Index

A

ABAQUS, 154–155
beam elements in, 151
input file, 124–127, 152–154, 234–236, 242–244, 282–284, 388–390
resonant frequencies of bridge, 129
Acceleration–time history, 245–246
Admissible displacement, 45
Advanced engineering systems, processes leading to fabrication of, 1–2
Aircraft for dynamic testing, 3
Aluminum
beam members, 155–156
material properties of, 118, 151
Amplitude curve, 243
Angular distortion, 307–308
ANSYS, 130–132, 415
analysis type in, 419
beam section properties in, 416
computation in, 420
geometry creation environment in, 417
material properties in, 417
meshing in, 418
screenshot of, 131
Anti-symmetric boundary condition, 315, 317
Arbitrary high orders, rectangular element of, 273
Area coordinates
definition of, 171
linear triangular elements, 170–172
Artificial damping elements, 297
Aspect distortion, 308
Aspect ratio distortion, 307
Asymmetric loading, symmetrical framework with, 319
Axial displacement, 82
Axisymmetric mesh, 7
Axisymmetric structures
2D planar problem using, 189
elements for, 188–191

B

Bar element, 81–82
Base state, 404
BC, Boundary conditions (BC)
Beams
constitutive equations, 31
dynamic equilibrium equations, 33–34
elements, 5–6, 111, 135–136
and local coordinate systems, 112
coordinate transformation, 118
in ABAQUS, 151
nodal force vector for, 116–118
equations, FEM for
element matrices, 116–118
shape function construction, 112–115
strain matrix, 115–116
moments and shear forces, 31–33
simply supported, 30
stress and strain, 30–31
∗BEAM SECTION keyword line, 406–407
BEM. See Boundary element method (BEM)
Bicycle frame
deformation plots of, 155–156
diamond-shaped, 150–151
finite element mesh of, 151–152
stresses in, 156
Bilinear shape functions, 375–376
Boundary conditions (BC), 91–92, 119–120, 359–361, 377–384
applying, 101–102
cards, 235–237, 243
constraints and, 281–282
equations for three-dimensional solids, 21–22
Boundary element method (BEM), 297–298, 397–398
coupling of FEM and, 297–298
Brick elements, 272–277

C

CAD. See Computer aided design (CAD)
Cantilever beam, 118, 405–411
Cartesian coordinate system, 250–255, 259
Central difference algorithm, 72–75
Chain rule of differentiation, 121
Clamped-clamped bridge structure, 122–123
geometrical dimensions of, 123
ten element mesh of, 128
Classical plate theory (CPT), 35
Commercially available software packages, 398
Compatible element, 47–48
Complete order of polynomial basis functions, 166–167
Composite wall, 365–366
Computational modeling
material/medium properties, 8
meshing, 6–8
of geometry, 5–6
physical problems in engineering, 4
solution procedure
discrete system equations, 8–9
equation solvers, 9–10
results visualization, 10–11
using FEM, 4–8
Computer aided design (CAD), 305–306
software packages, 5
Connectivity element, 47–48
Constant matrix, 374
Constant strain elements, 173–174, 255–256
Constant stress elements, 173–174
Constitutive equations
for beams, 31
for plates, 36
for three-dimensional solids, 18–19
for truss members, 27
for two-dimensional solids, 24–25
Constraints modeling by rigid body attachment, 335–336
Contact cards, 284
Contact modeling, 281–282
Control cards, 236, 244
Convection matrix, 358
Convective boundary conditions, 365
Conventional finite elements, 297–298
Conventional isoparametric 8-nodal element, 292–293
Coordinate mapping, 262–263
between coordinate systems, 184
linear quadrilateral elements, 183–186
Coordinate transformation process, 67–68, 140
Counter-clockwise manner, 261
Coupling effects, 233
CPT. See classical plate theory (CPT)
CPU time estimation, 303–304
Crack tip elements, 289–297
Cubic one-dimensional element, 105
Cubic tetrahedron element, 272
Cubic triangular elements, 194–195
Curvature distortion, 307–308
Curved edges, 2D solid elements with, 200
Curved surfaces, elements with, 277
Cyclic symmetry, 321–322
Cylindrical coordinate system, 188

