FiQuS Pancake3D

Sina Atalay

2023 – 2024

FiQuS Pancake3D, my technical studentship project at CERN under the supervision of Erik Schnaubelt, is an open-source 3D finite-element quench simulator for no-insulation high-temperature superconducting (HTS) pancake coils.

High-field magnets push the limits of technologies such as fusion reactors and particle accelerators: stronger fields enable smaller, cheaper, and more powerful machines. These machines use electromagnets, where electric current creates a magnetic field and more current makes the field stronger. Ordinary conductors would melt from resistive heating at the currents needed, so high-field coils use superconductors. The catch is that every superconductor stops superconducting above a certain critical magnetic field. For low-temperature superconductors (LTS), that limit is reached fairly quickly, which caps how strong their magnets can be. HTS materials keep working up to much higher fields, making stronger magnets possible, but reliable HTS magnets are still an active research problem.

The main obstacle is quench: a small part of the coil unpredictably becomes resistive. A small local disturbance can trigger it, but in practice you usually cannot determine why, when, or where it will begin. It therefore appears random. A quench can damage or destroy the magnet. Reliable quench protection for HTS magnets remains an unsolved engineering problem.

There is a promising approach to this problem: the no-insulation (NI) coil, which changes how the HTS tape is wound. An HTS coil consists of many touching turns that are normally electrically insulated from one another. In an NI coil, this insulation is left out, so neighboring turns remain in electrical contact. This gives current a path around a quenched spot, which can protect the coil from quench. But NI coils are still an active area of research.

Studying quench experimentally is difficult. Tests are expensive, a quench can destroy the coil, and the currents, magnetic fields, and temperatures are buried inside the tightly wound coil, where sensors can reach only a few locations. Simulation is therefore one of the main ways to investigate the problem: it exposes these quantities throughout the coil and allows the underlying physics to be analyzed. Pancake3D was built to do this for NI coils.

To simulate a quench, we must solve the physical equations governing the coil. Maxwell’s equations govern current and magnetic field, while the heat equation governs temperature. Together they form a coupled magneto-thermal problem.

Pancake3D solves these equations with the finite element method. For the electromagnetic problem, Pancake3D uses the $\vec{H}-\phi$ formulation. From an initial solution at $t=0$ , it finds the magnetic field strength $\vec{H} \in H_{\phi,I}(\operatorname{curl},\Omega)$ and temperature $T \in H^1_g(\Omega_c)$ for $t>0$ such that

\begin{aligned} &\int_\Omega \partial_t(\mu \vec{H}) \cdot \vec{H}'\,d\Omega \\ &\quad + \int_{\Omega_c} \rho\,(\nabla \times \vec{H}) \cdot (\nabla \times \vec{H}')\,d\Omega = 0, \\ &\forall \vec{H}' \in H_{\phi,0}(\operatorname{curl},\Omega) \end{aligned}

\begin{aligned} &\int_{\Omega_c} C_V\,\partial_t T\,T'\,d\Omega + \int_{\Omega_c} \kappa\,\nabla T \cdot \nabla T'\,d\Omega \\ &\quad - \int_{\Gamma_f} fT'\,d\Gamma = \int_{\Omega_c} \rho\,\vec{J} \cdot \vec{J}\,T'\,d\Omega, \\ &\forall T' \in H^1_0(\Omega_c) \end{aligned}

Here, $\vec{H}'$ and $T'$ are test fields, $\vec{J} = \nabla \times \vec{H}$ is current density, $\Omega$ is the full computational domain, and $\Omega_c$ is its conducting part. The material parameters $\mu$ , $\rho$ , $C_V$ , and $\kappa$ are magnetic permeability, electrical resistivity, volumetric heat capacity, and thermal conductivity. The function $f$ represents cooling on the boundary $\Gamma_f$ . In the non-conducting domain, $\vec{H}$ is curl-free and represented using the magnetic scalar potential $\phi$ and cohomology basis functions.

The electromagnetic and thermal equations are coupled in two ways: current heats the coil through Joule losses, $\rho\,\vec{J} \cdot \vec{J}$ , while temperature, magnetic field, and current density change the material properties used in both equations.

Simulating NI HTS pancake coils accurately is very difficult for three reasons:

Turn-to-turn contact is extremely thin. Meshing each resistive contact layer as a 3D volume would require impractically fine or poorly shaped elements. Pancake3D instead collapses it onto a surface using a thin-shell approximation.
HTS tape is a multilayer composite. Meshing its superconductor, copper, silver, and substrate separately is infeasible. An anisotropic homogenization replaces them with one effective material while preserving current sharing.
The system is strongly nonlinear. A superconductor’s electrical resistivity changes by many orders of magnitude during quench. At each time step, Pancake3D uses Picard iterations for the material parameters and Newton–Raphson linearization for the coated conductor’s current-dependent electrical resistivity.

Simulating the coil properly required techniques that were not available off the shelf. In particular, we developed a state-of-the-art thin-shell formulation that captures current and heat flowing between uninsulated turns through their electrical and thermal contact resistances, without meshing the contact layer as a 3D volume. Without it, resolving those contact resistances would require the impractically fine or poorly shaped elements described above.

Pancake3D resulted in three peer-reviewed publications, all open access:

Magneto-Thermal Thin Shell Approximation for 3D Finite Element Analysis of No-Insulation Coils (December 2023) derives the thin-shell approximation of the coupled magneto-thermal weak forms. 10.1109/TASC.2023.3340648
An Open-Source 3D FE Quench Simulation Tool for No-Insulation HTS Pancake Coils (May 2024) presents the implementation, how it works in practice, and simulation results. 10.1088/1361-6668/ad3f83
Influence of Critical Current Defect on Operation, Quench Detection and Protection of a Conduction-Cooled Pancake REBCO Coil (January 2025) studies a coil whose critical current is degraded along the length of the tape, and what that does to quench detection and protection. 10.1109/TASC.2025.3532246

The whole tool is open source.

Run the paper’s simulation: gitlab.cern.ch/steam/analyses/FiQuS_Pancake3D

Example simulation on GitHub: github.com/sinaatalay/fiqus-pancake3d-example

FiQuS source on CERN GitLab: gitlab.cern.ch/steam/fiqus