A Finite Element $a$-$h$-Formulation for the Reduced Order Hysteretic Magnetization Model for Composite Superconductors

The simulation of transient effects in large-scale superconducting systems with the finite element method is computationally expensive. A Reduced Order Hysteretic Magnetization (ROHM) model has been recently proposed for the computation of the magnetization and loss of composite superconductors. It accounts for the interplay between hysteresis, eddy, and coupling effects, without a need to model the detailed current density distribution, leading to a substantial reduction of simulation time. The ROHM model naturally fits in finite element formulations written in terms of the magnetic field such as the $h$-$\phi$- or $\phi$-formulation, but these formulations are not always the optimal choice. For example, in the presence of ferromagnetic materials, one may prefer formulations written in terms of the magnetic flux density. In this context, we introduce in this paper a mixed $a$-$h$-formulation that implements the ROHM model. The main advantage of this formulation is the direct use of the constitutive relation defined by the ROHM model, without the need for its inversion. We discuss the computational efficiency of the new formulation compared to a conventional $\phi$-formulation, based on a model of a composite superconducting strand.