Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 916

Fast Numerical Techniques forElectromagnetic Problems in

Frequency Domain

BY

MARTIN NILSSON

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2003

Dissertation at Uppsala University to be publicly examined in room 1211, Building 1, Polacks-backen, Friday, January 30, 2004 at 10:15 for the Degree of Doctor of Philosophy in NumericalAnalysis. The examination will be conducted in English

AbstractNilsson, M. 2003. Fast Numerical Techniques for Electromagnetic Problems in FrequencyDomain. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 916. 38 pp. Uppsala. ISBN 91-554-5827-0

The Method of Moments is a numerical technique for solving electromagnetic problems withintegral equations. The method discretizes a surface in three dimensions, which reduces thedimension of the problem with one. A drawback of the method is that it yields a dense systemof linear equations. This effectively prohibits the solution of large scale problems.

Papers I-III describes the Fast Multipole Method. It reduces the cost of computing a densematrix vector multiplication. This implies that large scale problems can be solved on personalcomputers. In Paper I the error introduced by the Fast Multipole Method is analyzed. Paper IIand Paper III describe the implementation of the Fast Multipole Method.

The problem of computing the monostatic Radar Cross Section involves many right handsides. Since the Fast Multipole Method computes a matrix times a vector, iterative techniquesare used to solve the linear systems. It is important that the solution time for each system is aslow as possible. Otherwise the total solution time becomes too large. Different techniques forreducing the work in the iterative solver are described in Paper IV-VI. Paper IV describes a blockQuasi Minimal Residual method for several right hand sides and a Sparse Approximate Inversepreconditioner that reduce the number of iterations significantly. In Paper V and Paper VI amethod based on linear algebra called the Minimal Residual Interpolation method is described.It reduces the work in an iterative solver by accurately computing an initial guess for the iterativemethod.

In Paper VII a hybrid method between Physical Optics and the Fast Multipole Method isdescribed. It can handle large problems that are out of reach for the Fast Multipole Method.

Keywords: Fast Multipole Method, Minimal Residual Interpolation, Sparse ApproximateInverse preconditioning, Method of Moments, fast solvers, iterative methods, multiple righthand sides, error analysis

Martin Nilsson, Department of Information Technology, Scientific Computing.Uppsala University. Box 337, SE-751 05 Uppsala, Sweden

c© Martin Nilsson 2003

ISBN 91-554-5827-0ISSN 1104-232Xurn:nbn:se:uu:diva-3884 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3884)

To my family

iv

List of Papers

The thesis is based on the following papers, which will be referred to in thetext by Roman numerals I-VII

[I] M. Nilsson, Stability of the Fast Multipole Method for Helmholtz Equationin Three Dimensions, IT Scientific Report 2003-054, ISSN 1404-3203, De-partment of Information Technology, Uppsala University, November 2003.Downloadable from Web sitehttp://www.it.uu.se/research/reports/2003-054/. Submitted for publication.

[II] M. Nilsson, A Fast Multipole Method Solver for Large Scale ScatteringProblems, In Fredrik Edelvik et al. editors, EMB 01 – ElectromagneticComputations – Methods and Applications, ISBN 91-631-1629-4, pages148–155, Uppsala University, November 2001. SNRV.

[III] M. Nilsson, A Parallel Shared Memory Implementation of the Fast Multi-pole Method for Electromagnetics, IT Scientific Report 2003-049,ISSN 1404-3203, Department of Information Technology, Uppsala Univer-sity, October 2003. Downloadable from Web sitehttp://www.it.uu.se/research/reports/2003-049/. Submitted for publication.

[IV] M. Nilsson, A Fast Multipole Accelerated Block Quasi Minimum Resid-ual Method for Solving Scattering from Perfectly Conducting Bodies, InMagdy F. Iskander editor, Proceedings of IEEE Antennas and PropagationSociety International Symposium, ISBN 0-7803-6369-8, volume 4, pages1848-1851, Salt Lake City, Utah, USA, July 2000. 1

[V] P. Lotstedt and M. Nilsson, A Minimal Residual Interpolation Method forLinear Equations with Multiple Right Hand Sides. This is a preprint of apaper that has been accepted and will appear in SIAM Journal of ScientificComputing. 2

1 c© 2003 IEEE. Reprinted, with permission.2 c© 2003 SIAM. Reprinted, with permission.

v

[VI] M. Nilsson, Rapid Solution of Parameter-dependent Linear Systems forElectromagnetic Problems in the Frequency Domain, IT Scientific Report2003-055, ISSN 1404-3203, Department of Information Technology, Upp-sala University, November 2003. Downloadable from Web sitehttp://www.it.uu.se/research/reports/2003-055/.

[VII] M. Nilsson, The Minimum Residual Interpolation Method Applied to Mul-tiple Scattering in MM-PO, In Jonathan D. Young and John L. Volakis edi-tors, Proceedings of IEEE Antennas and Propagation Society InternationalSymposium, ISBN 0-7803-7846-6, volume 3, pages 828-831, Columbus,Ohio, USA, June 2003. 3

3 c© 2003 IEEE. Reprinted, with permission.

vi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Numerical methods for electromagnetic problems . . . . . . . . . . . 11.2 Content of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 A note on GEMS and SMART . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Helmholtz’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Maxwell equations in a homogenous dielectric . . . . . . . . . . 62.3 Scattering from metallic objects . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 The far field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 The Fast Multipole Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Acceleration techniques for iterative methods . . . . . . . . . . . . . . . . . . 215.1 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 A hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.1 Paper VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

viii

Introduction

Products that depend on electromagnetics surround us. Everything from mo-bile telephones and televisions to large aircrafts and satellites use electric-ity. Many of these products have antennas that emit electromagnetic radiationwhen they are used. It is important to understand how the emitted radiationcan affect other products and most of all how they can affect us. The develop-ment and construction of new devices that depend on electricity are expensivebecause a lot of time has to be spent on measuring the devices. This is becauseone wants to check that the device works within specified parameters and thatit meets all requirements.

One way of reducing the development time is to use computers to predictthe electromagnetic fields. From the predicted fields an engineer can see wherea model should be changed in order to achieve a specified level of performance.Computer modeling can be used to give a first clue to which of the best modelsthat should be measured more closely. One can then proceed by construct-ing the models to see which one performs best. Earlier the engineers had toconstruct a new model every time they came up with a new idea. This is notnecessary with computers that test the models first.

Computer models are used for predicting the electromagnetic environmentin problems varying from Electromagnetic Compatibility (EMC), which dealswith the electromagnetic interference between different electric devices, toelectromagnetic scattering by large objects, which deals with how much anobject reflects from an incident plane wave.

The increased capacity of modern day computers means that real life prob-lems can be simulated on a regular PC. However, it is important to note thatthe simulation time still is several days with standard methods for the type ofproblems we want to solve here. This is a bottleneck in the design stage ofan electrical device. To address this problem one can either wait for a bettercomputer or develop new methods that reduce the computational time.

1.1 Numerical methods for electromagnetic problemsThere exist two main types of numerical methods for solving electromagneticproblems. The first type is Time-Domain methods (TD) [CJMS01, EL00,EL02, Ede02, Jin93, RT97, Ryl02, Taf95]. They solve Maxwell’s equations

1

in the time domain and are suitable for broadband applications. The solutionis obtained by stepping forward in time. Solutions for all frequencies that canbe resolved by the geometry are obtained at once. Time-domain methods usu-ally suffer from dispersion errors which arise because different frequenciesmove at different speeds on the computational grid. Methods in time-domaininclude the Finite-Difference Time-Domain method (FDTD) [Taf95], Finite-Volume Time-Domain methods (FVTD) [EL00, Ede00, Ede02, RT97] andFinite-Element Time-Domain methods (FETD) [EL02, Ede02, Jin93, Ryl02]and the integral equation methods in time called Marching-On-in-Time meth-ods (MOT) [CJMS01].

