Task 2.2: Formulation of Electric Field Integral Equation (EFIE) on fractal domains

• Deliverables: D4 T2.2 Final Task Report
• Outputs: General mathematical physical methods for fractal structures to be used in WP1 and WP3.

The Electric Field Integral Equation (EFIE) is the basis for the computation of antenna radiation and of most of the parameters of many microwave devices. It is therefore necessary to reformulate this equation in the case the induced electric and magnetic sources are localized on fractal structures. This will be developed in this task aimed at:

a) Solution of integral equations in the presence of almost-everywhere singular kernels (corresponding to current distribution localized on a fractal structure).

b) Formulation of EFIE and related methods (e.g. methods of moments) within the framework of the theory developed in Task 2.1

c) Parametric analysis of primary current distribution leading to multifractal electric and magnetic sources for simple model fractal antennas.

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 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.