Regularized integral equations and curvilinear boundary elements for electromagnetic wave scattering in three dimensions

The boundary integral equations (BIE's), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R(3)) or a Cauchy-singularity (1/R(2)), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. In this paper, regularized BIE's for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping [1] are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination [2] of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used in this paper to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions.