Well-Conditioned Volume-Surface Integral Equation for Scatterers with Positive-Negative Permittivity Regions

Heterogeneous scatterers containing positive- and negative-permittivity regions remain challenging for traditional surface integral equation (SIE) and volume integral equation (VIE) solvers, since SIE generally requires piecewise-homogeneous regions, whereas VIE can yield poorly conditioned systems. This work proposes a multi-region internally combined volume-surface integral equation (ICVSIE) solver for accurate electromagnetic analysis of such scatterers. In the proposed formulation, positive- and negative-permittivity regions are enclosed by equivalent surfaces, while polarization currents radiate in fictitious media selected to reduce permittivity contrast. Exterior and interior equations are combined through Galerkin testing. Numerical results show that ICVSIE remains well-conditioned and nearly insensitive to permittivity contrast, with condition numbers around 10² compared with 10⁵–10⁶ for VIE. It converges within a few hundred iterations, while VIE fails to converge within 15,000 iterations, and computed radar cross sections agree with FEKO and COMSOL, demonstrating a stable solver for mixed-sign-permittivity scatterers.