A Novel High-Capacity Electromagnetic Compression Technique Based on a Direct Matrix Solution

Electromagnetic solvers based on integral equations in conjunction with the method of moments or the partial element equivalent circuit method (PEEC) proved to be popular because of their efficiency and accuracy. There is one serious drawback of the integral equation approach: it often leads to a linear system involving a full matrix. Many efficient approaches have been proposed to overcome this, largely based on compressing the matrix-vector product operation and using an iterative solver. Iterative EM solvers, however, suffer from slow convergence, which does not have a totally reliable method to address; further, large multiple right-hand sides significantly increase the solving time. In this paper, we present a novel method to compress low rank sub-block matrixes into sparse matrix to be used with a direct sparse matrix solver to obtain an efficient high-capacity electromagnetic solver based on an integral equation formulation. The full-rank system matrix is represented in a hierarchical matrix format that has its sub-matrixes compressed with numerically controllable accuracy; it is then analytically converted to a sparse matrix which is further solved by a direct sparse matrix solver. Analytically this method results in O(N (log N)
²
) complexity for computing the inverse of a hierarchical matrix presented in Fig. 2 where N is the number of unknowns.