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Pragmatic Introduction To The Finite Element Method For Thermal And Stress Analysis, A : With The Matlab Toolkit Sofea - Petr Krysl

Pragmatic Introduction To The Finite Element Method For Thermal And Stress Analysis, A

With The Matlab Toolkit Sofea

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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.

Prefacep. v
Model of a Taut Wirep. 1
Deriving the PDE modelp. 1
Balance equationp. 2
Boundary conditionsp. 2
Boundary conditions (in space)p. 3
Initial conditions (boundary conditions in time)p. 4
Anything else?p. 5
The Method of Galerkinp. 7
Residual of the balance equationp. 7
Integral test of the residualp. 8
Test functionp. 8
Trial functionp. 10
Manipulation of the residualsp. 11
Stiffness and mass matrixp. 13
Piecewise linear basis functionsp. 15
How are the Galerkin and Finite Element Methods Relatedp. 17
Numerical quadraturep. 18
Putting it together: system of ODE'sp. 21
Exercisesp. 22
Statics and Dynamics Examples for the Wire Modelp. 27
Staticsp. 28
Statics: uniform loadp. 28
Free vibrationp. 32
Integration of transient motionp. 33
Using built-in Matlab solverp. 34
Using the Trapezoidal integratorp. 35
Exercisesp. 38
Boundary Conditions for the Model of a Taut Wirep. 41
Mixed essential and natural boundary conditionsp. 42
Essential boundary conditions onlyp. 43
Natural boundary conditions onlyp. 43
Overspecified boundary conditionsp. 44
Model of Heat Conductionp. 49
Balance equationp. 49
Constitutive equationp. 52
Boundary conditionsp. 53
On the sufficiency of boundary conditionsp. 54
Initial conditionp. 55
Summary of the PDE model of heat conductionp. 56
Exercisesp. 56
Galerkin Method for the Model of Heat Conductionp. 57
Weighted residual formulationp. 57
Reducing the model dimensionp. 59
Test and trial functions: basis functions on triangulationsp. 61
Basis functions on the standard trianglep. 63
Discretizing the weighted residual equationp. 66
Derivatives of the basis functions; Jacobianp. 70
Numerical integrationp. 74
Conductivity matrixp. 76
Surface heat transfer matrix and loadp. 80
Exercisesp. 86
Steady-state Heat Conduction Solutionsp. 89
Steady-state heat conduction equationp. 89
Thick-walled tubep. 89
Orthotropic insertp. 93
The T4 NAFEMS Benchmarkp. 96
Transient Heat Conduction Solutionsp. 101
Discretization in time for transient heat conductionp. 101
The T3 NAFEMS Benchmarkp. 104
Transient cooling in a shrink-fitting applicationp. 107
Expanding the Library of Element Typesp. 111
Quadratic triangle T6p. 112
Quadratic 1-D element L3p. 114
Point element P1p. 114
Integrating over n-dimensional domainsp. 115
Tetrahedron T4p. 120
Simplex elementsp. 122
Quadrilateral Q4p. 123
Hexahedron H8p. 124
Extracting the mesh boundaryp. 124
Exercisesp. 126
Discretization Error, Error Control, and Convergencep. 129
Interpolation errorsp. 129
Interpolation error for temperaturep. 129
Interpolation error for temperature gradientp. 132
Controlling the error; Convergence ratep. 134
Richardson extrapolationp. 136
The T4 NAFEMS Benchmark revisitedp. 138
Graded meshesp. 139
Shrink fitting revisitedp. 139
Representing functions by interpolationp. 141
Exercisesp. 143
Model of Elastodynamicsp. 145
Balance of linear momentump. 145
Stressp. 147
Balance of angular momentum and stress symmetry.p. 150
Local equilibriump. 152
Change of linear momentump. 152
Stress divergencep. 152
All together nowp. 156
Strains and displacementsp. 156
Constitutive equationp. 159
Boundary conditionsp. 161
Example: concrete damp. 161
Example: rigid punchp. 162
Formal definition of the boundary conditionsp. 163
Inadmissible "concentrated" boundary conditionsp. 164
Symmetry and anti-symmetryp. 166
Example: a pure-traction problemp. 168
Example: shaft under torsionp. 170
Example: overspecified boundary conditionsp. 172
Initial conditionsp. 172
Galerkin Formulation for Elastodynamicsp. 175
Manipulation of the residualsp. 175
The first two stepsp. 175
Step 3: Preliminariesp. 176
Step 3: The glorious conclusionp. 177
Method of weighted residuals as the principle of virtual workp. 179
Discretizingp. 179
The trial functionp. 179
The test functionp. 181
Producing the requisite equationsp. 182
The discrete equations: system of ODE'sp. 184
Inertial term: Mass matrixp. 185
Body loads and traction loadsp. 186
Resisting forces: Stiffness matrixp. 186
Summary of the elastodynamics ODE'sp. 187
Constitutive equations of linearly elastic materialsp. 188
General anisotropic materialp. 188
Orthotropic materialp. 188
Transversely isotropic materialp. 189
Isotropic materialp. 190
Imposed (thermal) strainsp. 191
Strain-displacement matrixp. 193
Transformation of basisp. 194
Stiffness matrixp. 197
Pure-traction problems and singular stiffnessp. 199
Exercisesp. 200
Finite Elements for True 3-D Problemsp. 201
Modal analysis with the tetrahedron T4: the drump. 201
Modal analysis with the tetrahedron T4: the composite rodp. 204
Tetrahedron T10p. 207
Example: the drum revisitedp. 208
The composite rod with the tetrahedron T10p. 209
Static analysis with hexahedra H8 and H20p. 210
Hexahedron H8p. 210
Dilatational lockingp. 211
Shear lockingp. 214
Thin clamped square plate with concentrated loadp. 215
Quadratic element H20p. 216
Quadratic element Q8p. 220
Pinched cylinderp. 221
Pinched spherep. 222
Beam deflection revisitedp. 223
Errors, validation, and verificationp. 224
Verification and Predictionp. 226
Validationp. 227
Errorsp. 227
Using modeling to make predictionsp. 227
Using benchmarksp. 228
Exercisesp. 230
Analyzing the Stressesp. 231
Singularitiesp. 231
Interpretation of stressesp. 234
Stress concentrationsp. 235
Plane Strain, Plane Stress, and Axisymmetric Modelsp. 237
Plane strain model reductionp. 237
Plane stress model reductionp. 240
Model reduction for axial symmetryp. 242
Material stiffness for two-dimensional modelsp. 245
Strain-displacement matrices for two-dimensional modelsp. 246
Integration for two-dimensional modelsp. 247
Thermal strains in two-dimensional modelsp. 249
Examplesp. 250
Thermal strains in a bimetallic assemblyp. 250
Orthotropic balloonp. 254
Transient dynamic analysisp. 257
Centered difference time steppingp. 257
Example: stress wave propagationp. 259
Exercisesp. 263
Consistency + Stability = Convergencep. 265
Consistencyp. 265
Completenessp. 265
Compatibilityp. 267
Stabilityp. 268
Conclusionp. 269
Exercisesp. 270
Bibliographyp. 271
Indexp. 273
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789812704115
ISBN-10: 9812704116
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 292
Published: 1st October 2006
Country of Publication: SG
Dimensions (cm): 21.59 x 14.61  x 1.27
Weight (kg): 0.42