Computational electrodynamics the finite-difference time-domain method
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| Format: | Buch |
| Sprache: | English |
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Boston [u.a.]
Artech House
2005
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| Ausgabe: | 3. ed. |
| Schriftenreihe: | Artech House antennas and propagation library
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| 100 | 1 | |a Taflove, Allen |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Computational electrodynamics |b the finite-difference time-domain method |c Allen Taflove ; Susan C. Hagness |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Boston [u.a.] |b Artech House |c 2005 | |
| 300 | |a XXII, 1006 S. |b Ill., graph. Darst. | ||
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| 490 | 0 | |a Artech House antennas and propagation library | |
| 650 | 4 | |a Electrodynamics - Mathematics | |
| 650 | 4 | |a Finite differences | |
| 650 | 4 | |a Time-domain analysis | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Electromagnetism | |
| 650 | 4 | |a Integro-differential equations |x Numerical solutions | |
| 650 | 4 | |a Maxwell equations |x Data processing | |
| 650 | 4 | |a Maxwell equations |x Numerical solutions | |
| 650 | 4 | |a Moments method (Statistics) | |
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| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Maxwellsche Gleichungen |0 (DE-588)4221398-8 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202/PHY 303 2015 B 1839(3) 0002/PHY 303 2006 B 2560(3) |
|---|---|
| DE-BY-TUM_katkey | 1560705 |
| DE-BY-TUM_media_number | 040006269230 040008192403 |
| _version_ | 1816712583205355521 |
| adam_text | Contents
Preface to the Third Edition xix
1 Electrodynamics Entering the 21st Century 1
1.1 Introduction 1
1.2 The Heritage of Military Defense Applications 1
1.3 Frequency Domain Solution Techniques 2
1.4 Rise of Finite Difference Time Domain Methods 3
1.5 History of FDTD Techniques for Maxwell s Equations 4
1.6 Characteristics of FDTD and Related Space Grid Time Domain Techniques 6
1.6.1 Classes of Algorithms 6
1.6.2 Predictive Dynamic Range 7
1.6.3 Scaling to Very Large Problem Sizes 8
1.7 Examples of Applications 9
1.7.1 Impulsive Around the World Extremely Low Frequency Propagation 10
1.7.2 Cellphone Radiation Interacting with the Human Head 11
1.7.3 Early Stage Detection of Breast Cancer Using an Ultrawideband Microwave Radar 11
1.7.4 Homing Accuracy of a Radar Guided Missile 12
1.7.5 Electromagnetic Wave Vulnerabilities of a Military Jet Plane 12
1.7.6 Millimeter Wave Propagation in a Defect Mode Electromagnetic Bandgap Structure 13
1.7.7 Photonic Crystal Microcavity Laser 14
1.7.8 Photonic Crystal Cross Waveguide Switch 15
1.8 Conclusions 16
References 16
2 The One Dimensional Scalar Wave Equation 21
2.1 Introduction 21
2.2 Propagating Wave Solutions 21
2.3 Dispersion Relation 22
2.4 Finite Differences 23
2.5 Finite Difference Approximation of the Scalar Wave Equation 24
2.6 Numerical Dispersion Relation 27
2.6.1 Case 1: Very Fine Sampling in Time and Space 28
2.6.2 Case 2: Magic Time Step 29
2.6.3 Case 3: Dispersive Wave Propagation 29
2.6.4 Example of Calculation of Numerical Phase Velocity and Attenuation 34
2.6.5 Examples of Calculations of Pulse Propagation 34
2.7 Numerical Stability 39
2.7.1 Complex Frequency Analysis 40
2.7.2 Examples of Calculations Involving Numerical Instability 43
2.8 Summary 45
Appendix 2A: Order of Accuracy 47
2A.1 Lax Richtmyer Equivalence Theorem 47
2A.2 Limitations 48
References 48
Selected Bibliography on Stability of Finite Difference Methods 49
Problems 49
v
3 Introduction to Maxwell s Equations and the Yee Algorithm
Allen Taflove and Jamesina Simpson 51
3.1 Introduction 51
3.2 Maxwell s Equations in Three Dimensions 51
3.3 Reduction to Two Dimensions 54
3.3.1 TMzMode 55
3.3.2 TEzMode 55
3.4 Reduction to One Dimension 56
3.4.1 x Directed, z Polarized TEM Mode 56
3.4.2 x Directed, y Polarized TEM Mode 57
3.5 Equivalence to the Wave Equation in One Dimension 57
3.6 The Yee Algorithm 58
3.6.1 Basic Ideas 58
3.6.2 Finite Differences and Notation 60
3.6.3 Finite Difference Expressions for Maxwell s Equations in Three Dimensions 62
3.6.4 Space Region with a Continuous Variation of Material Properties 67
3.6.5 Space Region with a Finite Number of Distinct Media 69
3.6.6 Space Region with Nonpermeable Media 71
3.6.7 Reduction to the Two Dimensional TMz and TEz Modes 73
3.6.8 Interpretation as Faraday s and Ampere s Laws in Integral Form 75
3.6.9 Divergence Free Nature 78
3.7 Alternative Finite Difference Grids 80
3.7.1 Cartesian Grids 80
3.7.2 Hexagonal Grids 82
3.8 Emerging Application: Gridding the Planet Earth 85
3.8.1 Background 85
3.8.2 The Latitude Longitude Space Lattice 86
3.8.3 The Geodesic (Hexagon Pentagon) Grid 99
3.9 Summary 103
References 104
Problems 105
4 Numerical Dispersion and Stability 107
4.1 Introduction 107
4.2 Derivation of the Numerical Dispersion Relation for Two Dimensional Wave Propagation 107
4.3 Extension to Three Dimensions 110
4.4 Comparison with the Ideal Dispersion Case 111
4.5 Anisotropy of the Numerical Phase Velocity 111
4.5.1 Sample Values of Numerical Phase Velocity 111
4.5.2 Intrinsic Grid Velocity Anisotropy 116
4.6 Complex Valued Numerical Wavenumbers 120
4.6.1 Case 1: Numerical Wave Propagation Along the Principal Lattice Axes 121
4.6.2 Case 2: Numerical Wave Propagation Along a Grid Diagonal 123
4.6.3 Example of Calculation of Numerical Phase Velocity and Attenuation 126
4.6.4 Example of Calculation of Wave Propagation 126
4.7 Numerical Stability 128
4.7.1 Complex Frequency Analysis 130
4.7.2 Example of a Numerically Unstable Two Dimensional FDTD Model 135
4.7.3 Linear Growth Mode When the Normalized Courant Factor Equals 1 137
4.8 Generalized Stability Problem 137
4.8.1 Absorbing and Impedance Boundary Conditions 137
4.8.2 Variable and Unstructured Meshing 137
4.8.3 Lossy, Dispersive, Nonlinear, and Gain Materials 138
4.9 Modified Yee Based Algorithms for Mitigating Numerical Dispersion 138
4.9.1 Strategy 1: Center a Specific Numerical Phase Velocity Curve About c 138
4.9.2 Strategy 2: Use Fourth Order Accurate Explicit Spatial Differences 139
4.9.3 Strategy 3: Use a Hexagonal Grid, If Possible 146
4.9.4 Strategy 4: Use Discrete Fourier Transforms to Calculate the Spatial Derivatives 150
4.10 Alternating Direction Implicit Time Stepping Algorithm for Operation
Beyond the Courant Limit 154
4.10.1 Numerical Formulation of the Zheng/Chen/Zhang Algorithm 155
4.10.2 Sources 161
4.10.3 Numerical Stability 161
4.10.4 Numerical Dispersion 163
4.10.5 Additional Accuracy Limitations and Their Implications 164
4.11 Summary 164
References 165
Problems 166
Projects 167
5 Incident Wave Source Conditions
Allen Taflove, Geoff Waldschmidt, Christopher Wagner, John Schneider, and Susan Hagness 169
5.1 Introduction 169
5.2 Pointwise E and H Hard Sources in One Dimension 169
5.3 Pointwise E and H Hard Sources in Two Dimensions 171
5.3.1 Green Function for the Scalar Wave Equation in Two Dimensions 171
5.3.2 Obtaining Comparative FDTD Data 172
5.3.3 Results for Effective Action Radius of a Hard Sourced Field Component 173
5.4 / and M Current Sources in Three Dimensions 175
5.4.1 Sources and Charging 176
5.4.2 Sinusoidal Sources 178
5.4.3 Transient (Pulse) Sources 178
5.4.4 Intrinsic Lattice Capacitance 179
5.4.5 Intrinsic Lattice Inductance 183
5.4.6 Impact upon FDTD Simulations of Lumped Element Capacitors and Inductors 183
5.5 The Plane Wave Source Condition 185
5.6 The Total Field / Scattered Field Technique: Ideas and One Dimensional Formulation 186
5.6.1 Ideas 186
5.6.2 One Dimensional Formulation 188
5.7 Two Dimensional Formulation of the TF/ SF Technique 193
5.7.1 Consistency Conditions 193
5.7.2 Calculation of the Incident Field 197
5.7.3 Illustrative Example 201
5.8 Three Dimensional Formulation of the TF/SF Technique 204
