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arxiv: 2402.19065 · v1 · pith:5B7H6YWKnew · submitted 2024-02-29 · 🧮 math.OC

Spline-Based Rotor and Stator Optimization of a Permanent Magnet Synchronous Motor

Michael Wiesheu , Theodor Komann , Melina Merkel , Sebastian Sch\"ops , Stefan Ulbrich , Idoia Cortes Garcia This is my paper

Pith reviewed 2026-05-24 03:50 UTC · model grok-4.3

classification 🧮 math.OC
keywords permanent magnet synchronous motorisogeometric analysisshape optimizationspline control pointstorque ripplegradient-based optimizationscaling lawsmotor design
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The pith

Spline modifications in isogeometric analysis enable efficient gradient-based optimization of permanent magnet synchronous motors, reducing material cost, torque ripple and losses across operating points.

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

The paper shows how rotor and stator shapes in a permanent magnet synchronous motor can be optimized by adjusting spline control points inside a 2D nonlinear isogeometric analysis simulation. Magnetism scaling laws extend the model to axial and radial dimensions, allowing simultaneous tuning of geometry, surface contours and scaling factors. A gradient-based optimizer then targets three objectives at once: lower material cost, smaller torque ripple and reduced losses. The method handles large numbers of design variables while improving every objective value.

Core claim

Rotor and stator designs of a permanent magnet synchronous motor are optimized for geometric parameters and surface shapes by modifying control points in an isogeometric analysis framework. Magnetism scaling laws permit axial and radial scaling so that all critical machine parameters can be adjusted. The gradient-based process, applied to multiple operating points, lowers motor material cost, torque ripple and losses simultaneously and remains computationally tractable even with many variables.

What carries the argument

Isogeometric analysis framework that represents rotor and stator surfaces via spline control points, combined with magnetism scaling laws that map 2D solutions to 3D machine dimensions.

If this is right

  • The same control-point approach can be applied to additional motor geometries without reformulating the underlying simulation mesh.
  • All three objectives (cost, ripple, losses) improve together rather than trading off against one another.
  • Multiple operating points can be included in a single optimization run once the scaling laws are active.
  • Gradient information from the isogeometric solver makes the process scale to dozens of design variables.

Where Pith is reading between the lines

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

  • The method could be tested on other machine topologies such as induction or reluctance motors to check whether the same spline-plus-scaling combination remains effective.
  • If the 2D-to-3D scaling accuracy holds, early-stage design loops could replace many full 3D finite-element runs with far cheaper 2D runs.
  • Adding manufacturing tolerance bounds as additional constraints would reveal how robust the spline-optimized shapes remain under realistic production variation.

Load-bearing premise

Two-dimensional nonlinear simulations plus magnetism scaling laws are assumed to capture the essential three-dimensional motor behavior across multiple operating points without important loss of accuracy.

What would settle it

Building a physical prototype from the optimized spline design and measuring its torque ripple, losses and material cost; if the measured values fail to show the predicted reductions relative to the baseline motor, the optimization claim is falsified.

Figures

Figures reproduced from arXiv: 2402.19065 by Idoia Cortes Garcia, Melina Merkel, Michael Wiesheu, Sebastian Sch\"ops, Stefan Ulbrich, Theodor Komann.

Figure 1
Figure 1. Figure 1: Parametrization of the PMSM geometry including parameter names, material definitions and boundary conditions. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of initial and optimized geometry. The yellow points show the control points of the motor, the red control [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the magnetic flux density. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of initial and optimized torque profiles for different operating points. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

This work features the optimization of a Permanent Magnet Synchronous Motor using 2D nonlinear simulations in an Isogeometric Analysis framework. The rotor and stator designs are optimized for both geometric parameters and surface shapes via modifications of control points. The scaling laws for magnetism are employed to allow for axial and radial scaling, enabling a thorough optimization of all critical machine parameters for multiple operating points. The process is carried out in a gradient-based fashion with the objectives of lowering motor material cost, torque ripple and losses. It is shown that the optimization can be efficiently conducted for many optimization variables and all objective values can be reduced.

