On the vector potential formulation with an energy-based hysteresis model and its numerical solution

Felix Engertsberger; Herbert Egger

REVIEW 2 major objections 6 minor 31 references

On the vector potential formulation with an energy-based hysteresis model and its numerical solution

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that a magneto-quasistatic field problem coupled to an energy-based magnetic hysteresis model is equivalent to one strongly convex minimization problem, and supplies two globally convergent iterative solvers for it.

desk verdict A clean variational formulation for vector-potential hysteresis that deserves referee time, but the Newton convergence claim is delegated to unpublished preprints and needs either proof or softening. read the letter →

arxiv 2507.14521 v1 pith:2XTS57GK submitted 2025-07-19 math.NA cs.NA

Herbert Egger , Felix Engertsberger This is my paper

classification math.NAcs.NA MSC 65N3065K1035Q6178M10

keywords ferromagneticmaterialsvectorpotentialformulationfiniteelementanalysisminimizationalgorithmsmagnetichysteresisconvexvariationalproblemNewtonmethodblock-coordinatedescent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that a magneto-quasistatic field problem coupled to an energy-based magnetic hysteresis model can be reformulated as a single convex minimization problem over the magnetic vector potential and the magnetic polarization. This makes it possible to prove existence and uniqueness of solutions by standard convex analysis, and to justify finite element discretization. The authors then propose two iterative solvers, a regularized Newton method and a block-coordinate descent method, and claim both converge globally at a linear rate. A numerical simulation of a transformer T-joint confirms mesh-independent iteration counts and shows the Newton method is substantially faster. A careful reader would care because most prior vector-potential hysteresis solvers lacked a rigorous convergence guarantee.

What carries the argument

The load-bearing object is the convex functional (9): $\int_\Omega \frac{\nu_0}{2}|\mathrm{curl}\,A-J|^2 - \langle H_s,\mathrm{curl}\,A\rangle + U(J) + \chi |J-J_p|_\varepsilon\,dx$, minimized over $A\in V\subset H_0(\mathrm{curl})$ and $J\in L^2(\Omega)^3$. Strong convexity in $(A,J)$ gives a unique minimizer and makes the first-order optimality conditions necessary and sufficient; those conditions are exactly the field equations (10)-(11) (or the inclusion (12) when $\varepsilon=0$). The regularization $|x|_\varepsilon$ of the pinning term makes the problem twice differentiable for $\varepsilon>0$, allowing the Newton method, while the block method minimizes alternately in $A$ and $J$. The splitting $H_0(\mathrm{curl})=\nabla H^1_0 \oplus V$ encodes gauging and boundary conditions.

What would settle it

Run the block-coordinate descent and the Newton method on the two-dimensional T-joint benchmark with a deliberately poor initial guess (e.g. $A_0=0$, $J_0$ far outside the polarization range), and check whether the iterates' residuals in the optimality system (10)-(11) decrease to zero at the claimed linear rate; if either method stalls or converges to a point that does not satisfy the first-order condition, the global convergence assertions would be contradicted.

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Extended reading notes

Core claim

On its own terms, the central discovery is Assertion 6: the unique minimizer $(A,J)$ of the strongly convex functional (9), with $J$ a polarization and $B=\mathrm{curl}\,A$, yields fields $B$ and $H=\nu_0(B-J)$ that satisfy the field equations (1)-(2) and reproduce the hysteresis minimization (3) for $J$. The equivalence is obtained from the first-order optimality conditions, and it extends to the non-smooth case $\varepsilon=0$ through subdifferential inclusions. Consequently the whole coupled problem is a variational problem with a unique solution, and its discretization by Nédélec and piecewise-constant finite elements inherits that well-posedness. The paper further asserts (Assertions 9 and 11) that both a damped Newton method and a block-coordinate descent converge globally, with linear rate, to that minimizer from any starting point.

Load-bearing premise

The proof that the regularized Newton method converges globally from any starting point relies on two external preprints by the same authors; if those results do not apply to the infinite-dimensional function spaces and nonsmooth pinning terms used here, the paper itself does not yet contain a complete proof of that convergence claim.

