Magnetodynamic H-ϕ 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–ϕ formulation has recently gained significant traction in order to speed up simulations. In our previous work, we implemented the H–ϕ 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–ϕ formulation in COMSOL by using Faraday's law in air domains to improve the coupling between the H and ϕ 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–ϕ formulation. The magnetodynamic H–ϕ 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.