5.8.1 Consistency Conditions 204
5.8.2 Calculation of the Incident Field 210
5.9 Advanced Dispersion Compensation in the TF/SF Technique 213
5.9.1 Matched Numerical Dispersion Technique 214
5.9.2 Analytical Field Propagation 218
5.10 Scattered Field Formulation 220
5.10.1 Application to PEC Structures 220
5.10.2 Application to Lossy Dielectric Structures 221
5.10.3 Choice of Incident Plane Wave Formulation 223
5.11 Waveguide Source Conditions 223
5.11.1 Pulsed Electric Field Modal Hard Source 223
5.11.2 Total Field / Reflected Field Modal Formulation 225
5.11.3 Resistive Source and Load Conditions 225
5.12 Summary 226
References 227
Problems 227
Projects 228
6 Analytical Absorbing Boundary Conditions 229
6.1 Introduction 229
6.2 Bayliss Turkel Radiation Operators 230
6.2.1 Spherical Coordinates 231
6.2.2 Cylindrical Coordinates 234
6.3 Engquist Majda One Way Wave Equations 236
6.3.1 One Term and Two Term Taylor Series Approximations 237
6.3.2 Mur Finite Difference Scheme 240
6.3.3 Trefethen Halpern Generalized and Higher Order ABCs 243
6.3.4 Theoretical Reflection Coefficient Analysis 245
6.3.5 Numerical Experiments 247
6.4 Higdon Radiation Operators 252
6.4.1 Formulation 252
6.4.2 First Two Higdon Operators 253
6.4.3 Discussion 254
6.5 Liao Extrapolation in Space and Time 255
6.5.1 Formulation 255
6.5.2 Discussion 257
6.6 Ramahi Complementary Operators 259
6.6.1 Basic Idea 259
6.6.2 Complementary Operators 260
6.6.3 Effect of Multiple Wave Reflections 260
6.6.4 Basis of the Concurrent Complementary Operator Method 261
6.6.5 Illustrative FDTD Modeling Results Obtained Using the C COM 267
6.7 Summary 270
References 270
Problems 271
7 Perfectly Matched Layer Absorbing Boundary Conditions
Stephen Gedney 273
7.1 Introduction 273
7.2 Plane Wave Incident upon a Lossy Half Space 274
7.3 Plane Wave Incident upon Berenger s PML Medium 276
7.3.1 Two Dimensional TE2 Case 276
7.3.2 Two Dimensional TMz Case 281
7.3.3 Three Dimensional Case 281
7.4 Stretched Coordinate Formulation of Berenger s PML 282
7.5 An Anisotropic PML Absorbing Medium 285
7.5.1 Perfectly Matched Uniaxial Medium 285
7.5.2 Relationship to Berenger s Split Field PML 288
7.5.3 A Generalized Three Dimensional Formulation 289
7.5.4 Inhomogeneous Media 290
7.6 Theoretical Performance of the PML 291
7.6.1 The Continuous Space 291
7.6.2 The Discrete Space 292
7.7 Complex Frequency Shifted Tensor 294
7.7.1 Introduction 294
7.7.2 Strategy to Reduce Late Time (Low Frequency) Reflections 296
7.8 Efficient Implementation of UPML in FDTD 297
7.8.1 Derivation of the Finite Difference Expressions 298
7.8.2 Computer Implementation of the UPML 301
7.9 Efficient Implementation of CPML in FDTD 302
7.9.1 Derivation of the Finite Difference Expressions 302
7.9.2 Computer Implementation of the CPML 307
7.10 Application of CPML in FDTD to General Media 310
7.10.1 Introduction 310
7.10.2 Example: Application of CPML to the Debye Medium 310
7.11 Numerical Experiments with PML 313
7.11.1 Current Source Radiating in an Unbounded Two Dimensional Region 313
7.11.2 Highly Elongated Domains and Edge Singularities 317
7.11.3 Microstrip Patch Antenna Array 320
7.11.4 Dispersive Media 322
7.12 Summary and Conclusions 324
References 324
Projects 327
8 Near to Far Field Transformation
Allen Taflove, Xu Li, and Susan Hagness 329
8.1 Introduction 329
8.2 Two Dimensional Transformation, Phasor Domain 329
8.2.1 Application of Green s Theorem 330
8.2.2 Far Field Limit 332
8.2.3 Reduction to Standard Form 334
8.3 Obtaining Phasor Quantities Via Discrete Fourier Transformation 335
8.4 Surface Equivalence Theorem 338
8.5 Extension to Three Dimensions, Phasor Domain 340
8.6 Time Domain Near to Far Field Transformation 343
8.7 Modified NTFF Procedure to More Accurately Calculate Backscattering from
Strongly Forward Scattering Objects 348
8.8 Summary 351
References 351
Project 352
9 Dispersive, Nonlinear, and Gain Materials
Allen Taflove, Susan Hagness, Wojciech Gwarek, Masafumi Fujii, and Shih Hui Chang 353
9.1 Introduction 353
9.2 Generic Isotropic Material Dispersions 354
9.2.1 Debye Media 354
9.2.2 Lorentz Media 354
9.2.3 Drude Media 355
9.3 Piecewise Linear Recursive Convolution Method, Linear Material Case 355
9.3.1 General Formulation 356
9.3.2 Application to Debye Media 358
9.3.3 Application to Lorentz Media 358
9.3.4 Numerical Results 360
9.4 Auxiliary Differential Equation Method, Linear Material Case 361
9.4.1 Formulation for Multiple Debye Poles 361
9.4.2 Formulation for Multiple Lorentz Pole Pairs 363
9.4.3 Formulation for Multiple Drude Poles 365
9.4.4 Illustrative Numerical Results 367
9.5 Modeling of Linear Magnetized Ferrites 369
9.5.1 Equivalent RLC Model 370
9.5.2 Time Stepping Algorithm 371
9.5.3 Extension to the Three Dimensional Case, Including Loss 373
9.5.4 Illustrative Numerical Results 374
9.5.5 Comparison of Computer Resources 375
9.6 Auxiliary Differential Equation Method, Nonlinear Dispersive Material Case 376
9.6.1 Strategy 376
9.6.2 Contribution of the Linear Debye Polarization 377
9.6.3 Contribution of the Linear Lorentz Polarization 377
9.6.4 Contributions of the Third Order Nonlinear Polarization 378
9.6.5 Electric Field Update 380
9.6.6 Illustrative Numerical Results for Temporal Solitons 381
9.6.7 Illustrative Numerical Results for Spatial Solitons 383
9.7 Auxiliary Differential Equation Method, Macroscopic Modeling of Saturable,
Dispersive Optical Gain Materials 387
9.7.1 Theory 387
9.7.2 Validation Studies 390
9.8 Auxiliary Differential Equation Method, Modeling of Lasing Action in a
Four Level Two Electron Atomic System 394
9.8.1 Quantum Physics Basis 394
9.8.2 Coupling to Maxwell s Equations 398
9.8.3 Time Stepping Algorithm 398
9.8.4 Illustrative Results 400
9.9 Summary and Conclusions 402
References 404
Problems 405
Projects 406
10 Local Subcell Models of Fine Geometrical Features
Allen Taflove, Malgorzata Celuch Marcysiak, and Susan Hagness 407
10.1 Introduction 407
10.2 Basis of Contour Path FDTD Modeling 408
10.3 The Simplest Contour Path Subcell Models 408
10.3.1 Diagonal Split Cell Model for PEC Surfaces 410
10.3.2 Average Properties Model for Material Surfaces 410
10.4 The Contour Path Model of the Narrow Slot 411
10.5 The Contour Path Model of the Thin Wire 415
10.6 Locally Conformal Models of Curved Surfaces 420
10.6.1 Yu Mittra Technique for PEC Structures 420
10.6.2 Illustrative Results for PEC Structures 421
10.6.3 Yu Mittra Technique for Material Structures 424
10.7 Maloney Smith Technique for Thin Material Sheets 427
10.7.1 Basis 427
10.7.2 Illustrative Results 430
10.8 Surface Impedance 432
10.8.1 The Monochromatic SIBC 434
10.8.2 Convolution Based Models of the Frequency Dependent SIBC 436
10.8.3 Equivalent Circuit Model of the Frequency Dependent SIBC 442
10.8.4 Sources of Error 445
10.8.5 Discussion 446
10.9 Thin Coatings on a PEC Surface 447
10.9.1 Method of Lee etal. 447
10.9.2 Method of Karkkainen 450
10.10 Relativistic Motion of PEC Boundaries 450
10.10.1 Basis 451
10.10.2 Illustrative Results 454
10.11 Summary and Discussion 458
References 458
Selected Bibliography 460
Projects 461
11 Nonuniform Grids, Nonorthogonal Grids,
Unstructured Grids, and Subgrids
Stephen Gedney, Faiza Lansing, and Nicolas Chavannes 463
11.1 Introduction 463
11.2 Nonuniform Orthogonal Grids 464
11.3 Locally Conformal Grids, Globally Orthogonal 471
11.4 Global Curvilinear Coordinates 471
11.4.1 Nonorthogonal Curvilinear FDTD Algorithm 471
11.4.2 Stability Criterion 477
11.5 Irregular Nonorthogonal Structured Grids 480
11.6 Irregular Nonorthogonal Unstructured Grids 486
11.6.1 Generalized Yee Algorithm 487
11.6.2 Inhomogeneous Media 491
11.6.3 Practical Implementation of the Generalized Yee Algorithm 493
11.7 A Planar Generalized Yee Algorithm 494
11.7.1 Time Stepping Expressions 495
11.7.2 Projection Operators 496
11.7.3 Efficient Time Stepping Implementation 498
11.7.4 Modeling Example: 32 GHz Wilkinson Power Divider 499
11.8 Cartesian Subgrids 501
