  For finite element (FE) analysis of no-insulation (NI) high-temperature
superconducting (HTS) pancake coils, the high aspect ratio of the turn-to-turn
contact layer (T2TCL) leads to meshing difficulties which result in either poor
quality mesh elements resulting in a decrease of the solution accuracy or a
high number of degrees of freedom. We proposed to mitigate this issue by
collapsing the T2TCL volume into a surface and using a so-called thin shell
approximation (TSA). Previously, two TSA have been introduced, one to solve the
heat equation and the other for an $\vec{H}-\phi$ magnetodynamic formulation.
  In this work, we propose to combine the magnetodynamic and thermal TSA to
create a coupled magneto-thermal TSA for three-dimensional FE analysis.
Particular attention is paid to the detailed derivation of the coupling terms.
In the context of NI HTS pancake coils, the TSA represents the electric and
thermal contact resistance of the T2TCL. For the HTS coated conductor (CC)
itself, an anisotropic homogenization is used which represents its
multi-layered structure. In axial and azimuthal direction, it resolves the
current sharing between the HTS and other layers of the CC. The coupled TSA
formulation is verified against a reference model with volumetric T2TCL. The
coupled TSA is shown to significantly reduce the solution time as well as the
manual effort required for high-quality meshes of the T2TCL. The implementation
is open-source and a reference implementation is made publicly available.
