Wavelets in electromagnetics and device modeling

Wavelets in 
Electromagnetics 
and Device Modeling......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 17
ACKNOWLEDGMENTS......Page 18
1.1 NOTATIONS AND ABBREVIATIONS......Page 21
1.2.1 Functions and Integration......Page 22
1.2.3 Regularity......Page 24
1.2.4 Linear Spaces......Page 27
1.2.5 Functional Spaces......Page 28
1.2.6 Sobolev Spaces......Page 30
1.2.7 Bases in Hilbert Space H......Page 31
1.2.8 Linear Operators......Page 32
BIBLIOGRAPHY......Page 34
2.1.1 Historical Development......Page 35
2.1.2 When Do Wavelets Work?......Page 36
2.1.3 A Wave Is a Wave but What Is a Wavelet?......Page 37
2.2.1 Potential Bene . ts of Using Wavelets......Page 38
2.2.2 Limitations and Future Direction of Wavelets......Page 39
2.3 THE HAAR WAVELETS AND MULTIRESOLUTION ANALYSIS......Page 40
2.4 HOW DO WAVELETS WORK?......Page 43
BIBLIOGRAPHY......Page 48
3.1 MULTIRESOLUTION ANALYSIS......Page 50
3.2.1 Franklin Scalet......Page 52
3.2.2 Battle ¨C Lemarie Scalets......Page 59
3.2.3 Preliminary Properties of Scalets......Page 60
3.3 WAVELET ¦×( ¦Ó)......Page 62
3.4 FRANKLIN WAVELET......Page 68
3.5 PROPERTIES OF SCALETS .( . ¦Ø)......Page 71
3.6 DAUBECHIES WAVELETS......Page 76
3.7 COIFMAN WAVELETS ( COIFLETS)......Page 84
3.8.1 Construction of Scalets......Page 89
3.8.2 Construction of Wavelets......Page 94
3.9.1 Basic Properties of Meyer Wavelets......Page 95
3.9.2 Meyer Wavelet Family......Page 103
3.10.1 Reconstruction......Page 112
3.10.2 Decomposition......Page 113
3.11.2 Exercise 2......Page 115
3.11.4 Exercise 4......Page 117
BIBLIOGRAPHY......Page 118
4.1 WAVELETS IN ELECTROMAGNETICS......Page 120
4.2 LINEAR OPERATORS......Page 122
4.3 METHOD OF MOMENTS ( MoM)......Page 123
4.4 FUNCTIONAL EXPANSION OF A GIVEN FUNCTION......Page 127
4.5 OPERATOR EXPANSION: NONSTANDARD FORM......Page 130
4.5.1 Operator Expansion in Haar Wavelets......Page 131
4.5.2 Operator Expansion in General Wavelet Systems......Page 133
4.5.3 Numerical Example......Page 134
4.6.1 Construction of Periodic Wavelets......Page 140
4.6.2 Properties of Periodic Wavelets......Page 143
4.6.3 Expansion of a Function in Periodic Wavelets......Page 147
4.7 APPLICATION OF PERIODIC WAVELETS: 2D SCATTERING......Page 148
4.8.1 Discretization of Operation Equations......Page 153
4.8.2 Fast Algorithm......Page 154
4.8.3 Matrix Sparsi . cation Using FWT......Page 155
4.9.1 Formulation......Page 160
4.9.2 Circuit Parameters......Page 161
4.9.3 Integral Equations and Wavelet Expansion......Page 163
4.10 INTERVALLIC COIFMAN WAVELETS......Page 164
4.10.1 Intervallic Scalets......Page 165
4.10.2 Intervallic Wavelets on [ 0, 1]......Page 174
4.11.1 Lazy Wavelets......Page 176
4.11.2 Lifting Scheme Algorithm......Page 177
4.12 GREEN¡¯S SCALETS AND SAMPLING SERIES......Page 179
4.12.1 Ordinary Differential Equations ( ODEs)......Page 180
4.12.2 Partial Differential Equations ( PDEs)......Page 186
