thesis
TRANSCRIPT
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.
10 CHAPTER 2. GOVERNING FIELD EQUATIONS
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.
12 CHAPTER 2. GOVERNING FIELD EQUATIONS
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.
14 CHAPTER 2. GOVERNING FIELD EQUATIONS
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.
16 CHAPTER 2. GOVERNING FIELD EQUATIONS
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)
22 CHAPTER 3. SCATTERING ANALYSIS METHODS
−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)
24 CHAPTER 3. SCATTERING ANALYSIS METHODS
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
26 CHAPTER 3. SCATTERING ANALYSIS METHODS
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.
28 CHAPTER 3. SCATTERING ANALYSIS METHODS
−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.
30 CHAPTER 3. SCATTERING ANALYSIS METHODS
−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.
Bibliography
[Stratton 41] J.A. Stratton.Electromagnetic Theory. McGraw-Hill, 1941.
[Rumsey 54] V. H. Rumsey. Reaction concept in electromagnetic theory. PhysicalRev. (2), 94:1483 1491, 1954.
[Keller 62] Joseph B. Keller. Geometrical Theory of Diffraction. Journal of the Op-tical Society of America, Vol. 52(2), pp. 116-130, February 1962.
[Kouyoumjian 65] R.G. Kouyoumjian. Asymptotic High-Frequency Methods. Proc,IEEE, Vol.53, No. 8, pp. 864-876, August 1965.
[Ludwig 68] A. C. Ludwig, Computation of radiation patterns involving numericaldouble integration, IEEE Transactions on Antennas and Propagation, pages767–769, Nov. 1968.
[Harrington 68] R.F. Harrington. Field Computation by Moment Methods. NewYork: Macmillan, 1968.
[Thiele 75] G.A. Thiele and T.H. Newhouse, A hybrid technique for combining mo-ment methods with geometrical theory of diffraction, IEEE Trans. AntennasPropagat., vol. AP-23, pp. 62-69, Jan. 1975.
[Burnside 75] W.D. Burnside, C.L. Yu and R.J. Marhefka, A technique to com-bine the geometrical theory of diffraction and moment method, IEEE Trans.Antennas Propagat.,pp. 551-558, July 1975.
[deBoor 78] C. deBoor, A Practical Guide to Splines. Applied Mathematical Sci-ences 27, Springer-Verlag, New York, 1978.
[Rao 82] S. M. Rao, D. R. Wilton, and A. W. Glisson. Electromagnetic scatteringby surfaces of arbitrary shape. IEEE Transactions on Antennas and Propaga-tions, 30(3):409418, May 1982.
[Pathak 84] R.G. Kouyoumjian and P.H. Pathak. A Uniform Geometrical Theoryof Diffraction for an Edge in a Perfectly Conducting Surface. Proc. IEEE,Vol. 62, pp. 1448-1481, November 1984.
35
36 BIBLIOGRAPHY
[Burnside 87] I. J. Gupta, W. D. Burnside, A Physical Optics Correction forBackscattering from Curved Surfaces, IEEE Transactions on Antennas andPropagation, vol. 35, N0. 5, May 1987.
[Greengard 87] L. Greengard and V. Rokhlin, A fast algorithm for particle simula-tions, Journal of Computational Physics, 73(2):325-348, Dec. 1987.
[Bowman 87] J.J. Bowman, T.B.A. Senior, P.L.E. Uslenghi Electromagnetic andAcoustic Scattering by Simple Shapes, HPC, 1987.
[Farin 88] Gerald Farin. Curves and surfaces for computer aided geometric design:a practical guide. Academic Press Professional, Inc., San Diego, CA, 1988.
[McNamara 89] D.A. McNamara, C.W.I. Pistorius, J.A.G. Malherbe Introductionto the Uniform Geometrical Theory of Diffraction, Artech House, London,1989.
[Glassner 89] Andrew S. Glassner. An introduction to ray tracing. Academic PressLtd., London, UK, 1989
[Piegl 91] Les Piegl. On NURBS: A survey. IEEE Computer Graphics and Applic-ation, South Florida, 1991.
