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

An efficient numerical solver for the <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> formulation in electromagnetics based on discrete exterior calculus (DEC) is proposed in this paper. The <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> 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 <bold>A</bold> equation and the <inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> equation. The <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> formulation is discretized by using the DEC, which is the discretized version of 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 <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> solver is inherently stable, free of spurious solutions and can capture highly complex structures efficiently. In this paper, the background knowledge about the <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> formulation and DEC is introduced, as well as technical details in implementing the DEC <bold>A</bold>-<inline-formula><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> solver with different boundary conditions. Numerical examples are provided for validation purposes as well.

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