  The growing interest in the modeling of superconductors has led to the
development of effective numerical methods and software. One of the most
utilized approaches for magnetoquasistatic simulations in applied
superconductivity is the $H$ formulation. However, due to the large number of
degrees of freedom (DOFs) present when modeling large and complex systems (e.g.
large coils for fusion applications, electrical machines, and medical
applications) using the standard $H$ formulation on a desktop machine becomes
infeasible. The $H$ formulation solves the Faraday's law formulated in terms of
the magnetic field intensity $\mathbf{H}$ using edge elements in the whole
modeling domain. For this reason, a very high resistivity is assumed for the
non-conducting domains, leading to an ill-conditioned system matrix and
therefore long computation times. In contrast, the $H$-$\phi$ formulation uses
the $H$ formulation in the conducting region, and the $\phi$ formulation
(magnetic scalar potential) in the surrounding non-conducting domains,
drastically reducing DOFs and computation time. In this work, we use the
$H$-$\phi$ formulation in 2D for the magnetothermal (AC losses and quench)
analysis of stacks of REBCO tapes. The same approach is extended to a 3D case
for the AC loss analysis of a twisted superconducting wire. All the results
obtained by simulations in Sparselizard are compared with results obtained with
COMSOL. Our custom tool allows us to distribute the simulations over hundreds
of CPUs using domain decomposition methods, considerably reducing the
simulation times without compromising accuracy.