Editorial extensions

If this is right

Finite element discretization of (9) inherits well-posedness; any conforming discretization yields a unique discrete solution.
The Newton method's per-iteration cost is essentially that of a linear magnetostatic problem, since the polarization increment can be eliminated locally (Remark 10).
Since the functional already contains all information of the field equations, any a posteriori error estimate for the functional directly controls the error in the fields.
The equivalence gives a constructive route to hysteresis loss computation: the pinning term $\chi |J-J_p|_\varepsilon$ integrated over the load step yields the loss density.
The convergence theory likely extends to multiple pinning forces and to the higher-order and curved-element discretizations mentioned in the paper, because the variational structure is preserved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same variational characterization might be used to couple the energy-based hysteresis model to other dissipative effects, e.g. eddy currents or mechanical deformation, by adding further convex terms to the functional; the paper does not explore this.
The block-coordinate descent method here is a slight variant of the original iterative scheme (6)-(7); if the convexity arguments are robust, they may be adaptable to prove convergence of the original scheme under an additional convexity or coercivity condition, which the paper leaves open.
The equivalence suggests a natural multigrid or domain-decomposition solver based on the functional, since the block updates are local in $J$ and elliptic in $A$.
The paper's regularization error $\sqrt{\varepsilon}$ (Remark 1) provides a ready-made tolerance: one can choose $\varepsilon$ so that the error of the smooth model is below the discretization error, though the paper does not give such an adaptive strategy.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the core idea is a simultaneous minimization over A and J for the Henrotte–François-Lavet hysteresis model, and the equivalence proof (Assertion 6) is clean and genuinely useful. The paper is worth taking seriously. But the advertised global convergence result for the regularized Newton method is not proved here; it is outsourced to two self-cited preprints, and the stress-test note is right that the singular energy makes the transfer non-automatic. That is the main soft spot, and it is addressable.

What is actually new: the variational characterization (9) with the saturating energy and the regularized pinning term. The first-order conditions reproduce the field equations and the hysteresis minimization, so existence, uniqueness, and a finite element discretization follow by convexity. That is a real contribution. The block-coordinate descent interpretation (Assertion 11) is also sensible, and the numerical results show mesh-independent iteration counts, which is what you expect from a strongly convex problem.

The soft spots, in order of size. First and biggest: Assertion 9. The proof is one sentence delegating to a textbook (for the discrete case) and to preprints [28],[29]. The stress-test note identifies the real issue: the Hessian contains terms like tan(pi/2 |J|/Js), which blows up as |J| approaches Js, so the Hessian is not globally Lipschitz. Standard global Newton–Armijo theorems require bounded sublevel sets and Lipschitz Hessians, and the preprints are about nonlinear magnetostatics without this singular boundary. It is plausible the argument can be adapted, but that has not been shown here. The claim of two globally convergent methods is therefore overstated for the Newton method.

Second: the 2D benchmark uses the 5-pinning-force extension, but the theory section only covers a single internal variable. The paper says everything generalizes immediately, and it probably does, but there is no proof. Minor.

Third: the numerical section has no comparison with existing formulations, e.g., the inverse-operator approach of Jacques et al. or scalar-potential methods. The tables show iteration counts and losses, not accuracy against a reference. Acceptable for an illustration, but it limits what we can conclude about efficiency.

The citation pattern is fine: self-cites are used where prior work is needed, and the core derivation is self-contained. The central equivalence proof does not depend on the preprints, so the main formulation stands regardless.

Bottom line: this deserves a serious referee. The referee should push the authors to either prove the Newton convergence for this specific energy or downgrade the claim to local convergence or empirical observation. With that fixed, it is a solid methods paper for computational electromagnetics.

Referee Report

2 major / 6 minor

Summary. The paper proposes a variational reformulation of magnetoquasistatic field equations coupled to the energy-based vector hysteresis model of Henrotte et al. For a single load step, the coupled system (1)-(3) is shown to be equivalent to the minimization of a strongly convex functional (9) over the vector potential A and the magnetic polarization J. The authors claim well-posedness of the continuous and discrete problems, present a regularized Newton method and a block-coordinate descent method, and illustrate the approach with numerical experiments on the T-joint transformer benchmark.

Significance. The central variational equivalence is a genuinely useful contribution: if it holds, it provides a clean formulation that avoids the numerical inversion of the hysteresis operator needed in earlier scalar-potential or vector-potential approaches. The first-order optimality argument in Assertion 6 is self-contained and non-circular, and the discrete well-posedness in Assertion 7 is immediate. The two proposed solvers, if fully justified, would give practitioners two globally convergent options. However, the manuscript's support for the Newton method is only a reference to the authors' preprints, and the same is largely true for the block-coordinate descent method. Thus the practical significance of the paper depends on completing or making explicit those convergence proofs.

major comments (2)

