A variety of numerical methods have been developed for the computation of electromagnetic fields, including the finite-element (FEM), boundary element (BEM), and finite integration techniques. FEM packages for both electromagnetic and mechanical design are nowadays standard tools in the design of accelerator magnets. For iron-dominated magnets, these tools can be used as a “black box”, except for often negligable effects stemming from iron hysteresis, 3D eddy currents, magneto-striction, and magnetic after effects.
Superconducting magnets are a bit more demanding. The field quality in these coil-dominated magnets also strongly depends on dynamic effects such as SC magnetization and inter-strand coupling currents. In particular for the next generation of magnets, quench simulations have to be an integral part in the early design studies. This paper gives an overview of the numercial techniques used and discusses the limitations of electromagnetic design programs. An important aspect in this context is to find a consistent basis for beam-physics simulation, numerical field computation and magnetic measurements, in the different phases of design, prototyping, series production, and commissioning.
References: Stephan Russenschuck, Field Computation for Accelerator Magnets, Wiley-VCH, 2010

Uniform and high-performing high-temperature superconductor (HTS) tape has recently become available in long lengths and with decreasing unit price, which has made HTS devices reach a pre-commercial stage: numerous prototypes (including cables, fault current limiters, and transformers) have been built, installed and are currently being used around the world. In parallel to this development, numerical modeling of HTS has also been flourishing, thanks to the increased interest of having powerful tools to predict the behavior of HTS devices and – not to be forgotten – the availability of relatively cheap computing power.
In this contribution, I aim to give an overview of the situation of numerical modeling of HTS applications. I will not indulge (other than strictly necessary) on listing the numerous and different approaches for modeling HTS materials: this is the topic of another contribution in this session and has recently been the subject of different review articles. Instead, I will rather focus on the models that have been used to simulate actual devices, showing their degree of complexity and their accuracy. More specifically, I will consider models describing the electromagnetic and thermal behavior of cables and windings, as well as models investigating the magnetic flux dynamics in HTS bulks and stacks of tapes used as permanent magnets. Macroscopic models describing the behavior in superconducting grids will also be examined. The final part of this contribution is dedicated to a discussion on the availability and easiness-of-use of those models, which will hopefully serve as a basis for discussion within the community.

In this talk I will present a selection of recent developments in the field of computational electromagnetics, focusing on finite element methods and related computational tools. Using the open source environment ONELAB (http://onelab.info), I will show how fast-converging high-order discretizations can be used to reach high-accuracy solutions, how cohomology basis functions can be computed to solve complex field-potential problems in three dimensions, and how optimized domain decomposition methods can be applied to solve extreme-scale problems on massively parallel computers. Hands-on demonstrations will be given on practical examples, including e.g. the modeling of twisted superconducting fibers.

The high current capacity of superconducting tapes and wires has made them candidates of choice for designing compact and light cables and coils that can be used in large scale power applications. However, their performance is often limited by their critical current (Ic) and the AC losses they experience. In general, these characteristics are determined by several factors, including the conductor's material properties, the electromagnetic environment and the geometric layout of the device. Therefore, estimation of the Ic and AC losses in a particular device, although important, often requires a detailed analysis that considers the aforementioned factors. While several design and simulation tools have already been developed to estimate both the Ic and AC losses, it is a common occurrence that these tools remain kept as a “secret sauce”, are implemented only using expensive commercial software and in most cases only a mathematical description of the modeling technique is made available.

In this work, we present in detail the implementations of a recently proposed self-consistent stationary model used to estimate the critical current of superconducting devices. Said implementations are made in different programming environments, including finite-element-based and general numerical analysis programs, both commercial an open-source. A comparison of the accuracy and calculation speed of the different implementations of the model is carried out for the case of a Roebel cable. The model is also used to evaluate the importance of choosing a very accurate description of the angular Jc(B) dependence of the superconductor as input for the material’s property. Finally, although the model is primarily developed with the purpose of calculating the critical current of HTS devices, its applicability for a simple way of approximately estimating the AC losses is discussed. The numerical codes, two of which are open-source, are made freely available to interested users.