4.13 APPENDIX: DERIVATION OF INTERVALLIC WAVELETS ON [ 0, 1]......Page 192
4.14.3 Exercise 7......Page 205
4.14.4 Exercise 8......Page 206
BIBLIOGRAPHY......Page 207
5.1 BASIS FDTD FORMULATION......Page 209
5.2 STABILITY ANALYSIS FOR THE FDTD......Page 214
5.3 FDTD AS MAXWELL¡¯S EQUATIONS WITH HAAR EXPANSION......Page 218
5.4 FDTD WITH BATTLE ¨C LEMARIE WAVELETS......Page 221
5.5 POSITIVE SAMPLING AND BIORTHOGONAL TESTING FUNCTIONS......Page 225
5.6.2 Formulation......Page 235
5.7.1 Dispersion Relation and Stability Analysis......Page 239
5.7.2 Stability Analysis for the SBTD......Page 242
5.8.1 Numerical Dispersion......Page 243
5.8.2 Convergence Analysis......Page 245
5.9 NUMERICAL EXAMPLES......Page 248
5.10 APPENDIX: OPERATOR FORM OF THE MRTD......Page 253
5.11.1 Exercise 9......Page 256
5.11.3 Project 2......Page 257
BIBLIOGRAPHY......Page 258
6.1 VECTOR- MATRIX DILATION EQUATION......Page 260
6.2 TIME DOMAIN APPROACH......Page 262
6.3 CONSTRUCTION OF MULTISCALETS......Page 265
6.4 ORTHOGONAL MULTIWAVELETS ¦×( t)......Page 275
6.5 INTERVALLIC MULTIWAVELETS ¦×( t)......Page 278
6.6 MULTIWAVELET EXPANSION......Page 281
6.7 INTERVALLIC DUAL MULTIWAVELETS ¦×( t)......Page 284
6.8 WORKING EXAMPLES......Page 289
6.9 MULTISCALET- BASED 1D FINITE ELEMENT METHOD ( FEM)......Page 296
6.10 MULTISCALET- BASED EDGE ELEMENT METHOD......Page 300
6.11 SPURIOUS MODES......Page 305
6.12 APPENDIX......Page 307
6.13.1 Exercise 11......Page 316
BIBLIOGRAPHY......Page 317
7.1 SCATTERING FROM A 2D GROOVE......Page 319
7.1.1 Method of Moments ( MoM) Formulation......Page 320
7.1.2 Coi . et- Based MoM......Page 324
7.1.4 Numerical Results......Page 325
7.2.1 Intervallic Scalets on [ 0, 1]......Page 329
7.2.2 Expansion in Coifman Intervallic Wavelets......Page 332
7.2.3 Numerical Integration and Error Estimate......Page 333
7.2.4 Fast Construction of Impedance Matrix......Page 337
7.2.5 Conducting Cylinders, TM Case......Page 339
7.2.6 Conducting Cylinders with Thin Magnetic Coating......Page 342
7.2.7 Perfect Electrically Conducting ( PEC) Spheroids......Page 344
7.3 SCATTERING AND RADIATION OF CURVED THIN WIRES......Page 349
7.3.1 Integral Equation for Curved Thin- Wire Scatterers and Antennae......Page 350
7.3.2 Numerical Examples......Page 351
7.4 SMOOTH LOCAL COSINE ( SLC) METHOD......Page 360
7.4.1 Construction of Smooth Local Cosine Basis......Page 361
7.4.2 Formulation of 2D Scattering Problems......Page 364
7.4.3 SLC- Based Galerkin Procedure and Numerical Results......Page 367
7.4.4 Application of the SLC to Thin- Wire Scatterers and Antennas......Page 375
7.5 MICROSTRIP ANTENNA ARRAYS......Page 377
7.5.1 Impedance Matched Source......Page 378
7.5.2 Far- Zone Fields and Antenna Patterns......Page 380
BIBLIOGRAPHY......Page 383
8.1 SCATTERING OF EM WAVES FROM RANDOMLY ROUGH SURFACES......Page 386
8.2 GENERATION OF RANDOM SURFACES......Page 388
8.2.1 Autocorrelation Method......Page 390
8.2.2 Spectral Domain Method......Page 393
