# Geometrical MoM formulation for eigenmode analysis

## TL;DR

This paper introduces a geometrical approach using the Method of Moments to efficiently compute resonant frequencies of structures, reducing the need for repeated frequency evaluations and handling complex, dispersive materials.

## Contribution

The paper presents the Geometrical Method of Moments (GMoM), a novel technique that precomputes geometrical integrals to rapidly evaluate resonant frequencies across complex values.

## Key findings

- Accelerates eigenfrequency computation for structures.
- Handles dispersive materials with extrapolated permittivity and permeability.
- Reduces computational effort compared to traditional frequency-domain solvers.

## Abstract

The resonant frequencies of a structure and the associated field distributions are generally determined by solving a non-linear eigenvalue problem. Using frequency-domain solvers, the response of the structure needs to be evaluated at many different frequencies in order to solve the non-linear problem. Moreover, these frequencies may be complex, the inverse of the imaginary part physically corresponding to the e-folding time of the energy of the mode. In this paper, we propose to use the so-called "Geometrical Method of Moments" (GMoM) to accelerate the computation of the resonant frequencies of a structure using the Method of Moments. First, purely geometrical reaction integrals are precomputed, which do not depend on the frequency nor material parameters. Then, by summing these terms with proper weights, the impedance matrix can be obtained for any complex frequency. This method easily accommodates for dispersive materials provided that the permittivity and permeability of the material can be extrapolated to complex frequencies.