Numerical simulations of the penetration of magnetic field in a superconducting film : H-phi formulation with shell-transformation techniques

We consider the Finite Element Method (FEM) to solve numerically the problem of the penetration
of magnetic field applied perpendicularly to a superconducting thin film. Although theoretical
boundary conditions on electromagnetic fields are given at an infinite distance from sources, the FEM method applies boundary conditions at the surface of the mesh, and hence has to incorporate in the model a large volume of air outside the superconducting film. This not only represents an approximation with regard to the theoretical boundary conditions, but also increases the number of degrees of freedom and the computation time. The shell-transformation techniques [1] consists in enclosing the near-field FE model in a meshed shell region, and to map this shell region, which is of finite extension, onto the exterior of the FE model, which is of infinite extension. The shell region can have various shapes (e.g., hollow parallelepiped and hollow sphere in 3D), and the boundary condition can then be imposed on the outer surface of the shell with no approximation. We investigate the possibility to reduce the number of degrees of freedom with the help of the shell transformation techniques, in the framework of an H-phi formulation. To this aim, we compare the number of degrees of freedom, the computational time, and the accuracy of the solutions in two cases : (1) with a shell-transformation, and (2) : without shell-transformation, in a large non conducting domain. The obtained results are compared for different geometries (thin disk and thin strip in the critical state, square thin film). The influence of the order of the elements in the transformed region, the shape and size of the region where the shell-transformation is applied, and the mesh quality inside this region are also considered.