![A Multigrid Method for Nonlinear Unstructured Finite ... · AMGe framework are also relevant for the nonlinear multigrid method (NMGM) of Hackbusch [8]. Extending our work to this](https://static.fdocuments.net/doc/165x107/5b5f027a7f8b9a057e8d5c33/a-multigrid-method-for-nonlinear-unstructured-finite-amge-framework-are.jpg)
Multigrid Finite Element Methods for …...Multigrid Finite Element Methods for Electromagnetic...
30
Multigrid Finite Element Methods for Electromagnetic Field Modeling Yu Zhu Cadence Design Systems, Inc. Andreas C. Cangellaris University of Illinois at Urbana-Champaign IEEE Antennas and Propagation Society, Sponsor IEEE Press Series on ElectromagneticWave Theory Donald G. Dudley, Series Editor IEEE IEEE PRESS A JOHN WILEY & SONS, INC., PUBLICATION
Transcript of Multigrid Finite Element Methods for …...Multigrid Finite Element Methods for Electromagnetic...
Multigrid Finite Element Methods for Electromagnetic Field Modeling
Yu Zhu Cadence Design Systems, Inc.
Andreas C. Cangellaris University of Illinois at Urbana-Champaign
IEEE Antennas and Propagation Society, Sponsor
IEEE Press Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor
IEEE IEEE PRESS
A JOHN WILEY & SONS, INC., PUBLICATION
This Page Intentionally Left Blank
Multigrid Finite Element Methods for Electromagnetic Field Modeling
IEEE Press 445 Hoes Lane
Piscataway, NJ 08854
IEEE Press Editorial Board Mohamed E. El-Hawary, Editor in Chief
M. Akay T. G. Croda M. S. Newman J. B. Anderson R.J. Herrick F. M. B. Pereira R. J. Baker S. V Kartalopoulos C. Singh J. E. Brewer M. Montrose G. Zobrist
Kenneth Moore, Director of IEEE Book and Information Services PIS) Catherine Faduska, Senior Acquisitions Editor
IEEE Antennas and Propagation Society, Sponsor AP-S Liaison to IEEE Press, Robert Mailloux
Technical Reviewers Andrew F. Peterson, Georgia Tech
J0hn.L. Volakis, The Ohio State University
Multigrid Finite Element Methods for Electromagnetic Field Modeling
Yu Zhu Cadence Design Systems, Inc.
Andreas C. Cangellaris University of Illinois at Urbana-Champaign
IEEE Antennas and Propagation Society, Sponsor
IEEE Press Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor
IEEE IEEE PRESS
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 0 2006 by the Institute of Electrical and Electronics Engineers. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, NJ 07030, (201) 748-601 1, fax (201) 748- 6008, or online at http://www.wiley.com/go/permission.
Limit of LiabilityDisclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data is available.
ISBN-1 3 978-0-471-741 10-7 ISBN-10 0-471-741 10-8
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
To Jie and Helen, for being there when no one else is.
This Page Intentionally Left Blank
CONTENTS
List of Figures
List of Tables
Preface
Acknowledgments
1 Introduction
1.1 Statement of the Boundary Value Problem 1.2 Ritz Finite Element Method 1.3 Petrov-Galerkin’s Finite Element Method 1.4 Time-Harmonic Maxwell’s Equations and Boundary Conditions
1.4.1 Boundary conditions at material interfaces 1.4.2 Boundary conditions at the enclosing boundary 1.4.3 Uniqueness in the presence of impedance boundaries Present and Future Challenges in Finite Element Modeling 1.5
2 Hierarchical Basis Functions for Triangles and Tetrahedra
2.1 The Importance of Proper Choice of Finite Element Bases 2.2 Two-Dimensional Finite Element Spaces
2.2.1 Two-dimensional potential space 2.2.2 Two-dimensional field space
... X l l l
xxi
xxiii
xxvii
13
14 19 19 20
vii
viii CONTENTS
2.2.3 Two-dimensional flux space 2.2.4 Two-dimensional charge space
2.3 Relationship Among 2D Finite Element Spaces 2.4 Gradient, Curl and Divergence Matrices for 2D Finite Element Spaces 2.5 Three-Dimensional Finite Element Spaces
2.5.1 Three-dimensional potential space 2.5.2 Three-dimensional field space 2.5.3 Three-dimensional flux space 2.5.4 Three-dimensional charge space Relationship Among 3D Finite Element Spaces Gradient, Curl and Divergence Matrices for 3D Finite Element Spaces The Spaces ‘HP(curZ) and ‘HP(div) The Issue of Orthogonality in Hierarchical Bases
2.6 2.7 2.8 2.9
3 Finite Element Formulations of Electromagnetic BVPs
3.1 Electrostatic Boundary Value Problems 3.1.1 3.1.2 3.1.3
Governing equations and boundary conditions Weak statement of the electrostatic BVP The case of unbounded domains
3.2 Magnetostatic Boundary Value Problems 3.2.1 3.2.2 3.2.3 Existence of solution (solvability) 3.2.4 Uniqueness of solution
3.3 Magneto-Quasi-Static (Eddy-Current) Problems 3.3.1 3.3.2 Electric field formulation 3.3.3 Potential formulation
