Analysis of Dielectric Photonic-Crystal Problems With MLFMA and Schur-Complement Preconditioners

Abstract

We present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners.Index Terms-Multilevel fast multipole algorithm (MLFMA), photonic crystals (PhCs), Schur-complement preconditioners, surface integral equations (SIEs).

MLFMM-Accelerated Integral-Equation Modeling of Reverberation Chambers [Open Problems in CEM]

“…The tolerance was set to 6 1 10 − × to minimize the inaccuracy due to the iterative solver. The iterative solver may be used together with precon ditioners such as the block-diagonal preconditioner (BDP) [36] and the near-fi eld preconditioner (NFP) [44][45][46][47][48][49]. The near-fi eld preconditioner is expensive in terms of memory and CPU time, especially when the hybrid integral equation is adopted.…”

“…Hence, it is highly desirable to develop a preconditioner that reduces the condition number of the interaction matrix to normal levels. This will not only signifi cantly reduce the number of iterations, but also make the solution less sensitive to small errors [48,49].…”

Parallel Implementation of Mlfma for Homogeneous Objects With Various Material Properties

“…For these accuracy parameters and the given model of processors, the parallelization efficiency is around 85% leading to 54-fold speedup using 64 processes compared to the corresponding sequential solution [3,25]. Note that problems involving more complicated objects may require effective preconditioners, such as those based on the Schur-complement reduction for penetrable objects [5] and the two-level scheme for PEC objects [36] that are appropriate for MLFMA implementations. For more quantitative assessment of the accuracy and efficiency, Table 4 lists the number of iterations, total time (that is dominated by iterations), memory, and root-mean-square (RMS) error in computational values with respect to analytical values for the sphere problems.…”

“…Real-life electromagnetics phenomena, such as scattering from airborne targets [1], radiation from antennas [2], transmission through dielectric lenses [3], metamaterials [4], and photonic crystals [5], often involve large objects with respect to wavelength. Accurate discretizations of these objects lead to large matrix equations, even when they are formulated with the surface integral equations.…”

“…Accurate discretizations of these objects lead to large matrix equations, even when they are formulated with the surface integral equations. The resulting large-scale problems can be solved iteratively, where the required matrix-vector multiplications are performed efficiently with various acceleration methods, such as the fast Fourier transform (FFT) [6,7], the adaptive integral method [8], and especially the multilevel fast multipole algorithm (MLFMA) [9][10][11][12][13], which has been successfully used to solve many challenging problems in various application areas [1][2][3][4][5]14].…”