11.8.1 Geometry 502
11.8.2 Time Stepping Scheme 503
11.8.3 Spatial Interpolation 504
11.8.4 Numerical Stability Considerations 505
11.8.5 Reflection from the Interface of the Primary Grid and Subgrid 505
11.8.6 Illustrative Results: Helical Antenna on Generic Cellphone at 900 MHz 508
11.8.7 Computational Efficiency 510
11.9 Summary and Conclusions 510
References 511
Problems 514
Projects 515
12 Bodies of Revolution
Thomas Jurgens, Jeffrey Blaschak, and Gregory Saewert 517
12.1 Introduction 517
12.2 Field Expansion 517
12.3 Difference Equations for Off Axis Cells 519
12.3.1 Ampere s Law Contour Path Integral to Calculate er 519
12.3.2 Ampere s Law Contour Path Integral to Calculate e^ 521
12.3.3 Ampere s Law Contour Path Integral to Calculate ez 523
12.3.4 Difference Equations 525
12.3.5 Surface Conforming Contour Path Integrals 528
12.4 Difference Equations for On Axis Cells 529
12.4.1 Ampere s Law Contour Path Integral to Calculate ez on the z Axis 529
12.4.2 Ampere s Law Contour Path Integral to Calculate e^ on the z Axis 532
12.4.3 Faraday s Law Calculation of hr on the z Axis 534
12.5 Numerical Stability 535
12.6 PML Absorbing Boundary Condition 536
12.6.1 BOR FDTD Background 536
12.6.2 Extension ofPML to the General BOR Case 537
12.6.3 Examples 543
12.7 Application to Particle Accelerator Physics 543
12.7.1 Definitions and Concepts 545
12.7.2 Examples 547
12.8 Summary 550
References 550
Problems 551
Projects 552
13 Periodic Structures
James Maloney and Morris Kesler 553
13.1 Introduction 553
13.2 Review of Scattering from Periodic Structures 555
13.3 Direct Field Methods 559
13.3.1 Normal Incidence Case 559
13.3.2 Multiple Unit Cells for Oblique Incidence 560
13.3.3 Sine Cosine Method 562
13.3.4 Angled Update Method 563
13.4 Introduction to the Field Transformation Technique 567
13.5 Multiple Grid Approach 571
13.5.1 Formulation 571
13.5.2 Numerical Stability Analysis 573
13.5.3 Numerical Dispersion Analysis 574
13.5.4 Lossy Materials 575
13.5.5 Lossy Screen Example 577
13.6 Split Field Method, Two Dimensions 578
13.6.1 Formulation 578
13.6.2 Numerical Stability Analysis 580
13.6.3 Numerical Dispersion Analysis 581
13.6.4 Lossy Materials 582
13.6.5 Lossy Screen Example 583
13.7 Split Field Method, Three Dimensions 583
13.7.1 Formulation 584
13.7.2 Numerical Stability Analysis 589
13.7.3 UPML Absorbing Boundary Condition 590
13.8 Application of the Periodic FDTD Method 594
13.8.1 Electromagnetic Bandgap Structures 595
13.8.2 Frequency Selective Surfaces 597
13.8.3 Antenna Arrays 597
13.9 Summary and Conclusions 603
Acknowledgments 603
References 603
Projects 605
14 Antennas
James Maloney, Glenn Smith, Eric Thiele, Om Gandhi, Nicolas Chavannes, and Susan Hagness 607
14.1 Introduction 607
14.2 Formulation of the Antenna Problem 607
14.2.1 Transmitting Antenna 607
14.2.2 Receiving Antenna 609
14.2.3 Symmetry 610
14.2.4 Excitation 611
14.3 Antenna Feed Models 612
14.3.1 Detailed Modeling of the Feed 613
14.3.2 Simple Gap Feed Model for a Monopole Antenna 614
14.3.3 Improved Simple Feed Model 617
14.4 Near to Far Field Transformations 621
14.4.1 Use of Symmetry 621
14.4.2 Time Domain Near to Far Field Transformation 622
14.4.3 Frequency Domain Near to Far Field Transformation 624
14.5 Plane Wave Source 625
14.5.1 Effect of an Incremental Displacement of the Surface Currents 625
14.5.2 Effect of an Incremental Time Shift 627
14.5.3 Relation to Total Field / Scattered Field Lattice Zoning 628
14.6 Case Study I: The Standard Gain Horn 628
14.7 Case Study II: The Vivaldi Slotline Array 634
14.7.1 Background 634
14.7.2 The Planar Element 635
14.7.3 The Vivaldi Pair 637
14.7.4 The Vivaldi Quad 639
14.7.5 The Linear Phased Array 640
14.7.6 Phased Array Radiation Characteristics Indicated by the FDTD Modeling 641
14.7.7 Active Impedance of the Phased Array 644
14.8 Near Field Simulations 647
14.8.1 Generic 900 MHz Cellphone Handset in Free Space 647
14.8.2 900 MHz Dipole Antenna Near a Layered Bone Brain Half Space 649
14.8.3 840 MHz Dipole Antenna Near a Rectangular Brain Phantom 650
14.8.4 900 MHz Infinitesimal Dipole Antenna Near a Spherical Brain Phantom 650
14.8.5 1.9 GHz Half Wavelength Dipole Near a Spherical Brain Phantom 652
14.9 Case Study III: The Motorola T250 Tri Band Phone 653
14.9.1 FDTD Phone Model 654
14.9.2 Measurement Procedures 656
14.9.3 Free Space Near Field Investigations and Assessment of Design Capabilities 656
14.9.4 Performance in Loaded Conditions (SAM and MRI Based Human Head Model) 657
14.9.5 Radiation Performance in Free Space and Adjacent to the SAM Head 659
14.9.6 Computational Requirements 661
14.9.7 Overall Assessment 661
14.10 Selected Additional Applications 661
14.10.1 Use of Electromagnetic Bandgap Materials 662
14.10.2 Ground Penetrating Radar 663
14.10.3 Antenna Radome Interaction 667
14.10.4 Biomedical Applications of Antennas 669
14.11 Summary and Conclusions 671
References 671
Projects 676
15 High Speed Electronic Circuits with Active and Nonlinear Components
Melinda Piket May, Wojciech Gwarek, Tzong Lin Wu, Bijan Houshmand, Tatsuo Itoh,
and Jamesina Simpson (HI
15.1 Introduction 677
15.2 Basic Circuit Parameters for TEM Striplines and Microstrips 679
15.2.1 Transmission Line Parameters 679
15.2.2 Impedance 680
15.2.3 S Parameters 680
15.2.4 Differential Capacitance 681
15.2.5 Differential Inductance 682
15.3 Lumped Inductance Due to a Discontinuity 682
15.3.1 Flux /Current Definition 684
15.3.2 Fitting Z( o) or S(a) to an Equivalent Circuit 684
15.3.3 Discussion: Choice of Methods 685
15.4 Inductance of Complex Power Distribution Systems 685
15.4.1 Method Description 685
15.4.2 Example: Multiplane Meshed Printed Circuit Board 687
15.4.3 Discussion 688
15.5 Parallel Coplanar Microstrips 688
15.6 Multilayered Interconnect Modeling 690
15.7 5 Parameter Extraction for General Waveguides 692
15.8 Digital Signal Processing and Spectrum Estimation 694
15.8.1 Prony s Method 695
15.8.2 Autoregressive Models 697
15.8.3 Pad6 Approximation 702
15.9 Modeling of Lumped Circuit Elements 706
15.9.1 FDTD Formulation Extended to Circuit Elements 706
15.9.2 The Resistor 708
15.9.3 The Resistive Voltage Source 708
15.9.4 The Capacitor 709
15.9.5 The Inductor 711
15.9.6 The Arbitrary Two Terminal Linear Lumped Network 711
15.9.7 The Diode 714
15.9.8 The Bipolar Junction Transistor 715
15.10 Direct Linking of FDTD and SPICE 717
15.10.1 Basic Idea 718
15.10.2 Norton Equivalent Circuit Looking Into the FDTD Space Lattice 719
15.10.3 Thevenin Equivalent Circuit Looking Into the FDTD Space Lattice 721
15.11 Case Study: A 6 GHz MESFET Amplifier Model 723
15.11.1 Large Signal Nonlinear Model 723
15.11.2 Amplifier Configuration 725
15.11.3 Analysis of the Circuit without the Packaging Structure 726
15.11.4 Analysis of the Circuit with the Packaging Structure 728
15.12 Emerging Topic: Wireless High Speed Digital Interconnects Using Defect Mode
Electromagnetic Bandgap Waveguides 731
15.12.1 Stopband of the Defect Free Two Dimensional EBG Structure 732
15.12.2 Passband of the Two Dimensional EBG Structure with Waveguiding Defect 732
15.12.3 Laboratory Experiments and Supporting FDTD Modeling 734
15.13 Summary and Conclusions 736
Acknowledgments 737
References 737
Selected Bibliography 740
Projects 741
16 Photonics
Geoffrey Burr, Susan Hagness, and Allen Taftove 743
16.1 Introduction 743
16.2 Introduction to Index Contrast Guided Wave Structures 743
16.3 FDTD Modeling Issues 744
16.3.1 Optical Waveguides 744
16.3.2 Material Dispersion and Nonlinearities 747
16.4 Laterally Coupled Microcavity Ring Resonators 747
16.4.1 Modeling Considerations: Two Dimensional FDTD Simulations 748
16.4.2 Coupling to Straight Waveguides 750
16.4.3 Coupling to Curved Waveguides 750
16.4.4 Elongated Ring Designs ( Racetracks ) 752
16.4.5 Resonances of the Circular Ring 752
16.5 Laterally Coupled Microcavity Disk Resonators 756
16.5.1 Resonances 756
16.5.2 Suppression of Higher Order Radial Whispering Gallery Modes 760
