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Shape Optimization of Rotating Electric Machines using Isogeometric Analysis and Harmonic Stator-Rotor Coupling

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the rotor shape of a 6-pole permanent magnet synchronous machine can be optimized through isogeometric analysis with harmonic stator-rotor coupling, reducing the total harmonic distortion of the electromotive force…

desk verdict Solid IGA shape-optimization paper with real validation; the unspecified harmonic index set in the THD objective is the flaw to fix before trusting the 75% claim. read the letter →

arxiv 1908.06009 v2 pith:674PAV4L submitted 2019-08-15 cs.CE

Melina Merkel , Peter Gangl , Sebastian Schöps This is my paper · ORCID
classification cs.CE
keywords electricmachinesisogeometricanalysisshapeoptimizationharmonicstator-rotorcouplingtotaldistortionpermanentmagnetsynchronousmachinederivativeNURBS
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 works out a gradient-based shape optimization pipeline for a rotating electric machine entirely within isogeometric analysis. Because the geometry is represented by NURBS curves, moving control points reshapes the rotor without remeshing, and rotation is handled by coupling the stator and rotor through harmonic basis functions at the air gap. The authors minimize the total harmonic distortion of the electromotive force and report a reduction from 0.099 to 0.025 in 59 iterations, with an independent finite-element solver confirming a close drop. A nonlinear-material check still shows a 68 percent reduction, so the linear-model optimum is not an artifact of ignoring saturation.

What carries the argument

The central machinery is the harmonic stator-rotor coupling: the rotor and stator are separate multipatch IGA domains, and continuity of the magnetic vector potential across the air gap is enforced by Lagrange multipliers expanded in Fourier harmonics $e^{-i\ell_k\theta}$. This produces a saddle-point system whose interface part is a small Schur complement, $K_{\mathrm{int}}(\alpha) = G_{\mathrm{rt}}^H K_{\mathrm{rt}}^{-1} G_{\mathrm{rt}} + R(\alpha) G_{\mathrm{st}}^H K_{\mathrm{st}}^{-1} G_{\mathrm{st}} R(\alpha)$, where the rotation angle $\alpha$ enters only through the diagonal matrix $R(\alpha)$. For optimization, the shape derivative (equation 20) converts the THD objective into a descent vector field, and an $H^1$-type Riesz lifting with bilinear form $b(W,Z)=\int_D (DW:DZ + W\cdot Z)\,dx$ extracts the update direction. Only the small interface problem is re-solved for each of the 120 rotor positions, which is why the per-iteration cost stays low.

What would settle it

Recompute the THD of the original and optimized geometries with the Fourier sum extended to every harmonic resolvable by the 120-step time discretization, using the published machine geometry; if the reduction from 0.099 to 0.025 shrinks below 75 percent, or if an unlisted high-order harmonic dominates the optimized spectrum, the central quantitative claim fails.

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Extended reading notes

Core claim

The central discovery is that harmonic stator-rotor coupling makes shape optimization of a rotating machine tractable in an isogeometric setting. The simulation splits into an offline phase that factors the stator and rotor stiffness matrices once and an online phase where only a small interface system in the Fourier coefficients of the magnetic field is solved for each rotor angle. The shape derivative of the THD objective is derived through shape calculus with an adjoint equation, and the resulting descent field moves the NURBS control points of the rotor. With 59 iterations the IGA model lowers THD(E) from 0.099268 to 0.024793; an independent finite-element computation gives 0.099754 to 0.02234 with linear materials and 0.10639 to 0.034106 with nonlinear materials. The optimized geometry suppresses the dominant Fourier harmonics of the electromotive force while leaving the permanent magnets unchanged.

Load-bearing premise

The load-bearing premise is that the index set $I$ of harmonics used in the THD objective really covers the harmonics that matter; the paper defines $I$ in the model section but never states which set the numerical optimization used.

Editorial extensions

If this is right

  • If the central claim is right, gradient-based shape optimization of rotating machines no longer requires remeshing on every geometry update.
  • The optimized rotor cuts EMF total harmonic distortion by a factor of about four, and the independent validation shows the improvement is not a quirk of the IGA discretization.
  • With nonlinear magnetic materials the same optimized shape still gives a 68 percent THD reduction, suggesting the linear-model design remains effective under saturation.
  • Because the optimized design is delivered as a NURBS geometry, it can be exported directly into standard CAD and CAE workflows.

