investigation of near-field contribution in sbr for...

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Investigation of Near-Field Contribution in SBR for Installed Antenna Performance

HARALD HULTIN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Investigation of Near-Field Contribution

in SBR for Installed Antenna Performance

by

Harald Hultin

June 2019

Master of Science Thesis

School of Electrical Engineering and Computer Science

KTH Royal Institute of Technology

SE-100 44 STOCKHOLM

Undersokning av Narfaltsbidrag

vid SBR-berakning av

Installerad Antennprestandaav

Harald Hultin

Juni 2019

Examensarbete

Skolan for Elektroteknik och Datavetenskap

KTH Kungliga Tekniska Hogskolan

SE-100 44 STOCKHOLM

Master of Science Thesis

Investigation of Near-Field Contribution in

SBR for Installed Antenna Performance

Harald Hultin

Examiner: Supervisor:

Lars Jonsson Henrik Frid

Commissioner: Contact person:

Saab Henrik Frid

Abstract

To investigate near-field contributions for installed antennas, an in-house code is

written to incorporate near-field terms in Shooting and Bouncing Rays (SBR). SBR

is a method where rays are launched toward an object and scatter using Geomet-

rical Optics (GO). These rays induce currents on the object, from which the total

scattered field can be found.

To gauge the effect of near-field terms, the in-house code can be set to exclude

near-field terms. Due to this characteristic, the method is named SBR Including

or Excluding Near-field Terms (SIENT). The SIENT implementation is thoroughly

described. To make SIENT more flexible, the code works with triangulated meshes

of objects. Antennas are represented as near-field sources, allowing complex an-

tennas to be represented by simple surface currents. Further, some implemented

optimizations of SIENT are shown.

To test the implemented method, SIENT is compared to a reference solution and

comparable commercial SBR solvers. It is shown that SIENT compares well to the

commercial options. Further, it is shown that the inclusion of near-field terms acts

as a small correction to the far-field of the installed antenna.

Keywords: Shooting and Bouncing Rays, Geometrical Optics (GO), Physi-

cal Optics (PO), antenna modeling, computational electromagnetics (CEM), CST,

dipole, ray tubes

i

Examensarbete

Undersokning av Narfaltsbidrag

vid SBR-berakning av

Installerad Antennprestanda

Harald Hultin

Examinator: Handledare:

Lars Jonsson Henrik Frid

Uppdragsgivare: Kontaktperson:

Saab Henrik Frid

Sammanfattning

For att undersoka narfaltsbidrag for installerade antenner, har en kod skrivits for

att ta med narfaltstermer i Shooting Bouncing Rays (SBR). SBR ar en metod dar

stralar (”rays”) skjuts mot ett object och sprids via Geometrisk Optik (GO). Dessa

stralar inducerar strommar pa objectet, fran vilka det totala sprida faltet kan hittas.

For att undersoka bidraget fran narfaltstermer, sa kan koden exkludera dessa. Pa

grund av denna karaktar, kallas koden SBR Including or Excluding Near-field Terms

(SIENT). Implementationen av SIENT beskrivs utforligt. For att gora SIENT mer

flexibel, arbetar SIENT med triangulerade nat av objekt. Antenner representeras av

narfaltskallor, vilket later komplexa antenner representeras med enkla yt-strommar.

Implementerade optimeringar av SIENT visas ocksa.

For att testa den implementerade metoden, jamfors SIENT med en referenslosning

och jamforbara kommerciella SBR-losare. Det visas att SIENT overensstammer bra

med kommerciella alternativ. Det visas ocksa att narfaltstermer agerar som en

mindre korrektion till fjarrfaltet av den installerade antennen.

Nyckelord: Geometrisk Optik, Fysikalisk Optik, antennmodellering, elektro-

magnetism, CST, dipol

ii

Acknowledgments

I want to start by thanking my family, for their immense support in everything I

do. Second, I thank my supervisor Henrik Frid for not only supporting me with the

thesis, but also for introducing me to SAAB. For this, I also wish to thank Johan

Malmstrom, who have acted as a second supervisor.

I am also grateful to Lars Jonsson, my examiner, for introducing me to electro-

magnetics three years ago, and for his feedback.

Finally, big thanks goes to Saab and the section of Microwave & Antennas for

giving me the opportunity to make this thesis.

Harald Hultin

iii

Contents

Abstract i

Sammanfattning ii

Acknowledgment iii

Nomenclature vi

List of Variable Definitions vii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Purpose & Research Question . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Ethical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 5

2.1 Shooting and Bouncing Rays . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Huygens’ Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Field Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Full Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.2 The Far Field Approximation . . . . . . . . . . . . . . . . . . 8

2.3.3 Discretization of the Source . . . . . . . . . . . . . . . . . . . 9

2.4 Far Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Methodology 13

3.1 Triangulation of Surface . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Detecting Bounces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Rays and Ray Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.2 Ray Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.3 Surface Illumination . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.4 Partially Intersecting Ray Tubes . . . . . . . . . . . . . . . . . 16

3.3.5 Current Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.6 Divergence Factor . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.7 Detecting Caustics . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.8 Saving Ray Tubes . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.9 Enforcing Huygens’ Surface Boundary Condition . . . . . . . . 21

3.4 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Huygens’ Surface . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Surface Current Density . . . . . . . . . . . . . . . . . . . . . 22

iv

3.5 Simplifying the Huygens’ Surface . . . . . . . . . . . . . . . . . . . . 22

3.6 Far Field Phase Reference . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Space Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.8 Ray Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.9 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Results & Analysis 27

4.1 Verification: Far Field from Huygens’ Surface . . . . . . . . . . . . . 27

4.2 Verification: FIT Simulation . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Simplified Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Test Case: Cut Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4.1 Reference Solution from CST . . . . . . . . . . . . . . . . . . 31

4.4.2 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.3 Far Field Comparison . . . . . . . . . . . . . . . . . . . . . . 36

4.4.4 Surface Currents . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Space Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Test Case: Corner Reflector . . . . . . . . . . . . . . . . . . . . . . . 40




4.7 Test Case: Simplified SAAB 37 Viggen . . . . . . . . . . . . . . . . . 48




4.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 Other Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Discussion & Conclusion 59

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Recommendations for Future Work 61

References 62

Appendix 65

A Divergence Factor from Interpolated Curvature 65

B Other Simulation Example 66

v

Nomenclature

CAD Computer Aided Design

FDTD Finite-Difference Time-Domain

FEM Finite Element Method

FF Far-Field

FIT Finite Integration Technique

GO Geometrical optics

IH In-House

NF Near-Field

PEC Perfect Electrical Conductor

PFI short for CST option ”perform full integration close to structure”

PO Physical Optics

RAM Random Access Memory

RMS Root Mean Square

RMSD Root Mean Square Deviation

SBR Shooting and Bouncing Rays

SENT SBR Excluding Near-field Terms

SIENT SBR Including or Excluding Near-field Terms

SINT SBR Including Near-field Terms

vi

List of Variable Definitions

k direction of propagation

n normal vector

r unit vector in r-direction

λ wavelength

E electric field strength

F far-field amplitude

H magnetic field strength

J electric current density

Jm magnetic current density

m magnetic dipole moment

p electric dipole moment

r vector pointing to observation point

r′ vector pointing to source point

∇ nabla operator, x∂/∂x+ y∂/∂y + z∂/∂z

θ Angle in spherical coordinate system, defined as θ = arccos(z/(x2 + y2 + z2))

φ Angle in spherical coordinate system, defined as φ = arctan(y/x)

ejωt time convention in this thesis

G− Green’s function for an outwards propagating wave

k wavenumber

vii

Introduction

1 Introduction

1.1 Background

When designing or buying antennas, the performance is usually defined for the

antenna in isolation. However, once an antenna is placed on a platform, the perfor-

mance changes. Because of this, investigations of Installed Antenna Performance are

required. One common case is to find the radiation pattern of an installed antenna

[1], [2]. This could be the case for e.g. an airplane communicating with something

on the ground. In these cases, it is common to use a blade antenna, which is the

antenna used throughout this thesis.

Another example is direction finding, where correct phase is important [3]. But

installing the antenna on a platform will affect the phase and thus the direction

estimation. Apart from performance in the far-field, antenna isolation is also a

problem in installed antenna performance [4]. For small platforms, these cases can

be simulated using full-wave simulations. Full-wave simulations are an important

tool in antenna design, since early prototypes can be replaced by a simulated model.

However, they have their limitations. As an example, take the Finite Integration

Technique (FIT) or the Finite-Difference Time-Domain (FDTD) methods. For

decent accuracy far from exited sources, mesh elements must be on a scale of a

fraction of the wavelength [5]. Close to sources and boundaries, this can increase to

20 or more mesh cells per wavelength.

On electrically large platforms, such mesh densities will result in a large amounts

of cells which must be stored in memory. As an example, take a 15 m long fighter

aircraft with a 8 GHz radar. Say the region simulated is 20 m× 20 m× 10 m. As-

suming each cell is a cube with a side of 0.2λ (7.4 mm) on average, which is a

relatively sparse mesh, this would require 1010 cells. Further, assume both electric

and magnetic field are stored as 32 bit float values in each cell, totaling 64 bits or

8 bytes per cell. The total amount of memory required is thus approximated to

80 GB. Moreover, it would take significant computational time to find the solution

as the solver iterates over these cells.

As an alternative to full wave simulations, there are high frequency approxima-

tions, such as Physical Optics (PO) and Geometrical Optics (GO) [6]. Further, such

methods can be incorporated into Shooting and Bouncing Rays (SBR) [7]. Here,

GO rays are launched from a source and scatter on the surface of the object. Assum-

ing the body is a Perfect Electrical Conductor (PEC), these scattered fields induce

currents on the surface. From these currents, the scattered field can be computed

by PO, assuming the surface curvature is large compared to the wavelength [6], [8].

To compute fields in between rays in this thesis, ray tubes are tracked similar

to [9]. The advantage of using SBR compared to full wave simulations is the ability

to efficiently simulate multiple reflections on electrically large, complex geometries.

Another method for multi-bounce geometries is Iterative Physical Optics (IPO) [10].

1

Introduction 1.1

Further, there exists geometries where SBR alone does not take critical effects in

consideration. One such effect is edge diffraction, which be resolved with e.g. Phys-

ical Theory of Diffraction (PTD) [11]. Another is creeping waves, which can be

solved with e.g. Uniform Theory of Diffraction (UTD) [2].

Originally, SBR was a method for determining Radar Cross Section (RCS) [7].

The excitation is in that case a radar far away, modeled as an incoming plane wave.

However, when investigating installed antennas this model is not applicable. One

way of adapting the model is to represent it by a far-field source. A far-field source

is a point source with the same far-field as the antenna to be installed. The issue

with this model is the lack of near-field contributions. Taking the example of the

radar on a fighter from above, assume the radars largest dimension D to be 0.25 m.