D

Damping element, 297
Data lines, 401–402
Deformation plots, 155
from PATRAN, 412
of bicycle frame, 155–156
Degrees of freedom (DOFs), 112–113, 137, 142, 219–220, 249–250, 303–304, 306
at nodes, 327
frame element in space with, 143
of linear triangular element, 164
Delta function, 272
property, 167–168
Diamond-shaped bicycle frame, 150–151
Dimensional of heat, 350
Direct assembly, 98
procedure, 361–362
Discrete numerical methods, 297
Discrete system equations, 8–9
Discretized system, equations of, 354
Displacement boundary condition, 22, 119–120, 419
Displacement constraints, imposition of, 69
Displacement interpolation, 49–50
Displacement–time history, 245
Distorted elements, 307
Distributed external body force, 19–21
DOFs. See Degrees of freedom (DOFs)
Domain discretization, 4, 47–49
Dynamic equilibrium equations
for beams, 33–34
for plates, 38
for three-dimensional solids, 19–21
for truss members, 27–28
for two-dimensional solids, 25–26
Dynamic testing, aircraft for, 3

E

8-Nodal hexahedron elements, 261–262
8-Nodal isoperimetric quadratic element, 291–292
Eight-node rectangular thick plate element, 227
∗ELASTIC option, 407
Electrostatic micro-motor, 202
Element connectivity, 6, 234, 242
Element displacement vector, 142–144
Element distortion, 307–310
Element matrices, 116–118, 225–227, 256–261, 266–269
in global coordinate system, 97–98
linear quadrilateral, 187–188
linear rectangular, 179–180
linear triangular, 174–176
obtaining, 119
∗ELEMENT option, 406
Element sets (ELSET), 400–401
Elements with curved surfaces, 277
ENCASTRE, 408
∗END STEP option, 404
Enforcing compatibility, 334–335
Equation solvers, 9–10
Euler-Bernoulli beam theory, 30–31
for thin beams, 31
Explicit approaches, 9

F

FDM. See Finite difference method (FDM)
FE. See Finite element (FE)
FEM. See Finite element method (FEM); Finite element model (FEM)
FE matrix equation, 120–122
solving, 102–103
FGM. See Functionally graded material (FGM)
Field problems
acoustic problems, 353–354
heat transfer
composite wall, 351–352
in long two-dimensional body, 349–350
in one-dimensional fin, 350–351
in two-dimensional fin, 348–349
ideal irrotational fluid flow, 353
torsional deformation, 352–353
Field variable interpolation, linear triangular elements, 164–166
Finite difference method (FDM), 8–9, 44
Finite element (FE), 81–82, 219–220
approximation for one-dimensional case, 3
formulation, 294–295, 386
for stress analysis, 162
matrices, 111–112
mesh of bicycle frame, 151
Finite element (FE) mesh, 204, 236
of quantum dot heterostructure, 281
Finite element method (FEM), 1, 250–255, 293–294, 390
computational modeling using, 4–8
convergence property, 94–95
definition, 3–4
for beams equations
element matrices, 116–118
shape function construction, 112–115
strain matrix, 115–116
fundamentals for
free vibration, analysis of, 69–71
Hamilton’s principle, 45–47
minimum total potential energy principle, 47
problem formulation, 44–45
procedure, recap of, 47–69, 77–78
static analysis, 69
sufficient requirements for, 76–77
transient response, 71–76
mathematical models of, 4
procedure
constructing shape functions, 50–54
coordinate transformation, 67–68
displacement constraints, imposition of, 69
displacement interpolation, 49–50
domain discretization, 47–49
global FE equation, 68–69
local coordinate system, finite element equations in, 63–67
shape functions properties, 54–63
rate of convergence of, 95–97
reproduction feature of, 121
reproduction property of, 94
sufficient requirements for, 83–84
truss element. See truss element
two-dimensional solids. See two-dimensional solids
Finite element model (FEM)
ABAQUS input syntax rules, 401
basic building block, 399–400
cantilever beam problem, 405–411
analysis running, 410–411
results, 411
data lines, 401–402
general procedures, 411–414
history data, 404–405
in ABAQUS, 402–411
labels, 402
model data, 403–404
using GUI, ANSYS, 414–421
using sets, 400–401
Finite strip elements, 298
Finite volume method (FVM), 8–9, 397–398
First order differential operators, 19–21
First order shear deformation theory, 35
Flexural vibration modes, 233
Force loading conditions, 419
Force vectors, 382
Fourier series, 244
Fourier superimposition, 320
Fourier’s heat convection law, 379–381
4-Nodal tetrahedron element, 250–255
Four-node rectangular thick plate elements, 224–225
4-Node tetrahedron elements, 251
Frames
element
coordinate transformation for, 145, 152
three-dimensional orientation of, 147
equations for planar frames
idea of superimposition, 137
in global coordinate system, 140–142
in local coordinate system, 137–140
equations for space frames
in global coordinate system, 144–149
in local coordinate system, 142–144
finite element analysis of bicycle frame
ABAQUS input file, 152–154
modeling, 151–152
results and discussion, 155–157
solution processes, 154–155
formulation for, 118
made of three members, 159
Free vibration analysis, 69–71
symmetric and anti-symmetric conditions for, 320
∗FREQUENCY, 402–403
Functionally graded material (FGM), 106
FVM. See Finite volume method (FVM)