[Rahmat-Samii 91] C.S. Kim and Y. Rahmat-Samii, Low profile Antenna StudyUsing the Physical Optics Hybrid Method (POHM), IEEE Antennas andPropagation Society Int. Symp., vol.3, pp. 1350-1353, June 1991.
[Crabtree 91] Glenn D. Crabtree, A numerical quadrature technique for physicaloptics scattering analysis, IEEE Transactions on Microwave Theory and Tech-niques, 27(5), Sep. 1991.
[Rokhlin 92] N. Engheta, W. Murphy, V. Rokhlin and V. Vassiliou, The fast mul-tipole method (fmm) for electromagnetic scattering problems, IEEE Trans. onAntennas and Propagat., vol 40, pp. 634-641, June 1992.
[Numer. Recipes 92] Numerical Recipes in Fortran 77. Cambridge University Press,Volume 1, 1986-1992.
[Bouche 93] D.P. Bouche, F.A. Molinet and R. Mittra, Asymptotic and Hy-brid Techniques for Electromagnetic Scattering, Proceedings of the IEEE,vol.81(12), Dec. 1993.
[Rius 93] J.M. Rius, M. Ferrando and L. Jofre. GRECO: Graphical ElectromagneticComputing for RCS prediction in real time. IEEE Antennas and PropagationMagazine, vol. 35, No.2, pp. 7-17, April 1993.
[Leviatan 93] B.Z. Steinberg and Y. Leviatan, On the use of wavelet expansions inthe method of moments, IEEE Transactions on Antennas and Propagation,vol. 41, pp. 610–619, 1993
37
[Mittra 94] Z. G. Figan, R. Mittra, A. Boag and E. Michielssen, A technique forsolving scattering problems at high frequencies, Int. URSI Symp., Seattle WA,June 1994.
[Moreira 94] F.J.S Moreira and A. Prata. A self-checking predictor-corrector al-gorithm for efficient evaluation of reflector antenna radiation integrals, IEEETransactions on Antennas and Propagation, 42(2):246–254, Feb. 1994.
[Gordon 94] William B. Gordon, High Frequency Approximations to the PhysicalOptics Scattering Integral, IEEE Transactions on Antennas and Propagation,42(3), Mar. 1994.
[Catedra 95] M. Domingo, F. Rivas, J.Perez, R.P. Torres, M.F. Catedra. RANURS:Computation of the RCS of Complex Bodies Modelling Using NURBS Sur-faces. IEEE Antennas and Propagation Magazine, vol. 37, No. 6, December1995.
[Aberegg 95] K.R. Aberegg and A.F. Peterson, ”Application of the IntegralEquation-Asymptotic Phase Method to Two-Dimensional Scattering”, IEEETrans. on Antennas and Propagat., vol. 43, pp. 534-537, May 1995.
[Jakobus 95] U. Jakobus and F. M. Landstorfer, Improved PO-MM Hybrid Formu-lation for Scattering from Three-Dimensional Perfectly Conducting Bodies ofArbitrary Shape, IEEE Transactions on Antennas and Propagation vol. 43,pp. 162–169, February 1995.
[Taflove 95] A.Taflove. Computational Electro-dynamics: The Finite-DifferenceTime-Domain Method, Artec House, 1995.
[Asvestas 95] J. S. Asvestas, The Physical-Optics Integral and Computer Graphics,IEEE Trans. on Antennas and Propagat., vol.43(12), Dec. 1995.
[Altman 96] Z. Altman, R. Mittra, O. Hashimoto and E. Michielssen, Efficientrepresentation of the induced currents on large scatterers using the generalizedpencil of function method, IEEE Trans. on Antennas and Propagat., vol.44(1),Jan. 1996.
[Hodges 97] R.E. Hodges and Y. Rahmat-Samii, An iterative current-based hybridmethod for complex structures, IEEE Trans. Ant. Propagat. 45 1997.
[Jackson 98] J. D. Jackson. Classical Electrodynamics. 3rd ed. New York: Wiley,p. 177, 1998.