The Fraunhofer distance,

df =2D2

λ, (1)

gives a measure of how far the near-field has a substantial effect. In this case,

df = 3.4 m. As parts of the aircraft are inside this distance, higher order, near-field

terms must also be considered.

To an extent, near-field effects can be taken in to account by a near-field source.

A near-field source is a bounding box surrounding the antenna on which the complete

fields are given by the equivalence principle [12]. However, if the fields from this

source are evaluated using the far-field approximation [12], higher order terms will

be neglected.

Commercial SBR solvers for installed antenna performance are available, where

CST Microwave Studios (CST MWS) and HFSS SBR+ (previously Savant) are two

examples. The exact implementation of these SBR solvers is not described in the

literature, but both claim to take near-field effects into account. Describing this

is one novel aspect of this thesis, as the whole process of implementing SBR is

described. The literature found is usually papers of a few pages, describing parts

of, or improvements to, SBR. CST has an option named perform full integration

close to structure or PFI for short. This option includes radial parts of the field

impinging on the scatterer, if the source is closer than λ to the target [13]. Savant

always includes higher-order terms by default from version 19.1 [14].

The goal of this thesis is to apply a SBR method to solve a scenario like the fighter

airplane described above. Using this method, near-field contribution is investigated.

Functionality will be implemented and verified gradually. The near-field source is

imported from CST MWS.

The work on this thesis has consisted of many, smaller challenges. Each of

these challenges are described in the subsections of Section 2: Theory and Section

3: Methodology. Further, the subsections are mostly in the chronological order in

which their challenge occurred. The first challenge was to understand what SBR is.

Following this, the challenge was to import an antenna as some sort of source and

2

Introduction 1.2

describing the near-field of this source. With a source in the form of a Huygens’

surface and the complete field using the dipole discretization, the source was placed

on a platform. Section 3.3 is the most important part of the methodology. Here,

the challenge was in ray tracing and ”visibility”. While a ray in itself is simple

(defined by a starting point and direction), the whole system of ray tracing a field

becomes complex. While the method is the larger part of this thesis, the numerical

investigations in Section 4: Results & Analysis are necessary to prove that the

method works. As above, the subsections are chronological.

1.2 Purpose & Research Question

The first priority is to get a working SBR code that can take near-field into ac-

count. A working SBR code will track rays from a source and bounce them on a

body. At each bounce, the surface current is saved. From this surface current, the

corresponding far-field is determined. If the implementation is inaccurate, that is

also seen as an result. This has been broken down to two main research questions,

stated below.

As the implementation is compared to commercial software, it may give insight

to if and how such software take near-field in to consideration. This understanding

is not a primary goal and is rather included as a secondary research question.

Main Research Questions:

1. How can the SBR method be implemented for installed antenna performance?

2. What is the error for installed antenna performance SBR when neglecting

near-field contributions?

Secondary Research Question:

1. Does commercial SBR software take near-field contribution into account? If

so, how?

1.3 Limitations

In Computational ElectroMagnetics (CEM), SBR is a large field of study. This field

can not be covered completely in a master thesis. The following limitations will be

set to give the thesis a clear scope:

1. Importing CAD models is not a requirement. The code will accept a triangu-

lated surface, i.e. a triangular surface mesh. These meshes must have correct

normals. Meshes shall be supplied from another software.

2. Taking edge diffraction into account is not a requirement.

3

Introduction 1.4

3. Concave surface reflections are difficult to implement as the curvature infor-

mation of the surface is necessary to get a correct phase [15]. Concave parts

of the surface have a focal point where the reflected rays may intersect. If this

occurs, the fields phase shift as described in [15]. If these phase shifts prove

too difficult to implement, they may be omitted.

4. The code will be tested against reference solutions.

5. The code will work for a source box, which is simulated to be on an infinite

ground plane. Any errors due to in-homogeneous surroundings are ignored.

The effect of this is studied in [16].

6. Basis functions will be constant. Accuracy may be improved by implementing

RWG [17] or rooftop [18] basis functions.

1.4 Ethical Aspects

The thesis work in itself is quite harmless. It is carried out by simulations and

calculations on a computer, not having much of an effect on the outside world

except for some energy consumption.

However, the results are very versatile as they concern a method of installed

antenna performance evaluation. This evaluation is an important part in the defense

industry, which is why the thesis is carried out at SAAB. This means that the results

can contribute to the development of defense materials.

One potential customer is Forsvarsmakten (the Swedish armed forces), but the

defense material may be exported to other nations. For this export to comply with

any ethical regulations, the author trust the Swedish government through Forsvarets

Materielverk (FMV) and the Inspectorate of Strategic Products (ISP), which are

responsible for regulating and controlling this export [19] [20].

While the connection to the defense industry is clear, the results of this thesis

can be used in other sectors as well. For example, installed antenna performance is

important on civilian airplanes, cars, trucks and antenna masts.

Further reading on military funded research and ethics can be found in [21].

4

Theory

2 Theory

2.1 Shooting and Bouncing Rays

Shooting and Bouncing Rays (SBR) is a high frequency approximation in CEM.

The term was first coined in [7]. Using GO, rays are shot toward an object. If a

ray intersects with the body, it bounces following PEC boundary conditions. This

bounce creates a new ray, which is tracked in the same manner. Once all bounces

(or a set amount of bounces) have occurred, the resulting excitation is evaluated.

In this thesis, the excitation is the induced surface current. However, it can also be

the fields of an aperture [7] for example.

In [7], the fields of an aperture were evaluated to find the Radar Cross Section

(RCS) od a cavity. It has since also been used for installed antennas [1]. SBR is

based on geometrical and physical optics which are described in e.g. [6] and [8].

2.2 Huygens’ Surface

In this work, a method to represent an antenna shown in [16] will be used. The

antenna is bounded by a box. Using Huygens’ principle [12], the equivalent electric

and magnetic surface currents J and Jm are found on this box. These currents

create a Huygens’ surface, which is used as the source for SIENT. One advantage of

using a Huygens’ surface to represent the antenna, is that more complex geometries

like Figure 9 can be represented by simple surface currents. A downside is the error

this causes as described in [16].

To find the currents, the antenna in the bounding box is simulated using the

FIT solver from CST on an infinite ground plane1. Thus, we consider the fields E

and H to be known on the surface. Inside the box (denoted by index 2) there are no

fields, i.e. E2 = 0 and H2 = 0. Outside (denoted by index 1), E1 = E and H1 = H.

With the normal n pointing out of the box, the boundary conditions [6] are

n× (E1 − E2) = −Jm, n× (H1 −H2) = J. (2)

Thus, with the above assumptions E2 = 0 and H2 = 0,

n× E = −Jm, n×H = J. (3)

Note that different methods of handling the boundary condition at the PEC

surface with a Huygens’ surface exist. These are described in Figure 1.

1A method for curved ground planes exists in Savant (now HFSS SBR+).

5

Theory 2.3

Huygens’surface

Figure 1: Different placements for a Huygens’ surface on a ground plane. Savantuses the dashed line, where the Huygens’ surface is placed a small ε above the groundplane [14]. Another method is to place it on the ground plane, giving no magneticcurrent Jm on the bottom. In SIENT and CST the dotted line is used, where thebox is placed such that the bottom face is a small ε inside the ground plane, givingno currents on the bottom face.

2.3 Field Integrals

2.3.1 Full Fields

We name the Huygens’ surface S. From the surface currents on S, we want to calcu-

late the fields everywhere in space. Consulting [6] and [8], taking normal direction

into account, fields excited by the surface can be written as

− j ηk∇×

[∇×

∫∫S

G−(r, r′, k)(n×H(r′))dS ′]

+∇×∫∫

S

G−(r, r′, k)(n× E(r′))dS ′ =

E(r), r outside S

0, r inside S(4)

and

j1

kη∇×

[∇×

∫∫S

G−(r, r′, k)(n× E(r′))dS ′]

+∇×∫∫

S

G−(r, r′, k)(n×H(r′))dS ′ =

H(r), r outside S

0, r inside S.(5)

Here, η is the wave impedance and G−(r, r′, k) is the Green’s function for an

outwards propagating wave [6]:

G−(r, r′, k) =e−jk|r−r

′|

4π|r− r′|. (6)

Inserting the boundary conditions (3) in (4) and (5) gives

6

Theory 2.3

− j ηk∇×

[∇×

∫∫S

G−(r, r′, k)J(r′)dS ′]

−∇×∫∫

S

G−(r, r′, k)Jm(r′)dS ′ =

E(r), r outside S

0, r inside S(7)

and

− j 1

kη∇×

[∇×

∫∫S

G−(r, r′, k)Jm(r′)dS ′]

+∇×∫∫

S

G−(r, r′, k)J(r′)dS ′ =

H(r), r outside S

0, r inside S.(8)

Consider the H-field from an electric surface current, i.e. Jm = 0. Using the

identity

∇× (ab) = ∇a× b + a∇× b, (9)

the first term in (8) vanishes, giving

∫∫S

[∇G−(r, r′, k)× J(r′) +G−(r, r′, k)∇× J(r′)

]dS ′ =

H(r), r outside S

0, r inside S.

(10)

∇ can be moved inside the integral for smooth enough sources [6]. The absolute

position of the source does not vary with the position of the observer, i.e. J(r′) does

not depend on r. Thus ∇× J(r′) = 0, simplifying (10) to

∫∫S

∇G−(r, r′, k)× J(r′)dS ′ =

H(r), r outside S

0, r inside S.(11)

The gradient of the Green’s function is known to be

∇G−(r, r′, k) = − r− r′

|r− r′|

(1

|r− r′|+ jk

)G−(r, r′, k). (12)

Assuming r is outside S,2 the H-field becomes

−∫∫

S

(1

|r− r′|+ jk

)G−(r, r′, k)

r− r′

|r− r′|× J(r′)dS ′ = H(r). (13)

2I.e. we say implicitly that H(r) = 0 inside S.

7

Theory 2.3

2.3.2 The Far Field Approximation

Now, use the far-field approximation [8]. Far away from the source, it holds that

k|r− r′| � 1,r− r′

|r− r′|≈ r and r � r′. (14)

This implies that k ≈ r. Rewrite |r−r′| as√

(r− r′) · (r− r′) =√r2 + r′2 − 2rr′ cos θr,

where θr is the angle between r and r′.

