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DIAZ-FITZGERALD TIME DOMAIN (D-FTD) METHOD APPLIED TO DIELECTRIC AND LOSSY MATERIALS

We present a new approach to the discretized solution of Maxwell's Equations for a real medium. The problem is equivalent to the solution of the equations of motion of a mechanical system of spheres properly connected. This model is an extension of the model proposed by G. F. Fitzgerald in 1885. The method is different from the usual FDTD implementation in that it provides a condensed node representation for the electric field, and it also uses different leap-frog scheme to propagate the field. In addition it provides an essential pedagogical tool for further understanding of the solution of Maxwell's equations in the time domain. As a validation of the technique the scattered field from an infinitely long cylinder is calculated and is compared to the exact solution for the two cases with and without losses, excellent agreement is obtained. 1.Introduction The discretization of Maxwell's Equations can be carried out via the introduction of the vector potential. If space is discretized, the time evolution of electric and magnetic field in each mesh position can be correctly described by a leapfrogging scheme between the electric field and the vector potential, in distinction to the FDTD where the leapfrogging involves the electric and magnetic fields. The Hamiltonian of this system corresponds to that of an array of spheres with fixed axes all connected with rubber spheres. A similar mechanical model was first proposed by Fitzgerald[1] in 1885, and recently implemented as a numerical technique by Rodolfo Diaz[2]. Two of the principal strengths of the D-FTD are the fact it avoids convolution integrals in the description of dispersive media, and the fact that the electric field is localized at each node of the computational grid. 2. Mathematical model For the general case we report here the coupled equations between Electric Field and Vector Potential i.e.: ∂E ∂t = − 1 εμ ∇ A E = − ∂A ∂t − ∇ φ   