Modeling Electromagnetic Radiation in the Time Domain using the Finite Element Method

Most of my before this involved the solution of Maxwell's Equations in the frequency domain (which is generally simpler). Modeling the open region, where waves can radiate outwards, never to return, is one of the “Holy Grails” of computational electromagnetics. In this work, I start with Maxwell's equations (in two dimensions now, for simplicity)

The first expression is the electric field curl equation, the second and third equations form the components of the magnetic field curl equation. The last expression is the divergence of the electric field (which vanishes because we are assuming a charge-free region). Note that these equations are equivalent to solving a corresponding second order scalar wave equation for hz . The z direction is assumed to be out of the page (the x-y plane is in the plane of the page). This corresponds to the TM propagation modes. TE modes can be modeled by replacing the components of H with E and E with -H and the wave impedance Z0 with wave conductance G0. The first curl equation will will become inhomgeneous, with source term -Jz while the second and third equations' source terms vanish.

By discretising these equations in time (using a Crank-Nicolson or implicit Euler scheme) and constructing a high-order finite-element discretisation in space, one can generate accurate representation of large bandwidth electromagnetic pulses in the time domain. By assuming that the waves obey

on the exit surface (where p0, p1 are constants which govern the reflection from our wave from the surface and is the curvature of the boundary, the subscripts n and s indicate directions normal and tangential to the boundary surface), we can simulate, with varying degrees of success, the disappearence of wave energy into the exterior, free-space region.

Material available for download



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Stability.pdf (1.1MB)

Paper that appears in the Proceedings of the 2004 WSEAS conference on Applied Mathematics, Corfu, Greece.



A draft (unpublished) paper which covers aspects of constructing the absorbing boundary condition using the least-squares finite element method.


Talk1BW.pdf (1.3MB)

Talk presented to the 2004 WSEAS conference on Applied Mathematics.


Last updated 20050925, Bill Slade