If ε is small, approximate

(1 + ε)p ≈ 1 + εp. (15)

This gives

√r2 + r′2 − 2rr′ cos θr = r

√1 +

r′2

r2− 2

r′

rcos θr ≈ r

(1− r′

rcos θr +

1

2

r′2

r2

). (16)

Taking only the leading terms gives

|r− r′| ≈ r − r′ cos θr. (17)

Similarly,

1

|r− r′|=

1

r

(1 +

r′2

r2− 2

r′

rcos θr

)−1/2≈ 1

r

(1 +

r′

rcos θr −

1

2

r′2

r2

)≈ 1

r. (18)

Now, (13) can be approximated as

H(r) = −jk∫∫

S

e−jk(r−r′ cos θr)

4πrr× J(r′)dS ′, (19)

which is the far-field expression in [6] and [8]. This can be rewritten as

H(r) = −jk∫∫

S

e−jk(r−r·r′)

4πrr× J(r′)dS ′. (20)

In the far-field region E can be found as

E(r) = −η(k×H(r)), (21)

where k is the propagation direction. Analogously, the fields can be found for an

electrical current density via (7). However, in this thesis the far-field approximation

can not be made as parts of the platform of the installed antenna may violate the

assumptions made in (14). This has two consequences,

1. more terms than just leading order must be considered,

8

Theory 2.3

2. Equations (20) and (21) are not valid.

Hence, (20) and (21) will not be used. Instead, (7) and (8) will be evaluated

numerically without the far-field approximation.

2.3.3 Discretization of the Source

To discretize the problem, S is approximated by small, planar unit cells. The surface

currents on these cells is assumed to be constant over the cells area.

Assuming each cell is sufficiently small, it can be modeled as a dipole. This is

a method used for near-field analysis [22], [23] and installed antenna performance

[24]. The method also seems similar to the method described in the Savant (now

HFSS SBR+) documentation [14].

From [12], it is known that the electric dipole moment p is

p =1

jω

∫V

J(r′)dV ′ (22)

For the case with the planar unit cell number i, this can be rewritten as

pi =1

jω

∫∫Si

JidS′ (23)

We use a constant basis functions to model the currents. Hence Ji is constant

within each mesh element. Moving Ji out of the integral gives

pi =1

jωJiAi, (24)

where Ai is the area of element i.

In [6], magnetic current density is defined as Jm = jωµ0M. [12] gives the mag-

netic dipole moment as

m =

∫V

MdV ′. (25)

Analogous to the electric dipole, the magnetic dipole moment becomes

mi =1

jωµ0

Jm,iAi. (26)

For oscillating dipoles, the complete fields are known [12]. For an electric dipole,

the fields are

H(r) =ck2

4π(k× p)

e−jkr

r

(1 +

1

jkr

)(27)

and

E(r) =1

4πε0

e−jkr

r

{k2(k× p)× k +

[3k(k · p)− p

]( 1

r2+jk

r

)}. (28)

9

Theory 2.4

Note that this is an expression for a dipole in origo, i.e. we replace r → |r− r′|.Also note that Equations (27) and (28) are the full fields, without any far-field

approximation. For the magnetic dipole,

H(r) =1

4π

e−jkr

r

{k2(k×m)× k +

[3k(k ·m)−m

]( 1

r2+jk

r

)}(29)

and

E(r) = −ηk2

4π(k×m)

e−jkr

r

(1 +

1

jkr

). (30)

Now, using Equations (24), (26), (27), (28), (29) and (30), the field excited by J

and Jm can be approximated as

Htot(r) ≈∑i

1

4π

Aijω

e−jk|r−r′i|

|r− r′i|

(ck2(k× Ji)

(1 +

1

jk|r− r′i|

)+

1

µ0

{k2(k× Jm,i)× k +

[3k(k · Jm,i)− Jm,i

]( 1

|r− r′i|2+

jk

|r− r′i|

)})(31)

and

Etot(r) ≈∑i

1

4π

Aijω

e−jk|r−r′i|

|r− r′i|

(− ηk2

µo(k× Jm,i)

(1 +

1

jk|r− r′i|

)+

1

ε0

{k2(k× Ji)× k +

[3k(k · Ji)− Ji

]( 1

|r− r′i|2+

jk

|r− r′i|

)}). (32)

Here, r′i is the vector pointing to the center of cell i. Equations (31) and (32) can be

read as the summation of fields from several electric and magnetic dipoles positioned

at r′i, using Equations (24) and (26) to describe the dipole moments.

One can also observe that taking the leading order term for electric current, i.e.

that of order |r− r′i|−1, gives

Htot(r) ≈∑i

1

4π

Aijω

e−jk|r−r′i|

|r− r′i|(ck2(k× Ji)) (33)

which is a discretized form of Equation (20). Thus, we identify the leading order

terms of Equations (31) and (32) as far-field terms. By including higher order terms,

we say that we include the near-field terms.

2.4 Far Field

With a known surface current on discrete elements, we want to calculate the far-field.

Using the same discretization as earlier, rewrite (33) using the far-field approxima-

10

Theory 2.4

tion, giving

Hf (r) =ck2

4π(k× p)

e−jk(r−r·r′)

r. (34)

Compared to Equation (27), r′ is included as the dipole position. However, using

the far-field approximation r′ only affects the phase. The electric field is found by

using Equation (21). For the magnetic dipole,

Ef (r) = −ηk2

4π(k×m)

e−jk(r−r·r′)

r. (35)

Taking the dipole moment expressions from Equations (24) and (26), the total

electric field from all elements is then

Etot,f (r) = −∑i

ηk2

4π

Aijω

(ck× (k× Ji) +

1

µ0

k× Jm,i

)e−jk(r−r·r

′i)

r. (36)

[4] defines the far-field amplitude F(r) as

F(r) = limr→∞

rejkr

V0E(r), (37)

where V0 is a normalization factor with unit volt. Define the radiation intensity U

as

U(r) = limr→∞

r2

2η0|E(r)|2. (38)

Realized gain G is then found by

G(r) = 4πU(r)

Ps, (39)

where Ps is the stimulated power of the antenna. Ps is kept as its default value in

CST, 0.5 W [13]. Expressing G in terms of F gives

G(r) = 4π|V0|2|F(r)|2

2η0Ps. (40)

By setting

V0 =

√η0Ps2π

, (41)

the realized gain G is found as

G(r) = |F(r)|2. (42)

11

Theory 2.4

Then the far-field amplitude becomes

Ftot(r) = −∑i

ηk2

4πV0

Aijω

(cr× (r× Ji) +

1

µ0

r× Jm,i

)ejkr·r

′i . (43)

12

Methodology

3 Methodology

3.1 Triangulation of Surface

To triangulate, or mesh, the surface Gmsh is used. Gmsh is an open source meshing

tool intended for FEM and is available at [25].

For the models used in this thesis, the maximum allowed size3 of a cell is set to

0.1λ and the minimum to 0.01λ if nothing else is stated.

For simulation efficiency, the triangulation is divided into several partitions. This

is described in Section 3.7.

3.2 Detecting Bounces

Detecting if a ray intersects the surface can be formulated as follows:

• Take a triangular mesh cell.

• This mesh cell has three vertices, P1,P2,P3.

• Does a ray from origin O, in direction D, intersect the triangle spanned by

P1,P2,P3?

An efficient method to test the intersection criteria above is described by Moller in

[26]. This has been implemented for Matlab by [27]. For a triangulated surface, test

all triangles for intersection of a ray. If multiple intersections exist, select the closest

one.

Determining if a triangle can be intersected before testing can have a big impact

on computational time. Therefore, two different tests are performed before testing

triangle intersection.

1. All triangles are divided into several partitions. The algorithm tests which

partitions that are intersected by the ray, then test the triangles in these.

This is described in depth in Section 3.7.

2. The element must face O, i.e.

n ·D < 0, (44)

where n is the face normal going out of the body. If this is not fulfilled, the

face is facing away from O.

Further efficiency can be achieved by e.g. storing triangles in hierarchical trees

as in [28].

3Maximum side length of triangle.

13

Methodology 3.3

3.3 Rays and Ray Tubes

3.3.1 Rays

From each equivalent dipole on the surface, rays are launched in all directions. In

other words, iterate with dipole positions as O. How dense the rays are launched

is configurable, see Section 3.3.8. As a ray bounces on the object, the following is

saved:

• the equivalent dipole location r′,

• the original direction,

• reflected directions,

• distance between intersections.

Bounces are modeled as a plane wave impinging on a planar PEC surface. [9]

gives

kR = kI − 2(kI · n)n. (45)

where kR and kI are the directions of propagation for the reflected and incoming

plane wave.

Any intersection can now be found. For the first intersection, take the first

distance in the original direction. From r′, this vector points to the first intersection.

For the second intersection, take the second distance in the first reflected direction.

This vector points to the second intersection from the first intersection.

3.3.2 Ray Tubes

To track the field in-between rays, let a group of three adjacent rays create a ray

tube, as described by [9]. This is illustrated in Figure 2. The ray tube size is

determined as described in Section 3.2.

3.3.3 Surface Illumination

To find illuminated mesh cells for each dipole, i.e. cells in the ray tube, a recursive

algorithm is used. In order to avoid elements being excited by two adjacent ray

tubes, the center of the mesh cell must be inside the tube. An example of finding

illuminated elements is illustrated in Figure 3.

First, find where the three rays spanning the tube intersect the surface. Then

let the three intersect points of the ray tube rays span a triangle T . Lastly, take

a mesh cell tn. If a ray from the origin to the center of tn intersects with T , tn is

illuminated.

As a preprocessing step, a built-in Matlab function is used to make a list of all

neighboring elements of all cells. Each element on the body is assigned a tri-state

14

Methodology 3.3

Figure 2: Illustration off a ray tube (blue) spanned by three rays (red).

variable. This variable has the values of illuminated, non-illuminated or un-tested.

The basic function of the algorithm is as follows:

1. Pick an element

2. if illuminated:

(a) set element to illuminated

(b) reset search depth

(c) run algorithm on neighbors that are un-tested

3. else:

(a) set element to non-illuminated

(b) decrease search depth by 1

(c) if search depth = 0, return

(d) else, run algorithm on elements that are un-tested.

The first element tested is the one intersected by the mean ray. Here, the mean

ray is an average of the three rays spanning the tube, defined in [9]. Search depth

sets how long the algorithm searches. Take Figure 3c as an example. The four

bottom illuminated cells are connected. These cells can be found with a search

depth of 1, assuming one is the first element tested. On all adjacent cells ta, search

depth is set to 0 and the search stops. If the search depth is set to 2, all triangles

adjacent to ta are searched as well, which includes the last illuminated cell.

If the mean ray misses the surface, the search is initiated in the three hit points.

15

Methodology 3.3

(a) Three rays intersect thetriangulated surface.

(b) The three points spana triangular ray tube, cre-ating the triangle T on thesurface.

(c) Triangles with a centerinside the tube are illumi-nated.

Figure 3: Demonstration of illuminated triangles.

