Finite Elements : Computational Engineering Sciences

Finite Elements: Computational Engineering Sciences......Page 1
Contents......Page 9
Preface......Page 13
About the Author......Page 17
Notations......Page 19
1.1 Engineering Simulation......Page 23
1.2 A Problem-Solving Environment......Page 25
1.3 Weak Formulation Essence......Page 26
1.4 Decisions on Forming WSN......Page 28
1.5 Discrete WSh Implementations......Page 30
1.6 Chapter Summary......Page 32
References......Page 33
2.1 Engineering Simulation......Page 35
2.3 Continuum Conservation Principle Forms......Page 36
2.4 Constitutive Closure for Conservation Principle PDEs......Page 38
2.5 Engineering Science Continuum Mechanics......Page 42
References......Page 43
3.1 Example Linear Heat Conduction Solutions......Page 45
3.2 Multidimensional PDEs, Separation of Variables......Page 46
3.3 Mathematical Foundation Essence for GWSN......Page 50
3.4 A Legacy FD Construction......Page 51
3.5 An FD Approximate Solution......Page 53
3.6 Lagrange Interpolation Polynomials......Page 54
3.7 Chapter Summary......Page 55
References......Page 56
4.1 A Steady Heat Conduction Example......Page 57
4.2 Weak Form Approximation, Error Extremization......Page 58
4.3 GWSN Discrete Implementation, FE Trial Space Basis......Page 60
4.4 Finite Element Matrix Statements......Page 62
4.5 Assembly of {WS} to form Algebraic GWSh......Page 64
4.6 Solution Accuracy, Error Distribution......Page 66
4.7 Convergence, Boundary Heat Flux......Page 67
Reference......Page 68
5.1 Introduction......Page 69
5.2 Steady Heat Transfer, n =1......Page 70
5.3 FE k = 1 Trial Space Basis Matrix Library......Page 72
5.4 Object-Oriented GWSh Programming......Page 75
5.5 Higher Completeness Degree Trial Space Bases......Page 78
5.6 Global Theory, Asymptotic Error Estimate......Page 81
5.7 Nonsmooth Data, Theory Generalization......Page 85
5.8 Temperature-Dependent Conductivity, Nonlinearity......Page 87
5.9 Static Condensation, p-Elements......Page 90
5.10 Chapter Summary......Page 92
Computer Labs......Page 94
References......Page 95
6.1 Introduction......Page 97
6.2 The Euler–Bernoulli Beam Equation......Page 98
6.3 Euler–Bernoulli Beam Theory GWSh Reformulation......Page 103
6.4 Timoshenko Beam Theory......Page 109
6.5 Mechanical Vibrations of a Beam......Page 114
6.6 Fluid Mechanics, Potential Flow......Page 122
6.7 Electromagnetic Plane Wave Propagation......Page 126
6.8 Convection–Radiation Finned Cylinder Heat Transfer......Page 127
6.9 Chapter Summary......Page 135
Computer Labs......Page 137
References......Page 138
7.1 Introduction......Page 139
7.2 Multidimensional FE Bases and DOF......Page 140
7.3 Multidimensional FE Operations for {Nk(ζ α)}......Page 143
7.4 The NCk = 1,2 Basis FE Matrix Library......Page 146
7.5 NC Basis {WS}e Template, Accuracy, Convergence......Page 149
7.6 The Tensor Product Basis Element Family......Page 152
7.7 Gauss Numerical Quadrature, k = 1 TP Basis Library......Page 154
7.8 Convection–Radiation BC GWSh Implementation......Page 158
7.9 Linear Basis GWSh Template Unification......Page 162
7.10 Accuracy, Convergence Revisited......Page 163
7.11 Chapter Summary......Page 165
Exercises......Page 166
References......Page 167
8.1 The FD–FE Correlation......Page 169
8.2 The FV–FE Correlation......Page 172
8.3 Chapter Summary......Page 176
Exercises......Page 177
9.1 Introduction......Page 179
9.2 The GalerkinWeak Statement......Page 180
9.3 GWSh Completion for Time Dependence......Page 182
9.4 GWSh + θTS Algorithm Templates......Page 183
9.5 GWSh + θTS Algorithm Asymptotic Error Estimates......Page 185
9.6 Performance Verification Test Cases......Page 186
9.7 Dispersive Error Characterization......Page 189
9.8 A Modified Galerkin Weak Statement......Page 193
9.9 Verification Problem Statements Revisited......Page 196
9.10 Unsteady Heat Conduction......Page 199
9.11 Chapter Summary......Page 201
Computer Labs......Page 202
References......Page 203
10.1 The Problem Statement......Page 205
10.2 GWSh + θTS Formulation Reprise......Page 207
10.3 Matrix Library Additions, Templates......Page 208
10.4 mPDE Galerkin Weak Forms, Theoretical Analyses......Page 210
10.5 Verification, Benchmarking, and Validation......Page 214
10.6 Mass Transport, the Rotating Cone Verification......Page 215
10.7 The Gaussian Plume Benchmark......Page 218
10.8 Steady n-D Peclet Problem Verification......Page 219
10.9 Mass Transport, a Validated n = 3 Experiment......Page 222
10.10 Numerical Linear Algebra, Matrix Iteration......Page 225
10.11 Newton and AF TP Jacobian Templates......Page 232
10.12 Chapter Summary......Page 233
Computer Labs......Page 235
References......Page 236
11.1 Introduction......Page 237
11.2 Structural Mechanics......Page 238
11.3 Structural Mechanics, Virtual Work FE Implementation......Page 241
11.4 Plane Stress/Strain GWSh Implementation......Page 243
11.5 Plane Elasticity Computer Lab......Page 246
11.6 Fluid Mechanics, Incompressible-Thermal Flow......Page 248
11.7 Vorticity-Streamfunction GWSh + θTS Algorithm......Page 253
11.8 An Isothermal INS Validation Experiment......Page 256
11.9 Multimode Convection Heat Transfer......Page 260
11.10 Mechanical Vibrations, Normal Mode GWSh......Page 264
11.11 Normal Modes of a Vibrating Membrane......Page 267
11.12 Multiphysics Solid-Fluid Mass Transport......Page 270
11.13 Chapter Summary......Page 276
Computer Labs......Page 279
References......Page 280
12 Conclusion......Page 283
Index......Page 285