# Generalized Debye Sources Based EFIE Solver on Subdivision Surfaces

## TL;DR

This paper introduces a novel, well-conditioned GDS-based EFIE solver on subdivision surfaces, combining generalized Debye sources with isogeometric analysis for improved accuracy and efficiency in electromagnetic scattering problems.

## Contribution

It extends the GDS method to subdivision surface discretizations, enabling well-conditioned systems and efficient isogeometric analysis for electromagnetic scattering.

## Key findings

- Demonstrates improved conditioning of the EFIE system.
- Shows high accuracy and efficiency of the GDS-IGA framework.
- Validates the approach with numerous numerical results.

## Abstract

The electric field integral equation is a well known workhorse for obtaining fields scattered by a perfect electric conducting (PEC) object. As a result, the nuances and challenges of solving this equation have been examined for a while. Two recent papers motivate the effort presented in this paper. Unlike traditional work that uses equivalent currents defined on surfaces, recent research proposes a technique that results in well conditioned systems by employing generalized Debye sources (GDS) as unknowns. In a complementary effort, some of us developed a method that exploits the same representation for both the geometry (subdivision surface representations) and functions defined on the geometry, also known as isogeometric analysis (IGA). The challenge in generalizing GDS method to a discretized geometry is the complexity of the intermediate operators. However, thanks to our earlier work on subdivision surfaces, the additional smoothness of geometric representation permits discretizing these intermediate operations. In this paper, we employ both ideas to present a well conditioned GDS-EFIE. Here, the intermediate surface Laplacian is well discretized by using subdivision basis. Likewise, using subdivision basis to represent the sources, results in an efficient and accurate IGA framework. Numerous results are presented to demonstrate the efficacy of the approach.