3.3.4 Partially Intersecting Ray Tubes

A ray tube may partially intersect the body, i.e. one or two of the three rays

spanning the tube intersect the body. Due to this, the mean ray from [9] can not

be calculated.

To find illuminated surface elements, the mean ray is replaced by a substitute

mean ray. If only one ray intersects the surface, this replaces the mean ray. If two

rays intersect the surface, the mean of these rays replace the mean ray of the ray

tube.

In order to span the ray tube in Figure 3, substitute points are created for rays

that miss the body. Take the length lm of substitute mean ray. For each missed ray

with a direction rr, the substitute point is given by the vector lmrr.

Say the rays emanate from origo. Two rays, R1 and R2 intersect the scatterer.

The third ray in direction r3 misses. Then the substitute mean ray is

M =1

2(R1 + R2) (46)

and the substitute ray becomes

R3 =1

2|R1 + R2|r3. (47)

In the current implementation, ray tubes are not tracked after a partial intersect.

As these ray tubes likely illuminate an edge, their detection could be a step in a

future implementation of edge diffraction. However, this detection does not take in

to consideration how acute the edge is, which is required in e.g. PTD.

3.3.5 Current Evaluation

For all triangles illuminated by the ray tube, we wish to find the induced current on

the body JB using JB = 2n ×H. This current is saved as three complex variables

for each face of the platform. These three variables represent the surface current

16

Methodology 3.3

density in x-, y- and z direction.

In this thesis, all bodies are assumed to be PEC. Consequently, no magnetic

current is induced on the body, JB,m = 0. Thus, tracking the H-field is sufficient.

Substituting

|r− r′| → R. (48)

H is found from the source similar to (31),

H(R, k) =1

4π

A

jω

e−jkR

R

(ck2(k× J)

(1 +

1

jkR

)+

1

µ0

{k2(k× Jm)× k +

[3k(k · Jm)− Jm

]( 1

R2+jk

R

)})(49)

Here, the full expressions are used when calculating H and JB from J and Jm.

This is what we denote including near-field terms. If the higher order terms are omit-

ted, the same far-field approximation as in [9] is used. When the field is evaluated,

R should be the total mean distance from the source. For subsequent reflections, the

total distance is the sum of all previous mean ray distances and the distance between

the current intersection and the last Mi. For example, the distance of R1 = |R01|in Figure 4 is used for the first intersection. For the second intersection, use the

distance R2 = |R01|+ |R12|.

Figure 4: Example of ray intersecting with and reflecting off a surface

To approximate how the rays traveled before the reflection, the mean ray method

from [9] is used. For the incoming ray tube, find the mean intersection Mi. Then,

assume all reflections emanate from Mi, with the reflected coefficients aH, bH and

cH. Illuminated elements also trace back to this mean point.

17

Methodology 3.3

Introducing the coefficients aH, bH and cH, rewrite Equation (49) as

H(r) = DF1

4π

A

jωe−jkR

(aH

(1 +

1

jkR

)+

{bH + cH

(1

R2+jk

R

)}). (50)

DF , the divergence factor, is explained in Section 3.3.6. For a spherical wave,

DF = 1/R. Reflections are modeled as a plane wave impinging on a planar, PEC

surface. This model assumes that the curvature of the surface is smaller than the

wavelength. With subscript I as incoming and subscript R as reflected, [9] gives

HR = HI − 2(HI · n)n (51)

As the coefficients aH, bH and cH describe the fields by a linear combination,

the boundary conditions in (51) can operate on aH, bH and cH instead. At the first

intersection, aH, bH and cH can be found by comparing Equations (49) and (50).

For subsequent intersections, reflections must be considered.

3.3.6 Divergence Factor

The divergence factor DF describes how a ray tube ”opening angle” changes when

reflected on a curved surface. A first approach for implementing the divergence

factor was to interpolate the surface similar to [29] and from that get the curvature

and divergence factor as in [7] and [15]. However, the implementation of this was not

accurate enough when using ray tubes as T became large. This resulted in erroneous

surface currents, several magnitudes higher than adjacent ones. The implementation

is described in Appendix A.

Instead, differential ray tubes are tracked similar to [9]. If all rays of the tube

intersect the surface, save the area of the triangle T spanned by the points D1,

E1 and G1 defined in [9] and illustrated in Figure 5. If the reflection intersects

an element at point P, the equiphase plane cutting this point is at a distance of

(P −M1) · Rm1 [9]. Name the reflected directions at D1, E1 and G1 as RD1, RE1

and RG1.

Now the equiphase triangle T2 at P is approximately spanned by

D2 = D1 + ((P−M1) · Rm1)RD1, (52)

E2 = E1 + ((P−M1) · Rm1)RE1, (53)

G2 = G1 + ((P−M1) · Rm1)RG1. (54)

These points are also shown in Figure 5. Let A1 denote the area of T1 and A2 the

18

Methodology 3.3

O

E,E1

D1

D

G1

G2D2

E2

RD1

Rm1

RE1

RG1

Figure 5: Definition of points along the ray tube.

area of T2. As in [9], the Divergence Factor DF1,2 between these is approximated by

DF1,2 =

√A1

A2

. (55)

Further, save the divergence factor DFm for each mean point. Note that all

triangles for each illumination by the ray tube has a separate DF , DFm is one of

these. At the first intersection, DF1 = 1/R. For intersection n, the total divergence

factor is given by

DF = DFm1 DF

m1,2DF

m2,3 . . . DF

mn−2,n−1DFn−1,n. (56)

As mentioned in [7], this method alone may miss phase shifts due to caustics.

Thus a method for detecting caustics is proposed in the following section.

3.3.7 Detecting Caustics

If the rays spanning a ray tube cross, the crossing point is a caustic. These caustics

introduce effects such as phase shifts to the field [15]. To detect caustics, one can

consider how the surface area spanned by the ray tube varies. It is shown in [15]

that:

1. ray tubes have a zero area in focal points,

2. the ray tube surface area varies as a quadratic equation.

Say the surface area Amn between intersection m and n = m+ 1 is described by

the equation

Amn = c1x2 + c2x+ c3, (57)

19

Methodology 3.3

where x is the distance traveled from m. If there is a caustic between m and n, Amnwill become zero before the distance traveled becomes the distance between m and

n. Solving the equation, one can find the distance to the focal points df the two

solutions to

df = − c22c1±

√(c22c1

)2

− c3c1

(58)

If one positive real value of df is smaller than the distance between m and n,

caustic has been passed. If two positive real values of df are smaller than the distance

between m and n, two caustics have been passed. Otherwise, no caustic has been

passed. As the curvature has two focal planes, the maximum number of caustics

that can be passed are 2 [15].

To find the coefficients in Equation (57), sample Amn for three values of x and

solve the equation system.

3.3.8 Saving Ray Tubes

In order to track a ray tube, it is necessary to discern which rays that were launched

adjacent to one another.

To group rays that span a ray tube together, a method similar to that in [30]

and [31] is used. Around points from which rays should be launched, a triangulated

sphere is placed centered. Rays are then launched in the directions corresponding

to the vertices of the sphere, where faces represent the ray tubes. Now, first trace

all rays. Save the information in a table with the same indices as the vertices. Next,

iterate over all faces on the surface of the sphere, i.e. the ray tubes spanned by the

three rays with the same index as the vertices of the face. An illustration of this,

for a single face, is shown in Figure 6.

To get a homogeneous density of rays, the sphere is created with the Matlab

library4 available at [32]. This function returns a tessellated icosahedron5, with a

positive, non-zero input FACTOR deciding how many vertices the surface will have.

The number of vertices NV is given by

NV = 12 + 30(FACTOR− 1) + 10(FACTOR− 2)(FACTOR− 1) (59)

and determines how many rays are launched from each source.

4More specifically, the function sphere imp gridpoints icos1.5An icosahedron is a Platonic solid with 20 faces of equally sized equilateral triangles. Tessela-

tion is the process of tiling these faces with smaller, equilateral triangles.

20

Methodology 3.4

Figure 6: Ray launching sphere where vertices represent rays and faces represent raytubes. An example ray tube is spanned by the dotted line between three launchedrays. The surface setting is FACTOR = 3, giving 92 rays.

3.3.9 Enforcing Huygens’ Surface Boundary Condition

As stated in Section 2.2, SIENT and CST assume the bottom of the Huygens’ surface

to be inside PEC. Thus, the fields on the bottom are set to zero when generated by

CST. When evaluating the surface currents, this condition may be broken due to

numerical errors. To avoid this, the surface elements directly under the Huygens’

surface are set to have no induced current for the first intersection. However, if a

ray reflects off a surface and then illuminates this part, it is allowed to induce a

current.

Coordinates are given as float values. This may introduce errors in determining

whether a point lies on a plane. To remove these errors, the bottom of the Huygens’

Surface is represented by a thin box rather than a plane. Any point inside of this

box is said to lie on the bottom of the surface. In the current implementation, the

thickness of this thin bottom box is set to 1 mm.

3.4 Surface Area

3.4.1 Huygens’ Surface

On the Huygens’ surface, the current densities are defined in discrete points. How-

ever, to assign dipole moments to these points, an area is required as shown in e.g.

Equation (24). To this end, each node is designated a rectangular area. This area

has its boundaries half way to the closest neighboring node in that direction, see

Figure 7. Edge elements are cut as the area should not extend outside the surface.

21

Methodology 3.5

Figure 7: Division of the Huygens’ surface into elements. Points are the givencoordinates and dashed lines outline the are assigned to the points. Pictured is thelower left corner of a surface.

3.4.2 Surface Current Density

On the triangulated surface, a current density J is defined on each triangular cell

as described in Section 3.3.3. Thus, in Equations (24) and (26), the area of the cell

can be used.

3.5 Simplifying the Huygens’ Surface

As the near-field source is imported from CST, the surface element size is inherited

from the FIT mesh. The FIT mesh is denser around complicated parts of the

simulated structure inside the Huygens’ surface. However, the field is not necessarily

strongest there. Take Figure 8 for example. The field is stronger toward the edges

in x. But as the antenna simulated in FIT is positioned around x = 0 and y = 0,

the mesh is finest there.

Figure 8: Real part of Jx, i.e. x.component of J, on an imported Huygens’ surface.Note that mesh density does not correlate with current strength.

The issue with a dense mesh is the computational cost. For the sums in Equation

22

Methodology 3.6

(31) and (32), weak contributions from these small cells with low field strength does

not necessarily increase accuracy.

To simplify the Huygens’ surface and decrease computational cost the following

method is used:

1. Specify a maximum allowed size.

2. Go through each surface, bundling together elements to cells smaller than the

maximum size.

