An Air-Gap Element for the Isogeometric Space-Time-Simulation of Electric Machines

Melina Merkel; Michael Reichelt; Michael Wiesheu; Olaf Steinbach; Sebastian Sch\"ops

arxiv: 2501.16099 · v1 · pith:2KI3O3L2new · submitted 2025-01-27 · 🧮 math.NA · cs.CE· cs.NA

An Air-Gap Element for the Isogeometric Space-Time-Simulation of Electric Machines

Michael Reichelt , Michael Wiesheu , Melina Merkel , Sebastian Sch\"ops , Olaf Steinbach This is my paper

Pith reviewed 2026-05-23 05:01 UTC · model grok-4.3

classification 🧮 math.NA cs.CEcs.NA

keywords air-gap elementisogeometric analysisspace-time simulationelectric machinesrotor-stator couplingNURBSfinite element method

0 comments

The pith

An air-gap element allows isogeometric analysis and space-time simulation of electric machines with efficient angle updates.

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

The paper establishes that the air-gap element, which uses an analytic solution in the gap between rotor and stator, can be combined with isogeometric analysis to model electric machines. It shows that the resulting angle-dependent coupling matrices can be updated for new rotor positions without repeated expensive integration. This combination transfers directly to a space-time formulation of the time-dependent problem. A reader would care because it removes the requirement that rotor motion be known in advance for meshing and supports exact representation of circular boundaries.

Core claim

The air-gap element can be applied to isogeometric analysis and space-time simulation of electric machines. The angle-dependent coupling matrices can be updated efficiently without expensive quadrature. NURBS exactly represent the air-gap geometry, and the model including the air-gap element can be seamlessly transferred to the space-time setting. The originality lies in this application rather than in the air-gap element itself, which is already known in the literature.

What carries the argument

The air-gap element, which derives transmission conditions from the analytic Fourier-type solution in the air-gap region to couple the isogeometric rotor and stator subdomains.

If this is right

Coupling matrices for arbitrary rotor angles can be obtained from a base set by simple transformations instead of quadrature.
The full machine model, including the air-gap element, moves without change into a space-time discretization.
Isogeometric NURBS meshes of the rotor and stator remain compatible with the analytic coupling at every angle.
Time-domain simulations of electric machines no longer require the motion schedule to be fixed before meshing.

Where Pith is reading between the lines

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

The same transmission conditions could be used with other high-order spatial discretizations that exactly represent circles.
Variable-speed operation becomes feasible in space-time models because angle updates no longer require remeshing or re-integration.
The approach may extend to other rotating-interface problems where an annular region admits a closed-form solution.

Load-bearing premise

The air-gap region between rotor and stator admits an exact analytic solution that supplies transmission conditions for the isogeometric subdomains.

What would settle it

A direct numerical comparison showing that the efficiently updated coupling matrices at several rotor angles produce the same electromagnetic fields as matrices recomputed by full quadrature over the air-gap interface.

Figures

Figures reproduced from arXiv: 2501.16099 by Melina Merkel, Michael Reichelt, Michael Wiesheu, Olaf Steinbach, Sebastian Sch\"ops.

**Figure 2.** Figure 2: Comparison of condition numbers for unscaled and scaled systems. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Geometry model for the PSMS. The air-gap element connects the current [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Torque evaluation for different scenarios. The torque curves are compared for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

read the original abstract

Space-time methods promise more efficient time-domain simulations, in particular of electrical machines. However, most approaches require the motion to be known in advance so that it can be included in the space-time mesh. To overcome this problem, this paper proposes to use the well-known air-gap element for the rotor-stator coupling of an isogeometric machine model. First, we derive the solution in the air-gap region and then employ it to couple the rotor and stator. This coupling is angle dependent and we show how to efficiently update the coupling matrices to a different angle, avoiding expensive quadrature. Finally, the resulting time-dependent problem is solved in a space-time setting. The spatial discretization using isogeometric analysis is particularly suitable for coupling via the air-gap element, as NURBS can exactly represent the geometry of the air-gap. Furthermore, the model including the air-gap element can be seamlessly transferred to the space-time setting. However, the air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time.

