Index
A
resonant frequencies of bridge,
129
Acceleration–time history,
245–
246
Admissible displacement,
45
Advanced engineering systems, processes leading to fabrication of, –
Aircraft for dynamic testing,
Aluminum
material properties of,
118,
151
beam section properties in,
416
geometry creation environment in,
417
material properties in,
417
Anti-symmetric boundary condition,
315,
317
Arbitrary high orders, rectangular element of,
273
Area coordinates
linear triangular elements,
170–
172
Artificial damping elements,
297
Aspect ratio distortion,
307
Asymmetric loading, symmetrical framework with,
319
Axisymmetric mesh,
Axisymmetric structures
2D planar problem using,
189
B
BC, Boundary conditions (BC)
Beams
constitutive equations,
31
dynamic equilibrium equations,
33–
34
and local coordinate systems,
112
coordinate transformation,
118
nodal force vector for,
116–
118
equations, FEM for
shape function construction,
112–
115
moments and shear forces,
31–
33
∗BEAM SECTION keyword line,
406–
407
BEM. See Boundary element method (BEM)
Bicycle frame
finite element mesh of,
151–
152
Bilinear shape functions,
375–
376
equations for three-dimensional solids,
21–
22
C
CAD. See Computer aided design (CAD)
Central difference algorithm,
72–
75
Chain rule of differentiation,
121
Clamped-clamped bridge structure,
122–
123
geometrical dimensions of,
123
Classical plate theory (CPT),
35
Commercially available software packages,
398
Compatible element,
47–
48
Complete order of polynomial basis functions,
166–
167
Computational modeling
material/medium properties,
meshing, –
of geometry, –
physical problems in engineering,
solution procedure
discrete system equations, –
results visualization,
10–
11
using FEM, –
Computer aided design (CAD),
305–
306
software packages,
Connectivity element,
47–
48
Constant stress elements,
173–
174
Constitutive equations
for three-dimensional solids,
18–
19
for two-dimensional solids,
24–
25
Constraints modeling by rigid body attachment,
335–
336
Convective boundary conditions,
365
Conventional finite elements,
297–
298
Conventional isoparametric 8-nodal element,
292–
293
between coordinate systems,
184
linear quadrilateral elements,
183–
186
Coordinate transformation process,
67–
68,
140
Counter-clockwise manner,
261
CPT. See classical plate theory (CPT)
Cubic one-dimensional element,
105
Cubic tetrahedron element,
272
Cubic triangular elements,
194–
195
Curved edges, 2D solid elements with,
200
Curved surfaces, elements with,
277
Cylindrical coordinate system,
188
D
frame element in space with,
143
of linear triangular element,
164
Diamond-shaped bicycle frame,
150–
151
Discrete numerical methods,
297
Discrete system equations, –
Discretized system, equations of,
354
Displacement constraints, imposition of,
69
Displacement interpolation,
49–
50
Displacement–time history,
245
Distributed external body force,
19–
21
DOFs. See Degrees of freedom (DOFs)
Domain discretization, ,
47–
49
Dynamic equilibrium equations
for three-dimensional solids,
19–
21
for two-dimensional solids,
25–
26
Dynamic testing, aircraft for,
E
8-Nodal hexahedron elements,
261–
262
8-Nodal isoperimetric quadratic element,
291–
292
Eight-node rectangular thick plate element,
227
Electrostatic micro-motor,
202
Element connectivity, ,
234,
242
Element displacement vector,
142–
144
in global coordinate system,
97–
98
Elements with curved surfaces,
277
Enforcing compatibility,
334–
335
Euler-Bernoulli beam theory,
30–
31
Explicit approaches,
F
FDM. See Finite difference method (FDM)
FE. See Finite element (FE)
FEM. See Finite element method (FEM); Finite element model (FEM)
FGM. See Functionally graded material (FGM)
Field problems
heat transfer
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), –,
44
approximation for one-dimensional case,
mesh of bicycle frame,
151
Finite element (FE) mesh,
204,
236
of quantum dot heterostructure,
281
computational modeling using, –
convergence property,
94–
95
definition, –
for beams equations
shape function construction,
112–
115
fundamentals for
free vibration, analysis of,
69–
71
Hamilton’s principle,
45–
47
minimum total potential energy principle,
47
problem formulation,
44–
45
sufficient requirements for,
76–
77
transient response,
71–
76
mathematical models of,
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
cantilever beam problem,
405–
411
Finite strip elements,
298
Finite volume method (FVM), –,
397–
398
First order differential operators,
19–
21
First order shear deformation theory,
35
Flexural vibration modes,
233
Force loading conditions,
419