This work has been partly supported by the Helmholtz Association (Young Investigator Group grant VH-NG-617). The authors are very thankful to Prof. Frédéric Hecht who provided valuable support for using FreeFEM++

For high current REBCO superconductors in high magnet fields with currents in the order of 50 kA, single coated conductors must be assembled in a cable. The geometry of such a cable is mostly such that combined torsion, axial and transverse loading states are anticipated in the tapes and tape joints. A set of experimental setups as well as a convenient and accurate method of stress-strain state modeling based on Finite Element Method (FEM) has been developed. Systematic measurements on single REBCO tapes are carried out combining axial tension and torsion as well as transverse loading. Then the behavior of a single tape subjected to the various applied loads is simulated in the model. This paper presents the results of experimental tests and detailed FE modeling of the 3D stress-strain state in a single REBCO tape under different loads, taking into account the temperature dependence and the elastic-plastic properties of the tape materials, starting from the initial tape processing conditions during its manufacture up to magnet operating conditions. Furthermore a comparison of the simulations with experiments is presented with special attention for the critical force, the threshold where the tape performance becomes irreversibly degraded. We verified the influence of tape surface profile non-uniformity and copper stabilizer thickness on the critical force. The FE models appear to describe the tape experiments adequately and can thus be used as a solid basis for optimization of various cabling concepts. The first modeling results for CORC conductors are presented.

Analysis of vortex motions in superconductors is important for applications of superconductors. There are four levels of numerical simulation methods for vortex dynamics: Microscopic Bogolibov-de Gennes equations, phenomenological Ginzburg-Landau (GL) equations, molecular dynamics (MD) method and macroscopic method. In order to tread hundreds of vortices, we may use the MD method, in which vortices are treated as classical particles. This method is used for many studies for decades [1]. But in the MD method, several features in vortex dynamics cannot be described. For example, these are heat generation, which occurs inevitably when vortices moves, and relaxation of superconducting order parameter after a vortex moves.
In this study, we include these features into the MD method. First, in order to incorporates the heat generation, the heat transport equation is solved simultaneously with MD equations [2]. In this case, vortices receive the entropy force that is proportional to gradient of the temperature distribution.
Next, in order to incorporate the relaxation of the order parameter, we consider a condensation energy field, which is proportional to square of amplitude of the order parameter. This condensation energy field becomes a potential field for vortices, because core energies of vortices are proportional to the condensation energy [3]. And this potential energy causes retarded attractive interaction between vortices when the vortex motion is rapid.
We apply this extended MD method to vortex dynamics in a superconducting Corbino disk, where temperature may become non-uniform and velocities of vortices around the center of disk becomes large. We show some typical vortex motions in the Corbino disk.

[1] C. Reichhardt, C. J. Olson and F. Nori, Phys. Rev. Lett. 78 (1997) 2648.
[2] D.E. Fujibayashi, M. Kato, Physica C 484 (2013) 94.
[3] M. Kato and O. Sato, Physics Procedia, in press.

We thank T. Koyama, T. Ishida, T. Tamegai, S. Okuma, N. Kokubo, K. Hirata, M. Hayashi, H. Ebisawa for useful discussions. This work was supported by JSPS KAKENHI Grant Number 26400367.

It is well known that commercial second generation (2G) high temperature superconductor (HTS) coated conductors (CCs) possess a low normal zone propagation velocity (NZPV) in comparison with low temperature superconductor wires. One consequence of a low NZPV is that 2G HTS CCs must be over-stabilized in order to prevent the appearance of hot spots, which reduces their engineering critical current and their resistance per unit length. Recently, a tape architecture called “current flow diverter (CFD)” was proposed to enhance by orders of magnitude the NZPV of 2G HTS CCs [1]. In the CFD concept, the interfacial resistance between the HTS and silver layers is strongly increased at the center of the tape but kept low at the tape edges. The current flow is then diverted towards the tape edges when it transfers from the HTS layer to the stabilizer, which results in an important increase of the current transfer length and the NZPV. In this work, we developed an electro-thermal model using the finite element method to investigate the effects of various parameters, such as the temperature, critical current, stabilizer thickness, applied current, etc., on the performance of the CFD architecture in comparison with commercial 2G HTS CCs. Our results show that the CFD architecture increases the NZPV for the whole range of operating temperature (4 – 90 K). Also, we found out that the benefits of the CFD architecture, in terms of NZPV, are higher when the stabilizer is thinner and the critical current gets higher. An enhancement of the NZPV of more than two orders of magnitude is predicted for specific tape configurations. Finally, we intend to compare our numerical results with experimental ones and discuss the possible improvements to the model in order to fully reproduce the experimental behavior of CFD tapes.
[1] C. Lacroix and F. Sirois 2014 Supercond. Sci. Technol. v27 p035003

