Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$p$-adaptive algorithms in Discontinuous Galerkin solutions to the time-domain Maxwell's equations

Apurva Tiwari, Avijit Chatterjee

TL;DR

The paper addresses efficient $p$-adaptive refinement in discontinuous Galerkin time-domain solutions of the time-domain Maxwell equations. It introduces two drivers: a feature-based gradient of EM energy for heuristic refinement and a divergence-error indicator (with the Divergence Error Evolution Equation) that serves as a rigorous, computable proxy for truncation error. Through canonical EM scattering tests, the divergence-based driver consistently yields substantial DOF savings while preserving accuracy, and it localizes refinement near sources of divergence and leading wavefronts, unlike the gradient-based method which follows field gradients more broadly. The work demonstrates that divergence-error adaptivity is robust, easily integrated into existing codes, and particularly well-suited for non-FDTD DGTD solvers, offering practical impact for efficient high-frequency EM simulations.

Abstract

The Discontinuous Galerkin time-domain method is well suited for adaptive algorithms to solve the time-domain Maxwell's equations and depends on robust and economically computable drivers. Adaptive algorithms utilize local indicators to dynamically identify regions and assign spatial operators of varying accuracy in the computational domain. This work identifies requisite properties of adaptivity drivers and develops two methods, a feature-based method guided by gradients of local field, and another utilizing the divergence error often found in numerical solution to the time-domain Maxwell's equations. Results for canonical testcases of electromagnetic scattering are presented, highlighting key characteristics of both methods, and their computational performance.

$p$-adaptive algorithms in Discontinuous Galerkin solutions to the time-domain Maxwell's equations

TL;DR

The paper addresses efficient -adaptive refinement in discontinuous Galerkin time-domain solutions of the time-domain Maxwell equations. It introduces two drivers: a feature-based gradient of EM energy for heuristic refinement and a divergence-error indicator (with the Divergence Error Evolution Equation) that serves as a rigorous, computable proxy for truncation error. Through canonical EM scattering tests, the divergence-based driver consistently yields substantial DOF savings while preserving accuracy, and it localizes refinement near sources of divergence and leading wavefronts, unlike the gradient-based method which follows field gradients more broadly. The work demonstrates that divergence-error adaptivity is robust, easily integrated into existing codes, and particularly well-suited for non-FDTD DGTD solvers, offering practical impact for efficient high-frequency EM simulations.

Abstract

The Discontinuous Galerkin time-domain method is well suited for adaptive algorithms to solve the time-domain Maxwell's equations and depends on robust and economically computable drivers. Adaptive algorithms utilize local indicators to dynamically identify regions and assign spatial operators of varying accuracy in the computational domain. This work identifies requisite properties of adaptivity drivers and develops two methods, a feature-based method guided by gradients of local field, and another utilizing the divergence error often found in numerical solution to the time-domain Maxwell's equations. Results for canonical testcases of electromagnetic scattering are presented, highlighting key characteristics of both methods, and their computational performance.
Paper Structure (14 sections, 27 equations, 11 figures, 1 table)

This paper contains 14 sections, 27 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Discontinuous Galerkin time domain method
  3. Energy driven adaptivity
  4. Role of strain energy in solid mechanics
  5. EM energy as driver
  6. Asymptotic convergence of EM energy
  7. Divergence error based adaptivity
  8. Comparison between local gradient and divergence errors
  9. Results
  10. Scattering off a circular cylinder
  11. Semi-open cavity
  12. Two adjacent cylinders
  13. Computational performance
  14. Conclusion

Figures (11)

  • Figure 1: "Misaligned" neighbouring nodes at common edges in a $p$-anisotropic nodal DG framework
  • Figure 2: Enclosing surfaces (dashed) around scattering cylinder (solid)
  • Figure 3: Asymptotic convergence of time-averaged $\int_{\Omega_{cyl}}\left(E_z^s\right)^2$vs. no. of elements $K_{cyl}$ on the surface of the scattering cylinder.
  • Figure 4: Schematic for the circular cylinder scattering problem
  • Figure 5: RCS, circular cylinder scattering, size $2\lambda$, TM illumination
  • ...and 6 more figures