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135 Pages·2014·4.98 MB·English

HELMHOLTZ AND HIGH-FREQUENCY MAXWELL MULTILEVEL FAST
MULTIPOLE ALGORITHMS WITH SELF-TUNING LIBRARY
by
Abdulkadir C. Yucel
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Master of Science
(Electrical Engineering)
in The University of Michigan
2008

Thesis Committee:
Professor Eric Michielssen, Chair
Associate Professor Mahta Moghaddam
Assistant Professor Anthony Grbic
“Đlm kesbiyle pâye-i rifat, Arzuyu muhâl imiş ancak,
Aşk imiş her ne var âlemde, Đlm bir kîl-ü kâl imiş ancak.”
- Fuzûlî
ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor, Professor Eric Michielssen,
for his support and guidance. His sincerity, scientific enthusiasm and persistence have set
a strong scientific model in this student’s mind. I am also grateful to Professor Mahta
Moghaddam and Professor Anthony Grbic for serving on my committee.
Many special thanks go to Professor Ahmet Arif Ergin from Gebze Institute of
Technnology for his encouragements and close mentorship throughout the last seven
years of my life. In addition, I would like to thank to Turkish Fulbright Commission for
financially supporting my grad research.
Thanks to my collegeues in Professor Michielssen’s research group with whom
I’ve shared knowledge, experience, hard times, fun, and lunch/dinner tables. I owe
particular acknowledgement to Dr. Francesco Andriulli, Dr. Hakan Bagci, Xi Lin, Pelumi
Osoba, Felipe Valdes Valenzuela, and Onur Bakir. In addition, I am grateful to all
radlabers, particularly Scott Rudolph, Amit Patel, and Morteza Nick for their valuable
friendships.
Last but not least, I would like to thank to my father, Dr.T.Savas Yucel, from
whom I’ve learned how to struggle for what I set to my mind, to my mother, S.Sevtap
Yucel, who taught me how to be patient while she’s raising me and my brothers up with
her great endurance and love. Moreover, many thanks to my brothers Dr. Ali Bogachan
Yucel and Abdullah Yucel for their encouragements and support. I desire to acknowledge
some of friends who are like members of my family and always with me through their
existences and encouragements. A special mention goes to Seyit Ahmet Sis, Atila Ucar,
and Ali Burak Unlu.
ii
Ann Arbor
2008

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TABLE OF CONTENTS
ACKNOWLEDGEMENTS.............................................................................................ii
TABLE OF CONTENTS................................................................................................iv
LIST OF FIGURES.........................................................................................................vi
LIST OF TABLES...........................................................................................................ix
CHAPTER 1 INTRODUCTION.....................................................................................1
1.1 Background...............................................................................................................1
1.2 Motivation.................................................................................................................2
1.3 Organization of Chapters..........................................................................................4
CHAPTER 2 FMM AND MLFMA.................................................................................5
2.1 Introduction...............................................................................................................5
2.2 Integral Equations.....................................................................................................6
2.3 Approximating the Green’s Function.......................................................................9
2.4 Fast Multipole Method for CFIE............................................................................12
2.5 The Multilevel Fast Multipole Algorithm for CFIE...............................................16
2.6 Optimum Local Interpolation Scheme....................................................................19
CHAPTER 3 SCALAR AND VECTOR SPHERICAL FILTERS ...........................22
3.1 Introduction.............................................................................................................22
3.2 The Scalar Spherical Filter.....................................................................................23
3.2.1 Overview.......................................................................................................23
3.2.2 The Standard Scalar Spherical Filter............................................................24
3.2.3 The Fast Scalar Spherical Filter....................................................................29
3.3 The Vector Spherical Filter.....................................................................................31
3.3.1 Overview.......................................................................................................31
3.3.2 The Standard Vector Spherical Filter...........................................................31
3.3.3 The Fast Vector Spherical Filter...................................................................35
3.4 Implementation.......................................................................................................40
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CHAPTER 4 MLFMA SELF-TUNING LIBRARY ...................................................45
4.1 Introduction.............................................................................................................45
4.2 Truncation Number Estimator Algorithm...............................................................46
4.2.1 Overview.......................................................................................................46
4.2.2 The Nature of the Diagonal Addition Theorem............................................47
4.2.3 Algorithm......................................................................................................54
4.2.3.1 The Bracketing Algorithm....................................................................55
4.2.3.2 The Golden Section Search Algorithm.................................................61
4.2.3.3 The Bisection Root Finding Algorithm................................................65
4.2.3.4 The Accelerators...................................................................................68
4.2.4 Numerical Results.........................................................................................71
4.3 The Local Interpolation Parameters Estimator Algorithms....................................75
4.3.1 Overview.......................................................................................................75
4.3.2 The Number of Interpolation Points (p) Estimator Algorithm.....................76
4.3.2.1 Algorithm..............................................................................................76
4.3.2.2 Numerical Results.................................................................................78
4.3.3 The Over-Sampling Ratio (s) Estimator Algorithm......................................86
4.3.3.1 Algorithm..............................................................................................86
4.3.3.2 Numerical Results.................................................................................87
CHAPTER 5 NUMERICAL RESULTS.......................................................................96
5.1 Introduction.............................................................................................................96
5.2 One-Level FMM.....................................................................................................96
5.2.1 Results for Helmholtz MLFMA....................................................................97
5.2.2 Results for Maxwell MLFMA....................................................................102
5.3 Two-Level FMM..................................................................................................107
5.3.1 Results for Helmholtz MLFMA..................................................................108
5.3.2 Results for Maxwell MLFMA....................................................................112
SUMMARY AND FUTURE WORK..........................................................................117
REFERENCES..............................................................................................................119

