Simultaneous Multi-Scale Homogeneous H-Phi Thin-Shell Model for Efficient Simulations of Stacked HTS Coils

TL;DR

This work tackles the computational challenge of simulating large HTS magnets with multi-scale geometry by extending the simultaneous multi-scale homogeneous (SMSH) framework to an $h$-$\phi$ thin-shell (TS) formulation. By collapsing analyzed tapes into surfaces and expressing the external field as $-\nabla \phi$, the method reduces mesh complexity while preserving accuracy, enabling monolithic solving of macro- and mesoscopic scales. Benchmark results on a 2-D stack of HTS tapes demonstrate that the $h$-$\phi$ SMSH-TS model matches detailed $h$-$\phi$ and $t$-$a$ formulations for AC losses, turn voltages, and local current density, with about an order of magnitude speedup and improved robustness over fully homogenized models. The approach offers a scalable, open-source tool for efficient HTS coil simulations, with promising extensions to 3-D and no-insulation coil configurations.

Abstract

The simulation of large-scale high-temperature superconducting (HTS) magnets is a computational challenge due to the multiple spatial scales involved, from the magnet to the detailed turn-to-turn geometry. To reduce the computational cost associated with finite-element (FE) simulations of insulated HTS coils, the simultaneous multi-scale homogeneous (SMSH) method can be considered. It combines a macroscopic-scale homogenized magnet model with multiple single-tape models and solves both scales monolithically. In this work, the SMSH method is reformulated using the $h$-$φ$ thin-shell (TS) approximation, where analyzed tapes are collapsed into thin surfaces, simplifying mesh generation. Moreover, the magnetic field is expressed as the gradient of the magnetic scalar potential outside the analyzed tapes. The discretized field is then described with nodal functions, further reducing the size of the FE problem compared to standard $h$ formulations. The proposed $h$-$φ$ SMSH-TS method is verified against state-of-the-art homogenization methods on a 2-D benchmark problem of stacks of HTS tapes. The results show good agreement in terms of AC losses, turn voltage and local current density, with a significant reduction in simulation time compared to reference models. All models are open-source.

Figures (8)

• Figure 1: Detailed stack of $N_{\text{c}}$ tapes (left), with its equivalent in the SMSH model, which replaces non-analyzed tapes with homogenized bulks (right).
• Figure 2: Internal representation of the $N_{\text{v}}$ virtual elements inside one thin-shell $\Gamma_{\text{c},i}$, each of virtual height $\Delta y$. Adapted from deSousaAlves2021.
• Figure 3: Principle and notations of the SMSH-TS model, replacing non-analyzed tapes with homogenized bulks, in which the current density $\bm{j}_{\text{s}}$ is interpolated from neighbouring analyzed tapes (top). Focus on the unit cell around single analyzed tapes, represented with the $h$-$\phi$ TS model (bottom).
• Figure 4: Conceptual sketch of the cohomology basis functions (or cuts) involved in the discretizations of the reaction \ref{['eq:hphi-discrete']} and source magnetic fields \ref{['eq:hs-discrete']}.
• Figure 5: Three-dimensional model of the racetrack coil to be studied (top), together with the simplified quarter cross-section modeled for verification (bottom). Adapted from Berrospe-Juarez2021.