[Mittra 98] Mittra R., Peterson, A.F. and Ray S.L.. Computational Methods forElectromagnetics. IEEE Press, 1998.
[Catedra 98] M.F Catedra, J. Perez, I. Gonzalez, I. Gutierrez, F. Saez de Adana.FASANT (Version S.4) Theoretical Foundations. Signal Theory and Commu-nications Dept., University of Alcala. 1998.
38 BIBLIOGRAPHY
[Altman 99] Z. Altman and R. Mittra,A technique for Extrapolating Numeric-ally Rigourous Solutions of Electromagnetic Scattering Problems to HigherFrequencies and Thier Scaling Properties, IEEE Trans. on Antennas andPropagat., vol.47(4), April 1999.
[Fares 99] A. Bendali, M.B. Fares, CESC: CERFACS Electromagnetics SolverCode. Technical Documentation TR/EMC/99/52, Toulouse, France, 1999.
[Bendali 99] A. Bendali, The Rumsey reaction principle. Lecture notes 1999.
[Oppelstrup 99] R. Buch, J. Long, J. Oppelstrup PSCI Progress Report 1997-1999.PSCI, KTH, pp 70-82, 1999.
[Kwon 00] D. Kwon, Efficient Method of Moments formulation for large conductingscattering problems using asymptotic phasefront extraction, Ph.D. disserta-tion, Dept. Elect. Eng., The Ohio State University, Colombus, OH, 2000.
[Taboada 00] F. Obelleiro, J.M. Taboada, J.l. Rodriguez, J.O. Rubinos and A.M.Arias, Hybrid moment-method physical optics formulation for modeling theelectromagnetic behavior of on-boaerd antennas, Microwave Opt. Tech. Letter27,pp. 88-93, 2000.
[Nilsson 00] Martin Nilsson, A fast multipole accelerated block quasi minimum re-sidual method for solving scattering from perfectly conducting bodies, Ant. andPropag. Society International Symposium No 4, 2000.
[Andersson 01] U. Andersson. Time-Domain Methods for the Maxwell Equations.Ph.D Thesis, KTH, Stockholm, 2001.
[Ledfelt 01] G. Ledfelt. Hybrid Time-Domain Methods and Wire Models for Com-putational Electromagnetics. Ph.D Thesis, KTH, Stockholm, 2001.
[Edlund 01] J. Edlund. A Parallel, Iterative Method of Moments and Physical Op-tics Hybrid Solver for Arbitrary Surfaces. Licentiate Thesis, Uppsala Univer-sity, 2001.
[Kwon 01] D. Kwon, R.J. Burkholder and P. H. Pathak, Efficient Method of Mo-ments Formulation for Large PEC Scattering Problems Using AsymptoticPhasefront Extraction (APE), IEEE Trans. on Antennas and Propagat., vol.49, No.4, April, 2001.
[Taboada 01] J. M. Taboada, F. Obelleiro, and J. L. Rodrígeuz. Improvement ofthe hybrid moment method-physical optics method through a novel evaluationof the physical optics operator. Microwave and Optical Technology Letters.,30(5):357–363, 2001.
39
[Sarkar 02] C. Su, T.K. Sarkar, Adaptive Multiscale Moment Method (AMMM) forAnalysis of Scattering from Three-Dimensional Perfectly Conducting Struc-tures, IEEE Transactions on Antennas and Propagation, vol. 50, N0. 4, April2002.
[Edelvik 02] F. Edelvik. Hybrid Solvers for the Maxwell Equations in Time-Domain. Ph.D Thesis, Uppsala University, 2002.
[Sefi 03] S. Sefi. Ray Tracing Tools for High Frequency Electromagnetics Simula-tions. Licentiate thesis, KTH, Stockholm, 2003.
[Abenius 04] E. Abenius. Time-Domain Inverse Electromagnetic Scattering usingFDTD and Gradient-based Minimization. Licentiate Thesis, KTH, Stock-holm, 2004.
[Djordjevic 04] M. Djordjevic and B.M. Notaros, Higher Order Hybrid Method ofMoments-Physical Optics Modeling Technique for Radiation and Scatteringfrom Large Perfectly Conducting Surfaces, IEEE Transactions on Antennasand Propagation, Feb 2004.