3. Calculate area of new cells.

4. Calculate new surface currents for cells.

The point representing the new cell will be the center point. Taking all elements

l in a bundle and consulting (31), the new surface currents are

Jbundle =∑l

JlAl

Abundle

jkr·r′l(60)

and

Jm,bundle =∑l

Jm,lAl

Abundleejkr·r

′l , (61)

where Abundle is the sum of the area of all the bundled elements. This simplifica-

tion is dependent on r. To avoid this, assume each bundle is small, thus ejkr·r′l ≈ 1.

Now, each bundle is used as a cell, following the theory in Sections 2.3 and 2.4.

3.6 Far Field Phase Reference

Calculating the far-field is described in Section 2.4, with the element area given

in Section 3.4. However, to get a comparable phase, one must consider the phase

center. For example, in CST it is the center of the bounding box by default and is

changed to the origin.

Also, CST normalizes phase to a distance of 1 m, while SIENT takes the actual

value. To compensate for this, Equation (43) is multiplied by e−jk·1m giving

Ftot(r) = −∑i

ηk2

4πV0

Aijω

(cr× (r× Ji) +

1

µ0

r× Jm,i

)ejk(r·r

′i−1m). (62)

3.7 Space Partitioning

To make the code more efficient, the scattering surface is partitioned. The basis for

this partitioning is the Matlab code from [33]. This code partitions space heuristi-

cally6 with Axis Aligned Bounding Boxes (AABB). The definition of an AABB can

6Heuristic is here used in a computer science context. In short, a heuristic solution is a fastsolution but not the optimal one.

23

Methodology 3.8

be found in [34]. A maximum number of triangles per AABB is set, with two other

restrictions. The minimum side of the AABB is not allowed to be smaller than a

factor 3/4 of the maximum side and a split will still occur if the volume can be

reduced by 90 %.

Note that the tree-creating algorithm takes a maximum amount of triangles. If

a bounding box already contains less than this, but can be split without violating

the other restrictions, it will be split. Without restrictions, the algorithm will create

many small partitions which increases simulation time.

Each box is set to contain a maximum of√NT triangles, where NT is the total

amount of triangles. With this setting, the amount of boxes should be close to√NT . Now, using the ray-AABB intersection from [34], test which AABB that are

intersected by the ray. The triangles contained in these boxes can then be tested

for intersections.

This partitioning can be motivated by the reduction of test required. Only testing

the triangles requires NT tests. With partitioning, the AABB test requires around√NT tests and will hit M boxes. Now, the set of triangles that may be intersected

has been reduced to those contained in these M boxes. The following triangle

intersection will test M boxes, each containing approximately√NT triangles. Thus

the triangle intersection takes M√NT test. In total, (M + 1)

√NT tests. Assuming

the partitioning has suitable settings, M should be smaller than√NT , reducing the

amount of tests by a factor

Nwp

Nnp

= (M + 1)√NT/NT = (M + 1)/

√NT , (63)

where Nwp and Nnp are the number of test with partitions or with no partitions

respectively.

3.8 Ray Density

As stated by the Nyquist-Shannon sampling theorem, the current on the surface of

the model must be sampled with λ/2 spacing to give a correct representation. With

the backward tracing in Section 3.3.5, the mesh will define this. However, the rays

must still be dense enough to discern the object. This required accuracy will set the

ray density. Consider a sphere with radius Rm centered on the Huygens’ surface,

which encloses the whole platform. This sphere has a surface area of

As = 4πR2m. (64)

Now, say that on this sphere, the rays should be spaced with a Faλ distance. As

the rays are launched in a pattern of equilateral triangles, this is fulfilled if the side

24

Methodology 3.9

length of the triangles is Faλ. The equilateral triangle will then have an area of

At =

√3

4F 2aλ

2. (65)

The number of ray tubes Ntubes required can then be approximated by

Ntubes ≈AsAt

=16πR2

m√3F 2

aλ2. (66)

Looking back at the icosahedron in Section 3.3.8, the number of faces, which is

equal to Ntubes, is given by

Ntubes = 20(FACTOR)2. (67)

Equating Equations (66), (67) and solving for FACTOR gives

FACTOR =

⌈√4π√3 · 5

Rm

Faλ

⌉. (68)

The ceiling operator dxe is used to keep FACTOR as an integer.

3.9 Validation

To validate SIENT, it will be compared to CST solutions. Comparisons are made to

both the FIT and asymptotic solver. In some cases, it is also compared to ANSYS

Savant, another SBR solver. If the problem is electrically large, the FIT mesh

density is reduced which will reduce the accuracy of the solution.

To find an approximation of the CST simulation error, convergence analysis

is performed. This is done by increasing accuracy until the Root Mean Square

Deviation (RMSD) in a region of interest is below a desired threshold. The region

of interest varies, in the ray tracing case it is set as the non-shadow region. In the

region of interest, a subset of directions is chosen. When applicable, this subset is a

cut of the main lobe. On this subset, the RMSD of far-field amplitude and phase is

evaluated.

For small problems, it can then be evaluated how FIT, asymptotic (with/without

PFI) and SIENT compares for installed far-field calculations. Once the accuracy of

the code has been validated on small problems, it is validated against the asymptotic

solver on large problems. The FIT solution is also included, but with a coarser mesh.

3.10 Summary

SIENT is can either include near-field using (50) or exclude near-field by only taking

the leading terms. To simplify the referencing to these, the nomenclature SBR

Including Near-field Terms (SINT) and SBR Excluding Near-field Terms (SENT)

25

Methodology 3.10

is used in Section 4.

26

Results & Analysis

4 Results & Analysis

4.1 Verification: Far Field from Huygens’ Surface

In a first step to verify the code, the far-field from a Huygens’ surface is examined.

The blade antenna in Figure 9 is simulated in CST FIT solver. Loss-less FR-4 with

εr = 4.3 is used as dielectric. From this, a near-field source is exported. The source

is exported for a frequency of 400 MHz and is used in all test cases. A blade antenna

is interesting from an installed antenna perspective as it is placed on a surface,

generating surface currents. Other antennas, mounted in e.g. wing tips, are not as

influenced by the surface on which it is mounted.

Figure 9: Blade antenna used at 400 MHz. Dielectrics are made transparent forvisualization, PEC is shown as gray and the red cone at the bottom is a discrete port.The square meshed bottom boundary is an infinite ground plane. The square PECelement is 150 mm× 150 mm. The radome has an oval footprint of 50 mm× 200 mmand a height of 180 mm.

Second, this near-field source is simulated alone in the asymptotic solver of CST.

This gives the far-field of the equivalent surface currents. The phase center is set to

origo.

Third, the source is imported to Matlab. Using the method in Section 2.4, the

far-field is calculated. Directivity is normalized such that the maximum of the CST

simulation and the Matlab simulation are the same.

The resulting directivity is shown in Figure 10 and the phase in Figure 11.

For both directivity and phase the in-house code get similar results to CST.

Between 5° and 175° the root mean square deviation (RMSD) is 0.11 dBi for the

directivity and 0.36° for the phase. RMSD DR between series A and series B is

defined as

DR =

√√√√ 1

N

N∑i=1

(A(i)−B(i))2, (69)

where N is the length of A and B.

27

Results & Analysis 4.1

0◦ 45◦ 90◦ 135◦ 180◦−50

−40

−30

−20

−10

0

10

θ

Dir

ecti

vit

y/d

Bi

CSTin-house

Figure 10: Directivity for Huygens’ surface of a blade antenna, φ = 0°, θ polarization.For CST asymptotic solver and in-house simulation. Note that the Huygens’ surfaceis placed in free space. Deviations close to 0° and 180° come from sampling density.

0◦ 45◦ 90◦ 135◦ 180◦0◦

45◦

90◦

135◦

180◦

θ

Phas

e

CSTin-house

Figure 11: Far field phase for Huygens’ surface of a blade antenna, φ = 0°, θpolarization. For CST asymptotic solver and in-house simulation. Note that theHuygens’ surface is placed in free space. Deviations close to 0° and 180° come fromsampling density.

28


4.2 Verification: FIT Simulation

With a small modification to the code used in Section 4.1, It can be tested against

the FIT solver in CST. In the FIT simulation, the boundary condition for z = 0

is n × E = 0, i.e. PEC. This boundary condition can be achieved by mirroring

the currents on the Huygens’ surface in the z = 0 plane, i.e. using image theory

[35]. For electric currents, parallel components are duplicated to the mirror point

and change polarity. Normal components are duplicated and keep their polarity.

Mirrored magnetic currents normal component changes polarity and the parallel

components keep their polarity.

To summarize, the following is done for each surface element:

1. copy the element, all operations will act on the copy

2. set z → −z

3. set Jx → −Jx, Jy → −Jy

4. set Jm,z → −Jm,z

5. set nz → −nz, where nz is the z-component of the normal.

Further, the normalization voltage V0 is used to evaluate realized gain. This is

compared to the realized gain from CST in Figure 12. A phase comparison is made

in Figure 13. As image theory has been used, the solution is only valid for θ < 90°.To avoid any errors due to the null at 0°, the solutions deviation is evaluated in the

range θ ∈ [5°, 90°].

0◦ 30◦ 60◦ 90◦−50

−40

−30

−20

−10

0

10

θ

Rea

lize

dga

in/d

B

CSTin-house

Figure 12: Realized gain of blade antenna for FIT and image theory in-house soft-ware. φ = 0°, θ polarization. Deviation close to 0° due to sparse sampling.

Between 5° and 90°, the RMSD of realized gain is 0.02 dB and 0.15° for phase.

With the small error between the in-house code and commercial options in Sections

4.1 and 4.2, we conclude that this part of the code is functional.

29


0◦ 30◦ 60◦ 90◦220◦

225◦

230◦

235◦

240◦

θ

Phas

e

CSTin-house

Figure 13: Phase in the far-field of blade antenna for FIT and image theory in-housesoftware. φ = 0°, θ polarization. Deviation close to 0° is due to sparse sampling.

4.3 Simplified Mesh

Simplifying the mesh as described in Section 3.5, the mesh in Figure 14 is obtained.

The maximum side of an element is set to 0.05 m (0.07λ). Another mesh is also

made, with a maximum size of 0.15 m (0.2λ).

These different meshes are evaluated using the in-house code and the results are

compared. All errors are taken between 5° and 175° to avoid nulls. In Figure 15,

the directivity has a RMSD of 0.03 dBi for the 0.07λ mesh and 0.3 dBi for the 0.2λ

one. The phase, which is more sensitive to position, is shown in Figure 16. It has a

RMSD of 0.14° for the 0.07λ case and 3.4° for 0.2λ.

