Graphical representation of workpackages
WP0 Project Management
WP1: Theory of fractal electrodynamics
The theory of fractal electrodynamics is an emerging field in electromagnetics. The subject of study is the interaction of electromagnetic waves with fractal objects. In a general manner the scattering of electromagnetic waves by fractal boundaries. Radiation problems including antenna design, are just particular cases of scattering problems in which the electromagnetic wave is generated at the feeding point. While there is a sound mathematical framework to analyze electromagnetic scattering for Euclidean objects, there are not equivalent theoretical foundations for the analysis of scattering of fractal objects.
In understanding fractal electrodynamics phenomena there are two issues that must be tackled. First is the identification of key mathematical bottlenecks that appear in formulating the scattering by fractal objects. At the expense of the completion of the workpackage and just to name a few obvious ones the following items can be mentioned: Gauss and Stokes theorems in fractal boundaries, Neumann boundary condition in fractal boundaries, or line integral along a fractal curve. A second item of importance is the appropriate way to model the fractal structures. Fractal geometries, although have an apparent high degree of complexity, can be defined in an astonishing simple manner: namely through the Iterated Function System (IFS) approach. One of the goals is to understand how the simple formalism of the IFS can be incorporated into the mathematical formalism of electromagnetic scattering in order to use with advantage its simplicity in solving complex problems. All this effort is essential for the development of an efficient software simulation tool in WP3.
WP2 Vector calculus on fractal domains:
Recently, a self consistent vector calculus on discrete lattices has been developed [9] based on algebraic topology of simplicial complexes, and this approach is particularly suitable for treating linear vector field equations on fractals, as required for a rigorous mathematical approach to electromagnetic theory in systems possessing dilation symmetry. This approach can be usefully coupled with real-space renormalization techniques [10,11,12] in order to obtain, either analytically (whether possible) or numerically (with arbitrary numerical accuracy), the solution of linear field equations at a generic level of construction (iteration) of the fractal structure. Specifically, the renormalization approach developed in [12,13] is suitable to be applied in the optimization of current distribution in a fractal antenna, by considering e.g. multifractal distributions, which may further improve antenna performances. A significant effort should be spent in order to apply these approaches (vector calculus on simplicial complexes and Green function renormalization) to solve electromagnetic problems. The solution of these fundamental issues is neither formally nor computationally trivial and straightforward, and may lead to a consistent and new formulation of electromagnetic field equations on fractals which is grounded on the intrinsic iterative and dilation-symmetric nature of fractal structures.
| [9] | W.A. Schwalm, B. Moritz, M. Giona, M.K. Schwalm, “Vector difference calculus for lattice models”, Phys. Rev. E, Vol. 59, 1999, pp. 1217-1233. |
| [10] | M. Giona, W.A. Schwalm, M.K. Schwalm, A. Adrover, “Exact Solution of Linear Transport Equations in Fractal Media I - Renormalization Analysis and General Theory”, Chem. Engng. Sci., Vol. 51, 1996, pp. 4717-4729. |
| [11] | A. Adrover, W.A. Schwalm, M.Giona, D. Bachand, “Scaling and scaling crossover on anisotropic fractal structures”, Phys. Rev. E, Vol. 55, 1997, pp. 7304-7312. |
| [12] | M. Giona, “Transport Phenomena in Fractal and Heterogeneous Media – Input / Output renormalization and exact results”, Chaos, Solitons & Fractals, Vol. 7, 1996, pp. 1371-1396. |
| [13] | M. Giona, “Analytic expression for the structure factor and for the moment-generating function of fractal sets and multifractal measures”, J. Phys. A, Vol. 30, 1997, pp. 4293-4312 |
WP3 Software simulation tool:
Current numerical modeling methods are not adequate to tackle with fractal devices for the following reasons:
A large effort must be done in order to push forward numerical modeling theory and algorithms in order to develop special purpose advanced software tools, so that the aims of this project can be reached.
WP4 Fractal devices development:
One of the objectives of the project is to explore the technological limitations of fractal-shaped microwave devices, including minimum detail size and loss efficiency. As for manufacturing process standard photo-etching techniques on microwave dielectric substrates will be considered. Microwave resonators and antennas are considered as microwave devices to be developed. Microwave resonators are essential building blocks of microwave filters. The potentiality to reduce the size of the resonator and still maintain low losses, or alternatively a high Q will be explored. In the case of antennas the possibility to miniaturize antenna and not compromising its loss efficiency will be assessed. This is a terminal workpackage that heavily relies on results of the previous tasks.
Gantt chart
