PRAGMATIC INTR TO THE FINITE ELEMENT..,A : With The Matlab Toolkit Sofea - KRYSL PETR

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PRAGMATIC INTR TO THE FINITE ELEMENT..,A

With The Matlab Toolkit Sofea

By: KRYSL PETR

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Paperback | 26 October 2006

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292 Pages

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22.86 x 15.24 x 1.55

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This textbook provides an accessible and self-contained description of the Galerkin finite element method for the two important models of continuum mechanics, transient heat conduction and elastodynamics, from formulation of the governing equations to implementation in Matlab.The coverage follows an intuitive approach: the salient features of each initial boundary value problem are reviewed, including a thorough description of the boundary conditions; the method of weighted residuals is applied to derive the discrete equations; and clear examples are introduced to illustrate the method.

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Non-Fiction Computing & I.T.Business Applications Mathematical & Statistical Software Science Science in General Physics Classical Mathematics Engineering & Technology Mechanical Engineering & Materials Mechanical Engineering Civil Engineering

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

PRAGMATIC INTR TO THE FINITE ELEMENT..,A

With The Matlab Toolkit Sofea

At a Glance

Paperback

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