Advances in the Formulations and Accuracy of the Method of Moments Applied to Electromagnetics

Sommaire

CONTENTS ...............................................................................................................I
LIST OF PUBLICATIONS ............................................................................................V
LIST OF FIGURES ...................................................................................................VII
LIST OF TABLES .....................................................................................................XI
LIST OF ABBREVIATIONS .......................................................................................XIII
PREFACE.................................................................................................................1
PART I ......................................................................................................................3
1 Electromagnetics ................................................................................5
1.1 Maxwell's equations ......................................................................5
1.2 Harmonic waves ............................................................................8
1.3 The Stratton-Chu equations .......................................................10
1.4 Fields on the surface S of a sheet ...............................................12
1.4.1 Fully embedded sheet ............................................................12
1.4.2 Sheet in contact with more than one volume........................17
1.5 Generalized formulation for mixed materials ............................19
1.6 Equivalent surface current densities .........................................22
1.7 Boundary conditions....................................................................22
1.8 Relations between normal and tangential components.............24
1.9 First and second form of the integro-differential equations......25
1.10 Near and far scattered fields ......................................................26
1.10.1 Exact near field expressions ..................................................26
1.10.2 Far field approximation .........................................................27
1.11 Normal and tangential components of the integro-differential
equations on the surface Si .........................................................28
1.12 Perfectly conducting sheets.........................................................29
1.13 Sum of Fields on both sides of a flat perfectly conducting sheet
of arbitrary shape ........................................................................30
1.14 Canonical expressions .................................................................31
1.15 Summary .....................................................................................33
2 Method of Moments ..........................................................................37
2.1 Preamble......................................................................................37
2.2 Basis functions ............................................................................38
2.2.1 Scalar or vector basis functions .............................................38
2.2.2 Discretized equations.............................................................39
2.2.3 Entire Domain basis functions ..............................................39
2.2.4 Subdomain basis functions ....................................................41
2.2.5 Conforming functions.............................................................43
2.2.6 Pulse basis function ...............................................................45
2.2.7 Linear basis function..............................................................45
2.2.8 Rao Wilton Glisson (RWG) basis functions...........................48
2.2.9 Rooftop functions....................................................................50
2.3 Testing .........................................................................................50
2.3.1 Normal and tangential testing ..............................................51
2.3.2 General considerations about testing....................................54
2.3.3 Galerkin testing .....................................................................57
2.3.4 Point Matching.......................................................................57
2.3.5 Razor blade testing.................................................................58
2.4 Summary .....................................................................................58
3 Method of Moments Formulation for Multiple Regions...........63
3.1 Preamble......................................................................................63
3.2 Singular edges and branches ......................................................65
3.3 RWG scheme on singular edges..................................................66
3.3.1 Physical Continuity and Sector Concept...............................66
3.3.2 Solder line...............................................................................67
3.3.3 Sector table.............................................................................67
3.4 Local ZI=V systems of equations ................................................69
3.5 Two properties of RWG sectors...................................................71
3.6 Global ZI=V system of equations ................................................73
3.7 PEC / PMC case...........................................................................75
3.8 Summary .....................................................................................76
4 Method of Moments for perfectly conducting sheets ..............81
4.1 Preamble......................................................................................81
4.2 Theory ..........................................................................................82
4.2.1 tEFIE-f and tMFIE-nxf for PEC sheets ................................82
4.2.2 tEFIE-f applied to PEC sheets ..............................................85
4.2.3 tMFIE-nxf applied to PEC sheets..........................................86
4.3 Singularities arising with PEC sheets .......................................87
4.3.1 Singularity of ZE obtained with the tEFIE-f .........................87
4.3.2 Singularity of ZM obtained with the tMFIE-nxf....................89
4.4 E-MFIE for PEC sheets...............................................................90
4.4.1 The E-MFIE formulation .......................................................90
4.4.2 Efficient implementation of the E-MFIE ..............................91
4.5 Summary .....................................................................................93
