A Memory-Efficient Implementation of Perfectly Matched Layer With Smoothly Varying Coefficients in Discontinuous Galerkin Time-Domain Method

Liang Chen,Shehab Ahmed,Hakan Bagci,Mehmet Burak Ozakin

doi:10.1109/tap.2020.3037651

Abstract

Wrapping a computation domain with a perfectly matched layer (PML) is one of the most effective methods of imitating/approximating the radiation boundary condition in Maxwell and wave equation solvers. Many PML implementations often use a smoothly-increasing attenuation coefficient to increase the absorption for a given layer thickness, and, at the same time, to reduce the numerical reflection from the interface between the computation domain and the PML. In discontinuous Galerkin time-domain (DGTD) methods, using a PML coefficient that varies within a mesh element requires a different mass matrix to be stored for every element and therefore significantly increases the memory footprint. In this work, this bottleneck is addressed by applying a weight-adjusted approximation to these mass matrices. The resulting DGTD scheme has the same advantages as the scheme that stores individual mass matrices, namely higher accuracy (due to reduced numerical reflection) and increased meshing flexibility (since the PML does not have to be defined layer by layer) but it requires significantly less memory.

Highlights

P ERFECTLY matched layer (PML) [1], [2] is often used in finite difference [3], finite element [4], and discontinuous Galerkin time-domain (DGTD) methods [5]–[10] to imitate/approximate the radiation boundary condition while solving Maxwell equations or the wave equation.The theoretical performance of the PML depends on the attenuation coefficient and the thickness of the layer [1]
The results presented here are obtained with the Perfect electric conductor (PEC) boundary condition
A PML implementation that allows the attenuation coefficient to vary inside the discretization elements yields a smaller numerical reflection from the interface between the PML and the computation domain and significantly simplifies the meshing process

Summary

INTRODUCTION

P ERFECTLY matched layer (PML) [1], [2] is often used in finite difference [3], finite element [4], and discontinuous Galerkin time-domain (DGTD) methods [5]–[10] to imitate/approximate the radiation boundary condition (i.e., truncate a unbounded physical domain to a finite computation domain) while solving Maxwell equations or the wave equation. 3 × 5 × Np2 per element, where 3 is the number of the (x, y, z) components of the vector field, 5 is the number of the coefficients a(r), b(r), c(r), d(r), and κ(r) Note that this memory requirement is significantly higher than that of storing the unknowns coefficients of the basis functions, which scales with 12 × Np in the PML. Note that in the above SC-PML formulation, directly multiplying (5) and (6) with a−1 on both sides reduces the number of element-dependent mass matrices to 4 This would result in a non-conservative form, whose solution is neither provably energy-stable nor provably high-order accurate [8], [20]. Equations (26)-(29) can be implemented in a matrix-free manner just like it is done in classical DG implementations [8]–[10], [28]–[31]

Computational complexity

NUMERICAL EXAMPLES

CONCLUSION