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FRACTALCOMS Exploring the limits of Fractal Electrodynamics for

the future telecommunication technologies IST-2001-33055

T0+6 intermediary progress report

Deliverable reference: D16

Contractual Date of Delivery to the EC: June 1, 2002

Author(s): Juan M. Rius

Participant(s): UPC

Workpackage: WP0

Security: Public

Nature: Report

Version: 1.0 Date: 23 July 2002

Total number of pages: 26

Keyword list: FRACTALCOMS, progress report

Abstract:

T0+6 progress report of FRACTALCOMS project.

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TABLE OF CONTENTS

1 MANAGEMENT ISSUES....................................................................................... 3

1.1 Schedule ........................................................................................................... 3

1.2 Funding............................................................................................................. 3

1.3 Deliverables ...................................................................................................... 3

1.4 Consortium agreement...................................................................................... 3

2 TECHNICAL ISSUES ............................................................................................. 4

3 Kick-off meeting ...................................................................................................... 4

4 UPC-CIMNE workshop ........................................................................................... 5

5 Seminar at ROME .................................................................................................... 5

5 Seminar at ROME .................................................................................................... 6

6 Analysis of the fractal Koch monopole .................................................................... 6

7 Dissemination activities............................................................................................ 6

7.1 International conferences.................................................................................. 6

7.2 International Journals ....................................................................................... 7

7.3 FRACTALCOMS project web site .................................................................. 7

8 WP1: Theory of fractal electrodynamics.................................................................. 8

8.1 T1.1: Understanding fractal electrodynamics phenomena ............................... 8

9 WP2: Vector calculus on fractal domains .............................................................. 13

9.1 Task 2.1: Solution of EM simple problems on fractal domains ..................... 13

9.2 Task 2.2: Formulation of EFIE on fractal domains ........................................ 14

10 WP3: Software simulation tool .......................................................................... 16

10.1 Task 3.1: Advanced meshing of fractal structures ......................................... 16

10.2 Task 3.3: Simulation of fractal structures in the frequency domain............... 20

10.3 Task 3.4: Simulation of fractal structures in the time domain........................ 21

11 Conclusions ........................................................................................................ 24

12 Tasks that start between T0+6 and T0+11 ......................................................... 25

Disclaimer....................................................................................................................... 26

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1 MANAGEMENT ISSUES

In the first 6 months there have been no substantial changes:

1.1 Schedule

It was decided in the kick-off meeting that tasks

o T3.3: Simulation of fractal structures in the frequency domain

o T3.4: Simulation of fractal structures in the time domain

would start at T0+6 although they were originally scheduled to start later.

1.2 Funding

Funding was transferred to partners.

There have been no budget rebalances.

1.3 Deliverables

The following deliverables have been generated:

D3: Final task report T2.1

D4: Final task report T2.2

D13: Project presentation

D14: Dissemination and Use Plan

D15: Project web site (http://www.tsc.upc.es/fractalcoms)

D16: T0+6 intermediate report (this document)

The T0+6 intermediate report contains partner progress reports as annexes.

1.4 Consortium agreement

A draft of the Consortium agreement will be distributed by the coordinator before T0+12.

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2 TECHNICAL ISSUES

The following tasks started at T0:

WP1: Theory of fractal electrodynamics

o T1.1: Understanding fractal electrodynamics phenomena

WP2: Vector calculus on fractal domains

o T2.1: Solution of EM simple problems on fractal domains

o T2.2: Formulation of EFIE on fractal domains

WP3: Software simulation tool

o T3.1: Advanced meshing of fractal structures

o T3.3: Simulation of fractal structures in the frequency domain

o T3.4: Simulation of fractal structures in the time domain

3 KICK-OFF MEETING

The kick-off meeting was held on December 13-14 2002 at UPC, Barcelona. All partners were represented.

Several project management issues were agreed, the most important of which are:

Since all FRACTALCOMS deliverables are public domain, partners are allowed to submit papers for publication before the corresponding project deliverable is sent to the Commission.

A copy of all papers related to the project and submitted for publication will be send to the coordinator, who will post them in the project website information hub, when available, or distribute among all partners if the web site is not available.

The most important technical issues agreed where:

Restrict the research to Fractal geometries than can be generated by an Iterated Function System (IFS), with emphasis on wire antennas.

In task 2.2 (Formulation of Electric Field Integral Equation (EFIE) on fractal domains), the analysis will be restricted to wire antennas.

CIMNE will provide a license of GiD pre- and post-processing software to all partners.

