THE GENERALIZED METHOD OF MOMENTS FOR ELECTROMAGNETIC INTEGRAL
EQUATIONS: NEW FORMULATIONS AND APPLICATIONS
By
Daniel Lawrence Dault

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Doctor of Philosophy
2015

ABSTRACT
THE GENERALIZED METHOD OF MOMENTS FOR ELECTROMAGNETIC INTEGRAL
EQUATIONS: NEW FORMULATIONS AND APPLICATIONS
By
Daniel Lawrence Dault
The moment method is the predominant approach for the solution of electromagnetic boundary
integral equations. Traditional moment method discretizations rely on the projection of solution
currents onto basis sets that must satisfy strict continuity properties to model physical currents.
The choice of basis sets is further restricted by the tight coupling of traditional functional descriptions to the underlying geometrical approximation of the scattering or radiating body. As a result,
the choice of approximation function spaces and geometry discretizations for a given boundary
integral equation is significantly limited. A quasi-meshless partition of unity based method called
the Generalized Method of Moments (GMM) was recently introduced to overcome some of these
limitations. The GMM partition of unity scheme affords automatic continuity of solution currents,
and therefore permits the use of a much wider range of basis functions than traditional moment
methods. However, prior to the work in this thesis, GMM was limited in practical applicability because it was only formulated for a few geometry types, could not be accurately applied to arbitrary
scatterers, e.g. those with mixtures of geometrical features, and was not amenable to traditional
acceleration methodologies that would permit its application to electrically large problems.
The primary contribution of this thesis is to introduce several new GMM formulations that significantly expand the capabilities of the method to make it a practical, broadly applicable approach
for solving boundary integral equations and overcoming the limitations inherent in traditional moment method discretizations. Additionally, several of the topics covered address continuing open
problems in electromagnetic boundary integral equations with applicability beyond GMM. The
work comprises five broad contributions. The first is a new GMM formulation capable of mixing both GMM-type basis sets and traditional basis sets in the same discretization. The scheme

handles geometries with mixtures of smooth and sharp features using an expanded range of basis functions classes and geometry representations, thereby permitting the application of GMM
to arbitrary scatterers with high fidelity. Second, an acceleration method based on the Multilevel
Fast Multipole Algorithm is developed that is equally applicable to GMM and traditional higher
order moment methods. The third contribution is an interior penalty integral equation formulation that regularizes GMM surface currents and affords significant savings in computational cost.
Fourth, a technique for creating arbitrarily smooth GMM current representations based on conformal mapping is advanced. Finally, a GMM formulation for a new surface description for boundary
integral equations, the subdivision surface, is proposed. The resulting subdivision-based method
holds promise for optimization and uncertainty quantification of scattering and radiating structures
through rapid geometry morphing and subwavelength deformation; furthermore it is readily extensible to traditional moment methods. Taken together, these contributions address several open
problems and yield a GMM boundary integral equation solver that is highly accurate, flexible, fast,
and applicable to a wide range of electromagnetics engineering problems.

To my wife and children.

iv

ACKNOWLEDGEMENTS

First, I would like to deeply thank Professor Shanker Balasubramaniam for all of the ideas, encouragement, and support over the years of my graduate education. Without your guidance, support,
and many insights, both in and beyond the academic sphere, this work would not have been possible.
Next, I would like to thank my committee members, Professors Rothwell and Piermarocchi
and Drs. Chahal and Albrecht for their many constructive thoughts and guidance.
No good work is accomplished in a vacuum, and I have been honored to work with an outstanding group of fellow students at Michigan State. I owe a few special mentions. Thank you to Jie Li
for the collaboration and many conversations that have spawned new ideas and research directions,
and without whose insights this work would have been much poorer. Thank you also to Dr. Naveen
V. Nair for his original work on the Generalized Method of Moments and the many conversations
and much support in the years since I took up the GMM mantle. Thanks to Dr. Andrew Baczewski
for serving as a mentor in my early years as a graduate student. Many thanks also to Andrew Pray
and Zane Crawford, Steve Hughey, Scott O’Conner, and Conner Glosser.
A special thank you to Dr. Yiying Tong and his students Beibei Liu and Rundong Zhao for
introducing me to subdivision surfaces and helping with some of the initial conceptual work and
implementation.
Finally a profound thank you to my family, especially my wife Nicole and my children Maya
and James. I could not have done this without your unfailing support and care over the years.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
CHAPTER 1 INTRODUCTION AND MATHEMATICAL PRELIMINARIES .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Introduction to Mathematical Preliminaries . . . . . . . . . . . . . . .
1.3 Time Harmonic Maxwell’s Equations in Free Space . . . . . . . . . . .
1.3.1 The Surface Equivalence Principle . . . . . . . . . . . . . . . .
1.4 Derivation of Electromagnetic Boundary Integral Equations . . . . . . .
1.5 The Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Manifold Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Definition of a smooth manifold . . . . . . . . . . . . . . . . .
1.6.2 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.3 Tangent and cotangent bundles . . . . . . . . . . . . . . . . . .
1.6.4 Embedding in R3 and Riemannian manifolds . . . . . . . . . .

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1
2
3
5
6
8
10
11
12
13
16

CHAPTER 2 THE GENERALIZED METHOD OF MOMENTS . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Neighborhoods from Point Clouds . . . . . . . . . . . . .
2.3.2 Partition of Unity . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Local Geometry Parameterization . . . . . . . . . . . . .
2.3.3.1 Smooth Polynomial Geometry Parameterization
2.3.3.2 Handling of Geometrical Singularities . . . . . .
2.3.3.3 Definition of the Tangent Space T Ωi . . . . . .
2.3.3.4 Vector Calculus on Parametric Surfaces . . . . .
2.3.4 Local Basis Functions . . . . . . . . . . . . . . . . . . . .
2.3.4.1 Entire Patch Basis Functions . . . . . . . . . . .
2.3.4.2 Sub-patch Basis Functions . . . . . . . . . . . .
2.3.4.3 Default EP/SP Basis . . . . . . . . . . . . . . .
2.4 Implementation Details . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Matrix Elements . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Automated Patch Construction . . . . . . . . . . . . . . .
2.4.2.1 Algorithm for Patch Merging . . . . . . . . . .
2.4.2.2 Automated Geometry and Basis Assignment . .
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Complexity and Matrix Fill Times . . . . . . . . . . . . .

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26
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51

vi

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2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CHAPTER 3

3.1
3.2

3.3
3.4

3.5

3.6
3.7
3.8

A MIXED POTENTIAL MLFMA FOR HIGHER ORDER MOMENT
METHODS WITH APPLICATIONS TO THE GENERALIZED METHOD
OF MOMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Electromagnetic Problem Statement . . . . . . . . . . . . . . . . . . . .
3.2.2 The Multilevel Fast Multipole Algorithm . . . . . . . . . . . . . . . . .
3.2.3 Application to higher order . . . . . . . . . . . . . . . . . . . . . . . . .
Previous Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Point-Based Mixed Potential MLFMA . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Mixed potential farfield representation . . . . . . . . . . . . . . . . . . .
3.4.2 Point-based algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Mixed potential point-based MLFMA nearfield . . . . . . . . . . . . . .
Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Memory scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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56
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70
71
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78
88

CHAPTER 4
4.1
4.2

4.3
4.4

4.5
4.6
4.7

AN INTERIOR PENALTY METHOD FOR THE GENERALIZED METHOD
OF MOMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 The Generalized Method of Moments . . . . . . . . . . . . . . . . . . . . 93
4.2.2 Solution errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Penalty Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Implementation details for two dimensions . . . . . . . . . . . . . . . . . 96
Spectral Analysis: Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1 Low order analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.2 Higher order analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2D Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Results in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

CHAPTER 5
5.1
5.2
5.3
5.4
5.5
5.6

SCHWARZ-CHRISTOFFEL BASED PARTITIONS OF UNITY FOR THE
GENERALIZED METHOD OF MOMENTS . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Statement: Nonsmooth Partitions of Unity . . . . . . . . . . . . . . . .
Schwarz-Christoffel Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schwarz-Christoffel GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evaluation of Field Integrals via Generalized Quadrature . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

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113
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125

5.7

Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

CHAPTER 6
6.1
6.2
6.3
6.4
6.5
6.6
6.7

SUBDIVISION SURFACES FOR THE GENERALIZED METHOD OF
MOMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Loop Subdivision Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Evaluation of the Limit Surface and Derivatives . . . . . . . . . . . . .
GMM Patches and Local Parameterizations . . . . . . . . . . . . . . . . . . .
6.3.1 Merging of Regular Patches . . . . . . . . . . . . . . . . . . . . . . .
Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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131
131
132
134
139
144
146
149
153
158

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

viii

LIST OF TABLES

Table 2.1

Matrix fill times for CFIE-RWG code. . . . . . . . . . . . . . . . . . . . . . . . 53

Table 2.2

Matrix fill times for higher order CFIE-GWP & CFIE-GMM codes. . . . . . . . 54

Table 3.1

Discretization details for series of plates used in scaling demonstration. . . . . . 79

Table 3.2

Matrix vector product timings for dyadic kernel for the VFY-218 aircraft, CFIE. . 82

Table 3.3

Matrix vector product timings for mixed potential kernel for the VFY-218
aircraft, CFIE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Table 3.4

Storage for 10λ diameter sphere, p = 2, h = .8λ . . . . . . . . . . . . . . . . . . 85

Table 3.5

Timings for 10λ diameter sphere, p = 2, h = .8λ . . . . . . . . . . . . . . . . . . 86

Table 3.6

Storage for 10λ diameter sphere, p = 7, h = 2λ . . . . . . . . . . . . . . . . . . 86

Table 3.7

Timings for 10λ diameter sphere, p = 7, h = 2λ . . . . . . . . . . . . . . . . . . 86

Table 3.8

Storage and timings for 15λ ship. . . . . . . . . . . . . . . . . . . . . . . . . . 88

Table 3.9

Detailed Matrix Vector Product Timings for 15λ ship. . . . . . . . . . . . . . . 88

Table 4.1

Iterations to convergence (τ = 10−3 ) vs. βJ in GMRES solver. . . . . . . . . . . 111

ix

LIST OF FIGURES

Figure 1.1

Setup for scattering from a perfectly electrically conducting obstacle in three
dimensional free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Figure 1.2

Manifold Ω, coordinate charts (Ω1 , ϕ1 ) and (Ω2 , ϕ2 ), and transition maps
τ12 , τ21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 1.3

Tangent bundle for a subset Ωi of a two-dimensional manifold Ω. The bundle
T Ωi consists of a collection of copies of R2 associated with each pair (u2i , u2i )
in the coordinate chart (Ωi , ϕi ). Coordinates for the bundle are mapped into
˜ where ϕ˜ maps from the product
R4 via the coordinate chart (Ωi × R2 , ϕ),
2
space Ωi × R into local coordinates in R4 . . . . . . . . . . . . . . . . . . . . . 15

Figure 2.1

Blending of approximation spaces on overlapping domains via the partition
of unity for a 1D signal. (a) A trial function f (x) = .2(x − 2)3 + .3 on the
interval Ω : 0 ≤ x ≤ 4. (b) Interpolation of f (x) using linear interpolatory hat
functions Ti (x) = T (x − .5i) on subinterval Ω1 : 0 ≤ x ≤ 3 such that fˆ1 (x) =
χ1 (x) ∑i ai Ti (x), χ1 (x) = 1, x ∈ Ω1 ; 0, x ∈ Ω1 . (c) Interpolation of f (x) using
a third order Legendre polynomial set Pn (x), n = 0, 1, 2, 3 on subinterval Ω2 :
2 ≤ x ≤ 4 such that fˆ2 (x) = χ2 (x) ∑3n=1 an Pn (x − 3), χ2 (x) = 1, x ∈ Ω2 , 0;
x ∈ Ω2 . (d) Partitions of unity ψ1 (x) and ψ2 (x). (e) interpolations of f (x)
multiplied by partitions of unity as fˆ1 (x)ψ1 (x) in Ω1 and fˆ2 (x)ψ2 (x) in Ω2 .
(f) Reconstructed function fˆ(x) = fˆ1 (x)ψ1 (x) + fˆ2 (x)ψ2 (x) (left axis) and
absolute reconstruction error (right axis). Through the action of the partition
of unity, the representations fˆ1 (x) and fˆ2 (x) are smoothly blended in the
overlap region, 2 ≤ x ≤ 3, and the approximation error transitions smoothly
between the linear and Legendre interpolations. . . . . . . . . . . . . . . . . . 27

Figure 2.2

Smooth patch geometry parameterization procedure for patch Ωi : (note: for
clarity, subscripts i are dropped on vector quantities) (a) Start with an oriented point cloud. (b) Define a neighborhood Ni of nearest neighbors about
node i. (c) Define a projection plane Γi with normal equal to average of all
node normals and specified by a point rc that uniquely defines an origin of
coordinates. (d) Construct a local orthogonal u1 , u2 , u3 coordinate system.
(e) Take projection of points in Ni into Γi to define the support of the patch
in the u1 , u2 plane. This projection is denoted PΓi . (f) Obtain a smooth parameterization to define the local patch surface. . . . . . . . . . . . . . . . . . 29

Figure 2.3

Tangent and normal vectors associated with a parametric patch surface. . . . . . 33

Figure 2.4

Entire patch basis function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

x

Figure 2.5

Illustration of sub-patch basis function arrangement with projected PU. . . . . . 36

Figure 2.6

Design of local RWG basis set with normal continuity across patch boundaries: (a) RWG basis functions associated with internal edges, (b) Additional
half-RWG functions associated with exterior patch edges provide a normal
component at patch boundary, and (c) Simplex-based partition of unity forces
the half-RWG function to zero at patch boundary, removing the line charge. . . 37

Figure 2.7

Graphical depiction of partitions of unity for overlapped RWG and Legendre
patches. (a) Partition of unity ψRW G and half-RWG basis functions for an
RWG patch. (b) Partition of unity ψLeg for a Legendre patch. . . . . . . . . . . 39

Figure 2.8

(a) Graphical representation of PΓ (Nl ) and (b) construction of triangular
l
integration subdomains. The domain of integration is Dl = ∪i Ti . . . . . . . . . 41

Figure 2.9

Reconstruction error for a function f(x, y) = (x − .5)2 xˆ using three different basis sets: all-RWG, all-Legendre, mixed RWG/Legendre. (a) All-RWG
discretization, (b) All-RWG reconstruction error, (c) All-Legendre patch discretization (2nd order Legendre polynomials) (d) All-Legendre reconstruction error, (e) Mixed RWG/Legendre discretization (2nd order Legendre Polynomials), (f) Mixed RWG/Legendre reconstruction error. . . . . . . . . . . . . 45

Figure 2.10 Surface currents induced by an incident plane wave on a sphere discretized
with (a) overlapping GMM patches supporting a 4th order Legendre basis
and (b) an RWG basis for reference. Subfigure (c) shows the patches used
to obtain the GMM result. Magnitude of imaginary part of current ||Jim || is
shown; ||Jre || yields similar plots. . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 2.11 GMM discretization of a 4.5λ NASA almond. . . . . . . . . . . . . . . . . . . 47
Figure 2.12 NASA almond bistatic RCS, taken along the φ = 0◦ cut. . . . . . . . . . . . . . 48
Figure 2.13 GMM discretization of a 1.2λ × .4λ × .4λ conesphere. . . . . . . . . . . . . . 49
Figure 2.14 Bistatic RCS for the conesphere at 300MHz. Incident wave is traveling in
the θ = 0◦ , φ = 0◦ direction with −x-polarization. The RCS is taken along
φ = 0◦ cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Figure 2.15 GMM discretization of a 5λ × 1.5λ × .8λ arrow (a) top view and (b) bottom
view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 2.16 Arrow bistatic RCS at 64MHz. Wave is incident in the θi = 135◦ , φi = 20◦
direction and is x-polarized relative to incidence direction. The RCS is taken
along the φ = 0◦ cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 2.17 GMM discretization of a 4λ × .35λ Parabolic Reflector (a) top view and (b)
side view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xi

Figure 2.18 Parabolic reflector bistatic RCS. Wave is incident in the θi = 180◦ , φi = 0◦
direction with x-polarization. The RCS is taken along the φ = 0◦ cut. . . . . . . 52
Figure 3.1

Division of a large curvilinear basis function support Ωn into nearfield and
farfield for a source leaf box bl . Problematic contour integrals appear along
the contour Γ between nearfield and farfield regions. . . . . . . . . . . . . . . . 62

Figure 3.2

Illustration of traditional and point-based MLFMA trees. (a) Traditional:
MLFMA leaf box size (and therefore tree height) is limited by the size of the
largest patch. (b) Point-based/mixed potential: leaf box size and tree height
are not limited by patch size, and may be chosen for optimal MLFMA scaling
subject to the low frequency limit of MLFMA. . . . . . . . . . . . . . . . . . . 67

Figure 3.3

Point-based MLFMA groupings and nearfield/farfield interaction partitions
for interaction between two patches. Only direct interactions between the
bold red boxes contribute to Z N . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.4

Comparison of interactions dyadic and mixed potential MLFMA. (a) Limit
on non-point-based dyadic MLFMA leaf box size means that interactions
between large patches must be computed directly. (b) For point-based/mixed
potential MLFMA, interactions between large patches may be split between
nearfield and farfield so that favorable scaling is maintained. . . . . . . . . . . . 69

Figure 3.5

Local patch trees used to compute preconditioners. Each local tree is a subset of the global MLFMA octree. Interactions between two patches are computed using intersections of the global interaction lists. . . . . . . . . . . . . . 74

Figure 3.6

Graphical depiction of the computation of a single column of a precondiN consists of the precomputed nearfield interactions and
tioner block M˜ lk . M˜ lk
F (ei ) is an MLFMA product with the ith column of the identity matrix ei .
N with the F (e ) yields the ith column
The summation of the ith column of M˜ lk
i
of M˜ lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 3.7

Stencil of overlapped block preconditioner used in preconditioning scaling tests. 76

Figure 3.8

Convergence in relative (Frobenius) error of preconditioning matrix filled
using O(Nq log Nq ) MLFMA-accelerated algorithm vs. O(Nq2 ) direct fill algorithm. Geometry is a 2λ sphere with p = 4 and h = 1λ . . . . . . . . . . . . . 76

Figure 3.9

Comparison of MLFMA-accelerated preconditioner fill times with respect to
χ for a 2λ sphere with p = 4 and h = 1λ . . . . . . . . . . . . . . . . . . . . . 77

Figure 3.10 Convergence in relative (Frobenius) error of preconditioning matrix filled
using O(Nq log Nq ) MLFMA-accelerated algorithm vs. O(Nq2 ) direct fill algorithm. Geometry is a 2λ sphere with p = 9 and h = 2λ . . . . . . . . . . . . . 77

xii

Figure 3.11 Comparison of MLFMA-accelerated preconditioner fill times with respect to
χ for a 2λ sphere with p = 9 and h = 2λ . . . . . . . . . . . . . . . . . . . . . 78
Figure 3.12 Scaling of MLFMA matrix vector product timings vs. Nqtot for a series of
square plates of size 2λ -12λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 3.13 Scaling in nearfield and farfield memory for a series of square plates of size
2λ -12λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 3.14 Scaling of nearfield matrix fill vs. Nqtot for a series of square plates of size
2λ -12λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 3.15 GMM discretization of VFY-218 aircraft. . . . . . . . . . . . . . . . . . . . . . 82
Figure 3.16 Surface currents for the 30λ VFY-218 aircraft. . . . . . . . . . . . . . . . . . . 83
Figure 3.17 Bistatic RCS for VFY-218 aircraft, θ = 90◦ cut. . . . . . . . . . . . . . . . . . 83
Figure 3.18 Bistatic RCS for 10λ diameter sphere, φ = 0 cut. Two GMM discretizations,
disc. 1 (p = 2, h = .8λ ) and disc 2. (p = 7, h = 2λ ) for both the dyadic (“D”)
and mixed potential (“MP”) are compared with the analytical Mie series result. . 84
Figure 3.19 GMM mixed discretization for 15λ destroyer: 1st order smooth polynomial
patches (patch diameter 1λ ) and 4th order Legendre basis on smooth regions, triangular tessellation and RWG basis on tips/corners. (Model credit:
Nicholas Miller) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 3.20 Bistatic RCS for 15λ destroyer, θ = 90◦ cut, plane wave incident on bow. . . . 88
Figure 4.1

Patch overlaps γlk in (a) two and (b) three dimensions. . . . . . . . . . . . . . . 93

Figure 4.2

Illustration of low and high frequency current errors induced by insufficient
quadrature for the TEz EFIE solution on cylinder of radius a = .5λ . . . . . . . . 98

Figure 4.3

Generalized eigenvalues for an a = .5λ cylinder. . . . . . . . . . . . . . . . . . 101

Figure 4.4

High frequency current spectrum of compensated EFIE in visible region for
an a = .5λ cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 4.5

Current spectrum of compensated EFIE in visible region for an a = .5λ cylinder. 102

Figure 4.6

Comparison of analytical and reconstructed current obtained using an 8-point
integration rule. (a) ||J||, β = 0, (b) RCS, β = 0, (c) ||J||, β = 10, (d) RCS,
β = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.7

Convergence vs β in EFIE solution currents for a = .5λ cylinder under TEz
illumination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xiii

Figure 4.8

Pointwise absolute error over the cylinder scattering surface. . . . . . . . . . . 105

Figure 4.9

Convergence of absolution current error in patch overlaps vs. β . . . . . . . . . 105

Figure 4.10 GMM patches for a teardrop shaped scatterer. . . . . . . . . . . . . . . . . . . 106
Figure 4.11 Comparison of reference (i = 30) and reconstructed currents/fields obtained
using i = 10 rule for the 2λ teardrop scatterer. . . . . . . . . . . . . . . . . . . 106
Figure 4.12 Generalized eigenvalues for the uncompensated and compensated GMM TEz
operator discretized on the teardrop shaped scatterer. Higher order eigenvalues of the compensated operator are shifted, while lower order eigenvalues
(inset) are unaffected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 4.13 GMM patches for a 6λ × 2λ elliptical scatterer. . . . . . . . . . . . . . . . . . 108
Figure 4.14 Comparison of reference (i = 30) and reconstructed currents and RCS obtained using i = 10 rule for the 6λ × 2λ elliptical scatterer. . . . . . . . . . . . 108
Figure 4.15 Convergence vs. β for a 1 λ sphere, p = 4, i = 5 case. . . . . . . . . . . . . . . 109
Figure 4.16 Convergence vs. β for p = 7, i = 5 case. . . . . . . . . . . . . . . . . . . . . . 110
Figure 4.17 Comparison of analytical and reconstructed current obtained using 10-point
integration rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 5.1

Two GMM patches and local (u1 , u2 ) coordinate systems.

Figure 5.2

Image of a triangle Tl under a non-affine transition function τkl . . . . . . . . . . 117

Figure 5.3

Simplex-based PU and derivative with non-affine transition functions. . . . . . 117

Figure 5.4

Schwarz-Christoffel mapping from canonical disc D to polygon P (plot generated using Matlab SC Toolbox [1]). . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 5.5

Schwartz-Christoffel mapping in u1 , u2 plane permits definition of arbitrarily
smooth partitions of unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 5.6

Polynomial sub-partition of unity function. . . . . . . . . . . . . . . . . . . . . 120

Figure 5.7

Smooth sub-partition of unity mapped from D(z) to projection polygon P(w)
via Schwarz-Christoffel map. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 5.8

Smooth partition of unity ψ(u1 , u2 ), (u1 , u2 ) ∈ P constructed from smooth
sub-partitions of unity mapped using Schwarz-Christoffel mapping. (a) Partition of unity ψ. (b) Derivative ∂ ψ/∂ u1 . . . . . . . . . . . . . . . . . . . . . . 121

xiv

. . . . . . . . . . . 115

Figure 5.9

Illustration of reduction in number of quadrature abscissa between adaptive
and generalized quadrature schemes. (a) Adaptive quadrature nodes for a
patch. Convergence of integrations to relative error of 10−5 requires 2148
points. (b) Generalized PU-Chebyshev quadrature nodes for a patch. Only
117 points are required for same accuracy. . . . . . . . . . . . . . . . . . . . . 124

Figure 5.10 Convergence of integrations of simplex and Schwarz-Christoffel mapped partitions of unity vs. number of quadrature points Nq . (a) Simplex partition of
unity and divergence. (b) Schwarz-Christoffel partition of unity ψSC . (c)
∂ ψSC /∂ u1 . (d) ∂ ψSC /∂ u2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 5.11 Bistatic RCS (φ = 0◦ cut) for a .5λ sphere discretized with smooth SC-GMM
basis, MFIE solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Figure 5.12 Bistatic RCS (φ = 0◦ ) for a 1λ sphere, smooth SC-GMM basis, EFIE solution. 127
Figure 5.13 Surface current density (||J||) for a 1λ sphere illuminated by an x-polarized
plane wave travelling in the +z direction. (a) Mie series solution. (b) SCGMM
solution using the EFIE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure 5.14 Mixed Schwarz-Christoffel/simplex partition of unity scheme for the 2λ ellipsoid.129
Figure 5.15 Bistatic RCS (φ = 0◦ ) for the 2λ ellipsoid, MFIE solution. . . . . . . . . . . . 129
Figure 6.1

Subdivision of a spherical mesh. (a) Primal mesh, (b) k = 1, (c) k = 2, (d)
limit surface (k = ∞). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 6.2

Neighborhood of a triangle Tl . Vertex v1 has a valence of N, vertices v2 ,
vN+1 have valence six. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 6.3

Recursive subdivision procedure about an irregular vertex v00 . Each subdivision yields three regular triangles and one irregular triangle. . . . . . . . . . . . 136

Figure 6.4

Divergent behavior of subdivision surface partial derivatives about an irregular vertex of valence N = 15. The (u1 , u2 ) coordinate system for this case
is defined via direct projection into a common plane that shares the average
normal of all triangles incident on the irregular vertex. . . . . . . . . . . . . . . 139

Figure 6.5

Eigen-neighborhood for a vertex of valence nine. The characteristic map is
constructed upon subdivision of this neighborhood. . . . . . . . . . . . . . . . 141

Figure 6.6

Characteristic map for a vertex of valence N = 5. (a) Triangulation of an
N-gon, (b) a subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xv

Figure 6.7

Characteristic map for a vertex of valence N = 6. (a) Triangulation of an
N-gon, (b) a subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 6.8

Characteristic map for a vertex of valence N = 15. (a) Triangulation of an
N-gon, (b) a subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Figure 6.9

Coordinate charts for overlapping irregular (valence 5) and regular GMM
patches induced by the characteristic maps F(ξ 1 , ξ 2 ). . . . . . . . . . . . . . . 144

Figure 6.10 Primal mesh (a) and local coordinate domain (b) for a merged regular patch. . . 145
Figure 6.11 Basis functions and divergences for a valence N = 5 patch. . . . . . . . . . . . 148
Figure 6.12 Basis functions and divergences for a valence N = 15 patch. . . . . . . . . . . . 149
Figure 6.13 Subdivision of triangle around irregular vertex v00 for partition of unity definition.150
Figure 6.14 Subdivision simplex partition of unity. (a) PU on regular patch with regular
neighbors. (b) PU on irregular valence N = 5 patch. (c) PU on regular patch
neighboring irregular patch (shown schematically for clarity). . . . . . . . . . . 152
Figure 6.15 Regular patch support near an irregular vertex. . . . . . . . . . . . . . . . . . . 152
Figure 6.16 Regular patch supports adjacent to an irregular vertex of valence N = 5. . . . . 153
Figure 6.17 Sampled partition of unity ψ(u1 , u2 ) on a merged patch adjacent to an irregular vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure 6.18 Basis function and divergence for a merged regular patch. . . . . . . . . . . . . 154
Figure 6.19 (a) RCS for 1λ subdivision sphere (φ = 0◦ ). (b) RCS for 1λ subdivision
sphere with merged patches (φ = 0◦ ). . . . . . . . . . . . . . . . . . . . . . . . 155
Figure 6.20 Real part of MFIE solution surface currents for 1λ sphere using h = .5λ
merged patches. Incident plane wave is traveling in +ˆz direction. . . . . . . . . 155
Figure 6.21 Chmutov surface (a) Coarse primal mesh. (b) Mesh after one subdivision.
(b) Limit surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 6.22 Bistatic RCS for 2λ Chmutov geometry (φ = 0◦ ). . . . . . . . . . . . . . . . . 156
Figure 6.23 Torus mesh (a) front view, (b) side view, and twisted torus limit surface (c)
front view, (d) side view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 6.24 Bistatic RCS (φ = 0◦ cut) for (a) torus and (b) twisted torus geometries. . . . . 157

xvi

LIST OF ALGORITHMS

Algorithm 2.1

Automated patch merging. . . . . . . . . . . . . . . . . . . . . . . . . . 42

Algorithm 2.2

Automated geometry and basis assignment. . . . . . . . . . . . . . . . . 43

xvii

CHAPTER 1
INTRODUCTION AND MATHEMATICAL PRELIMINARIES

1.1

Introduction

The development of efficient, accurate discretizations for electromagnetic boundary integral equations has been a primary concern for the field of computational electromagnetics since its inception
in the middle of the twentieth century. Research in this area has proceeded along three interrelated
tracks: the development of integral equation formulations that admit accurate discretization and
solution, accurate representation of scattering geometries, and accurate representation of equivalent current densities on scattering boundaries. A fourth area, which has come to the fore in recent
decades, is the development of acceleration methodologies to reduce the computational cost of
solving electromagnetic integral equations for electrically large scattering bodies. The topic of this
thesis is twofold. First, it gives several advances in the Generalized method of moments (GMM),
a recently developed discretization method for integral equations that uses a partition of unity to
effect blending of surface currents. Second, through the lens of GMM, it addresses some open
problems in the four areas of electromagnetic boundary integral equation research. Some of these
solutions are specific to GMM, and some, such as acceleration methodologies for higher order moment methods and discretization of currents on novel surface descriptions, may also be applied to
address ongoing challenges in the broader context traditional boundary integral equation methods.
The thesis is structured as follows. The remainder of this chapter gives the necessary physical
and mathematical preliminaries to define the model physical problem under consideration, and
provides the mathematical machinery necessary to define the Generalized Method of Moments.
Chapter 2 gives an introduction and context for GMM, and provides a detailed prescription for the
construction of GMM discretizations for arbitrary perfectly electrically conducting scatterers that
may include curved regions, sharp regions such as corners or tips, regions that are representable

1

as bodies of revolution, etc. Chapter 3 develops a Multilevel Fast Multipole Algorithm (MLFMA)
for GMM that retains optimal scaling for any mixture of low and high order basis functions. The
method is directly applicable to the acceleration of other higher order moment methods, which has
been a continuing open problem. Chapter 4 proposes an interior penalty method for GMM-type
discretizations that provides accurate solutions with significant reductions in the cost of evaluating
integrations via numerical quadrature. Chapter 5 gives a formulation for smooth parameterizations
of polygonal subdomain supports using the Schwarz-Christoffel map; this method has promise not
only for defining smooth GMM discretizations, but also for defining well-behaved local Helmholtz
decompositions that are stable to low frequency. Finally, Chapter 6 develops a GMM discretization
on subdivision surfaces, a powerful and flexible surface description developed by the computer
graphics community that has not been applied to the discretization of electromagnetic integral
equations. Several challenges of applying boundary integral equations to subdivision surfaces are
solved in the context of GMM. The resulting algorithm is both accurate and flexible, and holds
promise for several applications including rapid design optimization via surface morphing. Due to
the range of topics covered, this thesis forgoes an extended monolithic introduction and literature
review; instead each chapter contains an introductory section that provides context and a literature
review specific to the topic at hand.

1.2

Introduction to Mathematical Preliminaries

The remainder of this chapter develops the mathematical and physical concepts required to formulate GMM under specialization to the case of a perfectly electrically conducting (PEC) structures
residing in free space, which is the topic of this thesis. It is important to note that GMM itself
is not only applicable to the PEC case, and can be applied to a wide array of electromagnetic and
acoustic integral equations. Section 1.3 reviews the time harmonic, free space Maxwell’s equations
and related concepts. Section 1.4 uses these concepts to derive the Electric Field Integral Equation (EFIE) and Magnetic Field Integral Equation (MFIE), the electromagnetic boundary integral
equations studied in this work. A formal statement of the method of moments, the most common
2

solution technique for electromagnetic integral equations, is given in Section 1.5. Finally, the theory of smooth manifolds is introduced in Section 1.6. Manifold theory serves as a framework for
understanding various aspects of GMM discretizations throughout this thesis.

1.3

Time Harmonic Maxwell’s Equations in Free Space

Consider an unbounded region in R3 in which the constitutive parameters are those of isotropic,
homogeneous free space, µ0 , ε0 . In the case where all fields are assumed to have time harmonic dependence with angular frequency ω (i.e. all quantities are proportional to e jωt ), the time harmonic
Maxwell’s equations may be written
∇ · ε0 E(r) = ρ(r),

(1.1a)

∇ · H(r) = 0,

(1.1b)

∇ × E(r) = − jω µ0 H(r),

(1.1c)

∇ × H(r) = J(r) + jωε0 E(r).

(1.1d)

Here, E(r), H(r) are the electric field and magnetic field, respectively, ρ(r) is the scalar electric
charge density, and J(r) is the vector electric current density. Associated with each of the equations
(1.1a)-(1.1d) is a corresponding boundary condition:

nˆ · (E1 (r) − E2 (r)) =

ρs (r)
,
ε0

(1.2a)

nˆ · (H1 (r) − H2 (r)) = 0,

(1.2b)

nˆ × (E1 (r) − E2 (r)) = 0,

(1.2c)

nˆ × (H1 (r) − H2 (r)) = Js (r),

(1.2d)

where ρs (r) is the surface electric charge density and Js (r) is the surface electric current density.
These equations are supplemented by the continuity equation (1.3), which specifies the relation

3

between the divergence of electric current and the time rate of change (multiplication by jω in
frequency domain) of the electric charge.
∇ · J(r) + jωρ(r) = 0.

(1.3)

By taking the curl of (1.1c) and substituting equation (1.1d), the vector Helmholtz equation for the
electric field may be obtained as
∇ × ∇ × E − k2 E = − jω µ0 J,

(1.4)

and reversing the substitution, the vector Helmholtz equation for the Magnetic field may be obtained as
∇ × ∇ × H − k2 H = ∇ × J.
Here, k is the wave number, given by k = ω/c, with c =

√

(1.5)

µ0 ε0 the speed of light in free space.

Each of these equations may be written in operator notation as DE = − jω µ0 J for (1.4) and DH =
∇ × J. If J is known, the solution to these partial differential equations corresponds formally to
inversion of the operator D as
E(r) = D −1 (− jω µ0 J)

(1.6)

H(r) = D −1 (∇ × J).
Even if J is not known, this construction is still useful. The operator D −1 is an integral operator,
which will be derived separately for E(r) and H(r) below.
The electric and magnetic fields may be expressed in terms of a set of two potentials, the
magnetic vector potential A(r) and the scalar electric potential Φ(r). These potentials are nonunique, and are only completely fixed by the specification of a gauge. In this work, the usual
Lorenz gauge is chosen, which is characterized by the relation
∇ · A(r) =

− jω
Φ(r).
c2

(1.7)

Under the Lorenz gauge, the magnetic field may be expressed in terms of a magnetic vector potential A(r) as
H=

1
∇ × A(r).
µ0
4

(1.8)

The electric field is also expressible in terms of the vector potential plus the gradient of a scalar
potential Φ(r),
E(r) = − jωA(r) − ∇Φ(r).