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

Straightforward port of the classical air-gap element to IGA plus space-time, with the practical payoff in the matrix-update step.

read the letter

This paper takes the standard air-gap element—analytic Fourier solution in the annular gap to couple rotor and stator without meshing the gap—and moves it into an isogeometric setting and then into space-time. The geometry match is the obvious win: NURBS represent the circular interface exactly, so the transmission conditions sit cleanly on the boundary. They also give an explicit procedure to update the angle-dependent coupling matrices for a new rotor position without redoing quadrature each time, which follows directly from the rotational invariance of the Fourier basis. That step looks useful for any time-dependent run where the motion is not known in advance. The rest is mostly the expected transfer: the same transmission operators are carried into the space-time weak form. The soft spot is that the core technique is explicitly labeled as well-known, so the advance is limited to the combination and the implementation details. No numerical verification appears in the abstract, and the derivation itself is only sketched, so the strength of the claims depends on how cleanly those steps are written out in the full text. Nothing in the stress-test note suggests hidden inconsistencies or non-standard assumptions. This is for people already doing IGA on electromagnetic problems or experimenting with space-time formulations for machines; it is not going to shift the broader field. I would send it to peer review—the extension is grounded and the efficiency claim is checkable, so referees can sort out whether the update procedure delivers in practice.

Referee Report

2 major / 1 minor

Summary. The paper proposes applying the classical air-gap element (Fourier expansion in the annular gap) to couple rotor and stator subdomains in an isogeometric analysis (IGA) discretization of electric machines, enabling space-time simulation without requiring the motion to be known a priori. It claims to derive the air-gap solution, obtain angle-dependent transmission operators, demonstrate an efficient (quadrature-free) update of the coupling matrices under rotation, and transfer the resulting formulation directly into a space-time setting, exploiting the exact circular geometry representable by NURBS.

Significance. If the derivation and update procedure are made explicit and the method is verified, the work would provide a practical route to space-time IGA simulations of rotating machines that avoids remeshing or pre-known motion. The seamless transfer to space-time and the exact-geometry advantage of NURBS are genuine strengths of the application, though the core technique is an extension of a well-established finite-element method rather than a fundamentally new formulation.

major comments (2)

[Abstract / §1] Abstract and §1: The central claim rests on deriving an explicit analytic solution in the air-gap and the resulting transmission operators for the IGA subdomains, yet neither the Fourier expansion nor the coupling matrices are supplied; without these the efficiency claim for angle updates cannot be assessed.
[Abstract] Abstract: No numerical verification, benchmark, or example is mentioned that would confirm the claimed efficient matrix update or the correctness of the space-time solution; this is load-bearing for a methods paper whose originality is stated to be the IGA/space-time application.

minor comments (1)

[Abstract / §1] The statement that 'the air-gap element is well known' should be supported by at least two or three key references to the classical literature so readers can locate the original transmission conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the potential of the IGA/space-time application. We address the two major comments point by point below.

read point-by-point responses

Referee: [Abstract / §1] Abstract and §1: The central claim rests on deriving an explicit analytic solution in the air-gap and the resulting transmission operators for the IGA subdomains, yet neither the Fourier expansion nor the coupling matrices are supplied; without these the efficiency claim for angle updates cannot be assessed.

Authors: We agree that the explicit Fourier expansion and coupling matrices are required for the reader to evaluate the quadrature-free update. Although Section 3 outlines the derivation of the air-gap solution and the angle-dependent transmission operators, the explicit series coefficients and matrix expressions were not written out in full. In the revised manuscript we will insert the complete Fourier-series solution inside the annular gap together with the resulting closed-form expressions for the rotor-to-stator and stator-to-rotor coupling matrices, thereby making the O(1) angle-update procedure transparent. revision: yes
Referee: [Abstract] Abstract: No numerical verification, benchmark, or example is mentioned that would confirm the claimed efficient matrix update or the correctness of the space-time solution; this is load-bearing for a methods paper whose originality is stated to be the IGA/space-time application.

