Thesis

Computational Electromagnetics:

Software Development and High Frequency

Modelling of Surface Currents on Perfect Conductors

SANDY SEFI

Doctoral Thesis

Stockholm, Sweden 2005

TRITA-NA-0541ISSN 0348-2952ISRN KTH/NA/R-05/41-SEISBN 91-7178-203-6

KTH School of Computer Science and CommunicationSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i numeriskanalys och datalogi fredagen den 27:e januari 2006 klockan 10.00 i sal E3, Lind-stedtsvägen 3, Kungl Tekniska högskolan, Stockholm.

© Sandy Sefi, December 2005

Tryck: Universitetsservice US AB

iii

Abstract

In high frequency computational electromagnetics, rigorous numerical methods be-come unrealistic tools due to computational demand increasing with the frequency. Insteadapproximations to the solutions of the Maxwell equations can be employed to evaluate theelectromagnetic fields.

In this thesis, we present the implementations of three high frequency approximatemethods. The first two, namely the Geometrical Theory of Diffraction (GTD) and thePhysical Optics (PO), are commonly used approximations. The third is a new inventionthat will be referred to as the Surface Current Extraction-Extrapolation (SCEE).

Specifically, the GTD solver is a flexible and modular software package which usesNon-Uniform Rational B-spline (NURBS) surfaces to model complex geometries.

The PO solver is based on a triangular description of the surfaces and includes fastshadowing by ray tracing as well as contribution from edges to the scattered fields. GTDray tracing was combined with the PO solver by a well thought-out software architecture.Both implementations are now part of the GEMS software suite, the General ElectroMag-netic Solvers, which incorporates state-of-the-art numerical methods. During validations,both GTD and PO techniques turned out not to be accurate enough to meet the indus-trial standards, thus creating the need for a new fast approximate method providing bettercontrol of the approximations.

In the SCEE approach, we construct high frequency approximate surface currents ex-trapolated from rigourous Method of Moments (MoM) models at lower frequency. Todo so, the low frequency currents are projected onto special basis vectors defined on thesurface relative to the direction of the incident magnetic field. In such configuration, weobserve that each component displays systematic spatial patterns evolving over frequencyin close correlation with the incident magnetic field, thus allowing us to formulate a fre-quency model for each component. This new approach is fast, provides good control of theerror and represents a platform for future development of high frequency approximations.

As an application, we have used these tools to analyse the radar detectability of a newmarine distress signaling device. The device, called "Rescue-Wing", works as an inflatableradar reflector designed to provide a strong radar echo useful for detection and positioningduring rescue operations of persons missing at sea.

Keywords: High frequency approximations, Maxwell’s equations, Method of Moments,

Physical Optics, Geometrical Theory of Diffraction, Extraction Extrapolation.

Preface

This thesis is focused on the development of high frequency techniques in compu-tational electromagnetics. It is composed of two parts: first, an introduction, givingbackground to the subject, followed by a list of papers, each describing relevantresults.

• Paper 1: Sandy Sefi, Fredrik Bergholm, “Modeling and Extrapolating High-frequency Electromagnetic Currents on Conducting Obstacles”. ProceedingsICNAAM 2005 International Conference of Numerical Analysis and AppliedMathematics, Rhodes, Greece, September 2005.

• Paper 2: Sandy Sefi, Fredrik Bergholm, “Extrapolation and Modelling of Met-hod of Moments Currents on a PEC surface”. Technical report, TRITA-NA-0539, Royal Institute of Technology presented at the MMWP05 Conference onMathematical Modelling of Wave Phenomena, Växjö, Sweden, August 2005.

• Paper 3: Sandy Sefi, Jesper Oppelstrup, “Physical Optics And NURBS forRCS Calculations”. Proceedings EMB04 Conference on Computational Electro-magnetics - Methods and Applications, Göteborg, Sweden, October 2004.

• Paper 4: Tomas Melin, Sandy Sefi, “The Rescue Wing: Design of a MarineDistress Signaling Device”. Proceedings OCEANS 2005 Conference on OceanScience sponsored by the Marine Technology Society (MTS) and the IEEEOceanic Engineering Society, Washington DC, United States, September 2005.

• Paper 5: Stefan Hagdahl, Fredrik Bergholm, Sandy Sefi, “Modular Appro-ach to GTD in the Context of Solving Large Hybrid Problems”. ProceedingsAP2000 Millennium Conference on Antennas & Propagation, Davos, Switzer-land, April 2000.

• Paper 6: Sandy Sefi, “Architecture and Geometrical Algorithms in MIRA, aRay-based Electromagnetic Wave Simulator”. Proceedings EMB01 Conferenceon Computational Electromagnetics - Methods and Applications, Uppsala,Sweden, November 2001.

v

Acknowledgements

This oeuvre has been carried out from August 2000 to September 2005 at theDepartment of Numerical Analysis and Computer Science (NADA), at the RoyalInstitute of Technology (KTH), in Stockholm, Sweden. During these years, it hasbeen a true privilege to study at NADA. For this, I am grateful to my supervisorProf. Jesper Oppelstrup who offered me such an opportunity. I thank him for histime, his constant encouragement and for always enlighting the positive side of asituation.

I am also grateful to my coauthor in half of the papers, Fredrik Bergholm,for providing invaluable assistance and ultimately made this thesis a reality. Hisinvolvement has considerably enriched this work and I deeply appreciate his willing-ness to help. I also thank Stefan Hagdahl for his very early efforts as a contributoras well as Tomas Melin for a pleasant and rewarding collaboration and Martin Nils-son for his comments on the early draft. Lastly, a very special thank you to myfiancée Eva for her encouragement and great support.

Financial support has been provided by NADA, KTH, PSCI and the NationalAeronautical Research Program (NFFP) within the General ElectroMagnetic Solv-ers (GEMS) and the Signature Modeling and Reduction Tools (SMART) projects.

vii

Vita

Sandy Sefi was born on March 11, 1974 in Grenoble, France where he graduated in1997 from the Joseph Fourier University (UJF) with a Degree in Applied Mathem-atics and Computer Science. He spent the last five months of his graduate diplomaworking with Prof. Patrick Chenin in the LMC Laboratory at the Institute for Ap-plied Mathematics of Grenoble (IMAG) on the development of a computer programfor 3D geometrical modelling. After graduation, he was awarded a scholarship fromthe French Government to pursue a Master of Science Degree (DESS) in IngénierieMathématiques at the same university.

In May 1998, he left the country to complete a training period at NADA, KTH,in Sweden and in December the same year he enlisted for a sixteen months Frenchnational service as Coopérant du Service National en Suède, for which he obtaineda scholarship from NADA to work on a survey of numerical methods in Computa-tional Electromagnetics. At the same time, he begun to work in the GEMS projectas well as on the Diplôme de Recherche Technologique (DRT) which he defended inautumn 2000 at UJF. The same year he was granted a Ph.D. position at NADAfocused on the development of high frequency methods under the supervision ofProf. Jesper Oppelstrup.

In June 2003, he obtained the Licentiate Degree in Numerical Analysis andComputer Science at KTH concluding his work on ray tracing in Electromagnetics.

ix

Contents

Contents xi

1 Introduction 11.1 Outline and main results . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Background: the GEMS project . . . . . . . . . . . . . . . . . . . . . 4

2 Governing Field Equations 72.1 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Boundary conditions on a Perfect Conductor . . . . . . . . . . . . . 102.3 Plane wave solution in free space . . . . . . . . . . . . . . . . . . . . 122.4 The vector Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . 122.5 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 The field integral equations . . . . . . . . . . . . . . . . . . . . . . . 15

3 Scattering Analysis Methods 193.1 Geometrical Theory of Diffraction . . . . . . . . . . . . . . . . . . . 193.2 Physical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Modelling the behavior of the surface currents . . . . . . . . . . . . . 26

4 Summary of the Papers 314.1 Paper 1: Modelling and Extrapolation, an Extended Abstract . . . 314.2 Paper 2: Modelling and Extrapolation, Continued . . . . . . . . . . 314.3 Paper 3: Physical Optics and NURBS . . . . . . . . . . . . . . . . 324.4 Paper 4: The Rescue Wing, an Engineering Application . . . . . . 324.5 Paper 5: MIRA, a Modular Approach to GTD . . . . . . . . . . . . 334.6 Paper 6: Architecture and Geometrical Algorithms in MIRA . . . . 34

Bibliography 35

xi

Chapter 1

Introduction

Computational electromagnetics (CEM) may be defined as the branch of electro-magnetics that involves the use of computers to simulate electric and magneticsources as well as the fields these sources produce in specified environments. In thiswork we address the CEM problem of predicting, as realistically as possible, theelectromagnetic behavior of a three-dimensional conducting structure subjected toan incident wave.

Such models can be applied to a large variety of industrial applications. Com-putation of mobile phone coverage, design and placement of transmitter or receiverantennas and stealth aircraft technologies, are a few examples. In the latter applic-ation, CEM simulations are conducted to assist the analysis of the radar signatureof an aircraft, either during the design or at manufacturing stages. Ultimately theyaim to replace very expensive and time consuming measurements.

In all these diverse applications the physics is the same: an incident field Einc

or current source1 induces a current Js on the surface of the conductor, which, inturn, radiates back a scattered field response Escat propagating throughout space.

A8−132

Figure 1.1: Field scattered by an aircraft. In the monostatic configuration we lookfor reflections in the radar direction.

1For example in the case of transmitting antenna.

1

2 CHAPTER 1. INTRODUCTION

Optimization

Measurements

RCS

Mesh

CAD

CEM Solver for Maxwell Equations

Prototype

E inc

Escat

Figure 1.2: Design chain leading to the radar signature.

This dynamic interaction can be modeled in several ways. Our methods workin the frequency domain where all the excitations and their responses are sinusoidalwaves of the same frequency. This allows removal of the time dependence in thevariables. For radar applications, we focus on the high frequency range, a regimewhere the wavelengths of the electromagnetic waves are much shorter than the typ-ical dimension of the conductor. We also assume that all the scatterers are perfectelectric conductors (PEC). Physically this means that the fields do not penetratethe surfaces of the conductor resulting in a null electric field inside the conductor.The excitations are all plane waves and we look in particular at the monostaticresponse, a typical configuration, exemplified in Figure 1.1, where sources and re-ceivers are located at the same position. Figure 1.2 illustrates the typical designloop applied to the radar cross section (RCS) optimization. The RCS characterizesthe effective area illuminated by the incident plane wave which is scattered back tothe radar. A popular description of the concept of RCS can be found in Paper 4.RCS is measured in square meters, often normalized and displayed using a logar-ithmic scale defined in decibel square-meters (dBsm), see Figure 1.3. The reference0dB corresponds to the return from a sphere of 1m2 cross section.

1.1 Outline and main results

The starting point for modelling the dynamics of the fields is provided by thefundamental Maxwell equations, to be discussed in detail in Chapter 2. Theirsolutions in the frequency domain give rise to a great variety of numerical methods.Three of them have been studied and implemented in this work. We classify theminto two main groups of frequency methods, the ray-based and the current-basedmethods, see Chapter 3.

High frequency approximations which employ “rays” for the computation of thefields will be referred to as ray-based. More precise definitions of rays and their

1.1. OUTLINE AND MAIN RESULTS 3

Figure 1.3: Typical RCS values in m2 and in dBsm for a range of different objects.

propagation will be given later. These methods require known asymptotic highfrequency solutions such as Geometrical Optics and its generalization, the Geo-metrical Theory of Diffraction (GTD, UTD) [Keller 62, Pathak 84, McNamara 89].Ray-based methods are fast and have the advantage to provide a good insight inthe physics of the fields. Paper 5 introduces the implementation of our GTDsolver called MIRA and presents some illustrations for the rays. Paper 6 furtherdescribes the software architecture of MIRA and shows how to design a flexiblearchitecture which can handle a complex CAD geometry [Farin 88]. The idea is toavoid simplifications of the geometry of the scatterers and to use, for computations,the same CAD geometry as used for manufacturing the prototype which is requiredfor measurements.

When solving the Maxwell equations for the currents on the conductors, themethod will be referred to as current-based. A classic example is the crude current-based approximation called Physical Optics (PO). Paper 3 presents the imple-mentation of our PO solver. The aim was to assess efficiency and error analysis byfocusing on efficient numerical evaluation of the PO integral, which will be definedin Chapter 3. An adaptive triangular subdivision scheme for the surface integral isalso provided. In Paper 4, we use this tool to assist the design of a new passiveradar reflector signaling device to be used in case of distress at sea. This represents

4 CHAPTER 1. INTRODUCTION

a good illustration of the potential of these methods to solve real life engineeringproblems.

Both GTD and PO can provide very fast results but are approximations thatdo not incorporate a complete modelling of the physics and thus, typically degradein accuracy. The last contribution is this work, introduced as an extended abstractin Paper 1 and further developed in Paper 2, draws the basis of a new approx-imate current-based approach. After a careful examination of the related work inthe introduction, Paper 2 will lead us to the study of the extraction-extrapolationmethods. These methods are based on a simple yet powerful idea: to numeric-ally model (extrapolate) the behavior of the surface current vector fields at highfrequency, using information extracted from lower frequency solutions.

Our extraction-extrapolation approach obtains the behavior of the surface cur-rent from Method of Moments (MoM) models at lower frequency. The MoM, see[Harrington 68, Fares 99], can been viewed as exact brute force technique providingaccurate induced currents, but which do not incorporate any knowledge about thegeometrical structure of the vector field involved. The MoM produces a dense mat-rix problem, see Chapter 3 for detail, limiting its use to electrically small problems.In such a context, our goal is to provide a solution when the MoM is no longer anoption and when conventional high frequency techniques fail. In this niche, our newapproach, detailed in Paper 2, is fast, easy to implement and represent a platformfor future development of high frequency approximations.

1.2 Background: the GEMS project

All the numerical methods implemented are now part of the industrial software suitecalled GEMS: General ElectroMagnetic Solvers. GEMS has been developed in co-operation with Uppsala University, Chalmers University, Ericsson Microwave Sys-tems AB and Saab AB (Aerotech Telub) during the time period 1998 - 2005, makingthe GEMS project the largest Swedish CEM project ever. In the Swedish industry,GEMS is used by Aerotech Telub and by the Swedish Defense Research Agency(FOI) to simulate the radar performance of advanced stealth aircraft designs. Anexample of such a design, the unmanned aerial vehicle code-named Eikon, is illus-trated in Paper 3.

The software suite now incorporates a large choice of numerical tools for variousCEM applications both in time and frequency domain. In the latter, the methodsare efficient for the simulation of single frequency but inefficient for the simulationof broadband phenomena. Large electrical size problems can be solved using highfrequency approximations. Time domain methods have the exact opposite property,they are efficient over broad bands but can not efficiently handle high frequency.

In the time domain, GEMS specifies Finite Elements, Finite Volumes and FiniteDifferences, see [Taflove 95], as well as hybrid methods between these [Edelvik 02,Andersson 01, Ledfelt 01, Abenius 04] . In the frequency domain GEMS specifiesMethod of Moments, Fast Multipole Method, Geometrical Theory of Diffraction,

1.2. BACKGROUND: THE GEMS PROJECT 5

Physical Optics and Surface Current Extraction Extrapolation, as well as hybridmethods between these [Nilsson 04, Edlund 01, Hagdahl 05, Sefi 03].

The software is implemented for a number of different computer platforms,Linux, Sun, IBM, SGI and others, both in serial and parallel versions and mainlywritten in the Fortran 90 and MPI programming language. The version controlsystem CVS [1] is used to keep track of the changes. We use the netCDF [2]format for output data which allows visualization in Matlab [3] and OpenDX [4].The geometries are generated by the CAD software CADfix [5]. A more detaileddescription of the GEMS software suite can be found in [Oppelstrup 99].

Chapter 2

Governing Field Equations

In this chapter we describe how the Maxwell equations can be derived from the lawsof physics that govern electric and magnetic phenomena, such as Coulomb’s law andFaraday’s law. We will see that electromagnetic waves are created by time-varyingcurrents and charges governed by the Maxwell equations. We will demonstrate thatit is possible to write down the fields at a frequency ω > 0

E = iωA −∇φas solutions to the Maxwell equations in form of a complex sum of a magnetic vectorpotential A and the gradient of a scalar potential ∇φ. These two potentials will befound as solutions to vector differential Helmholtz equations, which will be derivedon the surface of the conductors from the Maxwell equations given a sensible choiceof boundary conditions. The vector potential A explains mostly the variations inthe magnetic field, whereas the scalar potential ∇φ can be linked to the variationof charge. Such analogy, when applied to the tangential component of the magneticfield on the surface - the current, will come in handy in Paper 1 and Paper 2 toexplain, to decompose and to predict how the surface currents behave.

First, let’s introduce the basic concepts necessary for the derivation of the equa-tions.

Identities for Zero Divergence and Zero Curl of a Vector Field In thederivation of the equations for the fields, the following identities for the nulls ofdivergence and curl, as well as Helmholtz’s theorem will be useful tools.

Identity 1 The curl of the gradient of any scalar field φ is identically zero,

∇× (∇φ) ≡ 0

It is also true that if the curl of a vector field B is zero, it can be written as thegradient of a scalar field φ, if ∇ × B ≡ 0 → ∃φ,B = −∇φ. This type of field iscalled conservative. This is valid in any system of coordinates. Thus we say that aconservative field can always be written as the gradient of a scalar field.

7

8 CHAPTER 2. GOVERNING FIELD EQUATIONS

Identity 2 The divergence of the curl of any vector field is identically zero,

∇ · (∇× A) ≡ 0

It follows that if the divergence of a vector field B is zero, it can be written as thecurl of another vector field A, if ∇ ·B ≡ 0 → ∃A,B = ∇×A. This type of field iscalled solenoidal.

Helmholtz’s Theorem 2.1 A general vector field v, which vanishes at infinity,can be represented as the sum of two independent vector fields; one that is conser-vative (zero curl) and another which is solenoidal (zero divergence),

v = ∇× A −∇φ

In other words, a general vector field which is zero at infinity is completely specifiedonce its divergence and curl are given.

2.1 The Maxwell Equations

This section results from [Stratton 41], to which the reader can return for moredetails. The Maxwell equations involve linear operations on vectorial quantitiesE,D,H,B, functions of position (r ∈ IR3) and time (t ∈ IR). The vector functionsD and B will later be eliminated from the description of the electromagnetic fieldsvia suitable constitutive relations. For now, we have

∇× E = −∂B

∂tFaraday’s law

∇ · D = ρv Gauss’ law for electric fields

∇× H = J + ∂D

∂tAmpere’s law

∇ · B = 0 Gauss’ law for magnetic fields

where in SI units,

E Electric field strength (or intensity) [V olt/m]H Magnetic field strength [Ampere/m]D Electric flux density (or displacement) [Coulomb/m2]B Magnetic flux density [Weber/m2]J Current density [Ampere/m2]ρv Volume charge density [Coulomb/m3]∇· Divergence operator∇× Curl operator

The first equation is called Faraday’s law and expresses how changing magneticfields produce electric fields. The first divergence condition is Gauss’s law anddescribes how electric charges produce electric fields. The distribution of charges is

2.1. THE MAXWELL EQUATIONS 9

given by the scalar charge density function ρv. The next equation is Ampere’s lawas modified by Maxwell. It describes how currents and changing electric fields whichact like currents, produce magnetic fields. Finally the second divergence equationexpresses the experimental absence of magnetic charges (magnetic monopoles donot exist) resulting in the fact that the magnetic flux is solenoidal.

We can simplify the notations by considering time-harmonic fields where thetime-varying fields E oscillating at temporal frequency ω = 2πf , are replaced bytheir corresponding vector phasor E(r, t) = Re[e−iωtE(r)]. This leads to the time-harmonic Maxwell equations for single frequency environments,

∇× E = iωB,∇ · D = ρ

∇× H = J − iωD

∇ · B = 0

The continuity equations for the current The two divergence conditions (thetwo Gauss laws) in the Maxwell equations are consequences of the fundamental curlfield equations provided charge is conserved. This is shown by taking the divergenceof the two curl equations (Faraday’s and Ampere’s law), recalling the null identityfor any vector field A, ∇ · (∇× A) ≡ 0, and by introducing the physical law thatrelate the density of current J to the distribution of charge ρv,

∇ · J = −∂ρv

∂tConservation of charge. (2.1)

It is then enough to study the two curl equations augmented with the continuityequation Eq. 2.1, see [Stratton 41], in the derivations of the theorems. Our motiv-ation is to use as few concepts as possible to describe the fields. In addition, wewill see that the two curl equations can be written as one second order equation.

The constitutive equations for the medium The Maxwell equations must beaugmented by two constitutive laws that relate E and H to D and B respectively,

E = ǫD and B = µH (2.2)

where we denote the electric permittivity ǫ = ǫrǫ0 and the magnetic permeabilityµ = µrµ0. These variables depend on the properties of the matter in the domainoccupied by the electromagnetic fields. ǫ0 and µ0 are constants, ǫ0 = 8.854 ×10−12Farads/m and µ0 = 4π×10−7Newton/Ampere2. ǫr and µr are dimensionlessscalar functions characterizing a medium. Vacuum, free space, air and materialwith no magnetic or no electric properties have µr = 1 and ǫr = 1. As a result, thespeed of the wave in such conditions is given by c = 1/

√ǫ0µ0.

In GEMS, frequency dispersive materials with ǫ(ω) and metals with µr 6= 1which behave nearly like PECs at radio frequency are of main interest. However,ferromagnetic materials with µr ≫ 1, e.g. magnets, require different models.


EH

k

∂

Ω

Ω

Figure 2.1: Scattering configuration of a plane wave inducing surface currents on aperfect conductor Ω.

Before any attempt to write a solution, boundary conditions for both electricand magnetic fields on the bounding surface should be specified to form a completeboundary value problem. In particular we are interested in the situation illustratedin Figure 2.1, where a plane wave travelling in a source-free medium encounters aPEC scatterer.

2.2 Boundary conditions on a Perfect Conductor

When the space near a set of charges contains electric material, the electric field nolonger has the same form as in vacuum. The interactions between the charges andthe electric field obey the boundary conditions of the Maxwell equations. Togetherwith the constitutive relations, Eq. 2.2, the Maxwell equations, in presence of a

2.2. BOUNDARY CONDITIONS ON A PERFECT CONDUCTOR 11

prefect conductor Ω, can now be restated as a complete system of equations,

∇× E = iωµH in R3 − Ω∇ · E = ρ

ǫin R3 − Ω

∇× H = J − iωǫE in R3 − Ω∇ · H = 0 in R3 − Ω

n × E = 0 on ∂Ωn · E = ρs on ∂Ω

n × H = Js on ∂Ωn · H = 0 on ∂Ω

where n is the normal to the surface S = ∂Ω of the PEC. The expression n ·E = ρs

represents the component of E along the normal n, where ρs is the surface chargedensity. The tangential component of E is given by n × E = 0. In other words,the tangential E-field must go to zero on the surface of a conductor. The normalcomponent of the magnetic field is given by n · H = 0 and the surface currentsare the tangential component of the magnetic field n × H = Js. Other boundaryconditions, for instance for dielectric media, are beyond the scope of this thesis.At large distance, it is assumed that only outgoing waves are present and that thefields are bounded when r → ∞.

The energy carried by the field propagates along the Poynting vector k,

k =1

µ0E × H (2.3)

which has the dimension of a power density, watts per square meter. This representsthe direction of the GTD rays described in Paper 5.

We now introduce the wavenumber κ = |k| which satisfies the following relationin terms of frequency ω and wavelength λ,

κ =ω

c=

2π

λ. (2.4)

The second order Maxwell equation The two curl equations for the electricand magnetic fields

∇× E = iωµH∇× H = J − iωǫE

are first order in spatial derivatives. They can be combined into one second orderequation as

∇× (∇× E) − κ2E = iωµJ (2.5)

thus removing H. Eq. 2.5 will be used later for the derivation of Helmholtz’sequations.


2.3 Plane wave solution in free space

In a source-free medium, the divergence of the electric field is zero, ∇·E = 0. Using

∇× (∇× E) = −∇2E + ∇(∇ · E) in R3 − Ω, (2.6)

the second order Maxwell system Eq. 2.5 in a source-free medium,

∇× (∇× E) − κ2E = 0, (2.7)

simplifies to

∇2E + κ2E = 0 in R3 − Ω. (2.8)

Eq. 2.8 is established as Helmholtz’s equation (homogeneous, source-free) and isalso called the wave equation. A possible solution to Helmholtz’s equation is aplane wave.

E(r) = E0eik·r (2.9)

The constant term E0 must be orthogonal to k. Hence the concept of polarizationsets the actual direction of E. The associated magnetic field for a plane wave isobtained from Faraday’s law

H(r) =1

Zk × E0(r) (2.10)

where Z is the wave impedance

Z = Zvacuum

√

µ

ǫ; Zvacuum =

√

µ0

ǫ0= 377Ω (2.11)

2.4 The vector Helmholtz Equation

As mentioned in section 2.1, the two curl equations suffice together with the continu-ity equation. In the previous section we have seen that the two curl equations canbe replaced by one second order curl equation in E. In this section we demonstratethat the electric field E is completely determined by one magnetic vector poten-tial A together with the gradient of the scalar potential φ. We show that thesetwo quantities are derived from the second order Maxwell equation and, when theyfulfill the Lorenz conditions, they satisfy two inhomogeneous Helmholtz equations.First let’s decompose E in the following complex expression,

E = iωA −∇φ (2.12)

2.4. THE VECTOR HELMHOLTZ EQUATION 13

Proof: We start with ∇ × E = iωB. Since the magnetic flux is solenoidal∇ · B = 0. Then ∃A such that B = ∇ × A, thus ∇ × E = iω∇ × A, that werestate

∇× (E − iωA) = 0.

From the null identity of the curl, ∃φ such that E − iωA = −∇φ, that we restate

E = iωA −∇φ.

Such decomposition already appears in [Stratton 41]. However, the authors do notgive any physical interpretation, using it only as a convenient formulation to deriveequations. Note that in Paper 1 and Paper 2, a similar decomposition has beenapplied to the surface current vector field.

Let’s now express the quantities A and φ as functions of the surface currents.To do so, we start from the second order Maxwell system Eq. 2.5

∇× (∇× E) − κ2E = iωµJ.

We insert the electric field defined in Eq. 2.12 into Eq. 2.5 to obtain

∇×∇× (iωA −∇φ) − κ2(iωA −∇φ) = iωµJ. (2.13)

Because of the null identity ∇× (∇φ) ≡ 0, this reduces to

iω∇× (∇× A) − κ2iωA + κ2∇φ = iωµJ (2.14)

Imposing the condition,

φ = − iωκ2

∇ · A (Lorenz Gauge), (2.15)

so that the gradient κ2∇φ = −iω∇(∇ · A), we obtain

iω∇× (∇× A) − iω∇(∇ · A) − κ2iωA = iωµJ, (2.16)

which allows us to simplify the expression using the vector triple product

∇× (∇× A) −∇(∇ · A) = −∇2A.