(1.9)

The vector and scalar potentials may be obtained in terms of vector and scalar source densities as
A(r) = µ0
Φ(r) =

G(r, r )J(r )dr ,

(1.10a)

1
G(r, r )ρ(r )dr ,
ε0 Ω

(1.10b)

Ω

where J(r) is the electric current density, ρ(r) is the electric charge density and Ω is the support of
the source domain. The function G(r, r ) is the free-space Helmholtz Green’s function, given by
G(r, r ) =

e− jk||r−r ||
.
4π||r − r ||

(1.11)

Using the continuity equation (1.3), (1.10b) may be rewritten so that the ρ(r) dependence is replaced with a dependence on the divergence of J(r)
Φ(r) =

1.3.1

j
G(r, r )∇ · J(r )dr .
ωε0 Ω

(1.12)

The Surface Equivalence Principle

The surface equivalence principle is necessary to derive the electromagnetic integral equations
treated in this thesis. It is based on tangential boundary conditions at an interface. Given an
interface Ω between two domains D1 and D2 that is equipped with a unit normal pointing from
D2 into D1 , assume that the regions D1 and D2 respectively support field distributions E1 , H1 and
E2 , H2 . Then the surface equivalence principle states that an equivalent problem may be substituted
in either region in which additional currents are inserted to provide the same field discontinuities
at the interface. These equivalent currents are obtained via the tangential boundary conditions at
the interface:

nˆ × (E1 (r) − E2 (r)) = −Ms (r), r ∈ Ω,
5

(1.13a)

nˆ × (H1 (r) − H2 (r)) = Js (r), r ∈ Ω,

(1.13b)

where the boundary condition on tangential E given previously has been augmented with a (fictitious) magnetic surface current Ms (r). If one is solving a problem in D1 in the presence of field
distributions in D2 , an equivalent problem may be constructed using a different set of fields E2 , H2
in D2 as long as equivalent currents Ms , Js are inserted at the interface to maintain the same field
ˆ Js = Js + nˆ × (H2 − H2 ). Uniqueness
distributions E1 , H1 in D1 , i.e. Ms = Ms + (E2 − E2 ) × n,
theorems in electromagnetic theory [2] dictate that only nˆ × E or nˆ × H need be determined at
each point on the interface to uniquely specify fields everywhere, and not both simultaneously. In
the case where the interior fields are known to be zero (i.e. the problem of a PEC scatterer), the
problem reduces to finding an equivalent current Js = nˆ × H1 such that E2 = 0, H2 = 0. Applied
to scattering from a PEC obstacle, the equivalence principle permits the scattering object to be
replaced by a set of equivalent currents residing in free space. The free-space Green’s function
may therefore be used instead of a Green’s function that explicitly accounts for the presence of
the scatter; this greatly simplifies the problem as specialized Green’s functions that incorporate the
presence of arbitrary objects are typically difficult to obtain.

1.4

Derivation of Electromagnetic Boundary Integral Equations

With the above concepts defined, the two electromagnetic integral equations of interest in this study
may now be derived. The first, the Electric Field Integral Equation (EFIE) is derived in terms of
tangential boundary conditions on the electric field, and the second, the Magnetic Field Integral
Equation (MFIE) is defined in terms of boundary conditions on the magnetic field.
Consider a region in unbounded free space containing a perfectly conducting object that is
defined by the open volume D2 ⊂ R3 and bounded by the surface Ω, which is equipped with an
ˆ The exterior domain is defined as D1 = R3 \ (D2 ∪ Ω). For this work,
outward pointing normal n.
the constitutive parameters of the medium in the exterior domain D1 are assumed to be those of
free space. As the fields inside the PEC object are assumed to be zero, the superscripts on E1 , H1

6

Figure 1.1: Setup for scattering from a perfectly electrically conducting obstacle in three dimensional free space.

will be dropped for the remainder of this section. Assume that an electromagnetic plane wave with
associated fields Ei (r), Hi (r) is incident upon the scatterer. Fig. 1.1 depicts the setup for this
problem.
The goal of solving the EFIE and MFIE is to obtain the equivalent current J(r) = nˆ × H(r),
r ∈ Ω. Knowledge of the surface current allows one to use (1.9) with equations (1.10a),(1.12) to
obtain the scattered electric field Es (r), r ∈ D1 due to Ei (r) impinging on the object. Application
of the tangential boundary condition for the electric field at a perfectly conducting interface gives
nˆ × nˆ × (Ei (r) + Es (r)) = 0, r ∈ Ω.

(1.14)

It is customary to include an additional cross product of the unit normal with the usual tangential
boundary condition with to directly obtain tangential E, and to simplify subsequent formulas.
Using equations (1.9), (1.10a) and (1.12) and the continuity equation (1.3), (1.14) may be rewritten
as
nˆ × nˆ × Ei (r) = −nˆ × nˆ × jω µ0

G(r, r )J(r )dr
Ω

j
∇ G(r, r )∇ · J(r )dr .
−
ωε0 Ω
7

(1.15)

This is the EFIE. The MFIE uses tangential magnetic field boundary conditions on a perfect electrical conductor,
nˆ × (Hi (r) + Hs (r)) = J(r).

(1.16)

Substituting (1.10a) into the relation (1.8), equation (1.16) may be rewritten as
nˆ × Hi (r) = J(r) − nˆ × ∇ ×

G(r, r )J(r )dr .

(1.17)

Ω

Equation (1.17) is the MFIE. The EFIE and MFIE operators by themselves exhibit spurious interior
resonances at frequencies corresponding to the cavity modes of closed scattering objects. One
common method for removing these resonances is to use a linear combination of the EFIE and
MFIE, given by
ˆ × n(r)
ˆ × Ei (r) + (1 − α)n(r)
ˆ × η0 Hi (r) =
α n(r)

(1.18)

ˆ × T ◦ J(r) + (1 − α)η0 (I − K ) ◦ J(r).
−α n(r)
where I is the identity operator, 0 ≤ α ≤ 1 is a constant weighting factor, η0 =

µ0 /ε0 is the

intrinsic impedance of free space, and the operators T and K are given by
ˆ ×
T ◦J(r) = −n(r)
+

j
η
dr G(r, r )J(r )
k 0 Ω

jη0
dr ∇∇ G(r, r ) · J(r )
k Ω

ˆ ×
K ◦ J(r) =n(r)

(1.19)

dr ∇G(r, r ) × J(r ).
Ω

This equation is known as the Combined Field Integral Equation (CFIE). In this thesis the Generalized Method of Moments is applied to solve the CFIE.

1.5

The Method of Moments

The method of moments, or moment method, is the most common approach to solving electromagnetic integral equations of the type defined in the last section. The method of moments may be
derived in a variety of ways, using several different mathematical tools [2]. One straightforward
8

treatment is in terms of projection of solutions onto approximation function spaces. Consider a linear operator L whose operation on some unknown source function f is equal to a known function
h. This may be expressed as
L f = h,

(1.20)

where it is understood that the operator L is acting on f . The operator L can be any linear functional
operating on f , e.g. a differential operator or an integral operator. The problem is to find the
unknown function f that satisfies the equality, or, formally, to find f = L−1 g. The moment method
is one solution technique for such equations.
To discuss the moment method formulation, it is necessary to first define an inner product
between two functions f and g, denoted f , g . Although the choice of inner product is not unique
and may be chosen to suit the requirements of the problem, the standard inner product is defined
as
f,g =

w(x) f (x)g(x)dx,

(1.21)

Ω

where Ω is chosen as a suitable domain and w(x) is a weighting function.
The first step in forming a moment method system is to discretize the unknown function f in
terms of a set of approximation functions as
N

f (x) =

∑ anφn(x).

(1.22)

n=1

2
Here, the set {an } is a set of unknown coefficients that is presumed to satisfy ∑N
n=0 |an | < ∞, and

the set {φn } consists of admissible functions, e.g. {φn } ⊂ L2 , with L2 the set of square integrable
functions. It is now possible to rewrite (1.20) as
N

∑ anLφn = h.

(1.23)

n

If L is an integral operator, interchange of summation and integration is allowed because both the
sum and the integral are assumed to converge. The next step in constructing a moment method
system is to take inner products with another set of expansion functions {wm }:
N

∑ an wm, Lφn
n

9

= wm , h .

(1.24)

This procedure is often referred to as “testing” (1.23) with the set {wm }, a term that originates
in the so-called “reaction” interpretation of the moment method [3], in which the fields due to
the equivalent sources {φn } are observed, or “tested” by another set of equivalent sources, {wn }.
Equation (1.24) is a formal statement of the moment method system, and may be solved for the
unknown coefficients {an }. It may be rewritten in matrix notation as
ZI = V,

(1.25)

where Z is a matrix with elements Zmn = wm , Lφn , the right hand side V = [v1 , v2 , v3 , . . . , vN ]T ,
v j = wm , h consists of a vector containing the forcing function tested with each wm , and I =
[a1 , a2 , a3 , . . . , aN ]T is the vector of unknown scalar (generally complex) coefficients. The solution
is obtained by computing I = Z −1V using a matrix inversion procedure, either direct inversion,
matrix factorization, or some sort of iterative process. As long as Zmn and vm can be computed,
the unknown coefficients may be solved for and an approximate solution to (1.20) obtained. Discretization and solution of integral equations via the moment method is also subject to different
interpretations; e.g. as minimization of an energy functional via a Rayleigh-Ritz procedure [4].
These interpretations will be developed as needed in the chapters below.

1.6

Manifold Theory

The Generalized Method of Moments may be interpreted as a direct application of the modern
theory of manifolds to the computational solution of electromagnetic boundary integral equations.
This section briefly develops some concepts and nomenclature required to define functions and
perform calculus on the manifold structures used in GMM. Discussion here is restricted to smooth
manifolds. Some GMM formulations in thesis are rigorously described by smooth manifold theory,
and some relax certain conditions required to be classified as globally smooth in the mathematically rigorous sense. Even in these cases, the language and theory of smooth manifolds are still
useful conceptual tools for describing GMM discretizations. Additionally, many GMM manifolds

10

become smooth in the limit that tolerances in geometry representation go to zero. For a comprehensive introduction to manifold theory, see [5].

1.6.1

Definition of a smooth manifold

A smooth manifold is a mathematical entity, denoted in this work by Ω, upon which the operations
of calculus may be performed. Heuristically, this means that an n-dimensional (real) manifold
looks locally like Rn , i.e. a copy of n-dimensional real space. To make this definition explicit, and
since calculus is done with respect to coordinates, it is necessary to define a notion of coordinates
at each point on the manifold. This is done using the mechanism of coordinate charts, which are
defined as follows. Given a smooth manifold Ω, for an open (i.e. not containing its boundary)
subset Ωi ⊂ Ω, associate a smooth map ϕi : Ωi → Rn such that each point p ∈ Ωi may be obtained
.
in terms of local coordinates by the inverse map ϕi−1 (u) : Rn → Ωi , u ∈ Rn = {uk }, k = 1, .., n.
Throughout this and all subsequent sections, raised indices ui denote local coordinates. In analogy
to a set of local maps covering the earth, the set of all charts {(Ωi , ϕi )} that covers Ω (i.e. ∪i Ωi = Ω
is termed an “atlas” for Ω. The atlas concept provides a means of stitching the coordinate charts
{Ωi } together to obtain the original manifold, and allows locally performed calculus operations
to be cast in a global setting. Finally, for a pair of neighborhoods with nonempty intersection,
i.e. where Ωi ∩ Ω j = {0},
/ “transition maps” are defined between two charts (Ωi , ϕi ) and (Ω j , ϕ j )
as τi j : (Ω j , ϕ j ) → (Ωi , ϕi ) = ϕi ◦ ϕ −1
j . The key property of transition maps relevant to GMM is
that they allow calculus done on Ω j to be expressed in terms of calculus performed on Ωi . These
transition maps also carry a significant amount of information about the manifold. Crucially, they
may be used to determine whether the manifold is smooth: if the transition maps are smooth
up to Ck (i.e. support k bounded derivatives), then the manifold is also Ck smooth. Fig. 1.2
graphically depicts the quantities defined in this section for a two-dimensional manifold. Since
GMM is intended to solve electromagnetic scattering problems from two dimensional surfaces in
R3 , Fig. 1.2 and the remainder of this document will consider only the n = 2 case.

11

Figure 1.2: Manifold Ω, coordinate charts (Ω1 , ϕ1 ) and (Ω2 , ϕ2 ), and transition maps τ12 , τ21 .

1.6.2

Partitions of unity

In general, it is not possible to cover an arbitrary manifold with a single coordinate chart [5].
However, it is possible to stitch together the many coordinate charts (Ωi , ϕi ), and furthermore to
provide smooth transitions between functions defined on different charts. This is accomplished
through the construction of a partition of unity, which is defined as follows. Let the set {Ωi } be an
open cover for the manifold Ω. Then subordinate to each open subset Ωi ∈ {Ωi }, define a function
ψi with the following properties:
1. 0 ≤ ψi ≤ 1
2. supp(ψi ) ⊂ Ωi
3. ∑i ψi = 1

12

Then the set of functions {ψi } is said to form a partition of unity for Ω. The effect of the partition
of unity is to allow calculus done in local coordinate systems to be stitched together to cover the
entire manifold. Particularly relevant to GMM is that partitions of unity preserve continuity of
disparate function descriptions across patch overlaps: for two different functions f1 : Ω1 → R and
f2 : Ω2 → R that are not continuous with each other in the overlap region Ω1 ∩ Ω2 (i.e. f1 + f2 is
not continuous), the construction f˜ = ψ1 f1 + ψ2 f2 inherits the continuity (and differentiability) of
the partitions of unity.

1.6.3

Tangent and cotangent bundles

With each point p ∈ Ω on an n-dimensional manifold, there is an associated vector space called
the tangent fiber that contains n tangent vectors, and denoted Tp Ω. The entities Xl (p) ∈ Tp Ω,
l = 1, . . . , n that reside in this vector space are referred to as tangent vectors, and are specified
with a subscript. As in elementary linear algebra, the notion of tangent vectors naturally gives rise
to a dual space, the space of cotangent vectors, with the cotangent fiber at a point p denoted by
Tp∗ Ω. The cotangent vectors that reside in this this space are denoted by X l , l = 1, . . . , n, and are
specified with a superscript. Formally, the cotangent fiber is the space of linear functionals that
acts on the space of tangent vectors as X j : Tp Ω → R. A more concrete way to define the relation
between tangent and cotangent vectors is in terms of a basis. For example, let {ei }, i = 1, . . . , n be
the standard basis in Rn , and let {ε j } denote the standard dual basis. Then the dual basis {ε j } is
define through the relation
j

ε j ei = δi ,

(1.26)

j

where δi is the Kronecker Delta. The collection of all tangent (cotangent) fibers over Ω is called
the tangent (cotangent) bundle for the manifold. The tangent bundle is denoted T Ω and the cotangent bundle is denoted T ∗ Ω.
A crucial property of the tangent and cotangent bundles is that they are manifolds in their own
right, which implies that they may be parameterized in terms of local coordinates in precisely the
same manner as the underlying manifold. Additionally, partitions of unity may also be constructed
13

on T Ω and T ∗ Ω, although these partitions of unity are distinct from the partition of unity used on
the manifold Ω itself. The surface current spaces used in GMM are constructed using this property.
In order to define computations on T Ω and T ∗ Ω when they are expressed as smooth manifolds,
it is necessary to define coordinate charts in a manner analogous to the coordinate charts (Ωi , ϕi ) on
Ω. A thorough discussion of the construction of a basis for the tangent and cotangent spaces of an
arbitrary manifold is beyond the scope of the current discussion. Instead, it is sufficient to state that
there is a standard tangent basis dented {∂ /∂ xl }, l = 1, . . . , n that consists of the set of directional
derivatives along the coordinate directions {ul } so that a tangent vector may be expressed as
n

X=

∂

∑ vl ∂ ul .

(1.27)

l=1

In standard three dimensional Euclidean space, the basis {∂ /∂ ul } corresponds exactly to the three
canonical coordinate basis vectors with u1 = x, u2 = y, u3 = z:
∂
∂
=
= xˆ ,
∂ u1 ∂ x
∂
∂
= yˆ ,
=
2
∂y
∂u
∂
∂
= zˆ .
=
3
∂z
∂u

(1.28)

Thus, in analogy with the familiar Euclidean picture, the coordinates of a vector X k in the tangent
space are (v1 . . . vn ) ∈ Rn . The final step necessary of the construction of coordinates for the tangent space (which makes T Ω a manifold) is to pair the tangent vector coordinates with the local
coordinate chart (Ωi , ϕi ) that gives local coordinates for the points p ∈ Ωi . This composite coordinate map on the tangent space is defined as ϕ˜ i : Ωi × Rn → R2n . Using the notation previously
defined for coordinate charts, the coordinate charts for the tangent space are (Ωi × Rn , ϕ˜ i ). The
associated transition functions are τ˜i j : Ω j × Rn → Ωi × Rn are given by:


τi j 0 
τ˜i j = 
,
0 σi j

(1.29)

with the transition map σi, j the n × n transition matrix between the sets of tangent vectors on T Ωi
and T Ω j , which corresponds to the Jacobian of the transition map τi j . Partitions of unity may
14

be defined on the open subsets Ωi × Rn ⊂ T Ω in precisely the same way they are on the sets Ωi .
Fig. 1.3 shows the portion of the tangent bundle associated with a subset Ωi of a two dimensional
manifold Ω. In Fig. 1.3, the tangent fibers are visualized as copies of R2 ; they are actually vectors
expressed as in (1.27), but it can be shown that local tangent bundles T Ωi of the type used in GMM
may be mapped smoothly onto copies of real space of the same dimension as Ω.

Figure 1.3: Tangent bundle for a subset Ωi of a two-dimensional manifold Ω. The bundle T Ωi
consists of a collection of copies of R2 associated with each pair (u2i , u2i ) in the coordinate chart
˜
(Ωi , ϕi ). Coordinates for the bundle are mapped into R4 via the coordinate chart (Ωi × R2 , ϕ),
2
4
where ϕ˜ maps from the product space Ωi × R into local coordinates in R .

The development of a basis for the cotangent space proceeds in a similar manner to that of the
tangent bundle basis, with cotangent vectors expressed in the standard cotangent basis {dul } as
n

X∗ =

∑ vl dul ,
l=1

and dxl (∂ /∂ xk ) = δkl ,

as required by duality.
15

(1.30)

1.6.4

Embedding in R3 and Riemannian manifolds

Two final concepts are necessary to describe the Generalized Method of Moments in terms of
manifold theory. The first of these is the definition of local parameterizations induced by a smooth
embedding of the two dimensional manifolds {Ωi } into R3 . Under this embedding, explicit expressions for the tangent bases may be derived. The second concept is that of a so called “Riemannian
metric” g on the manifold, which specifies an inner product on each set Ωi and allows notions of
differential length and area to be concretely defined.
If there is map F : Ω → R3 that is smooth and invertible, then Ω is said to be a submanifold of
R3 , R3 is said to be the “ambient manifold” for Ω, and the “codimension” of the embedding Ω ⊂
R3 is dim R3 -dim Ω=1. This embedding may be naturally extended to the open sets {Ωi } that cover
Ω, and provides the link between the manifold theory discussed in the previous sections and the
standard picture of a scatterer bounded by a two dimensional surface residing in R3 . Specifically,
the mapping of local coordinates on a specific neighborhood Ωi is of the form (u1i , u2i ) → (x, y, z).
Embedding also facilitates explicit expressions for the manifold tangent vectors (∂ /∂ u1i , ∂ /∂ u2i )
in terms of the standard basis on R3 via a map dF : T Ωi → T R3 as
 
∂
 



∂
∂x
∂y
∂z 
∂x
 ∂ u1   ∂ u1 ∂ u1 ∂ u1   
∂ 
 i  i
i
i
 =  ∂x
 .

∂
y
∂
z
∂
 
∂y

 
∂ u2i ∂ u2i ∂ u2i  ∂ 
∂ u2i
∂z

(1.31)

Using the identifications from the previous section of (∂ /∂ x, ∂ /∂ y, ∂ /∂ z) = (ˆx, yˆ , zˆ ), these relations may be written as
∂
∂x
∂y
∂z
∂r
ˆ
ˆ
ˆ
=
x
+
y
+
z
=
∂ u1i
∂ u1i
∂ u1i
∂ u1i
∂ u1i
(1.32)
∂
∂x
∂y
∂z
∂r
ˆ
ˆ
ˆ
=
x
+
y
+
z
=
.
∂ u2i
∂ u2i
∂ u2i
∂ u2i
∂ u2i
This basis for the tangent space is utilized to construct current basis function sets in the Generalized
Method of Moments.
16

Finally, we briefly define the Riemannian metric, which provides the necessary structure to
compute inner products and other quantities of interest such as gradients, divergences, curls, and
integrals on the manifold Ω. A Riemannian metric g in local coordinates is defined as
g = gkl duk dul .

(1.33)

i.e. g is a rank two tensor field defined at every point p ∈ Ω with respect to the product of the
cotangent vectors duk and dul . A manifold endowed with a Riemannian metric g is called a
Riemannian manifold and is denoted (Ω, g). Properties of the metric for cases of interest in this
work are that it is positive definite (Det(g) > 0) and symmetric (gkl = glk ).
The global metric g defined everywhere on Ω is constructed using local metrics on the covering
set {Ωi } using a partition of unity as
g = ∑ ψi gi .

(1.34)

i

The inner product of two vectors w, v ∈ T Ω is dictated by the metric
w, v = ∑ gkl wk vl .

(1.35)

k,l

Although the Riemannian metric may be arbitrarily selected as any rank two covariant tensor
field that yields a positive definite inner product, the local parameterizations discussed in previous chapters induce a particular choice of the metric. Explicitly, for a two dimensional manifold
embedded in R3 , the components of the metric with respect to the basis of the tangent space are
gkl =

∂
∂
, l ,
k
∂u ∂u

(1.36)

which, under the map F : Ω → R3 may be computed easily using the standard metric on R3 ,
3

gR
µν = δµν , which is the usual vector dot product
gkl = δµ,ν

∂ x µ ∂ xν
∂r ∂r
= k· l
k
l
∂u ∂u
∂u ∂u

(1.37)

Because the metric g is defined with respect to the cotangent basis vectors, which act on tangent
vectors, it is therefore a covariant tensor. Also important is the contravariant tensor associated

17

with the metric, and defined as the inverse of g. Traditionally, the components of the contravariant
metric tensor are denoted with superscripts as gkl . The contravariant tensor acts on the cotangent
vector space, and plays a prominent role in the definition of derivative operators on Riemannian
manifolds. Operators of particular interest in this work are
2

∇φ =

∑

2

∑ gkl

k=1 l=1

2 ∂φ
∂φ ∂
=
duk
∑
k
l
k
∂u ∂u
k=1 ∂ u

2

∇·f =

1

∂

√
g fk ),

∑ √g ∂ uk (

(1.38a)

(1.38b)

k=1

where for a two dimensional manifold Ω, (1.38a) is identified with the surface gradient of scalar
functions defined on Ω and (1.38b) is identified with the surface divergence of vector functions
defined in the tangent space of Ω. Equation (1.38b) may equivalently be written
2

∇·f =

2

∂f

∂

∑ ∑ gkl ∂ uk · ∂ ul .

(1.39)

k=1 l=1

With the concepts developed in this chapter, the Generalized Method of Moments for electromagnetic boundary integral equations may be formulated in terms of the theories of classical electromagnetic and manifolds. This formulation is the subject of Chapter 2.

18

CHAPTER 2
THE GENERALIZED METHOD OF MOMENTS

2.1

Introduction

Significant effort has been exerted in recent years toward constructing discretizations of electromagnetic integral equations that model the underlying continuous problem more closely than traditional low order approaches. The aim of these efforts is to obtain solutions with better fidelity to the
continuous solution while significantly reducing the size of the descretized system and realizing
commensurate savings in computational cost. Most of the work to these ends has focused on one
(or a combination) of three directions: higher order current approximating function spaces that
require fewer than the traditional 10 degrees of freedom per wavelength, higher order geometry
descriptions that more closely model the true continuous geometry, and mixtures of basis sets, e.g.
mixing low order interpolatory basis sets with those derived from asymptotic wave representations.
Moment methods that are higher order in both current approximation spaces and geometry representation have been a subject of intense development in recent years. Methods for higher order
current representation have been traditionally based either on hierarchical higher order functions
residing in Nedelec Spaces [6–9], or on variations of mapped higher order polynomial tensor products [10–13]. By reducing the number of degrees of freedom per wavelength required to discretize
the integral operator, each of these methods realizes reductions in MoM system size. However,
the physical requirement that approximation functions must provide current continuity between
discretization subdomains places strict limitations on the types of basis functions that may be employed, and means that the boundary integral operators generally must be discretized using a single
class of basis function, e.g. only polynomials. Some methods for mixing higher order polynomial
bases with other basis classes have been developed (e.g. singularity type bases in [14]), but these
also require that the basis set be designed to enforce current continuity.

19

Closely related to the issue of continuity in basis function set is that of continuity in geometry
description. Higher order surface descriptions for moment methods are generally based on smooth
polynomial surface parameterizations [15], which have difficulty accurately modeling geometrical singularities such as edges and tips. Recently, representations using Non-rational B-Splines
(NURBS) have been developed [16, 17]. A bottleneck with both of these methods is enforcing geometrical continuity at abutting patch edges. As with current approximations, this generally forces
the use of the same type approximating function for the parameterization of each subdomain.
GMM is a partition of unity method that decomposes the scatterer surface into overlapping
subdomains, termed “patches”. As in the Generalized Finite Element Method [18, 19], the partition of unity serves to decouple approximation function descriptions in neighboring subdomains.
Consequentially, GMM is able to easily incorporate higher order basis sets, higher order/mixed
geometry descriptions, and arbitrary mixing of approximation function spaces. This flexibility in
approximation functions occurs because interpatch continuity is provided not by the approximating
functions themselves, but by the partition of unity. This lifting of the continuity constraint on the
basis functions permits substantially more freedom in the types of approximation function spaces
that may be employed; additionally, it allows mixing of different approximation spaces in adjacent subdomains over the surface of the scattering body. Thus, in GMM, basis sets may be freely
chosen to match local current ansatz. The partition of unity scheme also allows blending of different functional geometrical descriptions in the overlap region between subdomains, effectively
providing geometrical continuity and permitting the use of distinct geometry parameterizations
on neighboring subdomains. This implies that geometry parameterizations from entirely different
spaces, e.g. polynomials, conics, and flat tessellations, may be utilized in the same simulation
while maintaining effective geometrical continuity.
Although partition of unity methods have been widely employed in finite elements to address
some of the challenges in basis function and geometrical continuity [20–23], extension of these
methods to integral equations has been limited, in part because defining meaningful partitions of
unity on arbitrary two dimensional manifolds residing in R3 is a nontrivial problem. Nonetheless,

20

the Partition of Unity Boundary Element Method (PUBEM) [24] has been developed, primarily for
the solution of two dimensional acoustic Helmholtz scattering problems. In PUBEM, traditional
interpolating approximation function spaces are enriched with asymptotic plane wave bases for
electrically/acoustically large simple scatterers in two dimensions [25, 26], and a sphere in three
dimensions [27]. A similar approach is taken in [28,29], wherein stationary phase methods are applied to acoustics and electromagnetics problems using asymptotic representations on electrically
large smooth geometries. The approach in the present paper is significantly more general than
these approaches because it applies to vectorial electromagnetics problems on arbitrarily shaped
scatterers in three dimensions. The approximation spaces utilized may be arbitrary mixtures of
basis functions including, but not restricted to, the plane-wave and asymptotic type expansions
employed in the PUBEM and [29].
Finally, we note that the Generalized Method of Moments may be classified as a quasi-meshless
method. Meshless methods have traditionally been confined to the finite element community, especially in the field of mechanics and mechanical engineering (e.g. [30, 31]), although there has
been increasing interest toward applying such methods to computational electromagnetics, primarily in the context of quasi-static problems [32–34], and generalized finite elements, [35,36]. To the
authors’ knowledge, the only applications of meshless methods to high frequency electromagnetic
integral equations, excepting GMM and the references in the preceding paragraph, are implementations of the traditional Moving Least Squares (MLS) method [37, 38] and the work in [39], in
which both a MLS-based collocation scheme and an integral transform-based approach are given.
The Generalized Method of Moments is distinct from these methods in that it incorporates ideas
from mesh-free discretization (node-based primitives, partitions of unity) to effect localization of
geometry and approximation functions, but then marries these ideas with the large variety of moment method basis sets that have been developed by the electromagnetics community over the last
five decades.
The present work provides a unified prescription for the algorithmic development and implementation of the Generalized Method of Moments for arbitrary PEC scatterers. GMM for electro-

21

magnetic integral equations was first presented in [40] for piecewise flat tessellations and has been
extended to low-order (tessellated) Müller formulations for dielectrics [41, 42] and a higher order
smooth formulation for acoustics [43]. In this work, we extend GMM for electromagnetics to geometries composed of mixtures of features including flat regions, tessellations, polynomial smooth
patches, sharp tips, bodies of revolution, conic sections, etc. The inclusion of local smooth geometry descriptions removes one major hurdle encountered in the piecewise flat approach in [40],
which is the appearance of spurious line charges at boundaries between non-coplanar triangles. By
introducing either smooth local geometry parameterizations or subdomain basis sets that cancel
line charges by construction, the present work avoids line charges altogether. Preliminary work on
smooth local parameterizations and hybridizations with non-smooth geometrical descriptions for
GMM is contained in [44, 45], and [46].
Specific contributions of this paper are:
• A framework for arbitrarily mixing different classes of basis functions over the surface of a
PEC scatter.
• A geometrical hybridization scheme wherein various functional geometry descriptions may
be combined and blended in a single problem.
• A smooth, higher order geometry representation framework for representing arbitrary curved
geometries.
• Hybridization with tessellations and introduction of “sub-patch” basis sets capable of handling geometrical singularities.
• Algorithms for automatically assigning approximation function types and local geometry
descriptions based on local physical characteristics of the scattering geometry.
• Results demonstrating the flexibility of the method on several test scattering problems.

22

The scattering results presented are designed to demonstrate mixing of geometry descriptions and
basis function classes, and the corresponding reduction in system size. To address larger problems,
the method will be hybridized with the Multilevel Fast Multipole Method [47] in future work.
The remainder of the paper is organized as follows. Section 2.2 briefly outlines the problem
under consideration. The formulation of GMM for arbitrary PEC scatterers including construction
of patches starting from point clouds, the partitions of unity, local geometry parameterizations,
local basis functions, and the evaluation of matrix elements, is detailed in Section 2.3. Results
validating the method against a reference code, demonstrating its application to several representative geometries, and comparing matrix fill times for GMM relative to extant higher order methods
are presented in section 2.5. Finally, Section 2.6 provides some concluding remarks and future
directions.

2.2

Problem Statement

The problem of interest is the computation of the scattered fields {Es (r), Hs (r)} due to a plane
wave, characterized by the triad {kˆ i , Ei (r), Hi (r)}, impinging on a PEC object residing in free
ˆ
space. The scatterer boundary Ω is equipped with a unit normal n(r)
defined for all r ∈ Ω except
at a finite number of geometrical singularities (e.g. corners, tips, edges, etc.). The boundary
integral formulation we employ is the Combined Field Integral equation (CFIE):

ˆ × n(r)
ˆ × Ei (r) + (1 − α)n(r)
ˆ × Hi (r) =
α n(r)
(2.1)
ˆ × T ◦ J(r) + (1 − α)(I − K ) ◦ J(r)
−α n(r)

23

where I is the identity operator, 0 ≤ α ≤ 1 is a constant weighting factor, and the operators T and
K are given by:
ˆ × jkη0
T ◦J(r) = −n(r)
+

dr g(r, r )J(r )
Ω

jη0
dr ∇∇ g(r, r ) · J(r )
k Ω

ˆ ×
K ◦ J(r) =n(r)

(2.2)

dr ∇g(r, r ) × J(r )
Ω

ˆ
Here, j(r) = n(r)
× {Hi (r) + Hs (r)} is the induced surface current, g(r, r ) is the free-space
Helmholtz Green’s function, k is the propagation constant, and η0 is the intrinsic impedance of
free space. An e jωt dependence is assumed and suppressed. To construct a moment method syss
tem, the surface current is discretized in terms of a set of basis functions as j(r) = ∑N
n=1 an fn (r)

(Ns is the total number of degrees of freedom) and the discretized operators in (2.2) are tested with
a set of functions {fm (r)}, m ∈ [1, Ns ] in the usual fashion.

2.3

Formulation

Although many details of the construction of GMM for piecewise flat tessellation and scalar acoustic equations are given in [40] and [43], we include a detailed discussion here because these techniques have not been developed for electromagnetic integral equations on smooth surfaces, and
because the present method differs in several respects from previous work.
GMM is constructed via a decomposition of the manifold Ω into overlapping subdomains,
termed “patches”, and denoted by the set {Ωi }. Each patch Ωi is composed of four elements: 1) a
set of nodes Ni , 2) a local geometrical parameterization Gi and associated projection plane Γi , 3) a
partition of unity ψi , and 4) a local approximation function space {fk (r)}. In the following section,
we first develop patch definitions and define the partitions of unity, which decouple geometry and
approximation functions spaces on neighboring patches. We then describe in some detail the design
of the local geometry descriptions and basis sets. The set of patches {Ωi } may be constructed in

24

several ways depending on the level of a priori information that is known about the surface. We
employ the following approach, which only requires an oriented point cloud as a starting point.

2.3.1

Neighborhoods from Point Clouds

.
To begin, we require only that Ω be sampled with a discrete set of Nm nodes N = {ni }, i =
1, 2, 3, . . . , Nm residing in R3 and equipped with a connectivity map in the sense of nearest neighbors (note that standard simplicial tessellations implicitly contain this information). Denote by Ni
the neighborhood of the ith node, which is initially defined as the set containing ni and the set of
.
its nearest neighbors N i = Ni /ni as specified by the connectivity map. Additionally, we require
an orientation, specified by a normal defined at each ni . These normals may be a priori provided,
or may be obtained using a classical normal estimation routine, e.g. [48]. In the case where a node
occurs at a geometrical singularity such as a tip or corner, the normal is not uniquely defined; in
GMM such nodes are flagged as “singular” and are handled using a modified algorithm, which is
discussed in a later section. For the remainder of this section, we assume that the geometry is free
from singularities.
ˆi
Given the node list N and corresponding normals, for each ni we define a subdomain Ω
ˆ i is termed a “patch primitive”. We label the set
centered on ni and bounded by N i . Each Ω
. ˆ
ˆ i as Ω
ˆ
ˆ
of patch primitives that neighbor Ω
N = {Ω j |n j ∈ N i }, i.e. ΩN is the set of all patch
i

i

ˆ i . By definition each Ω
ˆi
primitives associated with nodes that neighbor ni , but not including Ω
ˆ
overlaps with all patches in Ω
N .
i

Although the initial set of patches {Ωˆ i } is defined about every node in N , it is often advantageous to merge neighboring primitives into larger patches, e.g., if the patch primitives are small
with respect to the characteristic scale of the problem or form part of the same surface of revolution. We denote the set of merged patches as {Ωi }, with each final patch defined as a union of
. Nk ˆ
ˆ k an Nk -dimensional subset of {Ω
ˆ i }. After merging, the
Ωk , with Ω
patch primitives: Ωi = ∪k=1
final patches may be associated with much larger collections of nodes, with neighborhoods Ni and
patch neighbor sets {ΩN } updated accordingly. As patch merging algorithms depend on local
i

25

geometric characteristics, we defer discussion of a sample merging algorithm until section 2.4.2.
Because there is no formal distinction between merged patches and patch primitives (they are two
instances of the same type of object), we drop the hat notation for the remainder of the paper and
use the set {Ωi } to denote the set of all patches for a given scatterer, primitive or otherwise.

2.3.2

Partition of Unity

Subordinate to each Ωi , we define a partition of unity function ψi (r) with supp{ψi (r)} = Ωi with
the property that ∑i ψi (r) = 1∀r ∈ Ω. These partitions of unity serve to decouple neighboring
patches so that any surface function φ (r), r ∈ Ω may be reconstructed as φ (r) = ∑i ψi φi (r), where
the subsectional functions φi (r) are given by φi (r) = χi φ (r), with χi the characteristic function of
patch Ωi . In GMM, partition of unity is enforced using an approach based on traditional Shepard
Interpolation [49], which takes the form:

ψi (r) =

ψˆ i (r)
ψˆ i (r) + ∑ j ψˆ j (r)

(2.3)

in which ψˆ j (r) is a local shape function with compact support on patch Ω j , and {ψˆ j (r)} is the set
of local shape functions defined on the set ΩNi of patches that neighbor Ωi . The choice of local
shape function depends on the problem type, and may include simplex type functions, smoothly
decaying exponentials, polynomial-based functions, etc. In the present work, a simplex-based
partition of unity is utilized. Figure 2.1 illustrates the blending of basis function spaces via the
partition of unity in one dimension. These concepts extend directly to functions and patches defined on two dimensional manifolds, and also effectively permit blending of different geometry
descriptions in the overlap region between patches.