5 Accuracy of the Method of Moments ............................................95
5.1 Approximations introduced by the Method of Moments............95
5.2 Numerical issues and limitations ...............................................97
5.2.1 Problem size – CPU time limitation......................................97
5.2.2 Condition number ..................................................................98
5.2.3 Resonance .............................................................................101
5.2.4 Low frequency breakdown ...................................................102
5.3 Numerical integration...............................................................103
5.3.1 Quadratures over triangles..................................................103
5.3.2 Practical integration within the MoM.................................105
5.3.3 1/R and 1/R³ singularities ....................................................106
5.4 Extraction of the 1/R and 1/R³ singularities for the free-space
Green’s function.........................................................................108
5.4.1 Singularity associated with ZHJ ...........................................109
5.4.2 Singularities associated with ZEJ ........................................114
5.5 High efficiency quadratures and 1/R³ singularity....................117
5.5.1 Analytical analysis – regular triangles ...............................117
5.5.2 Numerical example ..............................................................120
5.5.3 Analytical analysis - very elongated triangles....................122
5.6 Summary ...................................................................................127
PART II.................................................................................................................135
6 Numerical examples .......................................................................137
6.1 Introduction...............................................................................138
6.1.1 Parameters common to all examples...................................138
6.1.2 Variable parameters ............................................................138
6.1.3 Presentation of the results...................................................139
6.1.4 The er = 1 reference solution ................................................140
6.2 Sphere ........................................................................................141
6.2.1 tEFIE-f..................................................................................143
6.2.2 tMFIE-nxf .............................................................................147
6.2.3 Homogeneous and inhomogeneous mesh............................149
6.2.4 Mesh density ........................................................................151
6.2.5 Dielectric sphere...................................................................153
6.3 Sphere in a half spherical shell ................................................156
6.3.1 PEC sphere inside a er = 1 shell...........................................156
6.3.2 PEC sphere inside a er = 9 shell...........................................160
6.4 Rounded cube.............................................................................162
6.4.1 Dielectric rounded cube........................................................163
6.4.2 PEC rounded cube................................................................166
6.5 Cube : regular meshes...............................................................168
6.5.1 Dielectric cube ......................................................................168
6.5.2 PEC cube ..............................................................................172
6.6 Cube : Log-distributed meshes .................................................175
6.6.1 Dielectric cube : er = 1...........................................................176
6.6.2 Dielectric cube : er = 2...........................................................180
6.6.3 PEC cube ..............................................................................182
6.7 Thin plates : regular meshes.....................................................186
6.7.1 Dielectric plate : er = 1..........................................................187
6.7.2 PEC plate..............................................................................193
6.8 Thin plate : log-distributed mesh .............................................198
6.8.1 Dielectric thin plate : er = 1..................................................199
6.8.2 Dielectric thin plate : er = 9..................................................203
6.8.3 PEC thin plate......................................................................206
6.9 Very thin and infinitely thin PEC plate : edge currents .........208
6.9.1 Introduction..........................................................................208
6.9.2 EFIE / MFIE and E-MFIE comparisons on Log-distributed
meshes..................................................................................217
6.10 PEC sheet on a thin dielectric plate .........................................227
6.10.1 PEC sheet on a er = 1 substrate...........................................228
6.10.2 PEC sheet on a er = 2 substrate...........................................234
6.11 Solder Line.................................................................................238
6.12 Examples where a 16 nodes quadrature is necessary .............241
6.12.1 Thin dielectric er = 1 plate ...................................................241
6.12.2 PEC and dielectric pyramids ...............................................244
6.13 Summary ...................................................................................247
CONCLUSION.......................................................................................................249
APPENDIX A........................................................................................................251
APPENDIX B........................................................................................................253
APPENDIX C........................................................................................................255
APPENDIX D........................................................................................................257
APPENDIX E........................................................................................................259
APPENDIX F ........................................................................................................271
APPENDIX G........................................................................................................273
APPENDIX H........................................................................................................283