The second type of methods is Frequency-Domain methods (FD) [CJMS01,Edl01, Kel62, Nil02, PK74, PRM98]. They solve the time-harmonic Maxwellequations for one frequency at a time. Therefore, they are better suited fornarrow band applications. Frequency-domain methods are divided in low fre-quency, mid frequency and high frequency methods. The low frequency andmid frequency methods include the Method of Moments (MM). It is a methodthat given an external field finds the currents on the surface of an electromag-netic scatterer and then computes the field from the known surface currents.The Method of Moments defines an integral equation which is discretized andsolved. The solution converges to the exact solution if the resolution is re-fined. A drawback with this method is that the memory requirements effec-tively prohibits the possibility of solving the equations for high frequencies.This is what makes the Method of Moments a low frequency to mid frequencymethod. The high frequency methods are based on asymptotic expansions ofthe solution and are often of ray-tracing type. They include Physical Optics(PO) [Edl01, ELS03], which can be viewed as an asymptotic approximation tothe Method of Moments. Other methods are Geometrical Optics (GO), Geo-metrical Theory of Diffraction (GTD) [Kel62] and Uniform Theory of Diffrac-tion (UTD) [PK74], which are ray-based methods. The expansions are usuallyvalid when the object is much larger than the wavelength. Between the highfrequency methods and the mid frequency methods there is a gap which onecan not handle with the Method of Moments because of memory requirements.It is not possible to use the high frequency methods either because the asymp-totic expansions are too inaccurate. Hybrid methods try to cover this interme-diate frequency range. Volume integral methods [CJMS01, PRM98] and FiniteElement methods [CJMS01, PRM98] are also used in frequency domain.

1.2 Content of the thesisThis thesis deals with ways of shortening the solution time for the Methodof Moments and reducing its memory requirements. Instead of solving theequations by explicitly generating the system of linear equations we can use

2

approximations of them. The approximations are then solved with iterativemethods. The advantages with this approach are that we do not require thesame amount of memory and that the method is faster provided that the numberof iterations in the iterative method is low. This implies that the frequencyrange in which the Method of Moments can be used is increased.

The approximation to the linear system is achieved through the multilevelFast Multipole Method for Helmholtz’ equation presented in Paper I, PaperII and Paper III. It can reduce the time of computing a matrix vector multi-plication from O

(N2)

for a standard method to O (N logN), where N is thenumber of unknowns. It also reduces the memory requirement from O

(N2)

toO (N logN). The reason is that the multilevel Fast Multipole Method computesthe action of the matrix on a vector and therefore it does not need to store thematrix in memory.

The Fast Multipole Method reduces the time for solving a linear systemfrom O

(N3), the time if Gaussian elimination is used, to O (KMN logN).

Here, M is the average number of iterations per right hand side in an iterativemethod and K is the number of right hand sides. Clearly, the total number ofmatrix vector multiplications KM should be low in order for the Fast MultipoleMethod to be effective. In Paper IV, Paper V and Paper VI several methods aredescribed that reduce KM.

For very large problems or problems involving a large number of right handsides, not even the Fast Multipole Method is fast enough. In Paper VII a hybridmethod between Physical Optics and the Fast Multipole Method is described.

The outline of this thesis is as follows. In Chapter 2 we discuss the Maxwellequations in frequency domain and derive the integral equations that are usedin the Method of Moments. Chapter 3 discusses discretization of the integralequations into systems of linear equations. The Fast Multipole Method is de-scribed in Chapter 4, where a summary of Paper I, Paper II and Paper III isgiven. Chapter 5 gives a summary of the methods for reducing the number ofmatrix vector multiplications in Paper IV, Paper V and Paper VI and Chapter 6explains the hybrid method in Paper VII. We end with some conclusions inChapter 7.

1.3 A note on GEMS and SMARTThis project has obtained its financial support from the Computational Electro-Magnetics program at the Parallel and Scientific Computing Institute (PSCI),which is a competence center supported by VINNOVA, The Swedish Agencyfor Innovation Systems.

The project was first part of the research activities in the General Electro-Magnetic Solver code project (GEMS), funded by VINNOVA and the NationalAeronautical Research Program (NFFP). The aim of GEMS was to develop an

3

industrial class suite of hybrid electromagnetic solvers in both frequency andtime domains. Later the development was carried out in the project SignatureModeling and Reduction Tools (SMART), which was funded by NFFP viaPSCI. Here, the aim was to develop tools for analyzing and reducing the RadarCross Section. GEMS has continued in the projects GEMS2 and GEMS3.

In frequency domain the aim was to develop a hybrid between MM and POand GTD. In order to speed up the initial development phase a MM code calledCESC was bought from CERFACS, Toulouse, France [BF99] and a GTDcode called FASANT was bought from Cantabria University, Spain [PSG+99].These codes were used as a basis for the design of the new codes, which weredeveloped in FORTRAN 90. Some of the progress is summarized in [Edl01,Nil02, Sef03, Hag03].

In time domain a hybrid method between FDTD and FVTD and FETDwas developed. The hybrid method can handle arbitrarily oriented thin wires.FVTD or FETD is used close to a surface in order to get a smooth descriptionof the surface. Far from the surface the more efficient FDTD is used. Due toproblems with late time instabilities in FVTD and the development of a stablehybrid between FDTD and FETD in [RB00, RB02] the emphasis of the devel-opment was on a hybrid between FDTD and FETD. Some of the progress issummarized in [And01, Led01, Ede02, Joh03, Atl03].

4

Maxwell’s equations

In this chapter the Maxwell equations are formulated. They are solved usingintegral equation formulations. The integral equations are used in Chapter 3 toformulate boundary element methods for Maxwell’s equations. The emphasisof this thesis is on perfect electric conductors, so formulas are derived for them.

2.1 Helmholtz’ equationIn order to understand the nature of the integral equations for Maxwell’s equa-tions one solution of the Helmholtz equation is formulated. The inhomoge-neous Helmholtz’ equation is

∆Ψ(x)+κ2Ψ(x) = − f (x) (2.1)

where Ψ is the unknown function, κ is the wavenumber and f (x) is a sourceterm. The function

G(x,x′

)=

eıκ|x−x′|

4π |x−x′| (2.2)

where x and x′ are two points in space, is a fundamental solution to Helmholtz’equation called Green’s function. It represents outgoing solutions or solutionssatisfying the radiation condition of Helmholtz’ equation given by [Ned01]∣∣∣ ∂Ψ

∂r − ıκΨ∣∣∣≤ C

|x|2 r = |x| → ∞ (2.3)

The Green’s function satisfies

∆G(x,x′

)+κ2G

(x,x′

)= −δ

(x−x′

)(2.4)

where δ is the Dirac measure. The solution to Helmholtz’ equation can ac-cording to Huygens’ principle be expressed as a superposition of waves fromindividual point sources. The properties of the Green’s function implies thatthe solution to the inhomogeneous Helmholtz’ equation can be written

Ψ(x) =∫

R3f(x′)

G(x,x′

)dx′ (2.5)

5

The fundamental solution (2.2) is used later as the basis for the fundamentalsolutions of Maxwell’s equations.