[Nilsson 04] M. Nilsson. Fast Numerical Techniques for Electromagnetic Problemsin Frequency Domain. Ph.D Thesis, Uppsala University, 2004.
[Hagdahl 05] S. Hagdahl. Hybrid Methods for Computational Electromagnetics inFrequency Domain. Ph.D Thesis, KTH, Stockholm, 2005.
[Taboada 05] J. M. Taboada, F. Obelleiro, J. L. Rodrígeuz, I. Garcia-Tunon and L.Landesa, Incorporation of Linear-Phase Progression in RWG basis functions,Microwave and Optical Technology Letters., 44(2), Jan. 2005.
[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
[4] OpenDX, the Open Visualisation Data Explorer:http://www.opendx.org
[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
−3 Parallel Current Component: k=19 cut Phi=0.3927
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
−3 Perpendicular Current Component: k=19 cut Phi=0.3927
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
Department of Numerical Analysis and
Computer Science, NADA
Royal Institute of Technology, KTH
Stockholm, Sweden
Email: [email protected]
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
k Extract (All) Extract. (Lit) Ref. (All) Ref. (Lit)
10 – 51% 21% 0%19 51% 58% 21% 0%29 48% 47% 15% -1%
ϕ = π8
k Extract (All) Extract. (Lit) Ref. (All) Ref. (Lit)
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
−3 Parallel Current Component: k=10 cut Phi=0.3927
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
−3 Parallel Current Component: k=19 cut Phi=0.3927
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
−3 Perpendicular Current Component: k=10 cut Phi=0.3927
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
−3 Perpendicular Current Component: k=19 cut Phi=0.3927
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)
Fig. 11. Reference Model (no modifying coefficients) for the parallel component JRef
‖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)
Fig. 13. Reference Model (no modifying coefficients) for the parallel component JRef
‖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.
REFERENCES
[1] R.F. Harrington, Field Computation by Moment Methods, New York:Macmillian, 1968.
[2] J.J. Bowman, T.B.A. Senior, P.L.E. Uslenghi Electromagnetic and
Acoustic Scattering by Simple Shapes, HPC, 1987.[3] Joseph B. Keller, Geometrical Theory of Diffraction, Journal of the
Optical Society of America, Vol. 52(2), pp. 116-130, February 1962.[4] R.G. Kouyoumjian and P.H. Pathak, A Uniform Geometrical Theory of
Diffraction for an Edge in a Perfectly Conducting Surface, Proc. IEEE,Vol. 62, pp. 1448-1481, November 1984.
[5] D.A. McNamara, C.W.I. Pistorius, J.A.G. Malherbe Introduction to the
Uniform Geometrical Theory of Diffraction, Artech House, London,1989.
[6] G.A. Thiele and T.H. Newhouse, A hybrid technique for combining
moment methods with geometrical theory of diffraction, IEEE Trans.Antennas Propagat., vol. AP-23, pp. 62-69, Jan. 1975.
[7] W.D. Burnside, C.L. Yu and R.J. Marhefka, A technique to combine
the geometrical theory of diffraction and moment method, IEEE Trans.Antennas Propagat.,pp. 551-558, July 1975.
[8] S. Sefi, Ray Tracing Tools for High Frequency Electromagnetics
Simulations, Licentiate thesis TRITA-NA-0314, KTH, May 2003.[9] D.P. Bouche, F.A. Molinet and R. Mittra, Asymptotic and Hybrid
Techniques for Electromagnetic Scattering, Proceedings of the IEEE,vol.81(12), Dec. 1993.
[10] C.S. Kim and Y. Rahmat-Samii, Low profile Antenna Study Using
the Physical Optics Hybrid Method (POHM), IEEE Antennas andPropagation Society Int. Symp., vol.3, pp. 1350-1353, June 1991.
[11] U. Jakobus and F. M. Landstorfer, Improved PO-MM Hybrid Formula-
tion for Scattering from Three-Dimensional Perfectly Conducting Bod-
ies of Arbitrary Shape, IEEE Transactions on Antennas and Propagationvol. 43, pp. 162–169, February 1995.
