Numerical Solution of Multi-scale Electromagnetic Boundary Value Problems by Utilizing Transformation-Based Metamaterials

Abstract

We present numerical solution techniques for efficiently handling multi-scale electromagnetic boundary value problems having fine geometrical details or features, by utilizing spatial coordinate transformations. The principle idea is to modify the computational domain of the finite methods (such as the finite element or finite difference methods) by suitably placing anisotropic metamaterial structures whose material parameters are obtained by coordinate transformations, and hence, to devise easier and efficient numerical simulation tools in computational electromagnetics by allowing uniform and easy-to-generate meshes or by decreasing the number of unknowns. Inside the modified computational domain, Maxwell’s equations are still satisfied, but the medium where the coordinate transformation is applied turns into an anisotropic medium whose constitutive parameters are determined by the Jacobian of the coordinate transformation. In other words, by employing the form-invariance property of Maxwell’s equations under coordinate transformations, an equivalent model that mimics the original problem is created to get rid of mesh refinement around the small-scale features. Various numerical applications of electromagnetic scattering problems are illustrated via finite element simulations.