G

Galerkin method, 355, 370–371
Galerkin residuals, 365
Galerkin weakform, 299
Gauss elimination method, 9, 155
Gauss integration
linear rectangular elements, 180–183
points and weight coefficients, 180
scheme, 179–181, 183, 187, 226–227, 266–268, 284–285
Gauss’s divergence theorem, 371–372
General beam element, 127, 135–136
Geometry modeling, 304–306
Global coordinate system, 67–68, 87, 250–255
element matrices in, 97–98
boundary conditions, 91–92
planar trusses, 90–91
recovering stress and strain, 92
spatial trusses, 87–90
elements in, 232–233
equations for
planar frames, 140–142
space frames, 144–149
Global FE equation, 69
assembly of, 68–69
Global FE matrices, 98–101
Gradual damping elements, 297
Graphical user interface (GUI), 131, 397–398
Grids, 6
GUI. See Graphical user interface (GUI)

H

Hamilton’s principle, 8–9, 45–47, 121, 221–223, 372–373
Heat conduction, 348–349
Heat convection, 348–349, 380
Heated road surface, temperature distribution of
ABAQUS input file, 388–390
modeling, 387–388
results and discussion, 390
Heating cables, 386
Heat insulation boundary, 379
Heat sink, 384–385
Heat source, 384–385
Heat transfer
FEM
1D heat transfer problem, 355–370
2D heat transfer problem, 370–386
field problems, 348–354
heated road surface, temperature distribution of, 386–390
weighted residual approach, 354–355
through composite wall, 351–352
Helmholtz equation, 348, 353
Hexahedrons
element, 261–269, 271
tetrahedrons to, 269
Higher order 3D tetrahedron elements, 271
Higher order elements, 227, 269–277
High order 3D serendipity elements, 274
Hilber–Hughes–Taylor operator (1978), 244–245
History data, 401
Homogenous boundary condition, 21
Hooke’s law, 92
for beams, 31
for isotropic materials, 24
for 1D solids, 27
for plates, 36
for 3D anisotropic materials, 18–19

I

Implicit approaches, 9
InAs quantum dots. See Indium arsenide quantum dots
Incompatible mesh, 311
Independent stress components, 16–18
Indium arsenide (InAs) quantum dots, 279–281
Infinite domains, methods for, 293–294
Infinite elements, 293–294, 296
Infinite line, 295
Inhomogenous boundary conditions, 21
Internal nodes, vanish of, 358–359
Isolated beam cell, 32
Isolated plate cell, 36
shear forces and moments on, 37
Isoparametric element, 188
Isoparametric quadratic element, 292–293
Isotropic materials, 18–19

