Task 3.2: Formulation of numerical methods for fractal structures

• Deliverables: D6 T3.2 Final Task Report
• Outputs: Formulation to be used in tasks 3.3 and 3.4.

The conventional numerical methods, based on the Electric Field Integral Equation (EFIE), must be re-formulated for fractal structures. Also, the usual thin-wire approximations for the analysis of wire antennas are not valid when the wire is a fractal curve. The actions to do in this workpackage are:

a) Use the results from task 2.2 to develop EFIE discrete formulation and numerical integration on fractal domains.

b) Develop new approximations for thin-wire antennas along fractal curves.

c) Compare results in a) and b) with classical numerical schemes pushed, if possible, to the limit of complexity afforded by the technology.

This task is essentially aimed at analyzing critically the problems associated with the numerical simulations of fractal structures based on the results achieved in Tasks 2.1 and 2.2, and at developing efficient computational approaches for a reliable numerical simulation of high-order iteration prefractal wire antennas using the Electric Field Integral Equation (EFIE). It was agreed in the Kick-off meeting that the research would be concentrated in wire antennas, that can be either a cylindrical wire or a strip.

Three main items have been addressed:

Validity of the thin-wire approximation:

The accuracy of the commonly used approximations for the analysis of thin-wire antennas when applied to fractal structures has been evaluated. It is concluded that:

• The thin-wire kernel of the Electric Field Integral Equation (EFIE) can be used in low-iteration pre-fractals without introducing significant errors. Examples of computer codes based on thin-wire approximation that have been used in this project are: DOTIG (time domain), NEC (frequency domain) and FIESTA (frequency domain).
• When the EFIE is discretized by Method of Moments in subdomain basis functions, the thin-wire kernel with or without equivalent radius produces important errors in the computation of highly iterated pre-fractal antenna parameters, since the pre-fractal curve segments become comparable to the wire diameter. Even the full kernel fails because the assumption of constant current along the wire circumference is no longer valid and the wire cylindrical segments overlap at corners. A extrusion strip can be used in this case in order to obtain an accurate subdomain discretization of the EFIE.
• The results of extrusion strip models computed with FIESTA code have been compared with the simpler thin-wire models. The comparison results fully support the conclusions above (Fig. 26).

Figure 26: Resonant frequency of a 6cm-height Koch monopole of 1 to 5 IFS iterations, computed using different formulations of the EFIE. Only the surface formulation with a extrusion-strip mesh shows the expected behaviour for a pre-fractal structure. The thin-wire approximation fails for highly iterated structures.

Implementation of wire antenna analysis in FIESTA computer code:

The computer codes for the analysis of wire antennas available at the start of FractalComs project, namely DOTIG and NEC, do not use advanced techniques for the solution of very large systems of equations. On the other hand, FIESTA code is able to solve the EFIE with hundreds of thousands of unknowns, but at the project start date only the analysis of perfectly conducting surfaces was implemented.

In order to solve very large wire antenna problems, the Electric Field Integral Equation for wires has been implemented in FIESTA. Thee different formulations of the kernel have been programmed: thin-wire, thin-wire with equivalent radius and full kernel.

The full kernel has been implemented using a new approach that allows computation in a run time only twice the time required for the much simpler thin-wire approximation. The new approach is based in a polynomial approximation of the 2-D integrals that arise from the full-kernel formulation, and is valid for segment length to segment radius ratios larger than 10^-4. Previously, the evaluation of the full-kernel base on series expansions of the integrand was cumbersome and computationally expensive.

New approaches for solving the EFIE of wire antenna analysis:

Some numerical approaches based on the Galërkin-Petrov expansion for solving the EFIE in pre-fractal structures have been studied. The approach proposed proves to be very versatile and computationally efficient. Particular attention is oriented towards the analysis of the thin-wire approximation from the numerical point of view, by discussing critically its validity and pitfalls, in the light of the compactness property of the associated linear operator. Numerical simulations have been performed on different pre-fractal structures, highlighting some interesting properties of the current profiles.

It is clear from the analysis developed in WP2 that any numerical simulation of the EFIE equations on fractal structures cannot be conveniently grounded on point-matching approaches, but should be framed within a weak-formulation of the EFIE resulting from a Galërkin-Petrov expansion in a basis of entire-domain functions. The proposed technique is therefore based in the Method of Moments Galërkin discretization of the Pocklington thin-wire kernel in a set of complete-domain sinusoidal basis functions.

Although the Pocklington equation for dipolar antennas has been known and widely used for a long time, the computational issues associated with the analysis of thin wires are still an open problem. Recent articles conclude that the thin-wire approximation gives rise to an “ill-posed problem”, and that all the simulations based on this approach are scientifically unreliable.

Since the thin-wire approximation is one of most useful simplifying assumptions in applied electromagnetism, we have analyzed very carefully this issue, and we have shown that, albeit the criticism of some authors is motivated by a correct mathematical observation, in practical problems involving thin-wire antennas the entire-domain Galërkin approach furnishes reliable and convergent approximations for the computation of the current and the antenna parameters. Of course, the numerical problems are much more delicate in dealing with fractal wire antennas, since the geometric complexity of the structures is superimposed to the intrinsic difficulty of handling a singular integral equation.