Numerical approximation of a transient thermo-electromagnetic problem in axisymmetric geometries

B. L\'opez-Rodr\'iguez; D. G\'omez; P. Salgado; P. Venegas

arxiv: 2604.25261 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA

Numerical approximation of a transient thermo-electromagnetic problem in axisymmetric geometries

D. G\'omez , B. L\'opez-Rodr\'iguez , P. Salgado , P. Venegas This is my paper

Pith reviewed 2026-05-07 15:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords transient thermo-electromagnetic problemaxisymmetric geometriesinduction heatingvariational formulationweighted Sobolev spacesfixed-point argumentfinite element discretizationa priori error estimates

0 comments

The pith

A variational formulation in weighted Sobolev spaces proves existence and uniqueness for the transient thermo-electromagnetic induction heating problem and supports a convergent finite element discretization in axisymmetric domains.

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

This paper develops a complete mathematical and numerical treatment for time-dependent problems that couple thermal diffusion with electromagnetic fields through temperature-dependent conductivity and Joule heating. It reduces the three-dimensional setting to a two-dimensional meridional plane by exploiting cylindrical symmetry and azimuthal current density. A variational formulation is posed in appropriately weighted Sobolev spaces, with existence shown by a fixed-point argument and uniqueness obtained under reasonable assumptions on the material coefficients. The continuous problem is then discretized by finite elements in space and implicit time stepping, with a priori error estimates derived and confirmed by numerical tests. The resulting scheme is illustrated on an industrially relevant heating configuration.

Core claim

The authors introduce a variational formulation for the transient thermo-electromagnetic problem in appropriately weighted Sobolev spaces on a two-dimensional meridional section. They prove existence of a solution by a fixed-point argument and uniqueness under reasonable assumptions on the physical parameters. A finite element discretization combined with implicit time stepping is analyzed, with a priori error estimates derived and validated numerically. Numerical simulations demonstrate the method's effectiveness in an industrially relevant configuration.

What carries the argument

The variational formulation posed in weighted Sobolev spaces, shown to be solvable by fixed-point iteration, which reduces the coupled time-dependent problem to a two-dimensional axisymmetric setting while preserving the nonlinear temperature dependence.

If this is right

The finite element solutions converge to the exact weak solution at the rates stated in the a priori error estimates when the mesh size and time step are refined.
The scheme produces reliable temperature and electromagnetic field histories for induction heating processes in which electrical conductivity varies with temperature.
Uniqueness of the solution is guaranteed once the material parameters satisfy the stated reasonable assumptions.
The two-dimensional reduction yields computationally feasible simulations that still capture the essential coupling between heat and electromagnetics.

Where Pith is reading between the lines

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

The same fixed-point structure could be reused to analyze related nonlinear couplings, such as those arising when magnetic permeability also depends on temperature.
The derived error bounds indicate that local mesh refinement near the workpiece surface would be an effective way to control computational cost while maintaining accuracy.
Because the formulation is fully time-dependent, the method supplies the transient data needed for inverse problems that recover unknown conductivity functions from surface temperature measurements.

Load-bearing premise

The assumption of cylindrical symmetry and purely azimuthal current density that permits reduction to a two-dimensional meridional section, together with the conditions on physical parameters required for the fixed-point map to deliver uniqueness.

What would settle it

A computation in which the observed error between successive mesh refinements or time-step reductions fails to decrease at the rate predicted by the a priori estimates would show that the discretization analysis does not hold for the given nonlinear coupling.

Figures

Figures reproduced from arXiv: 2604.25261 by B. L\'opez-Rodr\'iguez, D. G\'omez, P. Salgado, P. Venegas.

**Figure 1.** Figure 1: Induction heating system including the workpiece and the coil (left). Meridional view at source ↗

**Figure 2.** Figure 2: shows log–log plots of the error in T measured in the discrete norm L2 (0, T ; H1 1 (Ω0 )) versus the number of degrees of freedom (d.o.f.) and versus ∆t (left and right, respectively). To study the error with respect to the number of degrees of freedom, we set the time step to a sufficiently small value so that the error depends almost exclusively on the mesh size h. In this case, a linear dependence on h… view at source ↗