J

Jacobian matrix, 186–187, 257–260, 266–268
Joints modeling, 328–332

K

Keyword lines, 401
Kirchhoff plate theory, 35

L

Labels, 402
Lagrange interpolants, 104, 195, 272
Lagrange multiplier method, 338
Lagrange type elements, 272–273
rectangular elements, 195–196
Lamb waves, 297
dispersive characteristic of, 297
Laplace’s equations, 353
Layered composite wall, 393
Linear elastic fracture mechanics, 290–291
Linear element, 85
Linear quadrilateral elements
coordinate mapping, 183–186
element matrices, 187–188
remarks, 188
strain matrix, 186–187
Linear rectangular elements
element matrices, 179–180
Gauss integration, 180–183
shape function construction, 176–179
strain matrix, 179
Linear triangular elements, 165
area coordinates, 170–172
element matrices, 174–176
field variable interpolation, 164–166
shape function construction, 166–170
strain matrix, 172–174
Load cards, 127, 244, 284
Local coordinate system, 49, 87, 137–140
elements in, 86–87, 228–232
equations for
planar frames, 137–140
space frames, 142–144
finite element equations in, 63–67
LU decomposition method, 9

M

Mapping, infinite elements formulated by, 294–297
Mass matrix for rectangular element, 231–232
Material cards, 235–236, 243, 282
Matrix of shape functions, 250–255
Mechanics for solids and structures, 14–16
equations for beams
constitutive equations, 31
dynamic equilibrium equations, 33–34
moments and shear forces, 31–33
stress and strain, 30–31
equations for plates
constitutive equations, 36
dynamic equilibrium equations, 38
moments and shear forces, 36–38
Reissner-Mindlin plate theory, 38–40
stress and strain, 35–36
equations for three-dimensional solids
boundary conditions, 21–22
constitutive equations, 18–19
dynamic equilibrium equations, 19–21
stress and strain, 16–18
equations for truss members
constitutive equations, 27
dynamic equilibrium equations, 27–28
stress and strain, 27
equations for two-dimensional solids
constitutive equations, 24–25
dynamic equilibrium equations, 25–26
stress and strain, 22–24
MEMS. See Micro-electro-mechanical systems (MEMS)
Mesh compatibility, 310–313
elements, different order of, 310–312
straddling elements, 312–313
Meshfree methods, 299–300
Meshing, 47, 279, 306–310
axisymmetric, 7
definition, 6
density, 306–307
of hinge joint, 7
stress distribution, 7
transition, 388
Micro-electro-mechanical systems (MEMS), 122
Micro-motor
side drive
ABAQUS input file, 204–207
modeling, 203–204
plan view (2D) of, 203
results and discussion, 208–211
solution process, 207–208
transient analysis of, 240–247
Micro-resonant transducer
resonant frequencies of
ABAQUS input file, 124–127
comparison with ANSYS, 130–132
modeling, 123–124
results and discussion, 128–130
solution process, 127–128
vs.ANSYS, 130–132
review questions, 132–134
Mid-node position distortion, 310
Mindlin plate, shear deformation in, 39
Mindlin plate theory, 221
Minimum total potential energy principle, 47
Mirror symmetry, 314–322, 333
Mode I fracture, 290–291
Model data, 400–403
Modeling offsets, 322–328
methods for, 322–325
Modeling techniques
axial symmetry, 318–321
constraints modeling, by rigid body attachment, 335–336
CPU time estimation, 303–304
cyclic symmetry, 321–322
element distortion, 307–310
geometry modeling, 304–306
joints, modeling of, 328–332
Lagrange multiplier method, 338
mesh compatibility, 310–313
elements, different order of, 310–312
enforcement of, 334–335
straddling elements, 312–313
meshing, 306–310
mirror/plane symmetry, 314–322
modeling offsets, 322–328
modeling offsets methods, 322–325
MPC equations
applications of, 332–336
creation of, 325–328
implementation of, 336–339
penalty method, 338–339
repetitive symmetry, 322
supports, modeling of, 328–330
symmetry, use of, 313–322
Moment matrix, 54
Moments and shear forces
equations for beams, 31–33
equations for plates, 36–38
MPC. See Multipoint constraints (MPC)
MPC equations, 324–326
Multipoint constraints (MPC), 312

N

Natural boundary condition, 379
Natural coordinate system, 112, 176–177, 184, 258–259, 262–263, 307–309
and local coordinate system, 112–113
Natural frequencies of micro-motor, 233–240
Newmark’s method, 75–76
∗NGEN, 406
Nodal cards, 234, 242
Nodal displacements, 165–166
vector, 250–255, 263–265
Nodal force vector, 268–269
for 3D solid elements, 260–261
Nodal heat vector, 358
Nodal temperatures of road surface, 391
∗NODE option, 406
Nodes, 3–4, 6
sets, 242
Normal stress, 32
NS-FEM, 300
Numerical integration scheme, 266–268

O

1D axisymmetric elements, cylindrical shell structure modeled using, 320
1D heat transfer problem
composite wall, 365–366
direct assembly procedure, 361–362
one-dimensional fin, 355–361
worked example, 362–364, 366–370
One-dimensional fin, 355–361
of rectangular cross-section, 363
Orthogonal matrix, 87–88
Output control cards, 235, 244, 284
Over-stiff behavior, 201

P

Pascal triangle of monomials, 51
Penalty method, 338–339
Perturbation parameter, 409
Physical coordinate system, 184, 277, 307–309
Physical trial-and-error design procedure, 150–151
Planar frames, FEM, 136
element and DOFs, 137
idea of superimposition, 137
in global coordinate system, 140–142
in local coordinate system, 137–140
Planar truss, 90–91
structure, 157
Plane strain solid, 23
Plane stress conditions, 162–163
Plane stress solid, 23
Plane symmetry, 314–322
Plate
constitutive equations, 36
dynamic equilibrium equations, 38
elements, 5–6, 220–227
moments and shear forces, 36–38
Reissner-Mindlin plate theory, 38–40
stress and strain, 35–36
structure, 34
Poisson effect, 27
Poisson’s equation, 353
Poisson’s ratio, 18–19
Polynomial function, 294–295
Polynomial shape functions, 298–299
Polysilicon, elastic properties of, 123, 202
Pre-processing, 6–7
Property cards, 234, 243, 282

Q

Quadratic convergence, 95
Quadratic elements, 163
Quadratic one-dimensional element, 104
Quadratic triangular elements, 193–194
Quadrilateral elements, 183–184, 237–240
meshes, 6–7
2D domain meshed, 184
unacceptable shapes of, 307–309
Quadrilateral shell elements, 236
Quantum dot heterostructure, stress and strain analysis of, 277–286
Quarter model, 233
of micro model, 241

R

Rectangular domain, meshed with triangular elements, 165, 176
Rectangular elements, 221, 375–377, 382–384
and coordinate systems, 177
domain, 176–177
Lagrange type elements, 195–196
9-node, 196
of arbitrary high orders, 196, 273
serendipity type elements, 196–200
Rectangular hexahedron element, 266–268
Rectangular shell element, 229
Refinement, 289–290
Reissner–Mindlin plate theory, 35, 221–223
equations for plates, 38–40
Repetitive symmetry, 322
Residual equation, 356–357
Residual method, 8–9
Resonant frequencies of micro-resonant transducer
ABAQUS input file, 124–127
comparison with ANSYS, 130–132
modeling, 123–124
results and discussion, 128–130
solution process, 127–128
Resonant micro-beam strain transducer, 122
Rigid element, 324
Rigid slab on elastic foundation, 336
Rule of thumb, 250

