Fast Power Series Solution of Large 3-D Electrodynamic Integral Equation for PEC Scatterers

TL;DR

This paper introduces a rapid algebraic power series method for solving large 3D PEC electromagnetic integral equations, converging in just two iterations and compatible with existing fast solvers like H-Matrix and MLFMA.

Contribution

The novel method achieves fast convergence using a diagonally dominant matrix scaling approach, reducing computational complexity and memory usage for large PEC structures.

Findings

Converges in only two iterations, faster than traditional methods.

Maintains O(N log N) complexity, suitable for large problems.

Validated through numerical experiments demonstrating efficiency.

Abstract

This paper presents a new fast power series solution method to solve the Hierarchal Method of Moment(MoM) matrix for a large complex,perfectly electric conducting (PEC) 3D structures. The proposed power series solution converges in just two iterations which is faster than the conventional fast solver-based iterative solution. The method is purely algebraic in nature and, as such applicable to existing conventional methods. The method uses regular fast solver Hierarchal Matrix (H-Matrix) and can also be applied to Multilevel Fast Multipole Method Algorithm(MLFMA). In the proposed method, we use the scaling of the symmetric near-field matrix to develop a diagonally dominant overall matrix to enable a power series solution. Left and right block scaling coefficients are required for scaling near-field blocks to diagonal blocks using Schur's complement method. However,only the right-hand scaling coefficients are computed for symmetric near-field matrix leading to saving of computation time and memory. Due to symmetric property, the left side-block scaling coefficients are just the transpose of the right-scaling blocks. Next, the near-field blocks are replaced by scaled near-field diagonal blocks. Now the scaled near-field blocks in combination with far-field and scaling coefficients are subjected to power series solution terminating after only two terms. As all the operations are performed on the near-field blocks, the complexity of scaling coefficient computation is retained as O(N). The power series solution only involves the matrix-vector product of the far-field, scaling coefficients blocks, and inverse of scaled near-field blocks. Hence, the solution cost remains O(NlogN). Several numerical results are presented to validate the efficiency and robustness of the proposed numerical method.