The Finite Element Method – A Practical Course

The Finite Element Method – A Practical Course
اسم المؤلف
C.R. Liu & S.S. Quek
التاريخ
14 أغسطس 2021
المشاهدات
التقييم
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The Finite Element Method – A Practical Course
C.R. Liu & S.S. Quek
CONTENTS
Biographical Information ix
Preface xi
1 Computational Modelling 1
1.1 Introduction 1
1.2 Physical Problems in Engineering 3
1.3 Computational Modelling using the FEM 4
1.4 Simulation 7
1.5 Visualization 9
2 Introduction to Mechanics for Solids and Structures 12
2.1 Introduction 12
2.2 Equations for Three-Dimensional Solids 13
2.3 Equations for Two-Dimensional Solids 19
2.4 Equations for Truss Members 22
2.5 Equations for Beams 24
2.6 Equations for Plates 28
2.7 Remarks 34
3 Fundamentals for Finite Element Method 35
3.1 Introduction 35
3.2 Strong and Weak Forms 36
3.3 Hamilton’s Principle 37
3.4 FEM Procedure 38
3.5 Static Analysis 58
3.6 Analysis of Free Vibration (Eigenvalue Analysis) 58
3.7 Transient Response 60
3.8 Remarks 64
3.9 Review Questions 65
vvi CONTENTS
4 FEM for Trusses 67
4.1 Introduction 67
4.2 FEM Equations 67
4.3 Worked Examples 76
4.4 High Order One-Dimensional Elements 87
4.5 Review Questions 88
5 FEM for Beams 90
5.1 Introduction 90
5.2 FEM Equations 90
5.3 Remarks 95
5.4 Worked Examples 95
5.5 Case study: Resonant Frequencies of Micro Resonant Transducer 98
5.6 Review Questions 107
6 FEM for Frames 108
6.1 Introduction 108
6.2 FEM Equations for Planar Frames 109
6.3 FEM Equations for Space Frames 114
6.4 Remarks 120
6.5 Case Study: Finite Element Analysis of a Bicycle Frame 121
6.6 Review Questions 127
7 FEM for Two-Dimensional Solids 129
7.1 Introduction 129
7.2 Linear Triangular Elements 131
7.3 Linear Rectangular Elements 141
7.4 Linear Quadrilateral Elements 148
7.5 Higher Order Elements 153
7.6 Elements with Curved Edges 160
7.7 Comments on Gauss Integration 161
7.8 Case Study: Side Drive Micro-Motor 162
7.9 Review Questions 171
8 FEM for Plates and Shells 173
8.1 Introduction 173
8.2 Plate Elements 173
8.3 Shell Elements 180
8.4 Remarks 184
8.5 Case Study: Natural Frequencies of Micro-Motor 185
8.6 Case Study: Transient Analysis of a Micro-Motor 192
8.7 Review Questions 198CONTENTS vii
9 FEM for 3D Solids 199
9.1 Introduction 199
9.2 Tetrahedron Element 200
9.3 Hexahedron Element 209
9.4 Higher Order Elements 216
9.5 Elements with Curved Surfaces 222
9.6 Case Study: Stress and Strain Analysis of a Quantum Dot
Heterostructure 223
9.7 Review Questions 232
10 Special Purpose Elements 233
10.1 Introduction 233
10.2 Crack Tip Elements 234
10.3 Methods for Infinite Domains 236
10.4 Finite Strip Elements 242
10.5 Strip Element Method (SEM) 245
11 Modelling Techniques 246
11.1 Introduction 246
11.2 CPU Time Estimation 247
11.3 Geometry Modelling 248
11.4 Meshing 250
11.5 Mesh Compatibility 254
11.6 Use of Symmetry 256
11.7 Modelling of Offsets 265
11.8 Modelling of Supports 270
11.9 Modelling of Joints 271
11.10 Other Applications of MPC Equations 274
11.11 Implementation of MPC Equations 278
11.12 Review Questions 280
12 FEM for Heat Transfer Problems 282
12.1 Field Problems 282
12.2 Weighted Residual Approach for FEM 288
12.3 1D Heat Transfer Problem 289
12.4 2D Heat Transfer Problem 303
12.5 Summary 316
12.6 Case Study: Temperature Distribution of Heated Road Surface 318
12.7 Review Questions 321viii CONTENTS
13 Using ABAQUS© 324
13.1 Introduction 324
13.2 Basic Building Block: Keywords and Data Lines 325
13.3 Using Sets 326
13.4 ABAQUS Input Syntax Rules 327
13.5 Defining a Finite Element Model in ABAQUS 329
13.6 General Procedures 339
References 342
Index 34
INDEX
