Finite Element Analysis Applications – A Systematic and Practical Approach

Finite Element Analysis Applications – A Systematic and Practical Approach
Zhuming Bi
Purdue University Fort Wayne,
Indiana, United States
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,
13. 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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