3.4 Full-Wave Boundary Value Problems 3.4.1 3.4.2 Electric field formulation 3.4.3 Potential formulation 3.4.4 Field-flux formulation Partial Element Equivalent Circuit Model 3.5.1 Electric field integral equation 3.5.2 Development of the PEEC model 3.5.3 The case of surface current flow 3.5.4 Low-frequency numerical instability
Governing equations and boundary conditions Weak statement of the magnetostatic BVP
Governing equations and boundary conditions
Governing equations and boundary conditions
3.5
23 24 24 26 27 28 31 39 48 49 51 52 54
61
63 63 65 68 71 71 74 77 82 85 86 87 88 91 91 92 93 94 96 96 99
101 103
4 Iterative Methods, Preconditioners, and Multigrid
4.1 Definitions
111
112
CONTENTS ix
4.1.1 Vector space, inner product, and norm 112 4.1.2 Matrix eigenvalues and eigenvectors 113 4.1.3 Properties of Hermitian matrices 114 4.1.4 Positive definite matrices 115 4.1.5 Independence, invariant subspaces, and similarity transformations 115
4.2 Iterative Methods for the Solution of Large Matrices 4.2.1 Stationary methods 4.2.2 Convergence of iterative methods 4.2.3 Non-stationary methods
4.3 Generalized Minimum Residual Method 4.4 Conjugate Gradient Method 4.5 The Preconditioner Matrix
4.5.1 The Jacobi preconditioner 4.5.2 The symmetric Gauss-Seidel preconditioner 4.5.3 Incomplete LU factorization Multigrid Process and Its Use as a Preconditioner 4.6.1 Motivation for multigrid 4.6.2 The two-grid process 4.6.3 The multigrid process
4.6
5 Nested Multigrid Preconditioner
5.1 Weak Statement of the Two-Dimensional Helmholtz Equation 5.1.1 Total field formulation 5.1.2 Scattered field formulation Development of the Finite Element System 5.2
5.3 Nested Multigrid Preconditioner 5.4 Intergrid Transfer Operators 5.5 Applications
5.5.1 5.5.2 5.5.3
Plane wave scattering by a PEC cylinder Plane wave scattering by dielectric cylinder Plane wave scattering by electrically large cylinders
6 Nested Multigrid Vector and Scalar Potential Preconditioner
6.1 'rho-Dimensional Electromagnetic Scattering 6.1.1 6.1.2 6.1.3 'rho-dimensional potential formulation 6.1.4 Nested multigrid potential preconditioner 6.1.5 lko-dimensional intergrid transfer operators 6.1.6 Applications
lko-dimensional field formulation - TE, case The finite element matrix and its properties
6.2 Three-Dimensional Electromagnetic Scattering
116 117 118 119 119 123 128 128 129 130 131 133 138 141
145
145 146 147 148 149 150 153 153 154 155
1 59
161 161 163 166 169 171 175 179
X CONTENTS
6.2.1 Three-dimensional field formulation 6.2.2 Three-dimensional potential formulation 6.2.3 Nested multigrid potential preconditioner 6.2.4 Three-dimensional intergrid transfer operators 6.2.5 6.2.6 6.2.7 Applications Finite Element Modeling of Passive Microwave Components 6.3.1 6.3.2 Transfinite-element boundary truncation 6.3.3 6.3.4 Applications Symmetry of the Nested Multigrid Potential Preconditioner 6.4.1 Potential smoothing operators 6.4.2 Symmetric nested two-grid potential preconditioner
Grid truncation via a boundary integral operator Approximate boundary integral equation preconditioner
6.3 Electromagnetic ports and associated boundary condition
Nested multigrid potential TFE preconditioner
6.4
7 Hierarchical Multilevel and Hybrid Potential Preconditioners
7.1 Higher-Order Field Formulation 7.2 Higher-Order Potential Formulation 7.3 Hierarchical Multilevel Potential Preconditioner 7.4 Hybrid MultileveVMultigrid Potential Preconditioner 7.5 Numerical Experiments 7.6 Symmetric Hierarchical Multilevel Potential Preconditioner
7.6.1 Potential smoothing operations 7.6.2 Key Attributes of Multigrid and Multilevel Potential Preconditioners
Symmetric hierarchical two-level potential preconditioner 7.7
8 Krylov-Subspace Based Eigenvalue Analysis
8.1 8.2
8.3
8.4
8.5
Subspace Iteration Methods Based on Krylov Subspace Projection 8.2.1 Arnoldi algorithm 8.2.2 Lanczos algorithm Deflation Techniques 8.3.1 8.3.2 Non-standard Eigenvalue Problems 8.4.1 Generalized eigenvalue problems 8.4.2 Quadratic eigenvalue problems Shift-and-Invert Preconditioner
Deflation techniques for symmetric matrices Deflation techniques for non-symmetric matrices
179 181 182 183 189 191 193 195 195 199 204 206 208 209 210
21 5
216 218 22 1 222 224 230 230 23 1 232
235
235 237 237 239 240 240 242 244 244 249 250
9.1
9.2 9.3
9.4
9.5
CONTENTS
9 Two-Dimensional Eigenvalue Analysis of Waveguides