Reading between the lines

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

  • An implication the authors leave implicit: the same adjoint-based shape derivative can be applied to other design objectives, such as torque ripple or cogging torque, without changing the coupling machinery.
  • A reader should not treat the 75 percent figure as the total harmonic distortion until the harmonic index set $I$ is specified; re-running with all harmonics included is the natural test.
  • A testable extension would start the descent from several different initial rotor shapes to see whether the optimized design is a stable local optimum or depends on the starting geometry.
  • The near-constant online cost in the number of IGA degrees of freedom suggests the method scales to finer discretizations and possibly 3D extensions where classical moving-band methods become expensive.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper develops and demonstrates an isogeometric-analysis-based shape optimization workflow for a 6-pole permanent-magnet synchronous machine. The stator and rotor are discretized as separate multipatch NURBS domains and coupled through harmonic basis functions on the air-gap interface, allowing efficient evaluation of many rotor positions. The optimization objective is the total harmonic distortion (THD) of the electromotive force, computed from the flux linkage over 120 rotor positions; the shape derivative is obtained by an adjoint approach, and descent directions are computed by solving an auxiliary vector problem. The authors report a reduction of THD by more than 75% after 59 iterations, and they validate the optimized geometry with independent JMAG simulations using both linear and nonlinear material models. The central claim is the method's ability to produce a robust, independently confirmed reduction in EMF distortion.

Significance. If the reported results hold, the paper makes a useful contribution: it combines exact IGA geometry, harmonic stator-rotor coupling, and adjoint shape calculus in a way that is substantially faster than solving the full system for each rotor position, and it demonstrates the workflow on a realistic machine with an independent commercial-solver check. The JMAG validation with linear and nonlinear materials is a particular strength, as is the availability of the geometry data via the Zenodo DOI in [28]. The main obstacle to accepting the quantitative claim is the unspecified harmonic index set I in the THD objective; once that is clarified, the contribution should be judged on its engineering value and the quality of the numerical comparison.

major comments (1)
  1. [II, Eq. (4); VI, Table I and Fig. 8] The optimization problem (8) minimizes THD_I(E(u(t,Omega))), where I is introduced in Section II as an index set of frequencies to consider, with the remark that one may disregard frequencies that cannot be diminished by shape optimization. Section VI and Table I report the THD values for the original and optimized designs and state that the THD is reduced by more than 75%, but they never specify which I was used in the optimization or in the reported numbers. Figure 9 shows Fourier coefficients only up to harmonic 19, whereas with N_alpha=120 the discrete Fourier representation contains coefficients up to order 60. If the implemented I is a proper subset, the claimed reduction applies to that restricted functional and not necessarily to the total THD, and the JMAG comparison in Fig. 8 validates only the same restricted functional. Please state I explicitly, and in addition report the total THD (with all available harmonics included) for the original and optimized geometries for both IGA and JMAG.
minor comments (5)
  1. [III-B, Eq. (19)] In Eq. (19), the coefficient B_k is written with (M_a)_{k,j}, but from the definition it should be (M_b)_{k,j}; the derivative formula below uses M_b correctly. Please correct the typo.
  2. [IV-B] Please state the harmonic orders l_k used for the N_Gamma=36 coupling basis. The number 36 alone does not determine the approximation space, and the choice may affect the accuracy of the Fourier coefficients that feed the objective.
  3. [V] The admissible set A and the set of control points that are allowed to move are not specified. For reproducibility, state which control points are design variables and how the descent field W from Eq. (39) is transferred to control-point displacements.
  4. [I] The sentence claiming for the first time shape optimization of a rotating electric machine discretized with IGA is stronger than reference [5] supports; please qualify this claim in light of [5] and related work.
  5. [VI, Fig. 9] The Fourier-coefficient plots show only odd harmonics up to 19; please indicate the truncation and whether the omitted coefficients are negligible in the reported THD.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the optimized geometry's THD reduction is independently reproduced by JMAG, so the central claim does not reduce to its inputs.

