2-D Thin-Shell Model Based on the $H$-$\phi$-Formulation for Modeling HTS Tapes in COMSOL Multiphysics

This article presents a finite-element thin-shell (TS) model and its application to 2-D electromagnetic problems involving superconducting tapes in COMSOL Multiphysics. The magnetic scalar potential (<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>) is the state variable in nonconducting regions surrounding of the tapes, which are represented as zero thickness objects in the calculus domain. Inside the tapes, an auxiliary 1-D problem formulated in terms of the tangential components of the magnetic field (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula>) takes into account the physics across their thickness. The final finite-element system of equations includes both the 2-D and 1-D discretized equations, which are solved simultaneously in a fully coupled manner and transparently for the user. The use of thin cuts is required to impose transport currents in the tapes. This procedure allows the simulation of problems comprising superconducting tapes in any geometrical configuration. We demonstrate that both the normal and tangential fields agree well with reference solutions obtained with the widely used <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$A$</tex-math></inline-formula>-formulation and with the more standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula>- and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-<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>-formulations with a full 2-D discretization of the tapes. The proposed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-<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> TS model estimates ac losses accurately while speeding up simulations. This makes this model ideal for simulating large-scale superconducting devices in 2-D, particularly when they comprise compact arrangements of high-temperature superconductor tapes carrying antiparallel currents.