The following tasks, originally scheduled to start at T0+6 or later, will effectively start at T0+6:

o Task 3.3: “Simulation of fractal structures in the frequency domain”.

o Task 3.4: “Simulation of fractal structures in the time domain”.

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4 UPC-CIMNE WORKSHOP

Date: February 27

A workshop with UPC and CIMNE partners was held at UPC in February 25 (see annex). The objectives of the workshop were:

The objectives of the workshop were:

• Let CIMNE know the adaptive meshing needs for Fractal Antennas.

• Present the most basic concepts of Iterated Function Systems (IFS) to CIMNE.

• See what is possible to implement in GiD within the framework of the project. GiD is an advanced modelling and meshing software tool developed by CIMNE, that is available to all partners in FractalComs project.

• Establish the geometry parameters input for the new code to be developed in GiD.

• Define actions to do in task 3.1 b).

It was agreed that:

• UPC will mesh strip-wire antennas (Fig. 1).

• CIMNE will implement a plug-in in GiD in order to mesh planar-strip (Fig. 2) and planar-surface antennas according to UPC specifications. The mesh must be adaptive and element numbering such that the resulting linear system matrix has multilevel block structure. In the case of planar-surface antennas, the mesh will include the electric connections between surface patches that share only one vertex.

• Cylindrical-wire fractal antennas are very difficult to model because a lot of surface-surface intersections must be computed. This kind of model is not essential for numerical simulations, since most fractal antennas are printed strips and cylindrical wires can be modelled by one-dimensional wires and a radius parameter if some approximations are made in the electromagnetic computation code. For that reason it was decided that meshing cylindrical wires would be the least priority task for CIMNE, and very possibly out of the framework of this project.

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Fig. 2. One iteration strip Koch antenna, discretized in triangular patches.

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0

0.01

0 0.01 0.02 0.03 0.04 0.05 0.06

-101 x 10 -3

0

0.01

0 0.01 0.02 0.03 0.04 0.05 0.06

Fig. 1.One iteration strip-wire Koch antenna, discretized in triangular patches.

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SEMINAR AT ROME

A seminar on “Fractal modelling with Iterated Function Systems” was held at Rome on February 13-15. The ROME group gave the other partners an in-depth introduction to fractal algebra. The seminar was important in setting-up a common terminological background for all partners and in pinpointing key mathematical issues. The documentation of the seminar is attached as an Annex to this report.

6 ANALYSIS OF THE FRACTAL KOCH MONOPOLE

A document was generated by UPC on March 11 (see Annex). The objective of this document was the definition of the different configurations of the Koch fractal monopole to simulate.

7 DISSEMINATION ACTIVITIES

7.1 International conferences

The following conference papers have already been submitted in the first 6 months of the project lifetime:

• “FractalComs Project: Meshing Fractal Geometries with GiD”, Josep Parrón, Juan Manuel Rius, Jordi Romeu, Alex Heldring and Gabriel Bugeda, 1st Conference on Advances and Applications of GiD, Barcelona, Spain, 20-22 February 2002.

Partners: UPC and CIMNE

Status: accepted and already presented at a conference session

• “Numerical analysis of highly iterated fractal antennas”, Josep Parrón, Juan M. Rius and Jordi Romeu, , Invited Communication, 2002 IEEE AP-S International Symposium, San Antonio, Texas, June 16-21, 2002.

Partners: UPC

Status: accepted and already presented at a conference session

• “Solving large electromagnetic problems in small computers”, Juan M. Rius, Josep Parrón, Alex Heldring, Eduard Úbeda, Jordi Romeu, Juan R. Mosig, Leo Ligthart, Invited Communication, 2002 IEEE AP-S International Symposium, San Antonio, Texas, June 16-21, 2002.

Partners: UPC, EPFL

Status: accepted and already presented at a conference session

• “Estudio en el dominio del tiempo de la antena tipo fractal monopolo de Koch”, F. J. García Ruiz, M. Fernández Pantoja, A. Rubio Bretones, R. Gómez Martín, Reunión Nacional del Comité Español de la URSI, Septiembre 2002

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Partners: UGR

Status: accepted and to be presented at a conference session

• “Evaluation of Pre-Fractal Antennas Performance”, Juan M. Rius, Josep Parrón, Alex Heldring, Jordi Romeu, José Maria Gonzalez Arbesú, Juan R. Mosig, Gabriel Bugeda, Rafael Gómez, Amelia Rubio, Mario Fernández Pantoja, Massimiliano Giona, Journées Internationales de Nice sur les Antennes (JINA'02), Nice, 12-14 November 2002.