[12] R.E. Hodges and Y. Rahmat-Samii, An iterative current-based hybrid
method for complex structures, IEEE Trans. Ant. Propagat. 45 1997.[13] F. Obelleiro, J.M. Taboada, J.l. Rodriguez, J.O. Rubinos and A.M.
Arias, Hybrid moment-method physical optics formulation for modeling
the electromagnetic behavior of on-boaerd antennas, Microwave Opt.Tech. Letter 27,pp. 88-93, 2000.
[14] J. Edlund, A Parallel, Iterative Method of Moments and Physical
Optics Hybrid Solver for Arbitrary Surfaces, Licentiate Thesis, UppsalaUniversity, 2001.
[15] F.X. Canning, The impedance matrix localization (IML) method for
moment-method calculations, IEEE Transactions on Antennas andPropagation, vol. 32, pp. 18–30, 1990.
[16] B.Z. Steinberg and Y. Leviatan, On the use of wavelet expansions in the
method of moments, IEEE Transactions on Antennas and Propagation,vol. 41, pp. 610–619, 1993
[17] C. Su, T.K. Sarkar, Adaptive Multiscale Moment Method (AMMM) for
Analysis of Scattering from Three-Dimensional Perfectly Conducting
Structures, IEEE Transactions on Antennas and Propagation, vol. 50,N0. 4, April 2002.
[18] W. Wiscombe, Improved Mie scattering algorithms, Appl. Opt., Vol.19, pp. 1505-1509, 1980
[19] N. Engheta, W. Murphy, V. Rokhlin and V. Vassiliou, The fast multipole
method (fmm) for electromagnetic scattering problems, IEEE Trans. onAntennas and Propagat., vol 40, pp. 634-641, June 1992.
[20] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations,Journal of Computational Physics, 73(2):325-348, Dec. 1987.
[21] Martin Nilsson, A fast multipole accelerated block quasi minimum
residual method for solving scattering from perfectly conducting bod-
ies, Ant. and Propag. Society International Symposium No 4, 2000.[22] S. M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic scat-
tering by surfaces of arbitrary shape, IEEE Trans. on Antennas andPropagat.,vol.AP-30(3), pp. 409-418, May 1982.
[23] D. K. Cheng, Field and Wave Electromagnetics, Addison Wesley,Second Edition, 1998.
[24] I. J. Gupta, W. D. Burnside, A Physical Optics Correction for Backscat-
tering from Curved Surfaces, IEEE Transactions on Antennas andPropagation, vol. 35, N0. 5, May 1987.
[25] A. C. Ludwig. Computation of radiation patterns involving numericaldouble integration. IEEE Transactions on Antennas and Propagation,pages 767–769, Nov. 1968.
[26] Glenn D. Crabtree. A numerical quadrature technique for physicaloptics scattering analysis. IEEE Transactions on Microwave Theory
and Techniques, 27(5), Sep. 1991.[27] F.J.S Moreira and A. Prata. A self-checking predictor-corrector algo-
rithm for efficient evalaution of reflector antenna radiation integrals.IEEE Trans. on Ant. and Propagat., 42(2):246–254, Feb. 1994.
[28] S. Sefi and J. Oppelstrup, Physical optics and NURBS for RCS
calculations, Proc. EMB 04 Computational Electromagnetics Methodsand Applications, pages 90-97. SNRV, October 2004.
[29] J. M. Taboada, F. Obelleiro, and J. L. Rodrıgeuz. Improvement ofthe hybrid moment method-physical optics method through a novelevaluation of the physical optics operator. Microwave and Optical
Technology Letters., 30(5):357–363, 2001.[30] J. M. Taboada, F. Obelleiro, J. L. Rodrıgeuz, I. Garcia-Tunon and L.
Landesa, Incorporation of Linear-Phase Progression in RWG basis
functions, Microwave and Optical Technology Letters., 44(2), Jan.2005.