While the results do differ, the amount of mesh cells has decreased from 16500 to

3114 for the 0.07λ simplification. Calculation time for the original mesh is around

7 s, while it is around 1.5 s for the 0.07λ mesh. The amount of mesh cells decrease

further to 306 and the time to 0.3 s for the 0.2λ mesh. Note that the error may be

larger in the near-zone. Overall, this simplification should be applicable as long as

the mesh size is set correctly.

An anomaly in Figure 16 is that the error increases toward 0° and 180°. This

is likely from the simplification of ejkr·r′l ≈ 1 in Section 3.5. Consider the antenna

inside the box, a monopole. In the coordinate system used, the electric current will

only have a z-component. This in turn causes the electrical currents on the Huygens’

surface to be mainly in the z-direction.

Now consider the case in Figure 17. On the left, there are four dipoles. On the

right, a simplified version. Here, the positions is the mean of the two pairs and the

amplitude is the sum. In the solid red direction, nothing will change for the phase

in the far-field as the distance to the source does not change.

However, in the dashed direction the distance and thus phase changes. This

30


Figure 14: Simplified version of current in Figure 8. Maximum mesh cell size setto 0.07λ. When bundling elements together, gaps appear in the corners due to plotpoints moving.

change could be a cause of the phase error.

4.4 Test Case: Cut Cylinder

As a simple 3D test case, a cut cylinder is used. It is 4000 mm, 5.3λ at 400 MHz,

long with a radius of 500 mm. To have a locally flat ground plane, a plane is cut

on a radius of 300 mm. This gives the geometry in Figure 18. The blade antenna

Huygens’ surface from Section 4.1 is used as the source at 400 MHz. For SIENT,

the mesh used for the geometry in Figure 18 has 48948 mesh cells.

4.4.1 Reference Solution from CST

To get results to validate the SBR code against, the FIT and SBR solver in CST

is used. To make the FIT solution more comparable, it uses the same Huygens’

surface as the SBR solvers as a source. The CST SBR solver will be run twice in

addition to convergence tests. Once with the setting ”perform full integration close

to structure” (referred to as PFI ) on and once with it off. In both cases, ”include

metallic edge diffraction” is turned off as SIENT does not have this implemented.

The field source setting Rays in CST is also disabled. Other settings are listed in

Table 1, Case 2. Note that the Rays option in the CST asymptotic solver is very

important. It seems to override other settings, causing no difference with or without

PFI when enabled. This has been reported to CST, who are investigating the issue.

While the cut cylinder is a simple shape, some difference between FIT and SBR

is expected. Cutting the cylinder creates sharp edges, which will radiate due to edge

31


0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

θ

Dir

ecti

vit

y/d

Bi

original meshmax size 0.07λmax size 0.2λ

Figure 15: Directivity of Huygens’ surface for different mesh sizes using in-housesoftware. φ = 0°, θ polarization.

0◦ 45◦ 90◦ 135◦ 180◦0◦

45◦

90◦

135◦

180◦

θ

Phas

e

original meshmax size 0.07λmax size 0.2λ

Figure 16: Phase in the far-field of Huygens’ surface for different mesh sizes usingin-house software. φ = 0°, θ polarization.

(a) Higher resolution. (b) Lower resolution.

Figure 17: On the left, a higher resolution case with four dipoles (black). On theright, a simplified case. Red arrows point to two possible observation directions.

32


Figure 18: Dimensions of the cut cylinder. The box containing the antenna (blue)is centered on the top surface.

diffraction. Further, to get the complete surface current, creeping waves must be

included, which could be achieved with e.g. Uniform Theory of Diffraction (UTD)

[2].

4.4.2 Convergence Analysis

To ensure correct results in CST, a convergence analysis is performed. The settings

that differ from defaults are listed in Table 1. RMSD is evaluated for φ = 90°, i.e.

the yz-plane in Figure 18.

Setting Case 1 Case 2Include metallic edge diffraction disabled disabled

Far field accuracy factor 100 200Sub source reduction error 0 0

Smallest sub source in wavelengths 0.1 0.05Ray spacing 1λ 1λ

Adaptive ray sampling enabled enabled

Table 1: Settings for convergence analysis on cut cylinder using CST SBR solver.

First, convergence for the case with PFI disabled is analyzed. The resulting

directivity is shown in Figure 19 and phase in Figure 20. The RMSD of directivity

between 5° and 175° is 0.019 dBi and 0.044° for the phase.

With PFI, directivity is shown in Figure 21 and phase in Figure 22. The RMSD

of directivity between 5° and 175° is 0.0077 dBi and 0.089° for the phase.

Convergence for the reference solution using FIT is also analyzed. Settings are

listed in Table 2 and directivity and phase are plotted in Figures 23 and 24. RMSD

of directivity and phase for θ ∈ [0°, 180°] is 0.12 dBi and 1.8° respectively.

33


0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

θD

irec

tivit

y/d

Bi Case 1

Case 2

Figure 19: CST Far field convergence for cut cylinder without PFI, φ = 90°, θpolarization. Shadow region shadowed gray. For settings see Table 1.

0◦ 45◦ 90◦ 135◦ 180◦−270◦−180◦−90◦

0◦90◦

180◦270◦

θ

Phas

e

Case 1Case 2

Figure 20: CST far-field phase convergence for cut cylinder without PFI, φ = 90°,θ polarization. Shadow region shadowed gray. For settings see Table 1.

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2

Figure 21: CST far-field convergence for cut cylinder with PFI, φ = 90°, θ polariza-tion. Shadow region shadowed gray. For settings see Table 1.

Setting Case 1 Case 2Cells per wavelength near model 15 25

Cells per wavelength far from model 15 25Cells per max model box edge, near model 20 20

Cells per max model box edge, far from model 1 1Accuracy −40 dB −40 dB

Table 2: Settings for convergence analysis on cut cylinder using CST FIT solver.

34


0◦ 45◦ 90◦ 135◦ 180◦−270◦−180◦−90◦

0◦90◦

180◦270◦

θ

Phas

e

Case 1Case 2

Figure 22: CST far-field phase convergence for cut cylinder with PFI, φ = 90°, θpolarization. Shadow region shadowed gray. For settings see Table 1.

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2

Figure 23: CST FIT far-field convergence for cut cylinder, φ = 90°, θ polarization.For settings see Table 2.

0◦ 45◦ 90◦ 135◦ 180◦−270◦−180◦−90◦

0◦90◦

180◦270◦

θ

Phas

e

Case 1Case 2

Figure 24: CST FIT far-field phase convergence for cut cylinder, φ = 90°, θ polar-ization. For settings see Table 2.

35


4.4.3 Far Field Comparison

The resulting far-field from different simulations for φ = 90° can be seen in Figures

25 and 26. RMSD is only evaluated between 5° and 90°. This removes the null

and the shadow region from evaluation. The comparisons of realized gain, far-field

amplitude and phase between different simulations are tabulated in Tables 3, 4 and

5. The same comparison was made for the cut of φ = 0°, with similar deviations.

As expected, the SBR solvers differ from FIT when looking at Table 5. This is

likely due to edge diffraction.

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Rea

lize

dga

in/d

B

CST no PFICST PFI

SENTSINTFIT

5

0

-545°

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Rea

lize

dga

in/d

B

CST no PFICST PFI

SENTSINTFIT

5

0

-545°

Figure 25: Realized gain for cut cylinder, φ = 90°, θ polarization. With/withoutPFI is the CST asymptotic solver. For with/without PFI and FIT, realized gain istaken as directivity minus total losses. Shadow region shadowed gray.

SINT CST no PFI CST PFI CST FITSENT 0.68 1.3 1.5 1.6SINT 1.3 1.3 1.7

CST no PFI 0.55 2.7CST PFI 2.8

Table 3: RMSD of realized gain in dB between simulations on cut cylinder. Evalu-ated in the range θ ∈ [5°, 90°] for φ = 90°.

4.4.4 Surface Currents

From both the FIT solver and SIENT, one essential result is the surface current on

the body. For CST FIT solver, it is shown in Figure 27. For SIENT, SENT is used

36


0◦ 45◦ 90◦ 135◦ 180◦0◦

30◦60◦90◦

120◦150◦180◦210◦240◦270◦300◦330◦360◦

θ

Phas

eCST no PFI

CST PFISENTSINTFIT

Figure 26: Far field phase for cut cylinder, φ = 90°, θ polarization. With/withoutPFI is the CST asymptotic solver. Shadow region shadowed gray.



Table 4: RMSD of absolute value of far-field amplitude (see Equation (37)) betweensimulations on cut cylinder. Evaluated in the range θ ∈ [5°, 90°] for φ = 90°.

SINT CST no PFI CST PFI CST FITSENT 14° 6.2° 15° 14°SINT 8.1° 2.3° 12°

CST no PFI 10° 13°CST PFI 14°

Table 5: RMSD of phase between simulations on cut cylinder. Evaluated in therange θ ∈ [5°, 90°] for φ = 90°.

37


in Figure 28 and SINT in Figure 29. In Figure 30 the difference between SENT and

SINT is calculated. In all cases, the real part value of the current is shown. While

Figures 28 and 29 are similar, one can see the difference close to the source.

In Figure 27, two observations can be made when comparing to the SIENT

results. At the edges of constant x, the edge is not very sharp, causing creeping

waves to occur on the curved part of the cylinder. Further, the current directly

below the source is not zero. Due to the shape of this current, it is likely that the

edges along constant x, which are closer to the source than those of constant y, cause

diffraction which induce a notable current.

Figure 27: Real part amplitude of surface current on cut cylinder from CST FITsolver.

Figure 28: Real part amplitude of surface current on cut cylinder from SENT.

In Figure 30, the difference between the far-field approximation and near-field

38


Figure 29: Real part amplitude of surface current on cut cylinder from SINT.

inclusion is shown. The difference ∆A is found by

∆A = 20 log10(|<(JSINT − JSENT )|), (70)

where JSINT and JSENT is the surface current for SINT and SENT. Real part is

plotted to include the wave pattern in the current.

Figure 30: Real part of the difference between SENT and SINT.

4.5 Space Partitioning

To test the speed increase from partitioning, the cut cylinder case is used. The field

evaluation part of the code is disabled, i.e. only ray tracing is performed. In both

cases the accuracy factor Fa is set to 1/2.

39


As described in Section 3.7, the time reduction should be of the same magnitude

as√NT . For the cylinder,

√NT =

√8954 ≈ 95.

Running without the partition code takes 1785 s. With the code, it decreases to

94.22 s. This is a reduction by a factor of around 19. The reduction is a fifth of√NT , which is expected due to the (M + 1) term in Equation (63).

The partitions generated is shown in Figure 31.

Figure 31: Partitions of the cut cylinder (yellow, translucent boxes).

