Accelerated Chebyshev-Based Nyström Scheme for Boundary Integral Equation for Electromagnetic Scattering Using Interpolated Factored Green Function - Paul, Jagabandhu; Sideris, Constantine (2025 International Applied Computational Electromagnetics Society Symposium (ACES), 2025)
Solution of Maxwell's equations is quintessential to many modern applications including antennas, microwave and photonic devices. The lack of analytical solutions for general domains makes fast and accurate numerical solution methodology for Maxwell's equations of utmost importance in such applications, particularly in inverse design, where accurate field and gradient information is needed in each iteration. In particular, the boundary integral equation (BIE) approach only requires discretizing the surfaces of domains unlike the finite-difference and the finite-element methods, which are volumetric in nature. In addition, BIE methods are almost dispersion-less due to analytically propagating information from sources to targets using Green's functions, unlike finite difference and finite element methods. The Method of Moments approach is the most popular approach for discretizing BIEs in the available literature. Recently, several works based on the Nyström scheme have been proposed, demonstrating significant improvements in both runtime and accuracy. One of the main challenges of the BIE method is that it produces dense linear system leading to a \mathcal{O}(N^{2}) computational complexity per iteration of a iterative linear solver. We leverage the high-order Chebyshev-based Nyström method to discretize the integral operators of the indirect N-Müller formulation. To accelerate the far interaction, we repurpose the recently introduced "interpolated factored Green function" (IFGF) method in the context of the scalar Helmholtz problem [JCP 2021, Bauinger and Bruno] for its extension to the vectorial 3D Maxwell problem. The multilevel IFGF method uses a hierarchical box-structure, similar to the FMM-based approaches. The IFGF method factorizes the Green's function into certain "centered factor" that is independent of the source points and an "analytic factor" that oscillates slowly. The IFGF method evaluates the contribution to the discrete sum coming from the source points within a given box using a coarse interpolation scheme facilitated by a hierarchical cone-structure while maintaining the desired accuracy, instead of relying upon use of the Fast Fourier Transformation (FFT). Thus the IFGF method avoids the parallelization bottleneck that FFT is known to have. Moreover, the IFGF works equally well for both low- and high-frequency regimes. A direct extension of the scalar IFGF approach to accelerate the vectorial electromagnetic problem could be to evaluate the discrete sum for each pair of density and kernel at a time. A different approach leading to further runtime improvement, while maintaining a \mathcal{O}(N\log N) computational complexity, is introduced to evaluate the vectorial discrete sum (1) comprising of all the discrete sums for the vectorial integral potential of all the density components: