Hierarchical Pattern Exploitation for Efficient Electromagnetic Analysis of Finite Periodic Array

The hierarchical pattern exploitation (HPE) method is proposed for the efficient electromagnetic analysis of finite periodic arrays. Instead of taking advantage of the Toeplitz symmetry by discretizing integral equations (IEs), HPE partitions the array hierarchically to form considerably larger geometrical repetitions, which result in identical interaction blocks inside the system matrix. These interaction blocks representing patterns are characterized and hashed to generate a directory, which maps the patterns to the references of the submatrices. Due to the large proportion of repetitions in hierarchical matrix (<inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula>-matrix), HPE accelerates the matrix assembly and reduces the required storage drastically. Numerical examples show that the HPE outperforms the classical <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula>-matrix by a large margin with much less storage and CPU time cost for the analysis of finite periodic arrays. Compared with the multilevel fast multipole algorithm (MLFMA) and the characteristic basis functions method (CBFM), HPE also demonstrates competitive performance, which validates the effectiveness and efficiency of the proposed method.