Partners: UPC, EPFL, CIMNE , UGR, ROME

Status: accepted and to be presented at a poster session

7.2 International Journals

The following international journal papers have already been submitted:

• “Method of Moments enhancement technique for the analysis of Sierpinski pre-fractal antennas”, Josep Parrón, Jordi Romeu, Juan M. Rius and Juan Mosig, IEEE Trans. on Antennas and Propagation.

Partners: UPC, EPFL

Status: accepted with minor modifications

7.3 FRACTALCOMS project web site

For efficient information flow within the consortium, a web based information hub was opened at UPC on May 31. All reports and data gathered during project development will be available in the information hub. The internet address of the project website is http://www.tsc.upc.es/fractalcoms.

In order to ease the dissemination of both the project results and the newly gained knowledge, all final versions of reports and accepted journal or conference publications will be of public access. Draft version of reports and manuscripts submitted for publication but not yet accepted will have access restricted only to consortium partners.

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8 WP1: THEORY OF FRACTAL ELECTRODYNAMICS

8.1 T1.1: Understanding fractal electrodynamics phenomena

8.1.1 UPC:

Study of the Koch fractal monopole in the frequency domain (see Annex WP1 T1.1 UPC Koch)

Some fractal geometries have complex, highly convoluted shapes. This property can be used to build antennas with pre-fractal shape that occupy the same volume as their conventional counterparts but much longer, and consequently, with a lower resonant frequency. The classical Koch fractal curve shown is an example of such objects where the resonant frequency of the monopole configuration decreases as the number of fractal iterations (K1, K2, K3...) increases. The understanding of the physic phenomena behind this behavior is the objective of the work done at UPC within this task. The Koch fractal monopole has been compared with a zig-zag monopole (non-fractal geometry) to evaluate which configuration is more suitable for the reduction of size of antennas.

The Koch fractal is a one-dimensional curve, but, unfortunately, the numerical simulation using a wire model is not accurate for highly iterated pre-fractals because the convoluted shape invalidates the thin-wire approximation. However, the antenna prototypes are built using printed strip technology and the strip can be modeled easily by a triangle mesh, that can be accurately analyzed by MoM codes for three-dimension arbitrary objects (i.e. FIESTA code created by UPC) if enough integration points are used on each triangle. For easier meshing, the strips have been obtained by extrusion of the curve in the direction perpendicular to the plane containing the curve.

The conclusions of the study are the following:

1. The resonance frequency of a Koch monopole decreases with the number of iterations, however it stagnates in a certain frequency. This is due to the coupling between corners since the signal does not follow the pre-fractal curve but takes shortcuts. This effect increases with the width of the strip-wire: in wider strips more power is coupled between corners and the length of the effective path traveled by the signal is shorter, resulting in a larger resonant frequency for the same IFS iteration (Fig. 3).

Fig. 3. The signal does not follow theconvoluted shape of the fractal but takesshortcuts between bends. Because of thisthe resonance frequency stagnates whenthe number of iterations grows.

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2. From the point of view of miniaturization, the Koch fractal monopole has shown a poor performance in terms of quality factor and minimum resonance frequency achievable, in front of a zig-zag monopole of the same length enclosed in the same rectangle (Fig. 4). The reason is obviously that the zig-zag antenna ocuppies more efficiently the area within the rectangle. This fact breaks the non-sustainable assumption that pre-fractal antennas fill better the space than the conventional ones.

3. The quality factor of the pre-fractal Koch antenna is similar to that of the (much shorter) zig-zag that resonates at the same frequency, and larger than the Q factor of the zig-zag antenna of same length (Fig. 5). The best quality factor achieved for both geometries is still quite far away of the fundamental limit for small antennas.

Fig. 4: Resonant frequency of Koch monopoles and zig-zag antennas of equal enclosing area as a function of strip length

Strip width 1 mm

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

5 10 15 20 25 30Strip length (cm)

Res

onan

ce fr

eque

ncy

(GH

z)

Zig-zagKoch

K0

K1

K2 K3 K4 K5

Z0

Z1

Z2

Z3

Z4 Z5

Z6 Z7 Z8

Fig. 5: Q factor of Koch monopoles (0 to 5 iterations) and zig-zag antenas with 3 and 8 meanders, compared with the fundamental limit for small antenas (h=antenna height).

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0

100

200

300

400

500

kh

Strip width 1 mm

Limit Z3 Z8 K0 K1 K2 K3 K4 K5

K0

K1

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Fractal Dimension versus Quality Factor: Modelling with NEC (see Annex WP1 T1.1 UPC Dimension NEC)

This set of activities is aimed to check through computer simulations the hypothesis that monopoles with high fractal dimension have lower Q. For carrying out the simulations the classical freeware Numerical Electromagnetics Code (NEC), based on the thin-wire approximation has been used. Proprietary software FIESTA (Fast Integral Equation Software for scaTterers and Antennas in 3D) has been used too to correlate some results.