4.6 Test Case: Corner Reflector

As a difficult test, the antenna is placed close to the corner of a corner reflector as

shown in Figure 32. The same blade antenna Huygens’ surface at 400 MHz as earlier

is used as the source. The SIENT mesh has 19118 mesh cells. With a lot of sharp

edges, it is an example of where edge diffraction is important. Further, it should

gauge how the SBR solvers handle a platform which is not locally flat close to the

antenna.


First, the CST convergence without PFI is analyzed. Settings for this is shown in

Table 6. The resulting directivity is shown in Figure 33 and phase in Figure 34.

The RMSD of directivity between 5° and 90° is 0.34 dBi and 2.1° for the phase.

For CST with PFI, the same settings in Table 6 are used. Directivity is shown

in Figure 35 and phase in Figure 36. The RMSD of directivity between 5° and 90°is 6.4 dBi and 54° for the phase. Thus it is decided that the solver with PFI is not

converging.

For FIT, the same settings as in Table 2 are used for convergence analysis. This

gives the plots in Figures 37 and 38, with a RMSD of 0.11 dBi for directivity and

0.82° for the phase.

40


Figure 32: Geometry of corner reflector

Setting Case 1 Case 2 Case 3Include metallic edge diffraction disabled disabled disabled

Calculate far-fields for PO disabled disabled disabledFar field accuracy factor 200 400 600

Sub source reduction error 0 0 0Smallest sub source in wavelengths 0.1 0.05 0.03

Ray spacing in wavelengths 0.5 0.25 0.1Adaptive ray sampling on on on

Table 6: Settings for convergence analysis of corner reflector.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20

−15

−10

−5

0

5

10

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2

Figure 33: CST far field convergence for corner reflector, φ = 0°, θ polarizationwithout PFI. Settings are given in Table 6.

41


0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦60◦90◦

120◦150◦180◦210◦240◦

θ

Phas

e

Case 1Case 2

Figure 34: CST phase convergence for corner reflector, φ = 0°, θ polarization withoutPFI. Settings are given in Table 6.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20−15−10−5

05

101520

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2Case 3

Figure 35: CST far field convergence for corner reflector, φ = 0°, θ polarization withPFI. Settings are given in Table 6.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−240◦−180◦−120◦−60◦

0◦60◦

120◦180◦240◦300◦360◦

θ

Phas

e

Case 1Case 2Case 3

Figure 36: CST phase convergence for corner reflector, φ = 0°, θ polarization withPFI. Settings are given in Table 6.

42


0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20−15−10−5

05

1015

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2

Figure 37: CST far field convergence for corner reflector, φ = 0°, θ polarization forFIT. Settings are given in Table 2.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦

360◦405◦450◦495◦540◦585◦

θ

Phas

e

Case 1Case 2

Figure 38: CST phase convergence for corner reflector, φ = 0°, θ polarization forFIT. Settings are given in Table 2.

43



Plotting the CST solutions together with the SIENT results yields Figures 39 and

40. Their RMSD together with RMSD for the linear far-field amplitude is shown in

Tables 7, 8 and 9.

The PFI solution differs from all other solutions, which is expected as the solution

does not converge. For realized gain, the solution most similar to FIT is CST without

PFI. Some similarities between FIT and SIENT can be seen.

Looking at the phase, SIENT is closer to FIT.

0◦ 45◦ 90◦ 135◦ 180◦−15

−10

−5

0

5

10

15

20

θ

Rea

lize

dga

in/d

B

CST no PFICST PFI

SENTSINTFIT

Figure 39: Realized gain for corner reflector, φ = 0°, θ polarization. With/withoutPFI is the CST asymptotic solver. For with/without PFI and FIT, realized gain istaken as directivity minus total losses. Shadow region shadowed gray.

0◦ 45◦ 90◦ 135◦ 180◦0◦

30◦

60◦

90◦

120◦

150◦

180◦

210◦

θ

Phas

e

CST no PFICST PFI

SENTSIENT

FIT

Figure 40: Far field phase for corner reflector, φ = 0°, θ polarization. With/withoutPFI is the CST asymptotic solver. Shadow region shadowed gray.

44


SINT CST no PFI CST PFI CST FITSENT 1.5 4.0 11 4.4SINT 3.8 11 3.9


Table 7: RMSD of realized gain in dB between simulations on corner reflector.Evaluated in the range θ ∈ [5°, 90°] for φ = 0°.



Table 8: RMSD of absolute value of far-field amplitude (see Equation (37)) betweensimulations on corner reflector. Evaluated in the range θ ∈ [5°, 90°] for φ = 0°.

SINT CST no PFI CST PFI CST FITSENT 9.9° 29° 190° 26°SINT 31° 190° 19°

CST no PFI 200° 32°CST PFI 190°

Table 9: RMSD of phase between simulations on corner reflector. Evaluated in therange θ ∈ [5°, 90°] for φ = 0°.

45



For the corner reflector, there are clear differences between FIT and SIENT. The

FIT solver current in Figure 41 has no currents on the corner in the z-direction.

This is not the case for SIENT in Figures 42 and 42. The interference patterns also

differ. The difference between SENT and SINT can be seen in Figure 44. The noisy

character of the current from SIENT seem to occur for high ray density. This should

be investigated further.

Figure 41: Surface current on corner reflector from CST FIT solver.

Figure 42: Surface current on corner reflector from SENT.

46


Figure 43: Surface current on corner reflector from SINT.

Figure 44: Surface current difference on corner reflector between SENT and SINT.

47


4.7 Test Case: Simplified SAAB 37 Viggen

To get a test case similar to a real implementation without too much detail, a model

of SAAB 37 Viggen from [36] is used. The model is shown in Figure 45. For this

test case, HFSS SBR+ (previously known as Savant) is used for comparison as well

as CST. The Huygens’ surface in Section 4.1 is used as the source at 400 MHz. For

SIENT, the geometry in Figure 45 is meshed with 375253 mesh cells.

Figure 45: A simplified model of Viggen with a blade antenna on the wing, repre-sented by a Huygens’ surface.


For the CST ray solver, the settings in Table 6 are used. Without PFI, plots are

shown in Figures 46 and 47. Comparing case 2 and 3, the RMSD between 5° and

90° is 1.0 dBi for the directivity and 4.6° for the phase. With PFI, plots are shown

in Figures 48 and 49. Except the similarities in phase between case 1 and 3, the

solution does not seem to converge.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20

−15

−10

−5

0

5

10

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2Case 3

Figure 46: CST far field convergence for viggen without PFI, φ = 90°, θ polarization.See Table 6 for settings.

48


0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−270◦−180◦−90◦

0◦90◦

180◦270◦

θ

Phas

e

Case 1Case 2Case 3

Figure 47: CST far-field phase convergence for viggen without PFI, φ = 90°, θpolarization. See Table 6 for settings.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20−15−10−5

05

1015

θ

Dir

ecti

vit

y/d

Bi Case 1

Case 2Case 3

Figure 48: CST far field convergence for viggen with PFI, φ = 90°, θ polarization.See Table 6 for settings.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦

−720◦−540◦−360◦−180◦

0◦180◦

θ

Phas

e

Case 1Case 2Case 3

Figure 49: CST far-field phase convergence for viggen with PFI, φ = 90°, θ polar-ization. See Table 6 for settings.

49


Setting Case 1 Case 2Fa 1 0.5

Body mesh max size 0.2λ 0.1λMaximum bounces 3 3

Table 10: Settings for SINT convergence analysis on viggen. Fa is described inSection 3.8.

SIENT convergence is shown in Figures 50 and 51. With the maximum size set

to 5 m, the settings in Table 10 were used. Convergence analysis is performed with

SINT, as this should include the SENT results

The body mesh density was also increased, but caused Matlab to run out of

memory on a machine with 128 GB of RAM during the recursive search described

in Section 3.3.3. Overall, the higher settings together with this large mesh seems to

be close to the limit of what SIENT can handle.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20

−15

−10

−5

0

5

10

θ

Rea

lize

dga

in/d

B Case 1Case 2

Figure 50: Far field convergence for viggen with SINT, φ = 90°, θ polarization.Settings in Table 10.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦

−360◦

−180◦

0◦

180◦

θ

Phas

e

Case 1Case 2

Figure 51: Far field phase convergence for viggen with SINT, φ = 90°, θ polarization.Settings in Table 10.

The convergence of savant is demonstrated in Figures 52 and 53, with the settings

in Table 11. Between case 3 and 4, RMSD of realized gain is 0.77 dB and 7.6° for

the phase.

50


0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20−15−10−5

05

1015

θ

Rea

lize

dga

in/d

B Case 1Case 2Case 3Case 4

Figure 52: Far field convergence for viggen with savant, φ = 90°, θ polarization. SeeTable 11 for settings.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−900◦−720◦−540◦−360◦−180◦

0◦180◦

θ

Phas

e

Case 1Case 2Case 3Case 4

Figure 53: Far field phase convergence for viggen with savant, φ = 90°, θ polariza-tion. See Table 11 for settings.

51


Setting Case 1 Case 2 Case 3 Case 4Sample density 15 20 20 30

Ray density 8 12 20 30

Table 11: Settings for convergence analysis of viggen using savant.

Overall, all solvers are slow to converge. Due to this a hypothesis was stated:

as this geometry contains many corners and shadowed areas, SBR converges slowly

due to not including edge effects (edge diffraction, creeping waves, etc.). To test

this, PTD was enabled in the savant solver Changing no other settings, case 1 and 2

in table 11 were evaluated including PTD. The resulting plots are shown in Figures

54 and 55. While similar to the plots in Figures 52 and 53, including PTD seems to

increase convergence. The RMSD in phase between case 1 and case 2 is 32° without

PTD. With PTD, the RMSD in phase between case 1 and 2 decreases to 11°.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦−20−15−10−5

05

1015

θ

Rea

lize

dga

in/d

B Case 1Case 2

Figure 54: Far field convergence for viggen with savant including PTD, φ = 90°, θpolarization. See Table 11 for settings.

0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦

−720◦−540◦−360◦−180◦

0◦180◦

θ

Phas

e

Case 1Case 2

Figure 55: Far field phase convergence for viggen with savant including PTD, φ =90°, θ polarization. See Table 11 for settings.

52



Plotting the CST solutions together with the SIENT and Savant7 results yields

Figures 39 and 40. Their RMSD in gain, linear far-field amplitude and phase are

shown in Tables 7, 8 and 9.

A lot of the behavior seen in the corner reflector case can also be seen in the

viggen case. CST with PFI does not converge, and gives very erroneous values,

while the solver without PFI performs very well, even better than SIENT. The

added savant solver predicts the far-field pattern well, but seems to fail for the

phase. Note that the FIT solution may not have converged. It is set to the highest

mesh possible with 128 GB of RAM, 15 cells per wavelength.