[31] M. Djordjevic and B.M. Notaros, Higher Order Hybrid Method of
Moments-Physical Optics Modeling Technique for Radiation and Scat-
tering from Large Perfectly Conducting Surfaces, IEEE Transactionson Antennas and Propagation, Feb 2004.
[32] K.R. Aberegg and A.F. Peterson, ”Application of the Integral Equation-Asymptotic Phase Method to Two-Dimensional Scattering”, IEEETrans. on Antennas and Propagat., vol. 43, pp. 534-537, May 1995.
[33] Z. Altman and R. Mittra,A technique for Extrapolating Numerically
Rigourous Solutions of Electromagnetic Scattering Problems to Higher
Frequencies and Thier Scaling Properties, IEEE Trans. on Antennasand Propagat., vol.47(4), April 1999.
[34] Z. Altman, R. Mittra, O. Hashimoto and E. Michielssen, Efficient
representation of the induced currents on large scatterers using the
generalized pencil of function method, IEEE Trans. on Antennas andPropagat., vol.44(1), Jan. 1996.
[35] Z. G. Figan, R. Mittra, A. Boag and E. Michielssen, A technique
for solving scattering problems at high frequencies, Int. URSI Symp.,Seattle WA, June 1994.
[36] D. Kwon, Efficient Method of Moments formulation for large conduct-
ing scattering problems using asymptotic phasefront extraction, Ph.D.dissertation, Dept. Elect. Eng., The Ohio State University, Colombus,OH, 2000.
[37] D. Kwon, R.J. Burkholder and P. H. Pathak, Efficient Method of
Moments Formulation for Large PEC Scattering Problems Using
Asymptotic Phasefront Extraction (APE), IEEE Trans. on Antennas andPropagat., vol. 49, No.4, April, 2001.
[38] L.Takacs, A parabolic equation technique for reducing physical optics
shadow boundary errors Antennas and Propagation Society Interna-tional Symposium, 1990, pp:136 - 139, vol.1, May 1990
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
−3 Error REF model to MoM, Error Model with Coefficient to MoM
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
−3 Approximations of MoM, Parallel component, k=19 Phi=0.3927
u
JParMoM
JParmodel
JParREF
JPO
0 50 100 150 200 250 300 350 400 450−4
−3
−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. 19. Case ka=19, J‖ for ϕ = π/8 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM and PO.
0 50 100 150 200 250 300 350 400 450−2
−1
0
1
2x 10
−3 Approximations of MoM, Perpendicular component, k=19 Phi=0.3927
u
JPerpMoM
JPerpmodel
JPerpREF
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. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF
Fig. 20. Case ka=19, J⊥ for ϕ = π/8 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM.
19
0 50 100 150 200 250 300 350 400 450−6
−4
−2
0
2
4
6x 10
−3 Approximations of MoM, Parallel component, k=29 Phi=0.3927
u
JParMoM
JParmodel
JParREF
JPO
0 50 100 150 200 250 300 350 400 450−4
−2
0
2
4
6x 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. 21. Case ka=29, J‖ for ϕ = π/8 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM and PO.
0 50 100 150 200 250 300 350 400 450−2
−1.5
−1
−0.5
0
0.5
1
1.5x 10
−3 Approximations of MoM, Perpendicular component, k=29 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
−3 Error REF model to MoM, Error Model with Coefficient to MoM
u
Error Model+stand. dev. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF
Fig. 22. Case ka=29, J⊥ for ϕ = π/8 using RI=1,2,3
‖(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
−3 Approximations of MoM, Parallel component, k=19 Phi=0.7854
u
JParMoM
JParmodel
JParREF
JPO
0 50 100 150 200 250 300 350 400 450−6
−4
−2
0
2
4x 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. 23. Case ka=19, J‖ for ϕ = π/4 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM and PO.
0 50 100 150 200 250 300 350 400 450−2
−1
0
1
2x 10
−3 Approximations of MoM, Perpendicular component, k=19 Phi=0.7854
u
JPerpMoM
JPerpmodel
JPerpREF
0 50 100 150 200 250 300 350 400 450−3
−2
−1
0
1
2
3x 10
−3 Error REF model to MoM, Error Model with Coefficient to MoM
u
Error Model+stand. dev. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF
Fig. 24. Case ka=19, J⊥ for ϕ = π/4 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM.