Authors: The abstract deliberately emphasizes the methodological novelty rather than the numerical results. The full manuscript already contains a dedicated numerical section (Section 5) that reports both the cost of the matrix update under successive rotor angles and a space-time verification against a reference time-stepping solution. To satisfy the referee’s request we will add a single sentence to the abstract that points to these benchmarks. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct application of established air-gap element

full rationale

The paper's derivation chain begins from the classical air-gap element (Fourier-type analytic solution in the annular region between rotor and stator) and applies it to couple IGA subdomains in a space-time setting. The abstract and description explicitly state that the air-gap element is well-known in the literature, with originality limited to the IGA and space-time extension. No equation reduces a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from self-citations, and no ansatz is smuggled via prior author work. The efficient angle-dependent matrix update follows directly from rotational invariance of the Fourier basis and exact NURBS representation of the circular geometry—standard properties invoked without internal redefinition or statistical forcing. The method is self-contained against external benchmarks (prior FE air-gap literature) and does not rely on load-bearing self-citations or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the classical air-gap element (analytic solution inside the annular gap) and the exact geometry representation property of NURBS; no new free parameters, ad-hoc constants, or postulated entities are introduced.

axioms (2)

domain assumption The electromagnetic fields inside the annular air-gap admit an exact analytic representation (typically Fourier series) that supplies transmission conditions between rotor and stator.
Invoked when the authors state they first derive the solution in the air-gap region and then employ it to couple the rotor and stator.
domain assumption NURBS basis functions represent the circular air-gap geometry exactly, incurring no geometric approximation error in the coupling.
Stated directly as the reason IGA is particularly suitable for the air-gap element.

pith-pipeline@v0.9.0 · 5736 in / 1502 out tokens · 69044 ms · 2026-05-23T05:01:13.974839+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Az|ΩA(r,φ)=α0+α′0ln(r)+∑cos(kφ)(αkrk+α′kr−k)+∑sin(kφ)(βkrk+β′kr−k)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

A. A. Abdel-Razek, J.-L. Coulomb, M. Feliachi, and J. C. Sabonnadi` ere. The calculation of electromagnetic torque in saturated electric machines within com- bined numerical and analytical solutions of the field equations. IEEE Trans. Magn., 17(6):3250–3252, 1981

work page 1981
[2]

A. A. Abdel-Razek, J.-L. Coulomb, M. Feliachi, and J. C. Sabonnadi` ere. Conception of an air-gap element for the dynamic analysis of the electromagnetic field in electric machines. IEEE Trans. Magn., 18(2):655–659, 1982

work page 1982
[3]

A. Arkkio. Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations . Phd thesis, Helsinki University of Technology, 1987

work page 1987
[4]

D. Bast, I. Kulchytska-Ruchka, S. Sch¨ ops, and O. Rain. Accelerated steady-state torque computation for induction machines using parallel-in-time algorithms. IEEE Trans. Magn., 56(2):1–9, 2020

work page 2020
[5]

Bolten, S

M. Bolten, S. Friedhoff, J. Hahne, and S. Sch¨ ops. Parallel-in-time simulation of an electrical machine using MGRIT. Comput. Visual. Sci , 23(14), 2020

work page 2020
[6]

Bontinck, J

Z. Bontinck, J. Corno, S. Sch¨ ops, and H. De Gersem. Isogeometric analysis and harmonic stator-rotor coupling for simulating electric machines. Comput. Meth. Appl. Mech. Eng., 334:40–55, 2018

work page 2018
[7]

Davat, Z

B. Davat, Z. Ren, and M. Lajoie-Mazenc. The movement in field modeling. IEEE Trans. Magn., 21(6):2296–2298, 1985

work page 1985
[8]

De Gersem, J

H. De Gersem, J. Gyselinck, P. Dular, K. Hameyer, and T. Weiland. Comparison of sliding-surface and moving-band techniques in frequency-domain finite-element models of rotating machines. COMPEL, 23(4):1006–1014, 2004

work page 2004
[9]

Egger, M

H. Egger, M. Harutyunyan, R. L¨ oscher, M. Merkel, and S. Sch¨ ops. On torque computation in electric machine simulation by harmonic mortar methods. J. Math. Ind., 12(6), 2022

work page 2022
[10]

M. J. Gander and M. Neum¨ uller. Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. , 38(4):A2173–A2208, 2016

work page 2016
[11]

Gangl, M

P. Gangl, M. Gobrial, and O. Steinbach. A space-time finite element method for the eddy current approximation of rotating electric machines. Comput. Methods Appl. Math., 2024

work page 2024
[12]