Thus A satisfies the differential equation of the vector potential,

∇2A + κ2A = −µJ (2.17)

a Helmholtz equation with source term −µJ. Taking the divergence

∇2(∇ · A) + κ2∇ · A = −µ∇ · J (2.18)

and applying the continuity equation ∇ · J = iωρ with κ2/ω2 = µǫ,

∇2φ+ κ2φ = −ρǫ

(2.19)

we see that φ also satisfies a Helmholtz equation but with source term −ρǫ.

In the next sections, the solutions A and φ of Eq.2.17 and Eq.2.19 respectively,will be expressed as integral equations using Green’s function.


2.5 Green’s function

To get the solution of Helmholtz’s equation with a source term, we introduce anauxiliary function called Green’s function. First, let δr′(r) = δ(r − r′) denote theDirac delta function such that for any continuous function f we have

∫ +∞

−∞

f(r)δr′(r)dr =

∫

r→r′

f(r)δr′(r)dr = f(r′). (2.20)

This has to be understood as the result of a limit-process, where the source isdistributed over a finite domain which shrinkes until, in the limit, the entire sourceis applied at the point r = r′. Green’s function represents the inverse of thedifferential Helmholtz operator [∇2 + κ2] which has the Dirac delta function assource term.

∇2G(r, r′) + κ2G(r, r′) = −δ(r − r′) (2.21)

∇2G+κ2G is zero everywhere, except possibly at r = r′. For r 6= r′ we can expressthe solution G(r, r′) of Eq. 2.21 as an outward spherical wave of origin r′,

G(r, r′) =eiκ|r−r

′|

4π|r − r′| (2.22)

Green’s method: Let ρ be a function with bounded support, then the solutionψ(r) to the general Helmholtz equation

∇2ψ(r) + κ2ψ(r) = −ρ(r); ψ → 0 as r → ∞ (2.23)

is given by

ψ(r) =

∫

Ω

G(r, r′)ρ(r′)dΩ′ (2.24)

Consequently, to get the solution to our problem, we multiply G(r, r′) by the sourcefunction ρ and integrate over the region of interest.

Far field approximation: If we introduce the unit vector r = r/|r| pointingfrom the origin of the coordinates to the observation point, the distance |r − r′|becomes

|r − r′| =√

r2 + r′2 − 2r · r′ = |r|√

1 + [r′2/|r|2 − 2r · r′/|r|] (2.25)

which can be approximated with respect to r2 and in the limit as r → ∞ using

|r − r′| ≈ |r|(1 + [r′2/|r|2 − 2r · r′/|r|]/2) (2.26)

Through the above Taylor expansion, the term in r′2 becomes negligable,

|r − r′| ≈ |r| − r · r′ (2.27)

2.6. THE FIELD INTEGRAL EQUATIONS 15

Green’s function in the far field can now be approximated as

G(r, r′) ≈ eiκ|r|

4π|r|e−iκr·r′ (2.28)

This approximation will be used later to derive the PO integral studied in Paper 3.The criterion for far field is that the distance R = |r− r′| between source point andobservation point, is much larger than both the size of the scattering object andthe wavelength λ.

2.6 The field integral equations

In previous sections we have seen that the field E can be decomposed into

E = iωA −∇φwhere A satisfies

∇2A + κ2A = −µJand φ satisfies

∇2φ+ κ2φ = −ρǫ

Then the electric field E can be computed from the surface currents by using Green’smethod applied to A and φ. We find the solution for the magnetic vector potentialA as

A(r) = µ

∫

S

J(r′)G(r, r′)dS′ (2.29)

and, since ρ = 1iω∇ · J , the scalar potential as

φ =1

iωǫ

∫

S

(∇′ · J(r′))G(r, r′)dS′ (2.30)

where S denotes the surface of the PEC with surface currents J and ∇′ denotes thevector derivative with respect to r′. With wµ = κZ and wǫ = κZ−1, we obtain theintegral formulation for the electric field,

E = iκZ

∫

S

J(r′)G(r, r′)dS′ − iZ

κ

∫

S

∇(∇′ · J(r′))G(r, r′)dS′, (2.31)

and the integral formulation for the magnetic field,

H = iκZ−1

∫

S

J(r′)G(r, r′)dS′ +iZ−1

κ

∫

S

∇(∇′ · J(r′))G(r, r′)dS′. (2.32)

Green’s function Eq. 2.22 substituted into Eq. 2.31 produce

E = iκZ

∫

S

J(r′)eiκ|r−r

′|

4π|r − r′|dS′ − iZ

κ

∫

S

∇(∇′ · J(r′))eiκ|r−r

′|

4π|r − r′|dS′, (2.33)

which only depends on the surface currents J. For further readings on the subjectsee [Bendali 99]. Now it remains to find an integral equation for the unknownsurface currents J.


Electric Field Integral Equation (EFIE) We denote the tangential compon-ent of the total field on the surface, Etan,

Etan = E − (n · E)n = n × E. (2.34)

If imposing the boundary condition that the total tangential electric field vanisheson the surface, see Section 2.2, written as

Etan = Einctan + Escat

tan = 0 (2.35)

where we define the incident electric field Einc as the field that would exist in theabsence of the scatterer. If assuming that the scattered field Escat also can beobtained from the solutions to Maxwell’s equations, see [Hodges 97], and writtenas

Escat(r) = iωA(r) −∇φ(r), (2.36)

then we obtain the famous EFIE (Electric Field Integral Equation) for a PEC as

[iωA(r) −∇φ(r)]tan = −Einctan. (2.37)

We plug the expression for A, Eq. 2.29, and φ, Eq. 2.30, into Eq. 2.37 and get

[iκZ

∫

S

J(r′)G(r, r′)dS′ − iκZ

κ2

∫

S

∇(∇′ · J(r′))G(r, r′)dS′]tan = −Einctan. (2.38)

In the next chapter we will see how this integral equation can be solved numericallyusing the Method of Moments (MoM) with the surface currents as unknown. Oncethe surface currents are determined the scattered field can be found as

Escattan = n ×

∫

S

[−iκZJ(r′)G(r, r′) +1

−iκZ−1∇J(r′) · ∇G(r, r′)]dS′. (2.39)

Magnetic Field Integral Equation (MFIE) Using the same procedure ap-plied to the magnetic field with the boundary condition of the magnetic fieldn × H = J, see Section 2.2, the MFIE (Magnetic Field Integral Equation) is ob-tained,

J(r)

2− n ×−

∫

S

J(r′) ×∇′G(r, r′)dS′ = n × Hinc(r). (2.40)

where −∫

is called the principal value integral with regions where the source and fieldpoints coincide have been excluded. The EFIE and the MFIE can be used eithercombined or separatly to solve for the currents, except for frequencies correspondingto interior body resonaces. As it will be shown in the next chapter, if one ignores thefield expressed by the integral term in the MFIE, one obtains the Physical Opticsassumption, that is the current is given by twice the tangential component of theincident magnetic field J(r) = 2n × Hinc, without the need to solve an integralequation.

2.6. THE FIELD INTEGRAL EQUATIONS 17

Far field In the far field, Green’s function is approximated by Eq. 2.28 andplugged into Eq. 2.33 where the divergence can be placed outside the integral usingthe definition of the Lorenz Gauge combined with Eq.2.17 and Eq.2.19

E = iκZ[1 − 1

κ2∇∇·]

∫

S

J(r′) · eiκ(|r|−r.r′)

4π|r| dS′, (2.41)

which can be written as

E = iκZ[1 − 1

κ2∇∇·]e

iκ|r|

4π|r|

∫

S

J(r′) · e−iκr·r′dS′. (2.42)

It is now suitable to simplify the notation by introducing the vector field definedby

K(r) =iκ2Z

4π

∫

S

J(r′) · e−iκr·r′dS′. (2.43)

Then the electric field is written as

E = [1 − 1

κ2∇∇·](e

iκ|r|

κ|r| K(r)). (2.44)

Under approximation in the far zone, the operator 1κ2∇∇· in Eq. 2.44 is approxim-

ated using the fact that components that vanish faster than 1/κ|r| are negligible,see [Rumsey 54], such that the dominating contribution to the scattered electricfield becomes

Escat =eiκ|r|

κ|r| [K(r) − r(K(r) · r)]. (2.45)

Using the triple vector identity, we finally get

Escat =eiκ|r|

κ|r| [r × (K(r) × r)] (2.46)

which is called the far field radiation integral. This expression is a function ofthe direction r to the observation point and of the surface currents only and theexpression [r × (K(r) × r)] represents the far field amplitude of the wave. Finally,the RCS σ is then computed as

σ =4π

λ2|Escat|2 (2.47)

The magnetic scattered field is found by using Faraday’s law ∇ × E = iκZH byneglecting the components that vanish faster than 1/κ|r| in the evaluation of thecurl operator,

Hscat ≈ Z−1 eiκ|r|

κ|r| r × [r × (K(r) × r)]. (2.48)

Chapter 3

Scattering Analysis Methods

3.1 Geometrical Theory of Diffraction

Exact solutions to the Maxwell equations are known for canonical scattering geo-metries such as an infinite cylinder, sphere or cone. For complicated geometries, ap-proximate methods can be derived from these analytical solutions [Kouyoumjian 65].

In the limit of vanishing wavelength (λ→ 0), a widespread approximate methodis the Geometrical Theory of Diffraction (GTD) [Keller 62]. It is based on thefact that diffraction phenomena exhibit local properties at high frequency. As aconsequence, the scattered field does not depend on the interactions from all pointson the surface of the conductor, but rather on the contributions from few pointslocated in the neighborhood of some special positions called diffraction points.

Under such assumptions, the complex scattered field is analytically approx-imated with a superposition of known solutions from simple canonical scatteringgeometries. The total scatter solution is expanded, see [McNamara 89], as

Escat(r) = eiκΨ(r)∞∑

n=0

(iλ)nEn(r) (3.1)

where Ψ(r) is the optical distance from the source point r and the amplitude En(r)is independent of the wavelength λ. Furthermore, GTD postulates that energypropagates along direct, reflected and diffracted rays. This enables the use of theray tracing algorithms introduced in Paper 5 and further detailed in the author’sLicentiate thesis [Sefi 03] and in [Catedra 98]. These algorithms draw (“trace”) thevarious dominant propagation paths between sources and receivers interacting withthe surface of the conductor, describing the following asymptotic phenomena:

• Direct field

• Reflected field

• Diffracted field

19

20 CHAPTER 3. SCATTERING ANALYSIS METHODS

• Multiple fields, e.g. double reflected, diffracted-reflected, ...

The ray trajectories are determined independently and the computational cost de-pends on the geometry of the obstacle instead of the electrical size.

Despite these nice features, the GTD has a few drawbacks which may some-times reduce the usefulness of the results. First, there may be many higher ordermultiple ray interactions which involved combinations of diffractions, increasing thecomplexity of the ray tracing task. This has been addressed in Paper 6. Second,the accuracy of the calculated field is relatively low since the theory will only yieldthe leading terms in the asymptotic high frequency solution of the Maxwell equa-tions.

3.2 Physical Optics

The Physical Optics (PO) technique is also a well-known and widely used highfrequency approximate technique for the calculation of the electromagnetic fieldscattered from a PEC illuminated by an incident electromagnetic field. It is a veryfast technique since it does not require an integral equation to be solved. The ideais to approximate the surface currents as if they were obtained on an infinite flatplate. Physically it can be interpreted as replacing locally the conductor by a flatplate and neglecting the contributions from sharp edges, corners and all mutualinteractions such as multiple reflections or creeping waves, what remains being themain reflection.

In order to derive the PO current, we first need to have a look at the fieldswhich induce it. As previously mentioned in Chapter 2, the total field E can bedecomposed in a sum of incident and scattered field

ETotal = EInc + EScat. (3.2)

The boundary conditions on a PEC impose that the tangential component of ETotal

vanishesn × ETotal = 0 on ∂Ω, (3.3)

which meansn × EScat = −n × EInc. (3.4)

Consecutively, the tangential component of the electric fields flips on reflection, asillustrated in Figure 3.1.

Figure 3.1 displays the two generic cases, the well-known TM- and TE-case,characterized by magnetic (or electric) incident fields transverse to the plane ofincidence, spanned by k and n. In the TE–case, there is no component normal tothe surface,

n · ETotal = 0, n · HTotal = 0 (3.5)

which means that there are no charges on the surface (ρs = 0). The normalcomponent of the incident and scattered magnetic fields cancel each other. Due to

3.2. PHYSICAL OPTICS 21

Htan

EH

total tangent H double total tangent H double

E

H

H

E

H

E

E

Hk k n

E=0E =0

tangent. comp. E flips

TM case: tangent. comp. E flips TE case: normal comp. H flips

Figure 3.1: TE, TM: case on a plate.

the dynamics in the Maxwell equations, what is lost in the normal component isgained by the tangential component making the tangential component for the totalmagnetic field double. Thus on the surface where Js = n × HTotal we get

Js = 2n × Hinc (3.6)

In the TM–case, on the surface, the total E–field has only a component normalto the surface,

n · ETotal = ρs (3.7)

which means that we have a maximal transverse charge. The tangential componentof the incident and scattered magnetic fields cancel each other making the tangentialcomponent of the total magnetic field to double, getting once more Eq. 3.6 for thecurrent.

On an infinite flat plate there is no concentration of charge ∇ · J = 0 in thelit region. On the shadow side of an infinite plate, no field can penetrate resultingin no current. This leads to the following surface currents referred to as the POcurrents:

JPO =

2n × Hinc (lit region)0 (shadow region)

(3.8)

The vector field of the PO currents has zero divergence making it solenoidal.The next step is to obtain the far field scattered field induced by JPO. From

Eq. 2.46, we directly get

Escat =iκZeiκR

4πR[r ×

∫

S

JPO(r′)e−iκr·r′dS′ × r)] (3.9)

Escat =iκZeiκR

4πR[r ×

∫

S

JPO(r′)e−iκ(kscat−kinc)·r′dS′ × r] (3.10)


−1 −0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

1.5

EK

H

x

y

Figure 3.2: Direction of PO currentson a sphere viewed from the top.

−1 −0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

1.5

E

K

H

x

z

Figure 3.3: Direction of PO currentson a sphere viewed from the side.

For a finite plate and for even moderately high frequencies JPO provides a reas-onably good approximation of the surface currents, at least away from diffractionand creeping wave phenomena generated by the sharp edges of the plate or whengrazing incidence occurs.

In the case of monostatic direction, we have kinc = −kscat thus Eq. 3.10 sim-plifies to

Escat =iκZeiκR

4πR[kinc ×

∫

S

[2n × Hinc]ei2κkinc·r′dS′ × kinc]. (3.11)

Using the associated magnetic field for a plane wave Eq. 2.10 (Hinc = 1Zkinc×Einc

0 ),we obtain

Escat =i2κeiκR

4πREinc

0

∫

S

[kinc · n]ei2κkinc·r′dS′. (3.12)

This is the PO integral which has been further studied in Paper 3. In particularwe observe that the amplitude of the integrand is proportional to cos(n, kinc) whichis a slowly varying function over the surface whereas the phase is an exponentialfunction rapidly varying. We will come back to this point later.

For a smoothly curved perfect conducting body, the PO current is an amazinglygood approximation of the induced surface currents, away from shadow boundariesand as long as the radius of curvature is larger than the wavelength (high fre-quency). The physical origin of the PO currents is clear – the magnetic field only.In Figure 3.2 and Figure 3.3 we see how the direction of the PO current on a sphereis everywhere perpendicular to the magnetic field. The properties of the directionof the PO currents have been further detailed in Paper 1 and Paper 2 leading tothe concept of PO streamline. A comparison between the RCS of a sphere obtainedwith PO, and the analytic solution for the RCS of a sphere, see Mie solution in[Bowman 87], can be seen in Figure 3.5.

3.3. METHOD OF MOMENTS 23

Concerning the implementation, two major difficulties need to be treated. First,the detection of the shadow regions for complex structures has to be taken care of.Shadowing algorithms for PO has been introduced in Paper 6 and a fast algorithmbased on ray tracing has been proposed in the author’s Licentiate thesis [Sefi 03].Other known solutions consist in processing the surfaces with a graphical engine.In [Rius 93] and [Asvestas 95], an image of the surfaces on the computer screen isused and the shadowing is processed efficiently by the hardware.

A second difficulty is to solve numerically the PO integral, in particular the

treatment of the oscillating phase factor ei2κk·r′ . There exist many different wayswhich more or less can be grouped into three main basic concepts:

• The most classic way is to evaluated the PO integral using quadrature for-mulas, for example Gauss quadrature. This requires a number of evaluationpoints, proportional to the wave number squared.

• In Gordon’s method [Gordon 94], the surface integral on special geometriescan be converted into simpler single integrals.

• Approximation methods such as stationary phase or linear phase approxima-tion, produce simplified expressions for the integral which can then be solvedanalytically. In the stationary phase methods [Catedra 95], an approxima-tion to the integral is found at some points where the phase is stationary.The problem then reduces to determine the positions of all the stationaryphase points using complex minimization techniques similar to the ones usedin GTD. In the linear phase approximation [Ludwig 68, Moreira 94], polyno-mial approximations to the phase is used, see Paper 3. Note that partialquadratic phase approximations which do not include any mixed terms havebeen tried in [Crabtree 91].

3.3 Method of Moments

This section explains the theoretical foundations of the very popular and the mostaccurate method to solve the integral equation for the surface current J, namelythe Method of Moments [Harrington 68]. To do so, the starting point is to use theEFIE Eq. 2.38 and to approximate J by the expansion

J(r′) =∑

q

jqfq(r′) (3.13)

where fq(r′) are chosen vector basis functions and jq are scalar coefficients to be

determined. A popular choice of linear basis functions is the Rao-Wilton-Glisson(RWG) basis functions [Rao 82], which are defined as

fq(r) =

+lq

2A+q

ρ+q , if r ∈ T+

q

− lq

2A−

q

ρ−q , if r ∈ T−

q

0, otherwize

(3.14)


where A±q denote the area of the triangles T±

q , lq the length of the qth edge andρ±q = ±(r±q − r) the vector from the free vertices opposite to the qth edge, r±q , see

Figure 3.4. The basis functions represent the current flowing through the qth edge

+T

-T

edgeth

q+n

-qr +

qr r

+q

ρ

-q

ρ

Figure 3.4: Rao-Wilton-Glisson (RWG) basis function.

from the triangle T+q to T−

q . A nice property is that the divergence of the basisfunction across the edge is a constant

∇ · f±q = ± lq

A±q

, (3.15)

so no charge can be accumulated along the edge. Multiplying the EFIE with a testcurrent fp and integrating results in a set of linear equations

[A]J = V (3.16)

where the elements of the excitation vector V are given by

Vp =

∫

S

−Einc(r) · fp(r)dS (3.17)

and J is the column of the unknown coefficients jq. In Paper 2, the basis functioncoefficients have been used as direct input for extrapolation to high frequency. The

3.4. HYBRID METHODS 25

matrix [A] is called the impedance matrix whose entries are given by

Apq = iκZ

∫

S

∫

S

fp(r)¯G(r, r′) · fq(r′)dSdS′, (3.18)

where¯G(r, r′) = [1 − 1

κ2∇(∇′·)]G(r, r′) (3.19)

3.4 Hybrid Methods

The MoM requires a number of unknowns N inversely proportional to the square ofthe wavelength λ2 which makes the size of the impedance matrix unmanageable athigh frequency. Solving the system in Eq.(3.16) with Gaussian elimination requiresO(λ−6) arithmetic operations and O(λ−4) for storage. To overcome this difficulty,many types of hybrid approaches have been proposed.

These methods can be classified into four groups. First the methods which focuson obtaining an approximate sparser impedance matrix using domain decomposi-tion between MoM and PO or MoM and GTD [Thiele 75, Burnside 75, Bouche 93].The idea is to split the large smooth geometry parts away from small details mod-elled by MoM. This leads to the popular MoM-PO hybrid solvers [Rahmat-Samii 91,Jakobus 95, Hodges 97, Taboada 00, Edlund 01]. In our experience, the main weak-ness of these methods is that of displaying large errors where PO fails [Burnside 87],i.e. in shadow regions or near edge discontinuities.

Second the methods which also aim to avoid expensive linear algebra but usinginstead efficient matrix algorithms such as Multi-grid [Sarkar 02], Wavelet compres-sion [Leviatan 93] or Fast Multipole Method (FMM) [Greengard 87, Rokhlin 92].FMM which is the most successful, uses block decomposition [Greengard 87] wherethe far field elements are regrouped into small low rank blocks [Nilsson 00]. How-ever, the technique still needs to represent the fastest variation of the phase thusrequiring the same large number of unknowns as in MoM.

In the third group, we have the methods which reduce the total number ofunknowns [Taboada 01, Taboada 05]. They typically require difficult alternativebasis functions [Djordjevic 04]. More References as well as descriptions of relatedwork can be found in the introduction of Paper 2.

Finally, the last group of methods is the Extraction-Extrapolation methods.They model the behavior of the surface currents by looking at currents obtainedat low frequency [Mittra 94, Aberegg 95] and then extrapolate them to higher fre-quencies [Altman 96, Altman 99], The latest development in this direction, see nextsection for more details, is the Asymptotic Phasefront Extraction (APE) technique[Kwon 00, Kwon 01], which gives a propagating ray-based description analogue toGTD of the surface currents.

APE-MoM Phasefronts In [Kwon 01], the goal is to reduce the total numberof unknowns in the MoM procedure by approximating the induced surface current


over large smooth regions with one or several linear phase currents,

J(r′) = Ceikm·r′ , (3.20)

where C is a complex vector amplitude assumed to be constant and km is thedirection of a phasefront travelling on the surface. At each point r′ on the surface,a local low frequency phase variation is split into several phasefronts of m differentdirections. The phasefront extraction is achieved using a multi-dimensional complexFourier transform j(ku, kv) on a rectangular grid of the low frequency complexcurrent as

j(ku, kv) =

∫

S

J(r′)eikinc·r′dS′, (3.21)

where ku, kv are the components of the projection of kinc on the surface S and J(r′)is obtained from a low frequency MoM solution. The dominating components in theFourier space are found using a peak-searching procedure for local maximum. Givenm extracted phasefront directions, the linear phase currents are then expanded as

Jh(r′) =∑

Nh

jhfh(r′) +∑

Nl

Km∑

p

jlfleik

m,p

h·(r′−rm), (3.22)

where the subscript l indicates elements evaluated over the low frequency sparsegrid and h over the high frequency dense grid away from smooth regions. On thelatter grid, which ideally should correspond to small regions, the basis functionsfrom MoM are used as usual. Over the large smooth regions, the low frequencybasis functions fl are multiplied by the built-in phase propagation factor to formthe high frequency basis functions. The modified basis functions inherit the quickvariation in the exponential to speed up the surface integration. The wavenumberis scaled linearly to the right frequency such that

km,ph =

λlow

λhigh

km,pl , p = 1, ...,Km (3.23)

The high frequency approximate current is then plugged into the MoM formulationwith fewer unknowns than the conventional MoM requires. A shortcoming of thistechnique is that it still requires computationally expensive numerical integrationswhen solving the matrix system.

Thus, one would like to devise a technique which does not require a second MoMsystem to be built and solved. In Paper 1 and Paper 2, this has been achieved byconstructing high frequency currents with behaviors obtained from current-basedmethods only (MoM and PO), instead of using a propagating ray-based descriptionof the surface currents.

3.5 Modelling the behavior of the surface currents

Let’s apply the Physical Optics and the Method of Moments techniques described inthe previous sections to the typical scattering problem of a metallic sphere centered

3.5. MODELLING THE BEHAVIOR OF THE SURFACE CURRENTS 27

0 50 100 150 200 250 300 350 400 450 500−35

−30

−25

−20

−15

−10

−5

0

5

10

15Normalized monostatic RCS of a PEC−sphere, r=1m

f [MHz]

σ [d

Bsm

]

POMie serie 100 terms (Exact)MoM−QMR solver

Figure 3.5: Monostatic RCS of a one meter sphere using MoM and PO, comparedto the analytical Mie solution.

at the point (0,0,0), illuminated by a plane wave propagating in (0,0,-1) direction.For a sphere, the numerical methods can be compared to an analytic solution knownas the Mie-series.

In Figure 3.5, we display the monostatic Radar Cross Section given by the Mie-series along with the solutions obtained with the Physical Optics and the Methodof Moments for various frequencies ranging from 1MHz to 500MHz. Three obser-vations can be made:

• (i) The MoM breaks down due to poor resolution when the wavelength in-creases. This happens around f=250MHz as expected since the number ofunknowns is kept fixed.

• (ii) For low frequency, all three methods give similar results. The sphere istoo small, compared to the wavelength, to be seen.

• (iii) For high frequency, the PO converges to the Mie-series. The radius ofcurvature becomes much bigger than the wavelength, getting closer to theconfiguration of a large plate.


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

K

E

y

Surface Currents on a Sphere, ka =0.1

Hz

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

K

E

y

Surface Currents on a Sphere, ka =1

Hz

Figure 3.6: Directions of the surface currents obtained using MoM on a 1m sphereviewed from the side, illuminated by a plane wave coming from the top, for k=0.1and k=1.0. Note that the triangulation is transparent, thus we see currents on theback and on the front currents simultaneously.

A closer look at the real part1 of the surface currents obtained from the MoM atlow frequency is provided in Figure 3.6. In agreement with the observation (ii) thatthe MoM and the PO solutions match, the induced MoM current vectors computedat the frequency k = 0.1 are aligned like the PO current vectors. Increasing thefrequency to k = 1.0, we observe that the direction of the MoM current vectorsdeviate from their PO positions. They tend to point towards the South pole as wellas getting smaller in the shadow zone. Our motivation in Paper 2 is to model thisdeviation as well as to predict its behavior at higher frequency.

One clue about the behavior of the deviation is given in Figure 3.7, whichdisplays the components of one particular vector on the sphere as a function ofthe frequency. We observe that each spatial component displays systematic sinuspatterns evolving smoothly over frequency, until the MoM breaks down. Suchpatterns describe in fact a rotation of the surface current vector with frequency.The rotation of the vector takes place in the tangent plane of the surface and,according to observation (ii), begins at low frequency with a vector aligned to thePO currents.