The main results are:

• Computer simulations for different pre-fractal antennas show that increasing the number of IFS iterations reduces the Q factor at frequencies much lower than the resonant frequency, where the antenna is electrically small. As usual, the Q factor grows very rapidly when the operating frequency decreases.

• However, increasing the number of iterations also reduces the resonant frequency, leading to Q factors at the resonant frequency that are larger for more fractal iterations.

• The radiation resistance at the resonant frequency is also reduced when the number of iterations increases, making highly-iterated pre-fractal wire antennas very inefficient and difficult to match to the feeding circuit.

• Generalized Koch monopoles with different fractal dimensions have been simulated. The Q factor depends on the number of corners in the pre-fractal curve and on the fractal dimension of the limiting fractal curve. Configurations of different number of iterations that have a similar number of corners have been compared, observing that configurations with larger fractal dimension have lower Q factor. However, the differences in the number of corners are too large to draw a conclusion regarding the relation of the Q factor with the fractal dimension of the limiting curve.

• The same as in other UPC and UGR simulations and in the ROME theory, it has been observed that the parameters of the antenna do not change after a few IFS iterations. This is explained by the corner-coupling effect, suggested by UPC and visualized at UGR simulations.

• When comparing different pre-fractal antennas, the ones with larger fractal dimension have smaller resonant frequency. However, in these simulations the wire length is much longer and the occupied area much larger for the configurations with higher fractal dimension, which easily explains the reduction of the resonant frequency.

• Comparing pre-fractal antennas with limiting curves of different fractal dimensions that have the same the resonant frequency: the antennas with larger fractal dimension have larger Q factor and smaller radiation resistance due to the fact that their size is smaller.

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Monopole Iteration Fractal Dimension

Height (cm)

Wire length at λ0

k0h Q0 Rin (Ω)

800 MHz

λ/4 - 1 9.0 0.24 1.51 7.7 30 Koch K-3 3 1.262 5.9 0.38 0.99 17.4 13

Generalized Koch

Variant 1 GK1-2 2 1.293 5.9 0.36 0.99 17.3 13

Sierpinski Arrowhead SA-4 4 1.585 3.1 0.41 0.51 65.3 3

Peano P-2 2 2 2.6 0.56 0.4 125.3 2

• The quality factor of a monopole depends highly on the topology. Preliminar results for a few configurations show that the higher the number of loops for the same fractal dimension, the smaller the quality factor.

Table I. Height and wire length for several fractal monopoles tuned at 800 MHz. Fractal dimension,radius of enclosing sphere and Q at resonance are shown too.

2.55 cm

λ/4

K-3 9 cm

GK1-2

SA-4P-2

Fig. 6: Monopoles that resonate at 800 MHz. The smaller the monopole, the larger Q factor andlower radiation resistance.

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8.1.2 EPFL:

The meander-line printed antenna has been carefully studied in order to see if it has high-gain localized modes, like the pre-fractal Koch-island patch, or not. The objective here is to assess if the localized modes are exclusive of pre-fractal structures, or not.

The meander antenna here is a rectangular printed patch with N gaps of infinitesimal width parallel to two sides of the rectangle. The N gaps transform the rectangular patch in a line with N meanders.

The meander antenna has been compared with the original patch without the gaps:

At low frequencies, below the patch resonance, the current follows the meander line. The input impedance of the meander behaves as that of the equivalent line. This is useful for antenna miniaturization, because it presents almost the same resonance frequency that unwrapped line and occupies much less space.

At high frequencies (for example 4th resonance of the patch) the meanders couple one with the other and the antenna behaves more like a patch with gaps. Hot spots or “localized modes” appear at the end of the gaps (Fig. 7). The ones at the top of the antenna are in phase, and the ones at the bottom are also in phase, but in counter-phase with the ones at the top.

For higher frequencies complex resonance phenomena appear due to EM coupling between the parallel lines. Further studies are needed in order to understand these complex phenomena.

Since localized modes have been known since the 60’s and outside the scope of fractal or pre-fractal structures, we can conclude that localized modes are not exclusive of pre-fractal antennas.

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0

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0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

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x [mm]

y [m

m]

Fig. 7: Higher order mode at the printed meander line antenna. The color scale shows the phase and the arrow length the magnitude of the imaginary part of the electric current.

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9 WP2: VECTOR CALCULUS ON FRACTAL DOMAINS

9.1 Task 2.1: Solution of EM simple problems on fractal domains

ROME

a) Connection between the topological formulation of field equations on simplicial complexes and the continuum limit.