2.2 The Maxwell equations in a homogenous dielectricIn this section the Maxwell equations are solved based on integral equationformulations. To obtain an integral equation, the concept of ficticious equiva-lent currents will be used. The theory is applicable to homogenous dielectricbodies and homogenous perfect electric conductors. The reader is referredto [HK97, Ned01, PRM98, Rum54, BFG99, Nil02] for more information.

Let the electric and magnetic fields be time-harmonic in a homogenous di-electric with the time dependence e−ıωt . The dielectric domain Ω∈R

3 is eitheran interior or an exterior domain. It is characterized by the relative permittivityε and relative permeability µ. The wavenumber is κ = ωc−1, where c = c0n−1.

Here, c0 =√

(ε0µ0)−1 is the speed of light and n =

√εµ is the absolute index of

refraction. The impedance of the medium is Z = Z0

√µε−1, where Z0 = 120π

in SI-units is the impedance in vacuum. Applied magnetic currents, denoted byMa, and applied electric currents, denoted by Ja describe the effect of sourcesin the domain Ω. The total electromagnetic field in domain Ω is then governedby the Maxwell equations

∇×E− ıκZH = −Ma x ∈ Ω∇×H+ ıκZ−1E = Ja x ∈ Ω

(2.6)

Define the normal n to be the one pointing outwards from region Ω. Byintroducing the ficticious equivalent currents J = −n×H and M = n×E onthe boundary Γ as in Figure 2.1, the equations are extended to R

3, assumed tobe filled by the same dielectric, by the equation

∇×E− ıκZH = −M−Ma x ∈ R3

∇×H+ ıκZ−1E = J+Ja x ∈ R3

(2.7)

The ficticious surface currents replace the fields in R3 \Ω by zero fields so that

E = 0 and H = 0 there.The ficticious currents that are defined on the boundary Γ of Ω are called the

electric current J and magnetic current M respectively. Sometimes it is moreconvenient to use the normal pointing into region Ω. In that case the equationsare adjusted appropriately.

Let x be a point in space. In the case of an exterior domain it is assumedthat the field satisfies the Silver-Muller radiation condition [Ned01]

∣∣√εE−√µH× n

∣∣≤ C|x|2 |x| → ∞ (2.8)

6

Figure 2.1: The principle of ficticious equivalent currents applied to an interiordomain Ωi (left) and an exterior domain Ωe (right).

This condition ensures that the fields are bounded when |x| → ∞ [Ned01]⎧⎪⎪⎨⎪⎪⎩

|E ·x| ≤ C|x|

|H ·x| ≤ C|x|

|H ·E| ≤ C|x|3

|x| → ∞ (2.9)

The Silver-Muller radiation condition is taken into account by the fundamentalsolution constructed from the Green’s function in (2.2) [Ned01].

The currents are related to charges through the assumption of conservationof charge. This assumption leads to a relationship between ∇ ·M, ∇ ·Ma, ∇ ·J,∇ ·Ja and the respective charges ρm, (ρm)a, ρe, (ρe)a of the form

∇ ·J+ iωρe = 0 (2.10)

Charges are only present on the boundaries between different regions. Theseobservations can be used to rewrite the Maxwell equations in a form similar toHelmholtz equation

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−(∆E+κ2E)

=ıκZ

(1κ2 ∇∇ ·J+J

)

+ ıκZ

(1κ2 ∇∇ ·Ja +Ja

)−∇×M−∇×Ma

x ∈ R3

−(∆H+κ2H)

=ıκZ−1(

1κ2 ∇∇ ·M+M

)

+ ıκZ−1(

1κ2 ∇∇ ·Ma +Ma

)+∇×J+∇×Ja

x ∈ R3

(2.11)

7

Assuming that the applied sources do not intersect the boundary Γ, onecan express the applied currents Ja and Ma in the fields Ea and Ha. From theproperties of the Green’s function associated with Helmholtz’ equation in (2.4)it follows that

E = Ea + ıκZT J+K M x /∈ ΓH = Ha + ıκZ−1T M−K J x /∈ Γ

(2.12)

where the three operators defined in [HK97] have been introduced in order tosimplify the notation

⎧⎪⎨⎪⎩

V J =∫

Γ G(x,x′)J(x′)dΓ(x′) x /∈ ΓT J =

(1κ2 ∇∇ ·+1

)V J x /∈ Γ

K J = −∇×V J x /∈ Γ(2.13)

2.3 Scattering from metallic objectsA nice theory that can be used to describe electromagnetic interactions be-tween complicated metallic and dielectric objects is the Rumsey reaction prin-ciple [Rum54]. For details on how it works, see for instance [BFG99, Nil02].

Although the methods can be extended to dielectric materials, this thesismainly concerns fast methods for scattering from metallic objects. Hence, thefocus is on integral equations for metallic surfaces. The ability of metals toconduct electricity is good. This implies that a good approximation to manymetals is the perfect electric conductor, which is characterized by the fact thatE = 0 inside it. The boundary condition for the perfect electric conductor isthen Et = n×E = M = 0.

Let J′ be a tangential testing current on the surface Γ. Multiplying the firstequation in (2.12) with J′ and using the fact that M = 0 yields the ElectricField Integral Equation (EFIE)

∫Γ

TJ ·J′dΓ = − 1ıκZ

∫Γ

Ea ·J′dΓ (2.14)

Here, T is the limits of T as the boundary is approached in Cauchy principalvalue sense

TJ =1κ2 ∇Γ

∫Γ

G(x,x′

)∇Γ ·J

(x′)

dΓ(x′)

+(∫

ΓG(x,x′

)J(x′)

dΓ(x′))

t

(2.15)

where ∇Γ denotes the surface divergence of a vector field tangent to Γ and ()tdenotes the tangential part. The hypersingularity in the first part of T is usually

8

handled by moving one derivative from the x′-variable to the x-variable byusing partial integration on the variational formulation (2.14) yielding∫

ΓTJ ·J′dΓ =−

∫Γ

∫Γ

G(x,x′

) 1κ2 ∇Γ ·J

(x′)

∇Γ ·J′ (x)dΓ(x′)

dΓ(x)

+∫

Γ

∫Γ

G(x,x′

)J(x′)

J′ (x)dΓ(x′)

dΓ(x)(2.16)

Another integral equation, which is called the Magnetic Field Integral Equa-tion (MFIE), can be derived from the second equation in (2.12). In the caseof a scattering problem the normal is defined as pointing into the region. Thedefinition of the surface current is then J = n×H. When the boundary is ap-proached in the Cauchy principal value sense, equation (2.12) together withthe boundary limit of K yields

J = n×H = n×Ha −(−1

2J+ n×KJ

)(2.17)

where K is defined by

KJ =(∫

Γ∇x′G

(x,x′

)×J(x′)

dΓ(x′))

t(2.18)

Multiplying with a test current J′ defined as before and integrating yield theMFIE equation

12

∫Γ

J ·J′dΓ+∫

Γn×KJ ·J′dΓ =

∫Γ

n×Ha ·J′dΓ (2.19)

The EFIE (2.14) and the MFIE (2.19) both have different advantages anddisadvantages. The EFIE has spurious solutions if κ2 is a corresponding eigen-value to the interior problem with perfect electric conducting walls. In case κ2

is an eigenvalue, the spurious modes will not radiate in the exterior. Thus, thefield is not corrupted outside of the object. The MFIE can also have nullspacesolutions in the interior. However, the spurious solutions of the MFIE do ra-diate in the exterior domain and corrupt the field. The EFIE can also handlethe case of an open object which the MFIE is not able to handle. Therefore,the EFIE would be the best choice, when solving the perfect electric conduc-tor case. The EFIE is an Fredholm integral equation of the first kind, whilethe MFIE is a Fredholm integral equation of the second kind [HK97]. Hence,the EFIE suffers from ill-conditioning, while the MFIE is better conditionedthanks to the term 1

2J in (2.17). Thus, the MFIE is more appropriate for aniterative solution method.