Natural Sciences and Engineering Research Council of Canada (NSERC)

Strain dependence of Nb3Sn superconducting properties (i.e. Bc2, Tc, Jc) is known to be responsible for the degradation of transport current capability (or temperature margin) of large steel jacketed cable-in-conduit conductors (CICCs) made with about one thousand strands. This phenomenon was particularly observed on the big ITER Nb3Sn CICCs, either in the model coils or in the conductor qualification samples, and must be anticipated for still larger current conductors expected in the future fusion reactors. The mechanical deformations of the strands under bending in the cables, due both to cool down after heat treatment and to Lorentz force during operation, are the main sources of strand-in-cable critical current degradation. In order to better understand this phenomenon, strand-scale experiments have been carried out at CEA/IRFM to characterize Nb3Sn strands under bending using dedicated VAMAS-like mandrels. These mandrels allow strand bending under the Lorentz force created during critical current measurements. The paper will report on the description of the models and the analyses performed using coupled mechanical and electrical modeling of these experiments. The key of this modeling lies in its ability to characterize the superconducting properties of the twisted Nb3Sn filaments inside a locally bent strand from its global mechanical deformation map and to take into account possible current redistribution among filaments all along the strand length. Results of simulations are compared to available experimental data. This work can be seen as the first stone of an integrated coupled mechanical-electrical modeling of large Nb3Sn CICC which would be also relevant to future expected high temperature superconductor CICCs made with strain sensitive strands.

This work is partly funded by the French “Agence Nationale de la Recherche” within the Cocascope project.

Connections between superconducting regions by normal metals creates coupling AC loss. This produces additional heating, complicating the cryogenics of superconducting applications and reducing their efficiency. Coupling currents present also 3D behavior, which is still not fully understood and is frequency dependent. Predicting the current paths of coupling currents in general 3D samples require numerical computations, in order to reduce (and maybe cancel) coupling loss by sample optimization.
In this work we model and measure coupling (and hysteresis) loss of 4 mm wide face-to-face soldered tapes and striated tapes. The striations of the latter are done by laser ablation on a tape with 5 um copper stabilization. Coupling loss is measured at 72 Hz for applied fields up to 100 mT amplitude and frequencies from 2.4 to 576 Hz for low applied magnetic fields around 10 mT. Modeling is based on a novel variational principle in the J-q formulation, where J and q are the current and charge densities, respectively. In this contribution, this model is also benchmarked with Finite Element Method computations in the H formulation, showing a good agreement. For both models, we assume an isotropic power-law E(J) relation, where E is the electric field. As input, the models use measured current-voltage relations for the superconductor and the resistance per unit length between the two superconducting regions.
We have found that the model agrees with the experiments. In addition, the coupling loss presents the expected peak in the frequency dependence. Finally, at high frequencies the AC loss becomes frequency independent, experiencing a hysteresis loss that is higher than the low-frequency limit.
In conclusion, the presented model is able to predict coupling currents and AC loss in 3D objects containing superconducting and normal parts. This settles the basis for future work on more complex situations, such as coated conductor cables and multi-filamentary MgB­2 wires.

The research leading to these results have received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n0. NMP-LA-2012-280432.

Investigation of the influence that a magnetic field has on materials or biological substances requires sometimes to create an experimental chamber where the studied object can be accommodated at ambient conditions except the magnetic field that should be reduced to zero. Magnetic cloak is a device that could be used for his purpose. On top of shielding the ambient magnetic field the cloak does not influence the field in space outside the chamber. Such property, called the magnetic invisibility, could represent an advantage in a complex experimental environment.
We have developed magnetic cloaks with the so called core-shell cylindrical geometry consisting of the inner superconducting layer from HTS material and the outer ferromagnetic layer with diameter typically below 15 mm. Bulk superconducting tube as well as set of overlapping coated conductor tapes have been tested in place of the superconducting layer. Ferromagnetic materials in shape of sheets and a ferrite powder/ glue composite tube has been used for the outer layer.
Numerical calculations are necessary to find the suitable materials and their optimal arrangement. We present the performance of some of the prepared cloaks and discuss the main issues in design of the cloak with 50 mm diameter of the inner chamber that would be suitable for experiments involving living animals.