FEM Formulations of the -0-Dimensional Eigenvalue Problem 9.1.1 Transverse-Field Methods Transverse-Longitudinal-Field Methods 9.3.1 Field TLF formulation 9.3.2 Potential TLF formulation 9.3.3 9.3.4 Numerical examples Transverse-Transverse-Field Method 9.4.1 Finite element formulation 9.4.2 Algorithms 9.4.3 Applications Equivalent Transmission-Line Formalism for Planar Waveguides 9.5.1 Multi-conductor transmission line theory 9.5.2 S-parameter representation of a section of an MTL
Mathematical statement of the 2D vector eigenvalue problem
Computer algorithms for eigenvalue calculation
10 Three-Dimensional Eigenvalue Analysis of Resonators
10.1 FEM Formulation of the Three-Dimensional Electromagnetic Eigenvalue Problem 10.1.1 Finite element approximation: The case of symmetric material
tensors 10.1.2 Finite element approximation: The case of Hermitian material
tensors 10.1.3 Lumped parallel resonant circuit
10.2.1 Elimination of spurious DC modes 10.2.2 Extraction of multiple modes
10.3.1 Elimination of spurious DC modes 10.3.2 Extraction of multiple modes
10.4 Multigrid/hlultilevel Eigenvalue Analysis 10.5 Numerical Validation
10.2 Eigensolver for Lossless Media
10.3 Eigensolver for Lossy Media
11 Model Order Reduction of Electromagnetic Systems
11.1 Asymptotic Waveform Evaluation 11.2 Krylov Subspace-Based Model Order Reduction
1 1.2.1 Pad6 Via Lanczos process 11.2.2 Arnoldi process: SISO system 1 1.2.3 Arnoldi process: MIMO system
xi
253
254 255 258 26 1 26 1 263 267 27 1 274 274 280 28 1 284 285 290
295
296
297
299 302 303 304 305 307 309 311 313 313
321
324 328 329 332 333
1 1.3 Passive Reduced-Order Interconnect Macromodeling Algorithm (PRIMA) 335
Xii CONTENTS
1 1.3.1 Preservation of moments in PRIMA 11.3.2 Preservation of passivity 11.3.3 Error estimate in model order reduction 11.3.4 Pole-residue representation of the reduced order model
1 1.4 Model Order Reduction of Electric Field-Based Finite Element Models 11.4.1 Adaptive Lanczos-Pad6 sweep
11.5 Maxwell’s Curl Equations-Based Model Order Reduction 11.5.1 Passivity of discrete model 11.5.2 Incorporation of lumped elements 11.5.3 PRIMA-based model order reduction
1 1.6 Applications
12 Finite Element Analysis of Periodic Structures
12.1 Finite Element Formulation of the Scattering and Radiation Problem 12.2 Computational Domain Truncation Schemes
12.2.1 Space harmonics expansion of EM fields 12.2.2 Grid truncation using transfinite flements 12.2.3 Grid truncation using anisotropic perfectly matched layers
12.3 Finite Element Approximation Inside the Unit Cell 12.4 Periodic Boundary Condition 12.5 MultileveVMultigrid Preconditioner 12.6 Applications 12.7 Finite Element Modeling of Periodic Waveguides 12.8 Application
Appendix A: Identities and Theorems from Vector Calculus
336 338 339 340 342 345 347 348 349 35 1 353
365
366 368 369 37 1 372 377 378 384 387 393 396
401
Index 404
LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2D TV-edge-elements (ref) and NV-edge-elements (right).
Visual interpretation of the gradient, curl, and divergence matrices.
The node representation for the pth-order, non-hierarchical scalar basis, (p = 3).
The four first-order, node-type scalar basis functions that form the subspace W&, are assigned to the four vertices of a tetrahedron.
Assignment of edge-type scalar basis functions in W&. (p = 3), to nodes along the edges of the tetrahedron.
Assignment of facet-type scalar basis functions of W:r for (JI = 3) to nodes on the facets of the tetrahedron.
The node representations for volume-type scalar basis functions in W& (p = 4).
Vector basis functions in the edge-type TV subspace associated with edge (m, n).
Vector basis functions in the facet-type TV subspace associated with facet (m, n, k).
Vector basis functions in the volume-type TV subspace.
The six edge-elements.
23
27
29
29
30
31
32
33
33
34
36
xiii
xiv LIST OF FIGURES
2.12
2.13
2.14
2.15
2.16
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.1 1
3.12
3.13
4.1
4.2
4.3
The six cross-products of (2.85) depicted for two different orientations of the tetrahedron.
Facet-type (left) and volume-type (right) three-dimensional normally continuous vector (NV) basis functions.
The four facet elements for the two different orientations of the tetrahedron.
Construction of the curl of an edge element from the linear combination of the two facet elements associated with the two facets that share the specific edge.
Visual aid for the interpretation of the way the gradient, curl, and divergence matrices are constructed.
Domain for the statement of the electrostatic BVP.
A two-dimensional domain consisting of two elements.
a: Pictorial description of an unbounded domain. b: Definition of buffer and absorption layers for the application of coordinate-stretching based domain truncation.
