A High-Order-Accurate 3D Surface Integral Equation Solver for Uniaxial Anisotropic Media

TL;DR

This paper presents a novel high-order surface integral equation method for accurately solving 3D electromagnetic scattering problems involving uniaxially anisotropic dielectric objects, demonstrating superior convergence and accuracy.

Contribution

It introduces the first high-order accurate surface integral equation solver specifically designed for uniaxially anisotropic media using a Nyström-collocation approach.

Findings

Achieves high-order convergence for complex geometries

Validates accuracy against commercial finite element solver

Successfully handles strongly singular dyadic Green's functions

Abstract

This paper introduces a high-order accurate surface integral equation method for solving 3D electromagnetic scattering for dielectric objects with uniaxially anisotropic permittivity tensors. The N-M\"uller formulation is leveraged resulting in a second-kind integral formulation, and a finite-difference-based approach is used to deal with the strongly singular terms resulting from the dyadic Green's functions for uniaxially anisotropic media while maintaining the high-order accuracy of the discretization strategy. The integral operators are discretized via a Nystr\"om-collocation approach, which represents the unknown surface densities in terms of Chebyshev polynomials on curvilinear quadrilateral surface patches. The convergence is investigated for various geometries, including a sphere, cube, a complicated NURBS geometry imported from a 3D CAD modeler software, and a nanophotonic silicon waveguide, and results are compared against a commercial finite element solver. To the best of our knowledge, this is the first demonstration of high-order accuracy for objects with uniaxially anisotropic materials using surface integral equations.