Quadrilateral Barycentric Basis Functions for Surface Integral Equations

A framework for solving surface integral equation on quadrilateral, triangular, and mixed quadrilateral-triangular meshes is presented. The initial meshes are represented in terms of quadrilateral barycentric meshes (QBMs), which are obtained by partitioning each initial quadrilateral and triangle into four and three barycentric quadrilaterals, respectively. Quadrilateral barycentric basis functions (QBBFs) are defined on QBMs. The QBBFs serve a dual role. First, they allow defining primary basis functions (PBFs), which are well suited for representing surface currents on quadrilateral, triangular, and mixed meshes. Second, the QBBFs are used to define dual basis function (DBFs), which are natural for using in conjunction with Calderón multiplicative preconditioners (CMPs). These QBBFs, PBFs, and DBFs result in a substantial reduction in the number of nonzero elements in the sparse projection matrices for PBFs and DBFs as well as reduction of unknowns and quadrature points. When these PBFs and DBFs are used in CMPs, they reduce the number of iterations and eliminate the dense mesh breakdown of the surface electric field integral equation. Numerical examples demonstrate the efficiency of using the introduced QBBFs with associated PBFs and DBFs for solving electromagnetic surface electric field integral equations on quadrilateral, triangular, and mixed meshes.