S

Sandwiched composite wall, 393
Second order differential operators., 19–21
SEM, 298–299
Semi-automatic mesh generator, 6
Serendipity type elements, 273–277
construction of 8-node, 198
high order, 197
rectangular elements, 196–200
S-FEM. See Smoothed finite element methods (S-FEM)
Shape functions, 224–226, 262–263, 292–293
construction, 82–85, 112–115
delta function property, 167
linear rectangular elements, 176–179
linear triangular elements, 166–170
trusses, FEM for, 82–85
matrix of, 224–225, 250–255
properties of, 54–63
Shear deformation in plate, 222
Shear equivalence, 16–18
Shear locking, 226–227
Shell elements, 2, 227–233
Shell structures, 228–229
Simply supported anti-symmetric beam structure, 316
Simply supported symmetric beam structure, 316
Single point constraint (SPC), 63, 314–315
Singularity elements, 289–290
Sinusoidal function, 241, 244
Skeletal-type truss structural systems, 81–82
SK growth mode. See Stranski-Krastanow (SK) growth mode
Smoothed finite element methods (S-FEM), 299–300
Smoothed particle hydrodynamics (SPH), 299
Smoothed Point Interpolation Methods (S-PIM), 299
Smoothing domains, 300
Solid finite elements, 249–250
Solids and structures, mechanics for. See mechanics for solids and structures
Solution process, 236–237
Solvers, equation, 9–10
SOR method. See Successive over-relaxation (SOR) method
Space frames, 136
structure, 136
Spatial frame structure, three-dimensional, 135–136
Spatial trusses, 87–90
SPC. See Single point constraint (SPC)
SPD. See Symmetric Positive Definite (SPD)
Special purpose elements and methods, 289
Specified heat flux, 381
SPH. See Smoothed particle hydrodynamics (SPH)
S-PIM. See Smoothed Point Interpolation Methods (S-PIM)
Sprocket-chain system, finite element mesh for, 307
Stabilizing matrix, 232–233
Standard finite element mesh, 295–296
State of stresses, equilibrium equations, 20
Static analysis, 69
∗STEADY STATE DYNAMICS, 402–403
Steady state equation, 355
∗STEP line, 402–403
∗STEP option, 404
Stiffened plates with offset, 327
Stiffness matrix, 116–118, 229–230, 365–366
Straddling elements, 312–313
Strain-displacement relationships, 16–18, 24
Strain matrix, 85–86, 115–116, 250–256, 261–266, 357–358
linear quadrilateral elements, 186–187
linear rectangular elements, 179
linear triangular elements, 172–174
trusses, FEM for, 85–86
Stranski-Krastanow (SK) growth mode, 277–278
Stress
distribution, 278, 285–286
in bicycle frame, 156
tensors, 16–18
Stress and strain equations
for beams, 30–31
for plates, 35–36
for three-dimensional solids, 16–18
for truss members, 27
for two-dimensional solids, 22–24
Stress components, 255–256
independent, 16–18
Strip element method, coupling of FEM and, 298
Structural components, types of, 15
Subparametric elements, 188
Sub-space iteration scheme, 237
Successive over-relaxation (SOR) method, 9
Superimposition
idea of, 228
of element matrices, 137
Superparametric elements, 188
Supports modeling, 328–330
Surface of solid, 16
Symmetric Positive Definite (SPD), 91
Symmetric quarter model, 281
Symmetrical quarter model, 233

