Farhad Sheikh Hosseini Lori, Anton Menshov, Vladimir Okhmatovski
F. Sheikh Hosseini Lori, A. Menshov, and V. Okhmatovski, “Novel surface-volume-surface electrical field integral equation for solution of scattering problems on penetrable objects,” in Int. Work. Finite Elem. Microw. Eng., Florence, Italy, May 2016.
Publication year: 2016


The integral equations (IE) of electromagnetics provide an alternative representations of the Maxwell equations which in many applications allow for more efficient computation of the electromagnetic fields than the method relying on direct discretization of the Maxwell equations. As such Method of Moments (MoM) solution of IEs became the primary numerical technique for electromagnetic analysis of microstrip antennas and microwave circuits, solution of electrically large scattering problems, and solution of various other applied problems.

Scattering problems on penetrable homogeneous objects can be cast into the form of Combined Field Integral Equation (CFIE), PMCHWT integral equation, or Muller integral equation. Each of the above IEs features two unknown vector functions (equivalent electric and magnetic currents) on the surface of the scatterer. These two vector functions are typically discretized with basis function on local support (such as Rao-Wilton-Glisson basis function) to obtain their numerical approximation. Since the number of the involved basis functions defines the size of the pertinent dense matrix equation which is solved numerically. A few techniques exist which eliminate one of the equivalent currents and, thus, reduce the size of the pertinent dense matrix equation. Among such methods are Single Source Integral Equation (SSIE) formulations and the methods utilizing complicated specialized Green’s function satisfying particular boundary conditions required for elimination of one of the unknown surface currents. While both these techniques can lead to relatively efficient numerical schemes, generally, they are significantly more complicated to implement and suffer from various numerical inefficiencies. Specifically, the SSIE formulations feature either large number of matrix-matrix products if a direct method is utilized for solution of the resultant matrix equation or a large number of matrix-vector products if the dense matrix equation is solved iteratively.

In this work we discuss a new type of SSIE integral equation which we recently developed. The new SSIE features only one unknown vector function on the surface of the scatterer and is simple in implementation. The new SSIE is derived from the traditional Volume IE (VIE) through representation of the field in the volume of the scatterer as a superposition of the waves emanating from the scatterer’s boundary and weighed with a single unknown surface vector function. Such field representation in the scatterer volume satisfies the same curl-curl wave equation as the true field. The surface based representation of the field in scatterer’s volume upon substitution into VIE and localization of the observation points to the scatterer boundary reduces VIE to a SSIE. Thus obtained SSIE, however, features integral operators effecting field translation from the scatterer boundary to its volume and subsequent translation of the volumetric field distribution back to the scatterer’s surface. Because of the above field translations the new SSIE is termed the Volume-Surface-Volume IE (SVS-IE). The new SVS-IE has been applied to solution of both scalar and vector scattering problems. In this presentation we discuss the MoM discretization of the SVS-IE and demonstrate various numerical results proving its validity.