One observation is that the near-field contribution to the phase seems to decrease

with increased platform size. RMSD for SIENT is shown in Table 15. As the size

increases from the cut cylinder to viggen, the phase RMSD seems to decrease. While

this is not proven, a hypothesis is the differing area radiating. For the cut cylinder,

almost all of the surface with currents in Figure 28 also has currents in Figure 30.

However, in Figure 61, the area with currents is much smaller than that in Figure 59.

Thus the near-field terms should have a relatively larger effect for the cut cylinder

compared to the Viggen.

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Rea

lize

dga

in/d

B

CST no PFICST PFI

SENTSINTSavant

FIT

5

0

-545°

0◦ 45◦ 90◦ 135◦ 180◦−20

−15

−10

−5

0

5

10

θ

Rea

lize

dga

in/d

B

CST no PFICST PFI

SENTSINTSavant

FIT

5

0

-545°

Figure 56: Realized gain for viggen, φ = 90°, θ polarization. With/without PFI isthe CST asymptotic solver. For with/without PFI and FIT, realized gain is takenas directivity minus total losses.

7without PTD

53


0◦ 45◦ 90◦ 135◦ 180◦−720◦

−540◦

−360◦

−180◦

0◦

180◦

360◦

θ

Phas

eCST no PFI

CST PFISENTSINTSavant

FIT

Figure 57: Far field phase for viggen, φ = 90°, θ polarization. With/without PFI isthe CST asymptotic solver.

SINT CST no PFI CST PFI Savant CST FITSENT 0.87 4.7 10 3.4 3.9SINT 4.9 10 3.6 3.7

CST no PFI 9.2 3.6 3.9CST PFI 8.7 8.5Savant 3.3

Table 12: RMSD error of realized gain in dB between simulations on viggen. Eval-uated in the range θ ∈ [5°, 90°] for φ = 90°. Note that only the green cell has twoconvergent solutions.

SINT CST no PFI CST PFI Savant CST FITSENT 0.04 0.31 1.3 0.26 0.33SINT 0.33 1.3 0.27 0.34

CST no PFI 1.2 0.19 0.27CST PFI 1.3 1.2Savant 0.25

Table 13: RMSD of absolute value of far-field amplitude (see Equation (37)) betweensimulations on viggen. Evaluated in the range θ ∈ [5°, 90°] for φ = 90°. Note thatonly the green cell has two convergent solutions.

54


SINT CST no PFI CST PFI Savant CST FITSENT 4.8° 294° 310° 201° 121°SINT 294° 311° 201° 124°

CST no PFI 565° 142° 286°CST PFI 142° 399°Savant 279°

Table 14: RMSD error of phase between simulations on viggen. Evaluated in therange θ ∈ [5°, 90°] for φ = 90°. Note that only the green cell has two convergentsolutions.


As for the corner reflector, there are clear differences between FIT and SIENT for

viggen. The FIT solver current in Figure 58 is continuous and is clearly present on

the front wing, which shadows GO rays. In Figures 59 and 60, the current is missing

in shadowed areas around the fin, the nose and the front wing. This can be solved

by including edge diffraction and creeping waves as in [2].

Figure 58: Surface current on viggen from CST FIT solver.

55


Figure 59: Surface current on viggen from SENT.

Figure 60: Surface current on viggen from SINT.

56


Figure 61: Surface current difference on viggen between SENT and SINT

4.8 Concluding Remarks

To get an understanding of the effect near-field terms have, the far-field of the

surface current difference in Figure 30 is calculated. The resulting far-field and

phase is plotted in Figure 62. Compared to the total realized gain in Figure 25, the

near-field contribution is small, around 20 dB lower. This is also reflected in Tables

3 and 4. Further, the far-field in Figure 62 is very uniform, with nulls at 0° and

90°. Superpositioning the weak, uniform pattern from the near-field terms over the

stronger pattern from the far-field terms will not have a large effect on the shape of

the pattern. For example, the ripple in the far-field of the cut cylinder case, shown

in Figure 25, have the same position and similar shape whether near-field terms are

included or excluded. But the amplitude of these change when including near-field

terms.

Looking at the realized gain, both CST SBR options seem to predict a higher

overall gain as compared to SIENT. For all SBR solvers (all solvers except FIT) the

θ-direction with the highest gain is shifted except for the viggen test case. For the

SIENT, ripples in the far-field are in the correct direction for the cut cylinder and

corner reflector, but does not get the correct amplitude for any test case.

Comparing the SENT surface current on the cut cylinder in Figure 28 to the

difference between SENT and SINT in Figure 30, the area with near-field surface

currents is slightly smaller than the far-field current. However, the near-field region

is very constant in size between the test cases. The near-field components of the

field from the Huygens’ surface will only affect surfaces within this near-field region.

As the platform grows, the proportion of the near-field region to the total region

57


will decrease. As in Figures 59 and 61, where the near-field contribution is limited

to a small portion of the platform.

Test Case RMSD realized gain/dB RMSD linear amplitude RMSD phaseCut Cylinder 0.68 0.080 14°

Corner 1.5 0.18 9.9°Viggen 0.87 0.04 4.8°

Table 15: RMSD between SINT and SENT for all test cases.

The phase of the cut cylinder case in Figure 26 seems to be more sensitive than

the amplitude to near-field terms. A part of this difference could be the phase of

the current close to the source. Comparing the currents in Figures 27 and 28, the

nulls close to the source box are not in the same position. However, the included

near-field terms in Figure 29 better replicate the phase in 27.

0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦−40

−35

−30

−25

−20

−15

−10

θ

Rea

lize

dga

in/d

B

135◦

180◦

225◦

270◦

315◦

360◦

405◦

450◦

495◦

Phas

e

Realized gain Phase

Figure 62: Realized gain and phase for cut cylinder near-field currents, i.e. Figure30, for φ = 90°, θ polarization. Huygens’ surface is not included in the far-fieldcalculation i.e. this is only the radiation of the induced surface currents.

4.9 Other Test Case

One cause of error in SIENT is edge diffraction. Due to this, another test case

without sharp edges is included in appendix B.

58

Discussion & Conclusion

5 Discussion & Conclusion

5.1 Discussion

A code implementing the SBR method including near-field effects has been pre-

sented. The implementation is described thoroughly. Considering the limitations

set, it was realized that most platforms of interest are not given as an analytical

expression. So rather early, it was decided to break limitation 1: ”Importing CAD

models is not a requirement”. This allowed for models to be either created in CST

and imported to Matlab (as was the case for the cut cylinder and corner reflector)

or for an external model to be imported to CST and Matlab (as was the case with

viggen). Meshing the surface was a simpler task than expected using gmsh.

Further, there were no requirements for the code to be optimized. Even so,

many solutions in Section 3.3 were iterated for better performance. While the goal

was to have a functioning code, a faster code allowed more iterations and a more

efficient work process. Considering that some ray tracing functions were called tens

of millions of times during a simulation, shaving of a microsecond due to memory

access or such can reduce simulation time by up to an hour.

To validate the code piece by piece, different validations and test cases include

different requirements. The first results are to prove that importing the source and

calculating the far-field works. Then the first test case is performed to test the

functionality of the then completed parts of the code. Note that any ray impinging

on the cut cylinder ”bounces off” i.e. only one bounce has to be tracked. This meant

that the code could be evaluated without full functionality. The corner reflector was

included to be a difficult test case, but also to verify the functionality not tested

by the cut cylinder. A corner reflector will have multiple bounces and is a concave

shape, requiring DF calculation including phase correction.

Finally, the viggen test case was included to give a more realistic case and to

examine the maximal capabilities of the code. This test case is similar to the pro-

posed fighter in Section 1.1. The problem that was described in the theoretical case

occurred, as the mesh of the FIT simulation could not be refined due to insufficient

RAM.

5.2 Conclusion

A SBR code which takes near-field effects in to account has been developed.

The first main research question of the thesis is How can the SBR method be

implemented for installed antenna performance? By consulting the literature, this

has been shown in Sections 2 and 3. The installed antenna is represented by an

Huygens’ surface. This surface is divided into small, equivalent dipoles, from which

rays are launched. Using ray tubes and the expression for the H-field in (49), the

surface current on the installed platform is found. This surface current gives a

59

Discussion & Conclusion 5.2

corresponding far-field pattern, which is evaluated against commercial software.

The second main research question is What is the error for installed antenna

performance SBR when neglecting near-field contributions? For the test cases in

this thesis, RMSD lies between 0.68 dB to 1.5 dB for gain and 5° to 14° for phase

in the lit region. As both SENT and SINT results differ from the used FIT solver

results, it can not be concluded that SENT or SINT is more accurate. Discussions

on how accuracy could be improved in future work are given in Section 6.

However, it is an important result in that near-field effects do have an impact

on installed antenna performance. Including near-field terms does not change the

overall shape of the far-field, but usually corrects it, making it more similar to the

FIT solution. This result is not fully conclusive, as excluding near-field terms gives a

result closer to FIT for viggen. However, this could be due to lack of convergence of

both the FIT solver and SIENT. The difference in phase when including near-field

terms is relevant for e.g. direction finding [3]. While SIENT could be optimized

more for performance and lacks a good user interface, it is interesting that SIENT

seem to perform on par or better when compared to some commercial options.

The secondary research question is Does commercial SBR software take near-

field contribution into account? If so, how? Judging by the Savant documentation

[14], it uses a method similar to SIENT. However, Savant is more accurate than

SIENT. It is inconclusive whether CST PFI includes near-field terms.

Due to the apparent difference when including near-field terms, and to bench-

mark accuracy, results derived from this thesis will be submitted to a scientific

journal.

60

6 Recommendations for Future Work

While a lot has been done in this thesis, the resulting code needs more work before it

is used for actual installed antenna performance calculations. The biggest reason for

this is the lack of edge diffraction and creeping waves. Looking at [2], UTD should

be possible to implement in order to combat these issues. Comparing Figures 58

and 60, many areas are not lit by SIENT. Another part of SIENT which should be

improved is the bottleneck in the recursive search for lit elements.

One aspect which would be interesting to investigate is the basis functions. While

the current use of a constant basis function is simple, better accuracy and thus faster

convergence may be achieved with a more suitable one. Another way to improve

convergence would be to implement adaptive ray density. Now, rays are launched

homogeneously, but the ability to increase density in directions where it is necessary

would be a great improvement. This might also help with the noise issue described

in Section 4.6.3. If it does not solve the noise issue, this should be investigated

further.

In a similar area to adaptive ray tracing, a variable mesh density would be a

good optimization step. Currently, meshes are generated by simply specifying an

allowed size range. By only having a denser mesh in critical areas, simulation times

would go down.

Finally, it would be interesting to do larger benchmarks on a dedicated cluster.

61

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Appendix

A Divergence Factor from Interpolated Curvature

65

B Other Simulation Example

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