In this paper, we consider finite element models for high-temperature superconductors and compare two dual formulations, either magnetic-field conforming or magnetic-flux-density conforming. The electrical resistivity of superconductors is described by a power law and is strongly nonlinear. We compare the accuracy and the efficiency of the dual formulations by starting from simple considerations on the concavity / convexity of the constitutive law involved in each case. We then study the numerical behavior of each formulation in 1D, 2D, and 3D problems and compare their results against benchmarks. We draw general recommendations for the choice of a formulation, an iteration scheme for treating the corresponding linearized constitutive law, and a time-stepping interpolation scheme. This approach is extended to soft ferromagnetic materials with a saturation law. Given that our analysis shows that recommended formulations for treating ferromagnets are just the opposite of those for treating superconductors, we suggest a coupled formulation for systems where both types of materials are present. The coupled formulation is shown to be accurate and more efficient than single formulations applied indistinctly to all materials.

We numerically investigate the effect of an edge indentation on the threshold field of thermomagnetic instabilities in superconducting films subjected to a ramping magnetic field, applied perpendicular to the plane of the film. In particular, we are able to address the question on whether edge indentations promote magnetic flux avalanches. For the magnetic field-independent critical current density model, the triggering of the first magnetic flux avalanche systematically occurs at the edge indentation. In contrast to that, for the more realistic field-dependent critical current density model the first flux avalanche can take place either at or away from the indentation. This selective triggering of magnetic flux avalanches is shown to result from two effects. Namely, (i) the variation of the threshold magnetic field for the first flux avalanche triggered at the indentation and (ii) the reduction of the critical current density by large local magnetic fields at the tip of the indentation which translates in a lower power density dissipated near the tip. We demonstrate that this interplay can be tuned by varying the indentation size, ramp rate of applied field ${\dot{H}}_a$, and working temperature $T_0$. We build up a phase diagram in the ${\mu }_0{\dot{H}}_a-T_0$ plane with well-defined boundaries separating three distinct regimes of thermomagnetic instability.

In this article, 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.

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 $\vec{H}$ −φ 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

High-temperature superconducting coils are used in various large-scale applications, like rotating machines and high-field magnets. However, modeling these coils is a complicated and time-consuming process, especially due to the non-linearity of the current–voltage characteristics of the superconductors and the complex multiphysics involved. In this work, we used a fast homogenized method to model the coupled electromagnetic and electrothermal properties of racetrack and pancake coils for different applications. For this purpose, various formulations wielding homogenization methods are used and benchmarked with each other, as well as with models considering the detailed structure of the HTS tapes. We observe a very good agreement between different models (homogenized and detailed), and we discuss the pros and cons of the inclusion of insulating layers between the turns in homogenization. This work was performed under the collaboration of the COST action modeling teams and can be used as a review of the state-of-the-art superconductor modeling techniques, and a source for the development and benchmark of future numerical methods.

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- ϕ 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.