An amplitude finite element formulation for electromagnetic radiation and scattering

Electromagnetic radiation and scattering in an exterior domain within the context of a finite element method has traditionally involved imposing a suitable absorbing boundary condition on the truncation boundary of the numerical domain to inhibit reflection from it. In this work, based on the Wilcox asymptotic expansion of the electric far-field, we propose an amplitude formulation within the framework of the nodal finite element method, whereby the highly oscillatory radial part of the field is separated out a-priori so that the standard Lagrange interpolation functions that are used have to capture a relatively gently varying function. Since these elements can be used in the immediate vicinity of the radiator or scatterer (with few exceptions which we enumerate), it is more effective compared to methods that impose absorbing boundary conditions at the truncation boundary, especially for high-frequency problems. The proposed method is based on the standard Galerkin finite element formulation, and uses standard Lagrange interpolation functions, standard Gaussian quadrature and the same degrees of freedom as a conventional formulation. We show the effectiveness of the proposed formulation on a wide variety of radiation and scattering problems involving both conducting and dielectric bodies, and involving both convex and non-convex domains with sharp corners.