[Section III-B, Assertion 9] The proof of Assertion 9 is a two-sentence reference to [27] and the preprints [28], [29], yet this assertion carries one of the two advertised global-convergence claims. The objective in (9) contains U(J) = -2 As Js / pi log(cos(pi/2 |J|/Js)), whose gradient is unbounded as |J| tends to Js; the energy is finite only for |J| < Js almost everywhere. Consequently the Newton operator (19) contains U''(J), which is not a bounded multiplication operator on L2 unless the iterates are pointwise bounded away from saturation. Standard damped-Newton global-convergence theorems require assumptions such as Lipschitz continuity of the Hessian on the relevant sublevel set and control of the Newton decrement; the manuscript does not verify these for (9), and it is not automatic that the magnetostatics results in [28] or [29] cover the singular boundary and the pinning term chi |J - Jp|_eps. Please provide a self-contained proof of the claimed global linear convergence, or state the theorem from [28]/[29] and verify its hypotheses step by step.
[Section III-C, Assertion 11] The proof of Assertion 11 likewise consists of a reference to [30] with the sentence 'The results follow immediately'. The abstract's second global-convergence claim therefore also rests on an external result whose hypotheses are not stated. In particular, [30] concerns a non-smooth convex minimization problem in plasticity, not the present coupled field problem; the linear-rate conclusion for the infinite-dimensional Hilbert space setting of (23) is not immediate. Please give the precise convergence theorem from [30] and check its assumptions, or prove the assertion directly.

minor comments (6)

[Assumption 2] The displayed formula for U(J) omits the absolute value; as written it defines a scalar function of a vector argument. It should read U(J) = -2 As Js / pi log(cos(pi/2 |J| / Js)).
[Remark 10 and equation (19)] The denominator in the pinning-term derivative is missing from the displayed formula: it should read chi / |J - Jp|_eps (I - v otimes v), not chi |J - Jp|_eps (I - v otimes v).
[Section IV-D] The phrase 'mesh-independent with constant average iteration counts' is presented as an expected and confirmed property, but Assertions 9 and 11 only state linear convergence and no mesh-independent rate is proved. Please either state a rate theorem or qualify this conclusion as an empirical observation.
[Introduction, second paragraph] The name 'Berqvist' should be 'Bergqvist'.
[Section IV] The numerical section uses a five-cell model with multiple pinning forces, whereas the analysis in Sections II and III is stated for a single internal variable. Please state explicitly that the uniqueness and convergence assertions extend to the multi-cell case, for example by noting that the functional (26) is again strongly convex and satisfies the same structural assumptions.
[References] Reference [29] is cited only as an arXiv preprint. Since the proof of Assertion 9 relies on it, please provide a full statement of the cited result or update the reference to a published version if one exists.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the variational equivalence is self-contained, and the only self-citation dependency (Newton convergence via authors' preprints) is a proof-gap rather than an input-output reduction.

full rationale

The core derivation chain is not circular. Assertions 3-6 derive the minimizer of the strongly convex problem (9) via first-order optimality conditions, and the resulting equations (10)-(12) are algebraically the field equations together with the stationarity condition of the hysteresis minimization (3); this is a genuine variational characterization, not a fitted-input/prediction reversal, and no constants are fitted to the reported iteration counts or loss values. The block-coordinate descent convergence (Assertion 11) is delegated to the external reference [30]. The only self-citation-adjacent element is Assertion 9 (Section III-B): the proof states 'The discrete setting is fully covered by the results in [27]. The arguments presented in [28], [29] allow to treat also the infinite dimensional problem in a rigorous manner.' Since [29] is the authors' own preprint and the assumptions of its convergence theorem are not verified for the saturating energy U(J) with gradient blowing up at |J|=Js, the infinite-dimensional global convergence claim is unsupported inside this manuscript. That is a completeness/correctness risk, not a circular reduction: the paper never defines Assertion 9 into the hypotheses of [28]/[29], so there is no exhibited equivalence-by-construction. Hence no circular step is identified; score 1 reflects the minor self-citation dependency without treating it as circular.