In addition, we know from the structure of the incident magnetic field, that thePO currents run along level curves, referred to in Paper 1 and Paper 2 as POstreamlines. Figure 3.8 displays the PO streamlines on a sphere for two differentincident illuminations. Thus, in order to estimate the difference between the total

1The imaginary part can be treated analogously at a quarter of period later.

3.5. MODELLING THE BEHAVIOR OF THE SURFACE CURRENTS 29

0 50 100 150 200 250 300 350 400 450 500−4

−3

−2

−1

0

1

2

3

4x 10

−3 Components of current Real(J) on triangle1

f [MHz]

Rea

l(Jx)

, Rea

l(Jy)

, Rea

l(Jz)

real(Jx)real(Jy)real(Jz)

Figure 3.7: Variations of the components of one current vector with respect tofrequency computed with the MoM. When the frequency gets too large, the MoMbreaks down requiring more elements per wavelength. Also, frequencies correspond-ing to interior body resonaces can be observed when all the three components thebecome singular.

and PO currents, it makes sense to decompose the total current into one componentalong the PO streamline and one component perpendicular to the streamline, asillustrated in Figure 3.9 and further described in Paper 1. More details are givenin the next Chapter and in Paper 1 and Paper 2.


−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

−0.5

0

0.5

1

yx

EK

H

z

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z

x

y

Figure 3.8: PO streamlines on a 1m sphere, illuminated from the top with the mag-netic field in y-direction (on the left), illuminated from the side with the magneticfield in z-direction (on the right).

mom

crossJ

JpoJ

Figure 3.9: Decomposition of the surface currents along a PO streamline.

Chapter 4

Summary of the Papers

4.1 Paper 1: Modelling and Extrapolation, an Extended

Abstract

In this paper we introduce the decomposition of the surface currents J, expressed inthe triangular-type Rao-Wilton-Glisson (RWG) vector basis functions in the MoMformulation, into parallel J|| and perpendicular J⊥ vector components, denotednxH-parallel currents and cross-currents respectively, relative to the direction of theincident magnetic field on the surface. These two currents yield systematic spatialpatterns evolving over frequency in close relation to the incident magnetic field,both on the illuminated and the shadow side of the body. From these observationswe derive two reference models for the scalar components J|| and J⊥ and comparethem to MoM for various frequencies.

The author of the thesis performed the numerical experiments and presented itat the ICNAAM 2005 International Conference of Numerical Analysis and AppliedMathematics, Rhodes, Greece, September 2005. The theoretical derivation as wellas the implementation has been made by both authors in cooperation. The ideaof the reference model for the parallel component and the fact that the incidentmagnetic field can be used both on the illuminated and the shadow side come fromthe first author. The idea for the model of the perpendicular component comesfrom the second author.

4.2 Paper 2: Modelling and Extrapolation, Continued

In this paper the analysis presented in Paper 1 is extended. The vector fielddecomposition is not standard and yields a better understanding of the structureof the surface currents as well as the underlying physics governing their spatialvariations, i.e. charge transport and charge accumulation. The oscillating behaviorof the current is strongly linked to that of the incident magnetic field. The referencemodels for the components J|| and J⊥ of Paper 1 are modified by coefficients

31

32 CHAPTER 4. SUMMARY OF THE PAPERS

calculated from MoM at low frequency. In the neighborhood of the vertices of thetriangles, the MoM current yield unreliable values. To remedy this, corrections andsmoothing have been implemented. Trends in the modified coefficients have beenstudied and have allowed us to better extrapolate the current components to higherfrequency. This approach offers a way of creating new approximated currents.

As a side result, we can observe that the currents on the shadow side of aroundish body can be modelled by these fairly simple models. Their values atthe transition region into the shadow zone are also well handled by the modifiedreference models. Finally, the perpendicular component in our decomposition canbeen seen as a measure of the error in the PO current. Thus this provides anumerical procedure to automatically control the PO error.

The development of the method and the numerical simulations have been donein close cooperation by both authors as a joined effort. The second author was themain responsible for the implementation of the modifying coefficients. The authorof this thesis was responsible for the literature studies, the PO streamline concept,the study of the TE-TM cases and the MoM simulations and presented this paper atthe MMWP05 Conference on Mathematical Modelling of Wave Phenomena, Växjö,Sweden, August 2005. A shorter version of the report will appear in the proceedingsof the conference.

4.3 Paper 3: Physical Optics and NURBS

In this paper we present an implementation of the Physical Optics (PO) techniquefor Radar Cross Section (RCS) application using an adaptive triangular subdivi-sion scheme for the surface integral which arises during the computation of theelectromagnetic scattered fields.

Our interest was to assess efficiency and error analysis. Classically, the POsurface integral is solved using quadrature formulas which require the wavelengthto be resolved, i.e. the number of elements becomes proportional to the square ofthe electrical size. Instead, if we assume linear phase, we can solve the integralanalytically on fewer larger elements, whose number grows only linearly with theelectrical size. The size of the elements is determined by an adaptive scheme whichsupplies control of the error during the integration. The solver includes the fasttreatment of the shadowing described in Paper 6 and uses the ray tracer on NURBSdescribed in Paper 5.

The author of the thesis implemented the PO solver, performed the computa-tions, wrote the paper and presented it at the EMB04 Conference on ComputationalElectromagnetics, Göteborg, Sweden, October 2004.

4.4 Paper 4: The Rescue Wing, an Engineering Application

Here we present a multi-disciplinary engineering analysis aiming at the design of anew marine distress signaling device. The device, called "Rescue-Wing", works as

4.5. PAPER 5: MIRA, A MODULAR APPROACH TO GTD 33

an inflatable radar reflector and is intended for personal use in marine environment,assisting in localization of persons missing at sea during rescue operations.

The Rescue-Wing is in fact an inflatable gas bag shaped like a wing providingaerodynamic lift. Tethered to the life jacket of a person in distress, it is filled withhelium to provide aerostatic lift. The Rescue-Wing has been designed to operateas a balloon in calm winds and, in windy conditions, like a kite. When deployed, itwill position itself in tethered flight 10-15 meters above the sea surface providing aradar reflector target as well as a strong visual cue for detection and positioning.The analysis has been focused on the multidisciplinary design task of combiningresults from aerodynamics, flight mechanics, structures and electromagnetics intoone design.

In order to assess the Rescue-Wing’s ability to reflect radar signals, electro-magnetics simulations have been conducted to predict its RCS. The computationsinvolved the use of the General Electromagnetic Solvers "GEMS", e.g. the POtechnique described in Paper 3 as well as the Method of Moments. We show howthese methods have been used to assist in the analysis and in the design decisionmaking process of the radar detectability of the Rescue-Wing, as well as how itsRCS compare to the most popular radar reflectors used on yachts and sailboats.

The first author was responsible for the aerodynamic part and presented the pa-per at the OCEANS 2005 Conference, Washington D.C., United States, September2005. The author of the thesis was responsible for the analysis and the computa-tions of the radar cross section. The introduction is a collaborative effort.

4.5 Paper 5: MIRA, a Modular Approach to GTD

In this paper we present the GTD solver MIRA: Modular Implementation of RayTracing for Antenna Applications. The low cost of GTD, compared to the Methodof Moments, is due to both the fact that there is no runtime penalty in increasingthe frequency and that the ray tracing, which GTD is based on, is a geometricaltechnique. The complexity is then no longer dependent on the electrical size of theproblem but instead on geometrical sub problems which are manageable.

For industrial applications the scattering geometries, with which the rays in-teract, are modelled by trimmed Non-Uniform Rational B-Spline (NURBS) sur-faces [Piegl 91, deBoor 78] the most recent standard used to represent complex free-form geometries. In this paper we focus especially on the architecture of MIRA thatseparates mathematical algorithms from their implementation details and model-ling.

The part concerning the geometry has been written by the first author, theray-tracing part and the pictures of the rays has been done by the author of thethesis, the application part, the introduction and the last picture has been createdby the second author.

34 CHAPTER 4. SUMMARY OF THE PAPERS

4.6 Paper 6: Architecture and Geometrical Algorithms in

MIRA

Due to the introduction of NURBS presented in Paper 5, the geometrical subproblems tend to be mathematically and numerically cumbersome and introducecomplications such as mathematical complexity and several representations of thesame curve.

We show how proper Object Oriented programming techniques has allowed usto restructure the ray tracer into a flexible software package. The keywords hereare portability and modularity. Implementation of these concepts was realized withFortran 90 because of its robustness and portability, although it does not featureObject Orientation naturally.

A well thought-out software design allows simple and efficient implementationof geometrical ray tracing algorithms. Our experience is that good software ar-chitecture leads to flexibility and modularity, essential in the support of futureenhancements.

As a consequence, the independent modules of MIRA make suitable platformsfor hybrid techniques in combination with other methods such as MoM and PO. Ina first innovative hybrid technique, a triangle-based PO solver uses the shadowinginformation calculated with the ray tracer part of MIRA. The occlusion is performedbetween triangles and NURBS surfaces rather than between pairs of triangles, thusreducing the complexity. Then the shadowing information is used in an iterativeMoM-PO process in order to cover higher frequencies, where the contribution ofthe shadowing effects, in the hybrid formulation, is believed to be more significant.

This paper was presented at the EMB01 Conference on Computational Electro-magnetics, Uppsala, Sweden, November 2001.

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[1] CVS, the Concurrent Versions System:http://www.cvshome.org

[2] netCDF, the network Common Data Form:http://www.unidata.ucar.edu/packages/netcdf

[3] Matlab, the Matrix laboratory:http://www.mathworks.com

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[5] CADfix: TranscenDatahttp://www.cadfix.com

ICNAAM – 2005 Extended Abstracts, 1 – 4

Modeling and Extrapolating High-frequency Electromagnetic

Currents on Conducting Obstacles

Sandy Sefi∗1 and Fredrik Bergholm1.1 Royal Institute of Technology, KTH, SE-10044 Stockholm, Sweden

Key words Numerical Methods, Electromagnetics, Method of Moments, Physical Optics, Surface Currents.

We present a current-based approach to high frequency approximate techniques in Computational Electromag-netics (CEM). Our goal is to numerically model the behavior of electromagnetic and surface current vectorfields at high frequency, using information extracted from lower frequency solutions.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The Method of Moments [1] (MoM) is the most common current-based method for solving electromagnetic

scattering problems. It involves the solution of a large linear system of equations that has to be computed for

each frequency with surface currents as unknowns. The MoM can be seen as an asymptotically exact brute

force method which does not incorporate any knowledge about the structure of the vector fields. It results in

a full dense system of equations, thus limiting its use from low to mid-range frequency. For high frequency,

approximate techniques are also available, either crude current based approximations such as Physical Optics, or

ray-based techniques [2] which require known asymptotic high frequency solutions such as Geometrical Optics

or the Geometrical Theory of Diffraction [3].

In this work we study the possibility of extracting, from lower frequency (MoM), information about the elec-

tromagnetic and surface current vector fields that are slowly varying over smooth bodies. Our goal is to extrapo-

late these surface currents to higher frequencies, thus avoiding the prohibitively high cost of solving for billions

of unknowns that would be required by MoM and yet keeping better accuracy than classical high frequency

methods.

Our method decomposes the low frequency surface currents, expressed as triangular-type Rao-Wilton-Glisson

[4] (RWG) vector basis functions in the MoM formulation, into parallel and perpendicular vector components.

These components are labeled nxH-parallel currents and cross-currents respectively and are defined relatively

to the direction of the incident magnetic field on the surface. The two currents show systematic spatial patterns

evolving over frequency in close correlation with the incident magnetic field, both on the illuminated and on the

shadow side of the body. Such a vector field decomposition is not standard and yields a better understanding

of the structure of the surface currents as well as the underlying physics governing their spatial variations, i.e.

charge transport and charge accumulation.

2 Vector field Decomposition - a background

We start with the most simple current vector field, Jpo from the Physical Optics (PO) technique. It is an approx-

imation of the induced surface currents1 on a smooth perfectly conducting (PEC) body whose dimensions are

large compared to the wavelength λ. It is induced only by the magnetic field Hinc:

Jpo(r′) = 2 · n(r′) × Hinc(r′) (r′ in lit region) (1)

Jpo(r′) = 0 (r′ in shadow region) (2)

∗ Corresponding author: e-mail: sandy@ nada.kth.se, Phone: +46 8790 62 29, Fax: +46 8790 64 571 We abbreviate “current density” by current.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

where r′ is the position vector for a given point on the surface and n the outward surface normal. For a high

frequency or a fairly high wavenumber k = 2πλ

, this approximation is sufficient for flat smooth areas away from

shadow boundaries or geometrical discontinuities. Sharp edges or corners leads to erroneous Jpo–currents.

From the accounts for the Method of Moments, which computes exact currents, it is evident that, on average,

Jpo explains large parts of the surface current pattern, almost irrespectively of the chosen spatial frequency k.

Even low frequencies with small k, exhibit Jpo–like surface currents.

It is intuitively clear from Eq. 2 that the source of the approximate Jpo–currents, the magnetic field, creates

J = n × Hinc on the illuminated side of the body. For an incident planar wave in the propagation direction

k = −z, where the shadow-lines n⊥k are along the equator, we have:

Einc(r′) = x · E0 · ejkz · e−jwt (3)

Hinc(r′) = −y · (E0/Z0) · ejkz · e−jwt (4)

where r′ = xx + yy + zz, Z0 = 377 ohms is the free space impedance and w is the angular frequency.

We note that, in such a configuration, there are other currents present as well, and they are not equally easily

“explained”. One important observation is nevertheless that the Jpo–currents have only a limited relation to

charge accumulation, which lead us to following definition:

Observation 2.1 For plane wave incidence where Einc,Hinc,k form a trihedron, the Jpo–currents run

along planar sections spanEinc,k ∩ body that we call streamlines.

Concretely, for a sphere in polar coordinates2 , the Jpo–streamlines are:

(x, y, z) = (cos ϕ cos θ, sin ϕ, cos ϕ sin θ) (5)

These streamlines run as parallel sets of curves – there is no convergent or divergent pattern of current lines.

There is only a periodic reversal of direction of current along these streamlines, due to the periodic Hinc, see Fig.

1(a), making in most places ∇ · Jpo 6= 0. This leads us to the following observation:

Observation 2.2 Apart from the periodic variation (spatially) along the Jpo–streamlines, the part of the sur-

face current field associated with charge accumulation is the non-Jpo field.

Hence, if decomposing the total surface current J into two vector fields, irrespective of k, it is natural to let

Jpo be one component in that field, J − Jpo be the other component. Observation 2.2 then tells us that charge

accumulation behavior must primarily be contained in J − Jpo = Jc.

We now make an assumption. Let J‖ be the component of J that is parallel to the Jpo–current streamline

(Eq. 5), which we denote by Ψ, a parametric curve Ψ(u) = (x(u), y(u), z(u)) on the surface.

Assumption 2.3 The parallel component J‖ of the surface currents behaves qualitatively like the Jpo–currents.

For that reason, we decompose the total current J in J‖ and J⊥ = J − J‖, where the latter reminds of Jc

which we expect is due to other time-varying charge accumulation patterns than those directly associated with

current variations along Ψ.

Definition 2.4 We decompose J as follows:

J‖ = (J · T)T : nxH-parallel currents. (6)

T =dΨ

du: Tangent along the streamline Ψ. (7)

J⊥ = J − (J · T)T : Cross currents perpendicular to the streamline Ψ. (8)

Consequently, we have defined a new entity, which we call “cross currents”, by the above equation. Corre-

sponding scalar quantities for the amplitudes of the components are:

J‖ = ‖J‖‖ · sgn(J‖) = J · T (9)

J⊥ = ‖J⊥‖ · sgn(J⊥) (10)

where the sign (sgn = ±1) of sgn(J‖) is determined by the direction of the traveling current (Eq. 7) and

J⊥ = J · nc, where nc is outward normal to the curve.

Our underlying strategy, from now on, is to model the entities J‖ and J⊥ spatially over the body, as well as

over frequency k.

2 Non-standard polar angles are used to create plane sections, see Fig. 1(a) for ϕ = 0, θ ∈ [0, 2π]

2

−1−0.5

00.5

1

−0.2

−0.1

0

0.1

0.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

HE

k

x

y

z

(a) Incident magnetic field Hinc along the ϕ = 0 streamline .

0 50 100 150 200 250 300 350 400 450 500−4

−3

−2

−1

0

1

2

3

4x 10

−3 Components of current Real(J) on triangle1

f [MHz]

Rea

l(Jx)

, Rea

l(Jy)

, Rea

l(Jz)

real(Jx)real(Jy)real(Jz)

(b) Evolution over frequency of the component of the MoM current

at a point of the streamlines.

Fig. 1 Evaluation of MoM currents on a sphere 1 meter in radius.

3 Modeling of J‖ and J⊥

For general roundish bodies, we suggest that cross currents follow the fluctuations of Hinc. Let Y = E0/Z0. We

model J‖ by :

J‖ = 2Y · cos(k cos ϕ sin θ) (11)

on Ψ, both on illuminated and shadow side and we expect a-priori that an approximate model of J⊥ is:

J⊥ =∂

dθ2Y cos(k cos ϕ sin θ) + C0 (12)

where C0 is a modelisation constant. The logic is that cross currents J⊥ are due to changes of charge and

accumulation of charge takes place at turning points of J‖, where∂J‖

dθis high.

4 Preliminary Results and Discussion

To get a better intuition of the structure of vector fields represented by the MoM currents, we have studied their

evolution at one point on the surface of a sphere of radius a = 1 meter. The result is displayed in Fig. 1(b),

where we observe how the current rotates slowly with frequency by reversing its components. In addition we see

that, at higher frequency (f > 350MHz in this case), the MoM requires more elements to resolve the wavelength

and without what, the MoM starts to predict erroneous values. Fig. 1(a) illustrates the behavior of the magnetic

field on the sphere along the ϕ = 0 streamline. The directions of the field on the surface oscillate following the

exponential term in Eq. 4.

The model of our new approximate currents has been implemented and the results along a Jpo–current stream-

line are displayed in Fig. 3, and compared to MoM. We note here that the MoM result contains discontinuities due

to the coarse triangulation used. Smoothing in the data is required to emphasize the pattern in MoM, especially

when looking at the cross current. We use the simple model defined in section 3 and get good agreement for the

parallel component J‖ at both low and higher frequency. The nxH-parallel component, like the Jpo–current, dis-

plays error which amplifies with k close to the shadow-lines. The perpendicular component J⊥ is more difficult

to model, as seen in Fig. 3(a) for low frequency ka = 10 and in Fig. 3(b) for higher frequency. To this end, we

intend to use extraction from low frequency MoM in order to improve the amplitude and phase in our model, in

particular around the shadow-lines.

3

0 50 100 150 200 250−6

−4

−2

0

2

4

6

8x 10

−3 Parallel Current Component: k=10 cut Phi=0.3927

θ

|J|

From MoMModeled

(a) Case ka=10

0 50 100 150 200 250−6

−4

−2

0

2

4

6x 10


θ

|J|

From MoMModeled

(b) Case ka=19

Fig. 2 Simulation results for the parallel component J‖ compared to MoM.

0 50 100 150 200 250−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3 Perpendicular Current Component: k=10 cut Phi=0.3927

θ

|J|

From MoMModeled

(a) Case ka=10

0 50 100 150 200 250−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10


θ

|J|

From MoMModeled

(b) Case ka=19

Fig. 3 Simulation results for the perpendicular component J⊥ compared to MoM.

Acknowledgements Financial support has been provided within the GEMS project by KTH and the Swedish Agency for

Innovation Systems (VINNOVA) as parts of a collaborative research center PSCI.

References

[1] R.F. Harrington. Field Computation by Moment Methods. New York: Macmillan (1968).

[2] S. Sefi, Ray Tracing Tools for High Frequency Electromagnetics Simulations, Licentiate thesis KTH (2003).

[3] Joseph B. Keller. Geometrical Theory of Diffraction. J. Opt. Soc. of Am. 52, 116–130 (Feb 1962).

[4] Rao, Glisson, Wilson, IEEE Trans. Ant. Prop. 30, pp. 409-418 (1982).

4

Extrapolation and Modelling of Method of Moments

Currents on a PEC Surface.

Sandy Sefi and Fredrik Bergholm1

TRITA: NA-0539

Stockholm 2005

Technical Report

Royal Institute of Technology

Department of Numerical Analysis and Computer Science

c© Sandy Sefi and Fredrik Bergholm1, November 2005

TRITA: NA-0539

Stockholm 2005

1Authors in random order.

Extrapolation and Modelling

of Method of Moments Currents on a PEC

Surface.

Sandy Sefi

Department of Numerical Analysis and

Computer Science, NADA

Royal Institute of Technology, KTH

Stockholm, Sweden

Email: [email protected]

Fredrik Bergholm




Stockholm, Sweden


Abstract— We present a current-based approach to high fre-quency approximation techniques. Our goal is to numericallymodel the behaviour of electromagnetic and surface current vec-tor fields at medium-range or high frequency, using informationextracted from lower frequency solutions.

I. INTRODUCTION

Method of Moments [1] (MoM) is the most common surface

current based method for solving electromagnetic scattering

problems. It involves the solution of a full large linear system

of equations for each frequency, with surface currents1, as

unknowns.

MoM can be seen as an exact brute force method which does

not incorporate any knowledge about the geometric structure

of the vector fields involved, thus limiting its use to electrically

small problems.

For high frequency electrically large problems, approxi-

mate techniques are also available, either crude current-based

approximations such as Physical Optics (PO) or ray-based

techniques which require known asymptotic high frequency

solutions such as Geometrical Optics or the Geometrical

Theory of Diffraction (GTD, UTD) [3]–[5]. These can provide

very fast results but are approximations that do not incorporate

a complete modelling of the physics and thus, typically

degrade in accuracy.

In the intermediate frequency range neither technique is

effective and in the past decade, efforts have been concentrated

on hybrid approaches, resulting in four main groups of hybrid

methods.

The first group uses domain decomposition combined with

high-frequency techniques. The strategy is to achieve a sparser

matrix by splitting large smooth geometry parts away from

small details modelled by the MoM. The pioneer schemes [6],

[7] were realized for specialized geometries, by combining

the MoM with GTD. For general geometries, GTD involves

more complex ray tracing [8], a non robust operation which

1We abbreviate “current density” by current.

also tends to be expensive when introducing more asymp-

totic phenomena such as multiple interactions or creeping

waves [9]. In contrast, a current-based PO is a preferable

choice by making easier the integration with the current-based

MoM. This explains the worldwide development of MoM-PO

hybrid solvers [10]–[14]. In our experience, the main weakness

of this method is that of displaying large errors where the

PO fails, i.e. in shadow regions or near edge discontinuities.

We believe that shadow regions must be properly avoided

when using PO, in such a manner that the PO domain has

to be reassigned for every angle of incidence, leading to more

complex implementation.

The second group of methods focuses on obtaining sparsity

of the full matrix of the MoM with efficient algorithms such

as matrix compression using Wavelet [16], Multi-grid [17] or

efficient matrix vector products in the Fast Multipole Method

[19] (FMM). The latter, being so far the most successful,

uses block decomposition [20] where the far field elements

are regrouped into small low rank blocks [21]. This allows for

larger scale problems, but the technique still needs to represent

the fastest variation of the phase thus requiring the same large

number of unknowns as in MoM.

Schemes to reduce the total number of unknowns first

appeared for PO, with linear phase approximation scheme [25]

and quadratic phase approximation [26]. A recent implementa-

tion can be found in [28] and the method described in [27] was

also extended to a MoM-PO linearly phased hybrid solver [29].

The method performs faster using larger elements, but, without

proper treatment of the shadow regions, it will still suffer

from degradation in accuracy like the standard MoM-PO.

However, in such a case, the reassignment of the PO regions

will necessitate remeshing of the new MoM domain, leading

to more expensive MoM-PO. Reduction of the number of

unknowns in the MoM formulation can be achieved in 2D with

the Integral Equation Asymptotic Phase (IE-AP) method [32],

or more recently in 3D [30], with alternate basis functions

which incorporate variation of phase. More complicated high-

order basis functions are also available for the MoM-PO [31].

Finally, the last group of methods is the Extraction methods.

They are based on extraction of characteristic properties of a

known current obtained from lower frequency solutions and

then extrapolation of the currents to higher frequency [33].

Such extraction-extrapolation approaches have not been much

explored yet but have been reported in [34] to be possible [35].

Recently, the Asymptotic Phasefront Extraction (APE) [36]

and its extension the APE-MoM [37] use a similar domain

decomposition to MoM-PO with high order basis functions.

APE uses GTD physical insight of the electromagnetic fields

to predict the behaviour of the geometrical optics induced

currents.

In this paper we focus on obtaining insight about the

behaviour of the surface current owing to the PO and the MoM

induced currents instead of GTD. Our method first decomposes

the low or mid-range frequency surface currents expressed

into triangular-type Rao-Wilton-Glisson [22] (RWG) vector

basis functions in the MoM formulation, into parallel and

perpendicular vector components relative to the direction of

the incident magnetic field on the surface that we call n×H-

parallel currents and cross-currents respectively.

These two currents yield systematic spatial patterns evolving

over frequency closely related to the evolution of the incident

magnetic field over frequency, both on the illuminated and

shadow side of the body. Such a vector field decomposition is

not standard, and yields a better understanding of the structure

of the surface currents as well as the underlying physics

governing their spatial variations, i.e. charge transport and

charge accumulation.

Section II presents the background of the decomposition on

which the method is based. Different models for the induced

currents are discussed in Section III for both components and

illustrated in the case of a sphere. Section IV shows the use

of the RWG basis functions and Section V to VI discuss the

extraction procedure and how to obtain frequency dependent

behaviour from low frequency MoM or FMM runs. Numerical

results are presented in Section VII for the extrapolation and

in Section VIII for our models, as well as in Appendix B.

Conclusions are drawn in the last section.

II. VECTOR FIELD DECOMPOSITION - A BACKGROUND

For a smooth perfectly conducting body, it is well-known

that the PO-currents:

JPO = 2 · n × Hinc (1)

JPO = 0 (shadow region), (2)

for sufficiently high wavenumber k = 2π/λ, are a reasonably

good approximation of the induced surface currents J ∈ IC3.

However, to our knowledge, there are no general error es-

timates telling how good this approximation actually is for

arbitrarily shaped bodies.

It is known that JPO–currents close to shadow bound-

aries are quite erroneous. To alleviate this problem, some

researchers in the nineties [38] apply diffusion to JPO–

currents to create a continuous transition into the shadow.

Another technique, [24], works instead with the scattered field

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

−0.5

0

0.5

1

yx

EK

H

zFig. 1. Streamlines are level curves transverse to H

inc.

obtained by integrating a smooth function of JPO as integrand,

extended into the shadow.

