An A-Phi Formulation Solver in Electromagnetics Based on Discrete Exterior Calculus

TL;DR

This paper introduces an efficient, stable, and broadband-compatible numerical solver for the A-Phi formulation in electromagnetics using discrete exterior calculus, capable of handling complex geometries with unstructured meshes.

Contribution

It develops a DEC-based A-Phi solver that is immune to low-frequency breakdown, compatible with unstructured meshes, and preserves geometric properties for accurate electromagnetic analysis.

Findings

Solver is stable and free of spurious solutions.

Capable of modeling complex structures efficiently.

Validated through numerical examples.

Abstract

An efficient numerical solver for the A-Phi formulation in electromagnetics based on the discrete exterior calculus (DEC) is proposed in this paper. The A-Phi formulation is immune to low-frequency breakdown and ideal for broadband and multi-scale analysis. The generalized Lorenz gauge is used in this paper, which decouples the A equation and the Phi equation. The A-Phi formulation is discretized by using the DEC, which is the discretized version of the exterior calculus in differential geometry. In general, DEC can be viewed as a generalized version of the finite difference method, where Stokes' theorem and Gauss's theorem are naturally preserved. Furthermore, compared with finite difference method, where rectangular grids are applied, DEC can be implemented with unstructured mesh schemes, such as tetrahedral meshes. Thus, the proposed DEC A-Phi solver is inherently stable, free of spurious solutions and can capture highly complex structures efficiently. In this paper, the background knowledge about the A-Phi formulation and DEC is introduced, as well as technical details in implementing the DEC A-Phi solver with different boundary conditions. Numerical examples are provided for validation purposes as well.