Fast Direct Solution of Composite Dielectric Objects by M-HODLR

A fast direct solver based on the modified hierarchically off-diagonal low-rank (M-HODLR) structure is developed for electromagnetic analysis of composite dielectric objects. Due to the presence of both $L$ and $K$ operators, the system matrix for a single dielectric object is usually composed of four matrix blocks in the PMCHWT formulation, and the matrix structure becomes more complex for a composite object. To facilitate the compression of off-diagonal low-rank blocks in HODLR according to different operators, the basis functions for both electric and magnetic currents are first sorted at the bottom level of the binary tree and subsequently resorted at upper levels. To address the instability issue of the adaptive cross approximation (ACA) algorithm for blocks with multiple zero matrix entries in the composite dielectric modeling, a prejudging and local reordering strategy is further applied to separate the nonzero sub-blocks from zero ones, and an improved ACA strategy is devised to ensure that all nonzero sub-blocks are searched to avoid an early iteration termination. Numerical results show the accuracy and efficiency of the proposed method in analyzing composite dielectric objects.