ABAQUS input file, 327, 329
calculate eigenvalues of beam, 100
calculate low speed impact on bicycle frame,
122
calculation of 2D heat transfer, 318
calculation of stress distribution in quantum
dot structure, 227
cantilever beam with downward force, 337
eigenvalue analysis of micro-motor, 186
static analysis of mico-motor, 165
transient analysis of micro-motor, 193
ABAQUS input syntax rules, 327
Acoustic, 287
Admissible displacement, 37
Area coordinates, 137–138
Axisymmetric
elements, 262
loading, 250
mesh, 8
model, 224
solids, 250
Bandwidth, 247
Beam element(s), 90–107, 108
Beam(s), 5, 24, 90
Bending of beams, 24
Bending of plates, 28
Boundary conditions
cyclic, 264
essential,13, 18–19, 309
for 2D heat transfer, 308,
homogenuous, 19
in abaqus input file, 330, 334
infinite, 245
inhomogenuous, 19
natural, 13, 18–19
symmetric, 258, 274
Boundary element method (BEM), 242, 324
Central difference algorithm, 60
Comment lines, 327
Compatibility
equations, 37
of displacements, 243
of mesh, 254, 274
Complexity of linear algebraic system, 247
Computational modelling, 4
Conforming element, 177, 180
Consistency, 44–46, 65
Constant strain element, 204
Constant stress element, 204
Constitutive equations
beam, 26
plate, 30
one-dimensional solid, 24
three-dimensional solid, 16
two-dimensional solid, 21
Constraints, 57, 330
Continuity, 36, 254–255
Convective boundary condition, 310
Convergence, 78–79, 100
Coordinate mapping, 142, 148–149
Coordinate transformation, 55, 71, 73–74,
112–113, 116–117
CPU time, 247
Crack-like behaviour, 254
Crack tip elements, 233–234
Cubic element(s), 88, 160, 218–219, 222
345346 INDEX
Damping, 242
coefficients, 60
Data lines, 325, 328
Delta function property, 46–52, 64
Direct assembly, 82, 84, 294
Direct integration method, 58, 60, 195–196
Discrete system equations, 8, 38
Discretization, 4, 38–39
Displacement constraints, 57
Displacement interpolation, 39
Displacement method, 41
Dynamic equilibrium equation, 17–18, 24, 28,
32
Effort to accuracy ratio, 246
Eigenvalue analysis, 58–60, 102, 186
Element distortion, 250–252
Element force vector, 55, 307, 308
Element mass matrix, 111, 140, 146–147, 152,
178
Element matrices
2D solid element, 140–141, 145, 152
3D solid element, 204–208, 213–215
beam element, 93–94
frame element, 109–112, 115–116
heat transfer element, 294, 296–297, 300,
302–303, 308
plate element, 178
shell element, 180–183
truss element, 71
Element stiffness matrix, 53, 71, 140, 145, 152,
178–179, 230, 251, 299, 305
Elements with curved edges, 160, 223
Elements with curved surfaces, 222
Euler–Bernoulli beam theory, 25, 121
Field problems, 282
Finite difference method, 8, 36, 60
Fin
one-dimensional, 284, 289
two-dimensional, 282–283
Finite element method (FEM)
equations, 67, 90, 109, 114
procedure, 38, 64
Finite strip elements, 233, 242
Finite strip method, 243
Finite volume method (FVM), 8, 324
Flexural vibration modes, 185–186, 188
Fluid flow, 287
Force boundary condition, 19
Fourier
series, 195–196
superimposition, 263
Frame element, 108
Frame structure, 108, 112
Free vibration, 58
Functional, 37, 54, 279
Galerkin method, 289
Gauss elimination, 9, 58
Gauss integration, 145, 161, 213
Gauss points, 145–146, 161
Gauss’s divergence theorem, 303
Geometry modelling, 248
Global coordinate system, 55–56, 71–74,
112–114, 116–117, 184
h-adaptivity, 79
Hamilton’s principle, 37–39
Heat insulation boundary, 310
Heat source/sink, 314–315
Heat transfer
1D problem, 289
2D problem, 303
across a composite wall, 285, 298
in a long two-dimensional body, 283
in a one-dimensional fin, 284, 289, 296
in a two-dimensional fin, 282
Helmholtz equation, 282
Hexahedron element, 209
