A parallelizable direct solution of integral equation methods for electromagnetic analysis

A parallelizable direct solution of integral equation methods is proposed for electromagnetic scattering analysis in low to intermediate frequency regime. There are mainly two parts of the proposed direct solution: forward decomposition and backward substitution. For the forward decomposition, the dense impedance matrix is decomposed of the product of several block diagonal matrices implicitly, which is shown to be O(Nlog2N) for both memory and CPU time cost. The final solutions are obtained with several matrix vector products (MVPs) in the part of backward substitution with O(Nlog2N) complexity as well. Both forward decomposition and backward substitution can be parallelized because of the group independence. Furthermore, an effective preconditioner with a reasonable selection criterion of the diagonal blocks region is proposed to accelerate the convergence of the iterative solver. The proposed solution is independent of the Green's function, and it is suitable for all the integral equation methods. Without loss of generality, the solution is proposed to solve the electric field integral equation (EFIE) in this work. Numerical tests demonstrate the effectiveness of the proposed solution for the electromagnetic analysis, especially for multiscale structures.