Table of Contents

Preface     v
Model of a Taut Wire     1
Deriving the PDE model     1
Balance equation     2
Boundary conditions     2
Boundary conditions (in space)     3
Initial conditions (boundary conditions in time)     4
Anything else?     5
The Method of Galerkin     7
Residual of the balance equation     7
Integral test of the residual     8
Test function     8
Trial function     10
Manipulation of the residuals     11
Stiffness and mass matrix     13
Piecewise linear basis functions     15
How are the Galerkin and Finite Element Methods Related     17
Numerical quadrature     18
Putting it together: system of ODE's     21
Exercises     22
Statics and Dynamics Examples for the Wire Model     27
Statics     28
Statics: uniform load     28
Free vibration     32
Integration of transient motion     33
Using built-in Matlab solver     34
Using the Trapezoidal integrator     35
Exercises     38
Boundary Conditions for the Model of aTaut Wire     41
Mixed essential and natural boundary conditions     42
Essential boundary conditions only     43
Natural boundary conditions only     43
Overspecified boundary conditions     44
Model of Heat Conduction     49
Balance equation     49
Constitutive equation     52
Boundary conditions     53
On the sufficiency of boundary conditions     54
Initial condition     55
Summary of the PDE model of heat conduction     56
Exercises     56
Galerkin Method for the Model of Heat Conduction     57
Weighted residual formulation     57
Reducing the model dimension     59
Test and trial functions: basis functions on triangulations     61
Basis functions on the standard triangle     63
Discretizing the weighted residual equation     66
Derivatives of the basis functions; Jacobian     70
Numerical integration     74
Conductivity matrix     76
Surface heat transfer matrix and load     80
Exercises     86
Steady-state Heat Conduction Solutions     89
Steady-state heat conduction equation      89
Thick-walled tube     89
Orthotropic insert     93
The T4 NAFEMS Benchmark     96
Transient Heat Conduction Solutions     101
Discretization in time for transient heat conduction     101
The T3 NAFEMS Benchmark     104
Transient cooling in a shrink-fitting application     107
Expanding the Library of Element Types     111
Quadratic triangle T6     112
Quadratic 1-D element L3     114
Point element P1     114
Integrating over n-dimensional domains     115
Tetrahedron T4     120
Simplex elements     122
Quadrilateral Q4     123
Hexahedron H8     124
Extracting the mesh boundary     124
Exercises     126
Discretization Error, Error Control, and Convergence     129
Interpolation errors     129
Interpolation error for temperature     129
Interpolation error for temperature gradient     132
Controlling the error; Convergence rate     134
Richardson extrapolation     136
The T4 NAFEMS Benchmark revisited     138
Graded meshes     139
Shrink fitting revisited      139
Representing functions by interpolation     141
Exercises     143
Model of Elastodynamics     145
Balance of linear momentum     145
Stress     147
Balance of angular momentum and stress symmetry.     150
Local equilibrium     152
Change of linear momentum     152
Stress divergence     152
All together now     156
Strains and displacements     156
Constitutive equation     159
Boundary conditions     161
Example: concrete dam     161
Example: rigid punch     162
Formal definition of the boundary conditions     163
Inadmissible "concentrated" boundary conditions     164
Symmetry and anti-symmetry     166
Example: a pure-traction problem     168
Example: shaft under torsion     170
Example: overspecified boundary conditions     172
Initial conditions     172
Galerkin Formulation for Elastodynamics     175
Manipulation of the residuals     175
The first two steps     175
Step 3: Preliminaries     176
Step 3: The glorious conclusion     177
Method of weighted residuals as the principle of virtual work     179
Discretizing     179
The trial function     179
The test function     181
Producing the requisite equations     182
The discrete equations: system of ODE's     184
Inertial term: Mass matrix     185
Body loads and traction loads     186
Resisting forces: Stiffness matrix     186
Summary of the elastodynamics ODE's     187
Constitutive equations of linearly elastic materials     188
General anisotropic material     188
Orthotropic material     188
Transversely isotropic material     189
Isotropic material     190
Imposed (thermal) strains     191
Strain-displacement matrix     193
Transformation of basis     194
Stiffness matrix     197
Pure-traction problems and singular stiffness     199
Exercises     200
Finite Elements for True 3-D Problems     201
Modal analysis with the tetrahedron T4: the drum     201
Modal analysis with the tetrahedron T4: the composite rod     204
Tetrahedron T10     207
Example: the drum revisited      208
The composite rod with the tetrahedron T10     209
Static analysis with hexahedra H8 and H20     210
Hexahedron H8     210
Dilatational locking     211
Shear locking     214
Thin clamped square plate with concentrated load     215
Quadratic element H20     216
Quadratic element Q8     220
Pinched cylinder     221
Pinched sphere     222
Beam deflection revisited     223
Errors, validation, and verification     224
Verification and Prediction     226
Validation     227
Errors     227
Using modeling to make predictions     227
Using benchmarks     228
Exercises     230
Analyzing the Stresses     231
Singularities     231
Interpretation of stresses     234
Stress concentrations     235
Plane Strain, Plane Stress, and Axisymmetric Models     237
Plane strain model reduction     237
Plane stress model reduction     240
Model reduction for axial symmetry     242
Material stiffness for two-dimensional models     245
Strain-displacement matrices for two-dimensional models     246
Integration for two-dimensional models     247
Thermal strains in two-dimensional models     249
Examples     250
Thermal strains in a bimetallic assembly     250
Orthotropic balloon     254
Transient dynamic analysis     257
Centered difference time stepping     257
Example: stress wave propagation     259
Exercises     263
Consistency + Stability = Convergence     265
Consistency     265
Completeness     265
Compatibility     267
Stability     268
Conclusion     269
Exercises     270
Bibliography     271
Index     273