Evolutionary Topology Optimization

TOPOLOGY OPTIMIZATION / RESEARCH ATLAS

Topology optimization considering
magnetic fields and mechanical stress

Explore the distribution of steel and voids in a motor while considering magnetic fields and stress.
Discover the final Pareto solutions obtained with the NGnet on/off method and the role of their geometry.

01 / PARETO EXPLORER

Optimization results for a motor sized for hybrid vehicles

Maximize mean torque · Minimize torque ripple
Constraint: maximum von Mises stress ≤ 300 MPa
Click a point to display its geometry.

Loading final-generation data…
● Final-generation Pareto solutions (212 points)Feasible solutions only

Focus a point and press Enter to select it. All individuals are also available in the list below.

Individual —
■ Steel■ Magnet□ Void
Mean torque—Nm
Torque ripple—Nm
Maximum stress—MPa

Magnetic flux density, stress distributions and torque waveforms for the representative solution are shown in the reference comparison below.

1,024 individuals · 200 generations completed. 80 Arms · Axial length 50 mm. Maximum von Mises stress ≤ 300 MPa.

Final Pareto solutions CSV ↓Previous results (Japanese)

OPTIMIZATION HISTORY

Watch the optimization evolve

A 24.5-second video shows the Pareto front and representative geometries across 200 generations. Each generation’s representative is selected near 110 Nm; this is not a parent–offspring lineage.

Labels within the video are in Japanese.

Play or download the video (MP4)

02 / METHOD

Material distribution with the NGnet on/off method

Instead of switching many elements directly ON or OFF,
the weights of Gaussian basis functions serve as design variables.

NGnet / NORMALIZED GAUSSIAN NETWORK
φ(x)=∑i=1Nwigi(x)∑i=1Ngi(x)
gi(x)=exp(−‖x−ci‖22σ2)
φ(x) ≥ 0 Steel ONφ(x) < 0 Void OFF

100 Gaussian basis functions; covariance parameter 2×10⁻⁶ m². Material distribution varies within the design region while fixed regions, including magnets, are preserved.

  1. 01

    Smooth design field

    Gaussian weights determine the positions, shapes, and connectivity of holes. The symmetric fundamental region is repeated to construct the full motor geometry.

  2. 02

    Evaluate magnetic fields and stress

    Nonlinear magnetic analysis gives torque at each rotor position. P2 elements evaluate the maximum von Mises stress caused by centrifugal force.

  3. 03

    Selection and update with NSGA-II

    Solutions balance maximizing mean torque and minimizing peak-to-peak torque ripple. Violations of stress and other constraints receive penalties.

Method background: Sato et al., Topology optimization of an IPM motor using NGnet (2015). The numerical values and geometries on this site are results of this project.

BOUNDARY RESOLUTION

Accurate material boundaries through remeshing

Geometry discretization affects local stress evaluation. The NGnet function is re-evaluated at added nodes to resolve material boundaries, rather than merely increasing the element count.

01Refine the background mesh near contours
02Re-evaluate NGnet at new nodes

Material presence is evaluated at the added nodes to locate the boundary.

03Fit the material boundary with MMG

Fillet radius: 0.5 mm; 16 segments per 90° arc. Bonded and separated interfaces are preserved.

04Inspect short edges before stress analysis

Only eligible short free edges near bonded endpoints are snapped, with connectivity and element-quality checks.

References on remeshing and stress evaluation

Takenouchi, Hiruma and colleagues report IPM motor optimization with stress, connectivity, and mesh-quality constraints, using boundary-fitted meshes for accurate magnetic and elastic analyses. We cite their work for its relevance to remeshing and stress evaluation.

K. Takenouchi, S. Hiruma, T. Mifune and T. Matsuo (2024), Topology and parameter optimization of IPM motor considering mechanical strength by stress and connection constraints, COMPEL, 43(4), 839–851.
DOI: 10.1108/COMPEL-11-2023-0551 · Hokkaido University Repository

The validation values below and the optimization results on this site are from this project, not from the cited paper.

03 / REFERENCE TOPOLOGY

Reference motor geometry and analysis conditions

The reference is an 8-pole, 48-slot interior permanent magnet motor. The V-shaped magnet arrangement is fixed while the rotor steel distribution is optimized.

The reference geometry on the right has flux barriers at both ends of each magnet. It was generated from the original CAD and analyzed for magnetic fields and stress at 80 Arms and an axial length of 50 mm.

Magnets
V-shaped arrangement / Remanent flux density 1.25 T
Stack length
50 mm
Magnetic analysis
Phase current 80 Arms · Phase 30° / 0–15°, 17 samples (first excluded)
Stress analysis
12,000 rpm / Plane strain / P2 elements
Magnet interfaces
Only the long face pressed against the steel by centrifugal force is bonded; the other faces are separated and free.

Rotor reconstructed by symmetry. Gray: design steel; red: magnets.

04 / REFERENCE COMPARISON

Optimized and reference geometries compared

A representative final-generation solution (mean torque 110.247 Nm, ripple 4.264 Nm) was selected
and re-evaluated with the reference geometry using 64 intervals.

The reference geometry follows the mesh figure of the IEEJ public motor model and retains the magnet-end flux barriers from the original CAD.

One-pole comparison of reference and optimized geometries
Re-evaluation at 80 Arms, axial length 50 mm, 64 intervals
GeometryMean torque
Nm		Ripple
Nm		Maximum stress
MPa
Reference (with flux barriers)109.07017.368327.916
Optimized geometry110.2634.526298.944

Maximum stress is nodally projected von Mises stress. Values above 300 MPa are highlighted in red.

Compare magnetic flux density, stress distributions, and torque waveforms
Magnetic flux density in reference and optimized geometries
Magnetic flux density at 80 Arms and phase 30°.
Nodally projected von Mises stress in reference and optimized geometries
12,000 rpm · Plane strain · P2 elements. Only the outer long face of each magnet is bonded; the other faces are detached. “Max” is the same nodally projected maximum stress as in the table. The reference uses the original CAD corners; the optimized geometry includes machining fillets.
64-point torque waveforms for reference and optimized geometries
Mean torque is the arithmetic mean; ripple is maximum minus minimum.

Comparison results CSV ↓　64-point torque waveforms CSV ↓

Performance across currents and advance angles

Cogging torque comparison

Cogging torque comparison
Torque waveforms are compared at zero current (0 Arms).

Torque–current advance comparison

Torque–current advance comparison
Mean torque is compared at matching currents over advance angles from 0° to 90°.

Torque ripple at maximum-torque points

Torque ripple at maximum-torque points
For each geometry and current, the maximum-torque point is selected from the evaluated advance angles. Ripple percentage is peak-to-peak torque ripple divided by mean torque, multiplied by 100.

Cogging torque CSV ↓　Torque–advance curves CSV ↓　Ripple at maximum torque CSV ↓

05 / INDUSTRY–ACADEMIA COLLABORATION

Industry–academia collaboration inquiries

We welcome inquiries about industry–academia collaboration on motor topology optimization and design considering magnetic fields and mechanical stress.

Indicative cloud computing costs

Each individual in this optimization required 17 magnetic-analysis steps, remeshing, and one stress-analysis step. Estimated cloud computing costs for 200 generations using services such as AWS are shown below.

Indicative computing costs for these analysis conditions
Individuals per generationGenerationsEstimated computing cost
640 individuals200 generationsApprox. JPY 30,000
1,024 individuals200 generationsApprox. JPY 50,000

These are indicative cloud computing costs. Actual costs depend on the analysis conditions, cloud configuration, and service rates.