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
results and discussion,
155–
157
made of three members,
159
Free vibration analysis,
69–
71
symmetric and anti-symmetric conditions for,
320
Functionally graded material (FGM),
106
FVM. See Finite volume method (FVM)
G
Gauss elimination method, ,
155
Gauss integration
linear rectangular elements,
180–
183
points and weight coefficients,
180
Gauss’s divergence theorem,
371–
372
element matrices in,
97–
98
boundary conditions,
91–
92
recovering stress and strain,
92
equations for
Global FE matrices,
98–
101
Gradual damping elements,
297
Grids,
GUI. See Graphical user interface (GUI)
H
Heated road surface, temperature distribution of
results and discussion,
390
Heat insulation boundary,
379
Heat transfer
FEM
1D heat transfer problem,
355–
370
2D heat transfer problem,
370–
386
heated road surface, temperature distribution of,
386–
390
weighted residual approach,
354–
355
through composite wall,
351–
352
Hexahedrons
Higher order 3D tetrahedron elements,
271
High order 3D serendipity elements,
274
Hilber–Hughes–Taylor operator (1978),
244–
245
Homogenous boundary condition,
21
for isotropic materials,
24
for 3D anisotropic materials,
18–
19
I
Implicit approaches,
InAs quantum dots. See Indium arsenide quantum dots
Independent stress components,
16–
18
Indium arsenide (InAs) quantum dots,
279–
281
Infinite domains, methods for,
293–
294
Inhomogenous boundary conditions,
21
Internal nodes, vanish of,
358–
359
shear forces and moments on,
37
Isoparametric element,
188
Isoparametric quadratic element,
292–
293
Isotropic materials,
18–
19
J
K
Kirchhoff plate theory,
35
L
Lagrange multiplier method,
338
Lagrange type elements,
272–
273
dispersive characteristic of,
297
Layered composite wall,
393
Linear elastic fracture mechanics,
290–
291
Linear quadrilateral elements
Linear rectangular elements
shape function construction,
176–
179
Linear triangular elements,
165
field variable interpolation,
164–
166
shape function construction,
166–
170
equations for
finite element equations in,
63–
67
LU decomposition method,
M
Mapping, infinite elements formulated by,
294–
297
Mass matrix for rectangular element,
231–
232
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
equations for plates
constitutive equations,
36
dynamic equilibrium equations,
38
moments and shear forces,
36–
38
Reissner-Mindlin plate theory,
38–
40
equations for three-dimensional solids
boundary conditions,
21–
22
constitutive equations,
18–
19
dynamic equilibrium equations,
19–
21
equations for truss members
constitutive equations,
27
dynamic equilibrium equations,
27–
28
equations for two-dimensional solids
constitutive equations,
24–
25
dynamic equilibrium equations,
25–
26
MEMS. See Micro-electro-mechanical systems (MEMS)
elements, different order of,
310–
312
axisymmetric,
definition,
of hinge joint,
stress distribution,
Micro-electro-mechanical systems (MEMS),
122
Micro-motor
side drive
results and discussion,
208–
211
transient analysis of,
240–
247
Micro-resonant transducer
resonant frequencies of
comparison with ANSYS,
130–
132
results and discussion,
128–
130
Mid-node position distortion,
310
Mindlin plate, shear deformation in,
39
Mindlin plate theory,
221
Minimum total potential energy principle,
47
Modeling techniques
constraints modeling, by rigid body attachment,
335–
336
Lagrange multiplier method,
338
elements, different order of,
310–
312
mirror/plane symmetry,
314–
322
modeling offsets methods,
322–
325
MPC equations
supports, modeling of,
328–
330
Moments and shear forces
equations for beams,
31–
33
equations for plates,
36–
38
MPC. See Multipoint constraints (MPC)
Multipoint constraints (MPC),
312
N
Natural boundary condition,
379
and local coordinate system,
112–
113
Natural frequencies of micro-motor,
233–
240
for 3D solid elements,
260–
261
Nodal temperatures of road surface,
391
Nodes, –,
Numerical integration scheme,
266–
268
O
1D axisymmetric elements, cylindrical shell structure modeled using,
320
1D heat transfer problem
direct assembly procedure,
361–
362
of rectangular cross-section,
363
P
Pascal triangle of monomials,
51
Perturbation parameter,
409
Physical trial-and-error design procedure,
150–
151
idea of superimposition,
137
in global coordinate system,
140–
142
in local coordinate system,
137–
140
Plane stress conditions,
162–
163
Plate
constitutive equations,
36
dynamic equilibrium equations,
38
moments and shear forces,
36–
38
Reissner-Mindlin plate theory,
38–
40
Polynomial shape functions,
298–
299
Polysilicon, elastic properties of,
123,
202
Pre-processing, –
Q