2.3.3

Local Geometry Parameterization

In this section, we develop methods for constructing local surface geometry descriptions on each
patch in {Ωi }. These local surface descriptions are superposed as

26

i Ωi

to recreate the original

1.5
1.0
0.5
0.5
1.0

1

2

3

2.0
1.5
1.0
0.5
0.0
x
4
0.5
1.0
1.5

1

2

(a)
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5

4

x

(b)
1.0
0.8
0.6

1

2

3

4

x 0.4
0.2
1.5

2.0

(c)

1

2

2.5

3.0

3.5

4.0

x

(d)

1.5
1.0
0.5
0.5
1.0

3

3

4

x

2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5

(e)

0

-5
-10
0

1

2

3

-15
4

(f)

Figure 2.1: Blending of approximation spaces on overlapping domains via the partition of unity
for a 1D signal. (a) A trial function f (x) = .2(x − 2)3 + .3 on the interval Ω : 0 ≤ x ≤ 4. (b)
Interpolation of f (x) using linear interpolatory hat functions Ti (x) = T (x − .5i) on subinterval
Ω1 : 0 ≤ x ≤ 3 such that fˆ1 (x) = χ1 (x) ∑i ai Ti (x), χ1 (x) = 1, x ∈ Ω1 ; 0, x ∈ Ω1 . (c) Interpolation of
f (x) using a third order Legendre polynomial set Pn (x), n = 0, 1, 2, 3 on subinterval Ω2 : 2 ≤ x ≤ 4
such that fˆ2 (x) = χ2 (x) ∑3n=1 an Pn (x − 3), χ2 (x) = 1, x ∈ Ω2 , 0; x ∈ Ω2 . (d) Partitions of unity
ψ1 (x) and ψ2 (x). (e) interpolations of f (x) multiplied by partitions of unity as fˆ1 (x)ψ1 (x) in
Ω1 and fˆ2 (x)ψ2 (x) in Ω2 . (f) Reconstructed function fˆ(x) = fˆ1 (x)ψ1 (x) + fˆ2 (x)ψ2 (x) (left axis)
and absolute reconstruction error (right axis). Through the action of the partition of unity, the
representations fˆ1 (x) and fˆ2 (x) are smoothly blended in the overlap region, 2 ≤ x ≤ 3, and the
approximation error transitions smoothly between the linear and Legendre interpolations.

27

geometry. This scheme admits a wide variety in the choice of local geometry parameterization. We
now describe local geometry construction for both smooth and nonsmooth regions of the scatterer.
From the definition of patches above, each patch Ωi contains a set of nodes Ni = {nk } located
at positions {rk } and equipped (in the smooth case) with normals {nˆ k }. Local smooth surface
parameterization for Ωi proceeds as a two-step process. First, a “projection plane” Γi is defined
such that the plane normal nˆ i is the average of all subordinate node normals:

nˆ i =

∑k nˆ k
| ∑k nˆ k |

(2.4)

The plane Γi is uniquely defined by its normal and a point rci (defined with respect to the global
origin) through which the plane passes; generally, this point is taken as the average of the locations
of {nk }, although it may be chosen otherwise depending on the requirements of different geometry
parameterizations.
Once the plane Γi has been defined, a local orthogonal coordinate system (u1 , u2 , u3 ) is constructed on Γi as follows. First, the altitude direction is taken parallel to nˆ i : uˆ 3 = nˆ i . The uˆ 1
direction is defined in the direction of maximum patch dimension, max{|rkb − r jb |}, where the
b subscript indicates a boundary node. The remaining coordinate direction is then defined as
uˆ 2 = uˆ 3 × uˆ 1 . All local coordinates are defined with respect to a local origin located at rci .
Upon establishment of the local coordinate system, a parameterized manifold Λi passing through
the nodes {nk } is defined. This parameterization takes the form:

.
Λi ={r ∈ R3 | r(u1 , u2 ) = ri (u1 , u2 ) + rci },

(2.5)

ri (u1 , u2 ) = u1 uˆ 1 + u2 uˆ 2 + wi (u1 , u2 )uˆ 3
where ri (u1 , u2 ) indicates a local position vector pointing from the (u1 , u2 , u3 ) origin to the parameterized surface. The domain of the parameterization is restricted to a polygon in the projection
plane Γi that bounds the projection PΓi (Ni ) of the node set Ni into Γi : u1 , u2 ∈ PΓi (Ni ). We
denote the type of parameterization on patch Ωi by Gi . The altitude function wi (u1 , u2 ) may be
taken as any well-behaved function of (u1 , u2 ). The conditions for wi (u1 , u2 ) to be well-behaved
28

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.2: Smooth patch geometry parameterization procedure for patch Ωi : (note: for clarity,
subscripts i are dropped on vector quantities) (a) Start with an oriented point cloud. (b) Define a
neighborhood Ni of nearest neighbors about node i. (c) Define a projection plane Γi with normal
equal to average of all node normals and specified by a point rc that uniquely defines an origin of
coordinates. (d) Construct a local orthogonal u1 , u2 , u3 coordinate system. (e) Take projection of
points in Ni into Γi to define the support of the patch in the u1 , u2 plane. This projection is denoted
PΓi . (f) Obtain a smooth parameterization to define the local patch surface.
are that 1) it is single-valued, 2) its Jacobian is amenable to integration through standard numerical
quadrature techniques, and 3) it maintains an appropriate degree of geometrical continuity with
neighboring patch descriptions. As long as these conditions are satisfied, different local geometry
parameterizations may be used on different patches, and the choice of parameterization Gi my be
matched to local geometric features, e.g. a BoR description for rotationally symmetric surface
regions, corner elements for edges and tips, Bezier surfaces, etc. The mixing of multiple surface
descriptions generalizes the work in [46], which hybridized polygonal tessellations with smooth
polynomial surface descriptions. Figure 2.2 summarizes patch neighborhood and local geometry
construction for smooth geometries.
To connect the local geometry parameterization process to manifold theory, one can identify
29

the local parameterization ri with a coordinate chart of the form (Ωi , ϕi ), where ϕi : Ωi → R2 is a
map from the manifold (surface) Ωi into the (u1 , u2 ) plane. This map is the inverse of the mapping
ri , and is given explicitly with respect to the embedding in R3 as:

ϕi (r) = uˆ 1 · (r − rci ), uˆ 2 · (r − rci )

(2.6)

i.e. it is the projection of a point r into the (u1 , u2 ) plane associated with the patch. Using the
correspondence of ri ↔ ϕi−1 , coordinate changes in overlapping regions between to patches r ∈
Ωi ∩ Ω j may be computed using the transition function τi j = ϕi ◦ ϕ −1
j as
τi j = uˆ i1 · (r j − rci ), uˆ i2 · (r j − rci ) .

(2.7)

Note that, if the geometry description is approximate, then the transition functions will τi j will be
approximate as well.

2.3.3.1

Smooth Polynomial Geometry Parameterization

Although any parameterization satisfying the conditions stated in the previous section may be utilized to define patch surfaces, it is desirable to have a default parameterization scheme for surfaces
where no a priori surface description is known apart from the node locations and normals. To
effect this scheme, following [50] we use a polynomial surface description for the parameterized
manifold Λi :

g

ri (u1 , u2 ) = u1 uˆ 1 + u2 uˆ 2 + Pi (u1 , u2 )uˆ 3

(2.8)

g

where Pi (u1 , u2 ) is as a degree g tensor product of polynomials in u1 and u2 :
g

Pi (u1 , u2 ) =

∑

cα (u1 )α1 (u2 )α2

(2.9)

|α|≤g

.
Here we have employed standard multi-index notation α = (α1 , α2 ), |α| = α1 + α2 for compactness. The coefficients cα are obtained via least-squares fit. The geometrical fidelity of Λi to the

30

original node locations is controllable by the degree g and the tolerance in the least squares algog

rithm. By defining Λi as a polynomial function Pi (u1 , u2 ), Λi is ensured to lie the space Cg of
g-differentiable functions. This property is important in evaluating field integrals because basis
functions in the GMM scheme inherit the differentiability of the geometry parameterization.

2.3.3.2

Handling of Geometrical Singularities

In the case where a patch contains nodes that lie on geometric singularities, a modified geometry
parameterization method is employed. If an analytical description of geometry in the neighborhood
of singularity is known, as with a conical tip or straight edge, a custom geometrical description that
correctly handles the singularity may be constructed. Alternatively, a standard triangular tessellation may be defined on the set of nodes adjacent to (and including) the singular nodes. Appropriate
basis sets are then defined on the tessellation that correctly capture the behavior of currents and
charges near the singular feature. Although the design of these functions is outside the scope of
this paper, several such basis functions have been developed (the most prevalent being the RWG
class of functions) that can be easily incorporated into the GMM framework.

2.3.3.3

Definition of the Tangent Space T Ωi

The basis functions used to discretize vector current distributions J(r), r ∈ Ω are defined in the
tangent space T Ωi , or “Tangent Bundle” on each patch Ωi . Section 1.6.3 of chapter 1 gives a
general discussion of the construction of tangent bundles under local parameterizations of the form
(2.3), and provides expressions for the tangent basis vectors, which for the local parameterizations
developed above are given by:

1 2
∂ ri
1 + ∂ w(u , u ) u
ˆ
ˆ3
=
u
u
∂ u1
∂ u1
∂r
∂ w(u1 , u2 ) 3
ri 2 = 2i = uˆ 2 +
uˆ
u
∂u
∂ u2

ri 1 =

31

(2.10)

Using this basis, any vector v in the tangent bundle of the patch Ωi at a point ri ∈ Ωi may be
expressed as:

v(u1 , u2 ) = v1 (u1 , u2 )ri u1 + v2 (u1 , u2 )ri 2
u

(2.11)

Thus, the coordinates of the tangent bundle T Ωi are given by (u1 , u2 , v1 , v2 ) ∈ R4 , and a coordinate
chart (T Ωi , ϕ˜ i ) may be defined for the local tangent bundle in the same way that a coordinate chart
(Ωi , ϕi ) is associated with each patch Ωi . Crucially, this also implies that the global tangent bundle
may be reconstructed using a partition of unity as:

T Ω = ∑ ψi T Ωi

(2.12)

i

The ability to subsect the global tangent bundle in this way is the basis for the discretization of
currents in GMM.
The map ϕ˜ i : T Ωi → R4 is simply (ϕi , Id2 ), where ϕi : Ωi → R2 is given by (2.6) and Id2 :
R2 → R2 is the identity map. Finally, transition functions τ˜i j : R4 → R4 between coordinate charts
of the tangent bundles T Ωi , T Ω j of two overlapping patches Ωi , Ω j is given by τ˜i j = ϕ˜ i ◦ ϕ˜ −1
j as:


τi j 0 
τ˜i j = 

0 σi j


j
j
i
i
r 1 ·r 
r 1 · r
σi j =  u u1 u u2 
j
j
ri 2 · r 1 ri 2 · r 2
u

2.3.3.4

u

u

(2.13)

(2.14)

u

Vector Calculus on Parametric Surfaces

Here, we briefly define quantities required to do vector calculus on the parametric surface specified
by r(u1 , u2 ). Vectors tangential to the surface are expressed as t = aru1 +bru2 , where a, b are scalar
.
coefficients and the basis vectors ru j , j = 1, 2 are defined by ru j = ∂ r/∂ u j . Note that the tangent

32

Figure 2.3: Tangent and normal vectors associated with a parametric patch surface.
basis vectors are not normalized, and in general do not need to be orthogonal. For the types of
parameterizations used in this paper, the basis vectors are given explicitly by:

∂ w(u1 , u2 ) 3
uˆ
∂ u1
∂ w(u1 , u2 ) 3
ru2 = uˆ 2 +
uˆ
∂ u2
ru1 = uˆ 1 +

(2.15)

The unit normal is given in terms of these tangent basis vectors as nˆ = (ru1 ×ru2 )/||ru1 ×ru2 ||.
Figure 2.3 shows ru1 , ru2 and nˆ on the parametric surface.
For the purposes of integrating and taking derivatives with respect to the parametric coordinates (u1 , u2 ), it is necessary to define notions of differential length and area on the parameterized
surface. To do so, a transformation matrix called the “metric tensor” is defined as:


r 1 · r 1 ru1 · ru2 
g= u u

ru2 · ru1 ru2 · ru2 .

(2.16)

Traditionally, components of the metric tensor are indexed using lowered indices gi j . The inverse
of the metric tensor g−1 is used in expressions involving derivatives, and the components of g−1 are
referenced using raised indices gi j . The differential surface area element is given by dS = Jdu =
Jdu1 du2 where the Jacobian is J =

det(g), with det(g) the determinate of the metric tensor.

Finally, the surface divergence of a vector function f(r(u1 , u2 )) with respect to the parametric
coordinates is:
2

∇s · f(u1 , u2 ) = ∑ gi j
i, j

33

∂f
·r j
∂ ui u

(2.17)

This expression is used to compute the divergences of surface currents required in the T operator
of equation (2.2). For further development of these concepts in the context of electromagnetics,
we refer the reader to [51, 52].

2.3.4

Local Basis Functions

Next, we develop local basis function descriptions. GMM patches may incorporate either entire
patch (EP) bases with support over the entire patch, or sets of sub patch (SP) bases with support
smaller than the entire patch.

2.3.4.1

Entire Patch Basis Functions

The general entire patch vector basis function is given in terms of local coordinates as:

fi,k (r) = fk,1 (u1 , u2 )ru1 (u1 , u2 )
+

(2.18)

fk,2 (u1 , u2 )ru2 (u1 , u2 )

so that the unknown surface current on patch Ωi is expressed as:
Ni

ji (r) = ψi (r)

∑ ak fi,k (r)

(2.19)

k=1

Here, ψi (r) is the partition of unity on patch Ωi , the vectors ru1 and ru2 are tangent vectors to
the surface as defined in section 2.3.3.4, and the weighting functions f1,k (u1 , u2 ) and f2,k (u1 , u2 )
may be chosen from any set of functions that lead to convergent integrals on Λi . Examples of such
functions are hierarchical polynomials, plane waves, body of revolution functions, etc.
To construct EP basis functions, first a canonical minimum bounding shape is defined that encloses the projection PΓi of the patch node set Ni onto the local projection plane Γi . The choice
of bounding shape depends on the patch type and shape. Rectangles are generally employed for
mapped polynomial functions, circles for BoR, etc. Once the appropriate bounding shape is assigned, the component basis functions fk,1 (u1 , u2 ) and fk,2 (u1 , u2 ) are defined on its interior and

34

restricted to the domain included in PΓi . Finally, these basis functions are lifted onto the parametric
surface as in (2.18). This arrangement is illustrated in figure 2.4.

Figure 2.4: Entire patch basis function.

2.3.4.2

Sub-patch Basis Functions

GMM can also easily incorporate multiple basis functions on a single patch with sub-patch support,
so called sub-patch (SP) bases. This may be interpreted as defining an EP basis function as f(r) =
∑ j f j (r), with {f j (r)} a set of approximation functions with support smaller than supp{Ωi }. The
requirements on SP bases are that they introduce no line charges interior to the patch and that they
possess a nonzero normal component at the patch edges. Due to the partition of unity, the normal
component is only required to be finite, and need not satisfy any explicit continuity constraint
across patch boundaries.
Sub-patch bases may be defined on the projection plane and projected onto the parameterized
surface as with EP bases; alternatively, SP functions may be defined directly on a tessellation of
the point cloud as in a traditional Moment Method discretization. Likewise, the partition of unity is
defined either on the projection plane Γi and then lifted onto the true surface, or is defined directly
on the tessellation itself. Defining sub-patch bases in this way allows most basis function types

35

from existing tessellation-based Moment Method codes to be directly used in the GMM scheme
with minimal alteration. Figure 2.5 shows an example sub-patch tessellation.

Figure 2.5: Illustration of sub-patch basis function arrangement with projected PU.

To illustrate the manner in which sub-patch bases are implemented in the GMM scheme, we
detail the construction of a tessellated patch with Rao-Wilton-Glisson (RWG) [53] basis functions.
RWG functions are defined as usual on the tessellated surface, or equivalently, on the projection
plane and lifted onto the surface. A simplicial partition of unity fits naturally into the RWG framework and is defined so that it takes a value of unity in all interior triangles and falls linearly to zero
in boundary triangles. To provide the necessary normal component at the patch edge, a half-RWG
function is defined across exterior patch boundaries. Although half-bases are generally avoided in
RWG discretizations because they introduce line charges, this difficulty is avoided in GMM because the PU forces the function to zero at the patch edge. This property of the PU applies to all
GMM basis functions, and renders the basis set on a patch charge neutral, regardless of whether
the functions are EP or SP. Figure 2.6 shows interior and boundary RWG basis functions, and the
removal of the boundary line charge by the PU.

36

(a)

(b)

(c)

Figure 2.6: Design of local RWG basis set with normal continuity across patch boundaries: (a)
RWG basis functions associated with internal edges, (b) Additional half-RWG functions associated
with exterior patch edges provide a normal component at patch boundary, and (c) Simplex-based
partition of unity forces the half-RWG function to zero at patch boundary, removing the line charge.
2.3.4.3

Default EP/SP Basis

As with local geometry descriptions, it is advantageous to have default basis function types that
are utilized when a current ansatz is not available or difficult to obtain. A straightforward choice
of an EP basis function is:

f1 (u1 , u2 ) = φβ (u1 , u2 )ru1

(2.20)

f2 (u1 , u2 ) = φβ (u1 , u2 )ru2
.
where φβ (u1 , u2 ) = Pβ (u1 )Pβ (u2 ) is a product of two Legendre polynomials of degrees β1 and
1

2

β2 , and the entire pth order representation on patch Ωi is constructed as:
2

ji (r) = ψi (r) ∑

∑

l=1 |β
β |≤p

37

aβ ,l φβ (u1 , u2 )rul

(2.21)

More complex polynomial bases may be constructed, e.g. one which explicitly enforces a
surface Helmholtz decomposition, as in [40]. However, we have found that the minimal tensor
product of Legendre polynomials in (2.21) generally provides better matrix conditioning than other
choices of polynomial basis functions for the same order. Furthermore, since GMM provides
current continuity between patches by design, it is not necessary to use the modified Legendre
bases or hierarchical bases often employed in higher order mapped methods [11, 15].
With the influence of the partition of unity, the surface current approximation for the default
EP basis spans the space:
f1 (u1 , u2 ) ∈ span{ψi (r)(φβ (u1 , u2 )ru1 )
+ψi (r)(φβ (u1 , u2 )ru2 )}

(2.22)

β| ≤ p
|β
As described in section 2.3.3, when geometrical singularities such as sharp tips or edges are
present, a smooth geometry parameterization and polynomial EP basis cannot be used. In these
situations, we default to a surface tessellation supporting a SP RWG basis as detailed in section
2.3.4.2. The properties of these functions have been extensively developed [53] and are not repeated here.
Finally, we explicitly show how the partitions of unity and half-RWG basis functions are
constructed in the case of Legendre (EP-type) and RWG (SP-type) patches that overlap. Figure 2.7(a) shows the partition of unity ψRW G = ψˆ RW G /(ψˆ RW G + ψˆ Leg ) and location of halfRWG basis functions for the RWG patch, and Figure 2.7(b) shows the partition of unity ψLeg =
ψˆ Leg /(ψˆ RW G + ψˆ Leg ) on the Legendre patch. The sub-partition of unity ψˆ RW G on the RWG patch
is defined on a triangle-by-triangle basis as:

ψˆ RW G =





1, nbd = 0




ξ , nbd = 1






1 − ξ , nbd = 2

38

(2.23)

(a)

(b)

Figure 2.7: Graphical depiction of partitions of unity for overlapped RWG and Legendre patches.
(a) Partition of unity ψRW G and half-RWG basis functions for an RWG patch. (b) Partition of unity
ψLeg for a Legendre patch.
where nbd is the number of triangle nodes on the patch boundary (in the overlap region) and ξ is
the simplex coordinate associated with the exterior node if nbd = 1 or the interior node if nbd = 2.
A similar definition may be used to construct ψˆ Leg .

2.4

Implementation Details

In this section, we discuss some practical details of evaluating GMM matrix elements and give
sample algorithms for patch merging and basis function assignment. Each of these tasks is nontrivial, and investigations of the most efficient and effective methods for specific types of problems
are subjects of ongoing investigation. The integration methodology presented is sufficiently general that it may be used for most GMM discretizations, although more specialized rules could be

39

developed for specific basis function types. The integrations and algorithms developed here are
used to generate all of the results in section 2.5.

2.4.1

Matrix Elements

We devote this section to a brief discussion of the practical evaluation of GMM matrix elements.
Using the surface parameterizations from Section 2.3.3 and the differential geometry quantities
from Section 2.3.3.4, the tested scattered electric field portion of (2.2) becomes :

− jkη0
·
Dj

Di

dui Ji (ui )ψi (ui )fi,m (ui )

du j J j (u j )g(ri (ui ), r j (u j ))ψ j (u j )f j,n (u j ))
jη
du J (u )∇s · (ψi (ui )fi,m (ui ))
+ 0
k Di i i i
Dj

(2.24)

du j J j (u j )g(ri (ui ), r j (u j ))∇s · (ψ j (u j )f j,n (u j ))

where we have employed the standard derivative transfers in the Φ portion of the T operator. The
tested magnetic field integral equation portion of (2.2) is:

Di
Di
Dj

dui Ji (ui )ψi (ui )fi,m (ui )ψ j (u j )f j,n (u j )−

ˆ
dui Ji (ui )fi,m (ui ) · n(r(u
i ))×

(2.25)

du j J j (u j )∇g(ri (ui ), r j (u j )) × ψ j (u j )f j,n (u j )

where ul = (u1 , u2 ) ∈ Γl denotes the local transverse coordinates for patch l, Jl is the Jacobian as
defined in section 2.3.3.4, and Dl ∈ Γl is the domain of integration in the projection plane. If the
projection PΓ (Nl ) of the nodes in Nl is a canonical integration domain, e.g. a regular polygon,
l

then an appropriate quadrature rule can be used to integrate over the entire projection. For instance,
radial/angular rules are used for BoR patches. In the general case where the projection is not
bounded by a canonical shape, we perform a triangulation of PΓ , e.g. using Delaunay or another
l

40

Figure 2.8: (a) Graphical representation of PΓ (Nl ) and (b) construction of triangular integration
l
subdomains. The domain of integration is Dl = ∪i Ti .
method. Figure 2.8 illustrates this process. Subdividing the domain of integration in this way
allows the use of well-developed integration rules for triangles (including singularity rules), and
the total integral is the sum of integrations over subtriangles. It is important to note that not every
node in PΓ needs to be used in defining the triangulation: if the patch is large, some subset of
l
the nodes may be used to lessen the integration cost as long as the patch boundaries are correctly
handled.

2.4.2

Automated Patch Construction

For large, complex geometries, it is necessary to automate the processes of patch construction
including merging, geometry parameterization, and basis function assignment. The following are
two sample algorithms for patch merging and geometry/basis selection.

2.4.2.1

Algorithm for Patch Merging

Using the procedure from section 2.3.1, patch primitives are assigned in the initial pass. To reduce
the number of degrees of freedom in the MoM system, it is often advantageous to merge patches
if neighboring primitives share common geometrical features, e.g. are part of the same body of
revolution or fall on a sufficiently smooth portion of Ω. Given the definition of Ωi as a collection
of nodes, merging and or splitting may be simply done by operating on the sets Ni . Algorithm
1 illustrates a procedure for merging; splitting may be done in a similar fashion. Criteria for
merging are tied both to patch size and patch mean curvature, which is approximated via deviation

41

1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:

Define maximum normal deviation εn tolerance
Define maximum average normal deviation εd tolerance
Define maximum patch diameter Dm
ˆ i ∈ {Ω
ˆ i } do
for each patch primitive Ω
ˆ j do
for each n j ∈ Ni (corresponding to primitive Ω
if nˆ j · nˆ i > (1 − εn ) then
Define candidate merged neighborhood: Nl = Ni N j . Designate the
number of nodes in Nl as M.
ˆ k /| ∑M
ˆ k|
Compute new average patch normal nˆ l = ∑M
k=1 n
k=1 n
Compute merged patch maximum diameter D = max{|rna −
rnb |}, na , nb ∈ Nl
if D ≤ Dm then
.
Compute average deviation ∆nˆ l from patch normal nˆ l as: ∆nˆ l =
1 M ˆ ·n
l ˆk
M ∑k=1 n
if ∆nˆ l ≤ εd then
ˆ i, Ω
ˆ j:
Merge Ω
Ni ← Nl
ˆ i } ← {Ω
ˆ i }\Ω
ˆj
{Ω
end if
end if
end if
end for
end for
Algorithm 1: Automated patch merging.

of normals as shown in algorithm 1.
At the end of the merging process, the final set of merged patches is designated {Ωi }. Algorithm 1 is a simplified version of the actual merging algorithm used in the present GMM code.
In practice, additional higher level merging conditionals are also used that, for instance, favor
convex patch boundaries over concave boundaries, attempt to maintain equal patch sizes over the
entire geometry, and group patches with similar characteristics, such as edge singularities or small
curvature.

2.4.2.2

Automated Geometry and Basis Assignment

Algorithm 2 illustrates automated assignment of local geometry descriptions G, basis function
types b and (where applicable) orders g using using tessellations, BoR surfaces, and polynomial
42

1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:

Define maximum patch normal deviation εd for smooth surface description
Define maximum patch normal deviation for plane wave basis ε p
Define minimum patch diameter for plane wave D p
for each final patch Ωi do
.
Compute average deviation ∆nˆ i from average patch normal nˆ i as: ∆nˆ i =
1 M ˆ ·n
i ˆk
M ∑k=1 n
if ∆nˆ i > εd then
G: triangular tessellation
b: RWG Basis
else if Rotationally Symmetric Patch then
G: Axisymmetric BoR representation
b: BoR Basis
else
G: Local smooth approximation to Ωi .
if ∆nˆ i < ε p && Diam{Ωi } > D p then
b: Plane Wave Basis
else
b: Polynomial Basis of order g
end if
end if
end for
Algorithm 2: Automated geometry and basis assignment.

smooth surfaces for geometry parameterization and RWG, BoR, polynomial, and plane wave bases.
Any geometry or basis function class valid for the GMM framework may be included in the algorithm with the proper conditionals. The present code implementing GMM is written in a highly
modularized fashion that facilitates easy addition of new geometry and basis function types.

2.5

Results

In this section, we provide results that illustrate hybridization of geometry and basis types and verify the accuracy and flexibility of the method. First, we utilize a representation test to demonstrate
the hybridization of multiple basis types in the reconstruction of an analytical function. We then
provide several scattering results, each with a different mixture of basis function types and orders
defined on patches with varying local geometry descriptions. Throughout the results section, the

43

algorithms detailed in Section 2.4.2 are used to automatically assign local geometrical descriptions
and surface current approximation spaces that are matched to solution ansatzes for the continuous
problem. The resulting discretizations give accurate solutions with significant reductions in the
numbers of degrees of freedom relative to a reference CFIE-RWG code.
We begin with a representation (Gram) test in which a quadratic vector function f(x, y) = (x −
.5)2 xˆ is reconstructed on a flat 1m×1m plate using a mixture of a 2nd order Legendre polynomials
defined on smooth patches and RWG functions defined on a triangular tessellation. Figure 2.9
shows the discretization and reconstruction error for three cases. Reconstruction error is taken as
the pointwise absolute error (abs err)= |˜f(x, y) − f(x, y)|, where ˜f(x, y) is the reconstructed function.
First, as a reference, we interpolate the analytical function f(x, y) using the standard linear
RWG basis on a triangular tessellation with maximum edge length .07m. This discretization is
shown in figure 2.9.a. Figure 2.9.b shows the reconstruction error using the all-RWG basis set. As
one would expect, the error is highest at the edges where the RWG functions cannot reconstruct
the normal component, and lowest where the argument of f(x, y) goes to zero (at x = .5m). The
purpose of this result is to provide a reference error level for RWG discretization for case 3, in
which RWG and Legendre bases are mixed.
Figures 2.9.c and 2.9.d show discretization and reconstruction error for an all-Legendre polynomial basis of second order. Because the basis set is polynomial complete to the same order as
f(x, y), the reconstruction is perfect to machine precision everywhere on the plate. Most significantly, the reconstruction is perfect even in the overlap regions between patches where the partition
of unity provides the continuity and blending between basis sets on neighboring patches. Without
the partition of unity, there would be significant error in the overlap regions.
In figure 2.9.e, a .25m × .25m tessellated patch with RWG functions is placed in the center
of the plate, while the outer boundary is discretized using flat patches supporting second order
Legendre polynomials. Figure 2.9.f shows the reconstruction error for the mixed RWG/Legendre
basis set. In this case, the error in the RWG patch matches that from the center of the plate in
figure 2.9.b. Outside of the RWG region, the error falls significantly, although due to the coupling

44

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.9: Reconstruction error for a function f(x, y) = (x − .5)2 xˆ using three different basis
sets: all-RWG, all-Legendre, mixed RWG/Legendre. (a) All-RWG discretization, (b) All-RWG
reconstruction error, (c) All-Legendre patch discretization (2nd order Legendre polynomials) (d)
All-Legendre reconstruction error, (e) Mixed RWG/Legendre discretization (2nd order Legendre
Polynomials), (f) Mixed RWG/Legendre reconstruction error.

45

(a)

(b)

(c)

Figure 2.10: Surface currents induced by an incident plane wave on a sphere discretized with
(a) overlapping GMM patches supporting a 4th order Legendre basis and (b) an RWG basis for
reference. Subfigure (c) shows the patches used to obtain the GMM result. Magnitude of imaginary
part of current ||Jim || is shown; ||Jre || yields similar plots.
between the RWG and Legendre patches, it does not decrease to the level of the all-Legendre
solution. Crucially, due to the blending effect of the partition of unity, no additional error is
introduced in the transition region between the two basis sets.
The next result shows currents induced on a spherical scatterer of radius a = 1λ due to an
incident x-polarized plane wave propagating in the −ˆz direction. Figure 2.10.a shows currents
obtained via solving the MFIE using overlapping GMM patches supporting a 4th order Legendre
basis, and figure 2.10.b shows the same result using RWG basis functions for reference. Figure 2.10.c shows the overlapping patches used to obtain the GMM result. The current densities
46

show good agreement, although the higher-order GMM patches give smoother results relative to
the RWG discretization. Due to the analytical form of the scatterer, the sphere may be exactly
represented using spherical local geometry descriptions in GMM.
The next scattering result demonstrates automated geometry and basis function assignment
for a 4.5λ × 1.5λ × .4λ NASA almond, with the corresponding RCS compared against a wellvalidated CFIE-RWG reference solution. The almond is illuminated by a unit amplitude x-polarized
plane wave of frequency 300MHz incident in the +z direction. Here, a Legendre basis (eq. 2.20)
of maximum order p = 3 is defined on 2nd order polynomial patches in the smooth regions, which
are detected using Algorithm 2. Figure 2.11 shows the discretization of the almond in terms of
overlapping smooth and tessellated patches. Tessellated patches supporting RWG basis functions
are used to capture regions of high curvature and the geometric singularity at the tip. One could envisage a tip basis function that does not rely on tessellation, but instead captures the tip singularity
exactly. While the design of such a basis is outside the scope of this paper, it could be seamlessly
integrated into the GMM framework. Figure 2.12 shows the bistatic RCS for the almond, which
is taken along the φ = 0◦ cut. The number of degrees of freedom for the CFIE-RWG solution is
Nre f = 3636 and the GMM result requires NGMM = 1861.

Figure 2.11: GMM discretization of a 4.5λ NASA almond.

47

RCS, dBsm

GMM
CFIE-RWG

30
20
10
0
-10
-20
-30
-40
−150◦−100◦−50◦ 0◦
θ

50◦ 100◦ 150◦

Figure 2.12: NASA almond bistatic RCS, taken along the φ = 0◦ cut.

The next result, scattering from a 1.2λ × .4λ × .4λ diffraction-matched conesphere illuminated
by a −ˆx polarized plane wave incident in the zˆ direction, illustrates the high degree of geometrical
accuracy that may be obtained when the analytical form of the underlying scatterer is known.
Figure 2.13 shows the GMM discretization for the conesphere. The cone portion of the geometry
(excluding the tip) is represented exactly through a single BoR patch which supports two types of
BoR basis functions, defined as:
fm,1 (r) = aPn (ρ)e jmφ rρ (u1 , u2 )

(2.26)

fm,2 (r) = aPn (ρ)e jmφ rφ (u1 , u2 )
where ρ(u1 , u2 ) =

(u1 )2 + (u2 )2 and φ (u1 , u2 ) = arctan(u2 /u1 ). To capture the spherical por-

tion of the scatterer to a high degree of precision, 5th order polynomial patches are employed with
3rd order Legendre basis functions. The tip region is modeled with a tessellation and RWG functions. Figure 2.14 shows a comparison of the GMM result against a reference CFIE-RWG result.
In this case, the GMM solution uses 126 Legendre functions on the spherical cap, 130 BoR basis
functions on the barrel, and 294 RWG functions on the tip, so that the total GMM system size is
NGMM = 550. The reference RWG solution requires a total of Nre f = 1584 degrees of freedom.
Agreement between the two results is quite good.
To demonstrate the reduction in system size that results from using a polynomial bases on a
48

Figure 2.13: GMM discretization of a 1.2λ × .4λ ×
.4λ conesphere.

10

GMM
CFIE-RWG

RCS, dBsm

5
0
-5
-10
-15
-20
-25 ◦
−180−135◦−90◦−45◦ 0◦
θ

45◦ 90◦ 135◦ 180◦

Figure 2.14: Bistatic RCS for the conesphere at 300MHz. Incident wave is traveling in the θ =
0◦ , φ = 0◦ direction with −x-polarization. The RCS is taken along φ = 0◦ cut.
faceted scatterer possessing large flat regions, we present a scattering result from an arrow geometry that possesses several sharp features in addition to large flat faces. The scatter measures
5λ × 1.5λ × .8λ at 64MHz. In this case the scattering body is discretized with a combination of
large flat patches supporting a 4th order Legendre basis on the flat sides and triangular tessellation/RWG on the corners and tips. Figure 2.15 shows the resulting patches and tessellation. The
bistatic RCS in Figure 2.16 due to an x-polarized plane wave incident along the θi = 135◦ , φi = 20◦
direction was obtained with the CFIE, α = .5, and closely matches the reference solution. The
number of unknowns is reduced from Nre f = 3549 for the CFIE-RWG solution to NGMM = 1830
using the GMM mixed Legendre/RWG solution.
Finally, we analyze the parabolic reflector in Figure 2.17, which is an open curved structure.
49

(a)

(b)

Figure 2.15: GMM discretization of a 5λ × 1.5λ × .8λ arrow (a) top view and (b) bottom view.

40

GMM
CFIE-RWG

RCS, dBsm

30
20
10
0
-10 ◦
−180 −135◦ −90◦ −45◦

0◦
θ

45◦

90◦ 135◦ 180◦

Figure 2.16: Arrow bistatic RCS at 64MHz. Wave is incident in the θi = 135◦ , φi = 20◦ direction
and is x-polarized relative to incidence direction. The RCS is taken along the φ = 0◦ cut.
50

The smoothly curved interior is modeled using 4th order smooth polynomial patches, and the
rim is captured by a piecewise flat triangular tessellation. A 5th order Legendre basis is used
on the polynomial patches, and an RWG basis is employed on the tessellated portion. The RCS
result in figure 2.18 shows scattering due to an x-polarized plane wave with incidence angles θi =
180◦ , φi = 0◦ . Again, the agreement in RCS between the reference EFIE-RWG and GMM solutions
is excellent. For the reflector, the EFIE-RWG code required Nre f = 4494 degrees of freedom, and
the GMM solution required NGMM = 1638 degrees of freedom.

;
(a)

(b)

Figure 2.17: GMM discretization of a 4λ × .35λ Parabolic Reflector (a) top view and (b) side
view.

