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

Abstract

An efficient numerical solver for the A-Φ formulation in electromagnetics based on discrete exterior calculus (DEC) is proposed in this paper. The A-Φ formulation, where A is the vector potential and Φ is the scalar potential, can be decoupled by using the generalized Lorenz gauge. Compared with the E-H formulation in Maxwell’s equations, the A-Φ formulation is immune to low-frequency breakdown thanks to the additional gauge term which removes the null space of the double curl operator. Thus, the A-Φ formulation is ideal for broadband and multi-scale analysis in electromagnetics. In addition, as the rapid development of quantum technology, the incorporation of quantum effects in computational electromagnetics is attracting increasing attention. The potential quantities A and Φ are natural bridges that connect classical electromagnetics with quantum effects, which fact gives another advantage in solving the A-Φ formulation over the E-H formulation. Exterior calculus is a concept in differential geometry, where physical quantities, such as the vector potential A and scalar potential Φ, are cast into differential forms and their physical equations are described by exterior calculus operators. Differential geometry and exterior calculus offer new insights into electromagnetic problems as well as how the A-Φ equations should be discretized. Discrete exterior calculus is the discretized version of exterior calculus for digital computations. In general, DEC can be viewed as a generalized version of the finite difference method, where the Stokes’ theorem and Gauss’s theorem are naturally preserved. The preservation of Stokes’ theorem and Gauss’s theorem can get rid of spurious solutions, which can improve computational accuracy. Furthermore, compared with finite difference method, where rectangular grids are applied, DEC can be implemented with more flexible meshes, such as tetrahedral meshes. It makes DEC be able to capture complicated structures more easily and efficiently. In this paper, the A-Φ formulation along with the generalized Lorenz gauge are presented, and its DEC matrix equations are introduced as well. The proposed algorithm is validated with numerical example at the end of the paper.