21
0 50 100 150 200 250 300 350 400 450−6
−4
−2
0
2
4
6x 10
−3 Approximations of MoM, Parallel component, k=19 Phi=1.0472
u
JParMoM
JParmodel
JParREF
JPO
0 50 100 150 200 250 300 350 400 450−6
−4
−2
0
2
4x 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. 25. Case ka=19, J‖ for ϕ = π/3 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM and PO.
0 50 100 150 200 250 300 350 400 450−1.5
−1
−0.5
0
0.5
1
1.5x 10
−3 Approximations of MoM, Perpendicular component, k=19 Phi=1.0472
u
JPerpMoM
JPerpmodel
JPerpREF
0 50 100 150 200 250 300 350 400 450−2
−1
0
1
2x 10
−3 Error REF model to MoM, Error Model with Coefficient to MoM
u
Error Model+stand. dev. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF
Fig. 26. Case ka=19, J⊥ for ϕ = π/3 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM.
22
0 50 100 150 200 250 300 350 400 450−6
−4
−2
0
2
4
6x 10
−3 Approximations of MoM, Parallel component, k=29 Phi=1.0472
u
JParMoM
JParmodel
JParREF
JPO
0 50 100 150 200 250 300 350 400 450−5
0
5x 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. 27. Case ka=29, J‖ for ϕ = π/3 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM and PO.
0 50 100 150 200 250 300 350 400 450−1.5
−1
−0.5
0
0.5
1
1.5x 10
−3 Approximations of MoM, Perpendicular component, k=29 Phi=1.0472
u
JPerpMoM
JPerpmodel
JPerpREF
0 50 100 150 200 250 300 350 400 450−2
−1
0
1
2x 10
−3 Error REF model to MoM, Error Model with Coefficient to MoM
u
Error Model+stand. dev. −stand. dev.Error REF+stand. dev. REF−stand. dev. REF
Fig. 28. Case ka=29, J⊥ for ϕ = π/3 using RI=1,2,3
‖(k) and R
I=1,2,3
⊥ (k) compared to MoM.
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
θ [degree] (φ = 0o)
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
θ [degree] (φ = 0o)
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
θ [degree] (φ = 0o)
# 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
Royal Institute of Technology, KTH
Stockholm, Sweden
Email: [email protected]
Sandy Sefi
Department of Numerical Analysis and
Computer Science, NADA
Royal Institute of Technology, KTH