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LIST OF FIGURES
Figure 2-1: The vector quantities illustrating the vector decomposition between the
source and the observer locations.....................................................................................10
Figure 2-2: The box surrounding the object is subdivided hierarchically in each direction.
Each numbered box contains basis functions...................................................................14
Figure 2-3: Multilevel interactions between an observer box and the remaining source
boxes. The interactions with near field neighbors are directly computed by conventional
MoM.................................................................................................................................17
Figure 4-1: Large buffer case is depicted for the worst case analysis in FMM (ten box
buffer). Two fictitious spheres, source sphere and observer sphere, enclose the boxes...50
Figure 4-2: Small buffer case is depicted for the worst case analysis in FMM (one box
buffer). Two fictitious spheres, source sphere and observer sphere, enclose the boxes...50
Figure 4-3: The relative error of the addition theorem for a source-observer configuration
is plotted with the crosses. Here, kd=20 and kX=40 (one box buffer case).....................51
Figure 4-4: Different source/observer distributions are presented to investigate the
relative error......................................................................................................................52
Figure 4-5: Relative errors of different distributions for one buffer case.........................53
Figure 4-6: Relative errors of different distributions for large buffer case.......................54
( )
Figure 4-7: A unimodal function whose minimum is bracketed by the triplet a,b,c ...56
Figure 4-8: Locations of the points in the interval bracketing the global minimum........61
Figure 4-9: A smooth continuous function has a root on the interval [a,b] . c is the
midpoint of the interval.....................................................................................................65
Figure 4-10: Truncation error plot for kd=20 and kX=40.................................................72
Figure 4-11: Truncation error plot for kd=20 and kX=220...............................................73
Figure 4-12: Truncation error plot for kd=40 and kX=80.................................................74
Figure 4-13: Interpolation errors are plotted for different desired error levels with
estimated p – parameters, L=40, and s=2..........................................................................80
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Figure 4-14: Interpolation errors are plotted for different desired error levels with
estimated p – parameters, L=400, and s=1.2.....................................................................81
Figure 4-15: Interpolation errors are plotted for different desired error levels with
estimated p – parameters, L=1000, and s=1.2...................................................................82
Figure 4-16: Interpolation errors are plotted for different desired error levels with the p –
parameters estimated by the proposed algorithm, L=100, and s=1.2...............................83
Figure 4-17: Interpolation errors are plotted for different desired error levels with the p –
parameters estimated by previously used formula, L=100, and s=1.2..............................84
Figure 4-18: Interpolation errors are plotted for different desired error levels with the p –
parameters estimated by the proposed algorithm, L=100, and s=2..................................85
Figure 4-19: Interpolation errors are plotted for different desired error levels with the p –
parameters estimated by previously used formula, L=100, and s=2.................................86
Figure 4-20: Interpolation errors are plotted for different desired error levels with
estimated s – parameters, L=40, and p=10........................................................................89
Figure 4-21: Interpolation errors are plotted for different desired error levels with
estimated s – parameters, L=400, and p=15......................................................................90
Figure 4-22: Interpolation errors are plotted for different desired error levels with
estimated s – parameters, L=1000, and p=15....................................................................91
Figure 4-23: Interpolation errors are plotted for different desired error levels with the s –
parameters estimated by the proposed algorithm, L=100, and p=8..................................92
Figure 4-24: Interpolation errors are plotted for different desired error levels with the s –
parameters estimated by previously used formula, L=100, and p=8................................93