16.6 Vertically Coupled Racetrack 761
16.7 Introduction to Distributed Bragg Reflector Devices 765
16.8 Application to Vertical Cavity Surface Emitting Lasers 765
16.8.1 Passive Studies 766
16.8.2 Active Studies: Application of the Classical Gain Model 767
16.8.3 Application of a New Semiclassical Gain Model 769
16.9 Quasi One Dimensional DBR Structures 770
16.10 Introduction to Photonic Crystals 772
16.11 Calculation of Band Structure 774
16.11.1 The Order AT Method 775
16.11.2 Frequency Resolution 778
16.11.3 Filter Diagonalization Method 780
16.11.4 The Triangular Photonic Crystal Lattice 782
16.11.5 Sources of Error and Their Mitigation 784
16.12 Calculation of Mode Patterns 787
16.13 Variational Approach 790
16.14 Modeling of Defect Mode Photonic Crystal Waveguides 791
16.14.1 Band Diagram of a Photonic Crystal Slab 793
16.14.2 Band Diagram of a Photonic Crystal Waveguide 795
16.14.3 Intrinsic Loss in Photonic Crystal Waveguides 798
16.14.4 Transmission in Photonic Crystal Waveguides 803
16.14.5 Aperiodic Photonic Crystal Waveguides 806
16.14.6 Photonic Crystal Waveguide Extrinsic Scattering Loss from the Green Function 806
16.15 Modeling of Photonic Crystal Resonators 807
16.16 Modeling Examples of Photonic Crystal Resonators 810
16.16.1 Electrically Driven Microcavity Laser 810
16.16.2 Photonic Crystal Cross Waveguide Switch 812
16.17 Introduction to Frequency Conversion in Second Order Nonlinear Optical
Materials 813
16.18 PSTD 4 Algorithm 813
16.19 Extension to Second Order Nonlinear Media 814
16.20 Application to a Nonlinear Waveguide with a QPM Grating 814
16.21 Application to Nonlinear Photonic Crystals 817
16.22 Introduction to Nanoplasmonic Devices 820
16.23 FDTD Modeling Considerations 820
16.24 FDTD Modeling Applications 821
16.25 Introduction to Biophotonics 822
16.26 FDTD Modeling Applications 822
16.26.1 Vertebrate Retinal Rod 822
16.26.2 Precancerous Cervical Cells 824
16.26.3 Sensitivity of Backscattering Signatures to Nanometer Scale Cellular Changes 827
16.27 PSTD Modeling Application to Tissue Optics 828
16.28 Summary 830
Acknowledgments 830
References 830
17 Advances in PSTD Techniques
Qing Liu and Gang Zhao 847
17.1 Introduction 847
17.2 Approximation of Derivatives 847
17.2.1 Derivative Matrix for the Second Order Finite Difference Method 848
17.2.2 Derivative Matrices for Fourth Order and ATth Order Finite Difference Methods 849
17.2.3 Trigonometric Interpolation and FFT Method 850
17.2.4 Nonperiodic Functions and Chebyshev Method 851
17.3 Single Domain Fourier PSTD Method 854
17.3.1 Approximation of Spatial Derivatives 855
17.3.2 Numerical Stability and Dispersion 856
17.4 Single Domain Chebyshev PSTD Method 857
17.4.1 Spatial and Temporal Grids 857
17.4.2 Maxwell s Equations in Curvilinear Coordinates 858
17.4.3 Spatial Derivatives 860
17.4.4 Time Integration Scheme 861
17.5 Multidomain Chebyshev PSTD Method 861
17.5.1 Subdomain Spatial Derivatives and Time Integration 862
17.5.2 Subdomain Patching by Characteristics 863
17.5.3 Subdomain Patching by Physical Conditions 864
17.5.4 Filter Design for Corner Singularities 864
17.5.5 Multidomain PSTD Results for 2.5 Dimensional Problems 866
17.5.6 Multidomain PSTD Results for Three Dimensional Problems 868
17.6 Penalty Method for Multidomain PSTD Algorithm 868
17.7 Discontinuous Galerkin Method for PSTD Boundary Patching 87 3
17.7.1 Weak Form of Maxwell s Equations 873
17.7.2 Space Discretization and Domain Transformation 873
17.7.3 Mass Matrix and Stiffness Matrix 874
17.7.4 Flux on the Boundary 876
17.7.5 Numerical Results for DG PSTD Method 876
17.8 Summary and Conclusions 879
Appendix 17A: Coefficients for the Five Stage, Fourth Order Runge Kutta Method 879
References 880
18 Advances in Unconditionally Stable Techniques
Hans De Raedt 883
18.1 Introduction 883
18.2 General Framework 883
18.3 Matrix Exponential Concepts 884
18.4 Product Formula Approach 887
18.4.1 The Classic Yee Algorithm as a Particular Realization 887
18.4.2 The ADI Method as a Second Realization 888
18.4.3 Unconditionally Stable Algorithms: Real Space Approach 889
18.4.4 Unconditionally Stable Algorithms: Fourier Space Approach 891
18.5 Chebyshev Polynomial Algorithm 892
18.6 Extension to Linear Dispersive Media 895
18.7 Extension to Perfectly Matched Layer Absorbing Boundary Conditions 898
18.8 Summary 899
Appendix 18 A: Some Technical Details 900
Appendix 18B: Stability Analysis of Equation (18.17) 902
Appendix 18C: Stability Analysis of Equation (18.19) 904
References 904
Projects 905
19 Advances in Hybrid FDTD FE Techniques
Thomas Rylander, Fredrik Edelvik, Anders Bondeson, and Douglas Riley 907
19.1 Introduction 907
19.2 Time Domain Finite Elements 910
19.2.1 Coupled Curl Equations 910
19.2.2 Wave Equation 913
19.2.3 Equivalences Between Finite Elements and FDTD 917
19.3 Tetrahedral, Hexahedral (Brick), and Pyramidal Zeroth Order Edge and Facet Elements 918
19.3.1 Tetrahedral Finite Elements 919
19.3.2 Hexahedral (Brick) Finite Elements 921
19.3.3 Pyramidal Finite Elements 922
19.4 Stable Hybrid FDTD FE Interface 924
19.4.1 Spatial Discretization 924
19.4.2 Time Stepping on a Hybrid Space Lattice 927
19.4.3 Generalized Newmark Scheme 928
19.4.4 Proof of Stability 929
19.4.5 Alternative Time Stepping Schemes 930
19.4.6 Extensions of the Hybrid FDTD FE Concept 931
19.4.7 Reflection at the Interface of FDTD and FE Regions of a Hybrid Space Lattice 931
19.4.8 Scattering from the PEC Sphere 933
19.5 Mesh Generation Approaches 935
19.6 Subcell Wire and Slot Algorithms for Time Domain Finite Elements 936
19.6.1 Modeling Thin Wires 936
19.6.2 Modeling Thin Slots 939
19.6.3 Numerical Results for Thin Wires and Slots 941
19.7 Application to Advanced Scattering and Radiation Problems 943
19.7.1 Monostatic RCS of the NASA Almond 943
19.7.2 Bistatic RCS of the Saab Trainer Aircraft 945
19.7.3 Input Impedance of the Four Arm Sinuous Antenna 948
19.8 Summary 949
Acknowledgments 950
References 950
20 Advances in Hardware Acceleration for FDTD
Ryan Schneider, Sean Krakiwsky, Laurence Turner, and Michal Okoniewski 955
20.1 Introduction 955
20.2 Background Literature 956
20.3 Fundamental Design Considerations 957
20.4 Conceptual Massively Parallel FPGA Implementation 958
20.5 Case Study of Using the FPGA as a Coprocessor 962
20.6 Performance of Custom Hardware Implementations 964
20.7 Fundamentals of Graphics Processor Units 965
20.7.1 Overview 965
20.7.2 Graphics Pipeline 965
20.7.3 Memory Interface 967
20.7.4 Programmable Fragment and Vertex Processors 968
20.8 Implementing FDTD on a Graphics Processor Unit 969
20.8.1 Initialization 969
20.8.2 Electric and Magnetic Field Updates 970
20.8.3 Boundaries 972
20.8.4 Source Excitation 974
20.8.5 Archiving Observation Nodes 975
20.8.6 Multipass Rendering 975
20.8.7 Display 977
20.9 Performance Measurements of the GPU Accelerator 977
20.10 Summary and Conclusions 978
References 978
Acronyms and Common Symbols 981
About the Authors 985
Index 997
|
| adam_txt |
Contents
Preface to the Third Edition xix
1 Electrodynamics Entering the 21st Century 1
1.1 Introduction 1
1.2 The Heritage of Military Defense Applications 1
1.3 Frequency Domain Solution Techniques 2
1.4 Rise of Finite Difference Time Domain Methods 3
1.5 History of FDTD Techniques for Maxwell' s Equations 4
1.6 Characteristics of FDTD and Related Space Grid Time Domain Techniques 6
1.6.1 Classes of Algorithms 6
1.6.2 Predictive Dynamic Range 7
1.6.3 Scaling to Very Large Problem Sizes 8
1.7 Examples of Applications 9
1.7.1 Impulsive Around the World Extremely Low Frequency Propagation 10
1.7.2 Cellphone Radiation Interacting with the Human Head 11
1.7.3 Early Stage Detection of Breast Cancer Using an Ultrawideband Microwave Radar 11
1.7.4 Homing Accuracy of a Radar Guided Missile 12
1.7.5 Electromagnetic Wave Vulnerabilities of a Military Jet Plane 12
1.7.6 Millimeter Wave Propagation in a Defect Mode Electromagnetic Bandgap Structure 13
1.7.7 Photonic Crystal Microcavity Laser 14