**Figure 3.** Figure 3: Steel properties: B-H curve (left) and σ(T) (right) view at source ↗

**Figure 4.** Figure 4: Evolution of the temperature (◦C) in the workpiece. ED431C 2025/09. P. Venegas was partially supported by ANID-Chile through FONDECYT grant 1211030 and Centro de Modelamiento Matem´atico (CMM), grant FB210005, BASAL funds for centers of excellence. B. L´opez-Rodr´ıguez was partially supported by Universidad Nacional de Colombia through Hermes project 63726. References [1] G. Akrivis and S. Larsson. Linearl… view at source ↗

read the original abstract

This paper analyzes a transient thermo-electromagnetic problem arising in the modeling of induction heating processes. Unlike previous studies that focused on steady-state scenarios, we consider a time-dependent thermal problem coupled with a nonlinear time-harmonic electromagnetic problem through temperature-dependent electrical conductivity and Joule effect. Exploiting cylindrical symmetry and assuming a purely azimuthal current density, we formulate the problem on a two-dimensional meridional section. We introduce a variational formulation in appropriately weighted Sobolev spaces and prove existence of a solution by a fixed-point argument. Under reasonable assumptions on the physical parameters, we also prove uniqueness. A finite element discretization combined with implicit time stepping is used to compute the numerical solution. To evaluate the accuracy of the approximation, a priori error estimates are derived and validated by numerical experiments. Finally, numerical simulations illustrate the effectiveness of the proposed approach in an industrially relevant configuration.

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.

Desk Editor's Note private letter to a colleague

This extends steady-state thermo-electromagnetic analysis to the transient nonlinear case with existence/uniqueness proofs and error estimates, but uniqueness rests on untested parameter assumptions.

read the letter

The paper takes the standard numerical-analysis route for a coupled nonlinear system and applies it to the time-dependent thermo-electromagnetic problem that arises in induction heating. Prior work handled only steady state, so the new elements are the transient formulation, the coupling through temperature-dependent conductivity and Joule heating, a variational setting in weighted Sobolev spaces on the meridional section, a fixed-point argument for existence, uniqueness under extra assumptions on the data, FEM plus implicit Euler discretization, a priori error estimates, and numerical checks on an industrial geometry. That combination is a straightforward but useful increment for people who need to simulate these processes reliably. The discretization and error analysis follow the usual template without obvious internal contradictions, and the numerical experiments are presented as validation of the estimates. The axisymmetric reduction with purely azimuthal current is stated up front and fits the target application. The soft spot is the uniqueness claim. It holds only under “reasonable assumptions” on the physical parameters, yet the abstract gives no quantitative bounds and the stress-test note points out that these are not checked against the computed solutions. If the temperature dependence of conductivity in real materials violates those bounds, the uniqueness result does not apply to the discrete solutions they actually compute. A referee would want to see the precise conditions and whether they are satisfied in the examples. This is the kind of paper that belongs in a numerical PDE or computational electromagnetics journal. Readers working on multiphysics discretizations or induction-heating models would get value from the proofs and the error analysis. It has enough formal content and reproducible numerics to deserve peer review rather than a desk reject, though the parameter restrictions need clarification before the uniqueness statement can be taken at face value.

Referee Report

2 major / 2 minor

Summary. The paper analyzes a transient thermo-electromagnetic problem for induction heating, reduced to a 2D meridional section under cylindrical symmetry and azimuthal current density. It introduces a variational formulation in weighted Sobolev spaces, proves existence via fixed-point argument and uniqueness under assumptions on physical parameters (e.g., temperature-dependent conductivity), discretizes via FEM with implicit Euler time stepping, derives a priori error estimates, and validates them with numerical experiments on an industrial configuration.

Significance. If the fixed-point mapping and error analysis hold under the stated assumptions, the work supplies a rigorous existence-uniqueness framework together with computable a priori bounds for a coupled nonlinear problem that is standard in industrial modeling. The numerical validation of the error estimates is a concrete strength that can directly inform discretization choices.

major comments (2)

[Uniqueness theorem / parameter assumptions] Uniqueness result (following the fixed-point existence argument): uniqueness is asserted only under 'reasonable assumptions on the physical parameters' (bounds on σ(T) and related nonlinearities), yet these bounds are never quantified (no explicit Lipschitz constants, growth restrictions, or temperature-range conditions are given). The industrial numerical example does not verify whether the computed temperature field satisfies them, so the uniqueness claim does not necessarily apply to the reported solutions.
[Error analysis] A priori error estimates section: the constants in the error bounds inherit the same parameter restrictions used for uniqueness; if those restrictions fail for the nonlinear Joule-heating coupling, the estimates lose their justification. No sensitivity test or alternative analysis is provided when the assumptions are relaxed.

