Smooth Delaunay-Voronoï Dual Meshes for Co-Volume Integration Schemes

Yee’s scheme for the solution of the Maxwell equations [1] and the MAC algorithm for the solution of the Navier–Stokes equations [2] are examples of co-volume solution techniques. Co-volume methods, which are staggered in both time and space, exhibit a high degree of computationally efficiency, in terms of both CPU and memory requirements compared to, for example, a finite element time domain method (FETD). The co-volume method for electromagnetic (EM) waves has the additional advantage of preserving the energy and, hence, maintaining the amplitude of plane waves. It also better approximates the field near sharp edges, vertices and wire structures, without the need to reduce the element size. Initially proposed for structured grids, Yee’s scheme can be generalized for unstructured meshes and this will enable its application to industrially complex geometries [3]. Despite the fact that real progress has been achieved in unstructured mesh generation methods since late 80s, co-volume schemes have not generally proved to be effective for simulations involving domains of complex shape. This is due to the difficulties encountered when attempting to generate the high quality meshes that satisfying the mesh requirements necessary for co-volume methods. In this work, we concentrate on EM wave scattering simulations and identify the necessary mesh criteria required for a co-volume scheme. We also describe several approaches for generating two-dimensional and three-dimensional meshes satisfying these criteria. Numerical examples on the scattering of EM waves show the efficiency and accuracy that can be achieved with a co-volume method utilising the proposed meshing scheme.