While ITER is being built in Cadarache, France, a European “roadmap to fusion electricity by 2050” has been recently approved by the European Commission, which foresees the design of the DEMO reactor as the step following ITER for the production of electricity from fusion energy, strongly relying on technologies already well-developed for ITER. As a supplement and backup to this approach, new technologies are being explored to study potential alternative designs of critical and expensive components such as the superconducting magnets. In this framework, a novel cable-in-conduit conductor (CICC) has been recently proposed by ENEA, which foresees the use of 2nd Generation REBCO-based Coated Conductor tapes stacked into twisted grooves slotted into an Al core. A central hole in the Al core provides a low hydraulic impedance channel for the coolant, direct cooling of the tape stacks is also achieved through small auxiliary semi-circular channels at the bottom of the grooves. Such conductor can be cooled by either supercritical Helium, for high-field / high-current applications, as in fusion magnets, or by liquid nitrogen, for self-field / lower current requirements. While preliminary tests have already shown that the electrical performance of the cable is promising and the cable is industrially feasible, the issues of the thermal-hydraulic performance and of the corresponding optimization have not been addressed so far and are considered in this paper for the first time.
The commercial Computational Fluid Dynamics (CFD) software STAR-CCM+ is used to develop an adequate model for this complex 3D problem. The model is used first to evaluate the flow repartition among the different cooling paths in the CICC, which is then compared with the results of simple correlations. The results of the simulation are then validated against the results of thermal-hydraulic tests performed at cryogenic conditions at CRPP. The model can be used to perform parametric analyses, varying, e.g., the central hole diameter, as well as the localization and size of the grooves at the bottom of the tapes, within the margin allowed by structural constraints, to assess the best compromise between minimization of the pressure drop and maximization of the cooling capability.

Similar to other fields in research and engineering, the use of numerical models is a mandatory step in the design and analysis of superconducting magnet systems. Thanks to tools of increasing complexity, it is possible to forecast performance over the envelope of operating conditions, difficult to test even when a prototype is available. This can be used to optimize the system and reduce the engineering margin. The aim of this review is to describe the evolution of computational tools developed in the past 25 years for the design and analysis of superconducting magnet systems, with main focus on superconducting cable performance (margin, stability and quench), heat transfer in coils (cooling in steady state and transient), proximity cryogenics (process control), and magnet powering in inductively coupled systems. This wide spectrum of phenomena spans over the thermal, hydraulic and electromagnetic performance of superconducting magnet systems. We show how these building blocks have been attacked singularly at first, and presently are being integrated in design and analysis frameworks that provides a consistent multi-physics simulation. Applications to the analysis of laboratory solenoids, accelerator magnets and the ITER coils prove the potential of the method described.

Analytical technique is very useful for modelling of the electrodynamics of superconductors, because analytical formulae reveal information to understand the physics underlies the electrodynamic phenomena, and they suggest useful insights for the design of superconducting devices. Benchmarks of numerical modelling also require exact analytical formulae to check the validity of the numerical methods. Although analytical modelling are feasible only for simple symmetric geometries with restricted situation, many useful analytical formulae have already been obtained for thin and long superconducting tape wires and conductors with constant critical current density.
We investigate magnetic-field distributions for superconducting tapes, where the tape thickness is much smaller than the tape width, and the tapes are all parallel to the z axis with an infinite length. The two-dimensional magnetic field H_x and H_y are systematically analyzed by introducing the complex field, H_y + i H_x, which is an analytical function of the complex variable, z = x + i y, where i is the imaginary unit. The magnetic-field and current distributions are simply derived from the complex fields, and electromagnetic response (e.g., hysteretic ac losses) can also be investigated. We show several geometrical examples of superconducting tape wires (i.e., a single isolated tape, an infinite stack of tapes, curved tapes, superconducting tapes with soft magnetic substrates), for which analytical expressions of the complex field can be straightforwardly obtained by the Schwartz-Christoffel mapping from the H_y + i H_x plane to the x + i y plane.

[1] T. Lubin, K. Berger, and A. Rezzoug, “Inductance and Force Calculation for Axisymmetric Coil Systems Including an Iron Core of Finite Length”,. Progress In Electromagnetics Research B, EMW, vol. 41, pp. 377-396, 2012.

