Simultaneous Topology Optimization of Stator and Rotor of IPMSM

Takaya Furukawa, Hayato Minamoto, Mitsuru Endo, Yukio Tsutsui, Shimpei Tanaka

The 3rd IFToMM for Sustainable Development Goals Conference I4SDG2025

pp 128–135

Background

This study employs topology optimization to reduce the mass of motors. Minimizing the mass can decrease the required manufacturing resources and lower energy consumption during operation. This approach supports global environmental protection demands, particularly energy usage and metal resource conservation. Previous research has primarily focused on optimizing the cross-sectional shapes of the stator or rotor separately. However, it is essential to note that the optimum design for the stator or rotor is only ideal when considering the corresponding given rotor or stator conditions. Therefore, this study aims to optimize stator and rotor designs simultaneously, providing a more comprehensive solution.

Method

1　Optimization Problem and Optimization Method

Optimization involves two main components: the optimization problem, defined by design variables, constraints, and an objective function, and the optimization method used to solve this problem. The design variables transform into each individual’s motor structure through ``decoding.’’ Electromagnetic field analysis, using the motor structure, yields response values such as torque and mass for the decoded motor. Constraints are applied to these response values to identify the optimal solution, ensuring that only shapes meet or exceed specific thresholds. The objective function assesses each individual’s evaluation based on the response values, indicating how well the motor meets the required performance. The designer must carefully define the objective function to reflect the desired performance and structural characteristics accurately.

This study employs a coding method based on NGnet: Normalized Gaussian networks, which decodes design variables into motor structures[1]. Fig.1 illustrates the decoding process using NGnet in a 1D design area. First, the pink cells represent the mesh used in the Finite Element Method (FEM). Gaussian functions are placed at the representative points of each mesh. NGnet $\phi$ denotes the result of combining these Gaussian functions. Finally, the comparison between $\phi$ and the threshold determines the material distribution within the cells. As shown in the “structure,” the blue cells indicate meshes where Material A is present, while the white cells indicate meshes where Material B is present.

Fig.1 Decoding flow using NGnet on 1D design area

This study has extended a coding method of topology optimization for multiple materials by employing two NGnets with different design values[1]. Fig.2 illustrates the multi-material decoding process, where two NGnets are used simultaneously. With a single NGnet, only two materials can be distinguished based on the relationship between one function and a threshold. However, using two NGnets makes distinguishing up to $2\times2=4$ materials possible, as they can be differentiated based on the relationships between two functions and their respective thresholds.

Fig.2 Multi-material decoding

This study aims to create the lightest possible motor structure while ensuring that the average torque is comparable to the base model’s. The constraint is that the motor needs to produce an average torque exceeding the target torque $\tau_{\mathrm{des}}$ and expressed as: $\tau_{\mathrm{ave},i}>\tau_{\mathrm{des}}$ In optimization, constraints on response values must be assessed based on the values obtained during the optimization process, as the designer cannot directly control these response values. This study incorporates the torque constraint into the objective function when evaluating these constraints. The penalty function related to the torque constraint is expressed as: $f_p(\tau_{\mathrm{ave},i}(w_j)) = \begin{cases} 0 & \text{if } \tau_{\mathrm{ave},i}(w_j) \geq \tau_{\mathrm{des}} \\ k \left( 1 - \dfrac{ \tau_{\mathrm{ave},i}(w_j) }{ \tau_{\mathrm{des}} } \right) & \text{otherwise} \end{cases}$ where $\tau_{\mathrm{ave},i}(w_j)$ represents the average torque of individual $i$, ${\tau_{\mathrm{des}}}$ is the target torque, and $k$ is a coefficient related to convergence.

The objective function of this study is to minimize mass. $f(M_i(w_j))= \frac{M_i(w_j)}{M_{\text{ref }}}$ where $M_i(w_j)$ denotes the mass of the core and magnet for individual $i$. $M_{\text{ref }}$ represents the mass when the core occupies the entire design area. It is important to note that this optimization does not consider the mass of the coils, as their cross-sectional area must be constant.

To summarize the previous discussion, the optimization problem addressed in this study can be articulated as: $\begin{aligned} & \underset{w_j}{\textrm{minimize}} & & f(M_i(w_j))+f_p(\tau_{\mathrm{ave},i}(w_j)) \\ \end{aligned}$ Minimizing the objective with penalty functions results in a lightweight structure that generates an average torque that exceeds the target torque. The optimization method used in this study is the $(\mu/\mu_w,\lambda)\textrm{-CMA-ES}$, developed by Hansen et al. [2].