1.7.8 Photonic Crystal Cross Waveguide Switch 15
1.8 Conclusions 16
References 16
2 The One Dimensional Scalar Wave Equation 21
2.1 Introduction 21
2.2 Propagating Wave Solutions 21
2.3 Dispersion Relation 22
2.4 Finite Differences 23
2.5 Finite Difference Approximation of the Scalar Wave Equation 24
2.6 Numerical Dispersion Relation 27
2.6.1 Case 1: Very Fine Sampling in Time and Space 28
2.6.2 Case 2: Magic Time Step 29
2.6.3 Case 3: Dispersive Wave Propagation 29
2.6.4 Example of Calculation of Numerical Phase Velocity and Attenuation 34
2.6.5 Examples of Calculations of Pulse Propagation 34
2.7 Numerical Stability 39
2.7.1 Complex Frequency Analysis 40
2.7.2 Examples of Calculations Involving Numerical Instability 43
2.8 Summary 45
Appendix 2A: Order of Accuracy 47
2A.1 Lax Richtmyer Equivalence Theorem 47
2A.2 Limitations 48
References 48
Selected Bibliography on Stability of Finite Difference Methods 49
Problems 49
v
3 Introduction to Maxwell's Equations and the Yee Algorithm
Allen Taflove and Jamesina Simpson 51
3.1 Introduction 51
3.2 Maxwell's Equations in Three Dimensions 51
3.3 Reduction to Two Dimensions 54
3.3.1 TMzMode 55
3.3.2 TEzMode 55
3.4 Reduction to One Dimension 56
3.4.1 x Directed, z Polarized TEM Mode 56
3.4.2 x Directed, y Polarized TEM Mode 57
3.5 Equivalence to the Wave Equation in One Dimension 57
3.6 The Yee Algorithm 58
3.6.1 Basic Ideas 58
3.6.2 Finite Differences and Notation 60
3.6.3 Finite Difference Expressions for Maxwell's Equations in Three Dimensions 62
3.6.4 Space Region with a Continuous Variation of Material Properties 67
3.6.5 Space Region with a Finite Number of Distinct Media 69
3.6.6 Space Region with Nonpermeable Media 71
3.6.7 Reduction to the Two Dimensional TMz and TEz Modes 73
3.6.8 Interpretation as Faraday's and Ampere's Laws in Integral Form 75
3.6.9 Divergence Free Nature 78
3.7 Alternative Finite Difference Grids 80
3.7.1 Cartesian Grids 80
3.7.2 Hexagonal Grids 82
3.8 Emerging Application: Gridding the Planet Earth 85
3.8.1 Background 85
3.8.2 The Latitude Longitude Space Lattice 86
3.8.3 The Geodesic (Hexagon Pentagon) Grid 99
3.9 Summary 103
References 104
Problems 105
4 Numerical Dispersion and Stability 107
4.1 Introduction 107
4.2 Derivation of the Numerical Dispersion Relation for Two Dimensional Wave Propagation 107
4.3 Extension to Three Dimensions 110
4.4 Comparison with the Ideal Dispersion Case 111
4.5 Anisotropy of the Numerical Phase Velocity 111
4.5.1 Sample Values of Numerical Phase Velocity 111
4.5.2 Intrinsic Grid Velocity Anisotropy 116
4.6 Complex Valued Numerical Wavenumbers 120
4.6.1 Case 1: Numerical Wave Propagation Along the Principal Lattice Axes 121
4.6.2 Case 2: Numerical Wave Propagation Along a Grid Diagonal 123
4.6.3 Example of Calculation of Numerical Phase Velocity and Attenuation 126
4.6.4 Example of Calculation of Wave Propagation 126
4.7 Numerical Stability 128
4.7.1 Complex Frequency Analysis 130
4.7.2 Example of a Numerically Unstable Two Dimensional FDTD Model 135
4.7.3 Linear Growth Mode When the Normalized Courant Factor Equals 1 137
4.8 Generalized Stability Problem 137
4.8.1 Absorbing and Impedance Boundary Conditions 137
4.8.2 Variable and Unstructured Meshing 137
4.8.3 Lossy, Dispersive, Nonlinear, and Gain Materials 138
4.9 Modified Yee Based Algorithms for Mitigating Numerical Dispersion 138
4.9.1 Strategy 1: Center a Specific Numerical Phase Velocity Curve About c 138
4.9.2 Strategy 2: Use Fourth Order Accurate Explicit Spatial Differences 139
4.9.3 Strategy 3: Use a Hexagonal Grid, If Possible 146
4.9.4 Strategy 4: Use Discrete Fourier Transforms to Calculate the Spatial Derivatives 150
4.10 Alternating Direction Implicit Time Stepping Algorithm for Operation
Beyond the Courant Limit 154
4.10.1 Numerical Formulation of the Zheng/Chen/Zhang Algorithm 155
4.10.2 Sources 161
4.10.3 Numerical Stability 161
4.10.4 Numerical Dispersion 163
4.10.5 Additional Accuracy Limitations and Their Implications 164
4.11 Summary 164
References 165
Problems 166
Projects 167
5 Incident Wave Source Conditions
Allen Taflove, Geoff Waldschmidt, Christopher Wagner, John Schneider, and Susan Hagness 169
5.1 Introduction 169
5.2 Pointwise E and H Hard Sources in One Dimension 169
5.3 Pointwise E and H Hard Sources in Two Dimensions 171
5.3.1 Green Function for the Scalar Wave Equation in Two Dimensions 171
5.3.2 Obtaining Comparative FDTD Data 172
5.3.3 Results for Effective Action Radius of a Hard Sourced Field Component 173
5.4 / and M Current Sources in Three Dimensions 175
5.4.1 Sources and Charging 176
5.4.2 Sinusoidal Sources 178
5.4.3 Transient (Pulse) Sources 178
5.4.4 Intrinsic Lattice Capacitance 179
5.4.5 Intrinsic Lattice Inductance 183
5.4.6 Impact upon FDTD Simulations of Lumped Element Capacitors and Inductors 183
5.5 The Plane Wave Source Condition 185
5.6 The Total Field / Scattered Field Technique: Ideas and One Dimensional Formulation 186
5.6.1 Ideas 186
5.6.2 One Dimensional Formulation 188
5.7 Two Dimensional Formulation of the TF/ SF Technique 193
5.7.1 Consistency Conditions 193
5.7.2 Calculation of the Incident Field 197
5.7.3 Illustrative Example 201
5.8 Three Dimensional Formulation of the TF/SF Technique 204
5.8.1 Consistency Conditions 204
5.8.2 Calculation of the Incident Field 210
5.9 Advanced Dispersion Compensation in the TF/SF Technique 213
5.9.1 Matched Numerical Dispersion Technique 214
5.9.2 Analytical Field Propagation 218
5.10 Scattered Field Formulation 220
5.10.1 Application to PEC Structures 220
5.10.2 Application to Lossy Dielectric Structures 221
5.10.3 Choice of Incident Plane Wave Formulation 223
5.11 Waveguide Source Conditions 223
5.11.1 Pulsed Electric Field Modal Hard Source 223
5.11.2 Total Field / Reflected Field Modal Formulation 225
5.11.3 Resistive Source and Load Conditions 225
5.12 Summary 226
References 227
Problems 227
Projects 228
6 Analytical Absorbing Boundary Conditions 229
6.1 Introduction 229
6.2 Bayliss Turkel Radiation Operators 230
6.2.1 Spherical Coordinates 231
6.2.2 Cylindrical Coordinates 234
6.3 Engquist Majda One Way Wave Equations 236
6.3.1 One Term and Two Term Taylor Series Approximations 237
6.3.2 Mur Finite Difference Scheme 240
6.3.3 Trefethen Halpern Generalized and Higher Order ABCs 243
6.3.4 Theoretical Reflection Coefficient Analysis 245
6.3.5 Numerical Experiments 247
6.4 Higdon Radiation Operators 252
6.4.1 Formulation 252
6.4.2 First Two Higdon Operators 253
6.4.3 Discussion 254
6.5 Liao Extrapolation in Space and Time 255
6.5.1 Formulation 255
6.5.2 Discussion 257
6.6 Ramahi Complementary Operators 259
6.6.1 Basic Idea 259
6.6.2 Complementary Operators 260
6.6.3 Effect of Multiple Wave Reflections 260
6.6.4 Basis of the Concurrent Complementary Operator Method 261
6.6.5 Illustrative FDTD Modeling Results Obtained Using the C COM 267
6.7 Summary 270
References 270
Problems 271
7 Perfectly Matched Layer Absorbing Boundary Conditions
Stephen Gedney 273
7.1 Introduction 273
7.2 Plane Wave Incident upon a Lossy Half Space 274
7.3 Plane Wave Incident upon Berenger's PML Medium 276
7.3.1 Two Dimensional TE2 Case 276
7.3.2 Two Dimensional TMz Case 281
7.3.3 Three Dimensional Case 281
7.4 Stretched Coordinate Formulation of Berenger's PML 282
7.5 An Anisotropic PML Absorbing Medium 285
7.5.1 Perfectly Matched Uniaxial Medium 285
7.5.2 Relationship to Berenger's Split Field PML 288
7.5.3 A Generalized Three Dimensional Formulation 289
7.5.4 Inhomogeneous Media 290
7.6 Theoretical Performance of the PML 291
7.6.1 The Continuous Space 291
7.6.2 The Discrete Space 292
7.7 Complex Frequency Shifted Tensor 294
7.7.1 Introduction 294
7.7.2 Strategy to Reduce Late Time (Low Frequency) Reflections 296
7.8 Efficient Implementation of UPML in FDTD 297
7.8.1 Derivation of the Finite Difference Expressions 298
7.8.2 Computer Implementation of the UPML 301
7.9 Efficient Implementation of CPML in FDTD 302
7.9.1 Derivation of the Finite Difference Expressions 302