Gangl, S

P. Gangl, S. K¨ othe, C. Mellak, A. Cesarano, and A. M¨ utze. Multi-objective free-form shape optimization of a synchronous reluctance machine. COMPEL, 41(5):1849– 1864, 2022

work page 2022
[13]

Gyselinck, L

J. Gyselinck, L. Vandevelde, J. Melkebeek, P. Dular, F. Henrotte, and W. Legros. Calculation of eddy currents and associated losses in electrical steel laminations. IEEE Trans. Magn., 35(3):1191–1194, 1999

work page 1999
[14]

Henrotte, G

F. Henrotte, G. Deli´ ege, and K. Hameyer. The eggshell approach for the computation of electromagnetic forces in 2D and 3D. COMPEL, 23(4):996–1005, 2004

work page 2004
[15]

Henrotte, M

F. Henrotte, M. Felden, M. van der Giet, and K. Hameyer. Electromagnetic force computation with the eggshell method. In 14th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering , Graz, 2010

work page 2010
[16]

T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement.Comput. Meth. Appl. Mech. Eng., 194:4135–4195, 2005

work page 2005
[17]

J. D. Jackson. Classical Electrodynamics. Wiley & Sons, New York, 1998

work page 1998
[18]

Kapidani, M

B. Kapidani, M. Merkel, S. Sch¨ ops, and R. V´ azquez. Tree-cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains. Comput. Meth. Appl. Mech. Eng., 395:114949, 2022

work page 2022
[19]

S. Kurz, J. Fetzer, G. Lehner, and W. M. Rucker. Numerical analysis of three- dimensional eddy current problems with moving bodies by boundary element-finite- element method coupling. Surv. Math. Ind. , 9(2):131–150, 1999

work page 1999
[20]

P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford, 2003

work page 2003
[21]

Rodger, H

D. Rodger, H. C. Lai, and P. J. Leonard. Coupled elements for problems involving movement. IEEE Trans. Magn., 26(2):548–550, 1990

work page 1990
[22]

S. J. Salon. Finite Element Analysis of Electrical Machines . Kluwer, Norwell, 1995

work page 1995
[23]

Schmidt, O

K. Schmidt, O. Sterz, and R. Hiptmair. Estimating the eddy-current modeling error. IEEE Trans. Magn., 44(6):686–689, 2008

work page 2008
[24]

Sch¨ ops, I

S. Sch¨ ops, I. Niyonzima, and M. Clemens. Parallel-in-time simulation of eddy current problems using parareal. IEEE Trans. Magn., 54(3):1–4, 2018

work page 2018
[25]

Stipetic, D

S. Stipetic, D. Zarko, and M. Popescu. Scaling laws for synchronous permanent magnet machines. In 2015 Tenth International Conference on Ecological Vehicles and Renewable Energies (EVER) , pages 1–7, 2015

work page 2015

[1] [1]

A. A. Abdel-Razek, J.-L. Coulomb, M. Feliachi, and J. C. Sabonnadi` ere. The calculation of electromagnetic torque in saturated electric machines within com- bined numerical and analytical solutions of the field equations. IEEE Trans. Magn., 17(6):3250–3252, 1981

work page 1981

[2] [2]

A. A. Abdel-Razek, J.-L. Coulomb, M. Feliachi, and J. C. Sabonnadi` ere. Conception of an air-gap element for the dynamic analysis of the electromagnetic field in electric machines. IEEE Trans. Magn., 18(2):655–659, 1982

work page 1982

[3] [3]

A. Arkkio. Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations . Phd thesis, Helsinki University of Technology, 1987

work page 1987

[4] [4]

D. Bast, I. Kulchytska-Ruchka, S. Sch¨ ops, and O. Rain. Accelerated steady-state torque computation for induction machines using parallel-in-time algorithms. IEEE Trans. Magn., 56(2):1–9, 2020

work page 2020

[5] [5]

Bolten, S

M. Bolten, S. Friedhoff, J. Hahne, and S. Sch¨ ops. Parallel-in-time simulation of an electrical machine using MGRIT. Comput. Visual. Sci , 23(14), 2020

work page 2020

[6] [6]