The solution to these problems is to combine the two formulations into onecalled the Combined Field Integral Equation (CFIE). The CFIE is a linear com-bination of the EFIE (2.14) and the MFIE (2.19) defined as

9

CFIE = αEFIE+(1−α) ıκMFIE 0 < α < 1 (2.20)

With this choice there are no spurious modes. From experience it turns outthat this equation has better conditioning than both EFIE and MFIE [CJMS01].Since the formulation involves MFIE it can not be used for open objects. Thus,the CFIE is used for closed objects while EFIE is used for open objects involv-ing perfect electric conductors.

2.4 The far fieldThe behavior of the electromagnetic field far from the object is often importantin electromagnetic problems. Let x = rκ, where κ is a unit vector, and considerthe case when r → ∞. Through Taylor expansion

∣∣x−x′∣∣=√r2 −2rκ ·x′ + |x′|2

= r

√1− 2κ ·x′

r+

|x′|2r2

≈ r− κ ·x′

(2.21)

Applying this procedure to the operator V J yields

V J =∫

ΓG(x,x′

)J(x′)

dΓ(x′)

≈ eıκr

4πr

∫Γ

e−ıκ·x′J(x′)

dΓ(x′)

≡ eıκr

rFJ(κ)

(2.22)

Here, κ is the wavenumber and κ = κκ is a vector of length κ pointing in thedirection of κ.

Using the relation ∇x′e−ıκ·x′ = −ıκe−ıκ·x′ together with the definition of Tgives the far field for T

T J ≈ eıκr

r(FJ(κ)− κ(κ ·FJ(κ))) =

eıκr

rκ× (FJ(κ)× κ) (2.23)

The same relationship applied to K gives the far field generated by K

K M ≈ eıκr

r(−ıκ×FM(κ)) (2.24)

10

The equations (2.23) and (2.24) appear in the expressions for the Fast Multi-pole Method.

Consider the case of electromagnetic scattering from a plane wave withwavenumber κ described by

Ea (x, κa) = E0e−ıκκa·x

Ha (x, κa) = H0e−ıκκa·x (2.25)

The wave is traveling in the direction given by the unit vector −κa. LetEs (x, κa) and Hs (x, κa) be the corresponding scattered electromagnetic field.Then the bistatic Radar Cross Section (RCS) σ(κ, κa), in Figure 2.2, is definedas

σ(κ, κa) = limr→∞

4πr2 |Es (rκ, κa)|2|Ea (rκ, κa)|2

(2.26)

The special case σ(κa, κa) is called the monostatic RCS. The RCS is oftencomputed in decibels (dB) by the relation σdB (κ, κa) = 10log10 σ(κ, κa).

Figure 2.2: The scattered field from a source. The return direction is calledthe monostatic direction. The other directions are called bistaticdirections.

The Radar Cross Section is often used as a measure of the numerical errorsfrom different methods. It is the parameter that is minimized in the construc-tion of stealth targets.

The definition of the far field imply that one can express the scattered fieldas

Es (rκ, κa) ≈ eıκr

r A(κ,FJ) r → ∞ (2.27)

Here, A(κ,FJ) is defined by the behavior in the far field. It is derived fromthe equations (2.12), (2.23) and (2.24), where it is assumed that the electricsurface current is known. Thus, the RCS is given by

11

σ(κ, κa) = limr→∞

4πr2 |Es (rκ, κa)|2|Ea (rκ, κa)|2

= 4π|A(κ,FJ)|2

|E0|2(2.28)

The function A(κ,FJ) is often referred to as the far field pattern.

12

Boundary Element Method

Integral equations can be solved using Boundary Element Methods. Bound-ary Element Methods discretize the integral equations so that a dense linearsystems of equations is obtained. The discretization is achieved by a FiniteElement discretization of the surface integrals involved. In the case of theMaxwell equations, Boundary Element Methods are usually called Method ofMoments (MM).

An advantage with the Method of Moments is that because of their for-mulation they are free from dispersion errors, no outer artificial boundary isneeded for exterior problems and only surfaces have to be discretized in R

3,i.e. the dimension of the problem is reduced by one. The price of the dimen-sional reduction is a dense linear system matrix. Computationally this is nota big issue, because the computers of today are equipped with cache memoryhierarchy’s that are very effective for computing dense matrix vector multipli-cations. However, the available memory limits the size of the problems thatcan be solved.

3.1 DiscretizationThe integral equations EFIE (2.14), MFIE (2.19) and CFIE (2.20) are approx-imated numerically. The numerical approximation gives a dense complex sys-tem of linear equations in the unknown currents. The approximation requires asuitable representation of the geometry. Here, the surfaces are partitioned intotriangles, as in Figure 3.1. This gives a perfect approximation of flat surfaces,but introduces a geometrical error of O

(h2)

on curved surfaces, where h is inthe order of the length of the longest edge among all triangles.

The so called Rao-Wilton-Glisson (RWG) basis functions [RWG82] can beused to approximate the surface currents of the triangulated object. A RWGbasis function is defined over the common edge of two adjacent triangles. Ifthe two triangles are denoted by T+ and T−, then the RWG basis function isdefined as

j(x) =

⎧⎪⎨⎪⎩

12A+

(x−x+) x ∈ T+

− 12A− (x−x−) x ∈ T−

0 otherwise

(3.1)

13

Figure 3.1: An example of an object surface represented by triangles. In thiscase it is a generic aircraft model called RUND.

Here, x+ and x− are the two corners of the triangles not shared by the com-mon edge, as in Figure 3.2. The RWG basis functions ensure that there isno accumulation of line-charges at the outer boundaries of the triangle pair.The scaling makes the normal continuous across the common edge. Also, thedivergence on the triangle surface satisfies

∇Γ · j =

⎧⎪⎨⎪⎩

1A+

x ∈ T+

− 1A− x ∈ T−

0 otherwise

(3.2)

To discretize the integral equations RWG basis functions are used to expandthe currents as

J(x) =N

∑l=1

Iljl (x) (3.3)

Here, jl (x) is an electric current basis function and N is the number of electricbasis functions. The same basis functions are used in the testing procedure,resulting in a Galerkin method.

