Preface ................................................... xiii
1 Introduction ................................................. 1
1.1 Why FDTD? ............................................... 1
1.2 Other methods ........................................... 2
1.2.1 Finite volume time domain ........................ 2
1.2.2 Finite difference frequency domain ............... 3
1.2.3 Finite element methods ........................... 4
1.2.4 Spectral methods ................................. 5
1.3 Organization ............................................ 5
References ................................................... 6
2 Review of electromagnetic theory ............................. 8
2.1 Constitutive relations and material properties ......... 11
2.2 Time-harmonic Maxwell's equations ...................... 13
2.3 Complex permittivity: dielectric losses ................ 15
2.4 Complex permeability: magnetic losses .................. 17
2.5 Equivalent magnetic currents ........................... 17
2.6 Electromagnetic potentials ............................. 19
2.7 Electromagnetic boundary conditions .................... 21
2.8 Electromagnetic waves .................................. 23
2.8.1 The wave equation ............................... 25
2.8.2 Group velocity .................................. 27
2.9 The electromagnetic spectrum ........................... 28
2.10 Summary ................................................ 30
2.11 Problems ............................................... 32
References .................................................. 33
3 Partial differential equations and physical systems ......... 34
3.1 Classification of partial differential equations ....... 37
3.1.1 Elliptic PDEs ................................... 38
3.1.2 Parabolic PDEs .................................. 38
3.1.3 Hyperbolic PDEs ................................. 39
3.2 Numerical integration of ordinary differential
equations .............................................. 41
3.2.1 First-order Euler method ........................ 42
3.2.2 The leapfrog method ............................. 44
3.2.3 Runge-Kutta methods ............................. 45
3.3 Finite difference approximations of partial
differential equations ................................. 46
3.3.1 Derivatives in time ............................. 48
3.3.2 Derivatives in space ............................ 51
3.3.3 Finite difference versions of PDEs .............. 52
3.4 Finite difference solutions of the convection
equation ............................................... 53
3.4.1 The forward-time centered space method .......... 54
3.4.2 The leapfrog method ............................. 57
3.4.3 The Lax-Wendroff methods ........................ 60
3.5 Finite difference methods for two coupled first-order
convection equations ................................... 63
3.6 Higher-order differencing schemes ...................... 65
3.7 Summary ................................................ 67
3.8 Problems ............................................... 68
References .................................................. 71
4 The FDTD grid and the Yee algorithm ......................... 72
4.1 Maxwell's equations in one dimension ................... 74
4.1.1 Example ID simulations .......................... 77
4.2 Maxwell's equations in two dimensions .................. 78
4.2.1 Transverse electric (ТЕ) mode ................... 80
4.2.2 Transverse magnetic (TM) mode ................... 81
4.2.3 Example 2D simulations .......................... 82
4.3 FDTD expressions in three dimensions ................... 84
4.3.1 Example 3D simulation ........................... 87
4.4 FDTD algorithm for lossy media ......................... 88
4.4.1 ID waves in lossy media: waves on lossy
transmission lines .............................. 88
4.4.2 2D and 3D waves in lossy media .................. 90
4.5 Divergence-free nature of the FDTD algorithm ........... 92
4.6 The FDTD method in other coordinate systems ............ 93
4.6.1 2D polar coordinates ............................ 94
4.6.2 2D cylindrical coordinates ...................... 97
4.6.3 3D cylindrical coordinates ..................... 101
4.6.4 3D spherical coordinates ....................... 104
4.7 Summary ............................................... 107
4.8 Problems .............................................. 108
References ................................................. 112
5 Numerical stability of finite difference methods ........... 113
5.1 The convection equation ............................... 114
5.1.1 The forward-time centered space method ......... 115
5.1.2 The Lax method ................................. 116
5.1.3 The leapfrog method ............................ 119
5.2 Two coupled first-order convection equations .......... 120
5.2.1 The forward-time centered space method ......... 121
5.2.2 The Lax method ................................. 122
5.2.3 The leapfrog method ............................ 124
5.2.4 The interleaved leapfrog and FDTD method ....... 126
5.3 Stability of higher dimensional FDTD algorithms ....... 127
5.4 Summary ............................................... 128
5.5 Problems .............................................. 129
References ................................................. 131
6 Numerical dispersion and dissipation ....................... 132
6.1 Dispersion of the Lax method .......................... 133
6.2 Dispersion of the leapfrog method ..................... 136
6.3 Dispersion relation for the FDTD algorithm ............ 140
6.3.1 Group velocity ................................. 141
6.3.2 Dispersion of the wave equation ................ 143
6.3.3 Dispersion relation in 2D and 3D ............... 143
6.3.4 Numerical dispersion of lossy Maxwell's
equations ...................................... 145
6.4 Numerical stability of the FDTD algorithm revisited ... 147
6.5 Summary ............................................... 148
6.6 Problems .............................................. 149
References ................................................. 151
7 Introduction of sources .................................... 152
7.1 Internal sources ...................................... 152
7.1.1 Hard sources ................................... 152
7.1.2 Current and voltage sources .................... 153
7.1.3 The thin-wire approximation .................... 154
7.2 External sources: total and scattered fields .......... 156
7.2.1 Total-field and pure scattered-field
formulations ................................... 158
7.3 Total-field/scattered-field formulation ............... 159
7.3.1 Example 2D TF/SF formulations .................. 162
7.4 Total-field/scattered-field in three dimensions ....... 165
7.4.1 Calculating the incident field ................. 168
7.5 FDTD calculation of time-harmonic response ............ 168
7.6 Summary ............................................... 169
7.7 Problems .............................................. 170
