On the Stability of Mixed Finite-Element Formulations for High-Temperature Superconductors

TL;DR

The paper tackles the stability of two mixed finite element formulations for high-temperature superconductors: the h-a formulation for systems with superconductors and nonlinear ferromagnets, and the t-a formulation for thin superconducting tapes. By casting the discrete problems as perturbed saddle-point problems and applying inf-sup theory, it shows that spurious oscillations arise with naive, equal-order spaces and that stability is recovered by enriching only one of the coupled spaces with hierarchical (bubble) basis functions localized at the coupling interfaces. Detailed 2D analyses on stacked-bar and tape geometries reveal concrete guidance: for the h-a formulation, enrich either the h-space or the a-space (but not both); for the t-a formulation, enrich a on the tape with second-order hierarchy while keeping t at first order. These strategies yield mesh-independent stability across linear and nonlinear material regimes and provide practical guidelines for robust HTS simulations, with potential extension to 3D in future work.

Abstract

In this work, we present and analyze the numerical stability of two coupled finite element formulations. The first one is the h-a-formulation and is well suited for modeling systems with superconductors and ferromagnetic materials. The second one, the so-called t-a-formulation with thin-shell approximation, applies for systems with thin superconducting domains. Both formulations involve two coupled unknown fields and are mixed on the coupling interfaces. Function spaces in mixed formulations must satisfy compatibility conditions to ensure stability of the problem and reliability of the numerical solution. We propose stable choices of function spaces using hierarchical basis functions and demonstrate the effectiveness of the approach on simple 2D examples.

Figures (10)

• Figure 1: Stack of a superconducting bar (below, $n=20$, $j_{\text{c}}=3\times 10^8$ A/m$^2$) and a ferromagnetic bar (above, $\mu_\text{r} = 1000$), subjected to an external field ($b_{\text{ext}} = 0.4$ T). The thick curve is $\Gamma_\text{m}$.
• Figure 2: Conventions for the $t$-$a$-formulation derivation. (a) 2D case, a tape with current density perpendicular to the modeled plane. (b) 3D case with a tape loop, e.g., a racetrack coil. In 3D, the effect of an external voltage/current source is modeled on an arbitrary cross-section.
• Figure 3: Details of two solutions for the stacked bar problem, magnetic flux density near the material interface (arrows represent the average value in each element). (a) Unstable choice of function spaces, resulting in non-physical oscillations on $\Gamma_\text{m}^\delta$. (b) Example of a stabilized problem with hierarchical basis functions on $\Gamma_\text{m}^\delta$ for $\boldsymbol{h}^{\delta}$.
• Figure 4: Normal magnetic flux density distribution (horizontal position in abscissa) just above and just below the material interface for the stacked bar problem. (a) Unstable choice of function spaces, large spurious oscillations take place. (b) Stabilized solution, with higher order basis functions on $\Gamma_\text{m}^\delta$ for $\boldsymbol{h}^{\delta}$.
• Figure 5: Simple problem for the $t$-$a$-formulation: a superconducting tape in air, with an imposed total current intensity. (a) The problem geometry and domains. (b) Magnetic flux density in the neighbourhood of the tape, solution with first-order basis functions. Oscillations are not visible when looking at $\boldsymbol{b}^{\delta}$ only.