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A meshless method for computational electromagnetics with improved dispersion properties

Andrej Kolar-Požun, Gregor Kosec

TL;DR

The paper addresses grid-induced limitations of FDTD by introducing a meshless generalisation of Maxwell's equations using Radial Basis Functions. It develops two variants, RBF-FD and RBF-VFD, and stabilises explicit time stepping with hyperviscosity to achieve convergence on scattered nodes. The results show that increasing stencil size improves dispersion properties, and RBF-FD reduces dispersion anisotropy relative to FDTD, while RBF-VFD can be more dispersive. This work provides a practical, geometry-flexible, explicit EM solver with potential for irregular geometries and high-frequency applications, with future work on 3D extensions and divergence-free formulations.

Abstract

The finite difference time domain method is one of the simplest and most popular methods in computational electromagnetics. This work considers two possible ways of generalising it to a meshless setting by employing local radial basis function interpolation. The resulting methods remain fully explicit and are convergent if properly chosen hyperviscosity terms are added to the update equations. We demonstrate that increasing the stencil size of the approximation has a desirable effect on numerical dispersion. Furthermore, our proposed methods can exhibit a decreased dispersion anisotropy compared to the finite difference time domain method.

A meshless method for computational electromagnetics with improved dispersion properties

TL;DR

The paper addresses grid-induced limitations of FDTD by introducing a meshless generalisation of Maxwell's equations using Radial Basis Functions. It develops two variants, RBF-FD and RBF-VFD, and stabilises explicit time stepping with hyperviscosity to achieve convergence on scattered nodes. The results show that increasing stencil size improves dispersion properties, and RBF-FD reduces dispersion anisotropy relative to FDTD, while RBF-VFD can be more dispersive. This work provides a practical, geometry-flexible, explicit EM solver with potential for irregular geometries and high-frequency applications, with future work on 3D extensions and divergence-free formulations.

Abstract

The finite difference time domain method is one of the simplest and most popular methods in computational electromagnetics. This work considers two possible ways of generalising it to a meshless setting by employing local radial basis function interpolation. The resulting methods remain fully explicit and are convergent if properly chosen hyperviscosity terms are added to the update equations. We demonstrate that increasing the stencil size of the approximation has a desirable effect on numerical dispersion. Furthermore, our proposed methods can exhibit a decreased dispersion anisotropy compared to the finite difference time domain method.

Paper Structure

This paper contains 14 sections, 37 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Governing equations
  3. The Finite Difference Time Domain
  4. Shortcomings of FDTD
  5. Computation on scattered nodes
  6. Interpolation on scattered nodes
  7. Differential operator approximation
  8. Details of our setup
  9. Generalising the FDTD to a meshless setting
  10. Stabilising the method
  11. Convergence of the solution
  12. Dispersion
  13. Conclusion
  14. Naive instability on scattered nodes

Figures (12)

  • Figure 1: Yee grid with a spacing of $\Delta s$. An example domain boundary $\partial \Omega$ is also displayed.
  • Figure 2: Snapshot of the FDTD simulation at $n=400$, demonstrating anisotropic dispersion relation.
  • Figure 3: An example scattered discretisation, with the nodal spacing of approximately $h$. The domain boundary $\partial \Omega$ and the stencil for a chosen node are also displayed.
  • Figure 4: A visual demonstration of the virtual stencil approach for the case of a 5-point stencil with spacing $\delta$.
  • Figure 5: Behaviour of relative energy $u$ over time for the considered test case for different hyperviscosity parameters $\alpha$ and $c$.
  • ...and 7 more figures