minor comments (2)

[Variational formulation] The weighted Sobolev spaces are introduced in the abstract but their precise weights (and the associated function-space norms) appear only later; moving the definition to the formulation section would improve readability.
[Numerical results] Numerical experiments: the tables/figures reporting error decay should explicitly state the reference solution or manufactured-solution method used for validation, and include the observed convergence rates alongside the theoretical ones.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments on the uniqueness result and error analysis. We address each major comment below and indicate the corresponding revisions to the manuscript.

read point-by-point responses

Referee: Uniqueness result (following the fixed-point existence argument): uniqueness is asserted only under 'reasonable assumptions on the physical parameters' (bounds on σ(T) and related nonlinearities), yet these bounds are never quantified (no explicit Lipschitz constants, growth restrictions, or temperature-range conditions are given). The industrial numerical example does not verify whether the computed temperature field satisfies them, so the uniqueness claim does not necessarily apply to the reported solutions.

Authors: We agree that the assumptions underlying the uniqueness theorem should be stated more explicitly. In the revised manuscript, we will quantify the conditions by specifying that σ(T) is Lipschitz continuous with a given constant L and that the temperature remains in a bounded interval [T_min, T_max] ensuring the contraction mapping property. We will also add a post-processing check in the numerical section to confirm that the computed temperature field in the industrial example satisfies these bounds. revision: yes
Referee: A priori error estimates section: the constants in the error bounds inherit the same parameter restrictions used for uniqueness; if those restrictions fail for the nonlinear Joule-heating coupling, the estimates lose their justification. No sensitivity test or alternative analysis is provided when the assumptions are relaxed.

Authors: The error estimates are derived under the same assumptions as uniqueness, so the constants depend on those parameter restrictions. We will revise the manuscript to include an explicit remark on this dependence and to note that the observed convergence rates in the numerical experiments are consistent with the estimates holding in the tested regime. A full sensitivity analysis or alternative estimates under relaxed assumptions lies outside the current scope and would require a separate study. revision: partial

Circularity Check

0 steps flagged

No circularity: standard PDE analysis with external fixed-point and Sobolev theory

full rationale

The derivation chain consists of a variational formulation in weighted Sobolev spaces, existence proved via fixed-point argument, uniqueness under stated parameter assumptions, followed by FEM discretization, a priori error estimates, and numerical validation. None of these steps reduce by construction to the paper's own inputs or fitted data; the proofs invoke standard external theorems rather than self-citations or self-definitional loops. The reduction to axisymmetric meridional section is an explicit modeling assumption, not a hidden tautology. No load-bearing self-citation chains or renamed empirical patterns appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard tools from functional analysis and numerical PDE theory together with modeling assumptions about geometry and material behavior; no free parameters are fitted and no new physical entities are postulated.

axioms (3)

domain assumption The physical problem admits reduction to a two-dimensional meridional section under cylindrical symmetry and purely azimuthal current density.
Invoked at the start of the formulation to obtain the 2D problem.
standard math Existence of a solution follows from a fixed-point argument in appropriately weighted Sobolev spaces.
Standard application of fixed-point theorems for nonlinear elliptic-parabolic systems.
domain assumption Uniqueness holds under reasonable assumptions on the physical parameters.
Required for the uniqueness statement; the precise form of the assumptions is not given in the abstract.

pith-pipeline@v0.9.0 · 5457 in / 1594 out tokens · 42484 ms · 2026-05-07T15:36:36.000665+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Akrivis and S

G. Akrivis and S. Larsson. Linearly implicit finite element methods for the time-dependent Joule heating problem.BIT, 45(3):429–442, 2005

work page 2005

[2] [2]

Allegretto and H

W. Allegretto and H. Xie. Existence of solutions for the time-dependent thermistor equa- tions.IMA J. Appl. Math., 48(3):271–281, 1992

work page 1992

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Belhachmi, C

Z. Belhachmi, C. Bernardi, and S. Deparis. Weighted Cl´ ement operator and application to the finite element discretization of the axisymmetric Stokes problem.Numer. Math., 105(2):217–247, 2006

work page 2006

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Berm´ udez, D

A. Berm´ udez, D. G´ omez, M. C. Mu˜ niz, and P. Salgado. Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces.Adv. Comput. Math., 26(1-3):39–62, 2007

work page 2007

[5] [5]