Stockholm, Sweden
Email: [email protected]
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
!"#$&%#&'()!*!+,.-/+#!0)12 10&3%(4()*$5!768#9%*!(!*!+:;< !"#!&%= #;>?5!@ A#5!10&%=+5BDC9+0#9 #!!,FEG*!:(*!0)H*!(I5!!B00()0#D%JK%=#L&NMPOPQNRGS#5%=#! 0(+,2TU)4V1V%= CF6W:!(YXMZOPQNRGS#!5[]\^\\U%#! 05!+(N;8%*!_(!*0'#!5U`aMPOZQZRGS4;G%=*&1+=&3%0,bOZ()#!cB!! 5!#d%=@C6WaB! e%=5!+%0"#f#468*!1J*Y()!)g1+ad100h%J #:%=#! ZB@0 #!(W%=!,jif1]#:%*!0#Y+!k&10e%=*f#D%=+(+1+%#%=#! Vlm=]Cn68o%=*2#!I#D%0()01V%=#7CDlm(!hg10"(Y%=*$K%a%*!"#pdBq+_r #D%=0)(+1+%= #F10"!%05s (5!=&%1]& C5!+100(+5t,WuZ76rVv+r%=*! (W (b#!%W0#!*4( #!10Pa()10+#!e1]ƚ#D%=#"7v0Z7\\MPOPQNRGS(!hg10+(0wP()%= 7vU(@005plx!y%=*02&U%x6W6z]Cp(0 3I#D%=+(+1+%:]C|68o%=*(A;V6~MPOPQNRGS(@()(B HE5!F%=*! (c6rY68 z!(H;!IVv+("f1+10+0&%= #t YB@!#!5! #!2v !+(0wv ( B! %xC!+!p10+(( #!!wNg10 #!9#!5yB! 1J&0:!)0!)1+0()(#!, 3c05!10210()%Ya()g1+Vlx]C[#D%0()01V%=#t Vq+#!(ovq+=&%#,
7\
!apbj"! e8!0)a!_!_&) !G!)0
ebj"! 0z80_d+! q=V= t D8$&GWe]&t¡pJ¢£! q=&+8 ) =U D z!+t¤¥G!_@00p¦x!Y0$&04c§!=e;)0f ) =z+!)!+=§! ¤©¨eo+!G! ª+0!)08!0)a!_ !@$&=Y G§!NH )=&= $hª4@D0¤
««
¬®¯°±'²³Z´Gµ!±'¶0·¸!±'¹¶]ºK»»=±+°¹e»µ!±'¼$±+½¾?¶0·¿ ¸!®À;°)·¿Áº4¾!Â@·½±a¶0½ ·¹±a»=·4»µ!±d¶0·¸!±Ãf´Gµ!±c°º]Än¶+·¸D»=°ÆÅÇ ¯»= ·¸Y¶0·¿"±0¹GÀ;°)·¿È°)±+É$±+¶+»=±+¾±+ʱ0¸D»=¹+Ã
¬®¯°±Ë³aÌx¸»=µ¹_±Vͺ¿"Â!½±I»=µ±¹)¶]º&»)»=±0°)¸!®H¹¯°)Àgº¶+±0¹dº&°±"®°·¯Âq±+¾ ¸²H· ÇpÎ ±+¶+»=¹+Ã:´Ï·U¹Â@±0±+¾¯!»µ!±°=º]ÄI»=°=º&¶0¸®'º½ ®·°) »=µ¿ÐºÂ°±P°·p¶0±0¹)¹ ¸!®'·p¶0¶+½¯!¹)·¸I»=±+¹)»Gº&°±N°=º¸·¸»µ!±0¹)±ZÀ;·¯!°· ÇpÎ ±+¶+»¹0ÃÑZ±+°±Pµ®µ!±+°·°¾!±+°_±+Òq±0¶+»¹0ÓÔ¹¯¶Jµº¹f¾!·¯ Ç ½±c°)±+É$±+¶+»·¸!¹+Óº°±c ¿Â@·°)»=º¸D»e¹)¸!¶+±I»µ!±0°)±"º°±c¿º¸DÄ·À»=µ±0¿/º&¸!¾F»µ!±¾!¹h»Jº¸!¶+± Ç ±V»xÕr±+±0¸n»µ!±_¹·¯!°)¶0±º¸!¾Y°±+¶0±+ ʱ+°Z ¹G°±+½kº&» ʱ+½ Ľkº°)®±Ã
ÖØ×
ÙÚÛÜÝÞZßàáGâ!ÞZã=â!Ý)Þ0ÞPä!åæJçKåÛÞ0èzé_Þ+êë"Þ+ã=ÝhìíîZå]ìåï!ðñPä!äòÚæ0å&ã=Ú êïIâ$åèWóqÞ+Þ0ï4Ü!è)Þ0ð:Ú ïAåaâ£ìpó!Ý)ÚðYôHê£ôYõé_á8öÈæ0åòæ+Ü!òkåKã=Úêït÷AñZïåïDã=Þ+ï!ï$ånÚ èaë"êÜ!ïDãÞ0ðêïsåøùUúPûZîZüGýFåÚÝ)æ0Ýå&þ3ãåï!ðãâ!ÞAÝåð!Úå&ã=Þ+ðÿ$Þ+òðsÚ èæ0êë"ä!ÜãÞ0ð8â!Úò ÞeãJåç&Þ0ïnÚ ïDã=êIåæ+æ0êÜ!ïDãNã=â!ÞaåÚ Ýæ+Ý=å&þ3ã0÷
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.