The analysis electromagnetic fields on fractal supports leads to a topological formulation of the Maxwell equations. Since the domain in which the electromagnetic field propagate is fractal, the main issue is to reformulate the field equations in a way suitable to be applied for fractal lattices. This problem is extremely interesting for its physical and mathematical implications, but is rather aside the straight applications in engineering electromagnetic problems, especially as it regards antenna design. We think that this approach is not essential for the solution of antenna problems involving fractal structures. For that reason, the ROME reports corresponding to task 2.1 (Deliverable D3 and the Annex to this progress report) have reviewed the central ideas of this approach, without entering into the more technical details associated with the theory.

b) Solution of paradigmatic examples of electromagnetism in the presence of fractal structures

These problems involve the representation of sources radiating in free space and defined on fractal boundaries, which can be tackled by means of the formal apparatus developed in WP2 Task 2.2 (see Deliverable D4 and Annex to this progress report). This class of problems encompasses all the direct and inverse problems in antenna theory. The radiation problem of an antenna can be tackled by expressing the vector potential with respect to a current source distribution localized on a fractal support, the estimate of which can be obtained by solving the corresponding electromagnetic integral equation on a fractal support. The main issues are in this case the representation of scalar and vector fields on fractal structures, the setting of the integral equations on fractals and their solutions.

In the first place, direct electrostatic problems on fractals have been solved. By direct problems we mean that either the electric charge density or the electric current density are specified on a given structure. The classical integral equations involving charge/current density and scalar/vector potentials have been reformulated for fractal structures. The electrostatic potential of both a uniform and non-uniform parameterised charge distribution on a Koch curve and a non-uniform charge distribution within a Sierpinski gasket have been obtained.

In the second place, the solution of inverse electrostatic problem is addressed. Essentially, the inverse problem can be stated as finding the charge distribution on a fractal structure that produces a given scalar potential. It has been observed that, as the number of IFS iterations increases, the charge density –solution of the inverse problem- becomes a multifractal measure in the limit. Conversely, the charge measure –integral of charge density along the fractal curve- converges towards an invariant shape as the number of iterations increases.

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The conclusions are:

The charge density on a fractal structure displays a highly singular structure (resembling a multifractal measure). This is explained intuitively since fractal conductors display corners at all the lengthscales, and the singularity in the charge distribution reflects the presence of them.

The solution of the inverse problem is well posed in terms of a charge measure rather than by enforcing charge densities. A limit charge measure, solution of the inverse electrostatic problem is numerically attained after few iterations in the construction process of the fractal structure.

The highly singular structure of the charge measure motivates the use of collocation approaches which makes use of localized collocation functions.

The behaviour of pre-fractal condensers have been analysed as a function of the spacing between the two plates, and have been compared with the behaviour of parallel-plate condensers. Specifically, for the third iterate in the generation of the modified Koch curve, the fractal condenser holds a much larger (factor larger than 8) overall charge than a Euclidean one, possessing the same overall end-to-end length.

In the third place, the fully vector radiation problem is addressed. The vector potential and the radiation pattern have been obtained for uniform and sinusoidal current distributions on modified Koch pre-fractal antennas after 3 to 5 IFS iterations. It is remarkable that the radiation pattern converges towards a limit just after a few iterations.

9.2 Task 2.2: Formulation of EFIE on fractal domains

ROME

In this task, the formal mathematical apparatus for setting the integral equations of applied electromagnetics on fractal curves has been provided. The analysis starts from the parametric representation of fractal curves through Augmented IFS (AIFS), defines the concept of tangential sign measures, and develops the formulation of contour integrals over fractal curves through the use of these quantities.

Subsequently, the formulation of EFIE on fractal wire antennas has been developed using the thin-wire Pocklington equation. The numerical solution by method of moments has been thoroughly addressed. The extension of EFIE to more general fractal sets has been addressed by defining a parametrization of the fractal support through the use of space-filling curves.

These results are important because they represent a first systematic attempt towards a formulation of a vector field theory in true fractal domains in order to solve inverse electromagnetic scattering problems through EFIE on mathematical fractal supports, and not solely on finite approximations (pre-fractals) of the structure. This is, to our knowledge, the first theoretical formulation of the EFIE describing wire antennas on a fractal support, and is a significant improvement of the ideas and of the approaches proposed in a recent past by the members of the ROME group.

