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A discrete-topology method for bounded fractal domains' electromagnetic problems
No full textcomputational method called Topological Calculus, based on discrete mathematical tools known as
Algebraic Topology, is briefly introduced and its ability to represent a discrete counterpart to classical and
quantum field equations is shown.
First of all the Topological Calculus setting is that of a certain discrete class of n-dimensional lattices, called
simplicial complexes, whose constituting simplices are the n-dimensional analogue of triangles (0-simplices are
points, 1-simplices segments, 2-simplices triangles, 3-simplices tetrahedra, etc.). Topological Calculus
provides two nontrivial algebraic operators (cup ‘∪’ and cap ‘∩’ products) and two linear differential
operators (boundary ‘∂’ and coboundary ‘δ’), from the combinations of whom almost any
algebraic/differential equation can be imposed on the simplicial complex. They are, in fact, the discrete
counterpart of usual scalar ‘⋅’, cross ‘×’ and exterior ‘∧’ products, as well as divergence, gradient and curl Iorder
differential operators.
Once a suitable and accurate enough triangulation Σ of the continuum domain is performed, scalar, vector
and tensor fields (or differential p-forms, according to the exterior formulation of Field Theories) are
represented as p-chains: algebraic objects assuming different (e.g. time-dependent) values on each of Σ ’s psimplices.
The diagonalization of Σ ’s incidence and adjacency matrices allows to both compute the number of ‘holes’
inside the domain. As the simplicial Laplace (and Helmholtz)’s operators are correlated to the adjacency
matrices, this spectral decomposition also counts the number of linearly independent “harmonic functions”.
It is well known that the qualitative behaviour of electromagnetic fields on bounded domains strictly depends
on the topology of the region (i.e., roughly, on the number of ‘holes’ and connected components), for example
the number of a porous resonator’s static modes, or a multiply-connected cross-sectioned waveguide’s TEM
modes. Such information can be easily extracted with this method, and is refinement-independent from the
domain’s triangulation: it can be computed even from a “coarse” simplicial complex, as long as it is
topologically equivalent to the continuum domain.
Some examples are made with the spectral analysis of IFS prefractals of the Šerpinskij gasket and carpet. In
the case of self-similar simplicial complexes, adjaceny and laplacian matrices are recursively built using a
quasi-diagonal-block paradigm, physically interpreted as a renormalization of the multiscale electrodynamics
within. Self-similar eigenmodes and eigenvalue distribution is then observed, and compared with the wellknown
multi-band properties of fractal antennas and resonators
Proprietà spettrali di strutture elettromagnetiche frattali
No full textSome self-similar fractal sets may have simple closed-form Fourier transforms. This can be exploited in several Electromagnetics applications, as here suggested, such as respectively deriving either self-similar radiation or eigenmode patterns from (pre-)fractal apertures or resonators, as well as multifractal κ-space configurations of artificial, self-similar, crystal lattices or other electrodynamic hamiltonians
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