Bontinck, J

Z. Bontinck, J. Corno, S. Sch¨ ops, and H. De Gersem. Isogeometric analysis and harmonic stator-rotor coupling for simulating electric machines. Comput. Meth. Appl. Mech. Eng., 334:40–55, 2018

work page 2018

[7] [7]

Davat, Z

B. Davat, Z. Ren, and M. Lajoie-Mazenc. The movement in field modeling. IEEE Trans. Magn., 21(6):2296–2298, 1985

work page 1985

[8] [8]

De Gersem, J

H. De Gersem, J. Gyselinck, P. Dular, K. Hameyer, and T. Weiland. Comparison of sliding-surface and moving-band techniques in frequency-domain finite-element models of rotating machines. COMPEL, 23(4):1006–1014, 2004

work page 2004

[9] [9]

Egger, M

H. Egger, M. Harutyunyan, R. L¨ oscher, M. Merkel, and S. Sch¨ ops. On torque computation in electric machine simulation by harmonic mortar methods. J. Math. Ind., 12(6), 2022

work page 2022

[10] [10]

M. J. Gander and M. Neum¨ uller. Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. , 38(4):A2173–A2208, 2016

work page 2016

[11] [11]

Gangl, M

P. Gangl, M. Gobrial, and O. Steinbach. A space-time finite element method for the eddy current approximation of rotating electric machines. Comput. Methods Appl. Math., 2024

work page 2024

[12] [12]

Gangl, S

P. Gangl, S. K¨ othe, C. Mellak, A. Cesarano, and A. M¨ utze. Multi-objective free-form shape optimization of a synchronous reluctance machine. COMPEL, 41(5):1849– 1864, 2022

work page 2022

[13] [13]

Gyselinck, L

J. Gyselinck, L. Vandevelde, J. Melkebeek, P. Dular, F. Henrotte, and W. Legros. Calculation of eddy currents and associated losses in electrical steel laminations. IEEE Trans. Magn., 35(3):1191–1194, 1999

work page 1999

[14] [14]

Henrotte, G

F. Henrotte, G. Deli´ ege, and K. Hameyer. The eggshell approach for the computation of electromagnetic forces in 2D and 3D. COMPEL, 23(4):996–1005, 2004

work page 2004

[15] [15]

Henrotte, M

F. Henrotte, M. Felden, M. van der Giet, and K. Hameyer. Electromagnetic force computation with the eggshell method. In 14th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering , Graz, 2010

work page 2010

[16] [16]

T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement.Comput. Meth. Appl. Mech. Eng., 194:4135–4195, 2005

work page 2005

[17] [17]

J. D. Jackson. Classical Electrodynamics. Wiley & Sons, New York, 1998

work page 1998

[18] [18]

Kapidani, M

B. Kapidani, M. Merkel, S. Sch¨ ops, and R. V´ azquez. Tree-cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains. Comput. Meth. Appl. Mech. Eng., 395:114949, 2022

work page 2022

[19] [19]

S. Kurz, J. Fetzer, G. Lehner, and W. M. Rucker. Numerical analysis of three- dimensional eddy current problems with moving bodies by boundary element-finite- element method coupling. Surv. Math. Ind. , 9(2):131–150, 1999

work page 1999

[20] [20]

P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford, 2003

work page 2003

[21] [21]

Rodger, H

D. Rodger, H. C. Lai, and P. J. Leonard. Coupled elements for problems involving movement. IEEE Trans. Magn., 26(2):548–550, 1990

work page 1990

[22] [22]

S. J. Salon. Finite Element Analysis of Electrical Machines . Kluwer, Norwell, 1995

work page 1995

[23] [23]

Schmidt, O

K. Schmidt, O. Sterz, and R. Hiptmair. Estimating the eddy-current modeling error. IEEE Trans. Magn., 44(6):686–689, 2008

work page 2008

[24] [24]

Sch¨ ops, I

S. Sch¨ ops, I. Niyonzima, and M. Clemens. Parallel-in-time simulation of eddy current problems using parareal. IEEE Trans. Magn., 54(3):1–4, 2018

work page 2018

[25] [25]

Stipetic, D

S. Stipetic, D. Zarko, and M. Popescu. Scaling laws for synchronous permanent magnet machines. In 2015 Tenth International Conference on Ecological Vehicles and Renewable Energies (EVER) , pages 1–7, 2015

work page 2015