Numerical Electromagnetics: The FDTD Method

✍ Scribed by Umran S. Inan, Robert A. Marshall

• Publisher: Cambridge University Press
• Year: 2011
• Tongue: English
• Leaves: 406
• Edition: 1
• Category: Library

✦ Synopsis

Beginning with the development of finite difference equations, and leading to the complete FDTD algorithm, this is a coherent introduction to the FDTD method (the method of choice for modeling Maxwell's equations). It provides students and professional engineers with everything they need to know to begin writing FDTD simulations from scratch and to develop a thorough understanding of the inner workings of commercial FDTD software. Stability, numerical dispersion, sources and boundary conditions are all discussed in detail, as are dispersive and anisotropic materials. A comparative introduction of the finite volume and finite element methods is also provided. All concepts are introduced from first principles, so no prior modeling experience is required, and they are made easier to understand through numerous illustrative examples and the inclusion of both intuitive explanations and mathematical derivations.

✦ Table of Contents

CONTENTS......Page 9
Preface......Page 15
1.1 Why FDTD?......Page 17
1.2.1 Finite volume time domain......Page 18
1.2.2 Finite difference frequency domain......Page 19
1.2.3 Finite element methods......Page 20
1.3 Organization......Page 21
References......Page 22
2 Review of electromagnetic theory......Page 24
2.1 Constitutive relations and material properties......Page 27
2.2 Time-harmonic Maxwell’s equations......Page 29
2.3 Complex permittivity: dielectric losses......Page 31
2.5 Equivalent magnetic currents......Page 33
2.6 Electromagnetic potentials......Page 35
2.7 Electromagnetic boundary conditions......Page 37
2.8 Electromagnetic waves......Page 39
2.8.1 The wave equation......Page 41
2.8.2 Group velocity......Page 43
2.9 The electromagnetic spectrum......Page 44
2.10 Summary......Page 46
2.11 Problems......Page 48
References......Page 49
3 Partial differential equations and physical systems......Page 50
3.1 Classification of partial differential equations......Page 53
3.1.2 Parabolic PDEs......Page 54
3.1.3 Hyperbolic PDEs......Page 55
3.2 Numerical integration of ordinary differential equations......Page 57
3.2.1 First-order Euler method......Page 58
3.2.2 The leapfrog method......Page 60
3.2.3 Runge-Kutta methods......Page 61
3.3 Finite difference approximations of partial differential
equations......Page 62
3.3.1 Derivatives in time......Page 64
3.3.2 Derivatives in space......Page 67
3.3.3 Finite difference versions of PDEs......Page 68
3.4 Finite difference solutions of the convection equation......Page 69
3.4.1 The forward-time centered space method......Page 70
3.4.2 The leapfrog method......Page 73
3.4.3 The Lax-Wendroff methods......Page 76
3.5 Finite difference methods for two coupled first-order
convection equations......Page 79
3.6 Higher-order differencing schemes......Page 81
3.7 Summary......Page 83
3.8 Problems......Page 84
References......Page 87
4 The FDTD grid and the Yee algorithm......Page 88
4.1 Maxwell’s equations in one dimension......Page 90
4.1.1 Example 1D simulations......Page 93
4.2 Maxwell’s equations in two dimensions......Page 94
4.2.1 Transverse electric (TE) mode......Page 96
4.2.2 Transverse magnetic (TM) mode......Page 97
4.2.3 Example 2D simulations......Page 98
4.3 FDTD expressions in three dimensions......Page 100
4.3.1 Example 3D simulation......Page 103
4.4.1 1D waves in lossy media: waves on lossy transmission
lines......Page 104
4.4.2 2D and 3D waves in lossy media......Page 106
4.5 Divergence-free nature of the FDTD algorithm......Page 108
4.6 The FDTD method in other coordinate systems......Page 109
4.6.1 2D polar coordinates......Page 110
4.6.2 2D cylindrical coordinates......Page 113
4.6.3 3D cylindrical coordinates......Page 117
4.6.4 3D spherical coordinates......Page 120
4.7 Summary......Page 123
4.8 Problems......Page 124
References......Page 128
5 Numerical stability of finite difference methods......Page 129
5.1 The convection equation......Page 130
5.1.1 The forward-time centered space method......Page 131
5.1.2 The Lax method......Page 132
5.1.3 The leapfrog method......Page 135
5.2 Two coupled first-order convection equations......Page 136
5.2.1 The forward-time centered space method......Page 137
5.2.2 The Lax method......Page 138
5.2.3 The leapfrog method......Page 140
5.2.4 The interleaved leapfrog and FDTD method......Page 142
5.3 Stability of higher dimensional FDTD algorithms......Page 143
5.4 Summary......Page 144
5.5 Problems......Page 145
References......Page 147
6 Numerical dispersion and dissipation......Page 148
6.1 Dispersion of the Lax method......Page 149
6.2 Dispersion of the leapfrog method......Page 152
6.3 Dispersion relation for the FDTD algorithm......Page 156
6.3.1 Group velocity......Page 157
6.3.3 Dispersion relation in 2D and 3D......Page 159