The expansion ends up in an equation ZI = V, where Zkl is given by insert-ing (3.3) and letting J′ = jl in one of the equations EFIE (2.14), MFIE (2.19)or CFIE (2.20). The element Zkl is computed from the integral

Zkl = α∫

ΓT jl · j′kdΓ+(1−α)

ıκ

∫Γ

n×Kjl · j′kdΓ (3.4)

Here, α = 1 is the special case EFIE and α = 0 is the special case MFIE. The

14

Figure 3.2: The Rao-Wilton-Glisson basis function (RWG).

same testing procedure applies to the element Vk defined by the equation

Vk = −α1

ıκZ

∫Γ

Ea · j′kdΓ+(1−α)ıκ

∫Γ

n×Ha · j′kdΓ (3.5)

For different applied electromagnetic fields at a fixed wavenumber the onlydifference is in the right hand side V. The currents are given by the unknownvector I and the matrix Z is called an impedance matrix. In order for the ap-proximation to have any accuracy at least 2 edges per wavelength are needed.This is related to the ability of a general basis function to describe the functionsin(x). Experiments reveal that at least 5 edges per wavelength are needed toget any accuracy at all with RWG basis functions. At least 10 edges per wave-length are usually recommended. Since the object is a surface the requirementon the discretization density implies that N = O

(f 2)

where f is the frequency.The assembly of the matrix Z is carried out by calculating the contributions

from basis functions associated with each triangle and adding the contributionsto the impedance matrix. The simplest way to calculate the individual contri-butions from the basis functions is by numerical integration. Because of thesingular nature of the integrals associated with integral equations, numericalintegration is difficult for triangles close to each other. This problem can behandled in several ways. One way is to use analytical treatment of the singularintegrals to extract the singularity [YT03]. Another way is to transform the sin-gular integrals using Duffy’s transformation to remove the singularity [ES98].

15

Experiments in [YT03] show that this procedure is more sensitive to the shapeof the triangles.

16

The Fast Multipole Method

The equation from the Method of Moments discretization ZI = V is an N ×Ndense system of linear equations with Z complex. In the case of EFIE the ma-trix is complex symmetric, that is Z = ZT . The storage of the system requiresO(N2)

memory positions. Computing the solution requires O(N3)

arithmeticoperations when Gaussian elimination is used. Instead one can use iterativemethods like the Generalized Minimal RESidual method (GMRES) [SS86].Iterative methods require O

(MN2

)arithmetic operations, where M is the num-

ber of matrix vector multiplications required to meet some convergence crite-ria. The number of iterations depends on the condition number of the problem.For ill-conditioned problems an iterative method can be as slow as a directmethod or worse. A good preconditioner can reduce the number of iterations.

For the integral equations it is usually the memory that limits the size of theproblem one can solve. With about 10000 unknowns the storage requirementis larger than the memory size of most computers. One can resort to parallelcomputers to solve this problem, but in the order of 100000 unknowns thememory requirement is again prohibiting. In [RBBD01] it is reported that100000 unknowns can be solved with Gaussian elimination and requires 180Gb of memory and 43 hours of computing time on a supercomputer.

A parallel solver for boundary integral methods is described in [Edl99]. Ituses a dense block LDLT -solver to factorize the impedance matrix. This meansthat it only works for EFIE. A parallel iterative solver is described in [Nil99].It is also only applicable to EFIE and is based on a block symmetric matrixvector multiplication algorithm. It uses the symmetric Quasi-Minimal Resid-ual method (QMR) to solve the linear system and is preconditioned by a SparseApproximate Inverse preconditioner (SPAI) which is developed further in Pa-per IV and in [Nil02].

4.1 Paper IIt is evident that faster methods are needed in order to solve large scale prob-lems in reasonable time. The Fast Multipole Method [CRW93] is one such fastmethod. It is based on a diagonal approximation of the Green’s function

eıκ|x−x′|

4π |x−x′| ≈K

∑k=1

eıκk·(x−Xm)T Lk

(κ,Xm −X′

m

)eıκk·(X′

m−x′) (4.1)

17

where T Lk (κ,X) is called the translation operator and is defined by

T Lk (κ,X) =

ıκ16π2 wk

L

∑l=0

ıl (2l +1)h(1)l (κX)Pl

(κk · X

)(4.2)

Here, wk is the quadrature weight associated with plane wave direction κk,the function h(1)

l (x) is a spherical Hankel function and Pl (x) is a Legendrepolynomial of order l. The quadrature weights wk are usually chosen suchthat they integrate the 2L first spherical harmonics exactly. This implies thaterror in (4.1) only depends on the truncation number L in exact arithmetic. Aproblem with (4.1) is that the approximation becomes numerically unstable forlarge values on L. This is due to the finite precision in computer arithmetic andthe divergence of the spherical Hankel function for large order l and constantargument.

The Fast Multipole Method divides an object into a number of equally sizedboxes. Equation (4.1) approximates the Green’s function for two points x andx′, when the respective boxes they are in are far apart. The boxes that must bebetween the two points in order for them to be considered far apart are calledbuffer boxes.

In Paper I error estimates are given. Bounds on L where the approximationof the Green’s function is stable are also given. When the side length of a boxis a and the number of buffer boxes are n it is found that for

√3κa ≤ L <

(n+1)κa the relative error ε can be obtained if

L ≥√

3κa+1.8(− log10 ε)23

(√3κa) 1

3(4.3)

When L > (n+1)κa one should choose the truncation number as

L ≥ (n+1)κa+

log10 (ε)−(

(n+1−√3)κa

1.8(√

3κa)13

) 32

log10

√3

n+1

(4.4)

It is however important to note that the error level can only be achieved if

L < (n+1)κa+1.8

(log10

(ε

CnCεκa

))2/3

((n+1)κa)1/3 (4.5)

where Cn = 2(n+1+

√3)

and Cε is related to the relative error in the ap-proximations of the different parts in the Fast Multipole Method. Numericalexperiments validate the theory.

18

4.2 Paper IIThe Fast Multipole Method for electromagnetics uses the expansion in (4.1)to compute the interactions between the different basis functions. The interac-tions are divided into near field interactions and far field interactions. The farfield interactions are computed from

Zkl ≈K

∑j=1

Rk (κ j) ·T Lj

(κ,Xm −X′

m

)Fl (κ j) (4.6)

where, Fl (κ) is called the far field pattern and Rk (κ) the receiving pattern.They are defined by⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Fl (κ) = κ× ∫Γ eıκ·(X′m−x′)jldΓ(x′)× κ

Rk (κ) =ακ×∫

Γeıκ·(x−Xm)j′kdΓ(x)× κ

+(1−α)∫

Γeıκ·(x−Xm)j′k × n(x)dΓ(x)× κ

(4.7)

The far field pattern and the receiving pattern can either be stored or computedwhen they are needed.

The object is enclosed in a box which is then recursively divided into eightsmaller boxes until the smallest boxes have a prescribed size. Boxes that con-tain basis functions are stored in a tree structure together with the parents ofthe boxes. Interactions between basis functions that are close to each other iscomputed by the near field formula (3.4). The near field interactions are usu-ally stored in a sparse matrix structure. The far field interactions are computedfrom (4.6) on the level of the tree where the interactions are considered nearinteractions on the higher levels.

The Fast Multipole reduces the computational time for a matrix vector mul-tiplication from O

(N2)

to O (N logN). The memory consumption is reducedfrom O

(N2)

to O (N logN). In practice the memory consumption is reducedto O (N) because the memory usage on the lowest level is dominant.

In Paper II a brief description of the Fast Multipole Method is given andsome experiments are performed to validate the accuracy and efficiency of theFast Multipole Method.

4.3 Paper IIIIn an implementation of the Fast Multipole Method there are several potentialbottle necks that must be handled appropriately if one wants to solve largeproblems.