7.9.2 Computer Implementation of the CPML 307
7.10 Application of CPML in FDTD to General Media 310
7.10.1 Introduction 310
7.10.2 Example: Application of CPML to the Debye Medium 310
7.11 Numerical Experiments with PML 313
7.11.1 Current Source Radiating in an Unbounded Two Dimensional Region 313
7.11.2 Highly Elongated Domains and Edge Singularities 317
7.11.3 Microstrip Patch Antenna Array 320
7.11.4 Dispersive Media 322
7.12 Summary and Conclusions 324
References 324
Projects 327
8 Near to Far Field Transformation
Allen Taflove, Xu Li, and Susan Hagness 329
8.1 Introduction 329
8.2 Two Dimensional Transformation, Phasor Domain 329
8.2.1 Application of Green's Theorem 330
8.2.2 Far Field Limit 332
8.2.3 Reduction to Standard Form 334
8.3 Obtaining Phasor Quantities Via Discrete Fourier Transformation 335
8.4 Surface Equivalence Theorem 338
8.5 Extension to Three Dimensions, Phasor Domain 340
8.6 Time Domain Near to Far Field Transformation 343
8.7 Modified NTFF Procedure to More Accurately Calculate Backscattering from
Strongly Forward Scattering Objects 348
8.8 Summary 351
References 351
Project 352
9 Dispersive, Nonlinear, and Gain Materials
Allen Taflove, Susan Hagness, Wojciech Gwarek, Masafumi Fujii, and Shih Hui Chang 353
9.1 Introduction 353
9.2 Generic Isotropic Material Dispersions 354
9.2.1 Debye Media 354
9.2.2 Lorentz Media 354
9.2.3 Drude Media 355
9.3 Piecewise Linear Recursive Convolution Method, Linear Material Case 355
9.3.1 General Formulation 356
9.3.2 Application to Debye Media 358
9.3.3 Application to Lorentz Media 358
9.3.4 Numerical Results 360
9.4 Auxiliary Differential Equation Method, Linear Material Case 361
9.4.1 Formulation for Multiple Debye Poles 361
9.4.2 Formulation for Multiple Lorentz Pole Pairs 363
9.4.3 Formulation for Multiple Drude Poles 365
9.4.4 Illustrative Numerical Results 367
9.5 Modeling of Linear Magnetized Ferrites 369
9.5.1 Equivalent RLC Model 370
9.5.2 Time Stepping Algorithm 371
9.5.3 Extension to the Three Dimensional Case, Including Loss 373
9.5.4 Illustrative Numerical Results 374
9.5.5 Comparison of Computer Resources 375
9.6 Auxiliary Differential Equation Method, Nonlinear Dispersive Material Case 376
9.6.1 Strategy 376
9.6.2 Contribution of the Linear Debye Polarization 377
9.6.3 Contribution of the Linear Lorentz Polarization 377
9.6.4 Contributions of the Third Order Nonlinear Polarization 378
9.6.5 Electric Field Update 380
9.6.6 Illustrative Numerical Results for Temporal Solitons 381
9.6.7 Illustrative Numerical Results for Spatial Solitons 383
9.7 Auxiliary Differential Equation Method, Macroscopic Modeling of Saturable,
Dispersive Optical Gain Materials 387
9.7.1 Theory 387
9.7.2 Validation Studies 390
9.8 Auxiliary Differential Equation Method, Modeling of Lasing Action in a
Four Level Two Electron Atomic System 394
9.8.1 Quantum Physics Basis 394
9.8.2 Coupling to Maxwell's Equations 398
9.8.3 Time Stepping Algorithm 398
9.8.4 Illustrative Results 400
9.9 Summary and Conclusions 402
References 404
Problems 405
Projects 406
10 Local Subcell Models of Fine Geometrical Features
Allen Taflove, Malgorzata Celuch Marcysiak, and Susan Hagness 407
10.1 Introduction 407
10.2 Basis of Contour Path FDTD Modeling 408
10.3 The Simplest Contour Path Subcell Models 408
10.3.1 Diagonal Split Cell Model for PEC Surfaces 410
10.3.2 Average Properties Model for Material Surfaces 410
10.4 The Contour Path Model of the Narrow Slot 411
10.5 The Contour Path Model of the Thin Wire 415
10.6 Locally Conformal Models of Curved Surfaces 420
10.6.1 Yu Mittra Technique for PEC Structures 420
10.6.2 Illustrative Results for PEC Structures 421
10.6.3 Yu Mittra Technique for Material Structures 424
10.7 Maloney Smith Technique for Thin Material Sheets 427
10.7.1 Basis 427
10.7.2 Illustrative Results 430
10.8 Surface Impedance 432
10.8.1 The Monochromatic SIBC 434
10.8.2 Convolution Based Models of the Frequency Dependent SIBC 436
10.8.3 Equivalent Circuit Model of the Frequency Dependent SIBC 442
10.8.4 Sources of Error 445
10.8.5 Discussion 446
10.9 Thin Coatings on a PEC Surface 447
10.9.1 Method of Lee etal. 447
10.9.2 Method of Karkkainen 450
10.10 Relativistic Motion of PEC Boundaries 450
10.10.1 Basis 451
10.10.2 Illustrative Results 454
10.11 Summary and Discussion 458
References 458
Selected Bibliography 460
Projects 461
11 Nonuniform Grids, Nonorthogonal Grids,
Unstructured Grids, and Subgrids
Stephen Gedney, Faiza Lansing, and Nicolas Chavannes 463
11.1 Introduction 463
11.2 Nonuniform Orthogonal Grids 464
11.3 Locally Conformal Grids, Globally Orthogonal 471
11.4 Global Curvilinear Coordinates 471
11.4.1 Nonorthogonal Curvilinear FDTD Algorithm 471
11.4.2 Stability Criterion 477
11.5 Irregular Nonorthogonal Structured Grids 480
11.6 Irregular Nonorthogonal Unstructured Grids 486
11.6.1 Generalized Yee Algorithm 487
11.6.2 Inhomogeneous Media 491
11.6.3 Practical Implementation of the Generalized Yee Algorithm 493
11.7 A Planar Generalized Yee Algorithm 494
11.7.1 Time Stepping Expressions 495
11.7.2 Projection Operators 496
11.7.3 Efficient Time Stepping Implementation 498
11.7.4 Modeling Example: 32 GHz Wilkinson Power Divider 499
11.8 Cartesian Subgrids 501
11.8.1 Geometry 502
11.8.2 Time Stepping Scheme 503
11.8.3 Spatial Interpolation 504
11.8.4 Numerical Stability Considerations 505
11.8.5 Reflection from the Interface of the Primary Grid and Subgrid 505
11.8.6 Illustrative Results: Helical Antenna on Generic Cellphone at 900 MHz 508
11.8.7 Computational Efficiency 510
11.9 Summary and Conclusions 510
References 511
Problems 514
Projects 515
12 Bodies of Revolution
Thomas Jurgens, Jeffrey Blaschak, and Gregory Saewert 517
12.1 Introduction 517
12.2 Field Expansion 517
12.3 Difference Equations for Off Axis Cells 519
12.3.1 Ampere' s Law Contour Path Integral to Calculate er 519
12.3.2 Ampere's Law Contour Path Integral to Calculate e^ 521
12.3.3 Ampere's Law Contour Path Integral to Calculate ez 523
12.3.4 Difference Equations 525
12.3.5 Surface Conforming Contour Path Integrals 528
12.4 Difference Equations for On Axis Cells 529
12.4.1 Ampere's Law Contour Path Integral to Calculate ez on the z Axis 529
12.4.2 Ampere's Law Contour Path Integral to Calculate e^ on the z Axis 532
12.4.3 Faraday's Law Calculation of hr on the z Axis 534
12.5 Numerical Stability ' 535
12.6 PML Absorbing Boundary Condition 536
12.6.1 BOR FDTD Background 536
12.6.2 Extension ofPML to the General BOR Case 537
12.6.3 Examples 543
12.7 Application to Particle Accelerator Physics 543
12.7.1 Definitions and Concepts 545
12.7.2 Examples 547
12.8 Summary 550
References 550
Problems 551
Projects 552
13 Periodic Structures
James Maloney and Morris Kesler 553
13.1 Introduction 553
13.2 Review of Scattering from Periodic Structures 555
13.3 Direct Field Methods 559
13.3.1 Normal Incidence Case 559
13.3.2 Multiple Unit Cells for Oblique Incidence 560
13.3.3 Sine Cosine Method 562
13.3.4 Angled Update Method 563
13.4 Introduction to the Field Transformation Technique 567
13.5 Multiple Grid Approach 571
13.5.1 Formulation 571
13.5.2 Numerical Stability Analysis 573
13.5.3 Numerical Dispersion Analysis 574
13.5.4 Lossy Materials 575
13.5.5 Lossy Screen Example 577
13.6 Split Field Method, Two Dimensions 578
13.6.1 Formulation 578
13.6.2 Numerical Stability Analysis 580
13.6.3 Numerical Dispersion Analysis 581
13.6.4 Lossy Materials 582
13.6.5 Lossy Screen Example 583
13.7 Split Field Method, Three Dimensions 583
13.7.1 Formulation 584
13.7.2 Numerical Stability Analysis 589
13.7.3 UPML Absorbing Boundary Condition 590
13.8 Application of the Periodic FDTD Method 594
13.8.1 Electromagnetic Bandgap Structures 595
13.8.2 Frequency Selective Surfaces 597
13.8.3 Antenna Arrays 597
13.9 Summary and Conclusions 603
Acknowledgments 603
References 603
Projects 605
14 Antennas
James Maloney, Glenn Smith, Eric Thiele, Om Gandhi, Nicolas Chavannes, and Susan Hagness 607
14.1 Introduction 607
