Magnetodynamic H–$\phi$ Formulation for Improving the Convergence and Speed of Numerical Simulations of Superconducting Materials

The growing interest in fast and accurate simulation methods of the electromagnetic behavior of high-temperature superconducting materials has led to many exciting developments in the last decade. Although the H formulation implemented in the finite-element method has proven to be very robust for electromagnetic calculations, the dummy resistivity and vector-dependent variables required in air domains lead to long computation times and spurious currents in certain applications. Thus, the H–
$\phi$
formulation has recently gained significant traction in order to speed up simulations. In our previous work, we implemented the H–
$\phi$
formulation in COMSOL Multiphysics using the divergence-free condition in air domains and showed that the computation times were reduced by two and three in 2-D and 3-D, respectively. However, high-order shape functions were required in order to obtain sufficient accuracy when compared to the full H formulation. In this article, we implemented a magnetodynamic H–
$\phi$
formulation in COMSOL by using Faraday's law in air domains to improve the coupling between the H and
$\phi$
physics and correctly represent time-varying phenomena in superconducting materials. We show that this formulation leads to better convergence, more accurate solutions, and a slight speed advantage compared to the previously used H–
$\phi$
formulation. The magnetodynamic H–
$\phi$
formulation results are shown to be nearly identical to the full H formulation even with linear shape functions, while reducing the computation times by a factor of up to four for a given mesh.