It is evident that JPO explains large parts of the surface

current pattern, almost irrespective of the chosen frequency k.

Even low frequencies k, exhibit JPO–like surface currents2.

The physical origin of the JPO–currents is intuitively clear

– the magnetic field, e.g. an incident planar wave creates

before interactions with neighbouring surface elements Jinc =n × Hinc on the illuminated side of the body, where n is

the outward unit-length surface normal, and Hinc the incident

magnetic field. If we treat the sphere as transparent, then the

incident magnetic field, as seen in Fig. 2(b) along the whole

streamline, can be evaluated in the shadow too. This means

that the total induced surface current J = n×H is related to

Jinc everywhere. According to our experience, this is a good

way of capturing the oscillations of J.

To summarize, induced currents contain a large part of

PO–like currents. However, there are other currents present

as well, and they are not equally easily “explained”. One

important observation is nevertheless that JPO–currents have

only a limited relation to charge accumulation, to be discussed

below. For an incident planar wave in the propagation direction

k=(0, 0,−k), we have Einc(r) and Hinc(r):

Einc = x · E0 · eikz · eiwt (3)

Hinc = −y · (E0/Z0) · eikz · eiwt (4)

where r = xx+yy+zz, and Z0 = 377 ohms is the free space

impedance and w is the angular frequency. We refer to (0, 0, 1)as the North pole. The equator is the shadow boundary.

2For all variables in this text, vectors but not scalars are in boldface. Vectordimension is 3 unless stated otherwise.

2

In this case, the PO currents run along planar sections

y = Const ∩ S,

where S = (x, y, z) : f(x, y, z) = 0 is the set of points on

the surface S of the body.

Hence, the PO current streamlines are level curves, see

Fig. 1. Concretely, for a sphere:

(x, y, z) = (cos ϕ cos θ, sin ϕ, cos ϕ sin θ) (5)

are the PO current streamlines obtained analytically, with

θ as curve parameter along them, and θ ∈ [0, π], on the

illuminated side, as seen in Fig. 2(a). For complex shape, the

PO current streamlines can be obtained from the intersection

between the surface and the plane transverse to Hinc. The

set of intersecting points are then ordered clockwise to form

the discrete PO current streamlines. Fig. 3(b) illustrates the

result of this procedure on the small aircraft code-name Eikon

displayed in Fig. 3(a).

Observation 1: These streamlines run as parallel sets of

curves – there is no convergent, divergent pattern of current

lines. There is only a periodic reversal of direction of current

along these streamlines, due to the periodic Hinc as seen in

Fig. 2(b), making:

∇ · JPO 6= 0 (6)

in most places. The conclusion is:

Observation 2: Apart from the periodic variation (spatially)

along the PO current streamlines, the part of the surface current

field associated with charge accumulation is the non-JPO field.

Hence, if decomposing the surface currents into two vector

fields, irrespective of k, on the lit side, it is natural to let

JPO ∈ IC3 be one component in that field, and J−JPO be the

other component, Observation 2 then tells us that transverse

charge accumulation behaviour must primarily be contained in

J − JPO = Jc. (7)

In other words, we expect that time-varying charge accumu-

lation patterns, not associated with the PO streamlines, are

captured by Jc.

We now make an assumption. Let J‖ be the component of J

that is parallel to the PO current streamline (cf. Eq. 5), which

we denote by Ψ, and the curve is a function of the curve

parameter u:

Ψ(u) = (x(u), y(u), z(u)).

This component is parallel to n × Hinc. In Eq. 5, the curve

parameter u = θ.

Assumption 3: The n×H–parallel component J‖ behaves

qualitatively like the PO current JPO modulo a slowly varying

real function, on the illuminated side.

We decompose J in J‖ and J − J‖, see Fig. 4(a), where

we expect that Jc = J − JPO ≈ J − J‖ carries time-varying

charge not directly associated with current variations along Ψ.

Definition 4: The n×H–parallel component is:

J‖ = (J · T)T (8)

ϕ=0

ϕ

θθ=0

(a) Parameterization of the hemispherez = cos ϕ sin θ, −π < ϕ < π.

−1−0.5

00.5

1

−0.2

−0.1

0

0.1

0.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

HE

k

x

y

z

(b) Incident magnetic field Hinc,

Eq. 4, along one streamline.

Fig. 2. Streamlines parameterization and incident magnetic field.

(a) CAD description of the UAV.

5.3

5.3

5.3

5.3

5.3

−4−3−2−10123

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Streamline on Eikon

(b) Streamline ϕ = 0 on the UAV.

Fig. 3. Unmanned Aerial Vehicle (UAV) code-name Eikon.

where T is the unit tangent along Ψ. Furthermore,

J⊥ = J − (J · T)T (9)

which we call “cross currents”. Consequently we have defined

two new entities which have complex values. From now on,

we will only analyze the real part JRe of the current. The

imaginary JIm can be treated analogously at a quarter of

period later (ωt = π/2), since the current is defined by

J = JRe + iJIm (10)

and eiωtJ = eiπ/2J = iJ and furthermore

JRe = JRe‖ + JRe

⊥ = JRe‖ T + JRe

⊥ nc (11)

JIm = JIm‖ + JIm

⊥ = JIm‖ T + JIm

⊥ nc (12)

where T is the unit-length tangent and nc is the unit-length

normal to the curve in the tangent plane of the body. The

corresponding scalar quantities for the real components are:

JRe‖ = JRe · T (13)

JRe⊥ = JRe · nc (14)

Their signs are determined by the choice of the current curve

travelling direction.

Our underlying strategy, from now on, is to model the

entities J‖ and J⊥ (we drop the upper Re in the notations),

spatially over the body, as well as over frequency k.

3

A. The Functionality of Cross Currents

By running MoM for low and mid-range frequencies, on

a sphere, we find that on the sphere equator there are equa-

torial cross currents running from (e.g.) East to West poles

(r=(±1, 0, 0)), as well as cross currents at other latitudes

running from East to West or vice versa. The intuitive inter-

pretation behind this is that the cross currents J⊥ transport

charges back and forth between East and West parts on

either side of the closed body. The x-axis, connects East

and West, and the E-field runs in the x–direction. In line

with Observation 2, J⊥ performs other charge accumulation

patterns than those seen along the PO streamlines. Hence,

a charge transport appears to run along latitude circles with

respect to the North pole, as indicated in Fig. 4(b). Define yet

another component of J, namely transverse currents, denoted

Jφ:

Jφ = TL · J (15)

where TL is a unit–tangent vector along latitude circles with

respect to the North pole, it is clear that Jφ approaches J⊥

when θ approaches zero, and that Jφ and J⊥ are highly

correlated for θ ∈ [0, π/4].The standard analytical solutions, Mie solutions [2] for a

sphere (which is one of the few closed bodies for which there

are analytical solutions), are also expressed in latitudes and

longitudes φ with respect to the North pole.

B. What TM- and TE–polarization Tell us

A planar approximation of the scattered fields on the lit

side of a smooth body, leads to the PO current, see [23].

This is reasonably valid if the radius of curvature is small

compared to the wavelength. The approximating assumption

is simply that an incident planar wave gives rise locally to a

scattered planar wave (direct reflection). What is as interesting

is, however, that more information can be gained from this

simple approximation.

There are two generic cases, the well-known TM- and TE-

cases, characterized by magnetic (or electric) incident fields

transverse to the plane of incidence, spanned by k and n. See

figures in Appendix A.

The TE–case appears on the meridian θ = π/2 running

from the North pole through the forward pole (0, 1, 0). The

interesting thing is that for the scattered E–field there is no

component normal to the surface, n · E = 0, which means

no charge along this meridian. Physically, one would then

not expect any current running tangentially along the zero

charge meridian, having time invariant vanishing electric field.

The PO-current is perpendicular to that meridian for roundish

bodies. The conclusion is that for this kind of model in this

case the constraint

J⊥(θ = π/2) = 0 (16)

holds, and is expected to be valid for quite general roundish

bodies, and any frequency.

The TM–case appears on the meridian ϕ = 0 running from

the North pole through the West pole (1, 0, 0). Here, on the

JJ

J

(a) Decomposition along a streamlineof the MoM current J in nxH-paralleland cross current J⊥.

(b) Patterns on the hemisphere of nxH-parallel (rotating around y-axis) andtransverse currents (rotating around z-axis).

Fig. 4. Decomposition of the MoM current.

surface, the total E–field has only a component normal to the

surface, n·E = ρs where ρs is the charge density, which means

that we have a maximal transverse charge along this meridian,

i.e. maxima in a direction perpendicular to the meridian in

question. This is so, because in any other location you obtain a

weighted average of the TE- and TM-cases. Physically, with no

counter-gradient transport of charge, cross currents J⊥ should

be expected to be zero for ϕ = 0. The conclusion is here that

the constraint should be:

J⊥(ϕ = 0) = 0 (17)

Both these constraints are also valid for the analytical Mie

solutions, [2], for a sphere, so they are not just the consequence

of a planar wave approximation.

C. Charge Accumulation and Cross Currents

From a theoretical point of view, we have already mentioned

that the TE– and TM–polarizations with reflected planar

waves may be qualitatively valid local models for medium-

range and high frequency for smooth bodies, and that they,

as a byproduct, predict that there will be maximal charge

accumulations on both halves of the illuminated part of the

East–West meridian. This holds for any roundish body.

Observation 5: Local planar wave solutions (TM polar-

ization) predict maximal charge accumulation along the East-

West meridian of a roundish body, since the resulting normal

component of E is of maximum amplitude, in a transverse

direction (latitude circle Fig. 4(b)).

The conclusion is that J⊥ contributes to the transport

of charge between the curves of maximal charge, ϕ = 0,

θ ∈ [0, π/2] and with opposite charge, ϕ = 0, θ ∈ [π/2, π].There are transverse patterns of currents converging onto

the East–West meridian, hence zero current where they change

sign.

D. What 2D Analytic Solutions Tell us

For the TE-case on the zero meridian (from North pole by

way of Front pole to the South pole), the analytic solutions for

a 2D circular section of radius a are available and can be run

for high k-values, see Appendix D. The total E field oscillates

4

in the shadow region, but vanishes fairly quickly for ka = 29.

The higher k-values, the more of the shadow zone meridian

has E ≈ 0. Since J is related to ∇× E, we conclude that for

this particular meridian the currents tend to the PO-currents

quickly.

III. MODELLING OF SURFACE CURRENTS

For roundish bodies, we suggest that both parallel and cross

currents follow the fluctuations of Hinc,Re. Denote the inverse

free space impedance by Y = 1/377. Set E0=1. We will

concentrate on closed smooth bodies.

On the shadow side, it is not clear how to specify the

influence of Hinc, which is why simple approximations such

as JPO with zero values on the shadow side are used, or

smoothing the JPO across the shadow boundary. However, it

seems reasonable that Hinc still is a major factor behind the

induced surface current pattern, and in a crude approximation,

it may be worthwhile to use Hinc in the model of surface cur-

rents also on the shadow side. We investigate this empirically

by studying MoM-results for low or mid-range frequencies.

Such a study, resulted in the heuristic formulas given below,

the so-called reference model equations.

It should be noted that these reference models are not

necessarily realistic models, but rather reference points for

further modifications, as described in later sections.

A. Reference Model for Parallel Currents

We model J‖ on the previously mentioned streamline curves

Ψ by:

J‖ = 2Y · cos(kz), z = z(u), (lit side) (18)

which can be used for any smooth closed body. The formula

can be derived as follows: Since ‖Hinc‖ = Y ·cos(kz) and by

Assumption 3 with JPO = 2n×Hinc on the illuminated side

of the closed body, the above equation is obtained. However,

and this is an important point, we consider Hinc to be a major

explanatory factor behind the induced currents also on the

shadow side, and with some boldness use Hinc also on the

shadow side, however, not using the factor 2 since that factor

is motivated by direct reflection on the lit side. Whether this is

a good or bad model is partly an empirical story. Our study of

MoM-results indicates that this is a fairly reasonable model,

abstracting away from details. The reference model for the

shadow side is thus:

JRef‖ = Y · cos(kz), z = z(u), (shadow side). (19)

In what follows, our goal will be to model and fit an amplitude

function R(k) in

Jmodel‖ = R(k)Y · cos(kz), z = z(u), (shadow side).

(20)

We look upon the reference model as a simple tool for obtain-

ing currents in the shadow region at intermediate frequency.

For high frequency we expect R to vanish as k increases which

motivates the zero PO currents in the shadow regions.

The same model for the complex numbers for the phasor

yields (i =√−1):

JRef‖ = 2Y · exp(ikz(u)) (lit side). (21)

JRef‖ = Y · exp(ikz(u)) (shadow side). (22)

This means the imaginary part of J‖ is also modelled here.

B. Parallel Currents on a Sphere or Ellipsoid

In the special case of a unit radius sphere, z = R sin θ,

R = cos ϕ, we obtain the reference models for the real part:

JRef‖ = 2Y · cos(kR sin θ), θ ∈ [0, π], (23)

JRef‖ = Y · cos(kR sin θ), θ ∈ [π, 2π], (24)

where the choice of ϕ determines which streamline to follow.

Note here that for an incident plane wave in the z-direction,

the same formula applies to an ellipsoid with semi-axis of unit

length in the z-direction. This is so because z = R sin θ still

holds.

The first equation is for the illuminated side, and the second

for the shadow side, as mentioned.

If giving names not only to the North and South poles of

the sphere (with the North pole in (0, 0, 1)) we also have

East and West poles at (±1, 0, 0) and Forward and Backward

poles at (0,±1, 0). We will speak of N-,S-,E-,W-,F-, and

B-poles in what follows. The curves for ϕ = Const are

latitude curves with respect to the F-pole. R= radius of such

latitude circles. Hence ϕ = 0 is the latitude circle crossing the

illuminated N-pole running from E-pole to W-pole. We will

tabulate modelling results with respect to ϕ in a later section.

The significance of the F- and B-poles lies in that they are the

singular cuts for the planar sections of the closed body for the

PO current streamlines.

A good model for the cross currents is a harder feat, but

even some quite crude models seem to be of some descriptive

value for simple closed bodies.

C. Reference Model for Cross Currents

On Ψ, we expect a-priori that an approximate model of J⊥

is:

J⊥(u) = C1 ·d

duY cos(kz(u)) + C0 (25)

In practice, C1 is not a constant, but if we wish to to have

a simple reference model, one can choose some neutral value

of C1 .

The logic is that cross currents J⊥(u) vary periodically

along the curve Ψ(u), with local extrema for certain positions,

um,m = 1, 2, .... We expect J⊥ to be of larger magnitude

where accumulation of charge takes place.

Along the curve, we know that accumulation occurs at sign

changes of J‖, and more generally at any inflection points of

J‖ ( d2

du2 J‖ = 0). From this follows that ddu J‖ attains local

extrema at these inflection points. If using this function for

J⊥(u), and set um = u : d2

du2 J‖ = 0, we obtain expected

extrema for cross currents.

5

A simpler way of achieving something similar is to replace

the cos by sin in the formulas to move from J‖ to J⊥. We

have also tried this, but achieved better results with the above

formula.

It should also be remembered that the main point in this

simplistic reference model is that charge accumulation is

oscillatory, and that cross currents – as a consequence – are

oscillatory too. The phase of this variation is not expected to

exactly follow the reference model.

However, around θ = π/2 the function for cross currents

must be antisymmetric, due to the constraint Eq. 16.

D. Cross Currents on a Sphere

For a sphere, we set up a reference model from Eq. 25

without distinguishing between illuminated and shadow side.

We choose C0 = 0 and C1 = 1kR where the normalization

factor kR originates from the inner derivative. For a spheroid

type of body, our reference model for the perpendicular

currents is:

JRef⊥ (θ) = −Y sin(kR sin θ) · cos θ (26)

The curve parameter θ ∈ [0, 2π] completes a full circle.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

xaxis

Fig. 5. J⊥ from MoM at ka=10 on the lit side of 1/4 of sphere, top view.

E. Cross currents from MoM

If taking the currents J from MoM, and performing a

decomposition into J‖ and J⊥, how well are the reference

models and constraints, mentioned so far, reflected in such

data?

Let us study the example of a sphere of radius a, for two

electric sizes ka = 10 and ka = 19, on the lit side, as

illustrated in Fig. 5 and Fig. 6, respectively.

First, we clearly see that the cross currents are zero on the

meridian connecting the North and Forward poles, as stated

by the TE constraint, Eq. 16. Furthermore, in Fig. 6, the cross

currents diminish towards the farther-most East-West meridian

in accordance with the TM constraint, Eq. 17. Other noticeable

Fig. 6. J⊥ from MoM at ka=19 on the lit side of 1/4 of a sphere, side view.

structures are the equatorial cross currents close to the shadow

boundary and a couple of bands across the body. In the top

view in Fig. 5 the direction of a narrow bundle of equatorial

currents running clockwise can be seen, next to a broader band

of cross currents running counterclockwise.

When extracting components of surface currents using

MoM, to be described more in detail in the next section,

a general experience is that data are inherently noisy. The

calculated J⊥(u) do not form smooth profiles along the

streamlines, as a function of the curve parameter u. Such non-

smooth data are seen in Fig. 5.

F. GTD Currents in Shadow Zone Along Geodesic Paths

In GTD/UTD, an approximation for the surface currents

are the creeping wave currents [5]. They are calculated using

the GTD diffracted magnetic field and diffraction coefficients

based on canonical solutions on simple geometries. Con-

cretely,

Escat = D(k, sd).eiksd (27)

where sd is the distance travelled along the geodesic, D(k, sd)are the diffraction coefficients times the wavefront attenuation

times a complex vector. Thus, the creeping wave currents are

obtained from the associated magnetic fields as Jsd= n×H.

Looking at J⊥ and J||, we can see systematic deviations from

creeping currents being aligned with the geodesics, essentially

away from the East-West meridian. Some deviations from the

phasefront directions of the currents, see below, were also

mentioned in [36]. Furthermore, if disregarding cross current,

charge transport along PO streamlines could be an alternative

to geodesic wavefront transport.

G. APE-MoM Phasefronts

[37] proposed another representation of the surface cur-

rents, over large smooth regions, with one or several linear

phase currents as

J(r′) = Ceik(m)·r′ (28)

6

where C is a complex vector amplitude assumed to be constant

and k(m) is the direction of a phasefront travelling on the

surface. At each point r′ on the surface, a local low frequency

phase variation is split into m ≥ 1 different phasefronts. The

phasefront extraction is achieved using a multi-dimensional

complex Fourier transform j(ku, kv) on a rectangular grid of

the low frequency complex current as

j(ku, kv) =

∫

S

J(r′)eikinc·r′dS′ (29)

where ku, kv are the components of the projection of kinc on

the surface S and J ∈ IC3 is obtained from MoM.

The phasefront extraction is thus a way of describing prop-

erties of the surface currents and, in some cases, decomposing

them into phasefronts of different directions. From such a

representation, partial information about oscillations of the

currents can be obtained, in particular phasefront directions

k(m) which are not the same as directions of the currents.

IV. THE USE OF BASIS FUNCTIONS

This section explains how data needed from the MoM are

extracted. This is a crucial step for obtaining estimated currents

J, and components of currents, such as J⊥ and J‖. It is not

just a technicality involving basis functions. Some additional

treatment of data must also be done to avoid unneccesary

numerical errors.

Our need is that of evaluating currents J(r(u)) along a curve

Ψ(u) from MoM or from FMM, where we expect data from

FMM to be even more noisy. These currents will in subsequent

sections be used for modelling or extrapolation (to higher

frequency) of surface currents.

A. The RWG Basis Functions in MoM

The so-called RWG basis functions are briefly described

below. They work on a triangulated body and provide us with

the behaviour of the surface currents. We use them also for

the non–standard task of vector field decomposition into nxH-

parallel- and cross-currents along the streamlines, as presented

below.

The Method of Moments yields an estimate of the current

density J on q:th triangle ∆q using the Rao-Wilton-Glisson

(RWG) basis functions, [22]. The direct output3 of MoM is

just the coefficients jm of the RWG-basis functions fm(r),

J(r) =∑

m

jmfm(r), r ∈ ∆q (30)

Given a triangle ∆q, the index m runs over the three possible

common edges to the neighbouring triangles, thus m = 1, 2, 3.

The coefficients jm are associated with the edges of the

triangles. We assume here a Finite Element type triangulation

where two neighbouring triangles share only one common

edge, thus share one jm value. For each local m, denote the

3MoM yields complex phasors. We use the real part of the coefficients inthe experiments unless stated otherwise. The imaginary part of the surfacecurrents tells us what the pattern looks like quarter of a period later.

position of the triangle node located opposite to a common

edge by r3(m).In principle, the coefficients of the RWG-basis functions

may be used for calculating estimates of J everywhere on

each triangle ∆q. By construction, summations of RWG-basis

functions produce currents J in the tangent plane of each local

triangle.

Thereby, we replace the tangents to the JPO–current stream-

lines, T, by the approximation Tp, where Tp is T projected

orthogonally onto the local triangle along the triangle normal.

Then both J from the RWG-basis functions and Tp are in the

plane spanned by the edges of the local triangle. The cross

currents are then calculated as:

J⊥ = J − (J · Tp)Tp (31)

J(r) =

3∑

m=1

jm · (r − r3(m)) · em/(2Aq) (32)

r ∈ ∆q

where jm are the scalar coefficients of currents (from MoM)

for m = 1, 2, 3, at the positions r ∈ ∆q. The scalar em = ±1and Aq is the area of triangle ∆q.

For a given triangle, one must assign arbitrary signs em for

the direction of the current flow across the m:th edge. Then,

the edge of the neighbouring triangle automatically obtains the

opposite sign −em. Data must be provided with a list of em

values over the triangles obeying the sign rule.

Analogously, for the parallel component:

J‖ = (J · Tp)Tp. (33)

In the experiments reported below, we have run MoM with a

tessellation of 11.500 triangles, on a unit sphere. Up to k = 30,

this represents at least 8 elements per wavelength. Frequencies

higher than k ≈ 60 are expected to lead to rather poor currents,

not resolving wavelength, with this amount of triangles.

B. Avoiding Unreliable Currents

Observation 6: In practice the RWG basis functions do not

yield reliable values for J close to triangle nodes.

This is illustrated in Fig. 7(a), and it is intuitively clear, since

neighbouring triangles besides the four triangles involved in

the sums of Eq. 32 do not correctly influence values close

to triangle nodes. In particular, the component J⊥ is quite

sensitive, and tends to be more erroneous close to triangle

nodes. This should maybe not come as a total surprise, since

the computed current J =∑

m jmfm is not smooth. More

marked errors, when attempting to extract vector components

such as J‖ and J⊥ close to triangle nodes, are expected.

For that reason, we form a hexagon of the triangle edge

midpoints, see Fig. 7(b), given triangle nodes r1, r2, r3:

mij = (ri + rj)/2, (34)

i, j = 1, 2 i, j = 2, 3 i, j = 1, 3

7

(a) RWG basis functions arenot reliable close to trianglenodes.

(b) Hexagons of confidence.

Fig. 7. Errors when using the RWG basis functions and (b) confidenceregions.

and midpoints connecting the triangle barycenters rc with each

of the triangle nodes

mℓ = (rc + rℓ)/2, ℓ = 1, 2, 3 (35)

We define the inside of this hexagon to be data with

confidence equal to 1, and points outside the hexagon to be

of zero confidence. Hence:

confidence(J(r)) = 1, r ∈ Hexq ∈ ∆q (36)

confidence(J(r)) = 0, otherwise (37)

where J(r) is the RWG evaluation of Eq. 32 and Hexq is the

q:th hexagon corresponding to the q:th triangle ∆q. This means

that we only calculate J(r) values with non–zero confidence,

when r traverses the positions of the curve Ψ. As indicated in

Fig. 7(b), long J⊥(u) vectors often arise in the non–confident

regions of the triangles.

The omitted values of zero confidence are replaced by

values linearly interpolated from nearest values of confidence

equal to 1. J⊥ and J‖ should be interpolated separately. But

before the linear interpolation, some data smoothing takes

place, explained in the next subsection.

In all our experiments only J⊥ has been smoothed, inter-

polated and subjected to the confidence procedure. Data for

J‖ did not seem to be in need of much improvement. Some

minor jaggedness does arise, but only locally. If smoothing

can be avoided, less artificial errors are introduced.

C. Data Smoothing

Since the data for the component J⊥(θ) estimated from

RWG-basis functions is of fairly poor quality, in particular

for higher frequencies k, we also perform slight smoothing to

reduce the noise level.

The following smoothing (almost central averages) is done

in our experiments after omitting non-confident data (but

before the linear interpolation) for J⊥:

θj∗ = (θj−1 + θj+2)/2 (38)

J⊥(j∗) = ((J⊥)j−1 + (J⊥)j+2)/2 (39)

and analogously for J‖ which is less crucial to smooth.

Central averages are also possible. Here both j and j∗ are

integer indices for the subset of positions which has nonzero

confidence. For simplicity, no smoothing takes place at the

end-points.

After these formulas have been applied, we, as mentioned,

linearly interpolate data so that the original sample points θt,

t = 1, 2, ..N are associated with these interpolated values. The

number of sampling points on a curve is equal to N , whereas

j = 1, 2, ..Nj , where Nj is the number of confident sampling

points on that curve.

D. Use of Symmetric Data – ϕ–mixing

For roundish bodies that are symmetric (e.g. circles, el-

lipsoids, hyper-ellipsoids, etc.) one may use two symmetric

streamline curves Ψ(u;ϕ). (We denote Ψ with an extra

argument indicating the chosen streamline ϕ = Const.) The

point is that the data from MoM will not be the same on either

side of the body, because MoM does not ensure symmetry

in the estimated coefficients jm and the triangle tessellations

covering the level curves ϕ = ±ϕ0 are not identical.

In this case, it is possible to use the confidence formulas

twice, first for Ψ(u;ϕ0) obtaining a confidence vector conf1∈RN and, secondly, another confidence vector for Ψ(u;−ϕ0):conf2∈ RN , where N = number of sample points on the two

Ψ–curves. Let the index ℓ = 1, 2, ...N .

Let J(r+) be the value from ϕ = ϕ0 and let J(r−) be the

value from the symmetric position on ϕ = −ϕ0.

The following simple data fusion procedure we term ϕ–

mixing:

Definition 7: If conf1ℓ= conf2ℓ = 1

J = 0.5 · (J(r+) + J(r−)) (40)

else if conf1ℓ=1 and conf2ℓ = 0

J = J(r+) (41)

else if conf1ℓ=0 and conf2ℓ=1

J = J(r−), (42)

else omit data (and linearly interpolate from confident data as

before). This procedure substantially improves the quality of

the estimated J⊥ and J‖. For non–symmetric data other similar

techniques can be used, such as using adjacent streamlines.

