In this work, we present and analyze the numerical stability of two coupled finite element formulations. The first one is the h-a-formulation and is well suited for modeling systems with superconductors and ferromagnetic materials. The second one, the so-called t-a-formulation with thin-shell approximation, applies for systems with thin superconducting domains. Both formulations involve two coupled unknown fields and are mixed on the coupling interfaces. Function spaces in mixed formulations must satisfy compatibility conditions to ensure stability of the problem and reliability of the numerical solution. We propose stable choices of function spaces using hierarchical basis functions and demonstrate the effectiveness of the approach on simple 2D examples.

In this work, we present and analyze the numerical stability of two coupled finite element formulations. The first one is the h-a-formulation and is well suited for modeling systems with superconductors and ferromagnetic materials. The second one, the so-called t-a-formulation with thin-shell approximation, applies for systems with thin superconducting domains. Both formulations involve two coupled unknown fields and are mixed on the coupling interfaces. Function spaces in mixed formulations must satisfy compatibility conditions to ensure stability of the problem and reliability of the numerical solution. We propose stable choices of function spaces using hierarchical basis functions and demonstrate the effectiveness of the approach on simple 2D examples.

In applications requiring a large magnetic force, permanent magnets with non-parallel magnetization directions can be assembled in a Halbach array to generate a large gradient of magnetic flux density. The saturation magnetization of permanent magnets, however, brings a fundamental limit on the performance of this configuration. In the present work, we investigate experimentally the assembly of cuboid bulk, large grain melt-textured YBa2Cu3O7−x superconductors (∼14×14×14 mm3) with orthogonal c-axes so as to form a basic unit of Halbach array. The experiments are carried out at 77 K. The experimental distribution of the magnetic flux density above the array of trapped-field superconductors is compared to a similar array made of permanent magnets. A simple analytical model is developed and is shown to accurately reproduce the main experimental observations. The results suggest that a redistribution occurs in the current flowing in the central sample when the distance between the superconductors is reduced, whereas the neighbouring superconductors are unaffected. It is shown that this current redistribution yields a reduced contribution of the central sample to the magnetic flux density above the centre of the array and a new negative contribution associated with stray fields to the magnetic flux density at this location. This interpretation is confirmed by modelling of the distribution of transport currents in the superconductor using a 3D finite element model.

We discuss the relevance of several finite-element formulations for nonlinear systems containing high-temperature superconductors (HTS) and ferromagnetic materials (FM), in the context of a 3D motor pole model. The formulations are evaluated in terms of their numerical robustness and efficiency. We propose a coupled h-φ-a-formulation as an optimal choice, modeling the problem with an a-formulation in the FM and an h-φ-formulation in the remaining domains. While maintaining a low number of degrees of freedom, the h-φ-a-formulation guarantees a robust resolution and strongly reduces the number of iterations required for handling the nonlinearities of HTS and FM compared to standard formulations.

In this work we demonstrate the magnetic shielding ability of a stack of YBa2Cu3O7 (YBCO) tape annuli. The annuli are cut from a 46 mm wide second generation coated conductor deposited on a Ni-5at.%W alloy ferromagnetic substrate. The inner bore of the stacked tapes is 26 mm and the outer diameter is 45 mm. Three samples with different height (24 mm, 14.9 mm, 9.9 mm) are studied. All the experiments are carried out at both room temperature and liquid nitrogen temperature (77 K). The shielding efficiency is investigated when the magnetic field is applied either parallel to the axis of the stack (axial shielding) or perpendicular to it (transverse shielding). Under an axial field, magnetic shielding is found to be effective (SF > 3) up to magnetic flux densities of 0.67 T. The presence of the ferromagnetic substrates is found to have two important consequences. First, the stack of annuli is able to shield transverse flux densities in spite of its layered structure. Second, a finite magnetic shielding effectiveness is demonstrated at room temperature. In order to understand the contribution of the ferromagnetic substrates to the shielding mechanism, we use the experimental field dependence of the magnetic permeability as determined independently from hysteresis loop measurements on the same substrates. A finite-element homogenized model solved with an H-Φ formulation is shown to successfully reproduce the shielding factor of the stack at room temperature and 77 K, both under axial and transverse applied fields. These models are also used to assess the influence of the critical current density and the magnetic permeability on the shielding efficiency. Finally, the results are used to predict the magnetic shielding properties of higher stacks, demonstrating their significant potential to shield axial fields of ∼ 0.93 T (with SF > 10) at 77 K.