A Meshless Method for Solving Volume Integral Equations with Complex Materials

Meshless or meshfree methods use discrete points to replace meshes in the geometric description of solution domains and this replacement can significantly reduce the cost of discretizing geometries. The meshless methods are also desirable for solving electromagnetic (EM) problems but they have not received a sufficient attention. In this work, we propose a truly-meshless method for solving EM volume integral equations (VIEs) with complex materials which have not been addressed before. The method transforms the volume integrals of VIEs into the boundary or surface integrals based on Green-Gauss theorem so that volume integrals and volumetric discretization can be fully removed. To facilitate the calculation of boundary integrals, the original domain occupied by the object is expanded to a cylinder or cube circumscribing the object and the involved integrands are redefined. The singular integrals are excluded from the transformation and they are individually evaluated by a singularity treatment technique. A numerical example is presented to demonstrate the method and its good performance has been validated.