E. Improvement of MoM

The above mentioned procedure for symmetric roundish

bodies actually is of some interest in itself. The ϕ–mixing

actually can be used as a post-processing tool in order to

improve the MoM or the FMM currents.

V. DATA ANALYSIS

Here, data for a unit radius sphere are analyzed, by running

MoM for low and medium range coefficients, varying k from

10 to 29. Since k > 2π ≈ 6 the wave pattern performs more

than one wavelength when passing the spherical obstacle.

8

The reference models for J‖ and J⊥ may be modified as

follows:

Jmodel‖ = RI

‖ · 2Y · cos(kR sin θ), (43)

Jmodel⊥ = −RI

⊥ · Y sin(kR sin θ) · cos θ, (44)

where RI‖,RI

⊥ are examples of modifying coefficients. The

superscript I indicates what interval of the curve parameter

we refer to, the crudest possible interval decomposition being

the intervals corresponding to the shadow and illuminated

side, respectively, I = 1, 2. We omit the superscript except

when specifically distinguishing various intervals. Modifying

coefficients may be estimated from MoM data both for J‖

and J⊥, both being functions of θ given some PO streamline

ϕ = Const.

By estimating these coefficients for various frequency values

k, we see on one hand how large the systematic deviations

are from the reference models, and on the other hand, collect

information useful for extrapolation.

We present the estimation results first, and describe how the

modifying coefficients have been calculated, only thereafter.

A. Estimation of Modifying Coefficients

In Table I, we have, for a unit sphere, estimated R‖(k)and R⊥(k) for k = 10, 19, 29 for three different streamlines:

ϕ = π/3, ϕ = π/4 and ϕ = π/8.

The number of sample points is N = 200. The sampling is

equidistant in the curve parameter (θ) and it is convenient to

refer to an integer-valued curve parameter: t = 1, 2, ...N .

For k=29 cross current data are quite noisy and the earlier

described ϕ–mixing has been used, also. When using ϕ–

mixing the table value R1⊥(29) = 0.71 for ϕ = π/8 became

0.85. Hence, there is no verified downward trend over k.

B. Shadow Side

The second column of Table I shows that, in contrast

to the lit side, there seem to be clear trends over k for

the coefficients, such as decreasing cross currents causing

R2⊥(k)= 1.10, 0.91, 0.62 for k = 10, 19, 29.

It should be noted that the fit for J⊥ around the shadow

boundary, the equatorial zone, is poor, because the frequency

for the peaks of J⊥ does not follow k. Here, is an exception to

the rule of thumb that k and the incident H-field governs the

frequency of the variation of (components) of surface currents.

The equatorial parts of the streamlines Ψ, in particular on

the the shadow side, are not well-modelled by the reference

models, nor by the modified ones, Eqs. 43- 44.

C. Matching Peaks and Troughs

Here, some more details are given as an explanation how the

modifying coefficients of the previous section, concretely were

calculated. We have chosen a multiple scale peak matching

procedure.

The motivation for peak matching procedure instead of a

least square fit, is that, for the cross currents, it is not easy to

discern a clear correlation between the reference model and

TABLE I

RI=1,2

‖(k) AND R

I=1,2

⊥ (k) : ’MODIFYING COEFFICIENTS’ OVER

FREQUENCY k. ILLUMINATED SIDE (I = 1), SHADOW SIDE (I = 2).

ϕ = π3

k R1

‖(Lit) R

2

‖(Shadow)

10 0.54 -19 0.35 -29 0.32 -

k R1

⊥ (Lit) R2

⊥ (Shadow)

10 0.60 -19 0.60 -29 0.58 -

ϕ = π4

k R1

‖(Lit) R

2

‖(Shadow)

10 0.71 -19 0.51 -29 0.49 -

k R1

⊥ (Lit) R2

⊥ (Shadow)

10 0.82 -19 0.83 -29 0.85 -

ϕ = π8

k R1

‖(Lit) R

2

‖(Shadow)

10 0.96 0.8919 0.78 1.0029 0.76 1.11

k R1

⊥ (Lit) R2

⊥ (Shadow)

10 0.79 1.1019 0.78 0.9129 0.71 0.62

the MoM components, see an example of a cluster plot in

Appendix C. Yet, they do have qualitatively similar shape,

peaks and amplitudes.

The reference model functions JRef⊥ and JRef

‖ defined by

Eqs. 26 and Eqs. 23– 24, respectively, have easily defined

local maxima and minima, whereas J⊥ and J‖ from MoM

fluctuate a lot with many spurious local minima and maxima.

A subset of these are to be pairwise associated with the easily

defined local maxima and minima of the reference model.

A least squares fit of a line would require all the pairs of

(JRef⊥ , J⊥) to make a global fit of the slope of the line, see

Appendix C, thus increasing the influence of the noise for

estimating R⊥.

Consider the set of local maxima or minima in the dis-

cretized vectors of JRef⊥ (ui)i=1...N and JRef

‖ (ui)i=1...N ,

and denote the corresponding positions of these local extrema

by argmin and argmax. These positions are subsets of

t = 1, 2...N . With this notation, a list of t positions for the

extrema tM and tm for the reference model of cross currents

can be written:

tRefM = [argmax(JRef

⊥ )]M , M = 1, 2, .., NM (45)

tRefm = [argmin(JRef

⊥ )]m, m = 1, 2, .., Nm (46)

where NM and Nm are the numbers of local maxima, local

minima respectively for JRef⊥ and N is the number of samples

9

TABLE II

RI=1,2,3

‖(k) AND R

I=1,2,3

⊥ (k) : ’MODIFYING COEFFICIENTS’ OVER

FREQUENCY k. ILLUMINATED SIDE (I = 1), SHADOW SIDE (I = 2),

EQUATOR (I = 3).

ϕ = π3

k R1

‖(Lit) R

2

‖(Shadow) R

3

‖(Equator)

10 0.53 0.69 0.3019 0.43 0.79 0.3229 0.45 0.83 0.28

k R1

⊥ (Lit) R2

⊥ (Shadow) R3

⊥ (Equator)

10 0.86 1.52 0.3319 1.17 1.23 0.2529 0.79 0.56 0.39

ϕ = π4

k R1

‖(Lit) R

2

‖(Shadow) R

3

‖(Equator)

10 0.65 1.20 0.6319 0.66 1.05 0.4629 0.62 1.29 0.43

k R1

⊥ (Lit) R2

⊥ (Shadow) R3

⊥ (Equator)

10 1.09 – 0.7919 1.23 1.77 0.4329 1.49 0.77 0.52

ϕ = π8

k R1

‖(Lit) R

2

‖(Shadow) R

3

‖(Equator)

10 0.87 1.17 0.8919 0.90 1.13 0.7229 0.82 1.36 0.77

k R1

⊥ (Lit) R2

⊥ (Shadow) R3

⊥ (Equator)

10 1.12 2.33 0.6619 1.48 1.74 0.4229 1.09 0.70 0.51

of J⊥. The positions for the extrema of J⊥ from MoM are:

tM = [argmax(J⊥)]M , M = 1, 2, .., N ′M (47)

tm = [argmin(J⊥)]m, m = 1, 2, .., N ′m (48)

where M in Eq. 47 runs over all candidate maxima with

N ′M ≫ NM , normally.

Let us now match peaks. Troughs (local minima) are

matched analogously, in the same manner. Peaks in J⊥ are

matched with peaks (peak = 1, 2, ...NM ) in JRef⊥ by choosing

those M–values within a search distance S1 as

S1 =N

2(NM + Nm)(49)

which satisfy the inequalities:

∀peak : |tM − tRefpeak| < S1 (50)

|tM − tRefpeak| < 1.5 · S1. (51)

The second inequality is used only when no tM , for the given

peak, satisfies the first inequality. When several peaks satisfy,

e.g., the first inequality, the highest peak is chosen. If no peak

satisfies the first inequality, the second inequality applies, and

the highest peak there is chosen. Some obvious special cases

must also be taken care of. The quantities tM and tRefpeak are

defined by Eqs. 47 and 45. The output from this matching

procedure consists of two sets of extrema:

Mp = matched extrema, p = 1, 2, ...(Nm + NM ) (52)

MRefp = reference model extrema, p = 1, 2, ...(Nm + NM )

(53)

Finally, the modifying coefficients are calculated as:

RI⊥(k) =

P

p|Mp|

P

p|MRef

p |. , p ∈ I (54)

J‖ is treated analogously.

As a byproduct, the positions tM for the chosen peaks,

may also convey information on phase shifts in the sinusoidal

patterns.

D. Modifying Coefficients for Three Intervals

It makes more sense to use three intervals for the modifying

coefficients, since J‖ and J⊥ often exhibit different deviations

from the reference models in the equatorial regions (I = 3),

the deep shadow region (I = 2) and the central lit region

(I = 1). The equatorial regions are associated with relatively

lower magnitudes of R‖(k) and R⊥(k), normally without

strong trends. The three intervals used in Table II are

S1 = θ : θ ∈ [π/6, π − π/6] (lit zone), (55)

S2 = θ : θ ∈ [π + π/6, 2π − π/6] (deep shadow), (56)

and finally θ ∈ [0, 2π] − S1 − S2 (equatorial regions). The

interval termed ’Shadow’ in Table II means shadow zone,

excluding the equatorial parts around the shadow boundary,

and the same comment can be made for the lit zone.

The modifying coefficients are less exact on the shadow side

for J⊥ since, as mentioned, the frequency of the oscillations

is not correctly modelled. Then, the peak matching procedure

will for some items in the sum of the terms in the numerator

of Eq. 54 cause some bias in the resulting ratio R⊥(k). The

’-’ in Table II means that the automatically calculated R⊥(10)was erroneous due to poor peak matching. The peak matching

is intended for the situation when the number of peaks and

troughs of both signals are the same.

VI. CONCEPTS FOR MODELLING AND EXTRAPOLATION

In this and following sections all vectors are real. Hence, J

means in fact JRe and we have dropped the superscript Re.

We focus on comparisons between estimated surface currents

from MoM, and modelled surface currents by three models:

(a) standard PO, (b) our Reference model defined by Eq. 26,

Eq. 23, Eq. 24 and (c) our modified Reference Model, Eqs. 43–

44. A first test is to investigate the behaviour on a sphere,

which we use for illustrating our technique.

Consequently, three errors (deviations from MoM) are to

be compared. Let Jmodel be the estimated currents from

Eqs. 43–44, using some R⊥– and R‖–coefficients, and MoM

currents and reference model currents, as before, J and JRef ,

respectively.

We denote by [J] the concatenated vector containing all the

vector currents J over the streamlines of dimension 3N . To

10

avoid the use of absolute numbers, the following so–called

performances are defined:

Γmodel =

(

1 − ‖[Jmodel] − [J]‖‖[JPO] − [J]‖

)

· 100 (%) (57)

ΓRef =

(

1 − ‖[JRef ] − [J]‖‖[JPO] − [J]‖

)

· 100 (%) (58)

where JPO is the PO current of Eqs. 1–2. We use PO as a

reference for model errors. In order to compute the error, it is

not necessary to use J as a vector in IR3. Considering

J = J‖T + J⊥nc (59)

where T,nc ∈ IR3 which form an orthonormal basis, we

obtain the error vectors in PO: EPO = J−JPO and the error

in the reference models ERef = J − JRef as

EPO = (J‖ − JPO)T + (J⊥ − 0)nc (60)

ERef = (J‖ − JRef

‖ )T + (J⊥ − JRef⊥ )nc (61)

Emodel = (J‖ − Jmodel

‖ )T + (J⊥ − Jmodel⊥ )nc (62)

Then the Euclidean norm of Emodel is,

||Emodel(u)||2 = (J‖ − Jmodel‖ )2 + (J⊥ − Jmodel

⊥ )2 (63)

due to orthonormality. Thus, the scalar error is determined

solely by

Emodel‖ = J‖ − Jmodel

‖ (64)

Emodel⊥ = J⊥ − Jmodel

⊥ . (65)

Analogously for EPO‖ , EPO

⊥ and ERef‖ , ERef

⊥ .

For the global error, all the Euclidean errors must be

summed, yielding the norm of [Emodel] as

||[Emodel]|| =√

∑

j ||Emodel(uj)||2. (66)

In our experiments, we use a smoothed version for J⊥, Eq. 39.

VII. EXTRAPOLATION

One idea is to use the estimated coefficients R‖(k) and

R⊥(k) for “extrapolation” to higher frequency k. Denote the

two coefficients by Rn(k), n = ⊥, ‖. By extrapolation two

things may be meant. One possibility is to calculate some

average Rn(k)–value (averaging over k) in Eq. 43 or Eq. 44,

above, and use that approximate Rn–value for a higher k–

value in the equations. The other possibility is to extrapolate

Rn(k) as a function of k. The latter procedure is of course

preferable. But in some cases, trends in Rn are so hard to

calculate that it is probably safer to just use an average Rn–

value.

The following extrapolation from k = 10, 19, 29 to k = 60serves to illustrate the approach. The following averages were

used for the nxH parallel components:

RI=1,2,3‖ = 0.87 1.22 0.80 (ϕ = π/8) (67)

0 100 200 300 400 500 600 700−5

0

5x 10

−3 Extrapolation of MoM, Parallel component, k=60 Phi=0.3927

u

0 100 200 300 400 500 600 700−6

−4

−2

0

2

4

6

8x 10

−3 Error Model to MoM

u

Error Model+stand. dev. Model−stand. dev. Model

Fig. 8. Case ka=60, J‖ for ϕ = π/8 using extrapolated RI=1,2,3

‖(k)

and RI=1,2,3

⊥ (k) compared to MoM. Note that we do not extrapolate MoMcurrents in themselves but use low frequency values k = 10, 19, 29 toestimate the modifying coefficients for ka = 60. The deviations from MoM,called Error in the legend, is computed using Eq. 64.

by averaging the values in Table II, column-wise in rows 13

to 15. The other coefficients used for the same ϕ were:

RI=1,2,3⊥ = 1.23 0.58 0.47 (ϕ = π/8) (68)

by averaging column-wise over k, and by a very crude trend

extrapolation for the second value: 0.58 = 1/4 · R⊥(10),noting that R⊥(29) = R⊥(10)/3. The downward trend of

R⊥(k) is significant, in this case.

MoM for cross currents is inherently noisy, growing worse

for higher frequencies, so we used ϕ–mixing (Def.7) for

stabilizing the values somewhat. We thus use improved MoM.

When calculating the reference model and the modified

reference model using these coefficients for k = 60 in the

formulas, the following deviations from MoM arose:

‖[Jmodel] − [J]‖ = 0.0406 (69)

‖[JRef ] − [J]‖ = 0.0466 (70)

‖[JPO] − [J]‖ = 0.0557 (71)

and the previously defined performance value in percentage

relative to the PO technique for the whole streamline were

Γmodel = 27%, ΓRef = 16%, (72)

and for the lit side only:

Γmodel = 14%, ΓRef = −4%. (73)

Hence, the extrapolation yields about 30 % improvement over

PO, in modelling surface currents and does better than the

reference model.

The modifying coefficients have been modelled as piece-

wise constant. This is of course not necessary. One may as

11

well model smooth transitions between different intervals, for

example by using the constant levels in the interior of the

intervals and use splines for connecting these constant levels

in adjacent intervals. This is in our opinion a reasonable but

not crucial enhancement of the approach. Furthermore, when

extrapolating to the double electric size (using values from

ka=10,19,29 → ka=60), Fig. 8 shows that the resulting error,

cf. Eq. 64 using the modifying coefficients is of the same

magnitude as in many of the modelling cases, to be presented,

see for instance Fig. 21. The extrapolation does not introduce

more errors. Next section will go deeper into the modelling.

VIII. RESULTS OF MODELLING

The reference models can be directly applied as models

of surface currents without any knowledge of data from ex-

tracting algorithms. In that sense, they remind of PO currents.

Thus, it may be of interest to see whether there is any gain in

using the reference models Eq. 26 and Eqs. 23– 24 instead of

JPO.

Another alternative is that of using some data from MoM-

results for some k–values and use such data for modifying

the reference models exploiting the modifying coefficients Rn

which arise from comparing MoM–data with the reference

model for the chosen k–values. In the experiments, to be

presented below, we have chosen k = 10, 19, 29 for calculating

the three-interval modifying coefficients: RI=1,2,3n (k). The

first issue to clarify is whether these modified reference models

Eqs. 43, 44, using Table II, imply a visible improvement over

the original reference models.

In Table III, we compare on one hand reference model

currents JRef and J, and on the other Jmodel and J. We

measure the advantage of using JRef or Jmodel for mimicking

J, compared to JPO, using the performances defined by

Eqs. 57–58.

This technique is meant to be used for other k–values and

other ϕ–values, by defining functions RIn(k;ϕ) using sparse

data such as, e.g., k = 10, 19, 29 and ϕ = π/3, π/4, π/8 for

defining them.

A. Modelling of Currents on Sphere and Ellipsoid

For a sphere, the results along a PO current streamline are

displayed in Fig. 9-10 and compared to MoM, for ϕ = π/8.

Here, we note how non-smooth the results obtained from MoM

are. We present results for different k–values, by writing ka-

values, where a is the radius of the sphere.

When using our first reference model, good agreement for

the parallel component JRef‖ is obtained at both low and higher

frequency. The nxH-parallel component, like the JPO–current,

displays error which amplifies close to the shadow boundaries

located on the plots Fig. 9 and Fig. 11 around the end points

of the lit interval. The perpendicular component JRef⊥ is more

difficult to model directly, as seen in Fig. 10(a) for low

frequency ka = 10 and in Fig. 10(b) for higher frequency.

In the shadow region, the reference model for JRef‖ is in

phase with the MoM current, as seen Fig. 11(a) at ka = 10and Fig. 11(b) at ka = 19 but at the south pole the amplitude

TABLE III

PERFORMANCE WITH REFERENCE MODEL AND EXTRACTION METHOD

COMPARED TO PO, OVER k USING RI=1,2,3

‖(k) AND R

I=1,2,3

⊥ (k).

ϕ = π3

k Extract (All) Extract. (Lit) Ref. (All) Ref. (Lit)

10 55% 73% -5% -2%19 63% 74% 8% -1%29 63% 69% 0% -2%

ϕ = π4


10 – 51% 21% 0%19 51% 58% 21% 0%29 48% 47% 15% -1%

ϕ = π8


10 36% 11% 33% -2%19 36% 30% 26% 2%29 41% 17% 28% -2%

should in fact be higher. For JRef⊥ , modelling of phase is

needed since the reference model is out of phase, as seen in

Fig. 12.

For an ellipsoid

(x, y, z) = (cos ϕ cos θ, 2.0 sin ϕ, cos ϕ sin θ) (74)

the streamline is also an ellipse and the reference model for

JRef‖ works fine and no new pattern is introduced, see Fig. 13.

This is a good sign that our technique can be applied to more

general roundish bodies.

B. Performance Comparisons

In Table III, a series of experiments for frequencies k =10, 19, 29, for a sphere, are presented, in terms of perfor-

mances (Eqs. 57–58), relative to Physical Optics currents.

Of course, both J⊥ and J‖ influence the results, whereas

the latter component somewhat dominates by being roughly

3–4 times larger in magnitude.

Consider the reference model, first. When including the

shadow region, the reference model (column Ref. All, Ta-

ble III) is a 20% to 30% better model than PO, for ϕ = π/4or ϕ = π/8.

The last column of Table III tells us that JRef⊥ –estimates

have not contributed to improve the fit with MoM–data, com-

pared to PO. The dominant part of the signal J‖(u), is identical

for the lit side since JPO(u) = JRef‖ (u), so considering only

J‖(u) the performance percentage would be exactly zero. In

this case, the cross currents components are afflicted with too

much errors to improve over PO. Performance percentage for

the reference model for the cross current is about ±2%.

In general, for the coefficient model, if looking at the error

plots in Appendix B, for instance in Figures for ϕ = π/4k=19, or, ϕ = π/8, k = 29, we have quite a good model for

J⊥(u) away from the equatorial zone, with error magnitudes

around 0.3 · 10−3, but phase errors (three–four times larger)

in the sinusoidal signal in that zone outweigh what we gain

in the central lit part. The model with coefficients gives only

a slight improvement over PO, for the same reason.

12

0 50 100 150 200 250−6

−4

−2

0

2

4

6

8x 10


θ

|J|

From MoMModeled

(a) Case ka=10, ϕ = π/8 (lit side).

0 50 100 150 200 250−6

−4

−2

0

2

4

6x 10


θ

|J|

From MoMModeled

(b) Case ka=19, ϕ = π/8 (lit side)

Fig. 9. Reference Model (no modifying coefficients) for the parallel component JRef

‖compared to MoM. Good agreement in phase.

0 50 100 150 200 250−1.5

−1

−0.5

0

0.5

1

1.5x 10


θ

|J|

From MoMModeled

(a) Case ka=10, ϕ = π/8 (lit side)

0 50 100 150 200 250−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10


θ

|J|

From MoMModeled

(b) Case ka=19, ϕ = π/8 (lit side)

Fig. 10. Reference Model for the perpendicular component JRef

⊥ compared to MoM. Good agreement for the amplitude but slightly out of phase.

For the J‖(u)–part we have excellent fit in most cases, in

particular when using the Jmodel currents. This can be seen in

Figures in Appendix B corresponding to k = 19 and k = 29.

A general observation is that when gradually decreasing

ϕ, i.e., moving towards the equator, the performance of

the reference models or modified reference models increases

substantially. This is so, because PO is gradually a poorer

model (of the parallel component) the closer to the shadow

boundary in the lit zone the surface current measurements are

made. At ϕ = π/8, close to the East–West meridian in the lit

zone, it is not equally easy to improve over PO.

For the East-West meridian itself at ϕ = 0, we suggest using

J⊥ = 0, Eq. 16, as reference model. And J⊥ = 0 also when

ϕ → π/2.

In general, in the equatorial region, the model signals are

out of phase or even in counter phase compared to J⊥,

which deteriorates the achieved good result in non-equatorial

regions, both for the reference model and when modified with

coefficients.

C. Angle Measurements

The total mean angle on one streamline ϕ = π/8 (lit region

only) over frequency k for the PO, the reference model are

shown in the next table. It shows that the reference model gives

improved surface current directions on the lit side compared

to the PO-current directions. On the shadow side there is

nothing to compare since PO currents are zero and thus lacking

directions.

D. General Comments

It should be kept in mind that what Table III measures is

how much models deviate from MoM, rather than the error

relative to the true solution. For this the Mie solutions should

be used.

13

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Parallel Current Component: k=10 cut Phi=0.3927 (shadow)

u

|J|

From MoMModeled

(a) Case ka=10, ϕ = π/8 (shadow)

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Parallel Current Component: k=19 cut Phi=0.3927 (shadow)

u

|J|

From MoMModeled

(b) Case ka=19, ϕ = π/8 (shadow)


‖compared to MoM.

0 50 100 150 200 250 300 350 400−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3 Perpendicular Current Component: k=10 cut Phi=0.3927 (shadow)

u

From MoMModeled

(a) Case ka=10, ϕ = π/8 (shadow)

0 50 100 150 200 250 300 350 400−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3 Perpendicular Current Component: k=19 cut Phi=0.3927 (shadow)

u

From MoMModeled

(b) Case ka=19, ϕ = π/8 (shadow)

Fig. 12. Reference Model for the perpendicular component JRef

⊥ in shadow compared to MoM. Good agreement for the amplitude but out of phase.

TABLE IV

MEAN ANGLE AND PERFORMANCE PERCENTAGE.

ka 10 19 29PO 11.51o 23.85o 25.33o

Ref. model ϕ = π/8 12.93o 17.48o 22.35o

Perf. Ref-PO +12% +36% +13.0%

Since MoM is considered to be a fairly good algorithm, this

may not be so serious, but nevertheless some of the variation

of MoM are deviations from the true solutions. Locally, some

of the jaggedness of the MoM solution does not replicate the

true solution. Hence, some of the errors of Table III includes

implicitly variations in slightly erroneous MoM–data.

Good results for J‖, in particular for higher frequencies

are obtained for the coefficient model, as seen in Figures in

Appendix B, for k = 19, 29 for all angles ϕ.

A general lesson learned from Section B is that a better

reference model for J⊥(u) in the shadow zone is needed,

because the frequency k apparently does not coincide with

the actual local spatial frequency.

Moreover, phase errors of the fluctuations of J⊥(u) in the

lit zone should be possible to avoid with a somewhat improved

reference model. Generally, the modifying coefficients are

really useful only to the extent that they are stably defining

some fairly constant value or some simple trend function over

k, because it is desirable to infer Rn(k) values from a few

sparse frequencies. In our case, we chose k = 10, 19, 29because we noted that lower k–values are associated with

strong trends in the scattered E far–field. There is a special

dynamics over k for really low frequencies, but we are more

14

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10

−3 Parallel component, k= 10, Phi=0.3927, ellipsoid

from MoMModeled

(a) Case ka=10, ϕ = π/8 (ellipse, lit side)

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10

−3 Parallel component, k= 20, Phi=0.3927, ellipsoid

from MoMModeled

(b) Case ka=20, ϕ = π/8 (ellipse, lit side)


‖on an ellipse compared to MoM.

crossJJmodel

JmomJpo

Fig. 14. Mean angle as a measure of the error.

interested in the intermediate frequencies, or higher.

Another technique for extrapolation was presented in [37],

APE-MoM, where the MoM is used twice, once with standard

basis functions and once more multiplying the basis functions

by

ei

λlowλhigh

k(m)·(r′−rm)

(75)

where k(m) are extracted phasefront directions, Eq. 28.

IX. CONCLUSION

It has been shown that the vector field decomposition of

the surface currents yields a better understanding of charge

transport and charge accumulation. The oscillating behaviours

of the current have been shown to be strongly linked to

that of the incident magnetic H–field. The decomposition

of the surface currents into nxH-parallel and cross-current

vector components yield systematic spatial patterns evolving

over frequency closely related to the evolution of the incident

magnetic field over frequency. This is true both on illuminated

and shadow side of the body, allowing us to formulate a

reference model for each component over an entire roundish

body. Currents on the shadow side, wrongly assumed to be

zero for all meridians by the Physical Optics, can be modelled

by these fairly simple models without relying on a complex

procedure such as creeping rays in the GTD formulation.

However, for the TE-meridian, zero PO currents is a good

approximation, as seen in Appendix D.

As a side product, the perpendicular component J⊥ can be

interpreted as a source of error in the PO current. Thus, the

model for the cross-currents could be extended to a numerical

procedure for automatic control of the error growth in the PO

technique.

