Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a High-Frequency Analysis of Some Relevant Integral Equations in Electromagnetics

V. Giunzioni, A. Merlini, F. P. Andriulli

TL;DR

The work investigates how discretization of boundary integral operators for 2D PEC cylinder scattering behaves in the high-frequency regime. It establishes a formalism for the operators $S^k, D^k, D^{*k}, N^k$ and analyzes their continuous eigenvalues on circular geometry, then derives how discretized spectra differ and how this affects solution accuracy. Through asymptotic expansions, it reveals scaling laws for spectral and current errors, notably a $k^{1/3}$-type growth in transition regions and the beneficial effect of Calderón-based CFIE preconditioning. Numerical results corroborate the theory, showing reduced spurious resonances and improved high-frequency robustness for CFIE formulations, guiding practical choices in high-frequency BEM for electromagnetics.

Abstract

In this contribution we analyze the spectral properties of some commonly used boundary integral operators in computational electromagnetics and of their discrete counterparts, highlighting peculiar features of their spectra. In particular, a comparison with the eigenvalues of the continuous operators will be presented that highlights deviations in the high frequency regime and impacts, in a peculiar way, the accuracy of the numerical solutions of each formulation. A study and a proactive analysis of numerical results from standard boundary element solvers and the predictions from the theoretical analysis will corroborate the analytical framework employed and the validity of our observations.

On a High-Frequency Analysis of Some Relevant Integral Equations in Electromagnetics

TL;DR

The work investigates how discretization of boundary integral operators for 2D PEC cylinder scattering behaves in the high-frequency regime. It establishes a formalism for the operators and analyzes their continuous eigenvalues on circular geometry, then derives how discretized spectra differ and how this affects solution accuracy. Through asymptotic expansions, it reveals scaling laws for spectral and current errors, notably a -type growth in transition regions and the beneficial effect of Calderón-based CFIE preconditioning. Numerical results corroborate the theory, showing reduced spurious resonances and improved high-frequency robustness for CFIE formulations, guiding practical choices in high-frequency BEM for electromagnetics.

Abstract

In this contribution we analyze the spectral properties of some commonly used boundary integral operators in computational electromagnetics and of their discrete counterparts, highlighting peculiar features of their spectra. In particular, a comparison with the eigenvalues of the continuous operators will be presented that highlights deviations in the high frequency regime and impacts, in a peculiar way, the accuracy of the numerical solutions of each formulation. A study and a proactive analysis of numerical results from standard boundary element solvers and the predictions from the theoretical analysis will corroborate the analytical framework employed and the validity of our observations.
Paper Structure (7 sections, 19 equations, 2 figures)

This paper contains 7 sections, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. High-Frequency Analysis
  4. Spectral Error
  5. Current Error
  6. Numerical Results
  7. Conclusion

Figures (2)

  • Figure 1: Frequency dependency of $r_{H^{s}_k(\Gamma)}$ of the TM formulations: comparison between predicted (p.) values and numerical (n.) results from BEM.
  • Figure 2: Frequency dependency of $r_{H^{s}_k(\Gamma)}$ of the TE formulations: comparison between predicted (p.) values and numerical (n.) results from BEM.