EM modelling of arbitrary shaped anisotropic dielectric objects using an efficient 3D leapfrog scheme on unstructured meshes

Abstract

The standard Yee algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of the classical Yee scheme to 3D unstructured meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This allows the problem of accuracy losses, which are normally associated with the use of the standard Yee scheme and a staircased representation of curved material interfaces, to be circumvented. The 3D dual mesh leapfrog-scheme which is presented has the ability to model both electric and magnetic anisotropic lossy materials. This approach enables the modelling of problems, of current practical interest, involving structured composites and metamaterials.

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

“…(27) and (26) are not constant and are replaced by averaged values ̄a v , ̄a v , ̄a v and ̄m av . In this case, every component of the material parameter tensor is averaged following the approach employed earlier for the isotropic case [32]. To represent the vector-matrix multiplications that are required, we define…”

“…To perform the necessary matrix-vector multiplications in Eqs. (32) and 33, we need to obtain the field vectors scat and scat and associate them with Delaunay edge i and Voronoi edge j respectively. Unfortunately, we cannot get exact full field vector components directly from edge projections, but we are able to approximate the full field components at any location in the mesh.…”

“…. A similar process is adopted for the magnetic field, but in this case we have a point from the Delaunay mesh inside a Voronoi cell [32].…”

“…For higher chiralities, however, the numerical solution shows very strong fluctuations indicating an instability of the scheme. We performed a stability analysis with respect to variations in the time step, the spatial step and the real and imaginary part of the chirality [48]. The results showed that the stability is independent of the time step size.…”

EM modelling of arbitrary shaped dispersive chiral dielectric objects using a 3D leapfrog scheme on unstructured meshes

“…The unstructured mesh procedure is essentially an unstruc-tured implementation of the FDTD method, using a primal Delaunay and its Voronoi dual mesh [4]. The approach, which has already been employed for the analysis of scattering and transmission problems involving both isotropic and anisotropic materials, has been shown to require the use of meshes that are up to eight times coarser than those required by standard FDTD methods [5,6].…”