From an antenna engineering point of view, one of the most important results of task 2.2 is the following: In a pre-fractal structure generated by an IFS, when the

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number of iterations increases, the direction of the current flow changes many times inside a reduced volume. These rapid current variations do not converge with the number of iterations and are very difficult to model for point-based methods used to discretize the Electric Field Integral Equation (EFIE). However, the radiated field and input impedances are computed though a integration, or averaging, of the antenna current.

In this workpackage, the ROME group has demonstrated that the integrals of the current or charge along the pre-fractal antenna show a very fast convergence with the number of iterations. This has two consequences:

1. When the IFS iteration number increases beyond a certain threshold, the change in radiation patterns and input impedance of the antenna tend to zero. In other words, there is no use in increasing the number of IFS iterations. Convergence is usually achieved between 4 and 6 iterations. This value depends largely on the size and topology of the antenna, as can be demonstrated by the theory of mutual interaction between corners (suggested by UPC and numerically verified by UGR, see annexes WP1 T1.1 UPC Koch and WP3 T3.4 UGR to this T0+6 progress report).

2. The Galerkin (or weak) formulation of method of moments is the most convenient approach to discretize EFIE of fractal supports rather than point-based approaches like collocation and Nystrom methods. This is due to the fact that on a fractal structure it is not possible to enforce the boundary conditions in a point-wise way, while it is possible in a measure-theoretical (integral) sense, by making use of the tangential sign measures.

The ROME group suggests, as a first approach, the discretization of Pocklington EFIE using global (non-localized) basis functions in order to reduce the computational complexity.

The results so far obtained define the pathway and the program for the future research activity that will be performed by the ROME group within the FRACTALCOMS project, which will be oriented on the numerical application and exploitation of the theoretical tools described in this report by extensive simulation of EFIE for fractal wire antennas.

The use of a parametric representation of more complex geometrical sets through a space filling curve on them, coupled with the formulation of EFIE on fractal curves provides an interesting research field that will be also analyzed by means of numerical simulations by the ROME group in the remainder of FRACTALCOMS project.

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10 WP3: SOFTWARE SIMULATION TOOL

10.1 Task 3.1: Advanced meshing of fractal structures

10.1.1 CIMNE

An automation tool has been created to build Koch strip antennas and Sierpinski triangle antennas easily from within GiD. This window is invoked through a new menu in GiD, called “Antennas” (see annex WP3 T3.1 CIMNE to this T0+6 progress report).

Koch strip antennas: The maximum number of iterations is 10, which is more than enough for the current

state of electromagnetic simulation software.

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Sierpinski antenna: The maximum number of iterations is again 10, which is more than enough for the

current state of electromagnetic simulation software.

The meshing includes the connection triangles, but is not adaptive yet.

Future:

Presently, the meshing utilities developed by CIMNE are restricted only to Koch and Sierpinski pre-fractals.

A new UPC-CIMNE workshop will be held in order to discuss:

If CIMNE will develop a general tool for meshing any pre-fractal defined by an IFS or if many routines will be coded for different cases.

The detailed output requirements of the meshing software.

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10.1.2 UPC

One of the items that were scheduled in this task was the development by UPC of a Nystrom method discretization scheme for the EFIE in pre-fractal structures.

A first version of the formulation and code has been developed for the analysis of electrostatic problems. The kernel singularity has been corrected by Strain’s method, using polynomial testing functions. The charge is therefore represented by a underlying polynomial basis. The results for objects with open surfaces are very bad, since the charge is singular at edges and cannot be expanded by the underlying polynomial quadrature basis. On the contrary, when the unknown charge is a polynomial, Nystrom method is gives very small errors, up to machine precision (Fig. 11).

However, since the integrals of the current or charge along pre-fractal antennas converge very fast with the number of iterations (result of WP2), then it is possible to discretize the EFIE in highly-iterated pre-fractals using the conventional Method of Moments, which is in essence a weighted residual procedure.

On the other hand, the Nystrom method is not adequate for the analysis of pre-fractal antennas, since:

Nystrom method is based on quadrature integration rules that relay in underlying polynomial basis. Since the rapidly varying -almost discontinuous- current and charge in pre-fractals cannot be expanded by a polynomial basis, Nystrom method looses its power of accurate integrations with few samples and results will be very poor.

Nystrom method forces the EFIE equation in a point-matching fashion. Since the near field is also rapidly-varying, it is best to use a weighted residual procedure such as method of moments.

In conclusion, we think that the method of moments with Galerkin testing (weak formulation) is the most robust approach to discretize the EFIE in pre-fractals and there is no need to continue working in Nystrom method. This assumption has been corroborated by the excellent results of numerical simulations in tasks 3.3 and 3.4.