The multilevel version of the Fast Multipole Method requires that interpo-lation is performed between levels. The interpolation can be made spectrally

19

accurate using the filter in [JCA97]. For low accuracy settings it is enough touse Lagrange interpolation on the sphere. This is more efficient than a spectralfilter. In Paper III it is shown how the arithmetic complexity of the inter-polation can be reduced from O

(Kp2

)to O (Kp), where p is the number of

interpolation points.The values of the translation operator that should be stored is a memory

bottle neck. A naive way of storing the values is to store them separate for eachbox. Using symmetries the storage requirement can be reduced dramatically.The first symmetry comes from the fact that the same combination of (κ,X)appears several times. Hence, it is better to store the possible combinationsof (κ,X) in a table and let a box that needs a certain combination of (κ,X)fetch it from there. The second symmetry has to do with the fact that theonly important values in (4.2) are κX and κ · X. Since the κ-values are pickedfrom certain quadrature rules with symmetries on the sphere and the X-valuescomes from a Cartesian grid, the same combinations of κX and κ · X appearfor different X. This is explained in Paper III. An algorithm that uses this factis given in [Nil02]. These symmetries can also be used to reduce the memoryfor the exponential translation operator, which is used in the multilevel versionof FMM.

Another way to save memory is to use the symmetries for the far field andradiation patterns given in Paper III. They are present when the wavenumberis real.

A third way of saving memory is to prescribe the size of the box on thelowest level and construct a tree in an upward pass. The effect is that thenear memory, which is the dominant memory consumption is reduced. This isbecause the smallest box size is not always achieved in a downward pass. Asimple estimate of the impact on the memory consumption is given in PaperIII and validated by experiments. The price is a slight reduction in speed.

The parallelization of the Fast Multipole Method can be performed in twoways. Parallelization over boxes and parallelization over quadrature points.The parallelization over boxes is more advantageous on the lower levels be-cause the number of boxes is large, while the number of quadrature points islow. On the highest levels the situation is the other way around, so it is betterto parallelize over quadrature points there. A hybrid between the two methods,based on blocking, is proposed in Paper III.

A parallel shared memory implementation is described in Paper III. It com-bines the parallel strategies described. The experiments in Paper III show thatthe hybrid method improves the scalability of the parallel method.

20

Acceleration techniques for iterative methods

The total computational cost for solving one right hand side in an iterativemethod that uses the Fast Multipole Method is O (MN logN), where M is thenumber of iterations. Gaussian elimination requires O

(N3)

arithmetic opera-tions for the factorization phase, but only O

(N2)

arithmetic operations for thesubstitution phase. This implies that the advantage of any fast method that usesiterative techniques diminishes as the number of right hand sides is increased.

The computational cost of the iterative solver depends on the number of ma-trix vector multiplications. In order for the fast methods to compete with Gaus-sian elimination the number of matrix vector multiplications for each righthand side must be small.

5.1 Paper IVSince the EFIE is a Fredholm integral equation of the first kind it is not wellconditioned. This implies that the convergence rate of an iterative method likeGMRES or QMR is slow, so M is large in this case. An obvious solution isto use CFIE instead, but CFIE can only be used when the object is a closedsurface.

Block iterative methods are one way of reducing the number of iterationsper right hand side. They are based on block Krylov subspaces rather thanKrylov subspaces. In Paper IV a block Quasi-Minimal Residual (block-QMR)algorithm is used. It is demonstrated that for EFIE it can be very effective inreducing the work for each right hand side. The symmetric version withoutpreconditioner was given in [SG93]. The improvement is the ability to use asymmetric preconditioner. The theoretical proof of the correctness of the al-gorithm is given in [Nil02]. The algorithm is repeated in Algorithm 2 for thereader. It solves the system ZX = B, where Z is complex symmetric N ×Nmatrix. As input it takes Z, the complex symmetric N ×N preconditioner M,the N × s block of right hand side vectors B and the initial guess X. The ap-proximate solution X ≈ Z−1B is returned as output. In the algorithm the mod-ified Gram-Schmidt procedure in Algorithm 1 (gram M sym) and the blockversion of Givens rotations in [SG93, Nil02] (givens) are required. The modi-fied Gram-Schmidt procedure computes biorthogonal N× s vectors QT V = Is,where Is is the s×s identity matrix, given a block of vectors V and the complex

21

Algorithm 1 A Gram-Schmidt process modified to handle the case of con-structing M-biorthogonal vectors.

Require: V, MEnsure: V, Q, ρ, ψ, β

ρ = 0, ψ = 0, β = 0V = VQ = MVfor j = 1 : s−1 do

δ j = ||V:, j||2V:, j = V:, j

δ j

Q:, j = Q:, j

δ j

ρ j:s, j = QT:, j:s V:, j

β j, j = ρ j, jψ j, j+1:s = ρ j+1:s, jρ j, j = δ jρ j, jψ j, j = δ j

V:, j = V:, j

β j, j

V:, j+1:s = V:, j+1:s −V:, j ρTj+1:s, j

Q:, j+1:s = Q:, j+1:s −Q:, j ψ j, j+1:send forδs = ||V:,s||2V:,s = V:,s

δs

Q:,s = Q:,s

δs

βs,s = QT:,s V:,s

ρs,s = δsβs,sψs,s = δs

V:,s = V:,s

βs,s

symmetric N ×N matrix M. The output of the algorithm are the block vectorsQ and V and the s×s block matrices ρ, ψ and β. They are related to each otherby VρT = V, Qψ = MV and ψT = ρβ−1.

Another way of reducing the number of iterations is to use preconditionersto improve the condition number. Consider iterative solution of the systemZX = V. A natural idea is to solve an equation on the form M1ZM2X = M1V,where X = M2X. If M2M1 ≈ Z−1 the new problem is well-conditioned andconverges quickly since M1ZM2 ≈ I, where I is the identity matrix. The prob-lem is now to construct M1 and M2. The preconditioners are called left andright preconditioners respectively. Some natural requirements on a precondi-tioner is that it should not be more expensive to apply the preconditioner than

22

Algorithm 2 The Block-QMR algorithm for several right-hand sides.Require: B, X, Z MEnsure: X ≈ Z−1B

V0 = B−ZXX0 = X, R0 = V0

D0 = 0, U0 = 0G0 = I[V0,ρ0,Q0,ψ0,β0] = gram M sym(V0,M)P0 = Q0

τ0 = ψ0for k = 1 : maxiter do

Lk = ZPk−1

εk = PTk−1Lk

φk = β−1k−1εk

Vk = Lk −Vk−1εk

[Vk,ρk,Qk,ψk,βk] = gram M sym(Vk,M)ζk = Gk−1 (1 : s,s+1 : 2s)φkχk = Gk−1 (s+1 : 2s,s+1 : 2s)φk[Gk, τk,τk−1,γk] = givens(χk,ψk, τk−1)Dk = (Pk−1 −Dk−1ζk)γ−1

kXk = Xk−1 +Dkτk−1

Uk = (Lk −Uk−1ζk)γ−1k

Rk = Rk−1 −Ukτk−1

if Convergence is achieved thenX = Xk

Breakend ifδk = ε−1

k ρkPk = Qk −Pk−1δk

end for

to compute the original matrix vector product, it should not be more expensiveto store than the original matrix and the construction time of the preconditionershould be low compared to the work reduction in the iterative solver. Anothernatural requirement is that the construction of the preconditioner should paral-lelize easily, since a parallel solver is used.

