Closed-Form Expressions for Local Absorbing Boundary Conditions in Electromagnetic Scattering Problems and Their Implementation into Commercial FEM Software

When solving open-region electromagnetic scattering problems using the standard finite element method, the infinite region exterior to the scatterer must be truncated with an artificial outer boundary. In the interior or finite part of the domain, finite elements can be used, but a boundary condition must be introduced at this artificial boundary to obtain a unique finite element solution. One class of boundary conditions designed for this purpose is called absorbing boundary conditions (ABCs). Several ABCs have been reported in the literature. These can be broadly classified into two categories: local and non-local. In this paper, we present new closed-form expressions for Nth-order local ABCs, equivalent to the well-known sequences of the Bayliss, Gunzburger, and Turkel boundary operators. To the best of our knowledge, these expressions have not yet been reported in the literature. Two- and three-dimensional problems are considered in this study. We also discuss the problem of incorporating custom ABCs into commercial finite element method programs, which are usually closed-source software packages. Two 2D electromagnetic scattering and radiation problems with known analytical solutions are analyzed.