Fig. 11: Relative error in the computation of a polynomial charge distribution in a perfectlyconducting square plate from the electrostatic potential. Nystrom method with different ordersof quadrature has been used. In absisas, the number of triangles along the plate edge.

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10.1.3 EPFL

EPFL presented the development of a new set of basis functions defined on quadrangular regions. Particular cases of these functions are the well-known rooftop and Rao, Wilton and Glisson (RWG) basis.

These new functions will be very useful for the project because they simplify enormously the electric connection between different patches of the pre-fractal that share only one vertex. As a result, it will be easier to build adaptive meshes having less unknowns and modeling better the variation of the current in the pre-fractal geometry.

Three cases of quadrangular basis functions have been considered:

With constant normal component of the current at one edge. The divergence of the current (charge) is not constant.

With constant divergence of the current, but non-constant normal component of the current at the edge.

Having both properties of constant divergence of the current and constant normal component of the current at one edge.

The best results are obtained with the 3rd option. The agreement with the conventional rooftop and RWG functions is very good.

The potential of quadrangular basis functions for solving connectivity problems was shown in the simulation of a bow-tie antenna with different feeding zone sizes. The results were very promising.

Fig. 12: On the left, quadrangular mesh of a bow-tie antenna and, on the right, magnitude of surface density currentobtained with quadrangular basis functions.

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10.2 Task 3.3: Simulation of fractal structures in the frequency domain

10.2.1 UPC

Task 3.3 was originally scheduled to start at T0+6. However, in the kick-off meeting it was decided that task 3.3 started at T0 because UPC was interested in developing algorithms to reduce the computational cost of pre-fractal antennas simulation in the frequency domain.

The numerical analysis of highly iterated pre-fractal antennas by Method of Moments (MoM) involves many tiny subdomain basis functions, resulting in a very large number of unknowns. When these antennas can be defined by an Iterated Function System (IFS), the geometry has a multilevel structure with many equal subdomains. This property, together with a Multilevel Matrix Decomposition Algorithm (MLMDA) implementation in which the MLMDA blocks are equal to the IFS generating shape, has been used to reduce the computational cost of the frequency domain analysis of a Sierpinski based structure (see Annex WP3 T3.3 UPC Sierpinski to this T0+6 progress report).

It has been shown that the combination of the GMRES and the MLMDA scheme, together with the appropriate choice of the shape of the boxes in the multilevel subdivision, leads to a very efficient solution. Our best implementation produces a reduction by a factor of 20 in the total computation time and a factor of 10 in the total memory, compared with a direct application of MoM. The bottleneck in the MoM analysis of IFS defined geometries is now in the preconditioning, which has not been optimized for IFS structures yet.

MoM+GMRES GMRES+MLMDA optimized

Memory Time Memory Time

4 it., N=568 5.3 MB 5 s. 1.8 MB 1.1 s.

5 it., N=1702 48.0 MB 25 s. 7.9 MB 4.1 s.

6 it., N=5104 424.8 MB 219 s. 39.6 MB 25 s.

7 it, N=15310 - - 166.5 MB 176 s.

It has been observed, through numerical experiments, that the resonant frequency of the Sierpinski patch diminishes as the number of fractal iterations of the geometry increases. This feature can be used to build miniature antennas, however more research must be done in order to find out which are the limits for these pre-fractal structures.

Table II Computational requirements to analyze the Sierpinski antenna with the conventional method ofmoments (MoM) and GMRES iterative solver compared with the optimized approach developed in thisproject.

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10.3 Task 3.4: Simulation of fractal structures in the time domain

10.3.1 UGR

UGR group has a home-developed code Method of Moments in the time domain (MoM-TD) code, DOTIG, that is as an excellent tool for the visualization of electromagnetic scattering and radiation phenomena. DOTIG, has been extended in FRACTALCOMS project with non uniform segmentation, ground plane capability and the Prony and pencil parameter extraction techniques.

The current and near fields for the 1st and 2nd iteration of Koch monopoles in free space (K1 and K2) have been computed and presented in time-domain videos. In these videos the corner-coupling effect suggested by UPC is clearly visible when dB scale was used. This effect explains why the resonant frequency of pre-fractal antennas is much larger than what could be expected from the wire length only (see Annex to this T0+6 progress report).

UGR has also computed time-domain results with sinusoidal excitation.

UGR has compared the input impedance and Q factor results of time-domain simulations (DOTIG) with the well established NEC and WIPL frequency domain codes. The results agree very well for K1 and K2. For K3 there are important discrepancies between the three results. The three codes use different forms of the thin-wire approximation. This fact corroborates the UPC hypothesis that thin-wire approximations fail for highly-iterated pre-fractals (see report that will be prepared for T0+12 milestone). However, although the results are not very accurate, they are still useful for visualizing electromagnetic phenomena in the time domain and to draw conclusions about pre-fractal antennas behaviour.