In Paper IV a modified SParse Approximate Inverse preconditioner (SPAI)is used. The preconditioner is used as a right preconditioner, i.e. M1 = I,because the residual of the original problem remains unchanged in this case.The preconditioner was originally suggested in [Nil99] and is further analyzedin [Nil02]. It should be mentioned that a similar preconditioner have beendeveloped in [Car02].

23

Let M be the preconditioner generated by SPAI. It is the solution of a Frobe-nius norm minimization problem

minM∈A

‖I−ZM‖2F =

N

∑i=1

minM∈A

‖(I−ZM)ei‖22 (5.1)

where A is a constraint on the sparsity pattern of M. If the sparsity patternis chosen so that all entries in M are allowed to be nonzero M = Z−1. Theproblem is now to find a sparsity pattern that makes the minimization problemcheap to solve and gives a matrix M ≈ Z−1. The method is attractive since theproblem naturally reduces to N uncoupled problems, which makes paralleliza-tion simple.

The Fast Multipole Method only stores the near field of the matrix Z soone natural restriction on A is that the nonzero entries should be restricted tothe near field entries in Z. The number of nonzero entries can be made evensmaller by considering some distance measure to the center of the boxes in theFast Multipole Method.

Equation (5.1) is still difficult to solve. It requires that all values in Z fromthe columns needed in each minimization problem (5.1) are known. SinceG(x,x′) → 0 when |x−x′| → ∞ a simple modification is to only consider thevalues in the near field part instead. Some justification for this choice is givenin [Nil02]. With this choice one can prove that the construction time of thepreconditioner is O

(Nk2

)and that the application time is O (Nk), where k is

the average number of non zero values in each column of the matrix M [Nil02].The results in Paper IV show that for the EFIE a block-QMR method to-

gether with SPAI can have a huge impact on the number of iterations for eachright hand side.

5.2 Paper V

The solution time can be greatly reduced if the right hand sides are linearlydependent on each other. In this case it is enough to solve for vectors that spanthe space of the right hand sides and construct the solution to the right handsides from them. In Paper V it is shown that if the vectors depend smoothlybut nonlinearly on a parameter it is possible to find a subspace that accuratelypredicts the solutions.

Consider the problem Axi = bi for i = 1 . . .M. Assume m < M right handsides have been solved by an iterative method such that ri = bi−Axi = bi− si.Compute a guess to the next right hand side based on a linear combination ofthe previous right hand sides x0

m+1 = ∑mi=1 yixi = Xmym. If [s1 . . .sm] = QSmRSm ,

24

one can prove that the choice

x(0)m+1 = XmR−1

SmQH

Smbm+1

r(0)m+1 = (I−QSmQH

Sm)bm+1

(5.2)

minimize ‖r(0)m+1‖ the residual of the initial guess.

In Paper V we propose a simple strategy for picking the right hand sideswhen the dependence on the smoothness parameter is known. The right handsides are picked such that the space in the smoothness parameter is divided in abinary tree like fashion. The method is called the Minimal Residual Interpola-tion method (MRI). The analysis in Paper V show that it is an optimal method,in a certain sense. The main theorem in Paper V explains why this methodworks

Theorem 1 Assume that the components in the right hand side vectors bi =b(φi) have p continuous derivatives in φ, ‖ri‖ ≤ εI , and that an approximationto bα at φα is computed at level l by the minimization

miny

‖bα −p

∑i=1

siyi‖.

Then

‖bα −p

∑i=1

siyi‖ ≤ min(√

Nb(p)max∆φp

l−1 +√

p‖l‖εI,‖bα‖),

where b(p)max = max j maxφ |b(p)

j (φ)|, b(p)j is the p:th derivative of b j, and l con-

sists of the coefficients of the Lagrange polynomial at the point φα.

In fact, Theorem 1 suggests that if the right hand sides in a problem dependsmoothly on a parameter, a Singular Value Decomposition (SVD) [GvL96]can be used to find a subspace that approximates the right hand sides. How-ever, computing a SVD is expensive and not really necessary if the strategysuggested here is used instead.

In Paper V we also consider the special case of computing monostatic RCSat a fixed frequency. Although an approximate bound is given on the numberof right hand sides that span the subspace, a better estimate based on multipoletheory is given in Paper VI.

Several experimental results are presented that justify the method proposed.Specifically the convergence rate shows excellent agreement with the predic-tion in Theorem 1 when ∆φ → 0.

25

5.3 Paper VIAll aspects of the proposed methods in Paper V were not satisfactorily ad-dressed. In Paper VI some of these issues are dealt with in more detail.

A technique for handling the case when the matrix A depends on a smoothparameter was presented in Paper V. In paper VI the proposed method is im-plemented and tested on the case of electromagnetic scattering. Consider theproblem Aixi = bi for i = 1 . . .M and assume that m < M problems have beensolved. The change from Paper V is that we consider si,m+1 defined by

si,m+1 = Am+1xi = bi − ri +(Am+1 −Ai)xi, i = 1 . . .m (5.3)

and compute the initial guess and residual from

x(0)m+1 = XmR−1

Sm,m+1QH

Sm,m+1bm+1

r(0)m+1 = (I−QSm,m+1Q

HSm,m+1

)bm+1

(5.4)

A theorem related to the convergence rate is given in Paper VI. It is anextension of Theorem 1.

Theorem 2 Assume that the components in the right hand side vectors bi =b(φi) and the matrices Ai = A(φi) have p continuous derivatives in φ and letA = A(φ) and b = b(φ) , ‖ri‖ ≤ εI , |det(A(φ))| ≥C > 0 for some constant Cand φmin ≤ φ ≤ φmax, and that an approximation to bα at φα is computed bythe minimization

miny

‖bα −p

∑i=1

si,αyi‖.

Then

‖bα −p

∑i=1

si,αyi‖ ≤ min(√

N(AαA−1b

)(p)max ∆φp

l−1

+√

p‖l‖ max‖AαA−1‖εI,‖bα‖),

where(AαA−1b

)(p)max = maxi maxφ |∑N

j=1 (Aα)i j x(p)j (φ)|, x(p)

j is the p:th deriva-tive of x j = Dj/D where D = det(A) and Dj is the determinant of the matrixwith the j:th column of A replaced by b, and l consists of the coefficients of theLagrange polynomial at the point φα.

In the case of electromagnetic scattering the matrix A is the impedance ma-trix, which depends on the frequency. Usually one is interested in the monos-tatic RCS in a frequency band. In Paper VI, MRI is used to compute the initialguesses in the frequency sweep. It should be noted that since the Fast Mul-tipole Method is used to compute the solution, the matrix is not completely

26

smooth. This is because the number of terms used in the expansion varies withfrequency according to the estimates in Paper I.

Paper VI also discusses the possibility of using a version of the block-seedGMRES method to compute initial guesses. It is noted that if enough righthand sides are solved in a block, the initial guesses for the other right handsides will be accurate enough. However, one can only be certain that this oc-curs if the block size is as large as the number of vectors required in MRI toaccurately predict the remaining solutions. Since the Krylov subspace in GM-RES is larger than the block this strategy requires a lot of memory comparedto MRI.

For the case of monostatic RCS computations a formula is given to estimatethe number of right hand sides that must be solved before the remaining righthand sides can be predicted by MRI. It is K = O

(L2)

when the entire sphere ofmonostatic directions is considered and K = O (L) when a plane of monostaticdirections is considered. Here, the number L is computed from the formula

L ≈ κR+1.8(− log10 εI)23 (κR)

13 (5.5)

where R is the radius of the smallest sphere enclosing the object.There are still open issues from Paper V. The analyses there implies that

local interpolation can be used instead of global. A way of computing thebistatic RCS for all incident directions is also presented. These issues will beinvestigated in a separate paper.