2.5.1

Complexity and Matrix Fill Times

Although the computational costs associated with assembling and solving a GMM system are
highly dependent on the particular mixture of basis sets and patch sizes employed, the cost of the

51

40

GMM
EFIE-RWG

RCS, dBsm

30
20
10
0
-10 ◦
−180 −135◦ −90◦ −45◦

0◦
θ

45◦

90◦ 135◦ 180◦

Figure 2.18: Parabolic reflector bistatic RCS. Wave is incident in the θi = 180◦ , φi = 0◦ direction
with x-polarization. The RCS is taken along the φ = 0◦ cut.
algorithm in both complexity and storage is similar to that of extant Moment Methods. If currents
are predominantly discretized with entire patch basis functions, the cost scales as that of a mapped
higher-order Moment Method; alternatively, if sub-patch basis functions are the principle basis
type, the cost approaches that of traditional tessellation-based schemes such as RWG or rooftop
bases. The only added cost of evaluating matrix elements in GMM relative to traditional Moment
Method schemes is that of computing the partition of unity. Since the partition of unity on each
patch is non-unity only in the overlaps between patches, its evaluation adds very little to the overall
complexity.
To substantiate the qualitative scalings given in the preceding paragraph, we next provide a brief
comparison of matrix fill timings for GMM vs. a well-validated reference code that is higher order
in geometry and uses a higher order Graglia-Wilton-Peterson (GWP) [8] basis (note that the CFIERWG results generated above correspond to a first order geometry with a first order GWP basis).
We caution that drawing meaningful comparisons between codes with different implementations
is problematic due to variations in code optimizations and the large number of parameters that
may be tuned, e.g. surface order, basis function order, discretization rate, quadrature rule, etc.
Furthermore, it is difficult to directly compare low and high order codes because each solves the
problem on a fundamentally different surface with a different fidelity to the true surface, and total

52

Table 2.1: Matrix fill times for CFIE-RWG code.
Ns
576
1296
2304
3600
5184
7056
11664
14400

l2 err
Fill Time (s)
6.21e-02
6.052
2.66e-02
28.954
1.47e-02
100.164
9.30e-03
221.999
6.40e-03
457.722
4.68e-03
842.440
2.81e-03
2301.787
2.27e-03
3523.760

error in an observable is a composite of both geometrical and functional representation errors.
To ensure as fair a comparison as possible, we analyze scattering from a 1λ sphere due to an xpolarized plane wave traveling in the +ˆz direction. The sphere is selected because it is one of the
few geometries that admits comparison against an analytical solution. Both codes solve the CFIE
(α = .5). We fix the order of the basis function and geometry representation, and then find the
largest patch size that yields a solution convergent below an l2 relative error tolerance of 2.5×10−3
vs. the canonical Mie series solution in farfield data for an observation plane along the φ = 0◦
cut. All timings were run on a single 1.6GHz core of a Dell Optiplex 9010 8-core workstation
with 16GB of RAM, and solutions were obtained using the TFQMR iterative algorithm with a
convergence tolerance of 10−5 .
First, as a reference and to demonstrate typical convergence behavior of a lower order method,
we examine the time to fill the MoM impedance matrix for the CFIE-RWG case. Table 2.1 gives
convergence data and timings for a 1λ sphere at various discretization rates. To reach the desired
error tolerance, it is necessary to use Ns = 14400 degrees of freedom, with a matrix fill time of
almost 1 hour. The slow convergence is a primarily due to the poor fidelity of the low order mesh
to the true sphere.
Next we compare timings for the 1λ sphere using the higher order reference code with 3rd
order geometry and a 2nd order GWP basis compared with a corresponding GMM setup using a
3rd order polynomial local surface description and a 2nd order Legendre basis. Table 2.2 shows
the results for both GMM and the reference GWP code. The GWP result illustrates the reductions

53

Table 2.2: Matrix fill times for higher order CFIE-GWP & CFIE-GMM codes.
Code
GWP
GMM, All Leg
GMM, Mixed

Ns
l2 err Fill Time (s)
4320 1.17e-03
729.336
4488 1.95e-03
435.107
11532 4.57e-03
1790.287

in numbers of unknowns and fill time that is generally afforded by higher order Moment Methods
for a given accuracy. The first GMM result uses an all-Legendre basis set with patches sized to
give approximately the same number of degrees of freedom as the CFIE-GWP code, and exhibits
significantly faster fill time than the GWP code with comparable solution accuracy. To demonstrate
timings for a mixed GMM case, the third result in Table 2.2 is discretized with RWG on the top
half and the 2nd order Legendre basis on the bottom half. The same patch sizes, basis orders, and
integration rules as in the previous all-RWG and all-Legendre examples are utilized. The slightly
higher error in the mixed result is due to the use of the same integration rules as in the homogeneous
discretization examples; a slightly higher integration rule in the PU overlap region for the RWG
basis would be required to reach the prescribed tolerance. As expected, the fill time is between that
of the all-RWG and all-Legendre discretizations. These results demonstrate that GMM is quite
competitive in fill time with a comparable higher order code for the same solution accuracy and
support the qualitative scalings discussed above.
We do not include matrix-vector product timings or memory scalings in these tests as all matrices were fully computed without recourse to acceleration techniques such as MLFMA. Both the
matrix-vector product and storage costs therefore scale as Ns2 for all examples, and the relative
costs of different discretizations may be directly inferred from the reported numbers of degrees of
freedom.

2.6

Conclusion

This work extends the scope of the Generalized Method of Moments to include multiple geometry representations and mixed current approximation spaces on PEC scatterers. Decomposition

54

of the scatter surface via overlapping patches and the partition of unity allow local geometry descriptions capable of handling geometrical features including smooth regions, regions for which a
priori functional description is known or can be easily extracted, and regions with geometrical singularities. Furthermore, the partition of unity permits mixtures of multiple classes of Entire Patch
and/or Sub Patch basis sets within a single simulation. In particular, the introduction of subpatch
basis sets on traditional tessellations allows straightforward handling of geometrical singularities,
even in problems with smooth higher order geometry descriptions. Finally, the entire process of
geometry and basis assignment can be automated for complex structures. The resulting method
permits discretization the underlying integral operators in a manner that more closely matches the
physics and may result in significant reductions in the number of degrees of freedom required for
a given problem relative to traditional moment method solvers.
Future investigations will expand the types of basis functions and local geometry parameterizations used in GMM and apply GMM to dielectric problems. One interesting possibility is to
investigate GMM in the context of a domain decomposition method (DDM). GMM is distinct
from DDM in that it is a discretization method that stitches together different basis and geometry
descriptions to form a Moment Method system, whereas DDM is a solution method in which the
Moment Method system is solved by breaking the overall problem into subproblems and solving
each smaller problem individually, subject to global consistency constraints. The two methods
therefore address different aspects of the discretization and solution of electromagnetic integral
equations, and could naturally complement each other. Using a GMM discretization in a domain
decomposition framework would yield a method in which individual subdomains could be solved
independently as in [54] or the Equivalence Principle Algorithm in [55], but where continuous
transitions between non-conformal subdomains are provided by the partition of unity.
This chapter ©2014 reprinted, with permission, from Dault, D.L.; Nair, N.V.; Jie Li; Shanker,
B., "The Generalized Method of Moments for Electromagnetic Boundary Integral Equations," Antennas and Propagation, IEEE Transactions on , June 2014.

55

CHAPTER 3
A MIXED POTENTIAL MLFMA FOR HIGHER ORDER MOMENT METHODS WITH
APPLICATIONS TO THE GENERALIZED METHOD OF MOMENTS

3.1

Introduction

Higher order moment method discretizations are attractive alternatives to lower order moment
methods due to the potential reduction in number of degrees of freedom and higher order convergence afforded by higher order current representations. However, the practical application of
higher (and mixed) order discretizations to practical electrically large structures has been hampered
by the difficulty of accelerating higher order moment method representations using traditional acceleration methodologies. As the support of basis functions in higher order moment methods
becomes large with respect to a wavelength, the scaling of traditional implementations of the Multilevel Fast Multipole Algorithm (MLFMA) [52] rapidly degrades both in memory and in the cost
of computing MLFMA-accelerated matrix vector products.
The poor scaling of traditional MLFMA for higher order is caused by three main problems:
1. The larger support of higher order basis functions cannot be subdivided, which places restriction on leaf box size and severely limits the height of the MLFMA tree.
2. Because of the large leaf box size, the cost of filling the nearfield matrix can be as large or
greater than the solution time.
3. Large patches may not align well with large leaf level boxes, causing increased error in
MLFMA representation.
Additionally, traditional scaling analyses with respect to the number of basis functions can obscure
the true costs and scalings of MLFMA for higher order algorithms. Taken together, these issues
significantly hamper the application of MLFMA to the many higher order moment method for-

56

mulations that have been advanced over recent decades. The challenges become even more acute
when attempting to accelerate the recently developed Generalized Method of Moments [56], a new
moment method discretization scheme that permits mixtures of basis function types with supports
that may range from subwavelength up to many wavelengths.
This chapter presents an MLFMA implementation that maintains favorable MLFMA scaling and precomputation time for any mixture of basis function orders and supports; moreover,
MLFMA is used to accelerate filling of algebraic preconditioners, which are necessary for convergence in iterative solvers. We apply the method to solving large scattering problems using
the Generalized Method of Moments, although the method is equally applicable to other moment
methods using higher or lower order geometry and current representations.
Specific contributions of this chapter are:
• A mixed potential MLFMA formulation that retains optimal scaling for any mixture of basis
function types, orders, and patch sizes.
• An MLFMA-accelerated scheme for precomputing preconditioners that is significantly faster
than direct preconditioner fill, especially with large patches.
• Scaling analysis of the mixed potential MLFMA relative to extant methods.
• Results validating the scaling and accuracy of both the MLFMA and preconditioning schemes
applied to scattering problems discretized using the Generalized Method of Moments.
The remainder of this chapter is organized as follows. Section 3.2 gives preliminaries including
the electromagnetic problem statement, a short review of the dyadic form of MLFMA, and a brief
description of the Generalized Method of Moments, which serves as the test case for the scaling of
the mixed potential higher order MLFMA. Section 3.3 examines the challenges above in detail and
past approaches. Section 3.4 describes the mixed potential MLFMA formulation. Section 3.5 provides a complexity analysis of the new approach and compares it to past approaches. Section 3.6
gives an MLFMA-accelerated strategy for preconditioning higher order moment methods, and fi-

57

nally section 3.7 gives results demonstrating the scaling and efficiency of the MLFMA-accelerated
solution and preconditioning schemes.

3.2
3.2.1

Preliminaries
Electromagnetic Problem Statement

We seek to compute the scattered fields {Es (r), Hs (r)} due to a plane wave given by {kˆ i , Ei (r), Hi (r)}
that is incident on a perfectly electrically conducting scattering body residing in free space. The
scatterer boundary is defined by a two-dimensional bounding surface Ω embedded in R3 , and is
ˆ
equipped with a unit normal n(r)
defined for all points r ∈ Ω except a finite number of geometrically singular points (e.g. corners, tips, edges, etc.). The induced surface current may be obtained
via solution of the Combined Field Integral equation (CFIE):
ˆ × n(r)
ˆ × Ei (r) + (1 − α)η0 n(r)
ˆ × Hi (r) =
α n(r)
1
ˆ × T ◦ J(r) + (1 − α)η0 ( I − K ) ◦ J(r)
−α n(r)
2

(3.1)

with I the identity operator, η0 the impedance of free space, 0 ≤ α ≤ 1 a constant weighting factor,
and the operators T and K given by
ˆ × jkη0
T ◦J(r) = −n(r)

∇∇
dr I + 2 G(r, r )J(r )
k
Ω

(3.2)

ˆ × − dr ∇G(r, r ) × J(r )
K ◦J(r) = n(r)
Ω

ˆ
Here, J(r) = n(r)
× {Hi (r) + Hs (r)} is the induced surface current, G(r, r ) is the free-space
Helmholtz Green’s function, k is the propagation constant, I is the identity dyad, and the integral
in the K operator is taken in the Cauchy principal value sense. An e jωt dependence is assumed
and suppressed. To construct a moment method system, the surface current is discretized in terms
s
of a set of basis functions as J(r) = ∑N
n=1 an fn (r) and the discretized operators in (3.2) are tested

with a set of functions {fm (r)}, m ∈ [1, Ns ] in the usual fashion to obtain a matrix equation ZI = V ,
where V = [V1V2 . . .VNs ]T , with Vm = fm , nˆ × nˆ × Ei and I = [a1 , a2 , . . . , aNs ]T the vector of
unknown coefficients.
58

Although the method developed in this chapter works equally well for both the electric field
integral equation (EFIE) and the magnetic field equation (MFIE), the primary focus of the current
work is efficient acceleration of the EFIE, i.e. the T operator in (3.2); focus is therefore restricted
the case where α = 1. In the results section CFIE solutions are presented that combine both
the MFIE and EFIE. In the case where α = 1, the elements of the matrix Z (after application of
standard derivative transfers) are given by
S + IL ,
Zmn = Imn
mn

(3.3)

S and I L are
where the two components Imn
mn
S =
Imn

dr fm (r) ·
Ωm

G(r, r )fn (r )dr
Ωn

1
dr ∇ · fm (r)
G(r, r )∇ · fn (r )dr
− 2
k Ωm
Ωn
L = 1
Imn
dl nˆ γm (r) · fm (r)
G(r, r )∇ · fn (r )dr
k2 γm
Ωn
1
dr∇ · fm (r)
G(r, r )nˆ γn (r) · fn (r )dl
k 2 Ωm
γn
1
− 2
dl nˆ γm (r) · fm (r)
G(r, r )nˆ γn (r ) · fn (r )dl ,
k γm
γn

(3.4)

(3.5)

where Ωm is the support of fm , γm = ∂ Ωm is the boundary of the support of fm (r) and nˆ γm is
S term consists of surface source
a outward pointing normal to γm that is tangent to Ωm . The Imn
L term contains integrals in which at least one of the source and
and testing integrals, and the Imn

testing integrations is a contour integral. Contour integral terms in higher order moment methods
are problematic for two primary reasons. First, they may be difficult to parameterize and integrate
on higher order surfaces, and second, the term in which both the source and testing integrations are
contour integrals cannot be integrated if the contours γm and γn are coincident. The usual remedy
for for these issues is to design the basis sets {fm }, {fn } so that 1) nˆ γm,n · fm,n = 0 everywhere
on the boundary of Ωm,n and 2) any contour integrals arising at discontinuities in surface normals
interior to Ωm,n are explicitly canceled. As shown below, this places strict limitations on the
implementation of the MLFMA when patch sizes grow to be large relative to a wavelength.

59

3.2.2

The Multilevel Fast Multipole Algorithm

This section provides a brief review of MLFMA and establish the nomenclature that will be used
throughout the rest of the chapter. MLFMA relies on the partitioning of the moment method system
as:

ZI = (Z N + Z F )I = V,

(3.6)

where Z N consists of all “near” interactions, and Z F consists of all “far” interactions. Precise
definition of near and far interactions relies on a hierarchical partitioning of the problem domain
as follows. Consider a cubical domain BL that completely encloses the scattering surface Ω. This
box is termed the “root level” box. The root level box is recursively subdivided L times, with boxes
at the finest level termed the “leaf” boxes. The edge length of a single leaf box is denoted ∆x0 ,
and the edge length of the root level box BL is 2L−1 ∆x0 . Interactions between source functions
fn and testing functions fm that reside in boxes sharing a vertex are termed “nearfield”; all other
interactions are termed “farfield”. Nearfield interactions are computed directly using (3.4), where
it is assumed that the basis function is defined in such a way that the line integrals are canceled or
go to zero. Farfield interactions are computed using the MLFMA farfield representation:
Zmn =

S

ˆ · T (k,
ˆ ri j )Vni (k),
ˆ
d kˆ 2 V∗m j (k)

(3.7)

where ri , r j respectively denote the centers of the source an observer boxes, ri j = r j − ri is the
ˆ ri j ) is the translation
vector between the box centers, S is the unit sphere in k-space, and TN l (k,
operator [57], given by
Nl

ˆ ri j ) =
TN l (k,

(2)

∑ (− j)r (2l + 1)hr

(k|ri j |)Pr (kˆ · rˆ i j )

(3.8)

r=0

(2)

where hr (·) is an rth order spherical Hankel function of the second kind and Pr (·) the rth degree
Legendre polynomial. A key parameter is N l , the truncation order of the addition theorem, which
must be chosen large enough to ensure that (3.8) converges to a desired accuracy for boxes at level

60

l in the tree. The source (observer) multipoles (local expansions) are given by:
dre− jk·(r−rq ) I − kˆ kˆ · fr (r).

Vr,q =

(3.9)

Ωr

ˆ appears in the
We designate (3.7) as the “dyadic” version of MLFMA because the dyad [I − kˆ k]
equation. Due to the inherent separability of the exponential terms appearing in (3.9), multipoles
may be recursively aggregated about box centers at higher levels of the MLFMA tree. Using this
hierarchical structure, the MLFMA algorithm is divided into five steps:
1. Source to multipole (S2M) aggregation of sources into multipoles about leaf level box centers ,
2. Multipole to multipole (M2M) recursive interpolation of lower level multipoles into higher
level multipoles,
3. Multipole to local (M2L) translation using the translation operator,
4. Local to local (L2L) recursive anterpolation of higher level terms in terms of lower level
multipoles,
5. Local to observer (L2O) evaluation of potentials at observation locations due to local expansions residing at leaf box centers.
In dyadic MLFMA, the multipoles/local expansions Vr,q have two components, and each matrix
vector product requires two tree traversals.

3.2.3

Application to higher order

The challenge in using the dyadic MLFMA kernel with boxes sized smaller than a patch is illustrated by the following example (the terms “patch” and “basis function support” are used interchangeably in the following discussion). Consider a self-interaction Znn for a single higher order
basis function fn with diameter of support Diam(Ωn ) > 2∆x0 . For a given source leaf box bs , des that results from evaluation of interactions between the source
fine the partial matrix element Znn

61

Figure 3.1: Division of a large curvilinear basis function support Ωn into nearfield and farfield for
a source leaf box bl . Problematic contour integrals appear along the contour Γ between nearfield
and farfield regions.
fn localized to Ωn ∩ bs and observed by fn over its full support. Partition the element in the usual
s,F

s,N

s =Z
MLFMA manner as Znn
nn + Znn and denote by Γs the contour defined by the intersection

of the box faces delineating nearfield and farfield regions, with nˆ NF
Γs defined as the normal to Γs
pointing out of the nearfield region. Fig. 3.1 shows a schematic representation of the contour Γs induced by the split between nearfield and farfield. Summing over all source boxes with bs ∩ Ωn = 0,
/
the full matrix element is obtained as
N + ZF ,
Znn = Znn
nn

(3.10)

s,F

s,N

F =
Znn
∑ Znn .

N =
Znn
∑ Znn ,

s

s

N and Z F can be further partitioned as
Using the nomenclature of section 3.2.1, the interactions Znn
nn
S,N

L,N

N =I
Znn
nn + Inn
F
Znn
L,F

S,F
= Inn

(3.11)

L,F
+ Inn ,

L,F

where the contour terms Inn , Inn are aggregations of all line integrals along contours Γs and
L,F

L,F

Inn = −Inn

(3.12)
S,N

S,F

ˆ FF
because nˆ NF
Γs for all s. The equality (3.12) allows one to write Znn = Inn + Inn provided
Γs = −n
L,F

L,N

the terms Inn and Inn are correctly canceled.
S,F

L,F

Since both Inn and Inn are implicit in the dyadic MLFMA kernel and cannot be indepenL,N

dently extracted, the terms Inn must be explicitly computed to correctly recover Znn ; otherwise
62

S,N

S,F

L,F

the equivalent interaction is Inn + Inn + Inn = Znn . The same situation occurs for any Zmn in
which some portion of Ωm and Ωn fall within each others’ spatial nearfield. For higher order
surfaces, integrations along the contours Γs become quite difficult to parameterize and computationally burdensome to compute.
Due to these challenges, leaf boxes sizes in traditional MLFMA implementations cannot be
smaller than the size of the largest basis function supports so that each basis function is associated
with a single leaf box. For lower order bases with small supports relative to a wavelength, the basis
function size is generally smaller than the limit on leaf box size imposed by the low-frequency
breakdown of MLFMA and no splitting between nearfield and farfield occurs.

3.3

Previous Approaches

Before introducing the mixed potential MLFMA formulation, this section reviews previous approaches to accelerating higher order moment method problems with MLFMA. These fall into two
broad categories. The first type of approach maintains the large leaf box size and limited tree height
forced by large basis function supports, and uses a variety of strategies to ameliorate the various
costs in runtime and storage for different parts of the algorithm. One method [58–60] is to use
spherical harmonic expansions (SHE) to reduce storage at the leaf level. Recently, [61] adopted
this approach and introduced additional shifts at the leaf level to overcome the overlap problem.
Investigations of a similar approach was carried out in [62], which concluded that basis function
order and support were limited in higher order MLFMA implementations. In [63], an extended
tree was utilized in which the additional non-translating leaf levels were added to the MLFMA
octree to reduce storage costs and speed up the aggregation and disaggregation of multipoles at the
leaf level. While these approaches can reduce storage and computation time for aggregation and
disaggregation steps, they do not address the fundamental limitations of the high cost of nearfield
fill or limited tree height.
The second approach is the point-based method of [64], [65, 66], which treats samples of basis

63

functions at quadrature abscissa as individual radiating dipoles. MLFMA groupings and interaction
lists are defined based on the locations of these equivalent dipoles. This approach permits MLFMA
leaf box size to be much smaller than the support of a single higher order basis function, thereby
permitting the use of more levels the MLFMA tree and maintaining more favorable scaling of the
MLFMA algorithm. To avoid the problematic contour integrals that arise in dyadic MLFMA, [64]
defines nearfield matrix elements using the following subtraction scheme:

N = Z − Z FF
Zmn
mn
mn

(3.13)

FF correction term, which may be computed directly or
where Zmn is computed directly and the Zmn

using MLFMA, uses the dyadic kernel. Although the number of nearfield interactions may be less
than with the non-point-based approaches due to the reduced box size, (3.13) can still be costly to
compute since it requires 1) the full direct computation of entire patch-patch interactions for Zmn
and 2) the farfield subtraction step. As shown in the results section, for some configurations the
cost of these steps can dominate the overall cost of the algorithm at runtime, and furthermore, the
cost of the correction step can be approximately as costly as the direct evaluation of Zmn .
The algorithm presented in this work is related to the point-based MLFMA approach of [64] in
that the lowest level MLFMA groupings are indexed to discrete quadrature abscissas, and not basis
functions supports, with the result that the diameter leaf-level box size can be much smaller than
the extent of a basis function support. Instead of the dyadic Green’s function, a mixed potential
form of the EFIE is employed that does not suffer from the contour integration problem of the
dyadic Green’s function at box edges. The resulting algorithm is free of the overlap problem,
maintains MLFMA scaling regardless of basis function support size, and permits Z N to be filled
solely on the basis of a spatial nearfield criterion. To the authors’ knowledge, the only works that
have been published on mixed potential MLFMA in the context of high frequency electromagnetics
are [67], where it was applied to efficient representation using the SHE method applied to low order
discretizations, [68], where its accuracy was compared with that of the dyadic representation using
large leaf box sizes, and [69], where it was employed in a hybrid MLFMA-FFT method. None of

64

these works applied the mixed potential formulation to higher order MLFMA in the context of a
point-based method for higher order functions as is done in the present work.
An MLFMA implementation was developed for GMM in the conference work [50] using a tree
with non-uniform leaf box sizes. However, that work was based on a traditional dyadic MLFMA
implementation and suffered from the problems of large leaf box size, limited tree height, and the
overlap problem, although the latter problem was somewhat ameliorated by the non-uniform tree
employed.

3.4
3.4.1

Point-Based Mixed Potential MLFMA
Mixed potential farfield representation

As stated above, the bottleneck in accelerating the higher order moment method solution of the
EFIE using MLFMA is the appearance of uncompensated contour integrals at the boundaries between mixed potential nearfield and dyadic farfield representations. To remove this obstacle, the
mixed potential formulation (3.3)-(3.5) is used for both the MLFMA farfield interactions and the
nearfield interactions. Doing so permits the I L,N , I L,S integrals defined in the last section to be
analytically extracted and canceled so that the farfield portion of Zmn contains only the I S,F term
and the nearfield portion contains only the I S,N term. The mixed potential integral I S,F may then
be accelerated using MLFMA.
Equation (3.4) contains four independent terms that must be computed in the inner integral:
three corresponding to the vector potential, and one corresponding to the scalar potential. Each
basis function fk may therefore be identified with a four-component vector function:
˜fk = [ f 1 , f 2 , f 3 , j ∇ · fk ]T ,
k k k k

(3.14)

where [ fk1 , fk2 , fk3 ] are the x, y, z components of the basis function. Likewise, the potentials Φ(r)
and A(r) may be combined into a four-component potential vector as
S˜ = [A1 , A2 , A3 , Φ].
65

(3.15)

One may now compactly write
˜ ˜fn ) = ∑
Zmn = ˜fm , S(
l

Ωm

dr f˜ml Sl ( f˜nl )

(3.16)

where
Sl ( f˜nl ) = − jkη0

Ωn

dr G(r, r ) f˜nl (r ).

(3.17)

Equation (3.17) is nothing but the I S term defined in (3.4). Employing the MLFMA farfield representation on each of the four components, one may write
˜
dr ˜f(r) · S(r)
=
S

Ωn

˜ ∗ (k) · T (kˆ · ri j )V
˜ ni (k),
d kˆ 2 V
mj

(3.18)

dr e− jk·(r−rq ) ˜fr (r).

(3.19)

with the new multipoles defined as
˜ r,q (k) =
V
Ωr

Since the representation in the farfield is now identical to that in the nearfield and the problematic
contour integrals I L,N , I L,F analytically cancel, it is possible to arbitrarily subdivide the support of
a basis function into nearfield and farfield regions. The result is that the leaf box size may be much
smaller than the patch size, and discretizations containing any arbitrary mixture of patch sizes may
be accelerated without affecting the scaling of the method. Furthermore, as with the dyadic pointbased method, removal of the restrictions on box size permits the height of the MLFMA tree to be
chosen for optimal scaling, and not based on the diameter of the largest patch. Fig. 3.2b illustrates
these advantages.

3.4.2

Point-based algorithm

Practical evaluation of the spatial integrals in (3.16) for full patches is effected through numerical
quadrature as
Nqm Nqn

Zmn ≈ − jkη0

∑ ∑ wswt ˜fm(rt ) · G(rt , rs)˜fn(rs)

(3.20)

t=1 s=1

with ws the sth source quadrature weight, rs the location of the sth source quadrature abscissa in
R3 , and Nqn is the total number of quadrature points over Ωn , with corresponding definitions for
66

(a)

(b)

Figure 3.2: Illustration of traditional and point-based MLFMA trees. (a) Traditional: MLFMA leaf
box size (and therefore tree height) is limited by the size of the largest patch. (b) Point-based/mixed
potential: leaf box size and tree height are not limited by patch size, and may be chosen for optimal
MLFMA scaling subject to the low frequency limit of MLFMA.
the testing quadrature. Similar to the approach in [64], each source sample ws ˜fn (rs ) is interpreted
as an independent radiating dipole and each testing sample wt ˜fm (rt ) as an independent observing
dipole. These source/observer dipoles are then grouped into leaf level boxes in the usual fashion.
The source multipole for a single leaf box bq is given by:
˜ r,q (k) =
V

∑

ws e− jk·(rs −rq ) ˜fr (rs )),

(3.21)

s∈Λq

where Λq is defined as the set of quadrature points with rs ∈ bq . For a patch Ωr that is large
with respect to the leaf box size ∆x0 , dim(Λq ) << Nqr . Fig. 3.3 depicts the point-based grouping
scheme and interactions for two GMM patches, and Fig. 3.4 shows the interaction levels between
patches with various separations for traditional dyadic and point-based MLFMA. The point-based
mixed potential scheme permits MLFMA accelerated computation of (3.16) even for the case
where Ωm = Ωn , as long as the singular and near integrations are carefully handled.

3.4.3

Mixed potential point-based MLFMA nearfield

N are defined differently than in
In the mixed potential point-based MLFMA, the entries of Zmn

traditional and dyadic point-based MLFMA. If the support of ˜fn (r) is Ωn , denote as Bn the set of
leaf boxes indices such that bl ∩ Ωn = 0,
/ and assume that Bn contains Kn boxes. The set Bm for the

67

testing function ˜fm (r) with support Ωm is defined analogously. For each source leaf box bn , n ∈ Bn ,
define Θn as the set of indices m ∈ Bm of boxes with which bn experiences a nearfield interaction.
N is then given by
The nearfield element Zmn

− jkη0

∑ ∑

dr˜fm (r) ·

k∈Bn l∈ΘnΩ ∩b
m l

dr G(r, r )˜fn (r ).

(3.22)

Ωm ∩bk

For patches with diameter D > 2∆x0 , the set {Θn1 , Θn2 , . . . , ΘnK }, ni ∈ Bn , is smaller than Bn ×
n
N is only partially
Bm , sometimes considerably so if the patches are large. This implies that Zmn
F . The subtraction scheme of [64] effects
filled, with the remaining interactions accounted for in Zmn
N , but with the (often considerable) extra work of computing the full
a similar partial filling of Zmn

matrix element and then subtracting the MLFMA farfield contribution.
Evaluation of the partial integrations in (3.22) requires local self, near, and nonself source
quadrature rules, with the particular rule selected based on the proximity of a given interacting box
pair bl , bk . For instance, in the case that l = k, an appropriate singular integration rule is required.
In GMM, source integrations are defined on a subtriangulation of the patch support [56], with
the subtriangles generally smaller than the MLFMA leaf box size. Localized singular and near
integrations are therefore easily handled without modification to the GMM integration scheme.
The mixed potential MLFMA is also readily applicable to other higher order moment methods
that use similar subdivisions to compute near and self interactions [12, 15]. Schemes that use
whole-patch self rules over large patches, e.g. those in [11], do not fit immediately into the mixed
potential framework and would have to use modified quadrature schemes.

3.5

Complexity Analysis

Obtaining an accurate sense of scaling for GMM (and other higher order methods in general) is
somewhat more difficult than for lower order methods due to the potential collocation of many
higher order basis functions, as well as the relatively large spatial extent of basis function support. Furthermore, while asymptotic scaling is generally the metric of choice when discussing fast
methods, the constants that appear in front of the asymptotic scalings are often significantly larger
68

Figure 3.3: Point-based MLFMA groupings and nearfield/farfield interaction partitions for interaction between two patches. Only direct interactions between the bold red boxes contribute to
ZN .

(a)

(b)

Figure 3.4: Comparison of interactions dyadic and mixed potential MLFMA. (a) Limit on nonpoint-based dyadic MLFMA leaf box size means that interactions between large patches must
be computed directly. (b) For point-based/mixed potential MLFMA, interactions between large
patches may be split between nearfield and farfield so that favorable scaling is maintained.
for higher order methods than for lower order methods, and can have a more pronounced impact
on the relative cost of different formulations. Inspired by the point-based nature of the present
algorithm, the present scaling discussion is cast in terms of Nqtot , the total number of unique dipole
locations, and Ndip , the total number of dipoles that must be evaluated, instead of the traditional
measure in terms of the total number of basis functions Ns , which can obscure the true cost of setup
and execution time. Denoting the number of basis functions and quadrature abscissa on patch Ωr

69

as Nbr and Nqr respectively, one obtains:
Np

Nqtot

=

∑ Nqr ,

r=1
Np

Ndip =

(3.23)

∑ Nqr Nbr ,

r=1

with N p the total number of patches. For a discretization with the same number of basis functions
Nb per patch, Ndip = Nqtot Nb , i.e. Nb appears only as a scaling factor.
Using these definitions, the cost of directly evaluating the interaction between two patches Ωr
and Ωk is Nbr Nqr Nbk Nqk . (We exclude self-rules for the case r = k from this analysis since they
are restricted to a local neighborhood and scale linearly with respect to patch diameter.) When a
discretization is heterogeneous in the patch size and number of basis functions per patch, as is often
the case in GMM, determining scalings is quite complex; the following discussion assumes that all
patches are of approximately the same size and support the same numbers of basis functions (Nb )
and quadrature points (Nq ).

3.5.1

Computational complexity

Using the metric of Nqtot , the cost of filling the entire impedance matrix scales as O(Nb N p Nq )2 =
O(Nb Nqtot )2 . When MLFMA is employed using both the traditional and point-based dyadic methods, the cost of filling the nearfield matrix instead scales as C1 N p (Nb Nq )2 = C1 Nq Nb2 Nqtot , where
the constant C1 is determined by the number of nearfield interactions between patches, and the scaling proportional to Nb2 occurs because multiple basis functions must be evaluated at each quadrature point. For a dyadic method based on a traditional tree such as the SHE-based method in [58],
C1 may be quite large due to the large leaf box size and correspondingly large number of nearfield
interactions. For the point-based dyadic method, the smaller leaf box size can reduce the number of
interactions classified as nearfield, so C1 may be slightly smaller relative to the traditional method.
For the mixed potential point-based method, only interactions between quadrature points in each
others’ spatial nearfield must be evaluated, and the cost of filling the nearfield matrix scales proportional to N˜ q Nb2 Nqtot , with N˜ q the average number of quadrature points in each leaf box. For patches
70

with diameter D > 2∆x0 , N˜ q << C1 Nq for the mixed potential method relative to the point-based
dyadic method.
At runtime, the cost of the nearfield portion of a matrix vector product for the mixed potential
formulation and the dyadic point-based formulation are identical because both generate an identical
nonzero pattern for Z N . For the traditional dyadic MLFMA, the nearfield portion of the matrix
vector product is larger than the point-based method due to the higher density of Z N .
The dominant scaling of the farfield portion of a matrix vector product for both the mixed potential and dyadic point-based methods is O(C2 Ndip log2 Ndip + C3 Ndip ). Here, C2 is a constant
related to the number of tree traversals and the truncation limit of the translation operator (3.8),
and C3 is related to cost of aggregation/disaggregation of multipoles/local expansions at the leaf
level. Both the linear and logarithmic scalings are retained because in higher order discretizations where many basis functions are evaluated at each quadrature point, it is often the case that
C2 Ndip log2 Ndip <= C3 Ndip for reasonably sized octtrees. The scaling of the traditional dyadic
method degrades from Ndip log2 Ndip as patch size grows because only a few levels in the tree are
used due to the restrictions on leaf box size. Denoting by CkMP and CkDPB , k = 1, 2 the constants
for the mixed potential and dyadic point-based algorithms, one finds that C2MP ≈ 2C2DPB because
the mixed potential algorithm requires four MLFMA tree traversals instead of the two required for
dyadic MLFMA. However, as shown in the results section, C3MP < C3DPB because the mixed potential algorithm actually requires fewer flops per dipole in the aggregation/disaggregation steps.
The total time to compute a matrix vector product for higher order discretizations using the mixed
potential and dyadic point-based formulations is therefore often comparable.

3.5.2

Memory scaling

As partial matrix elements are computed for the nearfield in the mixed potential MLFMA, nearfield
storage for the point-based and mixed potential MLFMA implementations is the same and scales
as N p Nb2 , where N p is the number of patches in a discretization. Nearfield storage for traditional
approaches such as the SHE method can be higher because of the larger number of nearfield in-

71

teractions. Farfield memory requirements for the SHE method is less than that for the point-based
methods since the SHE affords both more efficient storage at the leaf level and also entails fewer
levels in the tree due to restrictions on leaf box size [58]. For the point-based and mixed potential
formulations, the farfield storage is O(C2 Ndip log Ndip ), where again C2MP = 2C2DPB .
From the scaling analysis presented here it is evident that for discretizations with large patches,
the mixed potential formulation provides speedups in precomputation time relative to both the
traditional and point-based dyadic formulations, while solution time is often comparable. As is
demonstrated in the results in section 3.7, the savings can be significant for mixed discretizations
commonly encountered in GMM, or for higher order methods when the patches are large.

3.6

Preconditioning

Practical implementation of MLFMA for the EFIE (and CFIE) requires preconditioning to speed
up convergence in iterative solvers; for example the MLFMA matrix vector product using a right
preconditioner M becomes
(Z F + Z N )M −1 MI = V.