Although we have studied a sphere, the approach should be

applicable to roundish smooth bodies rather generally. This

is because the nxH-parallel component J‖(u) can be well

modelled as a function of the curve parameter only along

PO streamlines. A procedure to determine the PO streamlines

has been given and illustrated on the complex shape of the

Eikon UAV. The reference model for the second component,

the cross currents J⊥, has been found to be not sufficient.

Although it qualitatively captures the oscillating behaviour of

that component of the current, the approximation is afflicted

with phase errors. In the lit zone, we can get phase shifting

errors, whereas the shadow zone also gives rise to fewer

oscillations of J⊥. The phase of J⊥ must be better modelled

but better understandings of the underlying physics are needed

for this endeavor. Nevertheless, the modelling of J⊥ is less

important than that of J‖, since the latter is about 4 times

larger as a rule.

Tentative extrapolation was achieved. This was done by the

use of reference models modified by coefficients calculated

from low frequency data from MoM. The coefficients act like

diffusion coefficients applied to the amplitude of each compo-

nent. A surprising result was that, in the neighborhood of the

vertices, the MoM current used to compute the coefficients

yield unreliable values. Corrections and smoothing of these

values have been implemented. Trends over frequency in the

coefficients have been studied, allowing us to extrapolate the

15

current to higher frequency. Our experiments show that we

could easily double the electrical size of the problem.

The whole approach presented in this study, as a stand alone

procedure, offers an alternative way to derive high frequency

approximate currents. As a processing tool plugged into an

MoM-PO hybrid solver, it could offer a way to control the

error of the Physical Optics current, with the potential to

efficiently improve the accuracy of the MoM-PO hybrid solver.

ACKNOWLEDGMENT

The authors would like to thank Jesper Oppelstrup for

his thoughtful recommendations and suggestions. Financial

support has been provided by NADA, KTH, PSCI and the

National Aeronautical Research Program (NFFP) within the

General ElectroMagnetic Solvers (GEMS) and Signature Mod-

eling and Reduction Tools (SMART) projects.

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16

APPENDIX A: TE-TM CASES

Htan

EH

total tangent H double total tangent H double

E

H

H

E

H

E

E

Hk k n

E=0E =0

tangent. comp. E flips

TM case: tangent. comp. E flips TE case: normal comp. H flips

Fig. 15. TE, TM: case on a plate.

HE

H

E

H

k n

E=0

EH

E

E

H

Hk

E =0tan

E

H

ϕ

θ

k

YX

TM TE

Fig. 16. TE, TM situations on a sphere.

APPENDIX B: MODELLING OF CURRENTS

The following figures present a series of experiments where J‖ and J⊥ are modelled for various frequencies k and different

streamlines ϕ = π/8, π/4, π/3. The first six pictures correspond to the streamline ϕ = π/8, close to the East-West meridian,

for k = 10, 19 and 29. Then, the two next figures represent J‖ and J⊥ evaluated on one intermediate streamline ϕ = π/4,

i.e. a section in the middle of the sphere, for one frequency k = 19. Finally, the last four figures represent J‖ and J⊥ for two

frequencies for k = 19 and 29 on the streamline ϕ = π/3 close to the forward pole. Each figure contains two pictures, the

component J‖ or J⊥ and their respective deviations from MoM, called Error in the legends and computed using Eq. 64 and

Eq. 65.

17

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6

8x 10

−3 Approximations of MoM, Parallel component, k=10 Phi=0.3927

u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2

3x 10

−3 Error REF model to MoM, Error Model with Coefficient to MoM

u

Error Model+stand. dev. Model−stand. dev. ModelError REF+stand. dev. REF−stand. dev. REF

Fig. 17. Case ka=10, J‖ for ϕ = π/8 using RI=1,2,3

‖(k) and R

I=1,2,3

⊥ (k) compared to MoM and PO.

0 50 100 150 200 250 300 350 400 450−3

−2

−1

0

1

2

3x 10

−3 Approximations of MoM, Perpendicular component, k=10 Phi=0.3927

u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−3

−2

−1

0

1

2

3x 10


u

Error Model+stand. dev. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF

Fig. 18. Case ka=10, J⊥ for ϕ = π/8 using RI=1,2,3

‖(k) and R

I=1,2,3

⊥ (k) compared to MoM.

18

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10


u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−4

−3

−2

−1

0

1

2

3x 10


u



‖(k) and R

I=1,2,3


0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2x 10


u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2

3x 10


u



‖(k) and R

I=1,2,3


19

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10


u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−4

−2

0

2

4

6x 10


u



‖(k) and R

I=1,2,3


0 50 100 150 200 250 300 350 400 450−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10


u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−3

−2

−1

0

1

2

3x 10


u



‖(k) and R

I=1,2,3

⊥ (k) compared to MoM. Here, the fit between J⊥ from MoM and J⊥ from the

coefficient model (JPerpmodel in the legend) is quite good.

20

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10


u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4x 10


u



‖(k) and R

I=1,2,3


0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2x 10


u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−3

−2

−1

0

1

2

3x 10


u



‖(k) and R

I=1,2,3


21

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10


u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4x 10


u



‖(k) and R

I=1,2,3


0 50 100 150 200 250 300 350 400 450−1.5

−1

−0.5

0

0.5

1

1.5x 10


u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2x 10


u



‖(k) and R

I=1,2,3


22

0 50 100 150 200 250 300 350 400 450−6

−4

−2

0

2

4

6x 10


u

JParMoM

JParmodel

JParREF

JPO

0 50 100 150 200 250 300 350 400 450−5

0

5x 10


u



‖(k) and R

I=1,2,3


0 50 100 150 200 250 300 350 400 450−1.5

−1

−0.5

0

0.5

1

1.5x 10


u

JPerpMoM

JPerpmodel

JPerpREF

0 50 100 150 200 250 300 350 400 450−2

−1

0

1

2x 10


u



‖(k) and R

I=1,2,3


23

APPENDIX C: MODIFYING COEFFICIENTS

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Cross current components (Ref)

Cro

ss c

urre

nt c

ompo

nent

s (M

oM)

Cluster plot

Mom versus Refleast sq. fitpeak matching

Fig. 29. Cluster of points (JRef

⊥ ,JMoM⊥ ) for the lit region only, ϕ = π

8, ka = 29, the slope of a fitted line in the lit interval giving the modifying

coefficients (R⊥. The line can be computed either by using least square fitting or peak matching fitting. The latter uses only a subset of the data rather thanthe whole cluster of points as in least square, thus removing noise.

APPENDIX D: 2D SOLUTION ON A CIRCULAR SECTION

Let u = Ez for E = (0, 0, Ez). Helmholtz’s equation

∇2u + k2u = 0, u ∈ IR2 − ∂S (76)

with boundary condition uinc = −uscat on the circular boundary ∂S of radius r = a, can be solved analytically for an incident

planar wave uinc to very high frequency for

uscat = −+∞∑

m=−∞

imJm(ka)

H2m(ka)

eimϑH2m(kr) (77)

where H2m is a Hankel function and Jm is a Bessel function of the first kind.

24

(a) Case ka=29, total E-field vanishes in the shadow zone (b) Case ka=29, scattered E-field

Fig. 30. 2D analytical solution on a circular PEC illuminated by a planar wave from right to left.

0 50 100 150 200 250 300 350 400−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Etot

(a) Case ka=29, total E-field vanishes to zero in the shadow region

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

1.5Escat

(b) Case ka=29, scattered E-field

Fig. 31. Circular section of E-fields close to the surface, E=0 exactly on the surface.

25

PHYSICAL OPTICS AND NURBS FOR RCS

CALCULATIONS

Sandy Sefi(I), Jesper Oppelstrup(II)

(I) Email: [email protected] Institute of Technology, Stockholm, Sweden

(II)Email: [email protected] Institute of Technology, Stockholm, Sweden

Abstract: This paper presents a Physical Optics method using an adaptive

triangular subdivision scheme for the surface integral in the computation of

monostatic RCS of NURBS surfaces.1

1 Introduction

The computation of electromagnetic radiation patterns from complex geometries withPhysical Optics (PO) involves the evaluation of a surface integral. Classically, the POintegral is evaluated numerically by quadrature formulas on triangular or quadrilateralpatches which require several evaluation points per wavelength in order to catch thehighly oscillatory behavior of the integrand. Alternative methods consist in decouplingthe integrand into an oscillatory phase factor from an amplitude factor which is smoother.Then, the integral can be solved analytically from few points per surface provided thatboth factors have been approximated separately using e.g. polynomials, linear as in [1],[2] and [3], or quadratic as in [4].

Typical geometrical scenes contain complex geometries like Unmanned Aerial Vehicles(UAV) such as Eikon in Figure 1. These are small aircraft which have 8 to 10 meters inlength. A simple CAD design for Eikon will consist in 109 NURBS surfaces along with264 trimming curves. Each NURBS patch can then be discretized into smaller triangleson which the PO integral is more easily evaluated.

Figure 1: Eikon UAV.

1This work is part of the CEM program at the Parallel and Scientific Computing Institute (PSCI)under the project GEMS: General Electromagnetic Solvers supported by the National Aeronautical Re-search Program (NFFP), the Swedish Agency for Innovation Systems (Vinnova), Saab AB and EricssonMicrowave Systems AB.

To do so without resolving the wavelength, a linear phase and constant amplitude ap-proximation to the PO integral is used for each triangle. But, to insure its accuracy, weadaptively adjust the triangle density based on local variation of the phase. This adaptivesubdivision scheme is the basis of the PO method used here and the gain resides in howfast it can be performed.

In addition, since working with general shape objects, the scheme automatically resolvesthe geometry, i.e. the subdivision criteria takes into account the curvature of the objectusing the NURBS description of the surface.

Furthermore, improvements to the basic PO formulation are made by including shadowing[5] and edge diffraction. For fast edge detection, we use the trimming curves topologyinformation available in the NURBS description.

Approximate shadow regions are obtained efficiently on the NURBS surfaces using ray-tracing techniques. Practically, a test on the outward normal at each triangle is combinedwith an occlusion test using ray tracing on NURBS, see [6]. One ray is launched from eachtriangle and traced toward the observer direction. As a result the triangle is consideredilluminated if the ray path is not blocked by any other surfaces. To be efficient, thismethod also requires the use of topological information between NURBS and triangles,see [6] for more details.

This set of algorithms can be used directly for an enhanced PO analysis on smoothcurved surfaces or can be applied in an iterative MoM-PO hybrid solver for more complexproblem, as in [3].

2 Evaluation of the radiation integral

In the frequency domain, the radiation process described by Physical Optics consists inthe approximation Jpo to the currents generated from a known incident field on a surfaceS,

Jpo =

2nscat(X

′) × H inc(X ′), on the lit portion of S0, on the shadowed portion of S

(1)

nscat is the unit normal vector to S at X ′, the position vector of a point on S and H inc isthe complex magnetic field incident on that point. PO assumes a zero surface current inshadow regions which neglects the creeping wave contribution.

The scattered fields Escat from the surface currents at an observation point X can beexpressed over the entire surface S by integrating the scattering component of the POcurrents applied to the Green’s function as follows,

Escat = jkZ0

∫

Slit

∫[Kscat × Kscat × Jpo]G(X, X ′)ds′ (2)

where ds′ is the infinitesimal element of area on S, Z0 the free space impedance, Kscat

the direction pointing from S to X and k = 2π/λ is the wavenumber where λ is thewavelength. The free space Green’s function is

G(X, X ′) =e−jk|X−X′|

4π|X − X ′| (3)

2

Under far field assumption and for monostatic emitter observer direction, the integralfrom a reflecting PEC surface reduces to

Escat =−j

λE0

e−jkR

R

∫

Slit

∫Kscat · nscate

2jk(Kscat.X′)ds′ (4)

where Slit is the illuminated portion of S which must be determined via a shadow regionprocedure. We call the PO integral the following complex function:

Ipo =1

λ

∫

Slit

∫Kscat · nscate

2jk(Kscat.X′)ds′ (5)

The exponential factor displays the rapidly varying phase. Finally we define σ, the RadarCross Section, as a normalization of the scattering integral Ipo such as

σ = 4π||Ipo||2 (6)

The numerical problem consists in evaluating the double integral in Equation (5). Wedecouple the phase factor which is highly oscillatory, from the amplitude factor which issmoother. Then the phase can be approximated by linear function Bc(x, y) over smallcells such as

Bc(x, y) = Bxx + Byy + B0 ≈ 2Kscat · X ′(x, y), (x, y) ∈ cellc (7)

The three coefficients Bα can be determined using triangular cells only which are mappedinto a reference configuration. The difference between the true integral and the numericalintegral can be measured as follow:

ǫ = |∫ ∫

A(x, y)ejkBc(x,y)dxdy −∫ ∫

A(x, y)ejk eBc(x,y)dxdy| (8)

|ǫKA+ ǫKB

| = |∫ ∫

(A − A)ejkBcdxdy +

∫ ∫Aejk eBc(1 − ejk(Bc− eBc))dxdy| (9)

|ǫKA| = |

∫ ∫(A − A)ejkBcdxdy| =

Ncell∑

cell=1

∫ ∫

Cell

|(A − A)ejkBc |dxdy (10)

|ǫKA| ≤ |A − A|KA ≤ KAh (11)

|ǫKB| = |

∫ ∫Aejk eBc(1 − ejk(Bc− eBc))dxdy| ≤

Ncell∑

cell=1

∫ ∫

Cell

|(1 − ejk(Bc− eBc))|dxdy (12)

ǫ ≤ KAh + KBkh2 (13)

where KA and KB are constant and h represents a measure of the size of the cell. Notefrom Equation (13) that ǫ → 0 when h → 0 and that we only need cell of size h ∼

√λ.

The surface normal is taken as the triangle normal such as the amplitude factor becomesconstant over a triangle and thus can be moved outside of the integrand. Then the linearphase approximate integral can be written

Ipo =1

λ

Ncell∑

cell=1

Ipocell(14)

3

Ipocell= 2Sc

∫ 1

0

∫ 1−x

0

ejk(Bxx+Byy+B0)dxdy (15)

where Sc is the area of the triangular cell. The coefficients are real numbers calculated tobest fit a plan from the values at the vertices of the triangle

Bx = 2(Bc,1 − Bc,3), By = 2(Bc,2 − Bc,3), B0 = 2Bc,3 (16)

where Bc,i are the phase values of the i-th vertex of the triangle. Equation (15) can thenbe evaluated analytically [1] yielding

Ipocell= 2Sce

jkB0

−j

kByej kBx

2 [ejkBy

2 sinc(kBx−kBy

2) − sinc(kBx

2)], if |Bx| > |By|

−j

kBxej

kBy2 [ej kBx

2 sinc(kBy−kBx

2) − sinc(kBy

2)], if |Bx| ≤ |By|

(17)

Special care must be applied to avoid large numerical errors when the Bα are both verysmall.

The size of each cell corresponding to the number of integration points is determinedon the NURBS by the subdivision scheme. The accuracy of the scheme is monitored asthe integration proceeds by an error estimate. For curved geometries, if X ′ do changemuch over the cell, due for instance to a change in curvature, then the cell will be furthersubdivided.

3 Adaptive Subdivision Scheme

Figure 2: Subdivision procedure necessary to resolve curved and non planar geometries.

The subdivision procedure, see Algorithm 1 below, is done recursively over each triangleand is stopped when a subdivision criteria reaches a desired tolerance. The subdivisioncriteria has been chosen carefully such that it insures that over stationary phase regions,where the phase is varying slowly [7], the density of triangle is sparse.

From an initial coarse triangle T with nodes on the surface, new nodes are inserted at themiddle of each edge of T . This creates four sub-triangles T1, T2, T3 and T4 as illustratedon the right picture of Figure 2. Special care is taken if the triangle lies on non planar

4

Algorithm 1 Integration (Triangle T, E , Tolerance)

Require: E ⇐ Solve-PO-Integral (Triangle T)use Linear Phase Approximationif Tolerance< ǫmachine then

Return E to avoid arithmetic exceptionselse

T1, T2, T3, T4 ⇐ Build-Sub-Triangles2 (Triangle T)for i = 1 to 4 do

Ei ⇐ Solve-PO-Integral (Triangle i) use Linear Phase Approximationend for

Enew = E1 + E2 + E3 + E4

if (||Enew − E ||2 <Tolerance) then

Return Enew Subdivision criteria is reachedelse

for i = 1 to 4 do

Ei ⇐ Integration (Triangle i, Ei, Tolerance/4) Recursive subdivisionend for

Enew = E1 + E2 + E3 + E4

Return Enew

end if

end if

Algorithm 2 Build-Sub-Triangles (Triangle T)

Require: Triangle T has nodes on NURBS, Topological information: T → NURBSnurbs S ⇐ Find-NURBS-Parent (Triangle T) use TopologyMiddlesOnT ⇐ Middles-Edges (Triangle T)MiddlesOnS ⇐ Project-On-NURBS (MiddlesOnT, nurbs S)use CGM if Edge-Is-On-Trim-Curve (Triangle T, nurbs S) then

MiddlesOnS ⇐ Local-Refinement (MiddlesOnT, nurbs S)end if

T1, T2, T3, T4 ⇐ Create-Sub-Triangles (Triangle T, MiddlesOnS)Return T1, T2, T3, T4

surfaces. The procedure resolves these geometries using local refinement and projectionof the new nodes on the NURBS surfaces, see middle and right pictures of Figure 2.

The projection is done by minimizing the distance between the node and the surface usingConjugate Gradient Method (CGM), see [6] for details. The subdivision does not changethe position of the initial nodes so only the three nodes at the middle of each edge needto be projected.

The local refinement corresponds to a displacement of the sub-nodes if the edges of Tlie on the trimming curve of the surface. In such case, the procedure projects the nodeson the curve itself. The difficulty here is to detect when a specific edge of a triangleis discretized one of the curves of the NURBS representation. To simplify the task,the initial triangulation has been connected with topological information [6] from theNURBS description so that each triangle knows on which surface or curve it belongs.Then a straight forward localization algorithm is applied in order to find the refinementpoint being the closest point on the curve.

When all four triangles have been defined, the next step is compute to PO integral on each

5

new triangles. As before we using linear phase approximation. Then the contribution ofthe new triangles to the scattered field is determined. The sum of their scattered fieldsEscat

Tnewis given by

EscatTnew

= EscatT1

+ EscatT2

+ EscatT3

+ EscatT4

(18)

The subdivision criteria is then the absolute error ǫabs in the square norm between thescattering field from the initial triangle Escat

T and the sum of scattering fields from thenew triangles Escat

Tnew:

ǫabs = ||EscatT − Escat

Tnew||2 (19)

If the error is not small enough, the subdivision proceeds recursively and the same func-tion call is applied to each sub-triangle with 1

4of the previous tolerance. The absolute

error insures that the subdivision will stop as soon as the scattering field does not vary(stationary) between two subdivision levels or if the contribution to the scattered fieldbecomes small.

Figure 3: One, four and five levels of subdivision applied to a circular plate.

4 Scattering from a circular plate

Figure 3 illustrates how the procedure works on a circular plate initially discretized ineight triangles. At the first level of subdivision all the eight initial triangles are cut infour such that 32 new triangles are created, see left plot of Figure 3.

The effect of local refinements can been seen immediately at this first level of subdivision.The sub-triangles extend outside the initial triangle since they have been projected onthe border of the circular plate.

Since the surface is plane, the integral is evaluated exactly on every triangle. The scatteredfield will only be affected by a change in the geometry of the triangles T2 and T4 whichhave edges on the borders of the circular plate. In opposition, for sub-triangles T1 andT3 the scattered field remains constant and they do not need to be further subdividedwhich explain the rose shape configuration after 4 levels of subdivision, see middle plotof Figure 3 and after 5 levels on the right plot of Figure 3.

This result was validated against analytic results of the PO integral for the same circularplate when the scattering direction Kscat is varying according to an angle θ ∈ [0, 90]

6

0 10 20 30 40 50 60 70 80 90−80

−60

−40

−20

0

20

40RCS of 1m radius circular plate at 300Mhz : λ = 1m

θ [degree] (φ = 0o)

σ [d

b]

PO−ADAPTIVEPO−ANALYTIC

0 10 20 30 40 50 60 70 80−4

−3

−2

−1

0

1

2

3

4erro in RCS of 1m radius circular plate at 300Mhz : λ = 1m

Figure 4: Adaptive PO on NURBS versus analytic results for a 1 meter radius circularplate at λ = 1 meter and corresponding error plot per angle of incidence.

0 10 20 30 40 50 60 70 80 90−80

−60

−40

−20

0

20

40RCS of 1m radius circular plate at 1Ghz : λ = 0.3m


σ [d

b]

PO−ANALYTICPO−ADAPTIVE ε = 1e−8

0 10 20 30 40 50 60 70 80−10

−5

0

5

10erro in RCS of 1m radius circular plate at 1.0Ghz : λ = 0.3m


Abs

olut

e E

rror

in σ

[db]

PO−ADAPTIVE ε = 1e−2PO−ADAPTIVE ε = 1e−4PO−ADAPTIVE ε = 1e−6PO−ADAPTIVE ε = 1e−8

0 10 20 30 40 50 60 70 80 900

500

1000

1500

2000

2500

3000

3500


# tr

iang

les

# triangles Created per PLW

PO−ADAPTIVE ε = 1e−8PO−ADAPTIVE ε = 1e−6PO−ADAPTIVE ε = 1e−4PO−ADAPTIVE ε = 1e−2

Figure 5: Adaptive PO versus analytic results for a 1m radius circular plate at λ = 0.3m,plus corresponding error plot per tolerence as well as total number of sub-triangles createdper angle and per tolerance.

degree. For each angle, the monostatic RCS for a circular plate of one meter radius wascomputed both analytically and using the adaptive subdivision scheme.

For the first test, the wavelength was 1 meter and 0.3 meter for the second test. Theresults are displayed respectively in Figure 4 and Figure 5. The computed curve whenusing adaptive subdivision follows well the behavior of the analytic one, except at somedips of low decibel, singularities which anyway give no contribution to the RCS. On theright plots we can see the error with the analytic RCS σ0.

With a smaller wavelength, the error remains small, see the top right plot of Figure 5, atleast for all the angles θ ∈ [0, 50] degree and decays when the tolerance is reduced up to1e−8. At such levels of accuracy, a maximum of 3.200 triangles is needed but this numberrapidly decays with the angle, see bottom right plot of Figure 5. In comparison, classicPO code will need at least twice as many triangles to resolve the same wavelength with10 triangles per wavelength and this regardless of the angle of incidence.

7

5 Conclusion

We have presented an adaptive triangular subdivision scheme for solving the PO integralwhich compute the integral on few points per surface. The innovation is that the numberof integration points and the accuracy of the scheme are monitored as the integrationproceeds by a measure of the local variations of the scattered fields.

To illustrate the software robustness, we have applied the scheme to the Eikon UAV, seeFigure 6. This shows the code works on complex geometries but results still remain to befurther investigated.

We have also shown how it is advantageous to preserve the native NURBS representation,given by the CAD design, along with the triangulation. In Particular, since each triangleknows on which surface it belongs, this is what allows the scheme to resolve the geometry.

Figure 6: Eikon triangulation initial and after adaptive subdivision.

References

[1] A. C. Ludwig. Computation of radiation patterns involving numerical double integra-tion. IEEE Transactions on Antennas and Propagation, pages 767–769, Nov. 1968.

[2] F.J.S Moreira and A. Prata. A self-checking predictor-corrector algorithm for efficientevalaution of reflector antenna radiation integrals. IEEE Transactions on Antennas

and Propagation, 42(2):246–254, Feb.. 1994.

[3] J. M. Taboada, F. Obelleiro, and J. L. Rodrıgeuz. Improvement of the hybrid mo-ment method-physical optics method through a novel evaluation of the physical opticsoperator. Microwave and Optical Technology Letters., 30(5):357–363, 2001.

[4] Glenn D. Crabtree. A numerical quadrature technique for physical optics scatteringanalysis. IEEE Transactions on Microwave Theory and Techniques, 27(5), Sep. 1991.

[5] Nazih N. Youssef. Radar cross section of complex targets. IEEE Transactions on

Antennas and Propagation, 77(5), May 1989.

[6] S. Sefi. Ray Tracing Tools for High Frequency Electromagnetics Simulations. Licentiatethesis No. 0314, Dept. of Numerical Analysis and Computer Science, KTH, June 2003.

[7] M. F. Catedra J. Perez. Application of physical optics to the rcs computation of bod-ies modeled with nurbs surfaces. IEEE Transactions on Antennas and Propagation,42(10), October 1994.

8

The Rescue Wing:

Design of a Marine Distress Signaling Device.

Tomas Melin

Department of Aeronautical and

Vehicle Engineering, AVE


Stockholm, Sweden


Sandy Sefi




Stockholm, Sweden


Abstract— We present a multidisciplinary scientific analysiscombining aerodynamics, flight mechanics and electromagneticsaiming at the design of a new marine distress signaling device.

We show how computational fluid dynamics (CFD) and com-putational electromagnetics (CEM) techniques have been usedto assist in the design of both the flight characteristics andthe radar performance of the device, as well as how its radarsignature compares to popular radar reflectors used on yachtsand sailboats.

I. INTRODUCTION

Marine distress signaling is of paramount importance for

persons in distress at sea, being the mean which lets rescue

teams be aware of the situation and deploy a search and rescue

mission.

Fig. 1. The Rescue Wing during test flight.

In this paper we focus on a personal balloon-type device

carried by the survivor, shown in Fig. 1 and called ”The Res-

cue Wing”. The Rescue Wing works as a passive radar reflector

and visual marker assisting in localization during search and

rescue operations of persons missing at sea.

Its design is the fruit of a multidisciplinary study combining

simulation results from aerodynamics, flight mechanics and

electromagnetics, as well as data from trial flights. For aero-

dynamics, one of the challenges was to create an inflatable

light-weight structure with adequate aerodynamic character-

istics. The design was implemented and modeled using the

commercial CFD software FLUENT [1].

In order to assess the Rescue Wing’s ability to reflect radar

signals, electromagnetics simulations have been conducted to

predict its radar cross section (RCS). The computations used

the General Electromagnetic Solvers GEMS [2], a software

suite developed at KTH for computational electromagnetics.

II. CHARACTERISTICS OF MARINE DISTRESS SIGNALING

On the 14:th of June 2003 a man fell overboard from a

small sailing yacht outside the Halland coastline in the south

of Sweden. The waves were about 1.5 to 2 meters high with a

wind speed of 10 m/s. In total, five surface vessels and three

helicopters were engaged in the search and rescue mission. The

search effort was hampered by the difficulty of determining

the position of the originating event. After three hours, one of

the helicopters spotted a wave crest apparently moving in the

wrong direction: the man wearing a white sweater over his

life jacket. He was suffering from beginning hypothermia but

was recovered unharmed. This example is typical for search

and rescue missions in that the time spent on search is often

many times larger than the time spent for the actual rescue. An

effective signaling/localization device would lessen the search

time and allow for more lives to be saved.