UGR has computed the current spectrum at different points of the antenna showing some interesting effects: the corners in the pre-fractal radiate the higher frequencies in the current spectrum. As a consequence, the current at the end of the antenna contains only low-frequency spectral components. It does not radiate at high frequencies. Therefore, a coarser mesh can be used and even some antenna segments can be dropped in the last part of the antenna, without altering antenna input impedance for operating frequencies more than 10 times above resonance. Unfortunately, this interesting conclusion is not useful for the miniaturization of fractal antennas, since the operating frequency must be equal or smaller than the resonant one.

The version of DOTIG code for the analysis of surface antennas modelled with a triangular mesh was also presented by UGR. Results were presented for the Koch wire-strip and the Sierpinski antennas.

The Koch strip width is twice the Koch wire radius. The input impedance versus operating frequency curves are different for the thin-wire and the wire-strip. However, the resonant frequency is the same. It has been observed by UPC group that the resonant frequency of Koch wire-strips is highly dependant on the strip width. A physical interpretation of why both antennas resonate at the same frequency when the width of the strip is twice the diameter of the wire must be found.

Future work: UGR suggested the following work to carry out for the T0+12 milestone:

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Analyse more pre-fractal structures

Solve the problem of wire attachments with non-uniform segmentation in DOTIG.

Search for new miniature antennas alternative to pre-fractals shapes.

Implement algorithms to reduce the computational cost of the solution of large problems.

All the results presented by UGR group where based on short-pulse or high-frequency sinusoidal excitations. Since a short pulse contains all the frequency spectrum from 0 up to a given frequency, the input impedance of the antenna can be computed in a large frequency band. In addition, since the pulse is much shorter than the antenna, different parts of the antenna are excited at different times and one can visualize the radiation of these parts separately. For that reason, the short-pulse results have been very useful to visualize the corner-coupling effect, which is perhaps the main contribution of the first 6 months of the project.

However, since the central point in this project is the miniaturization of pre-fractal antennas, it might be also useful to excite the antennas with wide pulses in order to visualize the simultaneous radiation of all parts of the antenna and how do they interact with each other.

Fig. 13: Near field for a one-iteration Kock monopole excited by a Gaussian short pulse. Notice the wave front with phase origin at the corner of the strip. Corners radiate and receive signal, thuscreating shortcuts between different parts of the antenna.

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Fig. 14: Space-time diagram for a two-iteration Koch dipole excited by a Gaussian short pulse. The arrows inthe zoomed rectangle clearly show the coupling between different parts of the antenna (shortcuts betweencorners).

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11 CONCLUSIONS

When the IFS iteration number increases beyond a certain threshold, the change in radiation patterns and input impedance of the antenna tend to zero. In other words, there is no use in increasing the number of IFS iterations. Convergence is usually achieved between 4 and 6 iterations. This value depends largely on the size and topology of the antenna, as can be demonstrated by the theory of mutual interaction between corners.

The current and near fields for the 1st and 2nd iteration of Koch monopoles in free space (K1 and K2) we presented in time-domain videos. In these videos the corner-coupling effect suggested by UPC was clearly visible. This effect explains why the resonant frequency of pre-fractal antennas is much larger than what could be expected from the wire length only and highly dependant on the width of the strip.

It seems that the high-gain localized modes than have been previously observed in the Koch-island printed patch antenna are not exclusive of pre-fractal antennas.

Highly-iterated pre-fractal wire antennas are not suitable for most practical applications of miniature antennas since they have smaller radiation resistance, larger ohmic losses and smaller Q factor at the resonant frequency than other miniature antennas.

Comparing pre-fractal antennas with limiting curves of different fractal dimensions that have the same the resonant frequency: the antennas with larger fractal dimension have larger Q factor and smaller radiation resistance due to the fact that their size is smaller. They are also not suitable for most practical applications of miniature antennas.

We have not observed yet any relation between the fractal dimension and the parameters of the antenna (resonant frequency and Q factor), when all the other geometry parameters (antenna size, wire length, occupied area) are equal.

It has been decided to drop Nystrom method from the workplan WP3 T3.1, since the conventional Method of Moments (MoM) is expected to give better results for pre-fractal antennas.

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12 TASKS THAT START BETWEEN T0+6 AND T0+11

WP3: Software simulation tool

o T3.2: Formulation of numerical methods for fractal structures

WP4: Fractal devices development

o T4.3: Prototype construction and measurement

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