Volume-Surface Integral Equations with hybrid curl-conforming and divergence-conforming basis functions

In this paper, the VSIE solved by a hybrid curl-conforming and divergence-conforming basis functions is presented. In this method, new EFIEs are formed first. Then curlconforming edge basis functions are used for the VIE with the electric field as the unknowns and divergence-conforming edge basis functions are used for the SIE on PEC surfaces with the surface electric current as the unknowns. Furthermore, we set the coefficient for the volume edge basis that are in contact with the conducting surface to zero. This is done because they are the bases which contribute tangential electric field on the PEC surface. It reduces the total number of unknowns, which will be very helpful when the object is a thin dielectric slab with PEC mounted on the surface such as occurs in considering microstrip circuits. To validate this method, a C++ program is implemented and simulation results are compared with the analytical solution for spherical scatterers. Moreover, the performance of this method is compared with the approach that is based on using only divergence-conforming basis functions. Throughout the paper, e^jωt time convention is assumed and it is suppressed in all equations. The objects´ permeability are assumed as constant μ₀.