Visualization of Fields and Applications in Engineering

Visualization of Fields
and Applications
in Engineering......Page 3
Contents......Page 7
Preface......Page 11
1.1 A General View......Page 13
1.2 Historical Development and Progress in Visual Science......Page 15
1.3 Scientific Visualization Philosophy, Techniques and Challenges......Page 19
2.1 Lagrangian/Eulerian Description and Transformation......Page 23
2.2 Curvilinear Coordinates......Page 27
2.2.1 Polar Coordinate......Page 36
2.2.2 Streamline (Flux Line) Coordinates......Page 41
2.2.3 Potential-Stream Function Coordinates......Page 55
2.3.1 Field Line Trajectory......Page 61
2.3.2 Field Line Integral Curves......Page 62
2.3.3 Field Lines, Material Lines and Path Lines......Page 66
2.3.4 Streamlines (Flux Lines)......Page 68
3.1 Field Models and Concepts......Page 75
3.2 Scalar Fields and Representation......Page 77
3.3 Vector Fields and Representation......Page 80
3.4 Vector Icons and Classifications......Page 81
3.4.2 Classification Based on Information Levels......Page 82
3.5 Scalar Potential......Page 83
3.6 Vector Potential......Page 86
3.7.1 Helmholtz’s Theorem......Page 89
3.8 Tensor Contraction and Transport Process Visualization......Page 91
3.8.1 Mechanical Energy Function and Heatfunction......Page 92
3.8.2 Strain Energy Trajectory and Strain Function......Page 96
3.9 Multiple Fields......Page 97
4.1 Complex Variables/Functions and Applications......Page 109
4.2 Complex Analysis and Cauchy–Riemann Equation......Page 112
4.3 Differentiation of Complex Function......Page 113
4.4 Integration of Complex Functions......Page 116
4.5.1 Trajectory Method......Page 119
4.5.2 Method of Curvilinear Squares......Page 120
4.5.3 Transfer Characteristics and Field Property Evaluation......Page 123
4.6 Example 4.1a Visualization of Heat and Fluid Transport in a Corner......Page 126
5.1 Introduction......Page 139
5.2.1 Congruent Mapping......Page 141
5.2.2 Similitude Mapping......Page 143
5.2.3 Affine Mapping......Page 144
5.3 Inversion Mapping......Page 145
5.3.1 Circle Inversion......Page 146
5.4 Mapping with Complex Functions......Page 147
5.5 Conformal Mapping and Applications......Page 149
5.6 Hodograph Method and Mapping......Page 159
5.6.1 Conjugate Hodograph......Page 160
5.7 Hodograph Representations and Applications......Page 161
5.7.1 Straight Boundaries......Page 168
5.7.2 Free Surface......Page 170
5.7.3 Special Field Patterns......Page 172
5.7.4 Projectile Trajectory in Constant Force Fields......Page 175
5.7.5 Motion Trajectory in Central Force Fields......Page 181
5.7.6 Trajectory of Charged Particles in Uniform Magnetic Fields......Page 191
5.8 Example 4.1b Mapping of Field Patterns and Image Warping......Page 195
6.1 Introduction......Page 211
6.2.2 Tensor Field Line Trajectories (Lines of Principal Stress)......Page 212
6.3 Tensor Description and Representation......Page 213
6.4 Tensor Decomposition and Tensor Rank Reduction......Page 216
6.4.1 Strain Tensor and Stress Tensor......Page 218
6.4.2 Rotation Tensor......Page 219
6.4.3 Rate of Strain Tensor and Viscous Stress Tensor......Page 220
6.4.4 Vorticity Tensor......Page 222
6.4.5 Tensor Contractions: Tensor Vector on a Reference Plane......Page 225
6.5.1 Tensor Invariants......Page 226
6.5.3 Principal States and Eigenanalysis......Page 229
6.5.4 Hybrid Method of Tensor Visualization......Page 239
6.6.1 Vorticity Concepts and Dynamics......Page 240
6.6.2 Forced Vortex......Page 244
6.6.3 Free Vortex......Page 249
6.6.4 Vortices Transport and Vorticity Function......Page 252
6.7 Example: 4.1c Convective Momentum Flux Tensor Visualization......Page 254
7.1 Introduction......Page 261
7.2 Complex Analysis of Critical Point......Page 263
7.3 Critical Point Theory and Classification......Page 269
7.3.1 Symmetric Tensor: [∇V] = [∇V]T ;
Im1 = Im2 = 0......Page 273
7.3.3 Asymmetric Tensor......Page 274
7.4 Example 4.1d Critical Point Topology......Page 275
7.5 Singular Point Visualization and Mapping......Page 277
7.6 Example 7.1 Mapping of a Point Source......Page 278
8.1 Example 8.1: Torsion of a Square Beam......Page 285
8.2 Example 8.2: Bending of a Cantilever Beam Subject to a Point Load......Page 314
8.3 Example 8.3: Squeezing Flow and Vorticity Transport......Page 335
8.4 Example 8.4: Groundwater Flows in an Anisotropic Porous Medium......Page 357
References......Page 377
Index......Page 381