Domain for the statement of the magnetostatic BVP.
A two-dimensional domain consisting of only two elements.
Geometry for the visualization of the numerical approximation of a solenoidal volume current density.
Domain for the statement of the magneto-quasi-static BVP.
A set of conductors in free space excited by current sources.
a: A row of incident matrix D. b: A column of loop matrix C.
Triangular mesh for a metal strip conductor.
Its corresponding circuit.
A conducting strip loop.
Its corresponding circuit.
Left column (from top to bottom): Plots of the smooth eigenvectors (modes) of A for a uniform grid, Rh. with 9 interior nodes. Right column (from top to bottom): Projection of the five modes on a coarser grid, 0 2 h of grid size twice that of G1.
Left column (from top to bottom): Plots of the oscillatory eigenvectors of A for a uniform grid, Rh, with 9 interior nodes. Right column (from top to bottom): Projection of these modes on a coarser grid, 522h of grid size twice that of Rh.
V-cycle: N = 4, Q = 1.
41
42
45
46
51
63
67
68
72
76
80
87
97
104
105
105
106
106
135
136
142
LIST OF FIGURES xv
4.4
5.1
5.2
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
W-cycle: N = 4, Q = 2.
Coarse and fine grids used for a two-level, nested multigrid method.
Convergence of the three-level V-cycle MCGC for scattering by a circular PEC cylinder.
Transition between basis functions in coarse and fine grids; edge element ( 1,2).
Transition between basis functions in coarse and fine grids; edge element ( 1,3).
Transition between basis functions in coarse and fine grids; edge element (2,3).
Iterative finite element solution convergence for TE, plane wave scattering by a circular PEC cylinder at 1.5 GHz. (After Zhu and Cangellaris [lo], 02002 IEEE.)
Convergence of NMGAV process for TE, plane wave scattering by a PEC circular cylinder at various frequencies. (After Zhu and Cangellaris [ 10],@2002 IEEE.)
/H,I at T = 1.0 m (outer boundary) at 1.25 GHz for the case of TE, plane wave scattering by a circular dielectric cylinder. (After Zhu and Cangellaris [lo], 02002 IEEE.)
Convergence of the NMGAV process in the solution of TE, plane wave scattering by the PEC inlet structure. (After Zhu and Cangellaris [ 101, 02002 IEEE.)
Decomposition of a coarse-grid tetrahedron into fine-grid tetrahedra.
Splitting of two pyramids sharing a common base into four sub-tetrahedra.
Edge-element (1,2).
Edge element (1,3).
Edge element (1,4).
Edge element (2,3).
Edge element (2,4).
Edge element (3,4).
An interior edge (i, j ) for a coarse-grid tetrahedron.
Convergence behavior of three CG-based iterative solvers. (After Yu and Cangellaris [24], 02002 AGU.)
Bistatic radar cross-section for the 1.5X PEC cube. (After Yu and Cangellaris [24], 02002 AGU.)
142
150
156
172
173
173
176
177
178
179
184
184
185
186
186
186
187
187
188
194
194
XVi LIST OF FIGURES
6.19
6.20
6.21
6.22
6.23
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
9.1
9.2
9.3
9.4
9.5
A multiport electromagnetic structure.
S-parameter magnitude plots for a low-pass filter. (After Zhu and Cangellaris [25], 02001 IEEE.)
Convergence behavior of the preconditioned iterative solver for the microstrip filter depicted in the insert of Fig. 6.20. (After Zhu and Cangellaris [25], 02001 IEEE.)
S-parameter magnitude plots for an annular microstrip resonator. (After Zhu and Cangellaris [25], 02001 IEEE.)
Convergence behavior of the preconditioned iterative solver for the microstrip filter depicted in the insert of Fig. 6.22. (After Zhu and Cangellaris [25], 02001 IEEE.)
Flowchart of the hybrid multilevelhultigrid preconditioner.
Number of iterations and CPU time versus the length of the microstrip line. (After Zhu and Cangellaris [7], 02002 IEEE.)
Magnitude of scattering parameters for a waveguide filter. Measured data obtained from [8]. (After Zhu and Cangellaris [6], 02002 IEEE.)
(a) Convergence comparison of three CG solvers. (b) Convergence comparison of multilevel preconditioned CG and multilevel preconditioned GMRES. (After Zhu and Cangellaris [6], 02002 IEEE.)
Magnitude of the scattering parameters of the depicted low-pass filter. (After Zhu and Cangellaris [6], 02002 IEEE.)
Convergence comparison of three CG-based iterative solvers applied to the solution of the microstrip filter depicted in the insert of Fig. 7.5. (After Zhu and Cangellaris [6], 02002 IEEE.)
Magnitude of the scattering parameters of a rectangular waveguide band-stop filter. (After Zhu and Cangellaris [7], 02002 IEEE.)
Convergence performance of the iterative solver with hybrid preconditioning for the waveguide filter depicted in Fig. 7.7. (After Zhu and Cangellaris [7], 02002 IEEE.)
Generic geometry of a uniform electromagnetic waveguide.
The complex y plane for the illustration of the eigenvalue distribution of the matrix eigenvalue problem.
