Accelerated 3D Maxwell Integral Equation Solver using the Interpolated Factored Green Function Method

TL;DR

This work develops an accelerated 3D Maxwell boundary integral equation solver for dielectric scattering by coupling a high-order Chebyshev-based Nyström discretization with the Interpolated Factored Green's Function (IFGF) acceleration. The method extends IFGF from scalar to fully-vector Maxwell problems, enabling efficient evaluation of far-field interactions via a multilevel, cone-structured interpolation scheme and careful singular quadrature handling. Numerical experiments across spheres and complex CAD geometries demonstrate $O(N\log N)$ per-iteration cost and substantial speedups (up to 42x) over unaccelerated CBIE, while maintaining accuracy near $10^{-4}$. The results indicate significant potential for fast forward maps and design optimization in nanophotonic and EM scattering applications, with future work targeting GPU acceleration.

Abstract

This article presents an $O(N\log N)$ method for numerical solution of Maxwell's equations for dielectric scatterers using a 3D boundary integral equation (BIE) method. The underlying BIE method used is based on a hybrid Nyström-collocation method using Chebyshev polynomials. It is well known that such an approach produces a dense linear system, which requires $O(N^2)$ operations in each step of an iterative solver. In this work, we propose an approach using the recently introduced Interpolated Factored Green's Function (IFGF) acceleration strategy to reduce the cost of each iteration to $O(N\log N)$. To the best of our knowledge, this paper presents the first ever application of the IFGF method to fully-vectorial 3D Maxwell problems. The Chebyshev-based integral solver and IFGF method are first introduced, followed by the extension of the scalar IFGF to the vectorial Maxwell case. Several examples are presented verifying the $O(N\log N)$ computational complexity of the approach, including scattering from spheres, complex CAD models, and nanophotonic waveguiding devices. In one particular example with more than 6 million unknowns, the accelerated IFGF solver runs 42x faster than the unaccelerated method.

Figures (16)

• Figure 1: Illustration for EM scattering by dielectric medium.
• Figure 2: Mapping the square $[-1,1]\times [-1,1]$ in parametric $(u,v)$-space to a patch on the surface of a sphere in Cartesian coordinates.
• Figure 3: Source point factorization test. Consider an origin centered box of side $H$. In this illustration, the source point is positioned at ${\boldsymbol{x}}'=(0,0,H/2)$, and the point ${\boldsymbol{x}}$ varies on the line $t(1,0,0)$ for $t\in[3H/2,12H]$. The graph at the top displays the real part of the kernel $G({\boldsymbol{x}},{\boldsymbol{x'}})$, which illustrates a highly-oscillatory behavior even for the smaller size boxes. The bottom graph illustrates the slow oscillatory behavior of the analytic factor $g_{S}({\boldsymbol{x}},{\boldsymbol{x'}})$ for boxes as large as $H=20\lambda$.
• Figure 4: Illustration of a cone structure for a single box in 2D. The figure illustrates a cone-structure for the box $B({\boldsymbol{x}},H)$ shown in red, the neighborhood boxes in $\mathcal{N}B({\boldsymbol{x}},H)$ separated by dashed lines within thick black lines, and a $P_s\times P_a=3\times 3$ interpolation scheme in cone-segment $C_{2,1}$. The hollow blue star-shape points denote the source points within $B({\boldsymbol{x}},H)$, blue solid circles denote the interpolation grid points in $C_{2,1}$, and the points denotes by black-$+$ signs are the target points within $C_{2,1}$. The IFGF method does not interpolate contribution from $B({\boldsymbol{x}},H)$ at any target point that lies within $\mathcal{N}B({\boldsymbol{x}},H)$.
• Figure 5: Illustration for box-hierarchy in 2D for $D=3$. The level $1$ box (the largest box) is always relevant as it contains the set $\Gamma_{N}$. Here, the all four level-$2$ boxes (divided by the red lines) are relevant as well. On level $3$--On left: only the light shaded boxes on the left image above are relevant as the other boxes do not intersect with the surface $\Gamma$. On right: for the box $B_{(2,1)}$, only the light shaded boxes are in $\mathcal{N}B_{(2,1)}$ whereas the dot-pattern boxes are in the cousin box set $\mathcal{M}B_{(2,1)}$. Note that the dark shaded boxes $B_{(1,1)}$ and $B_{(3,1)}$ not being relevant are not considered neighbors of $B_{(2,1)}$\ref{['eq:box_neighbor']}.

Theorems & Definitions (3)

• Remark 1: Weak singularity in \ref{['eq:n_muller']}
• Remark 2: Choosing cone-structure sizes.
• Remark 3: On selection of IFGF approach