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The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.

In recent years, several Nb3Sn high field magnet prototypes have been designed and tested in preparation for the LHC Luminosity upgrade and also for the potential Future Circular Collider (FCC). In this paper we present a Microsoft Excel-based database tool for storing magnet design parameters and results from quench protection tests. The hierarchical and flexible structure of the relational database allows for systematic and coherent analysis of the test data from different magnet assemblies and works as a practical reference for magnet design evolution. Data from quench protection heater tests in several high-field Nb3Sn magnet prototypes has been stored in the database. We use this data to validate the quench simulation assumptions used in FCC 16 T dipole magnet design. acceptedVersion

The purpose of this paper is to estimate the magnetic field dependent critical current density scaling relation of NbTi used in a superconducting wire. The estimation problem is set using a finite element method based forward model solving the critical current of a wire in external magnetic fields. A sensitivity analysis is carried out to reveal the uncertainties in the process. It is shown that the methods provide accurate predictions of critical currents in a wire, but there are uncertainties related to the zero-field critical current density of the material. acceptedVersion

The Future Circular Collider (FCC-hh) project is a conceptual study whose goal is to design the successor of the Large Hadron Collider, increasing the collision energy from 14 to 100 TeV. The energy stored in the 16-T superconducting dipole magnets and the length of the sectors composing the 100-km FCC tunnel are considerably larger than those in present accelerators. This means that the energy stored in the FCC-hh dipole circuit is likely to be much higher than that in existing superconducting circuits. In the case of magnet quenches or faults, the circuit needs to be protected, i.e., its energy needs to be rapidly dissipated without inducing excessive voltages in the magnet chain. This article proposes a conceptual design for the FCC-hh dipole circuit, which satisfies the constraint of the maximum allowable voltage-to-ground and fulfills additional requirements related to the FCC-hh operation and tunnel layout. A compromise among the considered requirements leads to a relatively simple circuit layout and a large number of circuits for the entire machine. The behavior of the proposed circuit during the critical fast power abort phase is simulated through a numerical model, which covers the electrical circuit domain and the electrothermal magnet domain. Each FCC-hh dipole magnet is protected by means of the coupling-loss-induced quench (CLIQ) protection system, which also acts at the circuit level. The simulations predict severe voltage oscillations in the FCC-hh dipole circuits that may pose a problem for the quench detection system. The simulations also show that the severity of the oscillations is not due to the presence of CLIQ. This protection system can be integrated into the proposed circuit layout and represents an effective protection system for the entire string of FCC-hh dipole magnets. publishedVersion

The use of practical high temperature superconductors (HTS), REBCO tapes especially, in magnet applications has become possible thanks to the increasing interest of manufacturers. One difficulty has been the nonlinear material properties that are challenging to measure and model. To advance in such, demo systems are needed and they must be thoroughly analyzed. Recently, one of the first HTS dipole magnets was built to study the usability of REBCO Roebel cables in particle accelerator magnets. The prototype magnet Feather-M2 was designed, constructed and tested within EUCARD2 collaboration project at CERN in 2017. In the measurements, the magnet behaved in an unexpected way: the magnet was able to be operated at operation currents above the maximum current that was predicted based on short-sample measurements. Additionally, unexpectedly gradual dependency between magnet's resistive voltage and operation current was observed. In this work, a thermodynamical model is formulated in order to study the behavior of Feather-M2. The model was parametrized and the parameters were solved via inverse problem by finding the best match to experimental results. Thereby insight was gained on the prospects of the utilized thermodynamical model and also on the behavior and operation conditions of the magnet via the inverse problem solutions. To summarize, this paper presents a new methodology for analyzing magnets in operation and applies it to a state-of-the-art magnet. acceptedVersion

The short-sample critical current is only an indicative property for the maximum current a magnet can be continuously operated with. This was especially visible in the experiments of one of the world's first Roebel-cable-based high temperature superconductors dipole magnet prototype built and tested at CERN in 2017 where the thermal runaway developed very slowly in many cases. Consequently, the maximum stable operation current could be overstepped and stable operation could be recovered by lowering the current below the maximum of the stable range again. It is non-trivial to quantitatively predict this behavior from the critical current measurements which are observed under specific cooling conditions and based on an arbitrarily selected electric field criterion for the critical current. To make more rigorous predictions on the maximum stable operation current, one needs to consider in detail the interplay of cooling over the magnet surface and heat generation in the winding. This paper presents a methodology to determine the maximum stable operation current for a given magnet, as well as studies its mathematical background. Insight to this problem comes from the Roebel-cable-based dipole magnet studied at CERN during 2017. acceptedVersion

International audience; Part of the Future Circular Collider (FCC-hh) study is dedicated to the development of the 16 Tesla ${\rm Nb_3Sn}$superconducting dipole magnets. The design of the magnets was enabled by a cooperative effort of national research institutes, universities, and CERN. These actors tackled the problem from different sides, namely, the electromagnetic design, the mechanical design, the design of the quench protection systems, and the circuit design. The article deals with the design of the quench protection systems and provides solid motivations for the selection of the coupling-loss-induced quench (CLIQ) device as the baseline protection system for the FCC-hh main dipole magnets. The article shows that the design domains mentioned above are tightly interconnected and, therefore, the simulation of a quench event involves a complex multiphysics problem. The STEAM cosimulation framework, recently developed at CERN, is applied to address the complexity. The STEAM-SIGMA models are employed to simulate the CLIQ quench protection system applied to the FCC-hh dipole magnets. Dedicated CLIQ configurations are identified to protect the magnets in case of a quench. In addition, the possible implications of the CLIQ protection system on the mechanical design of the magnets are discussed. To this end, the article employs the co-simulation of different software platforms to calculate the mechanical stress during a quench. The results show that CLIQ does not produce additional stress.

Three different 16 T dipole magnet options for the Future Circular Collider (FCC) have been designed within the H2020 EuroCirCol collaboration, namely a cosθ-, a block-, and a common-coil -type of magnet. All magnets were designed using the same constraints related to magnetic field, cable parameters, mechanics, and quench protection. The quench protection analysis during the magnet design was centralized and done for all the options using the same tools and methods. This paper summarizes the final conceptual protection schemes. They are based either on the novel CLIQ-technology (Coupling Loss Induced Quench) or on traditional quench heaters. We compare the performance of the protection systems and show that while both can protect the magnet at nominal operation current, CLIQ is more efficient in reducing peak temperatures than heaters, and the system operation is simpler due to smaller amount of protection units. Therefore, CLIQ has been chosen as the baseline protection option for the FCC dipole magnets. The future development of heaters is considered as a back-up option. acceptedVersion

A quench protection concept based on coupled secondary coils is studied for inductively transferring energy out of a quenching superconducting dipole and thus limiting the peak hotspot temperature. So-called 'quench absorption coils' are placed in close proximity to the superconducting coils and are connected in series with a diode for the purpose of preventing current transformation during regular operation. During a quench, current is then transformed into the quench absorption coils so that a significant fraction of the stored magnetic energy is dissipated in the these coils. Numerical calculations are performed to determine the impact of such a concept and to evaluate the dimensions of the quench absorption coils needed to obtain significant benefits. A previously constructed 15 T Nb3Sn block coil is taken as a reference layout. Finite-element calculations are used to determine the combined inductive and thermal response of this system and these calculations are validated with a numerical model using an adiabatic approximation. The calculation results indicate that during a quench the presence of the quench absorption coils reduces the energy dissipated in the superconducting coils by 45% and reduces the hotspot temperature by over 100 K. In addition, the peak resistive voltage over the superconducting coils is significantly reduced. This suggests that this concept may prove useful for magnet designs in which the hotspot temperature is a design driver. acceptedVersion

Future accelerator magnets are pushed to their limits in terms of magnetic field, mechanical strength and from the quench protection point of view. This forces the magnet designers to re-think the quench modelling. One issue that has not so far been largely explored is the mechanical behaviour of the superconducting coils during a quench. This can cause limitations to the design of high field accelerator magnets. This paper focuses on mechanical behavior in the event of a quench of a Nb3Sn 16 T dipole magnet currently developed in the framework of the EuroCirCol project in view of the Future Circular Collider conceptual design study. The thermo-mechanical analysis is performed through finite element modeling. The analysis takes into account the Lorentz force and the thermal stress due to the non-uniform temperature distribution in the winding during a quench. acceptedVersion