Higher order elements
brick, 218
one-dimensional, 87
plate, 180
rectangular, 156
tetrahedron, 216
triangular, 153
History data, 329, 331
Homogeneous equation, 58
Implicit method, 63
Infinite domains, 233, 236INDEX 347
Infinite element, 237
Initial conditions, 4, 61, 246, 330
Integration by parts, 290
Irrotational fluid flow, 287
Isoparametric element, 152
Jacobian matrix, 151–152, 207, 212–213
Joints
modelling, 271
Keyword lines, 326, 328
Kinetic energy, 37, 53, 177
Lagrange interpolants, 87
Lagrange multiplier method, 279
Lagrange type elements, 156, 218
Lagragian functional, 37, 54
Lamb waves, 240
Linear element(s), 70, 130
Linear field reproduction, 47, 50–52, 64–65
Linear independence, 46
Linear quadrilateral elements, 148
Linear rectangular elements, 141
Linear triangular elements, 131
Mass matrix, 53
2D solid element, 140, 146–147, 152
3D solid element, 204–205, 213- 215
beam element, 94
frame element, 111, 116
plate element, 178
shell element, 183
truss element, 71, 74
Matrix inversion, 63
Mechanics for solids and structures, 3, 12
Membrane effects, 182–184
Mesh
compatibility, 254–255, 274, 276
density, 250, 276
generation, 6
Mindlin plate, 28, 32–33, 174–175
Modal analysis, 58–59
Model data, 329–330
Moments, 26–28, 30–31
Multi-point constraints (MPC), 267, 271,
273–279
Natural coordinates, 91, 142
Newmark’s method, 63
Nodal interpolation functions, see shape
functions
Non-conforming, 174
Order of elements, 254
Offsets, 265–269
p-adaptivity, 79
Partitions of unity, 47–49, 52, 64–65
Pascal pyramid, 42
Pascal triangle, 41–42
Penalty method, 279
Planar frame element, 109, 110
Planar truss, 74
Plane strain, 20–22, 129–130
Plane stress, 20–21, 129–130
Plate element(s), 5, 28–29, 173–180
Poisson’s equation, 286, 287
Polynomial
basis functions, 43
integrand, 145, 161
interpolation, 39
Potential energy, 37
Quadratic element(s), 130, 157, 216, 219
Quadrilateral element(s), 6, 148–149, 151
Rate of convergence, 79
Rectangular element(s), 141–148, 156–158,
307–308
Reproduction property, 44–45, 50, 65, 78
Serendipity type elements, 157, 219
Shape functions
properties, 44, 65
standard procedure for constructing, 41
sufficient requirements, 64
Single point constraint, 51, 258
Singularity point, 234348 INDEX
Singularity elements, see crack tip elements
Space frame, 108–109, 114–120
Space truss(es), 67
Static analysis, 58
Steady state heat transfer, 282, 289
Straddling elements, 255
Strain displacement relation/relationship(s), 16,
21, 23
Strain energy, 37, 52, 179
Strain matrix, 53, 291
beam element, 93
hexahedron element, 209
linear quadrilateral element, 151
linear rectangular element, 144, 307
linear triangular element, 139, 306
mindlin plate element, 179
tetrahedron element, 200
truss element, 70
Streamline function, 287
Stress intensity factor, 234
Strip element method (SEM), 240, 242, 245
Strong form, 36, 38
Subparametric elements, 153
Subspace iteration, 59
Superparametric elements, 153
Supports
modelling, 270
Symmetric positive definite (SPD), 57, 59
Symmetry, 256
axial, 262
cyclic, 264
mirror, 256
repetitive, 264
Tetrahedron element, 200–207, 216–218
Torsional
deformation, 282, 286
element, 115
Transformation matrix, 56, 72, 74, 113, 117, 184
Transient
analysis, 192
dynamic analysis, 336, 341
response, 60
Triangular element(s), 131–141, 153–156, 289,
305–306
Truss(es), 22–24, 67–89
Two-dimensional (2D) heat transfer, 303, 318
Variational principle(s), 36, 38
Vibration, 12
modes, 59
problems, 240
Visualization, 1, 9
Volume coordinates, 201–202
Wave propagation, 240, 242
Weak form, 19, 36
Weighted residual
approach, 305
method, 8, 36

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