Dispersion diagram for the first three modes of a simple microstrip line (€1 = ~ . ~ Q , w = ~ . ~ ~ I I u I I , ~ = 1.92mm,d=0.6411~11).
Dispersion curve for the fundamental mode of a pedestal-support stripline.
Dispersion curves for the first two modes of a dielectric image line.
196
206
207
207
208
223
224
226
226
227
228
229
229
254
268
272
272
273
LIST OF FIGURES xvii
9.6
9.7
9.8
9.9
9.10
9.11
10.1
10.2
10.3
10.4
10.5
10.6
11.1
11.2
11.3
11.4
11.5
Dispersion curves for the three fundamental modes of a three-conductor microstrip structure on an anisotropic substrate (ezl = ez2 = 9.460. cyl = fY2 = l l .6c0, mm, a = 10.0 mm, s = 2.0 mm).
= ~ , p = 9.460 d = w = 1.0 M, h = 4.0
Dispersion curves for the first five dominant modes of a rectangular waveguide.
Eigenvalue distribution for the first several modes of the waveguide at f = 40 GHz.
Dispersion curves for the first six modes in a femte-loaded rectangular waveguide.
Eigenvalues of the ferrite-loaded rectangular waveguide at koa = 2.
a) Multiconductor transmission line model of a uniform multi-conductor waveguide; b) Cross-sectional geometry of the multi-conductor waveguide.
Lumped parallel resonant circuit.
Poles.
Eigenvalues of a rectangular cavity with a lossless dielectric block in the middle. Left plot depicts the eigenvalues obtained using the E-B formulation; right plot depicts the eigenvalues obtained using the E-field formulation.
Eigenvalues of the rectangular cavity with a lossy dielectric block in the middle.
A circular cylindrical cavity, partially filled with a dielectric rod.
Convergence of the dominant mode of a half-filled, lossy resonator for different values of the conductivity of the filling dielectric. (After Zhu and Cangellaris [20], 02002, IEEE.)
Magnitude of the reflection coefficient of a microstrip patch antenna generated by means of the AWE process.
Magnitude of the input reflection coefficient of a microstrip patch antenna.
Real and imaginary parts of the input impedance of the microstrip patch antenna shown in the insert of Fig. 1 1.2. (After Zhu and Cangellaris, [27], 02001 IEEE.)
Error bound for the microstrip patch antenna macromodel, obtained after 10,20, and 30 iterations of PRIMA.
Geometry of a low-pass microstrip filter. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
273
282
283
283
284
285
302
302
314
315
315
317
328
354
355
356
356
XViii LIST OF FIGURES
11.6
11.7
11.8
11.9
11.10
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.1 1
Magnitude of the scattering parameters S11 and S21 of the microstrip filter of Fig. 11.5. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
IS11 I of the E-plane, slot-coupled, short-circuited T-junction of Fig. 7.3. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
I Szll of the E-plane, slot-coupled, short-circuited T-junction of Fig. 7.3. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
Geometry of a gap-coupled resonance filter. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
Magnitude of the scattering parameters of a gap-coupled resonance filter. (After Zhu and Cangellaris [30], @John Wiley and Sons Ltd. Reproduced with permission.)
The unit cell of a two-dimensional periodic structure of infinite extent.
Planar (a: - y) interface between free-space and an anisotropic PML.
Portion of a rectangular PML box enclosing the computational domain.
Graphical interpretation of Property I of the Pi,j = J' A x d, . di ds.
Illustration of Property I1 of the integral Pi3, = Js A x dj . d, ds.
Edge-elements on the four side edges of the unit cell.
Magnitude of the reflection coefficient for an array of rectangular waveguides with infinitesimally thin PEC walls. (After Zhu and Lee [ 13],@2000 EMW. Reproduced with permission.)
Magnitude of the reflection coefficient for an array of rectangular waveguides with PEC walls of finite thickness. (After Zhu and Lee [ 131, 0 2 0 0 0 EMW. Reproduced with permission.)
Reflection coefficient for a dielectric slab with periodically varying permittivity. €1 = 2.56; €2 = 1.44. dl = dz = d/2; h = lm; h/d = 1.713. The incident plane wave has a y-polarized electric field and impinges onto the slab at Oinc = 45O. (After Zhu and Lee [13], 02000 EMW. Reproduced with permission.)
Sum of the squares of the magnitude of the reflection and transmission coefficients for the scattering problem of Fig. 12.9. (After Zhu and Lee [ 13],@2000 EMW. Reproduced with permission.)
Power transmission coefficient for a plane wave incident normally onto a PEC mesh. g is the period along 2 and y; t is the thickness of the mesh; e is the length of the side of the square hole. Finite element solution is obtained using IHo(mrZ) elements. (After Zhu and Lee [ 131, 02000 EMW. Reproduced with permission.)
357
358
358
359
359
367
313
376
379
380
382
387
388
389
390
39 1
LIST OF FIGURES xix
12.12 Power transmission coefficient for a plane wave incident normally onto a PEC mesh. g is the period along 5 and y; t is the thickness of the mesh; e is the length of the side of the square hole. Finite element solution is obtained using 3-1'(curZ) elements. (After Zhu and Lee [ 131, 02000 EMW. Reproduced with permission.)
