Volume Integral Equations for the Study of Electromagnetic Scattering by Bi-anisotropic Objects

Abstract

In this Master’s thesis volume integral equations based on polarization current densities and potentials, i.e. JM-formulation and AVFU-formulation are developed for bianisotropic scatterers. The volume integral equations are discretized and converted into a system of linear equations using the method of moments. The system of equations for the JM-formulation is solved using the generalized minimal residual method (GMRES) and multilevel fast multipole algorithm (MLFMA) is also applied. Whereas the AVFU-formulation is still under development and GMRES and MLFMA are not implemented; thereby limiting the problem domain that can be analysed. Numerical results are compared to existing model problems solved using either the volume integral equations for the fields, i.e. the EH-formulation, flux densities, i.e. the DB-formulation or the Mie series solutions. The scattering cross-section of simple scatterers like an isotropic sphere to complex scatterers like double negative chiral and metamaterial spheres is investigated.

From the numerical point of view the essence of JM- and AVFU- formulation is discretization using low order basis functions. The polarization current densities in the JM-formulation are expanded using piecewise constant basis functions and Galerkin’s technique is applied to test the equations. The scalar and vector potentials in the AVFU-formulation are expanded using the scalar interpolatory nodal basis functions and point-matching technique is applied to test the equations.

An important purpose of the Master’s thesis is to provide an academic document that can be easily followed to formulate and numerically implement volume integral equations based on fields, flux densities, polarization current densities or potentials. To achieve this goal the overall implementation from scratch is presented for the AVFUformulation.