High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
SIAM Journal on Numerical Analysis
An efficient method for band structure calculations in 2D photonic crystals
Journal of Computational Physics
Journal of Computational Physics
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
Fast and Efficient Algorithms in Computational Electromagnetics
Fast and Efficient Algorithms in Computational Electromagnetics
A Method for Computing Nearly Singular Integrals
SIAM Journal on Numerical Analysis
Photonic band structure calculations using nonlinear eigenvalue techniques
Journal of Computational Physics
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
Journal of Computational Physics
On the evaluation of layer potentials close to their sources
Journal of Computational Physics
Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Photonic Crystals: Molding the Flow of Light
Photonic Crystals: Molding the Flow of Light
Integral equation methods for elliptic problems with boundary conditions of mixed type
Journal of Computational Physics
Boundary integral equations in time-harmonic acoustic scattering
Mathematical and Computer Modelling: An International Journal
Journal of Computational Physics
Helmholtz equation in periodic media with a line defect
Journal of Computational Physics
On the numerical evaluation of the singular integrals of scattering theory
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.47 |
In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.