2　Optimization conditions

Fig.3 illustrates the design areas and the placements of NGnets for both the rotor and stator. In this figure, the blue circles indicate the locations of the NGnets, while the dashed lines represent the symmetry axes.

Fig.3 design area and placement of NGnet

The mass $M_{\text{ref }}$ is $3.75\ \mathrm{kg}$ when the core occupies the entire design area. The torque of the base model, denoted as ${\tau_{\mathrm{des}}}$ is $ 2.01 \ \mathrm{Nm} $. The penalty function gains $k$ is considered for two cases: $k=5$ and $k=10$. Additionally, the population size $\lambda$ for each generation in the $(\mu/\mu_w,\lambda)\textrm{-CMA-ES}$ is examined for two scenarios: $\lambda=28$ and $\lambda=56$.

In electromagnetic field analysis, a forced three-phase alternating current of $3\ \mathrm{A_{rms}}$ flow through the coils, with the current phase angle set to $ 0 ^\circ $. The analysis calculated torque in 16 steps, ranging from an electrical angle of $ 0 ^\circ $ to $ 60 ^\circ $ in increments of $ 3.75 ^\circ $, and the average torque by averaging the results obtained from these steps.

Result

Table.1: Result of Optimization for core, magnet, and coil

Case	base	(a)	(b)	(c)	(d)
λ: Number of individuals for each generation	-	28	28	56	56
k: Gain of penalty function	-	5	10	5	10
EVAL: Evaluation value of objective function	-	0.293	0.295	0.293	0.293
Mass of core and magnet [kg]	3.75	1.14	1.14	1.14	1.14
($S_{\mathrm{coil}}\, [\mathrm{mm^2}]$): Area of coil	424.9	424.9	424.9	424.9	424.9
($\tau_{\mathrm{ave}}\, [\mathrm{Nm}]$): Average torque	2.01	2.01	2.01	2.01	2.01

base	(a)	(b)	(c)	(d)

Figure.4: Result of Optimization for core, magnet, and coil

Table.1 presents the evaluation values of the objective function, the combined mass of the core and magnet, the coil’s cross-sectional area, and the average torque under each optimization condition. Fig.4 illustrates the resulting motor structures, with red arrows and blue arrows indicating the magnetization direction of Magnet 1 and Magnet 2, respectively. Table.1 and Fig.4 display the most favorable results for each condition after multiple optimization attempts.

In all cases, the average torque successfully reaches 100\% of the target torque, and the cross-sectional area of the coil achieves 99.6\% of the target area. The combined mass of the core and magnets is $1.14\, \mathrm{kg}$, which is significantly lighter than the $3.75 \,\mathrm{kg}$ mass of the core and magnets in the base model.

The arrangement of magnets in the rotor resembles a Halbach array, where the magnets are oriented to concentrate flux on one side. This design directs the flux primarily towards the stator, resulting in minimal leakage on the shaft side. Consequently, high-density magnetic flux is effectively funneled to the stator, even without a rotor core on the shaft side, contributing to reduced weight.

The stator’s structure is designed like a coreless motor without teeth, allowing the magnetic flux to flow freely through the coils. While this structure often challenges effective flux transmission through a traditional stator core, the Halbach array allows for efficient flux transfer in this configuration. In fact, coreless motors have been practically implemented. However, incorporating Halbach array magnets into motors remains challenging due to manufacturing constraints.

Conclusion

This study proposes a simultaneous topology optimization approach for the rotor and the stator. As a result, the combined mass of the core and magnets is lighter than that of the base model and lighter than that of individually optimized motor structures. Furthermore, the study found that combining a rotor with a Halbach array of magnets and a stator with a coreless motor coil is highly effective for weight reduction.

参考文献 - References

[1] Sato, T., Watanabe, K., Igarashi, H.: Multimaterial topology optimization of electric machines based on normalized Gaussian network. IEEE Transactions on Magnetics 51(3), 1–4 (2015). 10.1109/TMAG.2014.2359972

[2] Hansen, N., Ostermeier, A.: Completely Derandomized Self-Adaptation in Evolution Strategies. Evolutionary Computation 9(2), 159–195 (2001). 10.1162/106365601750190398

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