full rationale

The derivation chain is self-contained in the relevant sense: the objective J(Ω) := THD_I(E(u(t, Ω))) in Eq. (8) is minimized subject to the forward magnetostatic model (9), with the shape derivative derived through the adjoint system (14) and given explicitly in (20). The reported reduction is not a renamed fit: no parameter is fitted to the target THD values, and the optimized geometry is checked by an independent JMAG model with 281198 elements. Table I and Fig. 8 show IGA linear THD dropping from 0.099268 to 0.024793, while JMAG linear confirms 0.099754 to 0.02234; even JMAG nonlinear shows a 68% reduction. The harmonic stator-rotor coupling from [19] and the shape-calculus template from [21] are prior works with overlapping authorship, but they are not used as uniqueness theorems nor as the justification for the central numerical result; the external JMAG validation stands independently of those citations. The only genuine weakness is that the index set I in the THD definition is not stated in Section VI, so the exact functional minimized is under-specified. That is a reporting completeness issue, not a circularity: nothing in the paper's equations makes the optimization result equivalent to its inputs by construction, and the independent validation would not be explained by a self-referential objective definition. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model uses standard magnetostatic assumptions plus several discretization choices. The main unstated modeling choice is the harmonic index set I; the paper defines it in Section II but never specifies which frequencies are included in the reported THD, which affects the objective value.

assumptions (5)
  • domain assumption Magnetostatic approximation: displacement currents and eddy currents are neglected and materials are linear in the optimization model.
    Section II, equation (1), states the magnetostatic formulation; Section VI uses linearized material laws for IGA optimization, with nonlinear validation only in JMAG.
  • domain assumption The deformation vector field W in the shape derivative vanishes on the stator-rotor interface Gamma and on the stator, and the permanent magnet region is unchanged.
    Section III.B before Eq. (20): this restricts the optimization to rotor iron away from the interface; the resulting search space limitation is not quantified.
  • domain assumption The harmonic stator-rotor coupling is stable and accurate for the chosen number of harmonics NGamma and degrees of freedom Nq.
    Section IV.B states stability follows from [19] if Nq and NGamma are chosen consistently; no numerical convergence study is provided for the optimization runs.
  • domain assumption The discrete Fourier transform over Nalpha=120 rotor positions correctly resolves all harmonics used in the THD objective.
    Section III.A and Section VI; aliasing and the precise harmonic set I are not discussed.
  • ad hoc to paper THD is evaluated on an unspecified index set I of harmonics.
    Section II defines I but the numerical section never states which frequencies are included; Fig. 9 shows harmonics 1,3,...,19, but the objective may have been computed on a subset.

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Cite this review

Pith. "Pith review of Shape Optimization of Rotating Electric Machines using Isogeometric Analysis and Harmonic Stator-Rotor Coupling." pith.science (2026). https://pith.science/paper/674PAV4L

@misc{pith2026190806009,
  author       = {Pith},
  title        = {Pith review of: Shape Optimization of Rotating Electric Machines using Isogeometric Analysis and Harmonic Stator-Rotor Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/674PAV4L}},
  note         = {Machine review of arXiv:1908.06009}
}
read the original abstract

This work deals with shape optimization of electric machines using isogeometric analysis. Isogeometric analysis is particularly well suited for shape optimization as it allows to easily modify the geometry without remeshing the domain. A 6-pole permanent magnet synchronous machine is modeled using a multipatch isogeometric approach and rotation of the machine is realized by modeling the stator and rotor domain separately and coupling them at the interface using harmonic basis functions. Shape optimization is applied to the model minimizing the total harmonic distortion of the electromotive force as a goal functional.

Figures

Figures reproduced from arXiv: 1908.06009 by the authors.

Figure 1
Figure 1. Example for a domain D with coil region Dc, permanent magnet region Dpm and domain boundary ∂D. Drt Dst er Γ |B| (T) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Multipatch model of one pole of a 6-pole permanent magnet syn [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Comparison of the computational time of the simulation of a rotating [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Visualization of the modification of the shape of a NURBS curve by [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Visualization of the mappings Fk from the reference domain Dˆ to the patches Dk of a multipatch geometry. geometry, i.e., the mesh. In isogeometric analysis the geometry is represented by a smooth mapping F : Dˆ → D, (38) using NURBS as basis functions, where Dˆ is the…
Figure 7
Figure 7. Figure 7: Results of shape optimization (8) in generator mode under no load [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 10
Figure 10. Figure 10: Fourier coefficients of the electromotive force for the original design [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 9
Figure 9. Figure 9: Fourier coefficients of the electromotive force for the original design [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

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