The 5th international workshop on numerical modelling of high temperature superconductors

In Ainslie et al [2], a 2D axisymmetric numerical modelling framework based on the H-formulation is implemented for investigating the 'giant field leap' effect observed during pulsed field magnetisation of cylindrical bulk HTS samples with large J_c values. The model is able to reproduce the observed results of magnetic flux suddenly intruding into the centre of the sample, without the need for any new physics to explain the physical mechanism. A bounded E–J power law, able to naturally link the flux creep and the flux flow regime, is used to increase the accuracy of the numerical solution and to improve the convergence properties of the model.

In Bykovsky et al [3], the magnetisation loss for stacks of ReBCO tapes are investigated numerically and compared with experiments. The influence on the magnetisation loss of the number of tapes, width of the tape, magnetic field orientation and tape manufacturer is studied. Based on the numerical results, an analytical approach for estimating the magnetisation loss in stacks of tapes with an arbitrary number of tapes under the assumption of the critical state model is developed.

In Ruuskanen et al [4], a 2D model for evaluating the hysteresis loss of Roebel-cable-based accelerator magnets during ramp-up is developed. The model exploits the critical state model of the material, combined with the minimum magnetic energy variation principle for calculating the numerical solution. An open-source software package, suitable for large-scale nonlinear optimisation problems, is used for minimising the functional. The use of the critical state allows one to avoid the overestimation of losses which is obtained by using the power law, due to the fact that a non-zero electric field is associated with any non-zero current.

In Campbell et al [5], the crossed field decay of the trapped magnetisation of thin strip superconductors, and stacks thereof, are investigated numerically using two finite element method-based modelling frameworks: FlexPDE with the A-formulation and COMSOL with the H-formulation. The simulations show good agreement with the relevant theory developed by Brandt and Mikitik and confirm that the decay in very thin sheets is slow, even when the applied transverse field is much greater than the transverse penetration field.

In Zermeno et al [6], a method is developed for extracting the dependence of the critical current density J_c of a superconductor on the local magnetic field B_loc, starting from the experimental data relating the critical current, I_c, of a tape to the external applied field B_a. The output of the method is a set of J_c-B_loc data that, once interpolated, can be used as the material's properties for successive simulations of low-field-operating HTS devices, such as resistive fault current limiter or power cables, for which the effect of the self-field could cause inaccuracy in results.

In Gömöry et al [7], a method is developed for calculating the total AC loss of a coil, based both on local quantities (scalar product of E and J) and on global quantities (overall current and voltage of the coil), the latter of which are observable during experiments. It is suggested that possible differences between the two approaches can be used for identifying numerical errors from the calculations. The approach is implemented both for the A-ϕ and the H-formulations. A model for calculating the voltage of the coil as a post-processing variable when using the H-formulation is developed.

In Escamez et al [8], a 3D numerical model for the calculation of AC losses in an MgB₂ multi-filamentary wire subject to AC transport current and applied field is developed. The wire is made of 36 twisted filaments embedded within a resistive and ferromagnetic matrix. The model exploits the finite-element approach based on the H-formulation and is able to take into account the coupling and eddy current losses, without considering, so far, the effect of the resistive barriers surrounding the SC filaments. Data on the electrical and magnetic properties of all constitutive materials of the wire, obtained by extensive characterisation, are also reported.

In Pardo et al [9], a method is developed for calculating the current and field distribution in homogeneous 3D bulks subject to an applied field and/or transport current. The method is based on the Minimum Electro-Magnetic Entropy Production approach in 3D (MEMEP3D), which allows dealing with the high number of degrees of freedom needed for the analysis of actual 3D problems. The method is used for investigating the current density, saturation field and magnetisation loops of rectangular prisms with a square base and for several height-to-width aspect ratios. Non-rectangular current paths are found for applied fields below the saturation value for cubic-like geometries, which have a significant influence on the magnetic field produced at the surface.

In Zhang et al [10], a numerical model for calculating the current distribution and AC loss of 2G HTS tapes or coils is developed. The model assumes the thin sheet approximation for the HTS tape and is based on a hybrid T–A formulation that allows an easy assignment of the boundary conditions. The model is validated against an analytical solution for a 1D case (magnetisation of a thin disk) and is successfully applied to complex geometries such as a twisted stacked tapes conductor subjected to an applied magnetic field and a racetrack coil with a transport current.