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

2 major / 2 minor

Summary. The manuscript presents a gradient-based optimization framework for the rotor and stator of a permanent magnet synchronous motor (PMSM). It employs 2D nonlinear isogeometric analysis (IGA) simulations in which control points are varied to modify surface shapes, combined with axial and radial scaling laws derived from magnetism principles. The objectives are simultaneous reduction of material cost, torque ripple, and electromagnetic losses evaluated at multiple operating points. The central claim is that the procedure remains computationally tractable for a large number of design variables and yields simultaneous improvement in all three objectives.

Significance. If the 2D IGA model with scaling laws is shown to be sufficiently predictive, the work would demonstrate a practical route to high-dimensional, multi-point motor shape optimization that exploits the exact geometry representation of splines. The explicit use of gradient information and the scaling-law extension to three-dimensional effects are methodological strengths that could be adopted in other electromagnetic design problems.

major comments (2)
  1. [Abstract and Results (optimization outcomes)] The central claim that all three objectives are simultaneously reduced rests on the fidelity of the 2D nonlinear IGA model plus axial/radial scaling laws. No section compares the predicted torque ripple or loss values against either full 3D finite-element simulations or measurements on a prototype; end-effects and axial leakage can alter ripple by tens of percent away from the rated point.
  2. [Methodology (scaling-law subsection)] The scaling laws are invoked to map 2D results to 3D performance at multiple operating points, yet the manuscript provides neither an explicit statement of the scaling relations (e.g., how eddy-current loss or radial flux density scales with axial length) nor a sensitivity study showing that the assumed scaling remains accurate when the optimized geometry deviates from the baseline.
minor comments (2)
  1. [Section 2] Notation for the control-point displacement variables and the IGA basis functions should be introduced once in a dedicated nomenclature table or at first use to avoid repeated re-definition across sections.
  2. [Figure 7] Figure captions for the optimized geometries should state the number of control points varied and the final objective values achieved, rather than only qualitative descriptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the methodological contributions of the gradient-based spline optimization framework. We address each major comment below, providing clarifications and committing to revisions where feasible.

read point-by-point responses

  1. Referee: [Abstract and Results (optimization outcomes)] The central claim that all three objectives are simultaneously reduced rests on the fidelity of the 2D nonlinear IGA model plus axial/radial scaling laws. No section compares the predicted torque ripple or loss values against either full 3D finite-element simulations or measurements on a prototype; end-effects and axial leakage can alter ripple by tens of percent away from the rated point.

    Authors: We acknowledge that the manuscript does not include direct comparisons of the optimized designs against 3D FEM simulations or physical prototypes. The 2D nonlinear IGA approach combined with scaling was selected specifically to enable tractable gradient-based optimization over a high-dimensional design space, as full 3D nonlinear simulations would render the procedure computationally infeasible for the reported number of variables. The scaling laws follow standard electromagnetic principles commonly applied in machine design literature. We will revise the manuscript to include an explicit discussion of the limitations arising from end-effects and axial leakage, along with references to prior validation studies of similar 2D-to-3D approximations. revision: partial

  2. Referee: [Methodology (scaling-law subsection)] The scaling laws are invoked to map 2D results to 3D performance at multiple operating points, yet the manuscript provides neither an explicit statement of the scaling relations (e.g., how eddy-current loss or radial flux density scales with axial length) nor a sensitivity study showing that the assumed scaling remains accurate when the optimized geometry deviates from the baseline.

    Authors: We agree that the scaling relations require explicit formulation. In the revised manuscript we will add a dedicated subsection that states the scaling laws in mathematical form, including the dependence of torque, flux density, and loss components on axial length and radial scaling factors. We will also report a sensitivity study evaluating the accuracy of these scalings for geometry perturbations around the baseline design at the considered operating points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; optimization outcomes independent of inputs

full rationale

The paper describes a gradient-based optimization of PMSM rotor and stator geometries using 2D nonlinear IGA simulations combined with axial/radial magnetism scaling laws. Objectives (material cost, torque ripple, losses) are minimized across operating points, with results presented as direct outputs of the numerical process. No equations or steps reduce by construction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The derivation chain relies on external simulation fidelity and scaling assumptions rather than tautological re-expression of inputs. This is the expected non-finding for an applied optimization study whose central claim is empirical improvement rather than a derived identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities. The scaling laws for magnetism and the sufficiency of 2D nonlinear models are implicit background assumptions whose validity cannot be audited from the provided text.

pith-pipeline@v0.9.0 · 5641 in / 1111 out tokens · 31129 ms · 2026-05-24T03:50:22.416506+00:00 · methodology

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