Finite-Element Formulations for Systems With High-Temperature Superconductors

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.

A Coupled A–H Formulation for Magneto-Thermal Transients in High-Temperature Superconducting Magnets

“…For this particular application range, the domain decomposition brings advantages with respect to the canonical methods by avoiding issues related to the numerical instability of the A-ϕ formulation due to the vanishing resistivity of the superconductor, the increased computational cost of the H formulation due to the introduction of unnecessary degrees of freedom [11], and the determination of the cohomology basis for the T-Ω formulation [11]. The validity of these choices are confirmed by a recent paper [12] dedicated to the comparison of formulations for modeling superconducting materials. Furthermore, a slab approximation is introduced for the source regions consisting of superconducting tapes with high aspect ratio, ensuring computational efficiency in solving the current sharing regime.…”

Hyperbolic Basis Functions for Time-Transient Analysis of Eddy Currents in Conductive and Magnetic Thin Sheets

“…The volume integrals terms in this expression have opposite signs than those presented in (28), since they are on the right side of (29). In fact, here we should consider the outward unit normal vector of the boundary of Ω s , i.e.…”

“…Inside the sheet, we assume that the local magnetic field is written as h x (x, y, z, t) = h x (x, z, t)ζ(y) and the test function as h x (x, y, z) = h x (x, z)ζ (y), with h x (x, z, t) and h x (x, z, t) tangential to Γ s , and ζ(y) and ζ (y) differentiable in the interval −d/2 ≤ y ≤ d/2. The volume integrals terms in (29) are then reduced to surface integrals terms as follows…”

“…where µ r0 is the relative permeability at origin and m 0 the saturation magnetic field in A/m. The differential permeability required for the application of the NR-scheme was defined as in [29]. We carried out simulations with µ 0 m 0 = 1.31 and µ r0 = 12500.…”

Finite-element modelling of no-insulation HTS coils using rotated anisotropic resistivity

“…Unlike the traditional approach, the total current in the coil is not explicitly constrained, rather the coil is connected to a current source, see figure 2 , which effectively applies a constraint only on the net radial current I r = 2πr current I op is transfered to the coil through a normal-conducting bus simulated as copper with a resistivity ρ cu = 1.67 × 10 −8 Ωm. This model has been implemented in COMSOL multiphysics 5.4 using the Hformulation [40,41], chosen for its stability when solving problems with power-law resistivities [42]. However, in principle the approach described above is independent of the numerical system used to solve it.…”