A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis

A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the
$O(N\log ^{2}N)$
construction cost of the HOD-BF-compressed system matrix and
$O(N^{1.5}\log N)$
inversion cost for the preconditioner, where
$N$
is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a
$\mathcal {H}$
matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.