T

Temperature distribution of cross-section of road, 391
10-nodal tetrahedron element, 269–272
Tensile stress in matrix, 285–286
Tetrahedron elements, 250–261, 269–272
Tetrahedrons to hexahedrons, 269
Thermal conductive properties, 351
Thick beams, 30–31
Thin beams, 30–31
Euler-Bernoulli assumption for, 31
32-node tri-cubic element, 276–277
3D element types, 284–285
3D mesh
of island, 280
of matrix, 280
3D solid elements, 249–250
Three-dimensional solid element mesh, 333
nodal force vector for, 260–261
with curved surfaces, 278
3D visualization, 10
Three-dimensional (3D) solids
boundary conditions, 21–22
constitutive equations, 18–19
dynamic equilibrium equations, 19–21
element, 21–22
stress and strain, 16–18
Three-dimensional spatial frame structure, 135–136
‘Tied’ contact condition, 281–282
Time stepping, implicit and explicit approaches, 9
T-meshes, 299
Torsional bar element, 142–144
Torsional deformation, 142–144, 352
Torsional state, 352–353
Transient analysis of micro-motor, 240–247
Transient response, 71–76
central difference algorithm, 72–75
Newmark’s method, 75–76
Translational displacements, 229–230
Transverse displacement components, 241
Trapezoidal cross-sections (TRAPEZOID), 406–407
Triangular elements, 164, 373–375
cubic, 194–195
general formulation of shape functions, 191–193
quadratic, 193–194
rectangular domain meshed with, 165, 176
Triangular truss structure, 95–96
Trilinear functions, 263
Truss element, 81–82
connected by ridged bar, 110
convergence property of, 94–95
coordinate system, 83
dimensions and properties of, 97
element matrices in global coordinate system
boundary conditions, 91–92
planar trusses, 90–91
recovering stress and strain, 92
spatial trusses, 87–90
element matrices in local coordinate system, 86–87
high order one-dimensional elements, 103–105
linear shape functions for, 85
local coordinates and degrees, 96
nodal force vector for, 86–87
rate of convergence of, 95–97
reproduction property of, 94
shape function construction, 82–85
strain matrix, 85–86
structure, 107–108
transformation matrix, 87–88
with 3 nodes, 109
Truss members
beam and, 158–159
constitutive equations, 27
cross-sectional dimension of solid, 27
dynamic equilibrium equations, 27–28
stress and strain, 27
typical structure of, 26
Truss structure, three member, 96
Turbine-blade, 332
and turbine-disc system, 331
20-Nodal tri-quadratic element, 273–276
20-Node serendipity element, 274, 276
20-Node tetrahedron element, 272
2D axisymmetric elements
formulation of, 319–320
3D structure modeled using, 321
2D domain of plate, 222
2D element mapping, 295
2D finite element mesh
with boundary condition, 387
2D frame elements, coordinate transformation for, 140
2D heat transfer problem
boundary conditions and vector b(e), 359, 377–384
element equations, 370–373
point heat source or sink, 384–386
rectangular elements, 375–377
triangular elements, 373–375
2D solids
constitutive equations, 24–25
dynamic equilibrium equations, 25–26
elements
for axisymmetric structures, 188–191
with curved edges, 200–201
Gauss integration, 201
linear quadrilateral elements
coordinate mapping, 183–186
element matrices, 187–188
remarks, 188
strain matrix, 186–187
linear rectangular elements
element matrices, 179–180
Gauss integration, 180–183
shape function construction, 176–179
strain matrix, 179
linear triangular elements
area coordinates, 170–172
element matrices, 174–176
field variable interpolation, 164–166
shape function construction, 166–170
strain matrix, 172–174
rectangular elements
Lagrange type elements, 195–196
serendipity type elements, 196–200
stress and strain, 22–24
triangular element family
cubic triangular elements, 194–195
general formulation of shape functions, 191–193
quadratic triangular elements, 193–194

V

Velocity–time history, 245–246
Very stiff element, 324
Virtually designed building, air flow field in, 10–11
Virtual reality, 10
Volume coordinates, 250–255
Volumetric distortion, 307–309
Von Mises stress distribution, 208, 211
using 24 bilinear quadrilateral elements, 208
using 96 bilinear quadrilateral elements, 209
using 144 bilinear quadrilateral elements, 209
using 24 eight-nodal, quadratic elements, 210
using 192 three-nodal, triangular elements, 210

W

Weak form formulation, 22
Weight coefficients, 180–181
Weighted residuals, 354

Y

Young’s modulus, 18–19
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