95e97
two-dimensional rectangle elements,
84e87
linear triangle element, 90e93
under natural CS, 94e95
quadratic rectangle element, 87e89
quadratic triangle element, 95e97
rectangle elements, 84e87
beam member in LCS, 314e315
element modeling of 2D equilibrium
problems, 135e139
rectangle element, 136e138
triangle element, 138e139
element modeling of 2D propagate
problems, 139e141
incompressible and irrotational flow,
392e398, 394fe395f
modeling of 2D frame
element, 315e316
members, 415e420, 415f
steady heat transfer problems, 355e365
example, 362e365, 363f,
363be365b
modeling of rectangle elements,
357e360
Index 503Two-dimension (2D) (Continued)
modeling of triangular elements,
360e362
structure with frame members, 319b
boundary conditions on, 318f
modal analysis of, 317e319
vibration modes of structure, 319f
system models from2D element,
179e182
truss member in LCS, 312e314, 313f
U
Uncertainties, 34e35, 34fe36f
quantification, 485e488, 487be488b
Uncoupled decomposition, 42
Unified modeling language (UML), 238
for OOP, 238e239
Unstructured mesh, 49e50
Unstructured programming, 234e235
V
V&V. See Verification and validation
(V&V)
Validation, 456e457, 477e489, 477f
benchmarking for, 488e489, 489f
engineering judgment, 488
modal validation, 478e484
sensitivity study, 488
uncertainty quantification, 485e488,
487be488b
Vectors, 187e203
Verification, 456e457, 464e477
benchmarking, 475e477
calculation verification, 468e471, 469f,
469be471b
code verification, 466e468, 466f,
467b, 468f, 468b
convergence study, 475
at different modeling stages, 465f
meshing verification, 471e474, 472f,
472be474b
subjects in finite element analysis
model, 465f
Verification and validation (V&V), 15,
26, 455e457
difference of, 458t
in numerical simulation, 457f
planning in FEA modeling, 458e459,
458f
sources of errors, 459e464
validation, 477e489, 477f
verification, 464e477
Vibrating membrane, 179be182b
Virtual Reality Modelling Language
(VRML), 264t
Virtual wall contact condition, 157,
158t
Viscosity, 439e440
VisualFEA Software, 230te231t
Volume force, 283
W
Weak-coupling, 153, 411
Wedge-film lubrication, 441f
Weighted residual method(s), 22,
122e130
comments on, 130,
131t
exacting and weak solutions to PDEs,
122e123
procedure to finding approximated
solution, 123e124
variety of, 124e129
collocation method, 126
Galerkin method, 129
least-squares method, 128e129
subdomain method, 127
Weldments, 292be297b
What-if
models, 154e155
simulation, 154e155
Wind forces, 327t
Y
Young’s module, 440
Z
Zeroth law of thermodynamics, 341
“Zig-zagging” process, 41e42
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