Method of moments solution of a nonconformal volume integral equation via the IE-FFT algorithm for electromagnetic scattering from penetrable objects

This thesis is aimed at developing a volume integral equation which (1) is well-posed, i.e., when successively finer meshes are utilized to produce more accurate results by the method of moments, the condition number of the coefficient matrix remains bounded and the eigenvalue distribution preserves itself; (2) leads to a symmetric coefficient matrix when solved by the method of moments provided that the medium that fills the object of interest is reciprocal; (3) is applicable to a nonconformal mesh where the nodes do not match along an interface.; Previous studies and developments in the method of moments solution of volume integral equations require a conformal mesh to discretize a penetrable object due to the mathematical constraints imposed on the basis and/or testing functions. In order to avoid any mathematical constraints on the basis and/or testing functions, we apply the gradient-gradient operator on the integral equation kernel, which is the free space Green's function, instead of splitting it to operate on the basis and/or testing functions. This gives us the flexibility to choose piecewise constant functions for expansion and testing. Hence the mesh to discretize the object of interest need not be conformal. Moreover, the coefficient matrix is symmetric for a reciprocal medium due to the symmetry of the volume integral operators in the formulation. However, applying the gradient-gradient operator on the Green's function leads to a hypersingular integral which needs to be regularized. In this study, we adopt a regularization scheme which adds and subtracts a function, which can be integrated analytically on the surface of a volume element, to the integral equation kernel. A mathematical proof is provided to show that the nonconformal volume integral operator is coercive; in other words, the condition number of the coefficient matrix is bounded. The numerical study supports the well-posedness of the formulation.; The analysis of penetrable structures has traditionally been carried out using partial differential equation methods due to the large computation time and memory requirements of integral equation methods. To alleviate this problem, this thesis extends the fast algorithm termed IE-FFT to the method of moments solution of nonconformal volume integral equations. Previous studies show that the IE-FFT algorithm reduces the computational time and memory requirement to O (N1.5logN) and O (N1.5), respectively, for surface integral equations to solve electromagnetic scattering problems where N is the number of unknowns. When the IE-FFT algorithm is applied to nonconformal volume integral equations, the computational complexity and memory requirement reduce to O (NlogN) and O (N), respectively.