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QR-Recursive Compression of Volume Integral Equations for Electromagnetic Scattering by Large Metasurfaces

Vincenzo Mottola, Antonello Tamburrino, Luca Bergamaschi, Andrea G. Chiariello, Emanuele Corsaro, Carlo Forestiere, Guglielmo Rubinacci, Salvatore Ventre

Abstract

In this paper, a novel QR decomposition-based compression scheme is combined with a volume integral equations method for the fast and efficient numerical computation of the scattering of electromagnetic fields from large scale metasurfaces, via an iterative approach. The underlying problem is of a multiscale nature. Indeed, these metasurfaces are made of a large collection of interacting sub-wavelength scatterers, thus making the numerical computation of the solution very challenging. More specifically, the paper proposes a tailored version of a QR decomposition-based compression for a volume integral equation, together with a proper preconditioner that exploits the geometrical structure of the array, in order to achieve a fast and accurate iterative solver, in view of realistic applications. Numerical examples prove the effectiveness of the method in efficiently modeling metasurfaces made by thousands of particles.

QR-Recursive Compression of Volume Integral Equations for Electromagnetic Scattering by Large Metasurfaces

Abstract

In this paper, a novel QR decomposition-based compression scheme is combined with a volume integral equations method for the fast and efficient numerical computation of the scattering of electromagnetic fields from large scale metasurfaces, via an iterative approach. The underlying problem is of a multiscale nature. Indeed, these metasurfaces are made of a large collection of interacting sub-wavelength scatterers, thus making the numerical computation of the solution very challenging. More specifically, the paper proposes a tailored version of a QR decomposition-based compression for a volume integral equation, together with a proper preconditioner that exploits the geometrical structure of the array, in order to achieve a fast and accurate iterative solver, in view of realistic applications. Numerical examples prove the effectiveness of the method in efficiently modeling metasurfaces made by thousands of particles.
Paper Structure (20 sections, 32 equations, 13 figures, 6 tables)

This paper contains 20 sections, 32 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Numerical model
  4. Iterative method and preconditioner
  5. The QR-recursive method for a meta-atoms array
  6. Far and near distance interactions
  7. The low-rank QR-decomposition
  8. The recursive scheme
  9. The metasurface
  10. Numerical validation
  11. Description of the cases of interest
  12. Consistency of the Method
  13. Performances of the Method
  14. Efficiency of the Parallelization Method
  15. Timing Information
  16. ...and 5 more sections

Figures (13)

  • Figure 1: Examples of loop (orange) and star (blue) basis functions for a single meta-atom.
  • Figure 2: Iterative solvers.
  • Figure 3: Left: the meta-lens together with the grid (black) defining the blocks at the first grid level. Right: matrix $\mathbf{Z}$ partitioned according to the 9 blocks of meta-atom shown at left. Far distance interactions are marked in green, while near distance interactions are marked in red.
  • Figure 4: The grids for level 1 (left) and level 2 (right).
  • Figure 5: Matrix $\mathbf{Z}$ partitioned according to the 36 blocks at the second grid level. The new far distance interactions are marked in green, the current near distance interactions are marked in red, and the far distance interactions compressed at the previous level are marked in white.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Definition