Berm´ udez, D

A. Berm´ udez, D. G´ omez, M.C. Mu˜ niz, P. Salgado, and R. V´ azquez. Numerical simulation of a thermo-electromagneto-hydrodynamic problem in an induction heating furnace.Appl. Numer. Math., 59(9):2082–2104, 2009

work page 2082

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Berm´ udez, D

A. Berm´ udez, D. G´ omez, and P. Salgado.Mathematical Models and Numerical Simulation in Electromagnetism. New York: Springer, 2014

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Berm´ udez and R

A. Berm´ udez and R. Mu˜ noz Sola. Existence of solution of a coupled problem arising in the thermoelectrical simulation of electrodes.Quart. Appl. Math., 57(4):621–636, 1999

work page 1999

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Berm´ udez, R

A. Berm´ udez, R. Mu˜ noz Sola, and F. Pena. A nonlinear partial differential system arising in thermoelectricity.European J. Appl. Math., 16(6):683–712, 2005

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Berm´ udez, R

A. Berm´ udez, R. Mu˜ noz Sola, and F. Pena. Existence of a solution for a thermoelectric model with several phase changes and a Carath´ eodory thermal conductivity.Nonlinear Anal. Real World Appl., 14(6):2212–2230, 2013

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Berm´ udez, C

A. Berm´ udez, C. Reales, R. Rodr´ ıguez, and P. Salgado. Numerical analysis of a finite- element method for the axisymmetric eddy current model of an induction furnace.IMA J. Numer. Anal., 30(3):654–676, 2010. 20

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Chaboudez, S

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Chovan, C

J. Chovan, C. Geuzaine, and M. Slodiˇ cka.A-φformulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field.Comput. Methods Appl. Mech. Engrg., 321:294–315, 2017

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Chovan and M

J. Chovan and M. Slodiˇ cka. Induction hardening of steel with restrained Joule heating and nonlinear law for magnetic induction field: solvability.J. Comput. Appl. Math., 311:630– 644, 2017

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H. Gao, W. Sun, and C. Wu. Optimal error estimates and recovery technique of a mixed fi- nite element method for nonlinear thermistor equations.IMA J. Numer. Anal., 41(4):3175– 3200, 2021

work page 2021

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G´ omez, D

B. G´ omez, D. L´ opez-Rodr´ ıguez, P. Salgado, and P. Venegas. A coupled steady thermo- electromagnetic problem in axisymmetric geometries. Mathematical and numerical analysis. IMA J. Numer. Anal., 45(4):2132–2170, 2024

work page 2024

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Gopalakrishnan and J

J. Gopalakrishnan and J. Pasciak. The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations.Math. Comp., 75(256):1697–1719, 2006

work page 2006

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Jensen, A

M. Jensen, A. M˜ alqvist, and A. Persson. Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions.IMA J. Numer. Anal., 42(1):199– 228, 2022

work page 2022

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B. Li, H. Gao, and W. Sun. Unconditionally Optimal Error Estimates of a Crank–Nicolson Galerkin Method for the Nonlinear Thermistor Equations.SIAM Journal on Numerical Analysis, 52(2):933–954, 2014

work page 2014

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Lions.Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires

J.-L. Lions.Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires. Dunod, Paris; Gauthier-Villars, Paris, 1969

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M. Mbehou. The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems.Math. Methods Appl. Sci., 41(4):1480–1491, 2018

work page 2018

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B. Mercier and G. Raugel. R´ esolution d’un probl` eme aux limites dans un ouvert ax- isym´ etrique par ´ el´ ements finis enr, zet s´ eries de fourier enθ.RAIRO, Anal. Num´ er., 16(4):405–461, 1982

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D. Shi and H. Yang. Superconvergent estimates of conforming finite element method for nonlinear time-dependent Joule heating equations.Numer. Methods Partial Differential Equations, 34(1):336–356, 2018. 21

work page 2018

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L. Wang, C. Yao, and Z. Zhang. Convergence analysis of finite element approximation for 3-D magneto-heating coupling model.Int. J. Numer. Anal. Model., 17(1):1–23, 2020

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