An <b>A</b>-$\Phi$ Formulation Solver in Electromagnetics Based on Discrete Exterior Calculus
Boyuan Zhang, Dong-Yeop Na, Dan Jiao, Weng Cho Chew
Abstract
An efficient numerical solver for the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> equation and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> equation. The <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> formulation and DEC is introduced, as well as technical details in implementing the DEC <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Phi$</tex-math></inline-formula> solver with different boundary conditions. Numerical examples are provided for validation purposes as well.