(3.24)

For good convergence, it is necessary that the elements of M be “fully” computed in the sense that
all quadrature points corresponding to a matrix element of M must be summed over. Failure to do
so generally destroys the benefit of the preconditioner and leads to poor convergence in iterative
solvers. In traditional MLFMA, the elements of M are generally taken as subsets of Z N , since
these elements are precomputed. The mixed potential Z N only stores partial matrix elements,
so entries of Z N are not immediately suitable for preconditioning. Depending on the stencil of
the preconditioner, direct evaluation of the missing interactions between quadrature points can
substantially erase the cost gains from the partial fill of Z N . Instead, the mixed potential MLFMA
is exploited to rapidly precompute elements of the preconditioner M in the following manner,
which works for algebraic preconditioners built on any stencil, e.g. diagonal, block diagonal, ILU,
etc.

72

Higher order basis sets from a source patch Ωl and observer patch Ωk whose interactions
are included in M contribute to a submatrix M˜ lk of dimension Nbk × Nbl which may be full or
sparse depending on the selected preconditioning strategy. Each subblock is separated into spatial
nearfield and farfield contributions as
N +M
˜ F,
M˜ lk = M˜ lk
lk

(3.25)

N contains the contributions from quadrature points that are in each others’ spatial nearfield
where M˜ lk
F contains the complementary farfield
as specified by the global MLFMA interaction lists, and M˜ lk
N as in traditional MLFMA. We
interactions. For basis functions with small support, M˜ lk = M˜ lk

begin by inserting into M the relevant portions of the precomputed Z N matrix that fall onto the
selected preconditioning stencil, i.e. M˜ lk is filled up for all interacting l,k. To complete the farfield
contributions of M, for each patch Ωl define Tl , a subset of the global MLFMA tree that completely
encloses the patch. The local patch tree Tl extends only up only to the first level containing a box
that entirely circumscribes the patch. Fig. 3.5 shows patch trees for two patches that contribute to
the preconditioner and experience both near and farfield interactions. Associated with each patch
tree is a corresponding list τl of all translations in which boxes in the patch tree participate. For
patches Ωl , Ωk that contribute to M, take an intersection of the translation lists on each patch
τlk = τl ∩ τk . Using precomputed global translation operators indexed to the translations contained
in τlk , M˜ lk is completed by computing
N (:, i) + F (e ), i = 1, N l ,
M˜ lk (:, i) = M˜ lk
lk i
b

(3.26)

where ei is the ith column of the Nbl × Nbl identity matrix and the notation Flk (ei ) indicates the
MLFMA-accelerated portion of the contributions, which equivalent to the matrix vector product
F e . For patches that are included in the preconditioner but who are completely in each others’
M˜ lk
i
F , and the submatrix is filled exclusively using MLFMA, often with only
spatial farfield, M˜ lk = M˜ lk

one or a small number of translations. Fig. 3.6 depicts the manner in which fill of a preconditioner
block is split between nearfield and farfield computations.

73

Figure 3.5: Local patch trees used to compute preconditioners. Each local tree is a subset of the
global MLFMA octree. Interactions between two patches are computed using intersections of the
global interaction lists.

Figure 3.6: Graphical depiction of the computation of a single column of a preconditioner block
N consists of the precomputed nearfield interactions and F (e ) is an MLFMA product with
M˜ lk . M˜ lk
i
N with the F (e )
the ith column of the identity matrix ei . The summation of the ith column of M˜ lk
i
˜
yields the ith column of Mlk .

74

It is often sufficient (and even desirable) that the elements of a preconditioner only approximate
those of the moment method impedance matrix; however, for a robust and reliable preconditioning
strategy it is necessary that the approximation error be controllable. To verify that a preconditioner
filled using MLFMA-accelerated patch-patch interactions has controllable accuracy, the accuracy
of an overlapped block diagonal preconditioner is compared for a 2-lambda sphere filled directly
vs. using acceleration in Figs. 3.8-3.11. The preconditioner M is overlapped in the sense that,
in addition to the diagonal block, nonself blocks on the superdiagonal and subdiagonal are also
computed. The accuracy of the matrix fill is controlled by the parameter χ, which specifies the
truncation order N l in the addition theorem for the MLFMA translation operator at level l so that
N l = χkal , where al = 2(l−1) ∆x0 is the radius of the smallest sphere that completely encloses a
box at level l. For all preconditioning results, the discretization is chosen to produce an error in the
bistatic RCS of εl2 ≤ 5 × 10−3 , where εl2 is the relative l2 error, defined as
εl =
2

||E˜ − EMie ||
,
||EMie ||

(3.27)

with || · || the l2 norm, E˜ the solution radiation field, and EMie the analytical radiation field obtained
from the Mie Series. Convergence is computed as εFro = ||M˜ − MDir ||F /||MDir ||F , where || · ||F is
the Frobenius norm, M˜ is the preconditioner obtained using the mixed potential formulation with
MLFMA acceleration, and MDir is the preconditioner obtained through direct computation. All
timings in this and the results section were performed on a single core of a Dell Precision T7600
workstation operating at 1.2 GHz with 12GB of RAM.
Figs. 3.8-3.9 show convergence and computation time for computing M˜ relative to the time
to compute MDir . For this test, the sphere is discretized using N p = 65 GMM patches of diameter h = 1λ , each of which supports a p = 4 Legendre basis (Ns = 4950), for a range of ∆x0 .
Comparison of the two figures reveals that an accuracy of εFro ≈ 10−3 may be obtained using
MLFMA in approximately one half the time of a direct fill. The slower convergence for smaller
∆x0 in Fig. 3.8 is a manifestation of the low frequency breakdown of MLFMA. We note that the
convergence for small leaf boxes could be significantly improved through hybridization with low
frequency MLFMA [70] or the method of Accelerated Cartesian Expansions (ACE) [71], which
75

Figure 3.7: Stencil of overlapped block preconditioner used in preconditioning scaling tests.
Convergence vs. χ
0

∆x0 = .15λ
∆x0 = .2λ
∆x0 = .25λ
∆x0 = .375λ
∆x0 = .5λ

-1
log10 (εFro )

-2
-3
-4
-5
-6
-7
-8

1

1.5

2
χ

2.5

3

Figure 3.8: Convergence in relative (Frobenius) error of preconditioning matrix filled using
O(Nq log Nq ) MLFMA-accelerated algorithm vs. O(Nq2 ) direct fill algorithm. Geometry is a 2λ
sphere with p = 4 and h = 1λ .

are low frequency methods complementary to high frequency MLFMA. Because the mixed potential MLFMA contains four components, it maps directly onto ACE, which uses the same mixed
potential formulation.
Figs. 3.10-3.11 show convergence and timings for the same 2λ sphere with a p = 9, h = 2λ
discretization which yields N p = 28, Ns = 3080. Computation of M for larger patches clearly shows
the advantage of using the mixed potential MLFMA formulation to precompute preconditioners.
For all values ∆x0 , the computation of M˜ is faster than direct computation, and accuracies of
εFro ≈ 10−4 can be obtained with a speedup of approximately 10 times over the direct fill time.

76

Farfield Timings vs. χ
8

∆x0 = .15λ
∆x0 = .2λ
∆x0 = .25λ
∆x0 = .375λ
∆x0 = .5λ

7
TMLFMA /TDir

6
5
4
3
2
1
0

1

1.5

2
χ

2.5

3

Figure 3.9: Comparison of MLFMA-accelerated preconditioner fill times with respect to χ for a
2λ sphere with p = 4 and h = 1λ .

Convergence vs. χ
0

∆x0 = .15λ
∆x0 = .2λ
∆x0 = .25λ
∆x0 = .375λ
∆x0 = .5λ

-1
log10 (εFro )

-2
-3
-4
-5
-6
-7

1

1.5

2
χ

2.5

3

Figure 3.10: Convergence in relative (Frobenius) error of preconditioning matrix filled using
O(Nq log Nq ) MLFMA-accelerated algorithm vs. O(Nq2 ) direct fill algorithm. Geometry is a 2λ
sphere with p = 9 and h = 2λ .

77

Farfield Timings vs. χ
1

∆x0 = .15λ
∆x0 = .2λ
∆x0 = .25λ
∆x0 = .375λ
∆x0 = .5λ

TMLFMA /TDir

0.8
0.6
0.4
0.2
0

1

1.5

2
χ

2.5

3

Figure 3.11: Comparison of MLFMA-accelerated preconditioner fill times with respect to χ for a
2λ sphere with p = 9 and h = 2λ .

3.7

Results

This section presents results validating the scaling and accuracy of the mixed potential MLFMA
scheme. We also apply it to some relevant scattering geometries discretized with the Generalized Method of Moments using all-higher order discretizations and mixed high and low-order
discretizations. In all tests, the MLFMA leaf box size is fixed to ∆x0 = .25λ , the incident field
is a plane wave, and all RCS measurements are bistatic. For details on how GMM integrals are
evaluated, see [56].
The first set of results in Figs. 3.12-3.14 validates the theoretical scalings vs. the total number
of quadrature points Nqtot (in this case Ndip = Nb Nqtot , and scalings in Ndip and Nqtot are interchangeable). The scaling tests are performed for a series of square plates of side length a = 2λ to
a = 12λ residing in xy plane. Plates are chosen because they represent a worst-case scenario for
the scaling of MLFMA. The patch size is fixed at h = 1λ and a p = 4 tensor product Legendre
basis (30 basis functions/patch) is utilized. Table 3.1 lists the numbers of basis functions, patches,
and Nqtot for the series of plates.
Fig. 3.12 shows the scaling of matrix vector product timings for the series of plates using the
mixed potential MLFMA. The scaling is quite close to linear, with a slope of m = 1.018. Careful
checks were made to ensure that the primary scaling came from the farfield part of the matrix vector
78

Table 3.1: Discretization details for series of plates used in scaling demonstration.
a/λ
2
3
4
5
6
7
8
9
10
11
12

Ns
450
900
1560
2430
3540
4650
6060
7590
9450
11430
13410

Np
15
30
52
81
118
155
202
253
315
381
447

Nqtot
112140
251685
449505
708435
1026585
1392720
1823640
2311050
2860305
3468150
4118520

product. As shown below, this scaling is due to the dominant cost of the multipole aggregation
and local disaggregation steps, typical of higher order discretizations with many collocated basis
functions. Since these steps scale linearly with respect to the number of dipoles, the overall scaling
appears close to linear.
Scalings in storage for both Z N and the MLFMA tree are shown in Fig. 3.13. As expected,
the nearfield memory scales linearly, and the farfield memory scales as Nqtot log Nqtot . Note that the
jumps in the farfield memory correspond to the addition of a new level of the tree as the plate size
grows larger than (2n )∆x0 , where n is an integer. For the same number of levels in the tree the
scaling is as predicted by theory. The final scaling result for the plate geometries is shown in Fig.
3.14, which demonstrates that the fill time of the nearfield impedance matrix Z N scales linearly
with the number of quadrature points Nqtot , as expected.
The second result demonstrates that for a given geometry, because the leaf box size may be
fixed independent of the patch size or basis order, the only change in the cost of the farfield portion
of a matrix vector product for varying h and p occurs at the aggregation and disaggregation (S2M
and L2O) steps, which scale linearly in both the number of quadrature points Nqtot and the number
of basis functions per patch Nb . Fig. 3.15 shows a GMM discretization of a representative scattering geometry, a 15m VFY-218 aircraft discretized at 600 MHz using a mixture of higher order

79

log10 (T /Matvec)

Matrix Vector Product Scaling
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
3.8

MP-MLFMA
m = 1.018

4

4.2 4.4 4.6 4.8 5
log10 (Nqtot )

5.2 5.4 5.6

Figure 3.12: Scaling of MLFMA matrix vector product timings vs. Nqtot for a series of square
plates of size 2λ -12λ .

Memory Scaling

log10 (Mem(MB))

2.5

NF Mem
m = 1.06
2 FF Mem
x log2 x
1.5
1
0.5
0
-0.5
3.8

4

4.2 4.4 4.6 4.8 5
log10 (Nqtot )

5.2 5.4 5.6

Figure 3.13: Scaling in nearfield and farfield memory for a series of square plates of size 2λ -12λ .

patches and tessellations on edges/tips. The leaf box size is fixed at ∆x0 = .25λ , and h and p are
varied in tandem. Tables 3.2 and 3.3 show the timings for each step of the MLFMA matrix vector
product V F = Z F I for the mixed potential and dyadic point-based kernels, for a CFIE solution with
the coefficient α = .5. For both the dyadic and mixed potential kernels the MLFMA truncation
factor is fixed at χ = 1.4. We note that it was shown in [72] that the translation operator scalar
Green’s function can be truncated at one fewer harmonics than the dyadic translation operator;
however, because of the curl operator in the MFIE and the higher spectral content of the divergence of the basis function [69] required in the scalar potential contribution of the mixed potential

80

log10 (T )

Scaling of Z N Fill Time
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
3.8

NF Fill Time
m = 1.051

4

4.2 4.4 4.6 4.8 5
log10 (Nqtot )

5.2 5.4 5.6

Figure 3.14: Scaling of nearfield matrix fill vs. Nqtot for a series of square plates of size 2λ -12λ .

EFIE operator, the same number of harmonics is used for both.
Although the overall time to compute a mixed potential matrix vector product is slower, (as
expected) for the CFIE result, it is still quite competitive with dyadic kernel. The reason for this is
the number of floating point operations (flops) required to compute the aggregation/disaggregation
steps, which are the most costly steps in the algorithm. The innermost loop for evaluating the
dyadic multipole (3.9) at the ith sample point on the k-sphere requires computation of the terms
exp[ jki · (r − rq )]kˆ iθ · fn (r) and exp[ jki · (r − rq )]kˆ iφ · fm (r), for a total of 2 × 4 = 8 flops at each
sampling point on the k-sphere, where it assumed that the exponential term is computed in an
outer loop and therefore only introduces a single multiplication (additions are neglected). The
scalar kernel, on the other hand, does not contain dot products of the basis function with unit k
vectors, and therefore requires only 4 multiplications with the exponential term per sample point
on the k-sphere. The result is that, although the dyadic MLFMA is twice as fast above the leaf level,
the faster evaluation of multipole and local expansions in the mixed potential MLFMA renders that
matrix vector product only slightly slower.
Fig. 3.17 shows the RCS compared to a well-validated low-order RWG result for the VFY
illuminated by a zˆ -polarized plane wave traveling in the +ˆy direction. The GMM result used the
mixed potential CFIE algorithm with the p = 3 Legendre basis defined on h = .8λ patches from
Table 3.3. Fig. 3.16 shows the surface currents for the GMM result.
81

Figure 3.15: GMM discretization of VFY-218 aircraft.

Table 3.2: Matrix vector product timings for dyadic kernel for the VFY-218 aircraft, CFIE.
p
1
2
3
4
5
6

h/λ
Ndip
Ag.
0.30 2722734 6.97s
0.60 4643982 7.87s
0.80 6386674 8.53s
1.0
8340864 9.60s
1.25 10397394 10.86s
1.5 12660410 12.37s

Int.
3.08s
3.09s
3.09s
3.09s
3.09s
3.09s

Tr.
0.68s
0.68s
0.67s
0.67s
0.67s
0.67s

Ant.
Dis.
4.12s 8.65s
4.12s 10.78s
4.12s 12.41s
4.12s 14.70s
4.12s 17.44s
4.12s 20.05s

Total
23.53s
26.59s
28.87s
32.23s
36.23s
40.35s

Table 3.3: Matrix vector product timings for mixed potential kernel for the VFY-218 aircraft, CFIE.
p
1
2
3
4
5
6

h/λ
Ndip
Ag.
0.30 2722734 6.88s
0.60 4643982 7.73s
0.80 6386674 8.31s
1.0
8340864 9.32s
1.25 10397394 10.58s
1.5 12660410 12.02s

Int.
4.99s
5.01s
5.00s
5.01s
5.01s
4.99s

82

Tr.
1.21s
1.22s
1.21s
1.22s
1.22s
1.22s

Ant.
Dis.
5.90s 8.41s
5.92s 10.33s
5.92s 11.93s
5.90s 14.01s
5.90s 17.00s
5.90s 19.08s

Total
27.46s
30.28s
32.44s
35.53s
39.77s
43.27s

Figure 3.16: Surface currents for the 30λ VFY-218 aircraft.
50

MP GMM
RWG Ref

40
RCS, dBsm

30
20
10
0
-10
-20
-30

0◦

50◦ 100◦ 150◦ 200◦ 250◦ 300◦ 350◦ 400◦
φ (deg)

Figure 3.17: Bistatic RCS for VFY-218 aircraft, θ = 90◦ cut.

The next result directly compares timings and storage for the mixed potential and dyadic pointbased approaches in solving for scattering from a diameter 10λ sphere due to an xˆ -polarized plane
wave incident along the zˆ direction. A leaf box size of ∆x0 = .25λ yielded a 10 level MLFMA
tree, and χ = 1.4 was used as the MLFMA truncation parameter. Throughout the remainder of
the results section, the abbreviations “D” and “MP” respectively denote the dyadic and mixed
potential point-based kernels. Two discretizations are examined. The first divides the sphere into
1785 patches (average patch diameter h = .8λ ), each of which supports a p = 2 Legendre basis,
yielding at total of Ns = 21, 240 basis functions, and the second has 210 patches (average patch

83

50

D, disc 1
MP, disc 1
D, disc 2
MP, disc 2
Mie Series

45
RCS, dBsm

40
35
30
25
20
15
10
−180◦

−90◦

0◦
θ (deg)

90◦

180◦

Figure 3.18: Bistatic RCS for 10λ diameter sphere, φ = 0 cut. Two GMM discretizations, disc. 1
(p = 2, h = .8λ ) and disc 2. (p = 7, h = 2λ ) for both the dyadic (“D”) and mixed potential (“MP”)
are compared with the analytical Mie series result.

diameter h = 2.0λ ), each of which supports a p = 7 Legendre basis. Fig. 3.18 shows the bistatic
RCS obtained from both the dyadic and point-based methods compared to the analytical Mie series
result.
For the p = 2 discretization, storage and timings for various stages of the algorithm are given in
tables 3.4, 3.5. Both solutions converged in 80 steps of the FGMRES [73] iterative solver using a
block diagonal preconditioner, with each block corresponding to the interactions between all basis
functions on a given patch. Note that timings are only reported for for the costliest portions of the
algorithm; times for all other pre/post-computation steps such as geometry initialization, MLFMA
tree definition, RCS computation, etc. are common to both algorithms and are insignificant relative
to the steps shown.
Storage for Z N is the same since both methods use the same nearfield stencil. MLFMA storage
for the mixed potential algorithm is roughly double that of the dyadic potential since it requires
twice the number of MLFMA trees. However, fill time for Z N is considerably shorter for the mixed
potential algorithm because it only computes partial nearfield matrix elements, and furthermore,
the mixed potential approach avoids the farfield correction step, the costliest step for the dyadic
kernel. In the dyadic approach, full matrix elements of the block diagonal preconditioner are read-

84

Table 3.4: Storage for 10λ diameter sphere, p = 2, h = .8λ .
Kernel
D
MP

Z N mem.
72.85MB
72.85MB

MLFMA mem.
80.35MB
160.69MB

ily available from the nearfield matrix, but the mixed potential algorithm requires the additional
step to complete these matrix elements, incurring an additional cost of 756s. Because both algorithms converged in the same number of steps in the FGMRES solver, comparison of solution time
is meaningful. Here, the dyadic kernel has a slight advantage due to its only having two trees. The
total time to solution for the mixed potential algorithm is roughly one half that of the point-based
algorithm.
Finally, tables 3.6 and 3.7 show memory and timings for the same sphere discretized with .
Table 3.6 shows that, relative to the previous result, storage for Z N increases the same amount with
both methods due to the larger number of collocated basis functions on each patch, but MLFMA
storage stays the same since the MLFMA octree does not change. Because of the large patch size
and the necessity of directly computing complete nearfield matrix elements, the times required to
fill Z N and compute the farfield correction become very large and dominate the cost of the dyadic
point-based algorithm. Note that the farfield correction step was accelerated using the dyadic
MLFMA kernel for this discretization.
For the mixed potential point-based algorithm, the times to compute the nearfield matrix elements and farfield portion of the preconditioner grows far more modestly relative to the p = 2
result. Although solution time for the mixed potential algorithm is slightly higher than for the
dyadic algorithm, total time for the mixed potential algorithm is almost one order of magnitude
less than for the dyadic algorithm.
The final set of results, scattering from a mockup next generation navy destroyer, illustrates
the overall speedup in fill time for Z N and the block preconditioner that may be obtained using
the mixed potential MLFMA algorithm over direct evaluation and evaluation of MLFMA using
large boxes on the order of a wavelength, as in traditional MLFMA. The ship is 15λ in its longest

85

Table 3.5: Timings for 10λ diameter sphere, p = 2, h = .8λ .
Kernel
D
MP

ZN
fill
1,603s
442s

FF
Prec. Avg.
Soln.
Total
Corr. FF
Matvec
2,403s
– 14.34s 1177s 5,204s
– 756s 16.58s 1359s 2,557s

Table 3.6: Storage for 10λ diameter sphere, p = 7, h = 2λ .
Kernel
Z N mem. MLFMA mem.
D
336.18MB
80.35MB
MP
336.18MB
160.69MB
Table 3.7: Timings for 10λ diameter sphere, p = 7, h = 2λ .
Kernel
D
MP

ZN
fill
61,599s
4,129s

FF
Corr.
15,712s
–

Prec. Avg.
FF
Matvec
– 32.26s
2,248s 33.83s

Soln.

Total

2,002s
2,065s

79,313s
8,442s

dimension. A total of 2 RWG patches and 162 smooth polynomial patches (h = 1λ ) supporting
a p = 4 Legendre basis are used to discretize the problem, and the CFIE is solved with α = .5.
Note that each RWG patch supports many RWG functions. The resulting discretization, which
has a total of Ns = 25266 basis functions (NRW G = 25255, NLeg = 4860) is shown in Fig. 3.19.
Three configurations are compared: fill and solution of the full moment method system, MLFMA
with ∆x0 = 1λ , and MLFMA with ∆x0 = .25λ . The mixed potential algorithm was used for
both the ∆x0 = 1λ and ∆x0 = .25λ cases. All results converged to a relative residual error of
5.0 × 10−3 in 34 steps of the FGMRES iterative solver with a Sparse Approximate Inverse (SAI)
[74] preconditioner in the inner solver. Memory costs and timings are summarized in Table 3.8.
Direct fill time of the GMM matrix took 189,469s (52.6hr) and solution time of the full system
was 141s. For the ∆x0 = 1λ result, the fill time for Z N was 26,977s (7.49h) and solution time was
4436s. This illustrates one of the challenges of using MLFMA for GMM, in that the leaf box size
dictated by the large Legendre patches is deeply suboptimal for the RWG functions. Even without
the RWG functions, in higher order methods with large boxes much of the MLFMA cost can be

86

Figure 3.19: GMM mixed discretization for 15λ destroyer: 1st order smooth polynomial patches
(patch diameter 1λ ) and 4th order Legendre basis on smooth regions, triangular tessellation and
RWG basis on tips/corners. (Model credit: Nicholas Miller)
shifted to precomputation of the nearfield. Using the mixed potential MLFMA formulation with
a leaf box size of ∆x0 = .25λ , the fill time for Z N was 1842s (30.7min) and matrix solution time
was 440s (7.3min). The restricted spatial extent of the nearfield region for the ∆x0 = .25λ solution
also gives significant savings in memory storage for Z N relative to the ∆x0 = 1λ solution, which
contains many more nearfield interactions. Memory used for the MLFMA tree for ∆x0 = 1λ was
approximately half that used for the ∆x0 = .25λ solution. The reduced farfield storage for the
∆x0 = 1λ result is due to the smaller number of levels in the tree is smaller which means that fewer
radiation patterns and translation operators have to be stored.
Table 3.9 gives detailed matrix vector product timings for the three solution methods. The
matrix vector product for the direct approach is particularly fast since the final system size is not
large. However, as shown in table 3.8, faster solution time is far outweighed by the large setup
time. For the ∆x0 = 1λ MLFMA solution, the dominant cost (by approximately two orders of
magnitude) is in the aggregation/disaggregation steps at the leaf level. This is characteristic of
MLFMA with large boxes applied to higher order discretizations. Finally, although the aggregation/disaggregation steps are still the costliest part of the matrix vector product for the ∆x0 = .25λ
solution, they are greatly reduced relative to the ∆x0 = 1λ result due the smaller number of basis functions per box and the smaller number of sampling points on the k-sphere afforded by the
smaller box size. Even with the larger farfield storage cost, the smaller box size wins in setup
time, solution time, and storage over the ∆x0 = 1λ result; furthermore, both MLFMA results are

87

Table 3.8: Storage and timings for 15λ ship.
System type
Full Z
∆x0 = 1λ
∆x0 = .25λ

MLFMA
ZN
storage storage
9760MB
–
1921MB 18.52MB
142MB 39.33MB

Solution
Z N Precond Avg.
Matvec Time
fill FF
189,469s
–
3.7s
141s
26,977s
31s
122s
4436s
1842s
276s
12s
440s

40

Total
Time
191,688s
31,444s
2,118s

∆x0 = .25λ
∆x0 = 1λ
RWG Reference

30
RCS, dBsm

20
10
0
-10
-20
-30

0◦

45◦

90◦ 135◦ 180◦ 225◦ 270◦ 315◦ 360◦
φ

Figure 3.20: Bistatic RCS for 15λ destroyer, θ = 90◦ cut, plane wave incident on bow.
Table 3.9: Detailed Matrix Vector Product Timings for 15λ ship.
System type
ZN
Ag.
Int.
Tr.
Ant.
Dis.
Total
Full Z
3.70s
–
–
–
–
–
3.70s
∆x0 = 1λ
9.38e-1s 44.33s 2.20e-1s 2.35e-2s 2.71e-1s 76.42s 122.20s
∆x0 = .25λ 7.77e-2s 3.84s 7.31e-1s 1.17e-1s 8.08e-1s 6.48s 12.05s

significantly faster than the full matrix fill approach.

3.8

Conclusion

This chapter has detailed a mixed potential MLFMA formulation that has the advantage that the
large support of higher order basis functions may be arbitrarily subdivided down to the theoretical
limit of high frequency MLFMA. As a consequence, height of the MLFMA tree is not limited by
the patch size, as it is when traditional dyadic MLFMA is applied to higher order moment methods,
but is only limited by the low frequency breakdown of MLFMA. Furthermore, because the farfield
88

and nearfield integral formulations are the same, nearfield matrix elements can be only partially
filled based only on a spatial nearfield criterion, with the remaining farfield contributions completed
through the MLFMA tree. This leads to far better efficiency in the cost of nearfield matrix fill than
previous point-based methods. We have also leveraged the scheme to effect MLFMA-accelerated
precomputation of algebraic preconditioners. The overall method retains optimal MLFMA scaling
for any mixture of basis function support size and order from low order discretizations up through
extremely high order and asymptotic-type basis functions. Several results with mixed discretizations using the Generalized Method of Moments demonstrated the scaling and accuracy of the
method.
In this chapter, only the high frequency variant of mixed potential MLFMA was developed.
However, the method is ideally suited to hybridization with the ACE method, which will extend the
accuracy and convergence into the low-frequency regime. Also, the SHE expansions in [58] could
be used to decrease the memory cost at the lowest level of the MLFMA tree. Finally, although
this work utilized relatively simple preconditioning strategies, it can be readily applied to more
sophisticated strategies such as the spectral method in [75]. Investigations of these avenues will be
the subjects of future publications.

89

CHAPTER 4
AN INTERIOR PENALTY METHOD FOR THE GENERALIZED METHOD OF
MOMENTS

4.1

Introduction

The GMM partition of unity-based representation provides smooth transitions of basis sets from
patch to patch. However, because partitions of unity are generally rational functions, integral
equation formulations that require derivatives of the basis set can be difficult to integrate. The
quality of the GMM solution therefore depends on (i) whether derivatives of basis functions (and
therefore of partitions of unity) are necessary in the integral equation formulation, (ii) the order
of the numerical integration rules used and (iii) the metric used to measure quality. For example,
the electric field integral equation (EFIE) requires derivatives of the basis functions whereas the
magnetic field integral equation (MFIE) does not, implying that accurate nearfield solutions for the
EFIE require higher order numerical integration rules than for the MFIE. Additionally, it is well
known that metrics that rely on farfield data (for instance, radar scattering cross-section) can give
high accuracy when compared to those that use nearfield data (for instance, impedance) for the
same level of error in the solution current. To reduce the computational burden associated with
high order quadrature for GMM discretizations of the EFIE, this chapter proposes the addition
of a penalty constraint to the usual integral operators to enforce smoothness of solutions between
overlapping patches. With the addition of the penalty term, smooth, accurate solutions may be
obtained with significantly lower order numerical quadrature than with the unmodified operator.
Although constraint methods such as Lagrange Multiplier and penalty-type formulations have
been widely used in the context of finite element methods [76, 77], they have received relatively
little attention from the integral equation community. This is primarily because many integral operators of interest do not satisfy the classical saddlepoint conditions [77, 78] that form the basis for
convergence proofs traditionally used for finite elements. Recently, a few formulations that im-

90

pose solution constraints for some low-order discretizations have been advanced. Both a Lagrange
multiplier and a penalty (Nitsche-type) method were developed in [79], [80] for a provably elliptic
hypersingular integral equation associated with the Laplacian. Both of these approaches were extended without proof to three dimensional electromagnetic scattering problems in a penalty method
for the three dimensional EFIE and MFIE [81] and a mortar element approach for the EFIE [82].
Although the discretizaed EFIE operator has not been proven to be stationary for all basis
sets admissible in the GMM framework, the discretized integral operators used in this chapter
satisfy a saddlepoint condition away from resonant frequencies because the scattering surfaces
are everywhere smooth and the smooth divergence-conforming basis sets employed admit a local
Helmholtz decomposition [83, 84]. If the Moment Method solution to the EFIE is interpreted as
the Rayliegh-Ritz algorithm that correctly minimizes the associated functional in the variational
sense [85], the addition of a penalty constraint to the original functional should have the effect of
driving the minimization toward a constrained solution that possesses the desired characteristics.
This work shows the effects of the penalty formulation on through analysis of the spectrum of
the discretized operator on a circular cylinder. This analysis shows that the primary function of the
penalty method is to regularize high frequency surface current modes on the scatterer. The analysis
is far more straightforward and illuminating in two dimensions, and therefore restricted to the two
dimensional EFIE under TEz excitation. Results are furnished that show the same behavior in three
dimensions.
The content of this chapter is as follows. Section 4.2 formulates the electromagnetic scattering
problem in two and three dimensions, give a brief statement of the GMM discretization, and state
the problem addressed in this work. Section 4.3 furnishes details of the proposed penalty function
formulation, and section 4.4 provides a spectral analysis of effect of the operator on the discretized
operator. Finally, section 4.5 gives results showing convergence of solutions obtained using the
stabilized two dimensional EFIE operator, and section 4.6 shows results in three dimensions.

91

4.2

Problem Statement

We develop the penalty method in both two and three dimensions. The d dimensional model problem is that of the scattering of an incident electromagnetic plane wave due to a perfect electrically
conducting (PEC) object residing in Rd . In what follows, the notation rd ∈ Rd refers to the position vector; unless the dimensionality of the domain is not clear from the context, the subscript is
i

dropped on the position vector. The incident plane wave is given by Ei2 (r) = E0,2 e− jk ·r for d = 2
i

and Ei3 (r) = E0,3 e− jk ·r , where E0,d is a constant vector, and k = kkˆ i with k = ω/c (ω is the
angular frequency and c is the ambient wave speed), and kˆ i the direction of propagation. Throughout this chapter, an e jωt time dependence is assumed and suppressed. We denote the scattering
ˆ
surface by Ωd ⊂ Rd , and assume that an outward pointing normal n(r)
is defined ∀r ∈ Ωd . Using
the boundary conditions for tangential electric field at a PEC interface, the standard EFIE may be
written as:
Td (J(r)) = gi (r)

(4.1)

with the Td operators for d = 2 and d = 3 given by
ˆ × n(r)
ˆ × jkη0
T2 (J(r)) = n(r)
+

dr G(r, r )J(r )
Ω

jη0
∇ dr Gd (r, r )∇ · J(r ) ,
k
Ω
(4.2)

jkη0
ˆ ×
T3 (J(r)) = −n(r)
dr G(r, r )J(r )
4π Ω
+

jη0
dr ∇∇ Gd (r, r ) · J(r )
4πk Ω
ˆ × n(r)
ˆ × Eid (r).
gi (r) = n(r)

92

(4.3)

(a)

(b)

Figure 4.1: Patch overlaps γlk in (a) two and (b) three dimensions.
where it has been implicitly assumed that J accrues no contour charges on the boundary of Ω. In
(4.2), η0 is the intrinsic impedance of free space and the Green’s functions Gd (r, r ) are given by
j (2)
G2 (r, r ) = H0 (k||r − r ||),
4
e− jk||r−r ||
G3 (r, r ) =
.
4π||r − r ||

(4.4)

(2)

In (4.4), H0 (·) is the zeroth order Hankel function of the second kind and || · || denotes the
standard L2 norm.

4.2.1

The Generalized Method of Moments

This section introduces some slightly altered notation for the Generalized Method of Moments to
facilitate statement of the penalty formulation. The scattering surface Ω is subdivided into a set
of N p overlapping patches {Ωl } such that ∪l Ωl = Ω. The set of patch overlaps Ωl ∩ Ω j = 0/ is
denoted by γ = {γlk : r ∈ Ωl ∩ Ω j , l > k}. Note that the last inequality removes the redundancy
of the form γlk = γkl and also excludes the case of a patch overlapping itself. In the case where
three patches overlap the same region of Ω the three dimensional problem, the overlaps γlk are
taken pairwise. Figure Over the set Ωl , a partition of unity {ψl } is defined with the properties that
0 ≤ ψl ≤ 1, supp(ψl ) ⊂ Ωl , and ∑l ψl = 1.
Using the partition of unity, the surface current may be expanded as J(r) = ∑l ψl Jl (r), where
Jl (r), r ∈ Ωl is the portion of the global current J that is localized to patch Ωl via the action of the
partition of unity. Each local Jl (r) is expressed in terms of a set of local approximating functions

93

N

l a f (r). Discretization with this set of approximating functions and testing (4.1)
as Jl (r) = ∑n=1
ln ln

with a set of suitable similarly constructed functions yields a moment method system

∑ aln ψk fkm, T (ψl fln)

= ψk fkm , gi (r) ,

(4.5)

Λ

where k, l = 1, . . . , N p and for each k, m = 1, . . . , Nk , for each l, n = 1, . . . , Nl , and Λ is an appropriate
index set.

4.2.2

Solution errors

In GMM, ideally the solution currents and charges (reconstructed via the appropriate basis sets)
on two overlapping patches should be identical in the region of their mutual overlap, thus enabling
the recovery of the current description in all of Ω. In practice, when solving (4.5), the degree
to which this condition is realized is determined by the accuracy to which the integrations of
products of the basis functions with the partitions of unity can be evaluated. While the integrals
involving the current are well behaved and easily evaluated, those involving its derivative (i.e.
charge) are significantly more difficult to evaluate and require much higher numerical integration
rules. Insufficient numerical integration of these terms leads to solution errors in the patch overlaps.
The fundamental cause of this difficulty is easily understood by examining the divergence of the
current in (4.6), which consists of a desirable term given by the divergence of the current localized
by the partition of unity, and the second an undesirable term involving the gradient of the PU
function:
∇ · J(r) = ∑ ψl (r)∇ · Jl (r) + ∑ ∇ψl (r) · Jl (r) .
l

(4.6)

l
Desired Term

Undesirable Term

If Jl (r) = Jk (r), r ∈ γkl , then the second term vanishes by the property that ∇ ∑l ψl (r) = 0. However, if the reconstruction is not perfect, then upon summation of the currents, if the derivative of
the partition of unity is large (which may be the case), the difference Jl (r) − Jk (r) may be greatly
magnified in the solution, leading to spurious features in the solution currents. It is therefore de-

94

sirable to minimize the discrepancy in the overlaps by enforcing the condition that
Jl (r) − Jk (r) = 0.

(4.7)

In the limit that (4.7) is satisfied, any errors due to insufficient integration of the gradient of the
partition of unity are antisymmetric between the overlapping patches, and consequently precisely
cancel in the reconstructed solution.

