A Novel Direct Solver for Subwavelength Periodic Problems via the Multilevel Green’s Function Interpolation Method

In this article, a novel direct solver via multilevel Green’s function (GF) interpolation method (DSMLGFIM) is proposed for efficient analysis of 3-D composite dielectric and metallic structures with subwavelength double periodicity. In DSMLGFIM, the method of moments (MoM) matrix is quickly calculated via the newly devised multilevel interpolation scheme for both the far- and near-field interactions and then explicitly stored in order to apply the LU decomposition for fast solving the MoM matrix equations. Compared with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{2}$ </tex-math></inline-formula>) complexity for evaluations of the periodic GFs (PGFs) in the conventional MoM, the corresponding complexity of PGFs evaluations in DSMLGFIM is only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N \,log \,N)$ </tex-math></inline-formula>, where N is the number of unknowns. In addition, the direct evaluation of the gradient of PGFs is avoided by performing the gradient operation on the interpolation formula of the PGFs. The numerical simulations on the reflection and transmission characteristics of three composite dielectric and metallic subwavelength structures are presented to demonstrate the efficiency and accuracy of the proposed method. Up to 82 times speedup is achieved in our simulated examples compared with the conventional MoM.