Combining perturbation theory and transformation electromagnetics for finite element solution of Helmholtz-type scattering problems

A numerical method is proposed for efficient solution of scattering from objects with weakly perturbed surfaces by combining the perturbation theory, transformation electromagnetics and the finite element method. A transformation medium layer is designed over the smooth surface, and the material parameters of the medium are determined by means of a coordinate transformation that maps the smooth surface to the perturbed surface. The perturbed fields within the domain are computed by employing the material parameters and the fields of the smooth surface as source terms in the Helmholtz equation. The main advantage of the proposed approach is that if repeated solutions are needed (such as in Monte Carlo technique or in optimization problems requiring multiple solutions for a set of perturbed surfaces), computational resources considerably decrease because a single mesh is used and the global matrix is formed only once. Only the right hand side vector is changed with respect to the perturbed material parameters corresponding to each of the perturbed surfaces. The technique is validated via finite element simulations.