8.3.1 Moment Method Formulation of 2D Scattering......Page 396
8.3.2 Wavelet- Based Galerkin Method for 2D Scattering......Page 400
8.3.3 Numerical Results of 2D Scattering......Page 401
8.4 3D ROUGH SURFACE SCATTERING......Page 407
8.4.1 Tapered Wave of Incidence......Page 408
8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets......Page 411
8.4.3 Numerical Results of 3D Scattering......Page 414
BIBLIOGRAPHY......Page 419
9 Wavelets in Packaging, Interconnects, and EMC......Page 421
9.1.1 What Is Quasi- static?......Page 422
9.1.2 Formulation......Page 423
9.1.3 Orthogonal Wavelets in L 2([ 0, 1])......Page 426
9.1.4 Boundary Element Method and Wavelet Expansion......Page 428
9.1.5 Numerical Examples......Page 432
9.2 SPATIAL DOMAIN LAYERED GREEN¡¯S FUNCTIONS......Page 435
9.2.1 Formulation......Page 437
9.2.2 Prony¡¯s Method......Page 443
9.2.3 Implementation of the Coifman Wavelets......Page 444
9.2.4 Numerical Examples......Page 446
9.3 SKIN- EFFECT RESISTANCE AND TOTAL INDUCTANCE......Page 449
9.3.1 Formulation......Page 451
9.3.2 Moment Method Solution of Coupled Integral Equations......Page 453
9.3.3 Circuit Parameter Extraction......Page 455
9.3.4 Wavelet Implementation......Page 457
9.3.5 Measurement and Simulation Results......Page 458
9.4.1 Basic Formulation......Page 460
9.4.2 Wavelet Expansion and Matrix Equation......Page 464
9.4.3 Evaluation of Sommerfeld- Type Integrals......Page 467
9.4.4 Numerical Results and Sparsity of Impedance Matrix......Page 471
9.5 FULL- WAVE EDGE ELEMENT METHOD FOR 3D LOSSY STRUCTURES......Page 475
9.5.1 Formulation of Asymmetric Functionals with Truncation Conditions......Page 476
9.5.2 Edge Element Procedure......Page 480
9.5.3 Excess Capacitance and Inductance......Page 484
9.5.4 Numerical Examples......Page 486
BIBLIOGRAPHY......Page 489
10.1 PHYSICAL MODELS AND COMPUTATIONAL EFFORTS......Page 494
10.2 AN INTERPOLATING SUBDIVISION SCHEME......Page 496
10.3 THE SPARSE POINT REPRESENTATION ( SPR)......Page 498
10.4 INTERPOLATION WAVELETS IN THE FDM......Page 499
10.4.1 1D Example of the SPR Application......Page 500
10.4.2 2D Example of the SPR Application......Page 501
10.5 THE DRIFT- DIFFUSION MODEL......Page 504
10.5.1 Scaling......Page 506
10.5.2 Discretization......Page 507
10.5.3 Transient Solution......Page 509
10.5.4 Grid Adaptation and Interpolating Wavelets......Page 510
10.5.5 Numerical Results......Page 512
10.6 MULTIWAVELET BASED DRIFT- DIFFUSION MODEL......Page 518
10.6.1 Precision and Stability versus Reynolds......Page 519
10.6.2 MWFEM- Based 1D Simulation......Page 522
10.7 THE BOLTZMANN TRANSPORT EQUATION ( BTE) MODEL......Page 524
10.7.2 Spherical Harmonic Expansion of the BTE......Page 525
10.7.3 Arbitrary Order Expansion and Galerkin¡¯s Procedure......Page 529
10.7.4 The Coupled Boltzmann ¨C Poisson System......Page 535
10.7.5 Numerical Results......Page 537
BIBLIOGRAPHY......Page 544
Index......Page 547