Figure 4-25: Interpolation errors are plotted for different desired error levels with the s –
parameters estimated by the proposed algorithm, L=100, and p=25................................94
Figure 4-26: Interpolation errors are plotted for different desired error levels with the s –
parameters estimated by previously used formula, L=100, and p=25..............................95
Figure 5-1: One-level FMM test configuration................................................................97
Figure 5-2: Relative errors of field values at observer points at one-level FMM for the
configuration d =4l and X =1000l .............................................................................98
Figure 5-3: Relative errors of field values at observer points at one-level FMM for the
configuration d =4l and X =16l .................................................................................99
Figure 5-4: Relative errors of field values at observer points at one-level FMM for the
configuration d =40l and X =1000l .........................................................................100
Figure 5-5: Relative errors of field values at observer points at one-level FMM for the
configuration d =150l and X =1000l ........................................................................101
Figure 5-6: Relative errors of field values at observer points at one-level FMM for the
configuration d =4l and X =1000l (with formulas in literature)..............................102
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Figure 5-7: Relative errors of field values at observer dipoles at one-level FMM for the
configuration d =4l and X =1000l ...........................................................................103
Figure 5-8: Relative errors of field values at observer dipoles at one-level FMM for the
configuration d =4l and X =16l ...............................................................................104
Figure 5-9: Relative errors of field values at observer dipoles at one-level FMM for the
configuration d =40l and X =1000l .........................................................................105
Figure 5-10: Relative errors of field values at observer dipoles at one-level FMM for the
configuration d =65l and X =1000l ..........................................................................106
Figure 5-11: Relative errors of field values at observer dipoles at one-level FMM for the
configuration d =4l and X =1000l (with formulas in literature)..............................107
Figure 5-12: Two-level FMM test configuration............................................................108
Figure 5-13: Relative errors of field values at observer points at two-level FMM for the
configuration d =4l and X =1000l ...........................................................................109
Figure 5-14: Relative errors of field values at observer points at two-level FMM for the
configuration d =4l and X =38.62l ..........................................................................110
Figure 5-15: Relative errors of field values at observer points at two-level FMM for the
configuration d =40l and X =1000l .........................................................................111
Figure 5-16: Relative errors of field values at observer points at two-level FMM for the
configuration d =100l and X =1000l ........................................................................112
Figure 5-17: Relative errors of field values at observer dipoles at two-level FMM for the
configuration d =4l and X =1000l ...........................................................................113
Figure 5-18: Relative errors of field values at observer dipoles at two-level FMM for the
configuration d =4l and X =38.62l ..........................................................................114
Figure 5-19: Relative errors of field values at observer dipoles at two-level FMM for the
configuration d =40l and X =1000l .........................................................................115
Figure 5-20: Relative errors of field values at observer dipoles at two-level FMM for the
configuration d =60l and X =1000l ..........................................................................116

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LIST OF TABLES
Table 4-1: Truncation number values corresponding to desired error level and
computational time spent in the estimator routine for the configuration kd=20, kX=40..72
Table 4-2: Truncation number values corresponding to desired error level and
computational time spent in the estimator routine for the configuration kd=20, kX=220.73
Table 4-3: Truncation number values corresponding to desired error level and
computational time spent in the estimator routine for the configuration kd=40, kX=80..74

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