The model underneath
Simulation reads the material assignment. Optimization writes it. Everything else is a question of who may write where.
A rotor in this workflow is a finite element mesh. Every element carries one material: permanent magnet, electrical steel, or air. The electromagnetic solver reads that assignment and returns torque, induced voltage and losses; a mechanical solver reads the same assignment and returns stresses and displacement.
Topology optimization is the process that writes into that assignment and keeps whatever scores well. So the design space is not an abstract setting. It is the concrete list of elements the optimizer may touch, and the set of materials it may put there.
Widening that list is the single biggest lever on what a search can find, and on what it costs to run. Below are the three settings we use, in the order we usually try them.
Why air counts as a material
The pockets are doing work.
Before the three variants make sense, one piece of motor physics has to be on the table: the empty regions in a rotor are there on purpose, and getting their shape right is most of what the first variant is for.
A buried magnet drives flux that is supposed to cross the airgap and do work against the stator. Some of it takes a shortcut instead, curling around the ends of the magnet through the surrounding steel and returning to the magnet without ever leaving the rotor. That share produces no torque. It is paid for in magnet material and delivered nowhere.
Cutting air into the steel at the magnet ends closes the shortcut. Air has roughly the permeability of vacuum, so flux will not cross a pocket if a steel path exists anywhere else. Place the pockets well and a larger share of the same magnet ends up in the airgap.
The pockets cannot simply be extended to the rotor surface. A thin steel bridge has to survive between the pocket and the outside, or the pole separates under centrifugal load. In the research case behind this method, every candidate had to hold at 120 percent of maximum speed.
So bridge thickness is a direct trade: thinner leaks less and breaks sooner. It is a small number with a large effect, it interacts with pocket shape, and it is the kind of decision an optimizer settles more convincingly than a rule of thumb carried over from the last machine.
Three design spaces
Same model, same physics, three different permissions.
All three run on one rotor model and one set of requirements. What changes is the list of things the search may write to. Grey is electrical steel, violet is magnet, white is air.
Magnet locked
Fixed magnet
Your magnet, its supplier and its tooling stay untouched. The optimizer reshapes only the steel around it, opening air pockets that steer the flux into work instead of leakage. The cheapest efficiency you can buy once the line is running.
Fastest search
Magnet adjustable
Parameterized magnet
The magnet stays a rectangle but may move, widen or flatten while the steel and air follow it. That opens the trade nobody can judge by hand: how little magnet volume still delivers the torque you need.
Wider search
Everything open
Free material layout
Nothing is prescribed. Magnet, steel and air settle wherever they serve the target best, and the magnet takes the shape the physics rewards, usually one no catalog has a name for. The widest search, and the one with the most room to win.
Widest search · most compute
01
Fixed magnet
Design variables
One binary choice per element: steel or air
The magnet keeps the position, size and shape it already has. Every other element inside the design region is a yes-or-no question: steel, or air.
This is the variant for a machine that exists. The tooling is paid off, the magnet is on a supply contract, and the lamination is the only part anyone is willing to reopen. The search still has somewhere to go. Pocket geometry at the magnet ends decides how much flux leaks, and leakage decides how much torque you get out of the magnet volume you are already buying.
It is also the cheapest of the three to run, which makes it a reasonable first attempt even on projects that could afford more.
02
Parameterized magnet
Design variables
The same, plus four continuous magnet variables
The magnet stays rectangular and gains four numbers the optimizer may set: two for position, two for size. The steel and air around it are optimized as before, and the mesh is rebuilt to fit whatever rectangle the magnet has become.
That lets the search ask a question the first variant cannot: how little magnet volume still meets the torque requirement at every operating point. Permanent magnets are the most expensive material in the machine, so the answer tends to justify the extra compute on its own.
Moving the magnet radially does something less obvious as well. It changes how much steel sits between the magnet and the airgap, which shifts saliency and with it the machine’s behavior under field weakening. The rectangle survives, so the result is still something a supplier can quote and a press can seat in a slot.
03
Free material layout
Design variables
A three-way choice at every element
There is no magnet primitive left. Every element in the design region may become magnet, electrical steel or air, and the boundaries between them are wherever the optimization puts them.
The magnet stops being an object that gets moved and becomes a region that gets drawn. Results curve, because magnet material placed along the flux path contributes more than the same volume squared off into a block. Air arranges itself into barrier shapes that would take an experienced designer several attempts to guess and a junior one considerably longer.
This is the largest space and the most compute per useful answer. It is also the only one of the three that can return a geometry nobody on the project had thought of, which is usually the reason it gets used at all.
Choosing between them
A wider space is a better space. It is not automatically a better result.
The third variant contains every design the second one contains, and the second contains every design the first one contains. Its best possible answer can therefore never be worse. That is a fact about the space, and it says nothing about the search that has to cross it.
Optimization runs on a budget. A stochastic optimizer given the same number of evaluations has to spread them over a much larger set of possibilities, and a free layout that ran out of budget halfway can easily hand back a worse candidate than a narrow, well-chosen space that converged. More freedom buys a higher ceiling and a slower climb.
Which is why we start small. Pick the narrowest space that could plausibly contain the target and run it. If the result presses against the edge of what it was allowed to change, that is the signal to open the next variant on the same model, and the earlier run has already paid for itself as a benchmark.
The reverse case is worth naming too. A free layout that produces something close to a rectangle is not a wasted run. It is evidence that the geometry your project already builds was a good choice, which is a useful thing to be able to demonstrate.
Remeshing
Every candidate gets its own mesh.
Density-based topology optimization on a fixed grid leaves elements at intermediate values. An element that comes out 40 percent magnet is not a manufacturing instruction, and the filtering needed to force those values back to something real blurs the boundaries it acts on.
Rebuilding the mesh for each candidate keeps the material boundaries where the optimizer put them. It also keeps thin features resolved, which matters more than it sounds: the bridges and webs between pockets are exactly where the mechanical check either passes or fails, and a mesh too coarse to represent a bridge cannot tell you whether it holds.
The cost is that meshing sits inside the loop rather than before it. In exchange, a candidate that scores well is a candidate you can send to a laser cutter without rediscovering it from a density field first.
What none of the three may touch
The design space gets wider. The requirements do not get softer.
Opening a variant changes what the optimizer may propose. It changes nothing about what a proposal has to survive before it reaches an engineer.
Interfaces stay where the project fixed them: shaft, airgap, stator bore, package. Every candidate has to reach the required torque at every operating point, stay inside the voltage and current the inverter can supply, hold mechanically at overspeed, and survive the transient demagnetization case.
Geometry has to be producible: minimum feature sizes, radii a cutter can actually follow, pockets that do not collapse a lamination. Candidates that break one of these are scored out inside the loop. What arrives at the end of a run is a shortlist of designs that already passed, not a gallery of shapes waiting to be checked.
In practice
Where this has already run.
The doctoral work behind this method applied it to the rotor of a permanent-magnet synchronous machine, optimized for efficiency across a full vehicle drive cycle and for magnet volume at the same time. The selected geometry used 10 percent less magnet material and 2.4 percent less rotor mass than the reference, at the same simulated drive-cycle efficiency.
Two variants were then laser cut, bonded into stacks, fitted with magnets and measured on a test bench. That last step is what separates a design space from a drawing exercise.
Read the full research case
About the author
Dr. Alexander Schugardt
Alexander is CTO and co-founder of Neuraway AI. His doctoral work covered multi-criteria shape and topology optimization of permanent-magnet synchronous machines across drive cycles.
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