Assumptions & free parameters 3 free parameters · 9 assumptions · 0 invented entities

The central variational equivalence is derived from first-order optimality conditions and standard convex analysis; no new physical entities or fitted parameters are introduced by the authors. The material parameters (As, Js,k, chi_k) are taken from [9] and are not fitted in this paper. The main external dependencies are the convergence analyses of the Newton method in infinite dimensions ([28], [29], both by the same research group) and the energy-based hysteresis model itself ([7], [8], [9]).

free parameters (3)

regularization parameter epsilon = not specified (epsilon > 0)
Introduced to regularize the norm |J - Jp| and make the Newton method twice differentiable; its numerical value is not reported, and it affects the resulting hysteresis curves and iron losses.
time step tau for load cycle = 0.02 (200 steps over [0,2])
Discretization parameter for the load cycle simulation; chosen by hand, affects the reported iteration counts and iron losses.
Armijo line-search parameters q, sigma = q = 0.5, sigma = 0.1
Algorithmic choices for the backtracking line search in the Newton method; standard values, not fitted to data.

assumptions (9)

standard math The functional in (9) is strongly convex, coercive, and continuous; standard convex analysis guarantees a unique minimizer.
Used in Assertions 3 and 7; standard Ekeland-Temam theory.
standard math Minimizers of the convex functional are characterized by first-order optimality conditions with subdifferentials.
Used in Assertion 6 and Remark 5 to prove equivalence with (1)-(3).
domain assumption Omega is a bounded Lipschitz domain, topologically trivial, with a stable direct splitting H0(curl) = grad H1_0 + V.
Assumption 2; needed for the norm equivalence on V and the gauging of the vector potential.
domain assumption The source current satisfies js = curl Hs for some Hs in H(curl; Omega).
Assumption 2; compatibility condition for the variational formulation.
domain assumption The energy density U(J) has the specific log-cos form with parameters As, Js.
Assumption 2; taken from the energy-based hysteresis model of [9], not derived in this paper.
domain assumption The internal polarization is governed by the minimization problem (3) with pinning term chi |J - Jp|_epsilon.
Model definition from [9]; the paper analyzes this model and does not validate it experimentally.
standard math |x| <= |x|_epsilon <= |x| + sqrt(epsilon), so the regularization error is controlled.
Remark 1; used to justify replacing |x| by the regularized norm.
domain assumption The infinite-dimensional damped Newton convergence analysis of Heid [28] and Egger-Engertsberger-Radu [29] applies to the present problem setting.
Proof of Assertion 9 refers to these preprints for the infinite-dimensional treatment; this is a load-bearing external dependency.
standard math Alternating minimization converges for this strongly convex functional (Carstensen [30]).
Proof of Assertion 11 for the block-coordinate descent method.

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Pith. "Pith review of On the vector potential formulation with an energy-based hysteresis model and its numerical solution." pith.science (2026). https://pith.science/paper/2XTS57GK

@misc{pith2026250714521,
  author       = {Pith},
  title        = {Pith review of: On the vector potential formulation with an energy-based hysteresis model and its numerical solution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XTS57GK}},
  note         = {Machine review of arXiv:2507.14521}
}

read the original abstract

The accurate modelling and simulation of electric devices involving ferromagnetic materials requires the appropriate consideration of magnetic hysteresis. We discuss the systematic incorporation of the energy-based vector hysteresis model of Henrotte et al. into vector potential formulations for the governing magnetic field equations. The field model describing a single step in a load cycle is phrased as a convex minimization problem which allows us to establish existence and uniqueness of solutions and to obtain accurate approximations by finite element discretization. Consistency of the model with the governing field equations is deduced from the first order optimality conditions. In addition, two globally convergent iterative methods are presented for the solution of the underlying minimization problems. The efficiency of the approach is illustrated by numerical tests for a typical benchmark problem.

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**Figure 2.** Locus curves of the of the magnetic induction [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [28]

P . Heid. A short note on an adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations. Arch. Math. , 121:55–65, 2023

work page 2023

[2] [29]

Global convergence of iterative solvers for problems of nonlinear magnetostatics

H. Egger, F. Engertsberger, and B. Radu. Global converg ence of iterative solvers for problems of nonlinear magnetostatics. arXiv:2403.18520

work page Pith review arXiv

[3] [27]

Nocedal and S

J. Nocedal and S. Wright. Numerical Optimization . Springer, 2006

2006

[4] [30]

Carstensen

C. Carstensen. Domain decomposition for a non-smooth c onvex mini- mization problem and its application to plasticity. Numer . Lin. Algebra Appl., 4:177–190, 1997

work page 1997

[5] [1]

G. Meunier. The Finite Element Method for Electromagnetic Modeling . Wiley, 2008

2008

[6] [2]

S. J. Salon. Finite Element Analysis of Electrical Machines . Springer, New Y ork, NY , 2012

work page 2012

[7] [3]

D. C. Jiles and D. L. Atherton. Theory of ferromagnetic hy steresis. J. Magnetism. Magn. Materials , 61:48–60, 1986

1986

[8] [4]

I. D. Mayergoyz. Mathematical Models of Hysteresis . Springer-V erlag New Y ork Inc., 1991

work page 1991