The high current carrying capacity of MgB₂ strand, its round geometry and its low cost, make it a good candidate for compact high power cables. Such cables can be made of thousands of MgB₂ strands carrying in total over 120 kA. One way to obtain an accurate AC loss prediction tool for such cables is to model a single strand under the action of both transport current and external magnetic field, and then superpose the respective contributions of each wire with respect to the general magnetic distribution of the cable. The strand consists of 36 twisted MgB₂ filaments embeded in a pure Nickel matrix. Due to the high aspect ratio between the filaments diameter (100 µm) and the twist pitch of the conductor (around 12 cm), and due to the presence of a conductive and magnetic matrix, the numerical modeling of this problem is challenging, especially in terms of computational resources, and thus a good benchmark problem for modelling softwares. For the case of self-field AC losses, symmetry arguments allow reducing the problem to 2-D, but when the MgB₂ wire is submitted to an external field, the use of a full 3-D model is necessary in order to obtain the correct current and magnetic field distributions along the filaments, thus the correct AC losses. In this contribution, we compare the results of numerical simulations made on three different FEM software packages developed by different laboratories and companies worldwide. The first software package is DarylMaxwell, developed in Polytechnique Montreal, Canada. The second one is Flux, developed by CEDRAT and G2Elab (Grenoble, France). The third one is a dimension-independent finite element code developed at the Tampere University of Technology, Finland. The question of the metal matrix, which adds a second nonlinear material in the problem, is approached step by step, firstly considering air between the filaments, later substituting a conductive metal region, and finally by a nonlinear magnetic conductive region.

This work was supported by Nexans, the Natural Sciences and Engineering Research Council of Canada (NSERC), and The Academy of Finland project number 250652.

The development of high temperature superconducting (HTS) wires/tapes is now at a stage where long lengths of high quality are available commercially, and of these, (RE)BCO coated conductors show the most promise for practical applications. The most important characteristic of a superconductor for any practical superconducting device is the maximum electrical transport current density it can maintain without resistance, or its critical current density, Jc. However, these materials have presented a number of technical challenges for scientists and engineers alike, including grain boundaries and flux pinning, and the structural anisotropy leads to anisotropy in their superconducting properties, otherwise known as an angular dependence of Jc(B,θ). Therefore, one of the most crucial aspects for the numerical modelling of superconducting coils and devices is providing accurate data for this in-field performance.
In this paper, the in-field critical current density characteristics Jc(B,θ) of two commercial HTS coated conductor samples are measured experimentally, and based on this data, an engineering formula is introduced to represent this electromagnetic behavior as the input data for numerical modelling. However, due to the complex nature of this behavior and the large number of variables involved, the computational speed of the model can be extremely slow. Therefore, a two-variable direct interpolation method is introduced, which completely avoids any complex data fitting process for Jc(B,θ) and expresses the anisotropic behaviour in the model directly and accurately with a significant improvement in computational speed. The two techniques are validated and compared using numerical models based on the H-formulation by calculating the self-field and in-field DC critical currents and the AC loss for a single coated conductor.

Di Hu and Jin Zou would like to acknowledge the support of Churchill College, Cambridge, the China Scholarship Council and the Cambridge Commonwealth, European and International Trust. Dr Mark Ainslie would like to acknowledge the support of a Royal Academy of Engineering Research Fellowship. 

Russian Scientific R&D Cable Institute is involving in many projects on R&D HTS power cables including made of modern 2G HTS wires. Evaluation of AC losses and methods for the reduction of the AC losses are important for the design the cables. Fine experimental technique has been developed and implemented to measure AC losses with representative cable models with ~5 m length. But only calculation can provide ac loss prediction in cases when it could be difficult to make measurements. Thus, we need the comprehensive simulation model to calculate AC losses. In the 2G HTS cables main AC losses are a hysteresis loss and possible only numerical simulation is possible to predict them in full size 2G cables. We developed the FEM model to solve the electromagnetic problem for calculation AC losses. The model takes into account: strong nonlinear resistivity of the HTS materials, hysteretic behavior of the magnetic substrate, non-uniformity of critical current density across the tapes of cable and possible ferromagnetism of a substrate. The model takes into account any geometrical aspect in the 2G HTS cables also. The details of the model and comparisons of calculations with our measurements of the AC loss are presented.