14.2 Formulation of the Antenna Problem 607
14.2.1 Transmitting Antenna 607
14.2.2 Receiving Antenna 609
14.2.3 Symmetry 610
14.2.4 Excitation 611
14.3 Antenna Feed Models 612
14.3.1 Detailed Modeling of the Feed 613
14.3.2 Simple Gap Feed Model for a Monopole Antenna 614
14.3.3 Improved Simple Feed Model 617
14.4 Near to Far Field Transformations 621
14.4.1 Use of Symmetry 621
14.4.2 Time Domain Near to Far Field Transformation 622
14.4.3 Frequency Domain Near to Far Field Transformation 624
14.5 Plane Wave Source 625
14.5.1 Effect of an Incremental Displacement of the Surface Currents 625
14.5.2 Effect of an Incremental Time Shift 627
14.5.3 Relation to Total Field / Scattered Field Lattice Zoning 628
14.6 Case Study I: The Standard Gain Horn 628
14.7 Case Study II: The Vivaldi Slotline Array 634
14.7.1 Background 634
14.7.2 The Planar Element 635
14.7.3 The Vivaldi Pair 637
14.7.4 The Vivaldi Quad 639
14.7.5 The Linear Phased Array 640
14.7.6 Phased Array Radiation Characteristics Indicated by the FDTD Modeling 641
14.7.7 Active Impedance of the Phased Array 644
14.8 Near Field Simulations 647
14.8.1 Generic 900 MHz Cellphone Handset in Free Space 647
14.8.2 900 MHz Dipole Antenna Near a Layered Bone Brain Half Space 649
14.8.3 840 MHz Dipole Antenna Near a Rectangular Brain Phantom 650
14.8.4 900 MHz Infinitesimal Dipole Antenna Near a Spherical Brain Phantom 650
14.8.5 1.9 GHz Half Wavelength Dipole Near a Spherical Brain Phantom 652
14.9 Case Study III: The Motorola T250 Tri Band Phone 653
14.9.1 FDTD Phone Model 654
14.9.2 Measurement Procedures 656
14.9.3 Free Space Near Field Investigations and Assessment of Design Capabilities 656
14.9.4 Performance in Loaded Conditions (SAM and MRI Based Human Head Model) 657
14.9.5 Radiation Performance in Free Space and Adjacent to the SAM Head 659
14.9.6 Computational Requirements 661
14.9.7 Overall Assessment 661
14.10 Selected Additional Applications 661
14.10.1 Use of Electromagnetic Bandgap Materials 662
14.10.2 Ground Penetrating Radar 663
14.10.3 Antenna Radome Interaction 667
14.10.4 Biomedical Applications of Antennas 669
14.11 Summary and Conclusions 671
References 671
Projects 676
15 High Speed Electronic Circuits with Active and Nonlinear Components
Melinda Piket May, Wojciech Gwarek, Tzong Lin Wu, Bijan Houshmand, Tatsuo Itoh,
and Jamesina Simpson (HI
15.1 Introduction 677
15.2 Basic Circuit Parameters for TEM Striplines and Microstrips 679
15.2.1 Transmission Line Parameters 679
15.2.2 Impedance 680
15.2.3 S Parameters 680
15.2.4 Differential Capacitance 681
15.2.5 Differential Inductance 682
15.3 Lumped Inductance Due to a Discontinuity 682
15.3.1 Flux /Current Definition 684
15.3.2 Fitting Z( o) or S(a) to an Equivalent Circuit 684
15.3.3 Discussion: Choice of Methods 685
15.4 Inductance of Complex Power Distribution Systems 685
15.4.1 Method Description 685
15.4.2 Example: Multiplane Meshed Printed Circuit Board 687
15.4.3 Discussion 688
15.5 Parallel Coplanar Microstrips 688
15.6 Multilayered Interconnect Modeling 690
15.7 5 Parameter Extraction for General Waveguides 692
15.8 Digital Signal Processing and Spectrum Estimation 694
15.8.1 Prony's Method 695
15.8.2 Autoregressive Models 697
15.8.3 Pad6 Approximation 702
15.9 Modeling of Lumped Circuit Elements 706
15.9.1 FDTD Formulation Extended to Circuit Elements 706
15.9.2 The Resistor 708
15.9.3 The Resistive Voltage Source 708
15.9.4 The Capacitor 709
15.9.5 The Inductor 711
15.9.6 The Arbitrary Two Terminal Linear Lumped Network 711
15.9.7 The Diode 714
15.9.8 The Bipolar Junction Transistor 715
15.10 Direct Linking of FDTD and SPICE 717
15.10.1 Basic Idea 718
15.10.2 Norton Equivalent Circuit "Looking Into" the FDTD Space Lattice 719
15.10.3 Thevenin Equivalent Circuit "Looking Into" the FDTD Space Lattice 721
15.11 Case Study: A 6 GHz MESFET Amplifier Model 723
15.11.1 Large Signal Nonlinear Model 723
15.11.2 Amplifier Configuration 725
15.11.3 Analysis of the Circuit without the Packaging Structure 726
15.11.4 Analysis of the Circuit with the Packaging Structure 728
15.12 Emerging Topic: Wireless High Speed Digital Interconnects Using Defect Mode
Electromagnetic Bandgap Waveguides 731
15.12.1 Stopband of the Defect Free Two Dimensional EBG Structure 732
15.12.2 Passband of the Two Dimensional EBG Structure with Waveguiding Defect 732
15.12.3 Laboratory Experiments and Supporting FDTD Modeling 734
15.13 Summary and Conclusions 736
Acknowledgments 737
References 737
Selected Bibliography 740
Projects 741
16 Photonics
Geoffrey Burr, Susan Hagness, and Allen Taftove 743
16.1 Introduction 743
16.2 Introduction to Index Contrast Guided Wave Structures 743
16.3 FDTD Modeling Issues 744
16.3.1 Optical Waveguides 744
16.3.2 Material Dispersion and Nonlinearities 747
16.4 Laterally Coupled Microcavity Ring Resonators 747
16.4.1 Modeling Considerations: Two Dimensional FDTD Simulations 748
16.4.2 Coupling to Straight Waveguides 750
16.4.3 Coupling to Curved Waveguides 750
16.4.4 Elongated Ring Designs ("Racetracks") 752
16.4.5 Resonances of the Circular Ring 752
16.5 Laterally Coupled Microcavity Disk Resonators 756
16.5.1 Resonances 756
16.5.2 Suppression of Higher Order Radial Whispering Gallery Modes 760
16.6 Vertically Coupled Racetrack 761
16.7 Introduction to Distributed Bragg Reflector Devices 765
16.8 Application to Vertical Cavity Surface Emitting Lasers 765
16.8.1 Passive Studies 766
16.8.2 Active Studies: Application of the Classical Gain Model 767
16.8.3 Application of a New Semiclassical Gain Model 769
16.9 Quasi One Dimensional DBR Structures 770
16.10 Introduction to Photonic Crystals 772
16.11 Calculation of Band Structure 774
16.11.1 The "Order AT Method 775
16.11.2 Frequency Resolution 778
16.11.3 Filter Diagonalization Method 780
16.11.4 The Triangular Photonic Crystal Lattice 782
16.11.5 Sources of Error and Their Mitigation 784
16.12 Calculation of Mode Patterns 787
16.13 Variational Approach 790
16.14 Modeling of Defect Mode Photonic Crystal Waveguides 791
16.14.1 Band Diagram of a Photonic Crystal Slab 793
16.14.2 Band Diagram of a Photonic Crystal Waveguide 795
16.14.3 Intrinsic Loss in Photonic Crystal Waveguides 798
16.14.4 Transmission in Photonic Crystal Waveguides 803
16.14.5 Aperiodic Photonic Crystal Waveguides 806
16.14.6 Photonic Crystal Waveguide Extrinsic Scattering Loss from the Green Function 806
16.15 Modeling of Photonic Crystal Resonators 807
16.16 Modeling Examples of Photonic Crystal Resonators 810
16.16.1 Electrically Driven Microcavity Laser 810
16.16.2 Photonic Crystal Cross Waveguide Switch 812
16.17 Introduction to Frequency Conversion in Second Order Nonlinear Optical
Materials 813
16.18 PSTD 4 Algorithm 813
16.19 Extension to Second Order Nonlinear Media 814
16.20 Application to a Nonlinear Waveguide with a QPM Grating 814
16.21 Application to Nonlinear Photonic Crystals 817
16.22 Introduction to Nanoplasmonic Devices 820
16.23 FDTD Modeling Considerations 820
16.24 FDTD Modeling Applications 821
16.25 Introduction to Biophotonics 822
16.26 FDTD Modeling Applications 822
16.26.1 Vertebrate Retinal Rod 822
16.26.2 Precancerous Cervical Cells 824
16.26.3 Sensitivity of Backscattering Signatures to Nanometer Scale Cellular Changes 827
16.27 PSTD Modeling Application to Tissue Optics 828
16.28 Summary 830
Acknowledgments 830
References 830
17 Advances in PSTD Techniques
Qing Liu and Gang Zhao 847
17.1 Introduction 847
17.2 Approximation of Derivatives 847
17.2.1 Derivative Matrix for the Second Order Finite Difference Method 848
17.2.2 Derivative Matrices for Fourth Order and ATth Order Finite Difference Methods 849
17.2.3 Trigonometric Interpolation and FFT Method 850
17.2.4 Nonperiodic Functions and Chebyshev Method 851
17.3 Single Domain Fourier PSTD Method 854
17.3.1 Approximation of Spatial Derivatives 855
17.3.2 Numerical Stability and Dispersion 856
17.4 Single Domain Chebyshev PSTD Method 857
17.4.1 Spatial and Temporal Grids 857
17.4.2 Maxwell's Equations in Curvilinear Coordinates 858
