Discretization of Continuous Spaces using Barycentric Subdivision Method with Metric Space Constraints on the Nearest Neighbour Edges

Abstract

The Rao-Wilton-Glisson (RWG) is a commonly used basis function in the numerical solution of the electric field integral equation (EFIE)  using the Method of Moments (MoM) and Galerkin approach. This method relies on triangular patches to approximate the surface  current. Traditionally, barycentric subdivision of a primary triangle into n-sub-triangles has been used with RWG basis function to solve  the EFIE using MoM. This paper presents a method of approximating a surface using triangular patches by sub-dividing a primary  triangle into (2n-1) subtriangles. which creates a denser mesh than the widely used nine (9) points quadrature method. The structure is approximated by small square patches, which are further sub-divided into two primary triangles. By applying barycentric sub-division, the  primary triangles are decomposed into sub-triangles to create a mesh over the surface of the structure. Using graph theory, the  triangular meshes are defined by the function, G (V, E), where V and E are the vertices and edges of the triangles in the mesh space,  respectively. The connectivity matrix of shared edges is found by imposing a constraint on the edge length using metric spaces. This  method approximates a square patch with 32 scalene triangles and shows that it can be used to reconstruct equilateral patches and  doubly split double ring into mesh structures.