Quadratic convergence,
95
Quadratic one-dimensional element,
104
Quadratic triangular elements,
193–
194
meshes, –
unacceptable shapes of,
307–
309
Quadrilateral shell elements,
236
Quantum dot heterostructure, stress and strain analysis of,
277–
286
R
Rectangular domain, meshed with triangular elements,
165,
176
and coordinate systems,
177
Lagrange type elements,
195–
196
of arbitrary high orders,
196,
273
serendipity type elements,
196–
200
Rectangular hexahedron element,
266–
268
Rectangular shell element,
229
Reissner–Mindlin plate theory,
35,
221–
223
equations for plates,
38–
40
Residual method, –
Resonant frequencies of micro-resonant transducer
comparison with ANSYS,
130–
132
results and discussion,
128–
130
Resonant micro-beam strain transducer,
122
Rigid slab on elastic foundation,
336
S
Sandwiched composite wall,
393
Second order differential operators.,
19–
21
Semi-automatic mesh generator,
Serendipity type elements,
273–
277
construction of 8-node,
198
S-FEM. See Smoothed finite element methods (S-FEM)
delta function property,
167
linear rectangular elements,
176–
179
linear triangular elements,
166–
170
Shear deformation in plate,
222
Simply supported anti-symmetric beam structure,
316
Simply supported symmetric beam structure,
316
Single point constraint (SPC),
63,
314–
315
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
Solid finite elements,
249–
250
Solids and structures, mechanics for. See mechanics for solids and structures
SOR method. See Successive over-relaxation (SOR) method
Spatial frame structure, three-dimensional,
135–
136
SPC. See Single point constraint (SPC)
SPD. See Symmetric Positive Definite (SPD)
Special purpose elements and methods,
289
SPH. See Smoothed particle hydrodynamics (SPH)
S-PIM. See Smoothed Point Interpolation Methods (S-PIM)
Sprocket-chain system, finite element mesh for,
307
Standard finite element mesh,
295–
296
State of stresses, equilibrium equations,
20
∗STEADY STATE DYNAMICS,
402–
403
Steady state equation,
355
Stiffened plates with offset,
327
Strain-displacement relationships,
16–
18,
24
linear quadrilateral elements,
186–
187
linear rectangular elements,
179
linear triangular elements,
172–
174
Stranski-Krastanow (SK) growth mode,
277–
278
Stress
Stress and strain equations
for three-dimensional solids,
16–
18
for two-dimensional solids,
22–
24
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,
Superimposition
Superparametric elements,
188
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
Tetrahedrons to hexahedrons,
269
Thermal conductive properties,
351
Euler-Bernoulli assumption for,
31
32-node tri-cubic element,
276–
277
3D mesh
Three-dimensional solid element mesh,
333
nodal force vector for,
260–
261
with curved surfaces,
278
Three-dimensional (3D) solids
boundary conditions,
21–
22
constitutive equations,
18–
19
dynamic equilibrium equations,
19–
21
Three-dimensional spatial frame structure,
135–
136
‘Tied’ contact condition,
281–
282
Time stepping, implicit and explicit approaches,
Torsional bar element,
142–
144
Transient analysis of micro-motor,
240–
247
Transient response,
71–
76
central difference algorithm,
72–
75
Translational displacements,
229–
230
Transverse displacement components,
241
Trapezoidal cross-sections (TRAPEZOID),
406–
407
general formulation of shape functions,
191–
193
rectangular domain meshed with,
165,
176
Triangular truss structure,
95–
96
connected by ridged bar,
110
convergence property of,
94–
95
dimensions and properties of,
97
element matrices in global coordinate system
boundary conditions,
91–
92
recovering stress and strain,
92
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
transformation matrix,
87–
88
Truss members
constitutive equations,
27
cross-sectional dimension of solid,
27
dynamic equilibrium equations,
27–
28
Truss structure, three member,
96
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
3D structure modeled using,
321
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
point heat source or sink,
384–
386
2D solids
constitutive equations,
24–
25
dynamic equilibrium equations,
25–
26
elements
for axisymmetric structures,
188–
191
linear quadrilateral elements
linear rectangular elements
shape function construction,
176–
179
linear triangular elements
field variable interpolation,
164–
166
shape function construction,
166–
170
rectangular elements
Lagrange type elements,
195–
196
serendipity type elements,
196–
200
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
Virtually designed building, air flow field in,
10–
11
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
Y