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arxiv: 2504.11191 · v1 · pith:WCM6KT2Snew · submitted 2025-04-15 · 💻 cs.CE

Magnetic Field Conforming Formulations for Foil Windings

Louis Denis , Elias Paakkunainen , Paavo Rasilo , Sebastian Sch\"ops , Beno\^it Vanderheyden , Christophe Geuzaine This is my paper

Pith reviewed 2026-05-22 20:49 UTC · model grok-4.3

classification 💻 cs.CE
keywords foil windinghomogenizationmagnetic field formulationscalar potentialfinite element methodnumerical modelingeddy currents
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The pith

Magnetic field conforming models for foil windings match detailed conductor simulations while reducing the number of degrees of freedom.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends existing foil-winding homogenization techniques from magnetic-flux-density formulations to magnetic-field (H-conforming) formulations. It first writes a full H-based model by direct analogy, then introduces the magnetic scalar potential in non-conducting regions to shrink the discrete system, especially in three dimensions. Verification on two frequency-domain benchmarks and a transient high-temperature-superconducting coil shows that the reduced models reproduce the results of both B-conforming homogenized models and fully resolved turn-by-turn meshes.

Core claim

The homogenization relations previously derived for B-conforming formulations transfer directly to H-conforming formulations by simple analogy, and the addition of the magnetic scalar potential in non-conducting regions yields reliable results with a considerably smaller numerical problem.

What carries the argument

H-conforming foil-winding homogenization augmented by the magnetic scalar potential in non-conducting regions, obtained by direct analogy with prior B-conforming relations.

If this is right

  • The same reduction in degrees of freedom applies to both frequency-domain and transient simulations of foil-wound devices.
  • Three-dimensional problems become feasible that were previously limited by the size of the discrete system.
  • The method supplies reliable loss and field predictions for high-temperature superconducting coils without explicit discretization of every turn.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analogy approach may allow similar transfers to other vector-potential or mixed formulations used in eddy-current modeling.
  • The reduced models open the possibility of embedding foil-winding subdomains inside larger system-level simulations without prohibitive cost.
  • If the scalar-potential region can be extended to include thin insulating layers, the method could further simplify multi-turn coil geometries.

Load-bearing premise

The homogenization relations derived for magnetic flux density formulations carry over to magnetic field formulations by simple analogy without extra correction terms.

What would settle it

A new benchmark in which the foil winding experiences strong skin effect or highly nonuniform current distribution; if the H-based homogenized solution diverges from the resolved-conductor reference while the B-based version does not, the direct-transfer claim fails.

Figures

Figures reproduced from arXiv: 2504.11191 by Beno\^it Vanderheyden, Christophe Geuzaine, Elias Paakkunainen, Louis Denis, Paavo Rasilo, Sebastian Sch\"ops.

Figure 1
Figure 1. Figure 1: FW homogenization: Nc resolved conductors (left) replaced by a single homogenized bulk of thickness Lα (right). holds ∀a ′ , v′ , V ′ i , with ν the magnetic reluctivity and σ the electrical conductivity. Here, grad v = PNc i=1 Vi grad vs,i and the grad vs,i denote global basis functions [16], or winding functions [17]. Test functions are denoted · ′ , whereas (·, ·)Ω represents the volume integral over Ω … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of entities involved in the discretization of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometry (not to scale) of the 2-D axisymmetric verification [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Voltage per turn in the 20-foil winding computed with the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Foil winding inductor geometry for the 3-D verification [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Voltage per turn in the 30-foil winding computed with the [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Voltage curves obtained with the t-ω FW and h-ϕ FW models with various anisotropy ratios ra in (11), using a third-order global polynomial for Φ(α) and mesh M30. 0.000 0.005 0.010 0.015 0.020 0.025 Time (s) 0 2 4 6 8 10 12 AC losses per unit length (W/m) h-φ resolved j-a-v FW full-h FW h-φ FW 0.012 0.013 0.014 9.5 10.0 10.5 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of AC losses per unit length in the HTS coil [14] [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
read the original abstract

