Adaptive finite element analysis of unbounded electromagnetic wave problems

An adaptive finite element method is developed to solve two dimensional unbounded electromagnetic radiation and scattering problems.Though there has been considerable research in adaption in electromagnetics, to date there have been few attempts to deal with adaptively reducing the error introduced by the boundary that artificially truncates the unbounded domain in a finite element analysis. The technique proposed here adapts on this boundary error, as well as on the usual finite element discretization error. It combines three techniques: (i) p-adaptive hierarchial triangular finite elements, (ii) wave-envelope elements and (iii) absorbing boundary conditions.Hierarchial finite elements allow the polynomial orders of the elements to be used to control the distribution of degrees of freedom, and therefore they make p-adaption possible, i.e. adaption by varying the element orders. This is more efficient than h-adaption which requires an actual re-meshing of the finite elements.The wave-envelope method uses a change of dependent variable to remove the wave-like qualities of the solution and thereby permits the use of very large elements, i.e. elements much larger than a wavelength, in the external domain.An absorbing boundary condition is a boundary operator which approximately absorbs all the radiation incident on the boundary from within.In the new method, the scattering or radiating object itself, and its immediate surroundings, are meshed with hierarchial finite elements. Outside that, a thick layer of free space is meshed with hierarchial wave-envelope elements. The layer is thick enough that when an absorbing boundary condition is imposed on its outer surface, there is very small reflection from it. Such a thick layer can be meshed with relatively few wave-envelope elements. The boundary error seen by the finite element region is then a function of how well the wave-envelope region is discretized. During the p-adaption, increasing the order of the wave-envelope elements increases their ability to model the field accurately and therefore reduces the boundary error. Moreover, this reduction in boundary error is selective: in directions of strong radiation, the error reduction is greater.The new method is validated by comparison with exact solutions for a number of radiation and scattering problems over a range of frequencies