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28

A hybrid method

The Physical Optics approximation (PO) is a high frequency asymptotic ap-proximation of the solution to the MFIE. Consider the surface of an infiniteperfect electric conducting ground plane. On the surface of the plane the inte-gral operator KJ satisfies n×KJ = 0. Equation (2.17) implies that the electricsurface current J = 2n×Ha. For smooth electrically large surfaces a reason-able approximation is then J ≈ 2n×Ha, which can also be stated in a Galerkinform as in [Edl01, ELS03]

∫Γ

J ·J′dΓ ≈ 2∫

Γn×Ha ·J′dΓ (6.1)

This is the PO formulation used here. Discretizing with RWG-basis functionsleads to a sparse system matrix ZPO. It is positive definite and real since it isa mass matrix. This implies that the Conjugate Gradient method (CG) can beused to solve the problem iteratively. The experience in [Edl01, ELS03] andalso Paper VII is that the solution is obtained in a few iterations.

One way of improving the result from (6.1) is to apply shadowing. Theassumption is that the electric current J ≈ 0 if the current is not directly visibleto the source of the applied field. This implies that n×Ha ≈ 0, which is usedin the right hand side of (6.1). A simple way to determine if a patch is visibleto a plane wave traveling in direction −κa is to use the normal test

n×Ha =

n×Ha if n · κa ≥ 0

0 if n · κa < 0(6.2)

More refined methods are described in [Sef03].The PO method can be combined with the Method of Moments in a hybrid

method following [Edl01, ELS03]. This method is called the MM-PO hybridmethod. The surface is divided into two parts and solved by a Galerkin formu-lation. One part, which is considered smooth is solved with the PO-method.The other part, which yields a matrix ZMM is solved with the Method of Mo-ments. There are also matrices describing the couplings from PO to MM andMM to PO denoted by ZPOMM and ZMMPO. Since an approximation to MFIEis solved in the PO-part, it is natural to use MFIE in the Method of Momentspart as well. From experience CFIE or EFIE can also be used in the Method

29

of Moments part. The resulting system that should be solved is(ZMM ZPOMM

ZMMPO ZPO

)(IMM

IPO

)=

(VMM

VPO

)(6.3)

The advantage of the Galerkin formulation is that all the integrals appearingin (6.3) are already implemented in the Method of Moments part. It does notrequire any special treatment of the singularities either. On the other handwhen the MM and PO parts are distant from each other, the use of such rigorousapproximations in the coupling seem less motivated.

Shadowing can improve the solution and can be used on VPO, but also onZMMPO. An entry in ZMMPO is then approximated to zero if the triangles aredetermined not to be visible to each other.

6.1 Paper VIIThe matrices ZMM, ZMMPO and ZPOMM can be approximated by the Fast Mul-tipole Method. In Paper VII one such implementation is described. Equa-tion (6.3) is solved by iterative block Gauss-Seidel. The subsystems in itera-tion k, ZMMI(k)

MM = VMM −ZPOMMI(k)PO and ZPOI(k)

PO = VPO −ZMMPOI(k−1)MM , are

solved by iterative methods, where the initial guess is taken from the previ-ous iteration. Assume that the Gauss-Seidel method is required to fulfill theconvergence criteria ‖ZI−V‖ ≤ ε‖V‖. This is ensured if

‖ZMMIMM +ZPOMMIPO −VMM‖ ≤ 0.5ε‖V‖‖ZPOIPO +ZMMPOIMM −VPO‖ ≤ 0.5ε‖V‖ (6.4)

In Paper VII it is proven that when the convergence rate of the Gauss-Seidelmethod is fast, the method is faster than if the Fast Multipole Method is usedon the entire object.

For multiple right hand sides MRI in Paper V and Paper VI is used. The ex-periments in Paper VII show that MRI can reduce the solution time for sparseFinite Element type approximations like the PO approximation. Also, the hy-brid method benefits from MRI and a significant reduction of the solution timeis obtained. This demonstrates that MRI is independent of the iterative method.

30

Conclusions

This thesis presents fast methods for solving large scale electromagnetic prob-lems.

The Fast Multipole Method computes a dense matrix vector multiplicationin O (N logN) arithmetic operations and only requires O (N logN) memory.This increases the frequency range where the Method of Moments can be used.A scalable parallel implementation is described. It allows the solution of a1 million problem to be computed in a few minutes. A 5 million unknownproblem was solved on 24 processors of a SUNFire 15k server in 50 minuteswith this implementation. It required 27 Gb of memory. The error in theFast Multipole Method is analyzed and error estimates are given. A stabilitycriterion is also provided.

The convergence rate of iterative methods is improved by using block meth-ods and appropriate preconditioning. Minimal Residual Interpolation is a lin-ear algebra method for reducing the work needed in the iterative solver byfinding an appropriate subspace that accurately spans the space of right handsides and uses it as an initial guess. The method can also be used in frequencysweeps to compute an initial guess.

For very large problems where the Fast Multipole Method falls short, ahybrid method with Physical Optics is presented. The method is faster thanthe Fast Multipole Method. Minimal Residual Interpolation can be used in thiscase as well. This demonstrates its versatility.

Several open questions remain. Some of them are: How to pick the righthand sides in an optimal way in MRI. How to decide the areas that should beapproximated by Physical Optics in the MM-PO hybrid. How close to optimalspeed the implementation of the Fast Multipole Method is. How to reduce theconstant in front of the scaling in the Fast Multipole Method.

31

32

Acknowledgments

I wish to thank my supervisor Per Lotstedt for all his valuable advice duringthis research and always keeping an open door. Especially for the work onPaper V.

I would also like to thank Bo Strand at Aerotech Telub who has helped me alot over the years. He is the one who initiated the project and has always beensupportive of my work.

I also thank my previous colleague Johan Edlund. He has developed a largepart of the Method of Moments code and the Physical Optics code which mysolver is based on.

I would like to thank Fredrik Edelvik and Gunnar Ledfelt for helping meout from time to time and providing time domain results for comparison andAnders Alund for all the important debugging work.

I am thankful to all the people involved in GEMS and SMART that havecontributed to this work. A special thanks to Jonas Hamberg who providedseveral of the geometries.

I am grateful to Prof. Abderrahmane Bendali and Dr. M’Barek Fares atCERFACS, Toulouse, who taught us integral equation methods and providedthe original Method of Moments code. Without their initial help it is doubtfulthat I would have come this far.

Thanks to all the people at TDB. You make this department fun to work at.Finally, I want to thank my family for there love and support. A special

thanks to Helen and William. I love you.This work was supported by the CEM program at the Parallel and Scien-

tific Computing Institute (PSCI) through the General ElectroMagnetic Solversproject (GEMS) and the Signature Modeling And Reduction Tools project(SMART). Financial support was provided by the National Aeronautical Re-search Program (NFFP), Vinnova, Aerotech Telub and Ericsson MicrowaveSystems AB. During my research I have been able to participate in a numberof conferences and the Zurich Summer School thanks to: PSCI, a scholarshipfrom Prof. Stefan Sauter at Universitat Zurich, and Rektors reseanslag franWallenbergstiftelsen at Uppsala University.

33

34

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Acta Universitatis UpsaliensisComprehensive Summaries of Uppsala Dissertations

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