III. PURPOSE

Since the Titanic disaster, research in distress signaling

has been addressed and constantly updated by the SOLAS

(International Convention for the Safety of Life at Sea) [5].

These international treaties define a number of aids including

rocket flares, hand flares, buoyant smoke signals as well as

a number of active radio devices. The Rescue Wing can not,

strictly speaking, be classified as a signaling device as it only

provides a radar and an optical target, but it is intent to help

localizing a person in water. The Rescue Wing is an inflatable

gas bag, filled with helium to provide aerostatic lift. It is

shaped like a wing thus providing aerodynamic lift. As an

add-on device to standard life jackets, it will be at hand when

needed. In the case of a person falling overboard, or leaving

the ship due to an on-board emergency, the Rescue Wing

provides a mean for person in distress to communicate his

position. By a simple grip-and-twist mechanism, the inflation

of the envelope is engaged and Rescue wing is deployed, after

which no manual handling of the device is required.

The Rescue Wing has been designed to operate as a displac-

ing balloon in calm winds, and as a kite in windy conditions.

It hovers 10-15 meters above the sea surface providing a radar

reflector target as well as a strong visual cue for detection and

positioning.

In the next section, we will look at the constraints restricting

the design, as well as the functional specifications stipulating

the properties we expect from the device.

IV. CONSTRAINTS IMPOSED ON THE DESIGN

γ

Fig. 2. CAD model and triangulation of the Rescue Wing. γ is the anglebetween the main axis of the wing and the monostatic azimuth directions.

The design requires finding a good balance between dif-

ferent aspects and features of the device. These can range

from size, weight, portability, up to external constraints such

as environmental considerations, packaging, available mate-

rial, production methods, etc. A condensed wish-list of the

specifications is provided below:

• Inexpensive and easy to manufacture.

• In stores, presented as an add-on safety device.

• When packaged, fit in a cigaret box-size canister.

• In storage, have long shelf life and light weight.

• When not operating, waterproof, corrosion resistant.

• In stand-by operation, be attached to a life west or raft.

• In distress situation, easy to arm and to engage.

• When inflated, hover at 10-15 meters above sea.

• In calm waters, generate aerostatic lift as a balloon.

• In strong winds, remain in stable flight as a kite.

• At long range, reflect radar waves of X- or S-bands.

• At close range, provide a strong visual cue.

V. AERODYNAMICS

Due to the necessarily bluff body shape of the inflatable

device, standard aerodynamic conceptual design tools such

as handbooks [3] and panel methods [4] are not applicable.

Instead, a full Navier-Stokes solver had to be employed in

order to generate the aerodynamic database. Fortunately, the

simplicity of the design geometry (Fig. 2) led to easy grid

generation, thus enabling fast design loops.

As in airplane design, key parameters are lift and drag forces

L and D, and their ratio G, the glide slope. Aerodynamic

results were collected in the standard way into coefficients of

lift: CL, drag: CD and pitching moment CM as functions of

the angle of attack α. The coefficients are specified in equation

1, where q is the dynamic pressure and S the reference area.

CL =L(α)

qS, CD =

D(α)

qS, CM =

M(α)

qCMAC

(1)

where CMAC is the mean aerodynamic chord of the device,

see Fig. 3. The computed data was then curve fitted with

second degree polynomials to yield an expression suitable for

the flight mechanic analysis, equations 2 and 3. The numerical

error in this interpolation was small, as the aerodynamic

behavior of the device was smoother than that of ordinary

aircraft designs.

CL = CL0 + CLαα + CLα2α2 (2)

CD = CD0 + CDαα + CDα2α2 (3)

Together, these equations made it possible to define and

compute the glide slope G(α),

G =CL(α)

CD(α)(4)

As shown in Fig. 3, the glide slope determines the elevation

angle θ in steady flight when effects of gravity and buoyancy

have been neglected. AC is the aerodynamic center, b the

bridle point, where the leading and trailing edge tethers join

into the main tether of length l going down to the anchor point

A. V∞ is the free stream velocity, M is the moment around

the bridle point and s is the circle segment spanned by the

bridle point at different elevation angles θ.

The tangential plane to s, t is the reference plane when

measuring the angle of incidence, i of the CMAC . The

tangential force F acts in parallel with the plane t and exerts

a momentum in the bridle point when the moment arm a is

greater than zero.

θ = arctan(L/D) (5)

VI. FLIGHT MECHANICS

The flight mechanics of kites is somewhat different from

aircraft. In this paper only a brief overview of the method of

finding an appropriate longitudinal trim will be presented. For

the sake of simplicity, the weight of the line and the envelope

is neglected, as are the buoyant effects. This simplification is

2

Fig. 3. Variable definitions.

valid for a free stream velocity range giving a high dynamic

pressure resulting in the aerodynamic forces being much larger

than the mass effects, i.e. L ≫ mg.

The tangential force F must be zero at the equilibrium

elevation angle, as described in equation 6. Additionally, as

all aerodynamic forces are transmitted trough the bridle point

into the tether, the moment M at the bridle point must be zero

in trimmed flight according to equation 7.

F = L cos(θ) − D sin(θ) = 0 (6)

M = Fa = (L cos(θ) − D sin(θ))a = 0 (7)

The angle of attack α is a function of the incidence i and

the elevation angle θ according to equation 8

α =π

2+ i − θ (8)

Equations 6, 7 and 8 form a system of equations with

three unknowns: The incidence angle i, and the position of

the bridle point b in relation to the aerodynamic center AC.

This system is readily solved numerically, while keeping the

elevation angle θ or, the glide slope G as high as possible.

When plotting the tangential force against the elevation

angle as in Fig. 4 the dependence of the tangential force of

the incidence become clear. Of the three cases, the one with

i = +0.2 radians clearly never crosses the line F = 0 which

means that this configuration is not stable in elevation angle,

as the tangential force always is negative, thus forcing the

elevation angle to zero (giving zero altitude).

Decreasing the incidence to zero, gives a stable configu-

ration with a crossover F = 0 at θ ≈ 1. However, when

examining the glide slope at the crossover θ in the lower

graph, the glide slope there is still increasing with increasing

θ. Having a negative ∂G

∂θwould insure a stiffer system, and be

on the right side of Gmax with respect to the stall limit.

The third case, with an incidence of i = −0.2 radians has

a zero tangential force crossover elevation angle of slightly

above one and has the desired glide slope behavior.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1

−0.5

0

0.5

1

1.5

θ [rad]

F [N

]

Elevation trim graph

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1.5

−1

−0.5

0

0.5

1

1.5

2

θ [rad]

G [−

]

i = + 0.2 radi = 0 rad i = −0.2 rad

Fig. 4. Trimming the elevation angle using the incidence as parameter.

When a suitable angle of incidence had been attained, the

bridle point could be positioned using Fig. 5. The upper

panel shows the glide slope of the selected design with the

G = 1 design criterion and the maximum glide slope criterion

(Gmax). These boundaries limit the useful part of the glide

slope range. The G = 1 is set to ensure that the elevation of

the kite always is larger than 45 degrees. In Fig. 5 the vertical

distance between the aerodynamic center and the bridle point

was set to CMAC/2 while varying the lateral distance h. The

lower graph shows the moment coefficient around the bridle

point for three different lateral positions. For h = 0, the zero

moment crossover is at about 0.6 radians, which is over the

Gmax limit. For the h = 0.2 · CMAC , the crossover is at

about 0.22 radians which is within the cruise quadrant but off

the trim angle decided by equation 8 and the elevation trim.

Instead, setting the lateral distance h = 0.1 ·CMAC yields the

same trim glide slope as the elevation trim and thus positioning

the bridle point on the line connecting the aerodynamic center

and the anchor point.

A. Altitude Stability

We investigated the flow field over an assumed sea surface

in order to determine the behavior of the change in angle of

attack due to the shape of the surface. In this simulation we

assumed the waves being described by a series of cubic Bezier

curves. The wave height is 1.3 meters and wave length is

14 meters as shown in Fig. 6. Free stream velocity was set

to 10 meters per second in standard sea level atmosphere.

The simulation showed the formation of recirculation bubbles

in the wave troughs and an influence on the shape of the

streamlines above the wave crests.

3

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−2

−1

0

1

2

Alpha, Angle of Attack [rad]

G, G

lides

lope

, [−

]Pitch trim graph

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Alpha, Angle of Attack [rad]

Cm

, Brid

le M

omen

tum

Coe

ffici

ent,

[−]

h=0h=0.1h=0.2

G − Design minimum limit

G Max limit

Cruise quadrant

Fig. 5. Positioning the bridle point in relation to the aerodynamic center.

Fig. 6. Streamlines just above the sea surface. The wave height is 1.3 metersand the wave length is 14 meters.

In order to determine this influence quantitatively, we iso-

lated the vertical velocity component and computed the change

in angle of attack according to equation 9, where v is the

vertical speed component and V∞ is the free stream velocity.

In Fig. 7 we show the dependence of the change in angle of

attack with height over the sea surface, measured from the

crests, as a function of the wave phase.

∆α = arctan(v

V∞

) (9)

In correlation with our initial assumption, the influence of

the wavy surface diminishes with increasing altitude. At 10

meters, or about 7 wave heights the influence is negligible.

The offset from the zero line of the 10 meter curve is due to

boundary layer growth.

B. Operational considerations.

Radar wave clutter becomes an important consideration in

the design, and operational considerations of the Rescue Wing.

When investigating the radar range, the line-of-sight range R is

usually taken as equation 10, ignoring refraction effects. Here

0 2 4 6 8 10 12−4

−2

0

2

4

6

8Change in angle of attack with wave phase and height above the sea.

φ wave phase [rad]

∆ A

oA [d

eg]

h = 0.5 mh = 1 mh = 5 mh =10m

Fig. 7. Change in angle of attack (AoA) as a function of wave phase andthe height (h) above the wave crests parameter.

a is earth radius and h and H are the heights of transmitting

antenna and the target respectively [6].

R =√

2ah +√

2aH (10)

As the Rescue Wing has an radar cross section of about the

same size as the rough seas, the signal to clutter ratio (S/C)

becomes an important parameter in determining the possibility

of detection of a specific target. As shown in Fig. 8 the region

with low grazing angles, here called the clutter region, is where

the signal to clutter ration is the lowest. Beyond the line-of-

sight range, smaller waves disappear below the horizon, and

gradually the (S/C) increases. Assuming infinite power in the

transmitter, linear wave propagation and a sufficiently long

tether, the detectability increases for a larger range. However,

the Radar range also depends on the fourth power of the range

when discussing the received reflected energy. This means that

we would like to keep the far region as close to the transmitting

antenna as possible. One way of attaining this is to keep the

height of the transmitting antenna as low as possible, thus

favoring search from a small surface vessel rather than from

a helicopter.

C. Inflatability

In order for the wing to be buoyant in still air, the material

employed must be extremely light-weight and also work as an

efficient barrier for the gas inside. The choice fell on a compos-

ite multi-layer plastic: metallized polyethylene tereftelathe, or

PET. The buoyancy criterion also imposes harder constraint on

the weight of the device, as the buoyant force is much smaller

than the aerodynamic forces encountered.

In essence, the buoyant lifting force is proportional to the

volume of the envelope, while the weight is proportional to

the surface area (for a given material thickness).

4

Fig. 8. Sea clutter.

VII. THE RADAR CROSS SECTION

The strength of the radar reflection from the Rescue Wing

is measured by its monostatic radar cross section (RCS)

represented by the symbol σ. The RCS characterizes the

effective area illuminated by the incident plane wave which

is directly scattered back in the direction of the emitter.

Fig. 9. Typical RCS values in m2 and in dBsm for a range of differentobjects in size, including the Rescue Wing. The radar detectability thresholdis about +3dB.

RCS is measured in square meters, often normalized and

displayed using a logarithmic scale defined in decibel square-

meters (dBsm), see Fig. 9,

RCS[dBsm] = 10 log10

RCS[m2]

1[m2](11)

The reference 0dB corresponds to the return from a sphere

of 1m2 cross section.

It is often convenient to display RCS values using a polar

diagram. Fig. 11 shows a polar plot of a commercial radar

reflector (Fig. 10) and the response of the wing to signals

arriving from different azimuth direction θinc using a polar

plot are given in Fig. 13 and Fig. 14.

In common with aerodynamics design, a knowledge of the

factors which affect the radar performance of the system

is essential. The most significant factors are those which

influence the propagation of the radar wave such as directional

characteristics, radar operating frequency, polarization of the

incident and reflected waves, as well as size, shape and

material covering the surface of the object.

In following, we will have a closer look at these factors

in order to understand how they will affect the reflective

performance of the Rescue Wing. Also, we will see that the

influence on performance of constraints imposed on the design

is very significant.

A. Electrical Size and Shape

Due to a moderate size of 2 meters in diameter, a low wing

profile and an inflatable round shape, we expect the Rescue

Wing to have a low radar signature, in the range of the sphere,

as illustrated in Fig. 9.

This has been verified by the CEM simulation where the

Rescue Wing was modeled as a perfect electric conductor

2.0 m× 1.5 m× 0.5 m in dimensions, see Fig. 2. The results

given in Fig. 13 and Fig. 14 show that the RCS of the wing

displays mainly low values under 0dB.

B. Frequency Range

Rescue radar systems generally operate at frequency ranges

in the following bands:

• the S-band, ranging from 2 to 4 G-Hz with specific bands

at a frequency of 3.0 G-Hz (3000 MHz) which has a

wavelength λ of 10 cm,

• the X-band, ranging from 8 to 12 G-Hz with a typical

frequency of 9.4 G-Hz (9400 MHz), is characterized by

a smaller wavelength λ of 3.2 cm.

Ships and rescue helicopters will typically carry both the X-

and S-band while small vessels are limited to X-band units.

S-band gives a longer range, up to 24 miles. X-band radar

is more sensitive to interference from rain and waves (sea

clutter) but offers greater resolution and detection of smaller

targets such as the Rescue Wing. To be detected above the

sea clutter, in a moderate sea using X-band radar, a return of

+1dB to +5dB (≈ 3m2) must be reached as a threshold [7].

C. Radar Reflectors

Radar reflectors [10] are designed to increase the reflectivity

of buoys or small craft so that they become more visible on

radar. The key strategy in their design is to use flat plates which

produce strong backscatter at normal incidence. For instance,

the commercial Davis emergency reflector in Fig. 10, one of

the most popular radar reflectors, is composed by two vertical

flat disks butted together at 90o onto one horizontal disk to

form rectangular corners which maximize the reflections in

almost all directions.

Fig. 10. Typical geometry of a passive radar reflector (Corner reflector).

The RCS at X- and S-band are displayed in Fig. 11. Similar

results in agreement with our simulation have been previously

5

published in a study conducted by the US Sailing Safety-At-

Sea Committee [11] aiming to test the efficiency of various

radar reflectors. The small reflector displays a small radar

signature in accord with the tests reported in [11]. Inflatable

[8] [9], light and collapsible versions of such reflectors are

commercially available.

−40

−30

−20

−10

0

1030

210

60

240

90270

120

300

150

330

180

0 γ

Fig. 11. RCS of the Davis Emergency (5.7in) radar reflector.

If one or more units are inflated and attached to the tether,

their combined signatures would result in large RCS values

well above the +1dB detectability threshold.

D. Polarization

Typical radar units can send and receive in both vertical

and horizontal polarizations. In kite flight position, at angle

of attack α < 20o in horizontal elevation, or in balloon flight

position at angle of attack α < 45o as seen in Fig. 1, the low

wing profile will make horizontal polarized reflected waves

dominant. Thus, only horizontal polarization is of interest for

the simulation.

E. RCS Calculations Tools

In order to verify our assumptions on the Rescue Wing radar

detection, RCS simulations are required.

The GEMS [2] software suite covers a wide range of

numerical methods for predicting RCS of complex three-

dimensional geometries using numerical evaluations of the

Maxwell equations. For low frequencies and small objects

in term of wavelength, exact techniques such as Method of

Moments (MoM) combined with Fast Multipole Method [12]

(FMM) are available. These methods numerically solve the

exact integral equations for the electromagnetic fields using a

set of discrete basis functions on the boundary.

For high frequencies, i.e. electrically large targets, exact

methods become impractical either because of computer time

or memory requirement, so that approximate techniques must

be used.

It is well established that for monostatic RCS, the Physical

Optics (PO) based on surface current approximations [14] is

a good choice for computing the main reflection from large,

in term of wavelength, smooth surfaces.

Both PO and MoM take as input a discrete mesh of the CAD

geometry, where the elements must be small compared to the

wavelength. An example of such discretization for λ = 30 cm

is given in Fig. 2.

F. Physical Optics and Method of Moments

Since both low and high frequency methods operate on the

same geometry, we can compare them at mid-frequency (run

at 1 G-Hz) in order to investigate the error of the approximate

PO. The results are displayed in Fig. 12.

We see that the PO solution follows the MoM solution well

except at directions normal to the axis of the wing, γ ∈ [0o −20o] and γ ∈ [160o−180o], where PO predicts too high RCS.

This is mainly due to one wing shadowing the other, which is

badly modeled by the PO solution [13]. However, the results

show that away from normal axis incidence, i.e for about 2/3

of all aspect angles γ ∈ [20o − 160o], the PO solution is in

good agreement with MoM.

20 40 60 80 100 120 140 160 180−30

−25

−20

−15

−10

−5

0

5

10

15

20

RC

S [d

Bsm

]

Monostatic angle δ

Kite−40degree, Method of Moments vs Physical Optics 1.0Ghz, wavenumber:20.95

MoMPO

Fig. 12. Exact numerical solution versus fast approximate solution of theRCS in dBsm of the Rescue Wing in flight kite position, horizontal elevationof 40

o at a frequency of 1G-Hz, in function of δ. δ = 0o corresponds to a

radar waves coming from normal to the axis of the wing (γ = 90o).

G. RCS of the Rescue Wing at S-band and X-band

We look at the contribution from all 360o angles (γ) in an

horizontal plane where the wing is resting with the nose at 0o.

The RCS in all directions is quite low, as seen in Fig. 13 at

S-band and in Fig. 14 at X-band in horizontal polarization.The

long straight leading edges of the wing reflect the most. The

CAD geometry introduces a fictitious sharp edge at the nose.

The sharp edge breaks the reflection and that is why the RCS

for γ around 0o is small. The real shape is smoother so we

should expect the real wing to return slightly higher values at

this angle.

6

−40−30−20−10 0 10 20 30 30

210

60

240

90270

120

300

150

330

180

0 γ

Fig. 13. RCS of the Rescue Wing in dBsm for S-band.

−40−30−20−10 0 10 20 30 30

210

60

240

90270

120

300

150

330

180

0 γ

Fig. 14. RCS of the Rescue Wing in dBsm for X-band.

VIII. CONCLUSION

The design project has reached mid-term of its development.

The next stage in preparation is to launch further trial flights.

In order to reach production stage, auxiliary devices would

need to be added, such as helium valves and packaging

assembly.

Aerodynamics simulation proves that the flight dynamics

is stable but sensitive to large turbulence, to deformations in

the position of bridle point and to changes in the magnitude

of the incidence angle. Due to the inflatable nature of the

device, quite large deformations are possible. The rigidity of

the structure is dependent on the internal helium overpressure.

The higher overpressure, the higher rigidity.

However, limiting factors in overpressure are envelope

membrane strength and helium leak rate. The qualitative

relations between these factors are still to be found.

We have seen that the key factor influencing the radar

performance of a Rescue Wing is its size. However, constraints

limit size due to light weight requirement and that the deflated

wing has to fit into a cigaret box-size container. The RCS

can be effectively increased though the addition of a radar

reflector. The aluminum material covering the wing acts a

perfect electric conductor, so that radar reflectors inside the

wing will be ineffective.

To overcome this problem, a radar reflector could be at-

tached to the wire. It should be located at least 2 meters above

the see surface in order to filter out noise from the waves.

However this will result in more drag forces and could

lower the elevation angle of the wing, i.e. reducing its maximal

altitude.

ACKNOWLEDGMENT

The authors would like to thank Hans Sjoblom, who initi-

ated the work and is the holder of the Rescue wing related

patents. For the RCS calculation tools, financial support has

been provided within the GEMS project by KTH and the

Swedish Agency for Innovation Systems (VINNOVA) as a

part of a collaborative research center PSCI. For developing

the Rescue Wing, research grants has been financed by the

Carnegie foundation, Edvard Roses foundation and KTH, dept.

of Aeronautical and Vehicle Engineering.

REFERENCES

[1] Fluent Inc, http://www.fluent.com/, Fluent Inc, June, 2005.[2] GEMS, http://www.psci.kth.se/Programs/GEMS/,

NADA, KTH, June 2005.[3] Roskam, et.al., Airplane design,

Roskam Aviation and Engineering Corporation, Kansas, 1985.[4] Tomas Melin, http://www.ave.kth.se/divisions/aero/software/tornado/index.html,

KTH, June 2005.[5] International Maritime Organisation, International convention for the

safety of life at sea, SOLAS, IMO, London, 1986.[6] Skolnik, Introduction to radar systems, McGraw-Hill, 1962.[7] Kenneth Parker, Be Seen Or Be Sorry, Cruising Association, 2000.[8] Sidney Veazey, Inflatable radar reflectors,

United States Patent 5969660, October 1999.[9] James Schaff, Steven Ball, Emergency passive radar locating device,

United States Patent 6300893, October 2001.[10] John Briggs, Target Detection by Marine Radar,

IEE Radar, Sonar and Navigation series 16, 2004.[11] United States Sailing Safety at Sea Committee, Radar Reflector Test:

http://www.ussailing.org, Safety Studies, 1995.[12] Martin Nilsson, A fast multipole accelerated block quasi minimum

residual method for solving scattering from perfectly conducting bodies,Antennas and Propag. Society International Symposium No 4, 2000.

[13] Sandy Sefi, Ray Tracing Tools for High Frequency Electromagnetics

Simulations, Licentiate thesis No. 0314, Dept. of Numerical Analysis andComputer Science, KTH, June 2003.

[14] Sandy Sefi, Jesper Oppelstrup, Physical Optics and NURBS for RCS

calculations, EMB04 Computational Electromagnetics Conference Pro-ceedings, Gothenburg, Sweden, October 2004.

7

A Modular Approach to GTD in the Context of Solving

Large Hybrid Problems

Fredrik Bergholm1, Stefan Hagdahl1,2, Sandy Sefi1

1Department of Numerical Analysis and Computing ScienceRoyal Institute of Technology100 44 Stockholm, Sweden

E-mail: sandy/[email protected]

2Ericsson Saab Avionics AB, Electromagnetic Technology581 88 Linkoping, Sweden

E-mail: [email protected]

Abstract

In this paper we will present a ray tracer applied to Geometrical Theory of Diffraction (GTD). The solver is apart of the Swedish suite of CEM solvers called General ElectroMagnetic Solvers (GEMS)1. GTD is today a wellestablished semi analytical method for high frequency radiation problems. During the seventies efficient GTDtools were developed and used in the academy and industry for canonical geometries. Today an abundance ofGTD coefficients have been solved for simple geometries. The process in examine new geometries is still goingon. During the nineties GTD implementations that could treat industrial geometry, i.e. Non Uniform RationalBezier Spline (NURBS), were developed. To exploit all these new facilities and to hybridize them with otherCEM-methods there is a need for modern programming and efficient geometrical engines that treat industrialCAD in a robust way. We have developed a solver called MIRA (Modular Implementation of a GTD Ray tracerfor Antenna applications) that springs from a solver called FASANT [2] developed by The University of Alcala,Spain. The solver is written in the modern F90-language and a host of new opportunities that this languagegives are implemented, e.g modules, objects, derived types, overloading and linked chains.

Introduction

The solver is split up into three independent packages called Geometry, Ray and Application. The Geometrypackage can be executed as a stand alone solver that computes various kinds of geometrical entities. Any othersoftware that needs NURBS related data can make use of this module. In addition, the Geometry moduletogether with the Ray module build a stand alone solver that can serve any application that needs to access anypropagation based on rays such as light, sound wave, water wave. Today the application package do antennarelated computations but new application, e.g. Shooting and Bouncing Rays (SBR) via a new Application, mayeasily be added in the future. We have started to hybridize MIRA with a modern Method of Moment (MoM) [1]solver and it shows that the solver’s data structure well support the needs from the hybrid application.Since the solver is written in F90 language it is well prepared to be compiled on large parallel computers andthis will also be done in the near future. Below the three packages are discussed.

1This work was supported in part by the CEM program at the Parallel Scientific Computing Institute at the Royal Institute of

Technology.

Geometry

We here stress 5 features about the geometry. From a computational standpoint the object hierarchy, adaptivesampling and what we call a hole algorithm are important constituents. Furthermore, MIRA features multiplesurface types and an explicit Bezier buffer.Four basic structures in the object hierarchy are ‘scenes’, ‘objects’, ‘surfaces’ and ‘curves’. (‘Curve’ = curveon a surface patch, here.) Having structures representing groups of surfaces, means that whenever surfaces areused in the model, we either have a double loop over surfaces by objects, or, have a separate object loop beforedouble looping over surfaces by objects. This reduces computational time, e.g.: (1) In ray tracing, boundingboxes and spheres are important for reducing computations for intersection tests, and having such tests byobject bounding bodies before other tests, yields a speed-up in scenes with many surfaces. (2) For diffractionfield calculations in UTD, curvatures from the surfaces on either side of an edge are needed, and search forneighbouring surfaces may be limited to the same object (in most cases).Adaptive sampling refers to the need of (i) sampling entities such as normals and points on surfaces as pre-processing to avoid (many) function calls, (ii) make the sampling so that it is efficient, in the sense that, thesamples are sufficiently dense, e.g., inside a trim curve, and that surface curvature affects sampling to avoidproblems in finding reflection or diffraction points.Sometimes the physical material prevails outside a curve. Then the curve, whether delineating a hole or beinga rim curve for an indentation/cavity, is an important source for diffracted rays. Many types of antennaspossess narrow cavities or indentations. Mira, in contrast to FASANT, has a built-in hole algorithm, allowingan arbitrary number of holes, Fig. 2 App. B. The algorithm determines whether points on material surfacesare inside or outside a curve, in so general a fashion, that complex structures such as a ring-like hole withmaterial inside and yet another hole in that island of material, are permissible data. Each curve has a labeli or o, short for remove-inside (=i) and remove-outside. Let p ∈ R

2 be a map space (sample) point. Weperform the tests p ∩ E1 ∩ E2... ∩ Ek = p, p ∩ I1 ∩ I2... ∩ Iℓ = p, where E.. is a curve with label o and I.. withlabel i, and curves causing p ∩ Es = ∅, p ∩ Is = ∅ omitted. If p is in some hole(s), say Ij , j = 1, 2...ℓ, thenthe test Ij ∩ E1 ∩ E2... ∩ Ek = Ij is invoked, to find out if the hole is in innermost material, performed asq ∩ E1 ∩ E2... ∩ Ek = q, with one q ∈ Ij .When calculating some entity in Mira [3] we use a transformation of spline data into Bezier surfaces (or curves):For instance, calculating (∂/∂u, ∂/∂v)Cox(N(u, v)) to evaluate a B-spline N(u, v) ∈ R

3, where Cox is short forthe Cox-de-Boor algorithm [4]. Some afterthought reveals that the Bezier data form a 5-dimensional matrix, callit B, which sometimes is very large. When visiting a surface (or a curve) it is natural to store the coefficientsof B in a buffer. By requiring that the user may only use the buffer by making a non-nested begin-surfacecall and an end-surface call, one may do memory-saving in the begin-surface and end-surface subroutines, e.g.allocations and deallocations of B. Finally, there are 3 types of surfaces in MIRA: solid, thin and transparent,where the first is part of a body, with exterior/interior volume, the second has surface normal ambiguity, andthe third a surface not participating as a scatterer but just a mathematical entity, e.g., useful when makingGTD/MoM hybrids.