We extend the foil winding homogenization method to magnetic field conforming formulations. We first propose a full magnetic field foil winding formulation by analogy with magnetic flux density conforming formulations. We then introduce the magnetic scalar potential in non-conducting regions to improve the efficiency of the model. This leads to a significant reduction in the number of degrees of freedom, particularly in 3-D applications. The proposed models are verified on two frequency-domain benchmark problems: a 2-D axisymmetric problem and a 3-D problem. They reproduce results obtained with magnetic flux density conforming formulations and with resolved conductor models that explicitly discretize all turns. Moreover, the models are applied in the transient simulation of a high-temperature superconducting coil. In all investigated configurations, the proposed models provide reliable results while considerably reducing the size of the numerical problem to be solved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript extends foil winding homogenization to magnetic field conforming (H-conforming) finite-element formulations. It first derives a full H-field foil-winding model by direct analogy with prior B-conforming work, then replaces the vector potential with the magnetic scalar potential in non-conducting regions. The resulting models are verified on a 2-D axisymmetric frequency-domain benchmark and a 3-D benchmark, shown to reproduce results from both B-conforming formulations and fully resolved conductor models, and are applied to the transient simulation of a high-temperature superconducting coil, with the claim that they remain reliable while substantially reducing the number of degrees of freedom.

Significance. If the homogenization relations transfer without loss of accuracy, the work would provide a practical route to large-scale 3-D modeling of foil windings under H-conforming discretizations, especially useful for devices such as HTS coils. The explicit comparison against independent resolved-conductor models and the demonstration on a transient application case are positive features that support the efficiency claim.

major comments (1)
  1. [formulation derivation (by analogy)] The central derivation obtains the full magnetic-field foil-winding formulation 'by analogy' with B-conforming formulations (abstract and formulation section). Because tangential H and normal B continuity are enforced differently in H-conforming models, the averaged constitutive relations, current-sheet approximations, and power-loss expressions may require additional correction terms; the manuscript does not supply an explicit re-derivation or a priori error estimate for this substitution. This step is load-bearing for the claim that the models are reliable 'in all investigated configurations.'
minor comments (1)
  1. The abstract and verification sections would benefit from explicit statement of the quantitative error metrics (e.g., relative L2 differences in B or losses) used to declare agreement with the reference models.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the formulation derivation. We address this point below and have revised the manuscript to provide additional clarification while maintaining the focus on the practical verification and application of the models.

read point-by-point responses

  1. Referee: The central derivation obtains the full magnetic-field foil-winding formulation 'by analogy' with B-conforming formulations (abstract and formulation section). Because tangential H and normal B continuity are enforced differently in H-conforming models, the averaged constitutive relations, current-sheet approximations, and power-loss expressions may require additional correction terms; the manuscript does not supply an explicit re-derivation or a priori error estimate for this substitution. This step is load-bearing for the claim that the models are reliable 'in all investigated configurations.'

    Authors: We thank the referee for this observation. The H-conforming foil-winding model was initially obtained by direct analogy because the homogenization relies on averaging the fields and currents over the foil thickness, with the effective material properties derived from the same local 1-D problem as in the B-conforming case. The complementary continuity conditions (tangential H and normal B) are satisfied weakly through the choice of the field variable and the finite-element discretization, without requiring additional correction terms in the averaged relations. This is supported by the fact that the numerical results match both the B-conforming homogenized models and the fully resolved conductor models to within discretization error on the 2-D axisymmetric and 3-D benchmarks. In the revised manuscript we have expanded the formulation section with a short paragraph justifying the direct transfer of the relations and noting that the power-loss expressions follow from the same averaged Joule heating. A full a priori error estimate lies beyond the scope of this applied paper, but the extensive cross-verification provides empirical confirmation of reliability in the tested configurations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation verified against independent benchmarks

full rationale

The paper extends foil winding homogenization from B-conforming to H-conforming formulations by proposing a full magnetic field formulation by analogy, then inserting the magnetic scalar potential in non-conducting regions. It verifies the resulting models on two frequency-domain benchmarks against both B-conforming formulations and resolved-conductor models that explicitly discretize all turns, plus a transient HTS coil simulation. No load-bearing step reduces by construction to a fitted parameter, self-defined quantity, or unverified self-citation chain; the central claims rest on explicit numerical agreement with independent reference solutions rather than on renaming or re-deriving inputs as outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Ledger is necessarily sparse because only the abstract is available; the central extension rests on an unexamined analogy between two formulation families.

axioms (1)
  • domain assumption Homogenization of foil windings can be formulated by direct analogy for different field variables.
    The paper states it proposes the full magnetic field foil winding formulation by analogy with magnetic flux density conforming formulations.

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