17.4.3 Spatial Derivatives 860
17.4.4 Time Integration Scheme 861
17.5 Multidomain Chebyshev PSTD Method 861
17.5.1 Subdomain Spatial Derivatives and Time Integration 862
17.5.2 Subdomain Patching by Characteristics 863
17.5.3 Subdomain Patching by Physical Conditions 864
17.5.4 Filter Design for Corner Singularities 864
17.5.5 Multidomain PSTD Results for 2.5 Dimensional Problems 866
17.5.6 Multidomain PSTD Results for Three Dimensional Problems 868
17.6 Penalty Method for Multidomain PSTD Algorithm 868
17.7 Discontinuous Galerkin Method for PSTD Boundary Patching 87 3
17.7.1 Weak Form of Maxwell's Equations 873
17.7.2 Space Discretization and Domain Transformation 873
17.7.3 Mass Matrix and Stiffness Matrix 874
17.7.4 Flux on the Boundary 876
17.7.5 Numerical Results for DG PSTD Method 876
17.8 Summary and Conclusions 879
Appendix 17A: Coefficients for the Five Stage, Fourth Order Runge Kutta Method 879
References 880
18 Advances in Unconditionally Stable Techniques
Hans De Raedt 883
18.1 Introduction 883
18.2 General Framework 883
18.3 Matrix Exponential Concepts 884
18.4 Product Formula Approach 887
18.4.1 The Classic Yee Algorithm as a Particular Realization 887
18.4.2 The ADI Method as a Second Realization 888
18.4.3 Unconditionally Stable Algorithms: Real Space Approach 889
18.4.4 Unconditionally Stable Algorithms: Fourier Space Approach 891
18.5 Chebyshev Polynomial Algorithm 892
18.6 Extension to Linear Dispersive Media 895
18.7 Extension to Perfectly Matched Layer Absorbing Boundary Conditions 898
18.8 Summary 899
Appendix 18 A: Some Technical Details 900
Appendix 18B: Stability Analysis of Equation (18.17) 902
Appendix 18C: Stability Analysis of Equation (18.19) 904
References 904
Projects 905
19 Advances in Hybrid FDTD FE Techniques
Thomas Rylander, Fredrik Edelvik, Anders Bondeson, and Douglas Riley 907
19.1 Introduction 907
19.2 Time Domain Finite Elements 910
19.2.1 Coupled Curl Equations 910
19.2.2 Wave Equation 913
19.2.3 Equivalences Between Finite Elements and FDTD 917
19.3 Tetrahedral, Hexahedral (Brick), and Pyramidal Zeroth Order Edge and Facet Elements 918
19.3.1 Tetrahedral Finite Elements 919
19.3.2 Hexahedral (Brick) Finite Elements 921
19.3.3 Pyramidal Finite Elements 922
19.4 Stable Hybrid FDTD FE Interface 924
19.4.1 Spatial Discretization 924
19.4.2 Time Stepping on a Hybrid Space Lattice 927
19.4.3 Generalized Newmark Scheme 928
19.4.4 Proof of Stability 929
19.4.5 Alternative Time Stepping Schemes 930
19.4.6 Extensions of the Hybrid FDTD FE Concept 931
19.4.7 Reflection at the Interface of FDTD and FE Regions of a Hybrid Space Lattice 931
19.4.8 Scattering from the PEC Sphere 933
19.5 Mesh Generation Approaches 935
19.6 Subcell Wire and Slot Algorithms for Time Domain Finite Elements 936
19.6.1 Modeling Thin Wires 936
19.6.2 Modeling Thin Slots 939
19.6.3 Numerical Results for Thin Wires and Slots 941
19.7 Application to Advanced Scattering and Radiation Problems 943
19.7.1 Monostatic RCS of the NASA Almond 943
19.7.2 Bistatic RCS of the Saab Trainer Aircraft 945
19.7.3 Input Impedance of the Four Arm Sinuous Antenna 948
19.8 Summary 949
Acknowledgments 950
References 950
20 Advances in Hardware Acceleration for FDTD
Ryan Schneider, Sean Krakiwsky, Laurence Turner, and Michal Okoniewski 955
20.1 Introduction 955
20.2 Background Literature 956
20.3 Fundamental Design Considerations 957
20.4 Conceptual Massively Parallel FPGA Implementation 958
20.5 Case Study of Using the FPGA as a Coprocessor 962
20.6 Performance of Custom Hardware Implementations 964
20.7 Fundamentals of Graphics Processor Units 965
20.7.1 Overview 965
20.7.2 Graphics Pipeline 965
20.7.3 Memory Interface 967
20.7.4 Programmable Fragment and Vertex Processors 968
20.8 Implementing FDTD on a Graphics Processor Unit 969
20.8.1 Initialization 969
20.8.2 Electric and Magnetic Field Updates 970
20.8.3 Boundaries 972
20.8.4 Source Excitation 974
20.8.5 Archiving Observation Nodes 975
20.8.6 Multipass Rendering 975
20.8.7 Display 977
20.9 Performance Measurements of the GPU Accelerator 977
20.10 Summary and Conclusions 978
References 978
Acronyms and Common Symbols 981
About the Authors 985
Index 997 |
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| author | Taflove, Allen Hagness, Susan C. |
| author_facet | Taflove, Allen Hagness, Susan C. |
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| discipline | Physik Elektrotechnik Elektrotechnik / Elektronik / Nachrichtentechnik |
| discipline_str_mv | Physik Elektrotechnik Elektrotechnik / Elektronik / Nachrichtentechnik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV021702542 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:17:46Z |
| indexdate | 2024-11-25T17:26:05Z |
| institution | BVB |
| isbn | 9781580538329 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014916470 |
| oclc_num | 439544847 |
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| physical | XXII, 1006 S. Ill., graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
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| publisher | Artech House |
| record_format | marc |
| series2 | Artech House antennas and propagation library |
| spellingShingle | Taflove, Allen Hagness, Susan C. Computational electrodynamics the finite-difference time-domain method Electrodynamics - Mathematics Finite differences Time-domain analysis Datenverarbeitung Electromagnetism Integro-differential equations Numerical solutions Maxwell equations Data processing Maxwell equations Numerical solutions Moments method (Statistics) Zeitbereich (DE-588)4130720-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Maxwellsche Gleichungen (DE-588)4221398-8 gnd Elektrodynamik (DE-588)4014251-6 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd |
| subject_GND | (DE-588)4130720-3 (DE-588)4128130-5 (DE-588)4221398-8 (DE-588)4014251-6 (DE-588)4194626-1 |
| title | Computational electrodynamics the finite-difference time-domain method |
| title_auth | Computational electrodynamics the finite-difference time-domain method |
| title_exact_search | Computational electrodynamics the finite-difference time-domain method |
| title_exact_search_txtP | Computational electrodynamics the finite-difference time-domain method |
| title_full | Computational electrodynamics the finite-difference time-domain method Allen Taflove ; Susan C. Hagness |
| title_fullStr | Computational electrodynamics the finite-difference time-domain method Allen Taflove ; Susan C. Hagness |
| title_full_unstemmed | Computational electrodynamics the finite-difference time-domain method Allen Taflove ; Susan C. Hagness |
| title_short | Computational electrodynamics |
| title_sort | computational electrodynamics the finite difference time domain method |
| title_sub | the finite-difference time-domain method |
| topic | Electrodynamics - Mathematics Finite differences Time-domain analysis Datenverarbeitung Electromagnetism Integro-differential equations Numerical solutions Maxwell equations Data processing Maxwell equations Numerical solutions Moments method (Statistics) Zeitbereich (DE-588)4130720-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Maxwellsche Gleichungen (DE-588)4221398-8 gnd Elektrodynamik (DE-588)4014251-6 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd |
| topic_facet | Electrodynamics - Mathematics Finite differences Time-domain analysis Datenverarbeitung Electromagnetism Integro-differential equations Numerical solutions Maxwell equations Data processing Maxwell equations Numerical solutions Moments method (Statistics) Zeitbereich Numerisches Verfahren Maxwellsche Gleichungen Elektrodynamik Finite-Differenzen-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014916470&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT tafloveallen computationalelectrodynamicsthefinitedifferencetimedomainmethod AT hagnesssusanc computationalelectrodynamicsthefinitedifferencetimedomainmethod |