Ray

This section considers the contents of the Ray package and exposes its envisagable future components.In the package, a ray is defined by a starting point P and a direction D where both are in R

3. The first thingis to use the distance t from P in the ray’s direction as parameter. Then we are only interested in points Q =(x,y,z) given by: Q(t) = P + t · D. The internal data structure follows the above mathematical representationand allows to represent both a ray that start in the source and propagate as far as infinity (semi-line or FARfield radiation) or propagate to a finite point (segments, i.e., t belongs to a finite interval [a, b], or NEARfield radiation). This means that irrespective of whether the observer is infinitely far away or not, the datarepresentation of the ray stays the same.A second feature is that the rays are grouped in lists, thus preparing any distribution of illumination tasks ina distributed computing environment. Linked lists are used to allow dynamic memory allocation and flexibledata storage.The Ray package performs the computation of R

3 geometrical ray sheets that actually includes the followingfeatures: (i) reflection by convex smooth surfaces, (ii) diffraction by edges, (iii) multiple interaction and (iv)creeping rays launched from the shadow boundary of a convex smooth surface.

2

In a near future we will focus on two particular things. Firstly, to couple the Ray package with the NURBStrimmed representation of the surfaces in a more efficient way. Secondly we will focus on algorithms related to thefollowing topics: (i) intersection/reflection/diffraction on concave/convex surfaces, (ii) diffraction and creepingfrom an edge, (iii) multiple creeping-diffracted-creeping rays, (iv) dielectric materials, (v) localization of caustics(region with high concentration of energy where standard GTD prediction fails) and (vi) sources/observers onthe surface corresponding to on board antennas.The computation of characteristic points (intersection, reflection, diffraction, etc) is based on a numericaltechnique applied to the Generalized Fermat’s principle. This corresponds to trace a ray (Ray Tracing) betweentwo points (source and observer) such that the optical path length is an extremum (maximum or minimum).Numerical methods are in generally fast but not reliable. They often miss to converge to the proper solutionwhen a surface is both convex and concave or when a surface has more than one intersection with the ray. Thatis why we will investigate the possibility to couple a numerical method with a recursive subdivision method inorder to isolate multiple solutions as well as to get good starting point for the minimization method.In addition, the MoM/PO/GTD hybrid demands an efficient method to detect hidden surfaces, ray occlusionby surface patches and surface patch visibility tests, so in this case, the modularity should save a considerabletime during the assembly of all these technique together.

Application

The application package that we have implemented at present time considers antenna related computations.By convenience we also use the antenna package when we hybridize MIRA with a MoM solver. The mainconstituents of the application package is a set of receivers, sources and a module called attenuation thatcompute how much a GTD field is attenuated a long any type of ray path.The types of sources we have implemented are plane waves, spherical harmonics, dipoles, antenna diagrams andan intrinsic type of source we call simple sources. All of these types of sources have support in GTD if they aretreated in the proper way. The receivers are points, directions, antenna diagram and intrinsic simple receivers.The data structure in the package has been designed in such a way that one can easily build a complex antennasystem, e.g. built up by a combination of several antenna elements distributed over any platform where eachantenna element can in its turn be described by a collection of sub-sources. Receivers are built up in a similarway. When doing coupling calculations this is very useful since the user can easily build the receiving andemitting antenna systems.In addition the modularity permits one to easily add new types of sources and receivers without changinganything in the Geometry and Ray package.Probably we will in the future make use of the possibility to call other software packages inside the sourceroutines to be able to numerically compute the radiated field by a source. This may well come in hand whenmeasured data is not attainable or when one needs higher accuracy. It is important to notice that this will betransparent in any other part of the code since all calls are done via interfaces.The two main advantages of the above data structure is that it well supports the already started hybridizationwith a MM solver and that it will, in the future, make the adaptation of the code to parallel computers relativelyeasy.The input data needed for the Application package are given by an object called Event which makes the linkswith the Ray package. An Event describes an entire generic ray path (reflection path, diffraction path, ...) andcontains all the required geometrical information (normals, curvature at the characteristics points) needed tocompute the GTD field.The computationally most expensive part of the solver is to create these events. Having a list of all the eventsit is interesting to notice that they do not changes with frequency. This is in contrast to MoM which get a newset of equations for each frequency. On the other hand if the sources or receivers are moved the event data hasto be re-computed. In MIRA, the event object is stored and never re-computed when frequency is changed.While hybridizing a ray tracer with a Method of Moment solver it is important to take this in consideration.Also when doing coupling calculations sometimes the receiver geometry will change with frequency and thenthe event object has to be re-computed. This will make such computations a bit more expensive compared tofor example antenna diagram computations.

3

The global algorithm in MIRA (antenna application) constitutes of the following:

• Build the geometry.

• Build the sources and receivers.

For all, or for one event, do:

• Compute a certain type of ray, e.g. a reflection-diffraction ray.

• Build the event.

Do the field computations:

• Compute the GTD field from the source. Attenuate the GTD field along the ray with the help of theevent data and update the receiver.

Finally

• Deallocate the geometry, the antennas, the events and the rays..

All field computations are done in a module called attenuation. As argument is has an event, a source and areceiver.

Conclusions

We think that with the present MIRA one has a solver that is relatively robust and general both in terms ofgeometry and ray tracing.Thanks to the features presented in this article, MIRA can be used as a platform for future implementationsof various kinds of GTD coefficients. As a suggestion we think that it could be interesting to call eithernumerically constructed databases for different kinds of diffraction coefficients or to call numerical methods, e.gFDTD solvers, for computation of GTD coefficient.With the modern geometry data structure that well supports the visualization and industrial CAD tools, thatexist in the industry, MIRA will serve as an excellent starting point for new applications.

Acknowledgment

The authors would like to thank J. Gustafsson and C. Lundstedt at Ericsson Saab Avionics AB for providingus with interesting test geometries.

References

[1] J. Edlund et. al. An investigation of hybrid techniques for scattering problems on disjunct geometries. (ibid),2000.

[2] J. Perez et. al. Analysis of antennas on board arbitrary structures modeled by nurbs surfaces. IEEETransactions on Antennas and Propagation, AP-45:1045–1052, June 1997.

[3] M.F. Catedra et. al. Fasant theoretical foundations. technical report, signal theory and comm. dept., univ.of alcala, 1998.

[4] G. Farin. Curves and surfaces for computer aided geometric design. acad. press, 1998.

4

Appendix A:

Figure 1: Direct and reflected rays on a generic satellite. This model is composed of 90 NURBS surfacesand consisted of an union of simple objects, mostly cones and cylinders. A point source is situated above thesatellite (view on the right) . The shortest direct and reflected path between the source and the observer pointsunderneath the satellite are searched for in the ray tracer. The direct and reflected GTD field are summed upat the observer points.

5

Figure 2: Direct and reflected rays on four semi-transparent spheres modelled with 8 NURBS. One source pointis placed in the middle, four observer points behind each sphere. At each observer point it reaches three rays:one direct ray from the source passing through the semi-transparent surfaces of the sphere, and two indirectreflections coming from bounces on facing spheres.

6

Figure 3: Diffracted rays on a generic cube. The diffracted paths between one source point and one observerpoint above the cube occur at the middle of each edge. Diffraction from the corners of the cube can be alsoidentified in the ray tracer.

Figure 4: Diffracted rays on a generic cube, top view.

7

Figure 5: Near to Far field reflection on a generic Gripen-like aircraft built up with large and complex surfaces.The model is composed of 50 NURBS. An antenna has been placed under the aircraft’s left wing and act as asource point for the rays. The shortest reflected paths from the wing, from both the upper side and the bottomside of the fuselage are displayed for several far field directions.. Such a computation is performed by MIRA ina few minutes.

8

Figure 6: Near to Far field reflection on a generic aircraft, side view.

Appendix B:

9

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Architecture and Geometric Algorithms

in MIRA, a Ray-based Electromagnetic

Wave Simulator

Sandy Sefi, [email protected]

Department of Numerical Analysis and Computer Science, Royal Institute of Tech-nology KTH, SE:100 44 Stockholm, Sweden

Abstract

In this paper we describe the basic architecture of the user-oriented electromagneticsimulator called MIRA which determines the near and far field from transmitting anten-nas and plane waves. The simulator is a part of the Swedish code development projectGEMS: General ElectroMagnetic Solvers supported by PSCI, KTH, Saab Avionics AB,and NFFP and continues in the NFFP 473 project SMART.

Keywords: Design Software, Geometrical Theory of Diffraction, Ray Tracing, ShadowDetection

1 INTRODUCTION

An electromagnetic wave simulator is used in Aerospace and Telecom areas to assist inthe analysis of installed antenna performance and Radar Cross Section (RCS). Its aim isto predict the field emitted by antenna models on board large structures - buildings, air-crafts or ships - as realistically as possible. In the high-frequency band, asymptotic ap-proximations to the Maxwell equations are suitable to evaluate the electromagnetic field.One of the powerful asymptotic methods known is the Geometrical Theory of Diffraction(GTD). The GTD [1] is based on Geometric Optics and Diffraction Theory. It assumesthat all waves are “well-formed” and are locally plane waves. This enables Ray Tracingalgorithms to be used with the following advantages:

1. It supplies a method to asymptotically compute the interaction of an antenna witha structure, when classical integral formulation methods become computational-ly too expensive.

2. Ray-Tracing is geometric. The computational demands is not dependent on theelectrical size of the structure but only on the complexity of the geometry. Thereis no runtime penalty in increasing the frequency.

3. Ray-Tracing is not memory intensive.

4. It is relatively easy to parallelize the underlying algorithm.

MIRA is essentially a Ray-Tracer based on Fermat’s principle and works on complexCAD geometries where the objects are described by trimmed NURBS - Non UniformRational B-spline - and rational Bezier patches. NURBS is a standard for parametric sur-face representation [2]. Its Object Oriented design has been developed to support hybrid-ization techniques with other numerical methods in the frequency domain such asMethod of Moments (MoM) or Physical optics (PO).

Our first hybrid application uses the Ray-Tracer module of MIRA as a stand-alone solvercoupled with a PO/MoM solver [3] to determine accurately the shadow regions of the POdomain when it is illuminated by an antenna. The Method of Moments is used locally tocompute the input behavior around the antenna. A similar technique is used when thewave is scattered.

Below we will describe the different steps of the ray-based electromagnetic wave simu-lation, after that we will consider the basic features of the software architecture. Finally,we will give a brief overview of the Ray/PO hybrid.

2 THE RAY-BASED WAVE SIMULATION

The following gives a brief description of the physical process of the high-frequencywave propagation which is simulated:

1. Input of the geometry data (scene)consists of a collection of (i) surfaces ina trimmed NURBS format subdividedinto a combination of Rational BezierPatches by using the Cox De Boor algo-rithm [2], (ii) ElectroMagnetic (EM)sources and (iii) receivers.

2. The EM sources emit anincidentfield characterized by a certain direction,amplitude, phase and frequency. Thefield is represented by a wave.

3. The wave is absorbed, scattered, orreflected by the surfaces as it travelsthrough the scene. The EM field is atten-uated as the wave travels and the attenu-ation is proportional to the traveleddistances.

4. The wave reflected or diffractedfrom a surface depends on properties ofthe surface (curvature, transparency, sur-face material...) as well as on incident il-lumination.

Figure 1 High-frequency wave propagation.

5. The linearity of the Maxwell equations allows the superposition of the total EMfield from the sum of independent electromagnetic contributions.

Simplified Input Scene

Incident field

Wave Propagation

(1.)

(2.)

EijIncident

EijReflected Eij

Diffracted

Trimmed NURBS

(3.)

EijTotal

EijIncident

EijReflected

EijDiffracted

+ +=

6. The Total field will be found as sum of the direct contributions from thesourcei, reflected and diffracted contributions reaching the receiverj.

7. Each contribution is represented by rays traced from the sourcei to the receiverj

The determination of the rays is done following Fermat’s principle. This corresponds totracing a ray between the source and receiver such that the optical path length reaches anextremum (maximum or minimum). The characteristic points (intersection, reflection,diffraction, etc.) are found by numerical conjugate gradient techniques applied to Fer-mat’s principle. This part represents the most time consuming task in the solver since theray-path determination process requires both rational Bezier surface point evaluationsand computationally expensive evaluations of derivatives on NURBS surfaces.

3 PROGRAM DESIGN AND ARCHITECTURE

To design an architecture of a computer code is the task of choosing between variousavailable alternatives and at the same time considering possible future developments andtrends. The main problem is to design an architecture that is general enough to be usefulfor more than a single task, while providing an appropriate framework for developmentand efficient implementations. In this context, our aim is to develop a code capable ofperforming the calculations on fully realistic industrial models, to easily support futurerequirements or functionalities and to allow hybridization with other computational elec-tromagnetic tools.

HISTORY AND BACKGROUND

The starting point for the architecture of MIRA was the FASANT code which was devel-oped at Cantabria University by F. Cátedra and co-workers. From a computational elec-tromagnetic point of view, this is a good old - quite reliable - code. Its functionalities andcomparison with measurements can be found in [4]. A good comparison with theNEC/BSC code can be found in [5]. Apart from possessing these qualities, we found theFortran 77 FASANT code difficult to modify and we decided to redesign the code by go-ing back to the fundamental theory [6] and do a complete re-implementation using objectoriented design with a Fortran 90 implementation. However, it is important to remarkhere that in the course of redesigning FASANT, we also introduced new algorithms. Forexample, we added diffraction algorithms, based on a search of combinations of localminima and local maxima, diffraction corner detection, double diffraction path determi-nation, better visibility preprocessing taking into account transparent surfaces or dielec-tric sheets, etc.... The results and the description of the new features can be found in [7].

FEATURES OF THE ACTUAL ARCHITECTURE

The following design features were leading up to a wave simulator architecture powerfuland flexible enough to support hybrid applications:

1. Modular design

This work follows ideas from the Computer Graphics Society in the area of physical ren-dering processes [8]. These ideas [9] stipulate that a modular design should permit (i) thesame general architecture to be configured for use in various application areas, (ii) thedevelopment of new algorithms and (iii) the implementation of variations of existingtechniques reusing the already available environment. To fulfill it in our context, we havestructured the architecture by a clear division of the complete system into subsystemscalled modules. The different modules have to be as independent of each other as possi-ble and require well defined interfaces between them. The interfaces must be generalenough not to limit the range of possible implementations. One major obstacle on this

EijTotal

theme are dependencies between modules, which need to be minimized. The details ofthis modular approach can be found in [10].

Figure 2 Features of the software architecture

2. Object orientation.

Each object (3D surfaces, bounding curves, rays, antennas...) has well-defined interfacesconsisting of a set of operations or methods that may be invoked by other objects. It alsohas internal attributes that represent the state of the object. The set of methods and at-tribute variables is described by the class of the object. Classes are manually organizedin inheritance since automatic inheritance is not supported by Fortran 90 language. Aclass inherits behavior and attributes from its parents. In this way, the internal state of anobject is encapsulated by the interface and can be manipulated by other objects onlythrough the methods offered. This permits a clear separation between the interface of anobject and its implementation. It defers the handling of implementation details from theearly stages of the analysis and design process and therefore allows for better abstraction.

3. Good accuracy

The accuracy is only limited by the algorithms used and the parameters chosen, but notby the architecture itself. Since accuracy most often is coupled with the computationalcost of the calculations, the architecture allows various options for trading accuracyagainst computational complexity. One way to offer this is to allow for different solutionstrategies for various parts of the system (c.f. flexibility). The generic characteristics ofthe new structure permit the coexistence of different geometrical representation. For in-stance, the initial free-formed surfaces (NURBS, rational Bezier etc....) coexist with theset of the plane facets/triangles constructing the mesh. Each surface is labeled and each

HYBRID CODE

Software with Modular Design

Flexibilityswitch

Fast

Accu

rate

Limited modules

Other Modular Solver

dependencies

Object Orientation

Classes

inheritance

facet/triangle knows on which surface it belongs. All these links between data representan important step toward improving the accuracy. In fact, the solver will work faster withonly a facet description but will need to go back to geometrical information for tangents,radius of curvature, etc., to make precise field calculations.

4. Flexibility

Flexibility generally comes at a cost. In most cases a specific interface can be implement-ed with better performance than a general interface. In each case we have to decide if theadditional flexibility is worth the price imposed by it or if a restricted interface offers sub-stantially better performance. Sometimes, this might require the implementation of boththe general interface and an interface with better performance but restricted functionality.We believe that in many cases and in the long run it is more important to provide a flex-ible interface than the implementation with the highest performance.

5. Handling of complex models

To be useful for real world applications an architecture must handle large and complexdata sets. A fully realistic model is typically composed by more than one thousandNURBS surfaces. This requires the minimization of the storage requirements within thearchitecture. Practically, for the representation of the geometry data structure, linkedchains of buffers dynamically allocated are preferred to one huge list of surfaces storedin a fixed sized matrix.

6. Modern programing style

Good coding allows for more efficient research and development. Current coding prac-tices are in strict opposition to old fashion programming style characterized by “goto”loops, code duplication instead of subroutine calls and global variables used in many dif-ferent parts of the code. Instead we assist on variable names that mean something, paren-theses and white-space to make the code readable.

7. Robust design

We have significantly improved code quality and reliability thanks to systematic check-ing of input arguments, parameters and tolerances, division by zero etc....

8. Efficiency thanks to topological information

The topology of an object is the list of the connections between faces (surfaces), edges(bounding curves) and vertices (corners) of a geometrical object. Most recent CAD prod-ucts include topological facilities. A remaining problem is that engineers are used to ex-changing data using IGES1 formats likes which do not transfer the topology [11]. In thiscontext, we were faced with the task of extracting the necessary topological informationdirectly from the geometry. The topology is used to improve the global efficiency of thecomputations. For instance, when examining diffraction phenomena, the topology per-mits the following edge classification: an edge can be (i) free: a boundary without neigh-boring surface, (ii) C0: continuity everywhere between neighboring surfaces, (iii) C1: notthe same tangent plane for two surfaces or (iv) material discontinuity. This classificationpermits the removal of fictitious edges and speeds up the diffraction localization search[12] by directly focusing on the correct edge (C1). Edges are defined as Bezier curves.The classification is obtained from the study of normal discontinuities at the boundarycurves between NURBS. The topology is pre-computed in a table. The table contains la-beled information such as parentship relations with the surfaces and neighboring edge re-lations.

1 IGES, the Initial Graphics Exchange Specification, is a widely used CAD data exchange specification.

9. Performance

Maybe the most important factor is to obtain acceptable performance. The Ray tracerdoes this by using simple algorithms, avoiding redundant calculation and unnecessarycomputations. For example, the measured execution time of the determination of the di-rect illumination for 100.000 rays launched through a scene composed of 400 NURBSsurfaces, is less than 3 minutes on one node of an IBM RS6000 with 160 MHz, 256 MBof RAM. For comparison, on a very small example of a direct near field and diffractioncalculation at six receivers illuminated from one source around a cylinder (only 6NURBS), FASANT gets 0.37 Mflops (Million floating-point operations per second) fora total execution time (wall clock time) of 97,7 seconds. On the same problem, MIRAruns at 5.16 Mflops during a total execution time of 6.5 seconds, which makes it 15 timesfaster than FASANT. These results differ so much since we are using a new detection ofdiffraction path based on direct search of local minima and maxima (Fermat’s principle)instead of running a minima search and use an algorithm for discarding false solution. Inaddition this results also illustrates how important the topology can be to improve thecode performance.

4 HYBRIDIZATION: SHADOW DETERMINATION

The coupling of the Ray Tracer module of the wave simulator to a PO solver reuses theray module to determine accurately the parts of the non illuminated PO domain situatedin the shadow regions of an emitting source. The hybridization improves the PO approx-imation by removing from the study all the geometrical details residing inside the shadowcast by a PO object on itself (self shadow) or by any blocker (other eventual object) be-tween the PO domain and the emitting source (occlusion).

THE HYBRID ALGORITHM

To solve this problem, a test ray is launched from the barycenter of each PO facet/trian-gle. The ray is then traced back toward the point source. If the ray path to the source isnot blocked by other surfaces, then the facet is directly illuminated, the surface currentdistribution is computed with . A PO approximation of the scattered field canbe obtained from the surface current J by integrating the PO fields and finally computethe RCS of the object under study.

In our first version, only the incident field is used to illuminate (no reflection nor diffrac-tion). This task remains however highly time consuming since a huge number of rays hasto be traced. In the future, indirect illumination of the PO domain coming from the re-flection of a ray on a surface, can easily be added.

The two main characteristics of this algorithm are:

1. High computational complexity. There are (i x j) occlusion tests to perform. Duringone occlusion, several intersection tests must be executed with all the surfaces fac-ing the pointsi or j. Each intersection test requires heavy NURBS derivations andsurface point evaluations.

2. The independency of computations between each sourcei and receiverj gives nat-ural parallelization.The details of the parallelization realized with MPI library callsand implemented on the IBM SP with Power2 processors can be found in [13].

ILLUSTRATION

For the assumptions and algorithm presented above, the current distribution on a realisticaircraft is obtained. The tested geometries are realistic industrial models and have been

J 2nxH=

borrowed from Saab Avionics. The geometry is composed by 82.736 triangles/receiversthat facetize 368 trimmed NURBS surfaces.

Figure 3 represents the current distribution without taking into account the occlusion(this picture is given as reference to illustrate the PO solver working by its own).

Figure 4 represents the illumination taking into account self shadow and occlusion. Asingle source, a plane wave perpendicular to the nose of the plane, illuminates 82.000 re-ceivers placed on the aircraft surface.

Figure 3 PO only without Ray-tracing: current surface distribution on an aircraftfrom a plane wave pointing vector parallel to the aircraft center axis.

Figure 4 PO and shadow: illumination taking into account the self shadow of theaircraft. The dots represent triangles illuminated by a plane wave. In suchorientation, half of the wing surfaces are occluded.

5 CONCLUSION

In this paper we have proposed an architectural approach to the implementation of anhigh frequency electromagnetic wave simulator. Its Object Oriented design is powerfuland flexible enough to support a number of algorithms, yet can handle large and complexbodies. It has been developed to support hybridization techniques. In particular, we havegiven a detailed description of our recent Ray/PO hybrid. The method described com-bines several new concepts: use of topological information, double geometry representa-tion: NURBS & Triangulation and accurate shadow detection taking into account selfocclusion to a powerful Applied Electromagnetics software. Future improvements willfocus on efficiency aspects for industrial applications.

References

[1] J. B. Keller, “Geometrical Theory of Diffraction”, J. of Optical Soc. of Amer-ic., Vol. 52, No. 2, pp. 116-130, February, 1962.

[2] Gerald Farin, “Curves and Surfaces for Computer Aided Geometric Design”,Fourth Edition, Acad. Press, 1996.

[3] Johan Edlund, Stefan Hagdahl and Bo Strand, “An investigation of hybridtechniques for scattering problems on distinct geometries”, Technical ReportPSCI No. 2000:08, KTH, March, 2000.

[4] J. Pérez, F. Saiz, J., O. Gutierrez, I. Gonzalez, M.F. Cátedra, I. Montiel, J.Guzmán, “FASANT: Fast Computer Tool for the Analysis of Antennas On-Board Antennas”, IEEE Antennas and Propagation Magazine, Vol. 41, No. 2,June, 1999.

[5] J. Pérez, J.A. Saiz, O. Conde, R.P. Torres, M.F. Cátedra, “Analysis of Anten-nas on Board Arbitrary Structures Modelled by NURBS Surfaces”, IEEETransactions on Antennas and Propagation, Vol. 45, No. 6, June, 1997.

[6] M.F. Cátedra et. al., “FASANT (Version S.4) Theoretical Foundations”,Technical report, Signal Theory and Comm. Dept., Univ. of Alcalá, 1998.

[7] Sandy Sefi, “Design and Architecture of MIRA, a Ray-based ElectromagneticCode”, DRT’s thesis, University Joseph Fourier, Grenoble, France, October2000.

[8] Philipp Slusallek, Hans-Peter Seidel, “Vision: An architecture for global illu-mination calculations”, IEEE Transactions on Visualization and ComputerGraphics, 1(1):77--96, March, 1995.

[9] Philipp Slusallek, “Vision - An Architecture for Physically Based Render-ing”, PhD thesis, University of Erlangen, IMMD IX, Computer GraphicsGroup, April, 1995.

[10] Fredrik Bergholm, Stefan Hagdahl, Sandy Sefi, “A modular Approach toGTD in the Context of Solving Large Hybrid Problems”, proceeding publica-tion in AP2000 Millennium Conference on Antennas & Propagation, Davos,Switzerland, April, 2000.

[11] Patrick Chenin, “Geometric Modelling for ElectroMagnetic Simulation”,LMC-IMAG Unival S.A., August 1998.

[12] Fredrik Bergholm, “Locating Diffraction Points with Ray-based Methods”,Technical Report PSCI, KTH, December, 2000.

[13] Sandy Sefi, “Parallelization of a Ray-based Electromagnetic Wave Simulatorusing MPI”, PDC High Performing Computing Course Report, KTH, August,2001.

Thesis

Documents

society international

perfectly

thier scaling

nm nm

charge takes

dirac delta

electrically

ohio state