Power reflection coefficient for a uniform plane wave incident normally onto an infinite, two-dimensional planar patch antenna array. The electric field is polarized along y. (After Zhu and Lee [ 131, @2000 EMW. Reproduced with permission.) 392
Power balance for the (0,O) space harmonics for a plane wave incident normally onto a planar patch antenna array. The electric field is polarized along y. (After Zhu and Lee [ 13],@2000 EMW. Reproduced with permission.) 392
39 1
12.13
12.14
12.15 Generic geometry of an infinitely long periodic waveguide. 393
12.16 Two-dimensional corrugated waveguide. 396
12.17 Dispersion curves for the waveguide in Fig. 12.16, d = 2 0 , h = D, 1 = 0.50. 397
Calculated eigenvalues X = e Y D for the periodic waveguide of Fig. 12.16 at selected frequencies.
12.18 397
This Page Intentionally Left Blank
LIST OF TABLES
4.1
5.1
5.2
5.3
5.4
6.1
6.2
6.3
6.4
6.5
7.1
Numerical Eigenvalues of RJ, for Grid ah . with w = 0.5 136
Test I. Scattering by a PEC Cylinder (15032 Unknowns, v1 = w2 = 2) 154
Test I. Scattering by a PEC Cylinder (15032 Unknowns, V-cycle, SGS) 155
Test 11. Scattering by a Dielectric Cylinder (99361 Unknowns, v1 = v2 = 2)
Test 111. Scattering by a PEC Cylinder (18 1832 Unknowns, w1 = v2 = 2)
TE, Plane Wave Scattering by a PEC Circular Cylinder (44856 Unknowns). Iterative Solver Convergence Performance
TE, Plane Wave Scattering by a PEC Circular Cylinder (44856 Unknowns, V-cycle). Impact of Number of Smoothing Operations on Convergence
TE, Plane Wave Scattering by a Dielectric Cylinder (297 248 Unknowns). Convergence Performance at Different Excitation Frequencies
Numerical Computation Data for Scattering by PEC Cubes
Convergence Performance Comparison (1.5X Cube)
Convergence Comparison of Three CG-based solvers at 15.0 GHz
155
156
176
176
178
193
193
227
xxi
xxll CONTENTS
9.1 Normalized Eigenvalues (p/k,-,) of the Five Dominant Modes of a Metallic Waveguide at f = 40 GHz, Extracted Using ‘H1(curl) Basis Functions 282
10.1
10.2
Lowest Eigenvalues of a Partially Filled Circular, Cylindrical Cavity
Eigenfrequency of the Dominant Mode of a Partially Filled Rectangular Resonator 317
316
PREFACE
The central theme of this book is the development of robust preconditioners for the iterative solution of electromagnetic field boundary value problems (BVPs) discretized by means of finite methods. More precisely, the book provides a detailed presentation of our successful attempts to utilize concepts from multigrid and multilevel methods for the preconditioning of matrices resulting from the approximation of electromagnetic BVPs using finite elements.
Solution by iteration is the reluctant compromise of modem scientific computing. A brief review of the young history of numerical computation offers unequivocal support for Edward Teller's bleak conjecture that "A state-of-the-art culculution requires ZOO hours of CPU rime on the stare-ofthe-art computer; independent of the decade." Hidden in this statement is recognition and praise of our innate drive to advance our understanding of the physical world and, eventually, through its predictive modeling, establish our dominance over it. Yet, the deeper our understanding becomes the higher the complexity of the mysteries we need to unravel. And this complexity pushes our computing technology to its current limits.
In our desire to stay one step ahead of this seemingly perpetual complexity barrier, iteration seems to be an invaluable ally. Loosely speaking an iterative process should be interpreted as the attempt to solve a given problem by trial and error. In the context of the theme of this book, the pertinent problem is the solution of the system of linear equations resulting from the discretization of the electromagnetic BVP using finite elements. A guess for the unknown vector is made and the error (residual) is calculated between the forcing vector and the vector resulting from the multiplication of the trial vector by the matrix. A convergent iterative process is one through which the residual is iteratively reduced below some a-priori defined threshold.
Thanks to continuing advances in iterative matrix-solving methods, we are now equipped with solid knowledge about the right ingredients for the construction of an effective iterative
xxiii
XXiV PREFACE
solver, Among them, the availability of a good preconditioner stands out. Effective precon- ditioning of the iteration matrix expedites convergence and can enhance solution accuracy. Multigrid and multilevel methods have been shown to be most effective as preconditioners. Their success stems from the fact that the convergence of standard relaxation techniques can be accelerated by utilizing different spatial samplings for the damping of different compo- nents in the decomposition of the error in terms of the eigenvectors of the iteration matrix. In the context of finite methods spatial sampling is most typically effected through the size of the element in the numerical grid or the order of the interpolating polynomial.
The construction of an effective multigridmultilevel preconditioner requires the under- standing of the properties of the eigenvectors and eigenvalues of the iteration matrix. This, in turn, calls for a thorough insight in the way the development of the finite element ap- proximation of the boundary value problem maps the properties of the eigenpairs of the governing mathematical operator onto those of its finite element matrix approximation.