4.3

Penalty Formulation

Given the origin of the spurious current features associated with the divergence of ψl Jl , a formulation is desirable that will minimize the discrepancy Jl (r) − Jk (r), r ∈ γkl , ∀γkl ∈ γ. To do so, recall
that according to the reactance principle [3], (4.1) may be cast as a variational problem where the
functional of interest is given by
I(J, Jt ) = Jt , gi Ω + J, g∗ Ω − Jt , T (J) Ω

(4.8)

∗ is the tangential trace of a plane wave propagating away from the scatterer,
ˆ n×E
ˆ
In (4.8), g∗ = n×

the current Jt is the solution to the adjoint equation, and ·, · Ω is an inner product over Ω. A
minimization of (4.8) may be performed via discretization with the GMM basis set and the usual
Rayleigh-Ritz procedure. To enforce the constraint (4.7), the modified functional
˜ Jt ) = Jt , gi Ω + J, g∗ Ω
I(J,

(4.9)

− Jt , T (J) Ω + β [[Jt ]], [[J]] γ
is proposed, were [[Jt ]], [[J]] are the jumps in the testing (adjoint) and source current distributions
as defined by (4.7), β ≥ 0 is a real number, and ·, · γ denotes the inner product taken over the set
of patch overlaps. The purpose of the additional term in (4.9) is to penalize discrepancies between
patch currents in the overlap regions. Upon discretization with the functions given in section 4.2.1
and application of the Rayleigh-Ritz procedure, the following formulation of the discretized EFIE

95

is obtained.

∑aln

ψk fkm , T (ψl fln ) Ω
k

Λ

−β ∑ ∑ [[fklm ]]kl , [[fkln ]]kl γkl =

(4.10)

kl mn

ψk fkm , gi Ω ,
k
where the notation [[fklα ]]kl is understood to refer to the jumps in the testing (α = m) or source
(α = n) basis sets between patches Ωk and Ωl . We shall refer to (4.10) as the “compensated” form
of the discretized EFIE, and (4.1) as the “uncompensated” form. Equation (4.10) may be written
in matrix form as:
Z˜ = (Z MoM − β Z p )I = V,

(4.11)

where Z MoM is the uncompensated moment method matrix, Z p is a (block) tridiagonal matrix
corresponding to the penalty term, I is the vector of unknown coefficients, and V is the tested
forcing function.

4.3.1

Implementation details for two dimensions

The necessary general implementation details for three dimensional GMM has been developed
in Chapters 2 and 5. All two dimensional examples in this chapter are implemented using the
following general scheme. A scattering body is subdivided into N p patches of width W , and width
of the overlap region between patches is W /10. The partition of unity on patch Ωl is given by
ψl (ξl ) =

ψˆ l (ξ )
,
ψˆ l−1 (ξ ) + ψˆ l (ξ ) + ψˆ l+1 (ξ )

(4.12)

with ξ ∈ [−1, 1] a local coordinate frame on patch Ωl . The sub-partition of unity functions are
−1

2

ˆ ) = e 1−(ξ ) . Each patch supports a basis set of Legendre polynomials of order p.
given by ψ(ξ
To accurately evaluate the required integrations in (4.10), a 3-part composite integration rule is
constructed on each patch as follows. First, each patch Ωl is split into three regions: two overlap
.
− .
−
+
I .
I
regions Ω+
l = Ωl ∩ γl,l+1 , Ωl = Ωl ∩ γl,l−1 , and the patch interior Ωl = Ωl \ (Ωl ∪ Ωl ). In Ωl ,
where the value of the partition of unity ψl is 1, an order i point rule Gauss-Legendre is defined.
96

In each of the overlap regions, a Gauss-Legendre rule of order i + 1 is defined, where the higher
order is used because of the higher order behavior of ψl in patch overlaps. Finally, the three
rules are concatenated to form an integration rule for the entire patch. We have found that rules
constructed in this manner are far superior to a single Gauss-Legendre rule over the full patch. In
the subsequent discussion, a composite rule so constructed is referred to as an “ith order rule”,
where i is indexed to the rule defined in ΩIl . For singular integrals, a lin-log rule [86] is substituted
for one of the Gauss rules in the appropriate patch region. Integrations of the penalty term of (4.10)
use only the portion of the composite rule restricted to the patch overlaps.

4.4

Spectral Analysis: Two Dimensions

This section analyzes the spectral properties of the compensated and uncompensated EFIE operators in two dimensions. The two dimensional problem admits significantly clearer analysis than the
three dimensional case, which is far less tractable. This investigation demonstrates that insufficient
integration gives rise to both low and high frequency errors in the solution current, and that the
effect of the penalty function is to suppress the high frequency errors and correct low frequency
errors by smoothing lower order current modes. To simplify the discussion and admit analytical
treatment, the present analysis is restricted to a circular cylinder of radius a.
The effect of insufficient integration is shown in Fig. 4.2, where the solution currents are
obtained using the uncompensated operator discretized with a p = 3 order Legendre basis set. The
first trace shows the current obtained using a low-order integration rule (i = 8), the second shows
the current obtained using a high-order rules (i = 30), and the third shows the analytical Mie series
solution. The i = 30 rule sufficiently integrates the partition of unity and matches very closely with
the analytical solution. It is apparent that the low order i = 8 rule introduces two types of errors:
low frequency anomalies that corrupt the solution everywhere in the patch, and high frequency
errors that are manifested as sharp features in the patch overlaps.

97

10

Comparison of Solution and Mie Currents

log10 (|J |)

5
0
−5
Full Spect, i8
Full Spect, i30
Mie

−10
−15
0

100

200
φ (deg)

300

Figure 4.2: Illustration of low and high frequency current errors induced by insufficient quadrature
for the TEz EFIE solution on cylinder of radius a = .5λ .
4.4.1

Low order analysis

We begin our analysis with a low order discretization in which each patch supports a single basis
function, so that the total current discretization is given by J(r) = ∑n ψn (φ ) fn (φ )φˆ . The exact
form of the basis function is not necessary for this analysis, but it is required that fn (φ ) be shift
invariant, so that fn (φ ) = f (φ − nφ0 ), where φ0 = 2π/Ns and nφ0 is the center of the nth patch.
If Galerkin testing is used, the resulting impedance matrix of the uncompensated EFIE operator Z MoM is circulant, implying that its eigenvectors are sampled complex exponentials of the
form vq = [1, e jqφ0 , e j2qφ0 . . . e j(Ns −1)qφ0 ]. Using this property and following [87] the eigenvalues
λqMoM , q = 0, . . . , Ns − 1 of the discretized TEz EFIE operator on a cylinder of radius a may be
obtained as
λ MoM
=
q
2πη0 ka ∞
(2)
Jq+sNs (ka)Hq+sNs (ka)|Fq+sNs |2 ,
∑
2Ns s=−∞
(2)

where Jk (·), Hk

(4.13)

(·) are the derivatives of a kth order Bessel function of the first kind and a kth

order Hankel function of the second kind, respectively, and Fk is the kth Fourier coefficient of the
product of the partition of unity and the basis function ψ(φ ) f (φ ). Because the patches are evenly
spaced and because the penalty term is only nonzero in the overlaps of neighboring patches, the

98

penalty function matrix is also circulant. Using analysis similar to that of the moment method
matrix, one can obtain a formula for the qth eigenvalue λqP of the penalty matrix:
λqP = C1 −C2 cos(pφ0 ),

(4.14)

where
∞

C1 = 4πa

|F˜p |2 ,

∑

p=−∞
∞

C2 = 4πa

∑

(4.15)
|F˜p

|2 cos(qφ

0 ),

p=−∞

with F˜p is the pth Fourier coefficient of the f(φ ) restricted to the patch overlap. It can be shown
that although C2 ≤ C1 , generally C2 ≈ C1 . The eigenvalues of the penalty matrix therefore behave
as
λqP ∼ 2C1 (sin2 (qφ0 /2)) + δq (φ0 ),

(4.16)

with |δq (φ0 )| = |(C1 −C2 ) cos(qφ0 )/C1 | << |C1 |, |C2 |. From the leading term in (4.16), it is evident that for small q, the values of λqP are small, and then grow with increasing q until they reach
their maximum at q = Ns /2. This property is key to the regularizing effect of the penalty function
on the high frequency spectrum of the solution current, as is now demonstrated.
Because both Z MoM and Z p are circulant matrices of dimension Ns , they share the same eigenvector basis (sampled complex exponentials), and the eigenvalues of the compensated system Z˜
may be obtained through direct summation as λ˜ q = λqMoM − β λqP . If the elements of V are ex˜ one obtains a spectral representation of the solution current as
panded in the eigenbasis of Z,

J(φ ) = φˆ

∞

bq =aη0

∑
l=−∞

bq e jqφ
∑ MoM − β λ P
q
q=−∞ λq
∞

F−(q−lNs ) j−(q−lNs +1) Jq−lNs (ka).

(4.17)

(4.18)

Due to the spectral behavior of λqP , for large q and sufficiently large values of β , |β λqP | >> |λqMoM |
and |β λqP | >> |bq |. The penalty term therefore regularizes high order current modes by driving the
99

higher order current coefficients toward zero, significantly damping the spurious high frequency
features in the current.

4.4.2

Higher order analysis

When a higher order basis set is used with multiple basis functions per patch, both Z MoM and Z P
become block-circulant matrices and the analysis becomes considerably more involved. Nonetheless, similar spectral results hold as in the low order case. To demonstrate this, a numerical approach is employed in which the generalized eigenvalue problem
Z˜ v˜ q = λ˜ q Z G ,

(4.19)

is solved, where Z G is the gram matrix of the basis set and {λ˜ q , v˜ q } are the qth eigenvalue and
eigenvector. The geometry is the a = .5λ cylinder discretized with N p = 15 patches, each of which
supports a p = 3 Legendre basis set.
Figure 4.3 shows a comparison of the uncompensated and compensated (β = .01) eigenvalues
with the analytical eigenvalues of the continuous operator obtained via the Mie series. Two trends
are evident. First, it is clear that both the uncompensated and compensated systems correctly
capture most of the lower order eigenvalues. Second, the effect of the penalty function for the
higher order basis set is the same as that obtained analytically for the lower order basis set, i.e. to
add a significant real part to the higher order eigenvalues. In Fig 4.3, β = .01 is shown for plotting
purposes; in practice β ≈ 10 is used, which shifts the real part of the largest eigenvalues by ≈ 104 ,
driving the higher order current coefficients to values many orders of magnitude lower than the
lower order current coefficients.
Figures 4.4 and 4.5 show the corrective action of the penalty function on the spectrum of the
current when an i = 9 quadrature rule is used. Figure 4.4 shows the complete spectrum of the
uncompensated and compensated solutions out to very high frequency. The effect of the penalty
function with increasing β is to suppress the side lobes that introduce the spurious high frequency
components into the solution. Figure 4.5 shows the “visible spectrum” of the analytical current,

100

1

λan
λM oM
˜ β = .01
λ,

0
−1
−2
−3
−4
−0.2

0

0.2

0.4

0.6

0.8

Figure 4.3: Generalized eigenvalues for an a = .5λ cylinder.

Comparison of Jan (kφ ) and J (kφ )
β=0
β = 100
β = 102
Mie

log10 (|J (kφ )|)

0
−5
−10
−15
−5000

0
kφ

5000

Figure 4.4: High frequency current spectrum of compensated EFIE in visible region for an a = .5λ
cylinder.
i.e. harmonics that fall within the limit of q

ka + 5. The effect of the penalty function within

the visible region is to correct the erroneous eigenvalue corresponding to a low-order mode by
removing high frequency content that is aliased to the low order eigenmodes, thereby smoothing
the resulting modes and correcting their corresponding eigenvalues.
While the analysis in this section has been specialized to current solutions on a circular cylinder, the results section shows that the same regularizing behavior occurs for eigenvalues of the
compensated EFIE on arbitrarily shaped scatterers discretized with nonuniformly sized patches.

101

0
−0.5
−1
−1.5
−2
−2.5
−3

−5

0

5

Figure 4.5: Current spectrum of compensated EFIE in visible region for an a = .5λ cylinder.

4.5

2D Results

This section presents results demonstrating convergence of solutions using the penalty function
formulation. For all results, the exciting field is a −ˆy-polarized plane wave traveling the in the +x
direction, and the solution current is obtained using (4.10).
The first result in Fig. 4.6 shows the current for the circular a = .5λ cylinder with N p = 15
patches, each of which supports a p = 3 Legendre basis set. All integrals are evaluated with an
i = 8 point integration rule. The solution currents and bistatic radar cross section (RCS) for two
cases, β = 0 and β = 10, are shown. As in Fig 4.2, the currents with β = 0 exhibit spurious
high and low frequency features. In contrast, the RCS is quite smooth and accurate even for
β = 0. This highlights an important point: if the target metric is farfield computation, a low order
integration rule may suffice. If, however, the target metric is surface current (or a derived quantity
such as impedance that depends on surface current), then the low order integration rule is wholly
insufficient. When the same configuration is solved with β = 10 and a low order quadrature, the
solution matches the analytical current extremely well.
Next, convergence of the current error vs. β is investigated for the a = .5λ cylinder. Figure
4.7 shows l2 relative error of the solution obtained using GMM vs. the analytical Mie Series
solution for three integration rules, i = 8, i = 10, and i = 30. The error is computed as ||JGMM −

102

(a)

(b)

(c)

(d)

Figure 4.6: Comparison of analytical and reconstructed current obtained using an 8-point integration rule. (a) ||J||, β = 0, (b) RCS, β = 0, (c) ||J||, β = 10, (d) RCS, β = 10.
JMie ||/||JMie ||. For the i = 30 rule, nothing is gained by addition of the penalty term; in fact,
the error deteriorates as β is increased. For both the i = 8 and i = 10 rules, the solution error
converges until β ≈ 102 , and then diverges. The nonmonotonic convergence can be understood
by the spectral analysis in section 4.4. Initially, as β is increased, the higher order current modes
are suppressed and the solution is perturbed toward the smooth (correct) solution. Once β reaches
≈ 102 , however, the magnitude of the penalty term is on par with that of the operator, and lower
order eigenvalues of Z˜ begin to be perturbed from their correct values.
Figure 4.8 shows pointwise absolute error ||JGMM (φ ) − JMie (φ )|| of the compensated solution
vs. the analytical Mie solution for the i = 8 rule. As β is increased, the modified operator produces
successively smoother solutions. At β ≈ 102 , the penalty term begins to dominate the EFIE opera-

103

Convergence vs. β
8-pt rule
10-pt rule
30-pt rule

log10(err)

-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6

-4

-2

0
2
log10(β )

4

6

Figure 4.7: Convergence vs β in EFIE solution currents for a = .5λ cylinder under TEz illumination.

tor, and enforces a solution that is too smooth, i.e. sacrifices global error for continuity of currents
between patches.
To further confirm the efficacy of the penalty function, the error between current expansions
in patch overlaps for neighboring patches is examined next. Figure 4.9 shows convergence of the
average absolute overlap error, defined as || ∑γ (Jl (φ ) − Jk (φ ))||/(Nsamp N p ), ∀γlk ∈ γ, where
lk
Nsamp is the total number of points used to sample the error in a single overlap and N p is the total
number of patches. In this test, Nsamp = 100 was used. Unlike the total relative l2 error shown in
Fig. 4.7, convergence of the overlap error is monotonic. This further confirms that the action of
the penalty function is to perturb the solution of the compensated EFIE variational problem toward
a result that enforces continuity in patch overlaps.
Next, it is verified that the eigenvalues of the compensated moment method system follow the
same trend for a non-canonical scatterer as for the canonical circular cylinder. To do so, scattering
is computed from the 3λ × 1.25λ teardrop-shaped scatterer shown in Fig. 4.10, demonstrates
that the penalty method produces smoothed current solutions for non-canonical structures. The
teardrop is discretized with N p = 25 patches, each of which supports a p = 5 Legendre basis set.
Figures 4.11(a),(b) show the surface current and RCS obtained using the uncompensated operator
with an i = 10, and 4.11(c),(d) shows the same results for the compensated operator with β = 10.

104

log10 (|J(φ) − Jan (φ)|)

Pointwise Absolute Current Error

−1
β
β
β
β

−2
−3

=0
= 10−1
= 101
= 103

−4
0

100

200
φ (deg)

300

Figure 4.8: Pointwise absolute error over the cylinder scattering surface.

log10(err)

Convergence vs. β
-1
Avg Olap Err
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6 -5 -4 -3 -2 -1 0 1 2 3
log10(β )

4

5

Figure 4.9: Convergence of absolution current error in patch overlaps vs. β .

The reference result is obtained using the same discretization with a high order (i = 30) rule that
accurately integrates derivatives of the partition of unity. As with the circular cylinder result, the
uncompensated operator with the i = 10 rule gives erroneous surface currents that exhibit spurious
high and low frequency features, while the compensated operator gives results that are significantly
smoother and more accurate.
Figure 4.12 shows generalized eigenvalues for the uncompensated and compensated GMM TEz
EFIE operator discretized on the teardrop shaped scatterer. As in the cylindrical case, the higher

105

Scattering Geometry
1

y

0.5
0
−0.5
−1
−1

0
x

1

Figure 4.10: GMM patches for a teardrop shaped scatterer.
Surface Currents, β = 0

RCS, β = 0

0.5

60

0

RCS (dB)

log10 (||J||)

50

−0.5

2

4

30
20
β=0
Ref

10

β=0
Ref
−1
0

40

0
0

100

6

200
φ (deg)

300

t

(a)

(b)
RCS, β = 10

Surface Currents, β = 10

60

0.5

0

RCS (dB)

log10 (||J||)

50

−0.5
β = 10
Ref
−1
0

40
30
20

0
0
2

4

β = 10
Ref

10

6

t

(c)

100

200
φ (deg)

300

(d)

Figure 4.11: Comparison of reference (i = 30) and reconstructed currents/fields obtained using
i = 10 rule for the 2λ teardrop scatterer.

106

λMoM
˜ β = 10
λ,

0

ℑ(λ)

−0.2
ℑ(λ)

−0.4
−0.6

0.05
0
−0.05
0 0.05 0.1
ℜ(λ)

−0.8
−1
−1.2
0

200

400
ℜ(λ)

600

Figure 4.12: Generalized eigenvalues for the uncompensated and compensated GMM TEz operator
discretized on the teardrop shaped scatterer. Higher order eigenvalues of the compensated operator
are shifted, while lower order eigenvalues (inset) are unaffected.

order eigenvalues of the uncompensated operator are significantly shifted by the penalty function,
while the low order eigenvalues are quite stable (inset). The shifted eigenvalues effect the same
regularization of currents for the arbitrarily shaped scatterer as for the canonical circular scatterer.
The final result, scattering from a 6λ × 2λ ellipse, demonstrates that the penalty function formulation works for larger scatterers as well as the relatively small examples treated so far. The
ellipse is discretized with 25 patches, each of which supports a p = 5 basis set. Figure 4.13 shows
the patch discretization, and Fig. 4.14 shows the solution currents and RCS for the uncompensated
and compensated operators. The solution of the compensated system shows considerably better
matching in the currents and a more accurate RCS than does the uncompensated system.

4.6

Results in Three Dimensions

The conclusion that the penalty function compensates for insufficient integration is especially important when GMM is extended to the three dimensional case. This is because, in GMM, the
transition maps between patches τkl : Ωl → Ωk that are necessary to compute the partition of unity
are nonlinear for curvilinear scattering geometries, and are therefore much harder to integrate [56].
As the results in this section show, application of the penalty function to the three dimensional
107

Scattering Geometry
2

y

1
0
−1
−2
−2

0
x

2

Figure 4.13: GMM patches for a 6λ × 2λ elliptical scatterer.
Surface Currents, β = 0

RCS, β = 0

0.5

55

0

RCS (dB)

log10 (||J||)

β=0
Ref

−0.5

50

45

β=0
Ref
−1
0

2

4

40
0

6

100

t

(a)

200
φ (deg)

300

(b)

Surface Currents, β = 1

RCS, β = 1

0.5

55

0

RCS (dB)

log10 (||J||)

β=1
Ref

−0.5

50

45

β=1
Ref
−1
0

2

4

40
0

6

t

(c)

100

200
φ (deg)

300

(d)

Figure 4.14: Comparison of reference (i = 30) and reconstructed currents and RCS obtained using
i = 10 rule for the 6λ × 2λ elliptical scatterer.
108

Figure 4.15: Convergence vs. β for a 1 λ sphere, p = 4, i = 5 case.

case yields solutions that are both smoother and more accurate than for the EFIE alone. Investigation of the effect of the penalty function is performed for scattering from a sphere of radius 1λ
discretized with N p = 43 patches. Figure 4.15 shows convergence of surface current components
vs. β compared with the analytical Mie series solution for the case where each patch supports a
p = 4 Legendre basis set. The effect of the penalty constraint is clearly to drive the solution toward
the continuous analytical solution, however, as in the two dimensional case the convergence is not
monotonic. At approximately β = 10, the solution begins to diverge as the effect of the penalty
matrix dominates that of the original moment method system. Figure 4.16 shows convergence for
the same sphere, but with each patch supporting a p = 7 basis set. The penalty function again
drives the GMM solution currents toward the analytical currents, but is able to obtain higher accuracy due to the higher order basis employed. Interestingly, the divergence of the solution happens
at a higher value of β for the p = 7 case than for the p = 4 case.
The radiation from the spurious current features is negligible, as demonstrated the behavior
of fields in the very near-field region of a scattering sphere discretized with a p = 2 Legendre
basis. Figure 4.17 shows surface currents for the compensated GMM solution with β = .1 next to
the analytical Mie series solution currents, as well as a comparison of analytical and numerically
obtained fields at a height of d = .1λ above the surface of the sphere. The numerical fields are
computed using currents obtained via solution of the EFIE with β = 0 and β = .1. It is clear
that the fields are essentially the same, although the uncompensated result does show some slight

109

Convergence in Js vs. β

log10 (l2 rel err)

0.5
err(Jθ )
err(Jφ )

0
−0.5
−1
−1.5
−2

−10

0
log 10 (β)

10

Figure 4.16: Convergence vs. β for p = 7, i = 5 case.

(a) ||J||, β = .1
1.5

β=0
β = 10−1
Mie

1

|Eθ | (V/m)

|Er | (V/m)

1.5

(b) ||J||, Mie Series

0.5

0
−200

−100

0
θ (deg)

100

1

0.5

0
−200

200

(c) |Er |

β=0
β = 10−1
Mie

−100

0
θ (deg)

100

200

(d) |Eθ

Figure 4.17: Comparison of analytical and reconstructed current obtained using 10-point integration rule.

110

Table 4.1: Iterations to convergence (τ = 10−3 ) vs. βJ in GMRES solver.
βJ
0
10−1
100
101
102

Iterations
183
18
7
10
40

variation. This variation is, however, much smaller than the error in solution currents for the
uncompensated operator (cf. Figs. 4.15,4.16). So, while the apparent errors in surface currents
may be high, fields due to these errors are extremely tightly bound to the scatterer and effects on
fields even in the extreme nearfield of the scatterer are minimal. Thus, if one is only interested in
fields, especially fields in the far zone, the uncompensated operator may suffice, while if one is
interested in current-dependent quantities such as feedpoint impedance, the addition of the penalty
term is necessary.
Not only does the regularization of the penalty function give better accuracy in surface currents, it can also significantly reduce the number of iterations required for convergence in iterative
solvers. To illustrate this, the EFIE was solved for a 1λ sphere with a high-order basis set of
p = 8 Legendre polynomials on 47 patches for a total of 4230 basis functions, illuminated by an
xˆ -polarized plane wave traveling in the +ˆz direction. Table 4.1 shows convergence in unpreconditioned GMRES (with convergence tolerance set to τ = 10−3 ) for various values of βJ . The penalty
function clearly influences the number of iterations, although, as with error convergence there is a
region of maximum benefit, in this case around β = 1.

4.7

Conclusion

The Generalized Method of Moments is a promising technique for discretizing electromagnetic
and acoustic integral equations, but some operators, particularly those containing derivatives of
the basis set, require high numerical quadrature orders. This chapter has investigated the effects
of insufficient numerical quadrature on GMM solutions, and have proposed a penalty function

111

constraint that gives smooth, accurate solutions with significantly lower order quadrature rules
than would otherwise be required. Through analytical and numerical examples, the effect of the
penalty function on the operator has been detailed. The analysis in this chapter has been restricted
to two dimensions to enable transparent analytical and numerical analysis, although the penalty
method exhibits the same regularizing behavior in three dimensions as well. As is evident from
the results presented above, the approach works for a range of problem sizes from sub-wavelength
to many wavelengths. It seems that there is some dependence of the value of β on the wavelength,
and although the development of theory to explain the relationship between β , k, and patch size
is outside the scope of this chapter, it is a topic for future work. Other topics of future work
include investigation of the interior penalty method for mixed discretizations and on large practical
scattering targets using the Multilevel Fast Multipole Algorithm developed in Chapter 3, as well as
a detailed study of the effect of the penalty term on convergence of iterative solvers.

112

CHAPTER 5
SCHWARZ-CHRISTOFFEL BASED PARTITIONS OF UNITY FOR THE
GENERALIZED METHOD OF MOMENTS

5.1

Introduction

Past implementations of GMM have relied on nonsmooth partitions of unity that possess discontinuous derivatives. On curved surfaces, such nonsmooth partitions of unity lead to complicated
piecewise continuous domains that are difficult to integrate; furthermore, the limited smoothness
precludes the use of some basis sets (e.g. certain Helmholtz decompositions). This chapter proposes a means of defining arbitrarily smooth partitions of unity in the context of GMM through the
use of Schwarz-Christoffel mapping, a method for conformal mapping between the unit disk and
an arbitrary planar polygon.
Schwarz-Christoffel mapping is a classical technique first developed in the 1800s [88] for mapping between the half plane and regions bounded by a polygonal contour in the complex plane.
Only a few Schwarz-Christoffel maps can be evaluated analytically, and it was not until the advent of robust numerical methods for evaluating Schwarz-Christoffel mappings [89] that SchwarzChristoffel maps could be widely employed. Since then, the technique has been used to solve a
range of problems in areas as varied as electrostatics [90], design of resistors with arbitrary polygonal cross-section [91], and fluid dynamics [92–94]. Recently, Schwarz-Christoffel mapping has
been employed to evaluate integrations over irregular polygonal domains [95] with applications to
crack interface problems in the Extended Finite Element Method [96].
Nonsmooth partitions of unity have been required in past GMM implementations because
GMM patches are generally defined as projections from polygonal regions in a plane onto a higher
order surface. The vertices of the polygonal domain of support are geometrically singular, which
for the purposes of this chapter is defined as having a radius of curvature of ρ = 0. It is not generally possible to directly define smooth functions on a domain with a non-smooth boundary, so

113

GMM partitions of unity have hitherto been defined in a piecewise-smooth manner such that they
exhibit discontinuous derivatives on a finite number of contours interior to the polygonal support.
Schwarz-Christoffel mapping takes a function defined on a canonical smooth domain (in this case
the unit disc) into a polygon with n vertices, and is smooth everywhere on the open disk. This property permits smooth partition of unity functions defined on the open unit disk to be taken smoothly
into the polygonal support of GMM basis sets, yielding partitions of unity with continuous, smooth
derivatives up to any desired degree. In this chapter, a Schwarz-Christoffel mapping is employed to
effect smooth partitions of unity in the Generalized Method of Moments to solve both the Electric
Field Integral Equation (EFIE) and Magnetic Field Integral Equation (MFIE) for electromagnetic
scattering from arbitrary perfectly conducting objects. Because the EFIE discretized via GMM
includes derivatives of the GMM basis set, the smooth partitions of unity are particularly useful.
The rest of the chapter is structured as follows. Section 5.2 gives a formal statement of the
problem . Section 5.3 defines the Schwarz-Christoffel map, which is then used in a new a GMM
formulation with arbitrarily smooth partitions of unity in section 5.4. Section 5.5 gives a generalized integration scheme that permits efficient evaluation of field integrals involving the smooth
local partition of unity functions. Results demonstrating and validating the method are presented
in section 2.5, and finally section 5.7 gives some concluding discussion and further directions.

5.2

Problem Statement: Nonsmooth Partitions of Unity

For reference, figure 5.1 recalls definitions and notations from Chapters 1 and 2 for the local
coordinate systems, projection polygons, and transition maps. The partitions of unity employed in
GMM are based on Shepard interpolation [49], in which the partition of unity on patch Ωl is given
by

ψl =

ψˆ l
∑k∈Λl ψˆ k

114

(5.1)

Figure 5.1: Two GMM patches and local (u1 , u2 ) coordinate systems.

with Λl an index set containing all patches that intersect Ωl , and ψˆ κ a local “sub-partition of unity”
function (sub-PU) on patch Ωκ . In previous versions of GMM (cf. Chapter 2), these sub-partitions
of unity were chosen to be linear nodal shape functions defined on a triangulation of the patch
support. These functions have the property that they are zero at all exterior nodes and unity at
all interior nodes, with linear variation from the edge to the first ring of interior nodes. Although
such functions are smooth within each triangle, they are nonsmooth between triangles, and their
derivatives are therefore discontinuous.
For the purpose of numerically integrating simplex partitions of unity, it is desirable that triangulations of overlapping patches are consistent in the sense that the image of one triangle under the
transition function between the patches coincides exactly with a triangle in the overlapping patch.
More formally, given two overlapping patches Ωl and Ωk , define a triangulation Tl in the reference
domain Pl and Tk in the reference domain Pk . Then for a triangle Tli ∈ Tl , it is desirable that the
following condition hold:
τlk ◦ Til = T jk , T jk ∈ Tk ,

(5.2)

i.e. a triangle in Tl maps exactly onto some triangle in Tk . If ϕl and ϕk are GMM-style projection
mappings, condition (5.2) can in general only be satisfied if the transition functions are affine

115

maps, e.g. they maintain straight lines and points. In GMM, the transition functions are only affine
if 1) the normals of the projection planes Pl and Pk coincide (i.e. nˆ l = nˆ k ), or if 2) the surface
defined by ϕl is itself a triangulation consisting of piecewise flat triangles. When higher order
surface parameterizations are used to define ϕ, the transition functions are in general not affine,
and τlk ◦ Til = T jk .
Non-affine transition functions in GMM are problematic because the simple piecewise linear
sub-partitions of unity exhibit complex non-smooth behavior when (5.1) is computed. To illustrate,
consider figure 5.2, which shows the image of the triangle Tl under a non-affine transition mapping
τkl . Because ψˆ k (uk ) is not smooth on the boundaries between triangles in Tk , ψˆ k (τlk ◦ Tl ) is also
non-smooth. The contours that define the non-smooth region in Pl are not linear, leading to a
function ψl (ul ) that is practically impossible to numerically integrate using standard Gauss-type
rules over the triangle Tl . The problem is particularly pronounced in the derivative of the partition
of unity, which is required when solving the EFIE. Figure 5.3 illustrates the non-smooth derivatives
that result from a nonlinear transition function.
If the transition maps are nonlinear but close to affine, then they may be approximated by affine
maps with reasonable control over error. Past implementations of GMM that employed higher
order surfaces utilized this approximation, i.e. that the higher order surface could be sufficiently
approximated by a low order surface so that the non-affine transition functions could be replaced
by approximate affine transition functions. If such a level of approximation is not permissible
given either the desired accuracy in solution or because the surface description in the vicinity of
a patch overlap is highly nonlinear, then the full nonlinear transition functions must be employed.
In this case it is not possible to utilize simple piecewise linear sub-partitions of unity due to their
nonsmooth behavior, and a means of defining smooth partitions of unity must be found. The
main contribution of this chapter is to develop globally smooth sub-partitions of unity that remain
smooth under any smooth transition mapping.

116

Figure 5.2: Image of a triangle Tl under a non-affine transition function τkl .
Complex discontinuities in
at integration subdomain
boundaries

(a)

(b)

Figure 5.3: Simplex-based PU and derivative with non-affine transition functions.

5.3

Schwarz-Christoffel Mapping

Due to the polygonal shape of the reference domain, Pl , it is not possible to directly define functions
that are smooth everywhere away from the polygon vertices. Instead, an additional mapping is
introduced for the purposes of defining smooth sub-partitions of unity ψˆ l , namely the SchwarzChristoffel map. A Schwarz-Christoffel map is a conformal mapping from a canonical domain
(e.g. the upper half plane or the unit disk) to a polygon P in the complex plane defined by set of
vertices {wk }, k = 1, . . . , n, with w = u1 + ju2 . This work utilizes the closed unit disk D¯ as the
canonical domain in the complex z, for which the Schwarz-Christoffel map w = f (z) is written as:
z n

w(z) = f (z) = C

∏

z0 k=1

1−

ζ 1−αk −1
dζ ,
zk

(5.3)

where C is a complex constant, z0 ∈ D¯ is an arbitrary starting point, and the set of prevertices {zk }
are the preimages of the vertices wk , i.e. f (zk ) = wk . The prevertices reside on the boundary of
117

the closed unit disk. Each of the terms αk in the exponents of (5.3) is defined as φk /π, with φk
the interior angle formed by the two polygon sides incident upon the vertex wk . For the polygons
that appear in GMM discretizations, 0 < αk < 2; i.e polygons with degenerate sides or vertices at
infinity do not occur. Furthermore, since GMM polygonal domains are closed,
n

∑ (1 − αk ) = 2.

(5.4)

n=1

Since the {αk } are in general not integers, the integrand of (5.3) contains branch-point singularities
at the prevertices {zk }; specialized Gauss-Jacobi quadrature rules have been developed to handle
the evaluation of (5.3) in the vicinity of these singularities [89]. For all but a few canonically
shaped polygons, a nonlinear optimization problem (the “parameter problem”) must be solved
once to extract the prevertex locations. Once the prevertices are obtained, they may be stored for
use in all subsequent evaluations of (5.3).
Also necessary for implementation in GMM is the inverse Schwarz-Christoffel map z = f −1 (w),
which is obtained by solving
1 n
z 1−αk
dz
= ∏ 1−
.
(5.5)
dw C k=1
zk
Equation (5.5) is a nonlinear ordinary differential equation. Efficient, stable numerical techniques
for evaluating (5.3), (5.5), and the parameter problem are well known [88] and implemented in the
software libraries SCPACK [89] for Fortran and the SC-Toolbox [1] in Matlab. Implementation of
the Schwarz-Christoffel map in this chapter us built upon a modified version of [89]. Figure 5.4
shows an example Schwarz-Christoffel mapping from the unit disk to a polygonal domain.

5.4

Schwarz-Christoffel GMM

The key property of the Schwarz-Christoffel for GMM is that it is conformal, and is therefore
analytic everywhere in the open unit disk D. Branch points occur at the prevertex locations {zk },
¯ and are therefore outside of the domain of GMM
but these are restricted to the boundary of D,
basis functions as long as the partition of unity and its first derivative are zero on the boundary of
¯ The branch cuts corresponding to the branch points located at {zk } are taken so that they do not
D.
118

1

3

0.8
2

0.6
0.4

1

0.2
0

0
−0.2

−1

−0.4
−2

−0.6
−0.8

−3

−1
−3

−2

−1

0

1

2

3

−1

−0.5

0

0.5

1

Figure 5.4: Schwarz-Christoffel mapping from canonical disc D to polygon P (plot generated using
Matlab SC Toolbox [1]).