References ................................................. 173
8 Absorbing boundary conditions .............................. 174
8.1 ABCs based on the one-way wave equation ............... 176
8.1.1 First-order Mur boundary ....................... 176
8.1.2 Higher dimensional wave equations:
second-order Mur ............................... 177
8.1.3 Higher-order Mur boundaries .................... 182
8.1.4 Performance of the Mur boundaries .............. 184
8.1.5 Mur boundaries in 3D ........................... 186
8.2 Other radiation operators as ABCs ..................... 188
8.2.1 Bayliss-Turkel operators ....................... 188
8.2.2 Higdon operators ............................... 191
8.3 Summary ............................................... 193
8.4 Problems .............................................. 195
References ................................................. 197
9 The perfectly matched layer ................................ 199
9.1 Oblique incidence on a lossy medium ................... 200
9.1.1 Uniform plane wave incident on general lossy
media .......................................... 202
9.2 The Berenger PML medium ............................... 205
9.2.1 Berenger split-field PML in 3D ................. 209
9.2.2 Grading the PML ................................ 210
9.2.3 Example split-field simulation ................. 211
9.3 Perfectly matched uniaxial medium ..................... 212
9.3.1 Berenger's PML as an anisotropic medium ........ 217
9.4 FDTD implementation of the UPML ....................... 218
9.5 Alternative implementation via auxiliary fields ....... 222
9.6 Convolutional perfectly matched layer (CPML) .......... 225
9.6.1 Example simulation using the CPML .............. 228
9.7 Summary ............................................... 231
9.8 Problems .............................................. 233
References ................................................. 235
10 FDTD modeling in dispersive media .......................... 237
10.1 Recursive convolution method .......................... 238
10.1.1 Debye materials ................................ 239
10.1.2 Lorentz materials .............................. 244
10.1.3 Drude materials ................................ 248
10.1.4 Isotropic plasma ............................... 250
10.1.5 Improvement to the Debye and Lorentz
formulations ................................... 252
10.2 Auxiliary differential equation method ................ 253
10.2.1 Debye materials ................................ 254
10.2.2 Formulation for multiple Debye poles ........... 254
10.2.3 Lorentz materials .............................. 257
10.2.4 Drude materials ................................ 259
10.3 Summary ............................................... 260
10.4 Problems .............................................. 262
References ................................................. 264
11 FDTD modeling in anisotropic media ......................... 265
11.1 FDTD method in arbitrary anisotropic media ............ 265
11.2 FDTD in liquid crystals ............................... 268
11.2.1 FDTD formulation ............................... 271
11.3 FDTD in a magnetized plasma ........................... 274
11.3.1 Implementation in FDTD ......................... 277
11.4 FDTD in ferrites ...................................... 281
11.4.1 Implementation in FDTD ......................... 284
11.5 Summary ............................................... 287
11.6 Problems .............................................. 288
References ................................................. 290
12 Some advanced topics ....................................... 291
12.1 Modeling periodic structures .......................... 291
12.1.1 Direct-field methods ........................... 292
12.1.2 Field-transformation methods ................... 296
12.2 Modeling fine geometrical features .................... 302
12.2.1 Diagonal split-cell model ...................... 302
12.2.2 Average properties model ....................... 304
12.2.3 The narrow slot ................................ 305
12.2.4 Dey-Mittra techniques .......................... 306
12.2.5 Thin material sheets ........................... 309
12.3 Bodies of revolution .................................. 311
12.4 Near-to-far field transformation ...................... 316
12.4.1 Frequency domain formulation ................... 317
12.4.2 Time domain implementation ..................... 320
12.5 Summary ............................................... 323
12.6 Problems .............................................. 324
References ................................................. 326
13 Unconditionally stable implicit FDTD methods ............... 327
13.1 Implicit versus explicit finite difference methods .... 328
13.1.1 The forward-time centered space method ......... 328
13.1.2 The backward-time centered space method ........ 329
13.2 Crank-Nicolson methods ................................ 331
13.3 Alternating direction implicit (ADI) method ........... 333
13.3.1 Accuracy of the ADI method ..................... 337
13.4 Summary ............................................... 339
13.5 Problems .............................................. 339
References ................................................. 340
14 Finite difference frequency domain ......................... 342
14.1 FDFD via the wave equation ............................ 342
14.2 Laplace matrix and Kronecker product .................. 345
14.3 Wave equation in 2D ................................... 349
14.4 Wave equation in 3D ................................... 351
14.5 FDFD from Maxwell's equations ......................... 352
14.6 Summary ............................................... 353
14.7 Problems .............................................. 354
References ................................................. 355
15 Finite volume and finite element methods ................... 356
15.1 Irregular grids ....................................... 356
15.1.1 Nonuniform, orthogonal grids ................... 357
15.1.2 Nonorthogonal structured grids ................. 360
15.1.3 Unstructured grids ............................. 360
15.2 The finite volume method .............................. 361
15.2.1 Maxwell's equations in conservative form ....... 362
15.2.2 Interleaved finite volume method ............... 364
15.2.3 The Yee finite volume method ................... 366
15.3 The finite element method ............................. 368
15.3.1 Example using Galerkin's method ................ 370
15.3.2 ТЕ wave incident on a dielectric boundary ...... 375
15.4 Discontinuous Galerkin method ......................... 377
15.5 Summary ............................................... 382
15.6 Problems .............................................. 383
References ................................................. 383
Index ...................................................... 385
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