It was primarily due to lack of such insight and understanding that the application of multigridmultilevel ideas to the development of effective preconditioners for finite element matrix approximations of vector electromagnetic BVPs was slow in coming. However, over the past ten years significant progress has been made toward filling this void. This book attempts to provide the reader with an elucidating description of the path that led us to today’s state-of- the-art in multigridmultilevel preconditioners for finite element-based iterative electromagnetic field solvers.
Chapter 1 begins with a brief overview of the finite element method and its applica- tion to the numerical solution of the electromagnetic (BVP). This overview is followed by a discussion of current and future challenges in finite element-based electromagnetic field modeling, which helps motivate the methodologies and algorithms presented in the following chapters.
Chapter 2 presents a complete and systematic development of spaces of functions suitable for the expansion of scalar potential fields, tangentially continuous vector fields, normally continuous vector fields, and scalar charge densities on the triangles and tetrahedra used for the discretization of the computational domain. The corresponding spaces for these four classes of quantities are often referred to as Whitney-0, Whimey-1, Whitney-2, and Whitney-3 forms. They are the natural choices for the representation of the scalar potential, the field intensities (2 and I?), the flux densities (d and g) , and the scalar charge density, respectively. The reason for this is that their development is guided by their physical attributes, as described mathematically by the governing equations and associated boundary conditions. It is shown in Chapter 2 that explicit, hierarchical basis functions of arbitrary order can be developed for all four spaces. In addition, the mathematical relations of the four spaces are presented and discussed. The proposed basis functions can be used, either directly or after a further partial orthogonalization post-processing, to improve the condition number of the finite element matrix.
Chapter 3 is devoted to the development of the finite element formulations of various classes of electromagnetic BVPs. In particular, formulations pertinent to static, quasi- static, and dynamic problems are presented, along with their subsequent reduction into matrix equation statements of their discrete approximation using the Galerkin process. The emphasis of the chapter is on the appropriate choice of basis functions for the expansion of pertinent unknown field quantities and their sources, in a manner such that the require- ments for solvability and uniqueness of the solution of the continuous problem are mapped correctly onto the finite element matrix approximation of the BVP. The chapter concludes with an investigation of the source of the so-called low-frequency numerical instability of integral equation-based approximations of electrodynamic problems, in the context of the
PREFACE XXV
properties of the basis functions used for the approximation of the unknown electric current and charge densities.
In Chapter 4 an overview of iterative methods and preconditioning techniques for sparse linear systems of equations is provided. The discussion leads to the motivation for multigrid methods and a brief overview of the fundamental steps in the development of a multigrid algorithm. This is followed, in Chapter 5, by the demonstration of the application of the nested multigrid process as an effective preconditioner for enhancing the convergence of the iterative solution of finite element approximations of the scalar wave equation.
Chapter 6 extends the application of multigrid processes as preconditioners for enhancing the convergence of the iterative solution of finite element approximations of vector elec- tromagnetic BWs. Applications of the proposed nested multigrid preconditioners to the solution of numerous electromagnetic BVPs are used to demonstrate the superior numerical convergence and efficient memory use of these algorithms.
The nested multigrid algorithms discussed in Chapters 5 and 6 can be viewed as an h- adaptive finite element method, since use is made of the lowest order basis functions, while the finite element mesh is refined in a nested fashion. However, padaptive techniques, where the order of the basis functions is increased while the finite element mesh is kept unchanged, are oftentimes more effective in reducing the discretization error, especially when used in regions in the computational domain where the field quantities exhibit smooth spatial variation. Thus, in Chapter 7 we present a robust, hierarchical, multilevel preconditioning technique for the fast finite element analysis of electromagnetic BWs.
As a precursor to the discussion of electromagnetic eigenvalue problems, Chapter 8 briefly introduces various Krylov-based techniques for the solution of large matrix eigen- value problems. In Chapter 9 the Krylov-based matrix eigenvalue solution machinery is streamlined for application to the solution of matrix eigenvalue problems resulting from the finite element approximation of two-dimensional electromagnetic eigenvalue problems. In particular, the emphasis is on the finite element-based eigenvalue analysis of uniform, inhomogeneous, anisotropic electromagnetic waveguides.
In Chapter 10 an efficient algorithm is developed for the robust solution of sparse matrix eigenvalue problems resulting from the finite element approximation of three-dimensional electromagnetic cavities, as well as unbounded and lossy electromagnetic resonators. The proposed algorithm is based on a field-flux formulation of the finite element approximation of Maxwell's equations. The special relationship between the vector bases u_sed for the expansion of the electric field vector E' and the magnetic flux density vector B is used to reduce the computational complexity of the numerical solution of the finite element matrices resulting from the proposed formulation.
The formulation introduced in Chapter 10 is further exploited in Chapter 11 for the establishment of a systematic methodology and associated computer algorithms for the generation of reduced-order matrix transfer function representations of multi-port electro- magnetic systems. The emphasis of the presentation in Chapter 10 is on two topics. First, it is shown that the field-flux formulation of the electromagnetic system leads to a passive discrete, state-space model, which is compatible with the well-established Krylov-subspace model order reduction techniques for reduced-order macromodeling of linear systems. Sec- ond, it is shown that through the proper selection of the expansion functions for the finite element approximation of the electric field and the magnetic flux density, a computationally efficient algorithm can be devised for the calculation of the broadband reduced-order matrix transfer function of the electromagnetic system.