Figure 5.5: Schwartz-Christoffel mapping in u1 , u2 plane permits definition of arbitrarily smooth
partitions of unity.
intersect D. Due to the analyticity of the Schwarz-Christoffel map, any smooth function defined
¯ retains its smoothness under the Schwarz-Christoffel mapping. This property
on D (instead of D)
allows the construction of a GMM based on arbitrarily smooth partitions of unity.
Smooth partitions of unity are defined using smooth sub-partitions of unity as follows. For a
patch Ωl with associated polygonal reference domain Pl , a Schwarz-Christoffel mapping is defined
˜
from the open unit disk D to Pl . Figure 5.5 illustrates the composition of maps. A function ψ(ρ),
ρ = |z| with local support and the desired degree of smoothness (up to k) derivatives is defined in
D. Any smooth function with local support may be utilized. The two functions used in this chapter
˜ ρ ) = h(ρ), with h(t) defined as
are a polynomial sub-partition of unity ψ(ρ

119

h(t) =




(1 − t β1 )β2 , 0 ≤ t < 1, β1 , β2 ∈ Z

(5.6)



0, t ≥ 1.
˜ ρ ) = g(ρ),
and a sub-partition of unity based on the exponentially decaying “bump function” ψ(ρ
defined as:

1
+1
2
1
−
(t)
, |t| ≤ 1
g(t) = e

(5.7)

In this work, (5.6) is primarily used, although (5.7) is included for completeness in convergence
tests in the results section. A plot of (5.6) with β1 = 5, β2 = 4 is shown in figure 5.6. Figure 5.7
shows the smooth sub-PU h(ρ) mapped from D to a polygonal domain P(w).
1.0
0.8
0.6
0.4
0.2
0.2

0.4

t

0.6

0.8

1.0

Figure 5.6: Polynomial sub-partition of unity function.
In GMM the approximation function spaces, geometrical parameterization, and integration

0.8
0.6
0.4
0.2
−0.4 −0.2

ˆ
ψ(z)

ˆ
ψ(w)

rules are computed with respect to the (u1 , u2 ) coordinate system. It is therefore necessary evaluate

0.2
0 0.2
0.4
u1

0
−0.2 2
u

0.8
0.6
0.4
0.2

0.5
−0.5

0

0.5
x1

0
−0.5 2
x

Figure 5.7: Smooth sub-partition of unity mapped from D(z) to projection polygon P(w) via
Schwarz-Christoffel map.

120

(a)

(b)

Figure 5.8: Smooth partition of unity ψ(u1 , u2 ), (u1 , u2 ) ∈ P constructed from smooth subpartitions of unity mapped using Schwarz-Christoffel mapping. (a) Partition of unity ψ. (b)
Derivative ∂ ψ/∂ u1 .
the partitions of unity as:
ψ(u1 , u2 ) = ψ(ρ(u1 , u2 )),

(5.8)

where ρ(u1 , u2 ) = |z(w)|, with z(w) obtained through the numerical solution of (5.5). While transition functions for the main GMM algorithm remain unchanged, augmented transition functions
for the partition of unity are given by:

τψ,lk = fl−1 ◦ τlk ◦ fk .

(5.9)

The T operator in (1.19) requires derivatives of the basis set, which must be evaluated with
respect to the u1 , u2 coordinate system. Derivatives of the partition of unity are constructed using
a quotient rule applied to (5.1), which requires derivatives of the sub-partitions given by:

 
x
∂ ψˆ
 ∂ u1 
  ˆ

 ˜ ρ  ∂ ψ(ρ)
 ∂ ψˆ  = J  y  ∂ ρ
ρ
∂ u2


(5.10a)

where J˜ is the Jacobian of the Schwarz-Christoffel map. Full evaluation of derivatives on the
scattering surface with respect to (u1 , u2 ) requires additional Jacobians of the mapping ϕl as well
121

as of the transition functions τlk between patches; these have been addressed in [56], and are
omitted here for clarity. The Jacobian J˜ of the Schwarz-Christoffel map is given explicitly by:




∂w
∂w
∂x
∂y
 ℜ{ ∂ z } ℑ{ ∂ z } 
 ∂ u1 ∂ u1 
= 1 
,
J˜ = 
(5.11)
 ∂x
∂w 2 
∂w
∂w 
∂y 
−ℑ{ } ℜ{ }
∂z
∂z
∂z
∂ u2 ∂ u2
where ∂ w/∂ z is the integrand of (5.3). The formula (5.11) may be verified using the CauchyRiemann relations [4] applied to the derivatives of (5.3). Although the derivatives ∂ w/∂ z in (5.11)
exhibit branch-type singularities and may be unbounded in modulus near irregular vertices , it
can be shown that as long as either (5.6) with β1 , β2 > 0 or (5.7) is used as the sub-partition of
unity, (5.11) is nonsingular. Since any practical implementation of the method uses β1 , β2 > 1, the
derivatives of the Schwarz-Christoffel partitions of unity are well-behaved everywhere on the open
unit disk D for all cases of interest.
Practical implementation of Schwarz-Christoffel GMM requires a precomputation step that
identifies patches with highly nonlinear transition functions and definition of the Schwarz-Christoffel
maps. A unique Schwarz-Christoffel map fl is constructed for each designated patch, including
solution of the parameter problem and storing the prevertices {zk }, the exponents {ak } and the
constant C. At run time, these parameters are recalled and passed to the appropriate SCPACK
routines to compute either the map or its inverse.

5.5

Evaluation of Field Integrals via Generalized Quadrature

The procedure outlined in the preceding section produces partitions of unity that are smooth, and
therefore integrable. However, because the partition of unity is a ratio of nonlinear functions, and
because the sub-partitions of unity are not polynomial with respect to the (u1 , u2 ) coordinate system (due to the Schwarz-Christoffel mapping), (5.1) can be costly to integrate using traditional
Gauss-Legendre integration. To remedy this problem, the present method adopts a modified approach based on [97] and [98] to precompute generalized quadrature rules on a patch-by-patch
basis.
122

Here the algorithm is briefly sketched; see [97] for more details for the 1 dimensional case.
For each triangle Til ⊂ Pl , first compute an adaptive quadrature rule for functions of the form
ψl φi (u1 , u2 ), with φi = (u1 )m (u2 )n , m = 0, . . . , M, n = 0, . . . , N and M, N are chosen so that the
elementary monomials (u1 )m (u2 )n interpolate the integrands in (1.19) of non-self integrations to
prescribed accuracy. For basis functions of non-polynomial classes, different elementary functions
may be chosen as appropriate. The adaptive rule is constructed by defining successively higher
order Gauss-Legendre rules on Til up to some maximum order Nmax , after which the triangle is
subdivided into four subtriangles and the procedure is repeated recursively on each subtriangle
until the desired convergence is obtained. Fig. 5.9(a) shows an adaptive rule defined for a patch
tessellated with 13 triangles. The adaptive rule converged to a relative error of ε = 10−5 , with
ε = (Ik − Ik−1 )/Ik−1 for the kth refinement step. The total order of the adaptive quadrature rule
over the patch is Nadapt = 2148.
To extract the generalized quadrature rule, the generalized Vandermonde matrix is constructed
from the elementary functions as
Vi j =

√
w j φi (u1j , u2j )ψ(u1j , u2j ), k = 1, 2.

(5.12)

with (u1j , u2j ) and w j the jth quadrature abscissa and weight, respectively. Because the dimension
of V is Nadapt × k, (with k = M × N), the matrix V contains at most M × N linearly independent
rows, which implies that a quadrature rule of order M × N may be constructed that yields precisely
the same result as the high order quadrature rule. In practice, this rule is extracted using a QR
decomposition [97]. It is possible to further reduce the dimension of this quadrature rule using
nonlinear optimization [97], but the savings are substantial enough using only the QR factorization
that implementation of the optimization step is unnecessary. Using the QR-based approach, the
number of quadrature points for the example in figure 5.9 is reduced from 2218 for the adaptive
rule to 117 (9/triangle) for the generalized quadrature rule with M, N = 3. For a given solution,
these quadrature rules are precomputed and stored for each triangle, and then accessed at run time.
If the EFIE is being solved it is necessary to repeat the procedure for the derivatives of the partition
of unity, i.e. rules must be computed using the Vandermonde matrices with entries
123

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1
−0.2 −0.1

0

0.1

0.2

−0.2 −0.1

(a)

0

0.1

0.2

(b)

Figure 5.9: Illustration of reduction in number of quadrature abscissa between adaptive and generalized quadrature schemes. (a) Adaptive quadrature nodes for a patch. Convergence of integrations
to relative error of 10−5 requires 2148 points. (b) Generalized PU-Chebyshev quadrature nodes
for a patch. Only 117 points are required for same accuracy.

√
V˜ikj = w j φi (u1j , u2j )

∂ ψ(u1j , u2j )
∂ uk

, k = 1, 2.

(5.13)

Forming the generalized quadrature rules on a triangle-by-triangle basis permits the use of standard triangle-based GMM integration strategies for singular self and near integrations [99, 100].
The cost of computing singular rules scales as O(N) in the number of basis functions, and is
therefore the same as the cost of computing generalized quadrature rules, so there is no advantage
to precomputing singular rules; furthermore, the storage for singular rules would be prohibitive.
Singular integrations are therefore performed at run time without recourse to generalized quadrature for source integrations, and generalized quadrature rules are used for testing integrals and for
non-self source interactions. When singularity cancellation techniques that require subdivision of
a triangle about the projection of testing point are used, the precomputed generalized testing rules
afford a significant benefit due to the large reduction in the number of testing points that must be
projected, which greatly reduces the number of singular rules that must be computed at run time.

124

5.6

Results

This section gives results demonstrating the proposed method for a variety of scattering geometries
and examining convergence of the generalized quadrature scheme relative to direct and adaptive
schemes. For all scattering results, the excitation is a unit amplitude x-polarized plane wave traveling in the +z direction.
The first set of results motivates the need for the smooth partition of unity by demonstrating
the failure of quadrature rules to correctly integrate the simplex partition of unity. Figure 5.10a
shows convergence in relative error in the integration of the partition of unity and its derivatives
(taken for the ith step as (Ii − Ii−1 )/Ii ), with Ii the integral of the partition of unity or one of its
derivatives with rule i, which is indexed to the total number of quadrature points Nq . Integrations
are performed on a single patch comprised of 10 triangles that resides on the south pole of a 1λ
sphere, and the tensor product type rules from [101] were employed. Although the integrations
initially appear to converge, as the order is increased, the integrations diverge at approximately
40-50 abscissas/triangle (i.e. a 6-7th order integration rule). The divergence occurs because of the
nonsmooth behavior of the partition of unity near the edges of the triangles; this behavior is missed
by the lower order rules, but is detected by the higher order rules with points close to the edges of
the triangles. The effect is particularly marked in the derivatives of the partition of unity, which are
completely corrupted by the discontinuities in the simplex partition of unity derivatives.
Figures 5.10b, 5.10c and 5.10d show the same test for the Schwarz-Christoffel based partition
of unity. Although convergence is slower for the smooth partition of unity and there is some
oscillation in convergence, there is overall no divergence of the integral value as the quadrature
rule is increased to the same orders as in the simplex partition of unity test. Several different
orders of polynomial sub-partitions of unity are used in this study as well as a partition of unity
based on the bump function. In practice, the use of generalized quadrature rules based on adaptive
rules greatly reduces the number of integration points required for field integrations to N points
per triangle (with N the number of basis functions supported on the triangle), so the total number
of quadrature points used in practice is significantly less than that used in figures 5.10b-5.10d.
125

Convergence vs. Quadrature order, Simplex PU

Convergence vs. Quadrature order, ψ(u, v)
2

ψ
∂ ψ/∂ u
∂
ψ/∂
v
0

0
log10 (err)

log10 (err)

-2
-5
-10

-4
-6
Bump PU
Poly PU, a1=a2=2
Poly PU, a1=a2=3
PU, a1=a2=4
-10 Poly
Poly PU, a1=a2=8
Poly PU, a1=a2=12
-12
0.5
1
1.5
2
log10 (Nq )
-8

-15
0.5

1

1.5
2
log10 (Nq )

2.5

3

(a)

3

(b)

Convergence vs. Quadrature order, ∂ ψ(u, v)/∂ u

Convergence vs. Quadrature order, ∂ ψ(u, v)/∂ v

2

2

1

1
0
log10 (err)

0
log10 (err)

2.5

-1
-2
-3

Bump PU
Poly PU, a1=a2=2
Poly PU, a1=a2=3
Poly PU, a1=a2=4
-5 Poly PU, a1=a2=8
Poly PU, a1=a2=12
-6
0.5
1
1.5
2
log10 (Nq )
-4

-1
-2
-3

Bump PU
Poly PU, a1=a2=2
Poly PU, a1=a2=3
Poly PU, a1=a2=4
-5 Poly PU, a1=a2=8
Poly PU, a1=a2=12
-6
0.5
1
1.5
2
log10 (Nq )
-4

2.5

3

(c)

2.5

3

(d)

Figure 5.10: Convergence of integrations of simplex and Schwarz-Christoffel mapped partitions
of unity vs. number of quadrature points Nq . (a) Simplex partition of unity and divergence. (b)
Schwarz-Christoffel partition of unity ψSC . (c) ∂ ψSC /∂ u1 . (d) ∂ ψSC /∂ u2 .
The first result is scattering from a .5λ sphere computed using the MFIE (α = 0.0 in (1.18)).
The sphere is discretized with .3λ spherical sector patches (so that the geometry is a perfect
sphere), each of which supports a p = 2 Legendre basis. The RCS shown in figure 5.11 is taken in
the φ = 0◦ cut. The RCS obtained using Schwarz-Christoffel GMM exhibits good matching with
the analytical Mie series result. Generalized quadrature is used for this result; the original adaptive
rule required 291654 points, while the total number of generalized quadrature points is 16272, a
reduction of 94.42%.
Figure 5.12 shows the bistatic RCS for a 1λ sphere computed using the EFIE (α = 1.0 in
(1.18)), and figure 5.13 shows a comparison of the analytical and numerically obtained surface
currents. In this case, the sphere was discretized using 306 patches and a p = 2 Legendre basis.

126

10

SC-GMM
Mie Series

8

RCS, dBsm

6
4
2
0
-2
-4
-6
-8
−180◦−135◦ −90◦ −45◦

0◦
θ

45◦

90◦ 135◦

Figure 5.11: Bistatic RCS (φ = 0◦ cut) for a .5λ sphere discretized with smooth SC-GMM basis,
MFIE solution.
Instead of utilizing generalized quadrature rules, this solution instead employs high-order GaussLegendre tensor product rules (256 points/triangle) and the constraint scheme developed in chapter
4 with a penalty factor β = 1 to smooth currents. Matching is again quite good between analytical
and the numerical result in both bistatic RCS (Fig. 5.12) and surface currents (Fig. 5.13). Total l2

RCS, dBsm

relative error in the currents is εrel = 5.39 × 10−2 , with εrel = ||JGMM − JMie ||/||JMie ||.
SC-GMM
Mie Series

30
20
10
0
-10
-20
-30
-40
−150◦−100◦−50◦ 0◦
θ

50◦ 100◦ 150◦

Figure 5.12: Bistatic RCS (φ = 0◦ ) for a 1λ sphere, smooth SC-GMM basis, EFIE solution.

The final scattering result is scattering from 2λ × 1λ × .5λ PEC ellipsoid, which contains
regions of higher curvature on the sides and the ends. The solution was obtained using the MFIE,
and the GMM discretization of the ellipsoid contains 147 3rd order polynomial patches, each of

127

(a)

(b)

Figure 5.13: Surface current density (||J||) for a 1λ sphere illuminated by an x-polarized plane
wave travelling in the +z direction. (a) Mie series solution. (b) SCGMM solution using the EFIE.
which supports a p = 2 Legendre basis. Two partition of unity configurations are used. The first
uses the Schwarz-Christoffel mapping everywhere on the scatterer, with 17,307 generalized points
required to accurately integrate the partition of unity to six significant digits, versus 305,613 points
for the adaptive rule. The second configuration uses a mixed partition of unity, with simplex
partitions of unity utilized on the flatter portions and Schwarz-Christoffel partitions of unity on the
curved sides and ends. Fig. 5.14 shows the locations of GMM patches with the different partition
of unity types for the mixed result. The mixed result uses a standard 12-point Gauss-Legendre
rule on patches that only overlapped simplex patches, and generalized rules on all other patches.
With a 43,488 standard + 6,435 generalized = 49923 total quadrature points, the mixed result uses
approximately three times the number of points for the all Schwarz-Christoffel partition of unity.
However, the time to precompute quadrature rules for the mixed scheme was 247s vs. 749s for the
all Schwarz-Christoffel discretization.
Fig. 5.15 shows the bistatic RCS for both GMM partition of unity configurations and an RWG
reference result. The matching is quite good, and it is evident that the mixed discretization introduces no additional error into the solution.

128

Figure 5.14: Mixed Schwarz-Christoffel/simplex partition of unity scheme for the 2λ ellipsoid.
20

SC-GMM
Mixed PU
RWG Ref.

RCS, dBsm

15
10
5
0
-5
-10 ◦
−180 −135◦ −90◦ −45◦

0◦
θ

45◦

90◦ 135◦

Figure 5.15: Bistatic RCS (φ = 0◦ ) for the 2λ ellipsoid, MFIE solution.

5.7

Discussion and Conclusions

When GMM transition maps are highly non-linear, and in particular, far from affine, integrals of
piecewise smooth partitions of unity are extremely difficult to evaluate using standard numerical
quadrature. This chapter has proposed the use of Schwarz-Christoffel maps to define partitions of
unity with any desired degree of smoothness. Results show that the method is capable of accurately
solving both the MFIE and the EFIE on PEC scattering targets; however, the method is quite
general and can be used for any integral equation formulation that admits a GMM discretization.
The cost of evaluating the smooth partitions of unity is necessarily somewhat higher than the

129

cost of the simplex partition of unity due to evaluation of the Schwarz-Christoffel mapping. This
suggests that the simplex-based partition of unity should be used when possible, and the SchwarzChristoffel partition of unity only where the curvature of the surface is high. Although these new
partitions of unity are smooth and therefore theoretically integrable using numerical quadrature, in
practice their rational form and nonlinearity require very high order quadrature rules for accurate
integration. To reduce this cost, generalized quadrature rules were employed that intrinsically
integrate the partition of unity to desired precision. These rules may themselves be relatively
costly to compute since they rely on an initial adaptive quadrature rule that can be quite dense.
However, the computation of the generalized rules scales linearly with the number of patches,
and therefore is asymptotically far less costly than using the dense quadrature rules to compute
field elements, which scales quadratically in both the number of patches and the number of basis
functions. If a mixed Schwarz-Christoffel/simplex partition of unity approach is adopted, the setup
cost is further reduced. It is possible that the cost of precomputing the rules could be significantly
reduced through hierarchical rule aggregation; this is a topic of future research.

130

CHAPTER 6
SUBDIVISION SURFACES FOR THE GENERALIZED METHOD OF MOMENTS

6.1

Introduction

The topic of this chapter is the formulation of the Generalized Method of Moments on subdivision surfaces. Subdivision surfaces are widely used in the computer graphics community to
construct globally smooth, easily deformable and intrinsically refinable surfaces, e.g. for character animation (Pixar’s Toy Story 2, Cars, etc.). Several characteristics of subdivision surfaces are
of interest in constructing boundary integral equation discretizations. First, subdivision surfaces
provide an efficient means for constructing higher order surfaces directly from low order meshes
without the need for additional interpolation points; furthermore, the resulting surface is globally
smooth to second order everywhere but a small number of vertices. Subdivision surfaces do not
suffer from problems with watertightness, can easily handle topologically complex geometries,
can be easily deformed, and can also handle sharp edges and tips where desired. Because of these
properties, subdivision surfaces hold promise for electromagnetics in optimization problems, uncertainty quantification, analysis of deformable scatters, etc. The subdivision approach has been
used in thin-shell deformation problems [102] and a boundary element method for elastics [103],
but not for electromagnetic or acoustic boundary integral equations. One reason for this is that
subdivision surfaces present some unique challenges in constructing well-behaved basis sets near
so-called “irregular” vertices that arise in subdivision parameterizations of arbitrary meshes. This
chapter solves these problems in the context of the GMM and gives a formulation for a boundary
integral equation discretization on subdivision surfaces that can be readily hybridized with more
traditional moment method discretizations. Furthermore, the continuity properties of subdivision
surfaces paired with carefully chosen local parameterizations and basis sets near irregular vertices
yield a GMM discretization that is significantly easier to integrate than for arbitrary surface descrip-

131

tions. Subdivision-based GMM is also free from the problem of noncongruent overlaps between
integration domains discussed in Chapter 5 and admits the use of a relatively simple partition of
unity. Specific contributions of this chapter are:
• Definition of a Generalized Method of Moments parameterization of the subdivision surface
tangent space.
• Development of subdivision vector basis functions that are well behaved and suitable for representing physical currents everywhere over the subdivision surface, including in the vicinity
of irregular vertices.
• A simplified, efficient partition of unity based on first order box splines and corresponding
integration scheme.
• A prescription for merging of GMM patches on subdivision surfaces.
The remainder of the chapter is structured as follows. Section 6.2 defines the Loop subdivision
surface [104] utilized in this work. Section 6.3 develops a local parameterization of subdivision
surfaces suitable for GMM, and Sections 6.4 and 6.5 respectively detail the construction of basis
functions and partitions of unity in subdivision GMM. Finally, 6.6 gives results and 6.7 gives
discussion and concluding remarks.

6.2

Loop Subdivision Surfaces

Subdivision surfaces are defined through recursive refinement of an initial first order “primal”
mesh, denoted in this work by M 0 . The primal mesh contains a collection of vertices {v0i },
i = 1, . . . , Nv0 as well as an associated collection of faces {Tl0 }, l = 1, . . . , N 0f . The initial mesh M 0
is refined into the M 1 mesh by application of a subdivision rule, which adds additional vertices
and faces and repositions the original set of vertices to create new sets {v1i }, i = 1, . . . , Nv1 and
{Tl1 }, l = 1, . . . , N 1f . This process may be applied recursively k times, and in the limit that k → ∞, a

132

smooth “limit surface” is obtained. The properties of the limit surface depend on the type of initial
mesh and the choice of refinement rules.
This work focuses on the Loop subdivision scheme [104], which is constructed upon primal
meshes with triangular faces; other element types such as quadrilaterals may also be treated [105].
Surfaces generated through Loop subdivision fall under the class of “approximating” subdivision
surfaces. Unlike “interpolating” subdivision schemes, approximating subdivision surfaces do not
exactly interpolate their original generating mesh [102]. However, approximating schemes such as
Loop, Catmull-Clark, and Doo-Sabin [106] are C2 almost everywhere while interpolating subdivision schemes are at best C1 [107], [108]. For certain discretizations of electromagnetic boundary
integral equations, the higher degree of continuity is of interest; furthermore approximating subdivision surfaces can be fitted to a desired sampled surface [109, 110]. Loop subdivision is chosen
for GMM because it yields surfaces with C2 continuity and, since the primal mesh elements are
triangles, the subdivision may be directly applied to existing low order meshes without the need
for remeshing or the insertion of additional interpolation nodes as is required in Lagrange-type
interpolation.
Some additional concepts are necessary to precisely describe the Loop subdivision process.
Each vertex v0i in M 0 possesses a neighborhood called a “one-ring”, defined as the set of vertices
with which v0i shares an edge. A key characteristic of each vertex is its valence N, which is defined
as the number of vertices in its one-ring. The valence plays a prominent role in the choice of
subdivision rules in the neighborhood of a vertex. Practical meshes contain a mixture of “regular”
vertices (N = 6), and a small number of “irregular” vertices (N = 6).
In the Loop scheme, subdivision of a level k mesh to a level k + 1 mesh about a vertex vk0
proceeds by inserting one new vertex on each edge incident on v0 and recomputing the vertex
positions, with the new level k vertex positions given by
vk+1
= (3vk0 + vki−1 + 3vki + vki+1 )/8
i

(6.1)

vk+1
= (1 − Nw)vk0 + wvk1 + · · · + wvk1
0

(6.2)

for the new vertices, and

133

for vertex vk0 , where w = (10 − (3/2 + cos(2π/N))2 )/(16N). This choice of rules produces a limit
surface that is C2 everywhere but at irregular vertices, where it is C1 [104]. Figure 6.1 shows a
coarse sphere primal mesh, two successive subdivisions, and the limit surface (k → ∞).

(a)

(b)

(c)

(d)

Figure 6.1: Subdivision of a spherical mesh. (a) Primal mesh, (b) k = 1, (c) k = 2, (d) limit surface
(k = ∞).

6.2.1

Evaluation of the Limit Surface and Derivatives

Since the computation of any point on the limit surface is accomplished through an infinite number
of subdivisions, it is clearly not feasible to evaluate the limit surface using the subdivision procedure. Fortunately, it has been established that the Loop subdivision limit surface may be exactly
parameterized about regular vertices by a set of smooth quartic box spline functions [104], and may
therefore be directly evaluated without recourse to recursive subdivision. Near irregular vertices,

Figure 6.2: Neighborhood of a triangle Tl . Vertex v1 has a valence of N, vertices v2 , vN+1 have
valence six.

134

a modified procedure is used. The following discussion elucidates the evaluation procedures for
both regular and irregular vertices, and identifies that challenge of defining well-behaved tangent
spaces near irregular vertices that is relevant to the discretization of boundary integral equations.
For a triangle T 0 ∈ M 0 with K = N + 6 vertices in its neighborhood, the Loop subdivision
limit surface can be evaluated as:
S(ξ 1 , ξ 2 ) =

K

∑ bi(ξ 1, ξ 2)vi

(6.3)

i=1

where bi (ξ 1 , ξ 2 ) are a set of smooth functions and vi , i = 1, . . . , K are the K vertices in the neighborhood V0 of T 0 . Figure 6.2 shows the neighborhood and indexing scheme for an arbitrary
triangle Tl with two regular vertices and one irregular vertex of valence N. Since the subdivision
process preserves local mesh topology, the irregular vertex will be incident upon N triangles at any
level of subdivision refinement. For conciseness, (6.3) may be written in the following notation:
S(ξ 1 , ξ 2 ) = (bK (ξ 1 , ξ 2 ))T V0

(6.4)

with bK (ξ 1 , ξ 2 ) = [b1 (ξ 1 , ξ 2 ), b2 (ξ 1 , ξ 2 ), . . . , bK (ξ 1 , ξ 2 )]T . In the case that K = 12 (i.e. all
vertices in V0 are regular), the functions bi (ξ 1 , ξ 2 ) are box splines [104]. For irregular vertices,
the functions bK are not analytically available, and a modified subdivision procedure is adopted.
Subdivision of the mesh M 0 once into the mesh M 1 for the triangle T 0 can be expressed as
the action of a matrix A¯ upon V0 , the neighborhood of T 0 in the primal mesh as
¯ 0,
V1 = AV

(6.5)

where the rows of the matrix A¯ contain appropriately shifted versions of the Loop subdivision rules
(6.1) and (6.2). The result of this operation is a new set of L vertices, subsets of which define level
k = 1 neighborhoods of the 4 new subtriangles Tl1 , l = 1, . . . , 4 within to the original triangle T 0 .
These subsets are obtained by the application of “picking matrices” Pl that select only vertices in
the neighborhood of Tl1 , so that V1l , the level k = 1 neighborhood of triangle Tl1 , is given by:
¯ 0,
V1l = Pl AV
135

(6.6)

Figure 6.3: Recursive subdivision procedure about an irregular vertex v00 . Each subdivision yields
three regular triangles and one irregular triangle.
and the limit surface in each subtriangle is given by:
S(ξ 1 , ξ 2 )| 1 =(bK ( fl1 (ξ 1 , ξ 2 )))T Pl V1
T
l

(6.7)

¯ 0
=(bK ( fl1 (ξ 1 , ξ 2 )))T Pl AV
where fl1 (ξ 1 , ξ 2 ) transforms from barycentric coordinates on the parent triangle to barycentric
coordinates on Tl1 . For a triangle with one irregular vertex (it is assumed that the mesh has been
subdivided at least once to ensure that each triangle has at most one irregular vertex), three of the
subtriangles T21 , T31 , T41 have regular neighborhoods and may therefore be evaluated in closed form
using box splines; only T11 possesses a neighborhood with an irregular vertex. T11 may again be
subdivided, yielding four more subtriangles, three of which have regular neighborhoods and one
of which retains the irregular valence. Since only T11 has to be subdivided, it is useful to define the
¯ Then the limit surface within T 1 may be evaluated as:
K × K matrix A = P1 A.
1


(bK ( fl2 (ξ 1 , ξ 2 )))T A2 V0 ,
l=1
1
2
S(ξ , ξ )| 2 =
T

l
(b ( f 2 (ξ 1 , ξ 2 )))T P AAV
¯ 0 , l = 2, 3, 4,
12 l
l

(6.8)

Since the limit surface cannot be evaluated in closed form for T11 , the process is again repeated,
yielding three more regular triangles Tl2 , l = 2, 3, 4, and on irregular triangle T12 . Fig. 6.3 illustrates the first two levels of the subdivision process near an irregular vertex v00 . Generalizing to k
subdivisions, the limit surface in the vicinity of the irregular vertex may be parameterized as:


(bK ( f k (ξ 1 , ξ 2 )))T Ak V0 ,
l=1
l
1
2
S(ξ , ξ )| k =
(6.9)
T

k−1 V0 , l = 2, 3, 4
l
(b ( f k (ξ 1 , ξ 2 )))T P AA
¯
12 l
l
136

To reach the irregular vertex, it is necessary to take Ak , k → ∞, which is not computationally
feasible. Stam [111] solved this problem by turning to eigenanalysis, where he defined:
A = CΛC−1 ,

(6.10)

with Λ = diag(λ1 , λ2 , . . . , λK ) a diagonal matrix containing the eigenvalues of A, and C = [c1 , c2 , . . . , cK ]
a matrix whose columns ci form a basis for the eigenspace of A. Explicit formulas for the eigenvalues of A have been derived [111]. Using (6.10), (6.9) may be rewritten as

S(ξ 1 , ξ 2 )| k
T
l


K


k 1 2 T
k −1 0


 ∑ (bK ( fl (ξ , ξ ))) ci λi C V ,

l=1

k 1 2 T ¯ λ k−1C−1 V0 ,


i i
 ∑ (b12 ( fl (ξ , ξ ))) Pl Ac

l = 2, 3, 4

i=1
=

 12

(6.11)

i=1

which may be evaluated much more rapidly since it involves only powers of a diagonal matrix. To
ˆ = C−1 V0 . Since V
ˆ = [C−1 v1 ,C−1 v2 , . . . ,C−1 vK ]T has the form
further simplify 6.11, define V
of a triangle neighborhood, it is designated the “eigen-neighborhood” of T 0 , and the formula for
ˆ is:
the limit surface in terms of V

K


k 1 2 T
k −1


 ∑ (bK ( fl (ξ , ξ ))) ci λi C vˆ i ,
i=1
S(ξ 1 , ξ 2 )| k =
T
12

l

k 1 2 T ¯ λ k−1 vˆ ,


i
i i
 ∑ (b12 ( fl (ξ , ξ ))) Pl Ac

l=1
(6.12)
l = 2, 3, 4

i=1

Final simplifications may be found by examining the properties of the eigenvectors and eigenvalues
of the matrix A. It has been shown [112] that if the eigenvalues are ordered so that |λ1 | > |λ2 | ≥
· · · ≥ |λK |, λ1 = 1. Indeed, if this were not the case, (6.12) shows that in the limit k → ∞ the value
of S at the irregular vertex would be zero. Additionally, it can be shown that |λ2 | = |λ3 | [111],
and the first eigenvector corresponding to λ1 is v1 = [1, 1, . . . , 1]T [112]. Because of this and the
fact that the box splines b12 (ξ 1 , ξ )2 form a partition of unity, vˆ 1 may be extracted from the sum
in (6.12) to give

137

S(ξ 1 , ξ 2 )| k
T
l

=


K



ˆ
v
+
(bK ( flk (ξ 1 , ξ 2 )))T ci λikC−1 vˆ i ,

1
∑


l=1



k 1 2 T ¯ λ k−1 vˆ ,


i i
i
vˆ 1 + ∑ (b12 ( fl (ξ , ξ ))) Pl Ac

l = 2, 3, 4

i=2
12

(6.13)

i=2

Due to the condition that λ1 = 1 is the largest eigenvalue, it is clear that the limit surface converges
to vˆ 1 at the irregular vertex.
Definition of a basis set on S that is suitable for discretization of electromagnetic boundary
integral equations requires parameterizations of the local tangent vectors in the neighborhood of
the irregular vertex. The partial derivatives of (6.13) yield expressions for the tangent vectors:
S(ξ 1 , ξ 2 )ξ j | k =
T
∂

l
k
1
2
12
fl (ξ , ξ )
¯ i λ k−1 vˆ i ,
(b12 ( flk (ξ 1 , ξ 2 )))T Pl Ac
i
∂ξ j
i=2

∑

(6.14)
l = 2, 3, 4, j = 1, 2

where the first sum in (6.13) has been omitted as it cannot be analytically evaluated except at the
irregular vertex. The transformation flk is given by:




(2k ξ 1 − 1, 2k ξ 2 ),
l=2




flk (ξ 1 , ξ 2 ) = (1 − 2k ξ 1 , 1 − 2k ξ 2 ), l = 3






(2k ξ 1 , 2k ξ 2 − 1),
l = 4.

(6.15)

Taken together with (6.14), (6.15) shows that the leading term of Sξ scales as 2(2λ2 )(k−1) . Therej

fore, if λ2 > 1/2, which is the case for triangles with neighborhoods of size K > 12 [111], the
magnitudes of the non-normalized tangent vectors tends to infinity in the limit k → ∞, i.e. as the
irregular vertex is approached. Likewise, if λ2 < 1/2, as it is for neighborhoods with K < 12,
the magnitudes of the non-normalized tangent vectors tend to zero. Fig. 6.4 shows the divergent
behaveior of the subdivision Jacobian J = ||∂ S/∂ ξ 1 × ∂ S/∂ ξ 2 || of the subdivision surface about
a valence N = 15 vertex.
The only stable case is K = 12, for which λ2 = 1/2, which in the k → ∞ limit gives (2λ2 )k−1 =
1. The divergent behavior of the tangent vectors in the neighborhood of irregular vertices makes
138

them unsuitable for representation of physical vector current quantities. Normalization would lead
to better behavior of the tangent vectors themselves, but yields divergent higher order derivatives
of S, which are required for evaluation of divergences of basis functions. It is therefore necessary
to reparameterize the subdivision surface in the neighborhood of the irregular vertex in a way that
exactly compensates for the behavior of flk . Fortunately the GMM discretization, which is built
upon local coordinate charts defined on subsets of the mesh, permits the seamless inclusion of such
a reparameterization.

Figure 6.4: Divergent behavior of subdivision surface partial derivatives about an irregular vertex
of valence N = 15. The (u1 , u2 ) coordinate system for this case is defined via direct projection into
a common plane that shares the average normal of all triangles incident on the irregular vertex.

6.3

GMM Patches and Local Parameterizations

Definition of a GMM discretization on the subdivision surface first requires the division of the
surface into overlapping subdomains Ωk , termed patches, and appropriate parameterizations of the
surface and tangent space within each patch. For a subdivision surface, an initial primitive patch
Ωl is identified with a single vertex v0l on the primal mesh, such that the number of patches is Nv0 .
The domain of the patch Ωl is taken as the union of the set of primal faces, denoted Tl , incident
on v0l . The solution may be constructed in terms of these primitive patches directly, or they may
be merged into larger patches as discussed below.