In Peña-Roche et al [11], an app is developed for the numerical investigation of 2D magnetic levitation systems. More specifically, well-established physical models are implemented in the form of a utility for portable devices. A hybrid strategy is used that combines, via wireless or cable communication, pre-processing in a high performance computer and the final post-processing on the portable device. The app is well suited to provide information on hysteresis effects, so as to contribute to training and dissemination and to complement computer-based tools for Maglev design.

In Bonnard et al [12], a multiscale model is developed to investigate behaviour of the resistive fault current limiter. In particular, depending on the discretisation used, the model is able to take into account the relevant phenomena affecting the performance of the device, e.g., hot spots and diffusion of heat across the thickness of the tape. Remarkably, the model is implemented in the EMTP-RV environment, which is a professional power system simulator. This enables the accurate evaluation of the transient response and the effect of the device when it interacts with real-world power grids.

In Shukrinov et al [13], resonance phenomena in a model of intrinsic Josephson Junctions shunted by LC-elements are studied. The phase dynamics and IV characteristics are investigated in detail when the Josephson frequency approaches the frequency of the resonance circuit. A realisation of parametric resonance through the excitation of a longitudinal plasma wave, within the bias current interval corresponding to the resonance circuit branch, is demonstrated. The authors found that the temporal dependence of the total voltage of the stack, and the voltage measured across the shunt capacitor, reflect the charging of superconducting layers, a phenomenon that might be useful as a means of detecting such charging experimentally. Based on the voltage dynamics, a novel method for the determination of charging in the superconducting layers of coupled Josephson junctions is proposed. A demonstration and discussion of the influence of external electromagnetic radiation on IV characteristics and charge-time dependence is given. Over certain parameter ranges the radiation causes an interesting new type of temporal splitting in the charge-time oscillations within the superconducting layers.

Towards the end of the workshop, participants formed groups to discuss current and future challenges for numerical modelling in a number of key areas: (1) fundamental aspects and physics, (2) tapes, cables and coils, (3) multi-physics and multiscale problems, and (4) mathematical tools and software applicable to HTS modelling.

The following highlights were presented in the summaries of the group discussions:

3D critical state modelling is still an open problem, especially concerning crossed-field effects and the constitutive law between E and J, such as the E–J power law, when E and J are not parallel to each other. Recent attempts, like the double critical-state model (CSM), do not appear to be a valid solution. A theoretical explanation for the time dependence of the trapped field in bulk superconductors acting as trapped field magnets remains open as well: experimental results have shown a plateau in the remanent magnetisation shortly after magnetisation that tends towards logarithmic or faster decay. A new benchmark (full 3D) problem—the magnetisation of an HTS cube—was established, and some solutions are already available on the HTS Modelling Workgroup's website (Benchmark #5) [14]. Finally, it would be interesting to attempt to recover the CSM limit, for the case of a certain number of flux lines, using calculations based on the Ginzburg–Landau formalism.

The issue of long computing times for simulations, even for large-scale coils, does not seem to be a problem any more thanks to the possibility of using the 'effective' turns instead of real turns, modification of the aspect ratio of coated conductors, and simultaneous computations using parallel computing. A material properties database for modellers could be a very important outcome from the community in the near future. Collated data relevant to the electrical and thermal properties of HTS coated conductors is now available on the HTS Modeling Workgroup's webpage. Several tailored simulation tools have been implemented in different institutions for various purposes. The community strongly supports the exchange of these, and the relevant documentation, including related publications, to make the tools/models available to other scientists in the field.

Analytical approaches and solutions to situations where they are applicable are now highly mature and it is difficult to push many new developments, especially for more complex problems. New, alternative numerical modelling techniques are being developed that challenge the commonly used, finite element method-based approaches. These include the boundary element method, the fast multipole method, and mixed approaches combining different methods. Meshless methods may be useful for HTS modelling, but this has not been explored in great detail to date. Again, there is a clear trend for publishing codes open source and this can fast-track the exchange of codes. 3D modelling for general simulation situations is still coming, and only specific tools can solve specific or superficial problems. It is strongly recommended that the community improve coupling with experts in numerical analysis to accelerate developments HTS modelling.

We would like to express our gratitude to the Department of Electrical, Electronic and Information Engineering of the University of Bologna, the European Society for Applied Superconductivity and the IEEE Council on Superconductivity for their generous economical support. We are also grateful to the moderators of the open discussion tables and the session chairs. Finally, we would like to acknowledge the precious assistance of Eleonora di Felice and Nadia Borrelli of the Fondazione Alma Mater in the organisation of the workshop.