Finally, in Chapter 12, we consider the application of finite elements to the modeling of periodic structures. In addition to the discussion of how the finite element formulation of
XXVi PREFACE
the electromagnetic BVP must be modified to incorporate the results of Floquet’s theory, the way the convergence of the iterative solution of the resulting finite element matrix can be enhanced via a multigrid/multilevel preconditioner is presented also.
To facilitate the reader with the computer implementation of the various methodologies presented in this book we have provided their algorithmic description in pseudo-code form. It is hoped that this feature will help expedite the incorporation of these algorithms in existing finite element-based solvers for wave problems.
The numerical examples presented in the applications section of each chapter are used to validate the presented methodologies and demonstrate their computational efficiency and robustness. As such, they tend to involve rather simple electromagnetic problems for which the answer is already known or can be obtained by other means. We hope the reader will appreciate the important role that such simple exercises play in building confidence in the tools we develop to tackle harder problems with yet unknown answers.
In all, this is a book about making iterative methods work well when applied to solving wave problems using finite methods. Our hope is that its readers will benefit from using some of the presented ideas to enrich their bag of tricks with new ones, and advance their wave field solvers to a new plateau of modeling versatility, computational efficiency and numerical robustness.
Y. ZHU
A. C . CANGELLAWS
ACKNOWLEDGMENTS
We would like to begin by singling out Prof. Robert Lee from the Ohio State University as the individual who deserves special credit for making this book happen. It was through Prof. Lee’s Ph.D. thesis advising that Andreas C. Cangellaris’s (A.C.C.’s) interest in finite element methods was shifted to a higher gear. It was through Prof. Lee’s supervision of Yu Zhu’s (Y.Z.’s) M.S. thesis that Y.Z. became interested in finite element methods. It was through Prof. Lee’s introduction that the Y.Z.-A.C.C. scientific partnership was formed. We are most thankfi~l to Prof. Lee, for his multi-faceted, catalytic role in the happening of this book.
Most of the material that appears in this book was the outcome of Y.Z.’s Ph.D. research work in the Center for Computational Electromagnetics (CCEM), in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign (UIUC). This research would not have been possible without the sponsorship of the Motorola Center for Communications at UIUC. We are particularly indebted to Dr. Nick Buris from the Motorola Corporate Research Labs, Schaumburg, Illinois, for his support and technical contributions.
We would also like to express our sincere thanks to Professors Weng Cho Chew, Jianming Jin, Eric Michielssen, and JosC Schutt-AinC who, in their role as colleagues (to A.C.C.) and as teachers (to Y.Z.) have contributed immensely to our understanding and further learning of applied and computational electromagnetics. Their presence makes CCEMNIUC a unique center of excellence in electromagnetic wave theory education and research.
We would also like to thank Mr. Hong Wu, who read through several of the chapters in early drafts of this book and made useful suggestions.
We are also indebted to the numerous experts from the applied mathematics and com- putational electromagnetics communities who, over the past ten years or so have been
xxvii
xxviii
aggressively pursuing the advancement and application of multigrid methods for wave field modeling. Among them, Prof. Jin-Fa Lee from Ohio State University deserves special credit for his inspirational, trend-setting contributions to the applications of multigridlmultilevel methods in finite element-based electromagnetic modeling. Even though every effort has been made to give reference to all those whose work has influenced the material presented in this book, we know that our list of references is incomplete. For this we are most regretful.
We are convinced that the multigrid/multilevel methodologies and algorithms discussed in this book have an important role to play in the way multi-scale finite element modeling of wave phenomena will be done in the years ahead. While mostly confined today in the academic environment in the form of prototype research codes, there exists nevertheless evidence of the growing appreciation of their unique attributes and their potential to enhance the modeling accuracy and efficiency of electromagnetic wave simulation in support of practical engineering innovation and design. We are indebted to Dr. An-Yu Kuo of Optimal Corporation, San Jose, California, for embracing the methodologies presented in this book, and for undertaking their implementation in some of Optimal’s electromagnetic design automation tools.
We would also like to express our deepest appreciation to Dr. Donald G. Dudley, Prof. Emeritus, University of Arizona, who in his role as Editor of the IEEE Press Se- ries on Electromagnetic Wave Theory stood behind this book project, kept us focused and motivated, and provided us with expert guidance throughout its completion.
Special thanks go to those colleagues from the computational electromagnetics commu- nity for their voluntary service as reviewers of the original book proposal and early drafts of the book. Their constructive criticism and expert feedback have been most helpful in enhancing the technical content and improving the clarity of its presentation. Along with them, Christina Kuhnen, Cathy Faduska, Anne Reifsnyder and Anthony Vengraitis of the editorial staff at IEEE Press contributed in their unique way to the successful completion of this book project. For their help with improving the quality of the book we are most grateful. For any remaining shortcomings and mistakes we are solely responsible and remain most thankful in advance to those readers who will bring them to our attention.
Y.Z. and A.C.C.