6.3.4 Numerical dispersion of lossy Maxwell’s equations......Page 161
6.4 Numerical stability of the FDTD algorithm revisited......Page 163
6.5 Summary......Page 164
6.6 Problems......Page 165
References......Page 167
7.1.1 Hard sources......Page 168
7.1.2 Current and voltage sources......Page 169
7.1.3 The thin-wire approximation......Page 170
7.2 External sources: total and scattered fields......Page 172
7.2.1 Total-field and pure scattered-field formulations......Page 174
7.3 Total-field/scattered-field formulation......Page 175
7.3.1 Example 2D TF/SF formulations......Page 178
7.4 Total-field/scattered-field in three dimensions......Page 181
7.5 FDTD calculation of time-harmonic response......Page 184
7.6 Summary......Page 185
7.7 Problems......Page 186
References......Page 189
8 Absorbing boundary conditions......Page 190
8.1.1 First-order Mur boundary......Page 192
8.1.2 Higher dimensional wave equations: second-order Mur......Page 193
8.1.3 Higher-order Mur boundaries......Page 198
8.1.4 Performance of the Mur boundaries......Page 200
8.1.5 Mur boundaries in 3D......Page 202
8.2.1 Bayliss-Turkel operators......Page 204
8.2.2 Higdon operators......Page 207
8.3 Summary......Page 209
8.4 Problems......Page 211
References......Page 213
9 The perfectly matched layer......Page 215
9.1 Oblique incidence on a lossy medium......Page 216
9.1.1 Uniform plane wave incident on general lossy media......Page 218
9.2 The Bérenger PML medium......Page 221
9.2.1 Bérenger split-field PML in 3D......Page 225
9.2.2 Grading the PML......Page 226
9.2.3 Example split-field simulation......Page 227
9.3 Perfectly matched uniaxial medium......Page 228
9.3.1 Bérenger’s PML as an anisotropic medium......Page 233
9.4 FDTD implementation of the UPML......Page 234
9.5 Alternative implementation via auxiliary fields......Page 238
9.6 Convolutional perfectly matched layer (CPML)......Page 241
9.6.1 Example simulation using the CPML......Page 244
9.7 Summary......Page 247
9.8 Problems......Page 249
References......Page 251
10 FDTD modeling in dispersive media......Page 253
10.1 Recursive convolution method......Page 254
10.1.1 Debye materials......Page 255
10.1.2 Lorentz materials......Page 260
10.1.3 Drude materials......Page 264
10.1.4 Isotropic plasma......Page 266
10.1.5 Improvement to the Debye and Lorentz formulations......Page 268
10.2 Auxiliary differential equation method......Page 269
10.2.2 Formulation for multiple Debye poles......Page 270
10.2.3 Lorentz materials......Page 273
10.2.4 Drude materials......Page 275
10.3 Summary......Page 276
10.4 Problems......Page 278
References......Page 280
11.1 FDTD method in arbitrary anisotropic media......Page 281
11.2 FDTD in liquid crystals......Page 284
11.2.1 FDTD formulation......Page 287
11.3 FDTD in a magnetized plasma......Page 290
11.3.1 Implementation in FDTD......Page 293
11.4 FDTD in ferrites......Page 297
11.4.1 Implementation in FDTD......Page 300
11.5 Summary......Page 303
11.6 Problems......Page 304
References......Page 306
12.1 Modeling periodic structures......Page 307
12.1.1 Direct-field methods......Page 308
12.1.2 Field-transformation methods......Page 312
12.2.1 Diagonal split-cell model......Page 318
12.2.2 Average properties model......Page 320
12.2.3 The narrow slot......Page 321
12.2.4 Dey-Mittra techniques......Page 322
12.2.5 Thin material sheets......Page 325
12.3 Bodies of revolution......Page 327
12.4 Near-to-far field transformation......Page 332
12.4.1 Frequency domain formulation......Page 333
12.4.2 Time domain implementation......Page 336
12.5 Summary......Page 339
12.6 Problems......Page 340
References......Page 342
13 Unconditionally stable implicit FDTD methods......Page 343
13.1.1 The forward-time centered space method......Page 344
13.1.2 The backward-time centered space method......Page 345
13.2 Crank-Nicolson methods......Page 347
13.3 Alternating direction implicit (ADI) method......Page 349
13.3.1 Accuracy of the ADI method......Page 353
13.5 Problems......Page 355
References......Page 356
14.1 FDFD via the wave equation......Page 358
14.2 Laplace matrix and Kronecker product......Page 361
14.3 Wave equation in 2D......Page 365
14.4 Wave equation in 3D......Page 367
14.5 FDFD from Maxwell’s equations......Page 368
14.6 Summary......Page 369
14.7 Problems......Page 370
References......Page 371
15.1 Irregular grids......Page 372
15.1.1 Nonuniform, orthogonal grids......Page 373
15.1.3 Unstructured grids......Page 376
15.2 The finite volume method......Page 377
15.2.1 Maxwell’s equations in conservative form......Page 378
15.2.2 Interleaved finite volume method......Page 380
15.2.3 The Yee finite volume method......Page 382
15.3 The finite element method......Page 384
15.3.1 Example using Galerkin’s method......Page 386
15.3.2 TE wave incident on a dielectric boundary......Page 391
15.4 Discontinuous Galerkin method......Page 393
15.5 Summary......Page 398
References......Page 399
Index......Page 401