Magnetic shielding of various geometries of bulk semi-closed superconducting cylinders subjected to axial and transverse fields

We consider the properties of bulk superconductors to be used as low-frequency passive magnetic shields. Although remarkable shielding properties have been recently achieved using high-temperature superconductors of various kinds, one current issue is to assemble medium-size superconducting parts to obtain large superconducting volumes. The aim of the present work is to understand how hollow, semi-closed superconductors can be combined to improve the shielding properties over sizeable volumes. In axisymmetric superconducting geometries subjected to an axial field, 2D modelling can be used to understand important features of the shielding properties. When finite-size superconductors are subjected to a transverse field, 3D modelling must be used. In this work, we use 3D finite-element modelling with an A-phgr formulation to investigate various geometries in which a tube is closed by a superconducting element shaped like a disk, a cup, or another cup-shaped superconductor that is coaxial with the first. The simulations help in revealing the most performant configurations to use as a function of the geometry of the applied field. Under an axial field, the type of closing is found to be irrelevant and the key ingredient to improve the shielding factor is to reduce the average field in the opening plane, e.g. by using a thicker superconductor near the open end. Under a transverse field, the difference between the shielding properties arise from the different routes taken by flux lines to penetrate the shield. In particular, the presence of flux lines channelled through the gap between a tube and a cup-shaped sample surrounding the tube are detrimental to the shielding properties. The configurations where the tube surrounds the cup-shaped sample are found to yield much higher shielding factors, whose field dependence is further improved when the tube extends slightly beyond the end of the cup. The values of the shielding factors that can be reached under a transverse field of low amplitude are discussed by comparing them to those predicted for an ideal perfectly diamagnetic superconductor of similar dimensions.