139

Although subdivision surfaces admit a natural parameterization of the surface Ωl in terms of
the barycentric (ξ 1 , ξ 2 ) coordinates on each face, in general this parameterization is only higher
order smooth within a face, and exhibits only C0 smoothness between faces [113] (note that the
surface itself parameterized in terms of barycentric coordinates is C2 almost everywhere). The only
case in which parameterization in terms of (ξ 1 , ξ 2 ) is higher order continuous between triangles is
the case where all triangles are equilateral, which occurs only when all triangles are coplanar and
the vertex v0l has valence six. For regular vertices, such a parameterization can be constructed by a
piecewise linear mapping of the triangles into a regular hexagon in the (u1 , u2 ) plane via the maps
ϕr : Tr → Γl , with Γl the regular hexagon with all vertices placed on the unit circle.
For vertices with valence N = 6, a linear piecewise mapping is not sufficient for higher order
continuity between triangles in the unit N-gon. Several possible maps can be used, for example
the conformal maps defined in [114], which are by definition C∞ . However, for reasons that will
become apparent below, this work uses the so-called characteristic map parameterization.
The characteristic map F : R2 → R2 , (ξ 1 , ξ 2 ) → (u1 , u2 ) is defined by performing subdivision
on an “eigen-neighborhood” of the irregular vertex. The coordinates of eigen-neighborhood vertices are given by v˜ 2 , v˜ 3 , the subdominant eigenvectors of the subdivision matrix BN about a vertex
of valence N. The matrix BN , which is related to but distinct from the matrix A above, is formed
by tabulating the Loop subdivision rules (6.1), (6.2) for a vertex v0l and all triangles incident on
its one- and two-rings (collectively referred to as the “two-neighborhood” of the vertex) such that
the application of BN upon v0l and its two-neighborhood yields the vertex v1l with a topologically
similar two-neighborhood at the k = 1 subdivision level. Details of construction and analytical
expressions for the eigenvalues of BN are given in [115].
Characteristic maps have been a primary analysis tool for subdivision surfaces and play a key
role in smoothness proofs for subdivision schemes about irregular vertices [112, 115, 116]. They
have been utilized for local reparameterizations leading to curvature continuous surfaces in [117,
118] for Catmull Clark and [113] for Loop subdivision schemes. For all cases of interest to this
work, it has been shown that the Loop subdivision characteristic map is C2 everywhere but at the

140

Figure 6.5: Eigen-neighborhood for a vertex of valence nine. The characteristic map is constructed
upon subdivision of this neighborhood.
irregular vertex, where it is C1 [115]. Figure 6.5 shows the (normalized) eigen-neighborhood for
a vertex of valence nine, where the (u1 , u2 ) coordinates are defined by [˜v2 , v˜ 3 ], the subdominant
eigenvector pair for the B9 subdivision matrix. The inner ring of the eigen-neighborhood for a
vertex of valence N is always a regular N-gon.
Since the characteristic map is defined via a Loop subdivision procedure, its limiting values
can be obtained by the same evaluation procedure outlined in section 6.2.1 as
[u1 , u2 ] = bTK (ξ 1 , ξ 2 )[˜v2 , v˜ 3 ];

(6.16)

specifically, the same recursive eigen-based approach illustrated in Fig. 6.3 is used to compute the
limit position in the (u1 , u2 ) plane. Figs. 6.6-6.8 show primal triangulations, subtriangulations,
and images of the subtriangulations under the characteristic map for valences five, six, and fifteen.
For valence six the characteristic map is the regular hexagon.
The key property of the characteristic map for this work is that it is defined using the same
subdivision procedure as for the surface. This means that the partial derivatives of the characteristic
map with respect to (ξ 1 , ξ 2 ) contain the same dependence on flk (equation (6.15)) as do derivatives
of the subdivision surface S. We exploit this property below to construct well-behaved basis sets
near irregular vertices.
The local coordinate chart for each GMM patch primitive Ωl is constructed using the character141

(a)

(b)

(c)

Figure 6.6: Characteristic map for a vertex of valence N = 5. (a) Triangulation of an N-gon, (b) a
subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ).

(a)

(b)

(c)

Figure 6.7: Characteristic map for a vertex of valence N = 6. (a) Triangulation of an N-gon, (b) a
subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ).
istic map Fl of the appropriate valence that maps from the subdivision surface into the local (u1 , u2 )
coordinate domain. Figure 6.9 shows the local coordinate charts defined by the characteristic maps
for overlapping patches of valence five and six. The global subdivision surface is completely described by the atlas (see Chapter 1) made up of the union of the patches Ωl , l = 1 . . . Nv0 and their
corresponding local coordinate charts defined via the characteristic maps.
The final consideration in constructing GMM parameterizations is the continuity and behavior
of the transition functions τlk between local coordinate charts Fl (Ωl ), Fk (Ωk ), Ωl ∩ Ωk = 0.
/ For
characteristic parameterizations on Loop subdivision surfaces, the transition functions are defined

142

(a)

(b)

(c)

Figure 6.8: Characteristic map for a vertex of valence N = 15. (a) Triangulation of an N-gon, (b)
a subtriangulation T of the N-gon, (c) a piecewise linear representation of F(T ).
.
as τlk = Fl ◦Fk−1 , with Fq the characteristic map associated with vertex vq . Since both Fl and Fq are
C2 (C1 at an irregular vertex) and invertible, the transition function τlk is also C2 (C1 at an irregular
vertex). The continuity of the characteristic map and transition functions between triangles permits
the definition of partitions of unity and basis sets with support larger than a single triangle that have
well-behaved (bounded) derivatives, a key characteristic for discretization electromagnetic integral
equations.
Evaluation of Fk−1 , the inverse of the characteristic map, is practically quite difficult. To avoid
this difficulty, practical implementations of GMM on subdivision surfaces are expressed entirely
in terms of the barycentric coordinates (ξ 1 , ξ 2 ), so that Fk−1 need never be explicitly evaluated.
Another feature of particular importance for GMM is that, given a triangle Tr ∈ Tl ∩ Tk , the images Fl (Tr ), Fk (Tr ) map precisely to a single “wedge” in their respective local (u1 , u2 ) coordinate
systems regardless of the valence of the two patches. GMM discretizations on subdivision surfaces therefore completely avoid the noncongruent overlap problem identified in Chapter 5 and the
associated integration issues.

143

Figure 6.9: Coordinate charts for overlapping irregular (valence 5) and regular GMM patches
induced by the characteristic maps F(ξ 1 , ξ 2 ).
6.3.1

Merging of Regular Patches

Defining an individual patch about every vertex in the primal mesh can lead to over-discretization
for large, dense meshes. Furthermore, many GMM basis sets, such as plane wave-type functions,
are designed for patches with large support. To employ such bases, and to reduce the number of
unknowns, it is desirable to merge the initial primitive patches into larger patches in the manner
of Chapter 2. For regular vertices, the regularity of the characteristic map yields transition functions that are isometric, i.e. images of equilateral triangles under the transition map between two
regular patches is identity up to a rotation. It is therefore possible to merge the local coordinate
systems of overlapping regular patches into larger regions with common support. Patches defined
about irregular vertices, however, cannot in general be merged with neighboring patches since the
transition functions are not isometric. Specifically, merging can only be performed under one of
two conditions. Either 1) both patches are regular, or 2) parameterization of a merged irregular
patch with a regular patch is performed using the characteristic map of the irregular patch, with
appropriate expansion of the irregular patch parameterization domain to include the regular patch.
In this work, only the first case is considered as it is by far the most common. The second is of
less interest because merging is limited to only the two ring of the irregular vertex and therefore
does not afford much benefit in terms of larger patch support. Irregular patches in the current
formulation are never merged.

144

Given a group of overlapping regular patches, the merging algorithm proceeds as follows. First,
the patch associated with the vertex nearest the geometrical center of the grouping is selected as
the reference vertex. The origin of the corresponding (u1 , u2 ) local coordinate system is fixed as
the merged patch origin. The transition functions between the remaining overlapping patches are
then recursively traversed, starting with the nearest neighbors of the reference patch. The local
coordinate system of each successive patch is assigned both a translational and a rotational offset
with respect to the coordinate system of the reference patch. Using the translational offsets, each
interior node on the merged patch is assigned a location in (u1 , u2 ) space relative to the origin,
and its coordinate system is rotated appropriately to align with the reference coordinate system.
Since all patches are regular and all primal coordinate domains are regular hexagons, the resulting
parameterization domain is a regular hexagonal lattice. The function defined by these translations
and rotations in (u1 , u2 ) space yields a local coordinate chart, designated F˜l , for the merged patch
Ωl .
Figure 6.10 shows the geometry and local parameterization domain of a merged patch. In this
case the reference origin is taken as vertex nine; after merging the local origin may be redefined
arbitrarily, e.g. as the geometric center of the (u1 , u2 ) parameterization domain.

(a)

(b)

Figure 6.10: Primal mesh (a) and local coordinate domain (b) for a merged regular patch.
Care must be taken when merging regular patches in the vicinity of irregular vertices. Although

145

the regular and irregular patches are never merged, if several regular patches adjacent to the irregular vertex are allowed to merge, self-intersections of the local coordinate system may result. This
is specifically a problem for irregular vertices with valence greater than six. Since all triangles in
the merged regular patch domain are equilateral and therefore have interior angles of π/3 if the
number of regular patches about an irregular vertex is allowed to be more than five, the coordinate
system wraps back upon itself and the map F˜l becomes non-unique. To avoid this complication,
no more than five regular primitive patches are allowed to merge into a single patch around any
irregular vertex.

6.4

Basis Functions

Basis sets for GMM are defined on the tangent space of the surface with respect to the local (u1 , u2 )
coordinate system. Expressed purely in terms of local coordinates, a basis function defined on
patch Ωl may be written
f = f1 (u1 , u2 )

∂S
∂S
+ f2 (u1 , u2 ) 2 .
1
∂u
∂u

(6.17)

However, because integrations are carried out with respect to the local (ξ 1 , ξ 2 ) coordinates, it is
more convenient to express the basis sets as
f(ξ 1 , ξ 2 ) = f1 (Fl (ξ 1 , ξ 2 ))

∂S
1 , ξ 2 )) ∂ S ,
+
f
(F
(ξ
2
l
∂ξ1
∂ξ2

(6.18)

with Fl (ξ 1 , ξ 2 ) the characteristic map associated with patch Ωl . This expression is also more
efficient to compute since the partial derivatives of the subdivision surface are expressed directly
in terms of (ξ 1 , ξ 2 ). The regularity of the partial derivatives ∂ S/∂ ξ i about regular vertices means
that basis sets defined on regular patches may be chosen with the usual freedom permitted by the
GMM scheme. As shown in section 6.2.1, the partial derivatives of the surface about irregular
vertices become singular in the limit that the irregular vertex is approached. Representation of
physical currents in the neighborhood of irregular vertices therefore requires carefully designed
basis sets that are smooth and non-singular regardless of valence.

146

Based on the conclusions of section 6.2.1, the partial derivatives of the subdivision surface S
behave as O((2λ )k ), with λ the subdominant eigenvalue of the subdivision matrix and k ∝ log2 (ρ),
ρ the radial distance from the irregular vertex. A well-behaved basis set therefore requires that the
coefficient functions f1 , f2 in 6.18 behave as O(2λ )−k as the irregular vertex is approached. To
effect this behavior, the basis set around the irregular vertex is defined in terms of the surface
gradient of a scalar function
f = ∇φ (u1 , u2 ).

(6.19)

Recalling the explicit form of the surface gradient from Chapter 1, (6.19) expressed in terms of the
ξ 1 , ξ 2 coordinate system is
2

2

∑∑

i=1 j=1

∂ φ (F(ξ
gi j

1 , ξ 2 ))

∂ξi

∂S
,
∂ξ j

(6.20)

where gi j is the (i, j)th component of the inverse of the metric tensor g. Components of g−1 are
proportional to (Det(g))−1/2 = J −1 = ||∂ S/∂ ξ 1 × ∂ S/∂ ξ 2 ||−1 ∼ O((2λ )−2k . Because the characteristic map uses the same subdivision scheme as the surface, ∂ φ /∂ ξ i = (∂ F/∂ ξ i )φ (F(ξ 1 , ξ 2 )) ∼
O((2λ )k ). The overall scaling of each term in ∇φ is therefore O(1) for patches of any valence.
A complete basis set is defined by including, in addition to the curl-free function ∇φ , the
divergence-free function
fi = nˆ × ∇φi .

(6.21)

The combined basis set forms a local surface Helmholtz decomposition on the patch. In this work,
a pth order basis is constructed using the scalar functions
φmn (u1 , u2 ) = Pn (u1 )Pm (u2 ), 0 < m + n ≤ p + 1

(6.22)

with Pq (·) a qth degree Legendre polynomial, although any suitably well behaved set of scalar
functions may be employed. Figs. 6.11 and 6.12 show a set of p = 2 basis functions and their
divergences for a valence 5 and a valence 15 patch. Because nˆ ×∇φ is divergence free, divergences
are only shown for the ∇φ functions. For the cases of irregular vertices both N < 6 and N > 6,
the basis functions are smooth everywhere across the patch, and the divergence is continuous
everywhere except the irregular vertex. This is a consequence of the continuity of the subdivision
147

surface and characteristic map, which are C2 everywhere but at the irregular vertex, where they are
C1 .
∇φ 0,2 (r)

∇φ 0,1 (r)

0.6
z

z

0.6
0.5
0.4
1.9
1.85
y 1.8

0.5
0.4
1.9
1.85
y 1.8

1.1
1
0.9

x

(a)

1.1
1
0.9

(b)

(c)

0.6
z

z

0.6
0.5

0.5

0.4

0.4
1.9
1.85
y 1.8

1.1
1
0.9

x

(e)

(f)

x

(h)

n
ˆ × ∇φ 0,1 (r)

0.6

n
ˆ × ∇φ 0,2 (r)

0.6

0.5
0.4

z

z

0.6
0.5
0.4

1.1
1
0.9

(i)

(j)

0.6

1.9
1.85
y 1.8

0.9

x

(l)

0.6

1.1
0.9

(m)

x

z

z

0.6

1

1.1
1
0.9

n
ˆ × ∇φ 2,0 (r)

n
ˆ × ∇φ 1,0 (r)

0.4

1.9
1.85
y 1.8

1.1
1

(k)

n
ˆ × ∇φ 1,1 (r)

0.5

0.5
0.4

1.9
1.85
y 1.8

x

z

z

1.1
1
0.9

(g)

∇φ 2,0 (r)

1.9
1.85
y 1.8

(d)

∇φ 1,0 (r)

∇φ 1,1 (r)

1.9
1.85
y 1.8

x

0.5

0.5

0.4

0.4

1.9
1.85
y 1.8

1.1
1
0.9

(n)

x

1.9
1.85
y 1.8

1.1
1
0.9

x

(o)

Figure 6.11: Basis functions and divergences for a valence N = 5 patch.

148

x

∇φ 0,2 (r)

∇φ 0,1 (r)

0

0

z

0.5

z

0.5

−0.5

−0.5
−0.5

0
x

−0.5

0.5

(a)

(b)

(d)

∇φ 1,0 (r)
0.5

0

0

z

0.5

z

0.5

(c)

∇φ 1,1 (r)

−0.5

−0.5
−0.5

0
x

−0.5

0.5

(e)

(f)

0
x

0.5

(g)

(h)
n
ˆ × ∇φ 0,2 (r)

n
ˆ × ∇φ 0,1 (r)

∇φ 2,0 (r)

0

0.5

0.5

0

0

z

z

0.5

z

0
x

−0.5

−0.5

−0.5
−0.5

0
x

−0.5

0.5

(i)

(j)

z

(m)

0.5

0.5

n
ˆ × ∇φ 2,0 (r)

0.5

0.5

0

0

−0.5

−0.5

0
x

(l)

z

z

0

−0.5

(k)

0.5

0
x

0.5

n
ˆ × ∇φ 1,0 (r)

n
ˆ × ∇φ 1,1 (r)

−0.5

0
x

−0.5
−0.5

0
x

(n)

0.5

−0.5

0
x

0.5

(o)

Figure 6.12: Basis functions and divergences for a valence N = 15 patch.

6.5

Partition of Unity

A key feature of the GMM discretization is that the basis set on each patch is multiplied by a
partition of unity that provides automatic current continuity between patches. The partition of unity
permits the use of different types of basis functions in the same discretization. For smooth non149

subdivision surface representations, the partition of unity is based on Shepard interpolation [49].
The local parameterization and patch definitions introduced in section 6.3 for Loop subdivision
surfaces admit the use of a partition of unity based on first order box spline for regular patches,
which are C0 functions that naturally form a partition of unity. Fig. 6.14(a) shows a first order box
spline about a regular vertex. In terms of the local barycentric coordinates in each triangle, the box
spline for a regular patch Ωl is simply defined as
φ l = ξl ,

(6.23)

with ξl the barycentric coordinate corresponding to vertex vl . This first order spline partition of
unity may be identified with the simplex partition of unity of Chapter 2.
In the vicinity of an irregular vertex, the partition of unity must be modified to ensure that basis
sets on neighboring regular patches are well behaved as the irregular vertex is approached. The
reason for this is that basis functions about regular vertices are parameterized with respect to a
regular parametric map, which does not provide adequate compensation for the divergent behavior
of the surface partial derivatives at the irregular vertex. Furthermore, unless the partition of unity
is expressed as a function of the (u1 , u2 ) coordinates generated by the irregular characteristic map,
its gradient will also diverge at the irregular vertex. To circumvent these issues, the partition of
unity is defined as follows.
First, the triangles in the 1-neighborhood of the irregular primal vertex v00 are subdivided once
as shown in Fig. 6.13. For the patch Ωl associated with the irregular vertex v0 in Fig. 6.13, the

Figure 6.13: Subdivision of triangle around irregular vertex v00 for partition of unity definition.

150

partition of unity is defined at a point p as




2ξ00




ψ0 = 1






0

, p ∈ T0 \ T11
, p ∈ T11

(6.24)

, O.W.

where the barycentric coordinate ξ00 is associated with the v00 node in Fig. 6.13. This definition
ensures that the derivative of the partition of unity is identically zero near the irregular vertex, leading to a partition of unity with a well-defined gradient that, while discontinuous, is non-divergent
and easily integrable over the entire support of the patch. Fig. 6.14(b) shows the partition of unity
function for a valence five vertex plotted versus (u1 , u2 ).
Finally, the partition of unity for the regular patch associated with vertex v01 a regular patch
overlapping the irregular patch is given by




ξ10 + .5ξ11 ,







.5ξ 1 ,
1
ψ1 =



.5ξ11 ,







0,

p ∈ T21
p ∈ T31

(6.25)

p ∈ T41
O.W.,

where the barycentric coordinate labels match the subdivided vertex labels in Fig. 6.13 and the
barycentric coordinates are associated with the subdivided (not primal) triangles. The appropriate
reflection of (6.25) gives the partition of unity function ψ2 for the regular patch associated with
v02 . The definition (6.25) ensures that gradient of the partition of unity is well-defined and nondivergent over its support. Furthermore, the partition of unity restricts the support of the basis set
defined on the regular patch with respect to the regular characteristic map so that the basis function
is well-behaved everywhere the partition of unity is nonzero. It can be readily verified that, for a
triangle T 0 with one irregular vertex and the functions ψl identified with vertices vl , l = 0, 1, 2, the
summation ψ0 + ψ1 + ψ2 = 1 everywhere in T 0 .
Fig. 6.14(c) shows a schematic depiction of the resulting partition of unity functions on a
regular patch neighboring an irregular patch. Fig. 6.15 shows the effective support of the partition
151

of unity for the regular patch, and Fig. 6.16 shows the effective supports of all regular patches
neighboring a valence five irregular vertex. In triangles shared with regular patches the usual first
order linear box spline partition of unity is used, but for triangles adjacent to the irregular vertex
(6.25) is used.

(a)

(b)

(c)

Figure 6.14: Subdivision simplex partition of unity. (a) PU on regular patch with regular neighbors.
(b) PU on irregular valence N = 5 patch. (c) PU on regular patch neighboring irregular patch
(shown schematically for clarity).

Figure 6.15: Regular patch support near an irregular vertex.
To accurately integrate the partition of unity in the neighborhood of an irregular vertex, it is
necessary to define triangular Gauss quadrature rules on each level 1 subtriangle in Fig. 6.13.
This incurs little extra cost since the subdivided integration rules need to be used only in triangles
incident on an irregular vertex. Using the partition of unity described in this section, the final
GMM basis set for patch Ωl is
ψl flmn,div (r) = ψl ∇φmn
ψl flmn,rot (r) = nˆ × ψl ∇φmn ,

152

(6.26)

Figure 6.16: Regular patch supports adjacent to an irregular vertex of valence N = 5.
and the divergences are given by
∇ · (ψl flmn,div (r)) = ∇ψl · ∇φmn + ψl ∇2 φmn

(6.27)

∇ · (ψl flmn,rot (r)) = ∇ψl · (nˆ × ∇φmn ).
In this work, the basis set (6.26) is used for both unmerged regular and irregular patches. For
merged regular patches, the Legendre basis set defined in Chapter 2 is utilized as there is no issue
with divergence of the subdivision tangent space on regular patches. A sampled partition of unity
on a merged patch next to an irregular vertex is shown in figure 6.17. The total partition of unity
over the patch is the summation of the local partitions of unity defined about each interior vertex.
Figure 6.18 shows for the same patch a basis function in R3 and its divergence in the local (u1 , u2 )
coordinate domain (shown sans partition of unity for clarity). Note the restriction of support for
both functions in the vicinity of the regular vertex.

6.6

Results

This section provides results demonstrating the viability and accuracy of the subdivision for some
electromagnetic scattering problems. The first result is scattering from a 1λ sphere due to an xˆ polarized plane wave incident along the +ˆz direction. No patch merging was employed and a p = 1

153

Figure 6.17: Sampled partition of unity ψ(u1 , u2 ) on a merged patch adjacent to an irregular vertex.

(a)

(b)

Figure 6.18: Basis function and divergence for a merged regular patch.
Helmholtz-decomposition (HD) basis with a total of Ns = 1736 GMM unknowns. Figure 6.19(a)
shows the bistatic RCS, taken in the φ = 0◦ cut, for the MFIE and EFIE compared with the Mie
series solution. Good agreement is evident for all cases. Relative l2 error in RCS for the MFIE
result is 3.8 × 10−3 , for the EFIE result is 5.8 × 10−3 and for the CFIE result is 3.1 × 10−3 .
Figure 6.19(b) shows the RCS for the same sphere computed using the MFIE, EFIE, and CFIE
(α = .3), but with merged patches and a mixture of a p = 2 HD bases on irregular patches and a

154

25

GMM-MFIE
GMM-EFIE
GMM-CFIE
Mie

GMM-MFIE
GMM-EFIE
GMM-CFIE
Mie

20
15
RCS, dBsm

RCS, dBsm

25
20
15
10
5
0
-5
-10
-15
-20
-25 ◦
−180 −135◦ −90◦ −45◦

10
5
0
-5
-10
-15

0◦
θ

45◦

-20 ◦
−180 −135◦ −90◦ −45◦

90◦ 135◦

(a)

0◦
θ

45◦

90◦ 135◦

(b)

Figure 6.19: (a) RCS for 1λ subdivision sphere (φ = 0◦ ). (b) RCS for 1λ subdivision sphere with
merged patches (φ = 0◦ ).

Figure 6.20: Real part of MFIE solution surface currents for 1λ sphere using h = .5λ merged
patches. Incident plane wave is traveling in +ˆz direction.
p = 2 Legendre bases on regular merged patches, for a total of Ns = 2208 GMM unknowns. The
maximum merged patch diameter is set to h = .5λ . Again, the results show good agreement, with
the l2 relative error in this case 1.9 × 10−3 for the EFIE result, 1.8 × 10−3 for the MFIE result,
and 1.9 × 10−3 for the CFIE result. Fig. 6.20 shows the real part of the currents for the merged
MFIE result. The smoothness of the currents everywhere indicates that the blending of functions
between patches of different valences is smooth and well-behaved.
The second result, scattering from a 2λ × 2λ Chmutov surface, shows the ease with which
subdivision can produce higher order surfaces from topologically nontrivial primal meshes. Fig.
155

(a)

(b)

(c)

Figure 6.21: Chmutov surface (a) Coarse primal mesh. (b) Mesh after one subdivision. (b) Limit
surface.
25
20

RCS, dBsm

15
10
5
0
-5
-10
-15

GMM Subdivision
RWG Reference
-20 ◦
−180 −135◦ −90◦ −45◦ 0◦
θ

45◦

90◦ 135◦

Figure 6.22: Bistatic RCS for 2λ Chmutov geometry (φ = 0◦ ).

6.21 shows the primal mesh, the k = 1 mesh, and the limit surface for the Chmutov, as well as
the incident plane wave. Despite the coarseness (and therefore sparseness) of the primal mesh
representation, the subdivision process yields a smooth limit surface suitable for discretization with
GMM. The Chmutov was discretized with a p = 1 mixed HD/Legendre GMM basis on h = .3λ
patches using the CFIE with α = .1. Fig. 6.22 shows the RCS compared with a reference first
order RWG CFIE implementation constructed on the k = 1 mesh. The results match well.
Finally, the scheme is applied to a shape deformation problem, scattering from a 2λ torus that
is deformed by twisting. Figure 6.23 shows the k = 1 mesh for the original torus and the smooth
subdivision representation of the twisted mesh, as well as the incident wave direction. The RCS
for the untwisted torus is shown in fig. 6.24a for a p = 2 mixed basis discretization with patches

156

(a)

(b)

(c)

(d)

Figure 6.23: Torus mesh (a) front view, (b) side view, and twisted torus limit surface (c) front view,
(d) side view.
15
10
5
0
-5
-10

5
0
-5
-10

-15
−180◦−135◦ −90◦

GMM-Subdivision
RWG Reference

10
RCS, dBsm

RCS, dBsm

15

GMM-Subdivision
RWG Reference

-15
−45◦

0◦

45◦

90◦

135◦

180◦

−180◦

θ

(a)

−90◦

0◦
θ

90◦

180◦

(b)

Figure 6.24: Bistatic RCS (φ = 0◦ cut) for (a) torus and (b) twisted torus geometries.
merged up to h = .5λ and an MFIE solution. The reference solution uses and RWG basis on a
low order triangular mesh. To obtain the high-order representation of the twisted torus, only the
primal mesh vertices were transformed; this is in contrast traditional higher order representations
in which all interpolation points must be properly transformed to obtain a well-behaved surface.
It is also more efficient than previous GMM higher order polynomial geometry representations
which relied on re-interpolation of the surface after any deformation. The ease with which the
GMM subdivision surfaces may be deformed, coupled with the accelerated solution method of
Chapter 3, provides a promising framework for tailoring electromagnetic response of scatterers
using rapid geometry optimization. Figure 6.24b compares the twisted torus RCS for the two
solution methods; subdivision exhibits good matching with the reference solution.

157

6.7

Discussion and Conclusions

This chapter has given a prescription for the construction of a GMM discretization on subdivision
surfaces, with a specific implementation of Loop subdivision, although other subdivision surfaces
could also be easily handled. The resulting GMM algorithm has several advantages: the surface description is always smooth and single-valued, the partition of unity is simple and easily integrated,
and integration domains are well-defined. A chief challenge of defining a well-behaved integral
equation discretization on subdivision surfaces is the handling of irregular vertices due to the singular nature of subdivision tangent vectors near irregular vertices. In this work this difficulty was
solved by using a local characteristic map parameterization of the subdivision surface and defining
basis functions using gradients of scalar functions, which are invariant under reparameterization.
Results furnished demonstrate the use of the subdivision-based scheme on a variety of surfaces,
and show that subdivision is a viable representation for a GMM discretization of electromagnetic
boundary integral equations. Although the scattering geometries shown in the results section are
all smooth, sharp tips, edges, and creases can be readily included in the subdivision surface description [119]. These capabilities and corresponding modifications to the GMM discretization
will be explored in future work.
Several features of GMM on subdivision surfaces make it attractive for an array of targeted
applications. Since subdivision is built upon a refinement of an initial coarse mesh into successively finer meshes, and because the mesh at any level of refinement converges to the same limit
surface, subdivision induces a natural h-adaptivity that can be exploited in refining solutions. Since
the support of GMM basis functions extends over multiple triangles, the method can easily handle meshes containing triangles with poor aspect ratios or with multiple length scales driven by
geometrical meshing constraints. Furthermore, due to the global smoothness of the subdivision
surface, it is possible to decouple the support of the basis functions from the support of faces in the
generating mesh. The ease with which subdivision surfaces can be deformed makes subdivisionbased integral equations attractive for uncertainty quantification and optimization applications in
which electromagnetic response is tailored via shape deformation. Finally, the method may be
158

readily extended to traditional moment method discretizations by using a hybrid approach with
isolated GMM patches about regular vertices and traditional basis sets everywhere else. All of
these avenues will be subjects of future research.

159

APPENDIX

160

EIGENVALUES OF THE GMM INTERIOR PENALTY SYSTEM.

This appendix gives the full derivations of equations (4.13) and (4.14) that underlie the spectral
analysis section of Chapter 4. Analytical expressions for the eigenspectra of the compensated and
uncompensated GMM operators are derived; the two eigenspectra are compared to deduce the
effect of the addition of the penalty term to the GMM system.
Under the assumptions that the basis set is shift invariant and that each patch supports a single
basis function, patch indices may be identified with basis function indices such that l → n, n =
1, . . . , Ns for source functions and k → m, m = 1, . . . , Ns for testing functions, with Ns the total
number of functions on the cylinder. The total current is then expanded as
Ns

ρ) =
Jφ (ρ

∑

Ns

ρ ) fn (ρ
ρ)
an ψn (ρ

n=1

∑ anψn(φ − nφ0) fn(φ − nφ0).

(28)

n=1

The source and testing functions are expanded into Fourier series as
∞

fn (φ ) = fn (φ − nφ ) =

∑

Fp e jqφ e− j pnφ0 ,

(29)

Tp e jqφ e− j pmφ0 ,

(30)

p=−∞
∞

tm (φ ) = t(φ − mφ ) =

∑

p=−∞

where Fp and Tp are the Fourier coefficients of f (φ ) and t(φ ), respectively, and the two dimensional Green’s function is expanded using the addition theorem for Hankel functions,
(2)

ρ − ρ ||) =
H0 (k||ρ

∞

∑

(2)
J p (kρ )H p (kρ )e− j p(φ −φ ) .

(31)

p=−∞

Using (29), (30), and (31), a spectral representation of the moment method matrix elements may
be obtained as [87]:
2πη0 ka ∞
(2)
J p (ka)H p (ka)T−p Fp e− jq(mφ0 −nφ0 ) .
Zmn =
∑
2Ns p=−∞

(32)

(2)

Here J p (ka) and H p (ka) are the derivatives of a first kind Bessel and second kind Hankel Function, respectively, and T−p , Fp are the Fourier coefficients For a circular cylinder, the impedance
161

matrix is an Ns × Ns circulant matrix, and therefore has eigenvectors given by e jlφ0 , l = 0, . . . , Ns −
1. The eigenvalues of a circulant matrix are given by the Fourier coefficients of the discrete Fourier
transform of the first row,
Ns −1

λq =

∑

Z0n e− jqnφ0 ,

(33)

n=0

which, upon substitution of the expression (32) yields
Ns −1

λq =

∑

n=0

e− jqnφ0

2πη0 ka ∞
(2)
J p (ka)H p (ka)T−p Fp e j pnφ0 ) .
∑
2Ns p=−∞

(34)

Rearranging summations, one obtains
λq =

Ns −1
2πη0 ka ∞
(2)
J
(ka)H
(ka)T
F
−p
p
p
∑ p
∑ e− jqnφ0 e j pnφ0 .
2Ns p=−∞
n=0

The rightmost term in (35) may be simplified as:



(−1)l Ns , q − p = lN, l ∈ Z
Ns −1
−
jqnφ
j
pnφ
0
0
e
=
∑ e


n=0
0, O.W.

(35)

(36)

Reindexing (35), the eigenvalues of the Z matrix are therefore given by:
λq =

2πη0 ka ∞
(2)
∑ Jq+lNs (ka)Hq+lNs (ka)T−q−lNs Fq+lNs
2Ns l=−∞

(37)

This equation matches that in [87], which was obtained by slightly different means.
The addition of the penalty function adds another matrix to the system as
Z˜ = Z MoM + Z p .

(38)

By virtue of the definition of the penalty function as a local operator that only operates on equispaced neighboring patches, the matrix Z p is circulant, with nonzero entries only in the ith,(i − 1)th,
and (i + 1)th slots of row i (where the arithmetic is taken modulo Ns ). Entries of Z p are given ex-

162

plicitly by:

2π
2π


+ (φ )t(φ − nφ ) f (φ − nφ )dφ +


P
Pn− (φ )t(φ − nφ0 ) f (φ − nφ0 )dφ , m = n
0
0
n


0
0



2π


−
Pn+ (φ )t(φ − (n − 1)φ0 ) f (φ − nφ0 )dφ , m = n + 1
p
0
Zm,n =

2π



−
Pn− (φ )t(φ − nφ0 ) f (φ − (n + 1)φ0 )dφ , m = n − 1


0





0, O.W.
(39)
where Pn± (φ ) = P(φ − (n ± 1/2)φ0 ) is a pulse function that restricts the integration to either
γn,n−1 or γn,n+1 depending on the sign, and the arithmetic m = ±1 is understood to be modulo
Ns − 1. (γl,k is the overlap between patches Ωl and Ωk ). Making use of the identity Pn± (φ ) =
(Pn± (φ ))2 and expanding Fourier series of the products yields
Pn± (φ ) f (φ

∞

∑

F˜p± e jqφ e− jq(n±1)φ0

∑

T˜p± e jqφ e− jq(n±1)φ0 ,

− (n ± 1)φ0 ) =

p=−∞
∞

Pn± (φ )t(φ

− (n ± 1)φ0 ) =

(40)

p=−∞

where F˜p± and T˜p± are understood to be the Fourier coefficients of the product of the pulse and
basis or testing function. Rewriting (39) using these expansions, one obtains




4πa ∑ T˜−p F˜p , m = n



p





jqφ

 −2πa ∑ T˜−p F˜p e 0 , m = n + 1
p

p

Zm,n =

(41)




−2πa ∑ T˜−p F˜p e− jqφ0 , m = n − 1



p






0, O.W.
Note that the ± superscript is dropped in (41) because the phase factors of the pulse functions
implicitly cancel under integration. These expansions may now be used to define the eigenvalues
by a process analogous to those of Z MoM via the DFT. In this case, since there are only three
nonzero terms in the first row of Z p , the DFT expression becomes
∞

λq = 2πa

∑

T˜−p F˜p (2 − e− j(q+p)φ0 − e j(q(Ns −1)+p)φ0 ),

p=−∞

163

(42)

where the phase of (q − (Ns − 1))φ0 comes from the fact that in the first row of the circulant matrix
Z p , the only nonzero terms are n = 0, 1, Ns − 1. Further manipulations yield
∞

λq =2πa

∑

j(q( N2s +1)+p)φ0
− j p N2s − j(q( N2s +1)+p)φ0
(e
−e
))
T˜−p F˜p (2 − e

∑

− jp 2
T˜−p F˜p (2 − e
cos

∑

T˜−p F˜p 1 − (−1)q cos (q + p)φ0 + qπ

p=−∞
∞

=2πa

p=−∞
∞

=4πa

Ns

q

Ns
+ 1 + p φ0
2

(43)

.

p=−∞

The definition of φ0 = 2π/Ns gives the ratio Ns φ0 /2 = π, and finally,
∞

λq =4πa

T˜−p F˜p 1 − (−1)q cos (q + p)φ0 cos(qπ) − sin (q + p)φ0 sin(qπ)

∑

p=−∞
∞

=4πa

∑

(44)
T˜−p F˜p 1 − cos (q + p)φ0

.

p=−∞

A few additional manipulations give more insight into the behavior of the eigenvalues λq . First,
the last line in 44 may be rewritten
∞

λq =4πa

∞

∑

∑

T˜−p F˜p − cos(qφ0 )

p=−∞
∞

=4πa

∞

T˜−p F˜p −

p=−∞

∑

T˜−p F˜p cos(qφ0 ) cos(pφ0 ) +

p=−∞

∑

T˜−p F˜p sin(qφ0 ) sin(pφ0 )

p=−∞
∞

∑

∞

T˜−p F˜p cos(pφ0 ) + sin(qφ0 )

p=−∞

∑

T˜−p F˜p sin(pφ0 ) .

p=−∞

(45)
Under the assumption that both f (φ ) and t(φ ) are purely real so that T˜−p F˜p = (T˜p F˜−p )∗ , then
T˜−p F˜p + T˜p F˜−p = 2ℜ{T˜p F˜−p } and T˜−p F˜p − T˜p F˜−p = 2 jℑ{T˜p F˜−p }. If the usual Galerkin testing
is used (T˜p = F˜p ), then T˜−p F˜p + T˜p F˜−p = 2|T˜p F˜−p |2 = 2|T˜−p F˜p |2 . Thus, for Galerkin testing all
eigenvalues λq are real since the sine term vanishes by odd symmetry. The sums over p are independent of q and may be assigned real values C1 ,C2 , so that the final expression for the eigenvalue
becomes
λq =4πa C1 −C2 cos(qφ0 )
2πq
) ,
=4πa C1 −C2 cos(
Ns

164

(46)

where
∞

C1 =4πa

T˜−p F˜p ,

∑

p=−∞
∞

C2 =4πa

∑

(47)
T˜−p F˜p cos(pφ0 ).

p=−∞

These are the expressions in (4.14) and (4.15).

165

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