Multi-material topology optimization of permanent magnet synchronous motors

Abstract

This paper presents a multi-material topology optimization for the design of permanent magnet synchronous motors (PMSMs). Specifically, structural shapes of permanent magnet (PM) and iron core in a PMSM rotor are simultaneously designed together with the orientation of PM magnetization. For a co-design of PM and iron core, relative permeability and residual magnetic flux density are interpolated by the three-field density approach based on the Helmholtz filtering and regularized Heaviside step function. Here, the Helmholtz filtering aims to attain smooth border in design results, and the Heaviside function enables us to acquire a clear border (i.e. zero-one design) without intermediate densities. The optimization goal is set as maximizing the average torque of PMSMs. The average torque is calculated by Maxwell stress tensor (MST) method considering a maximum torque per ampere (MTPA) control. To validate the effectiveness of the proposed multi-material topology optimization approach, a PMSM rotor with 4 poles and 12 slots is designed. In addition, design results at various settings of input current amplitude and PM strength are compared and discussed. When the input current is stronger than the PM strength, rotor PM and iron core are designed for utilizing both PM and reluctance torque components like V-shape interior PMSMs. On the other hand, in the case of stronger PM strength, PM is designed near the air-gap, which utilizes only PM torque component like surface PMSMs.

Keywords

Topology optimization permanent magnet synchronous motors (PMSMs)permanent magnets (PMs)iron finite element methods

1. Introduction

Topology optimization is a numerical design tool that aims to determine an optimal material distribution for a given design goal. It can find optimal structural shapes and configurations through numerical analyses and optimization algorithms. After first being proposed in [1] for structural stiffness problem, topology optimization has been successfully extended to vibration, thermal fluid and electromagnetic problems [2].

Recently, studies on topology optimization of electric motors has been actively carried out. A rotor shape of switched reluctance motors (SRMs) was designed for torque ripple reduction in [3]. In [4–8], structural rotor topologies of synchronous reluctance motors (SynRMs) were optimized for improving torque characteristics considering mechanical performance or iron losses. In interior or surface permanent magnet synchronous motors (PMSMs), rotor or stator iron cores were designed using topology optimization in [9–12]. Most of previous works on topology optimization of electric motors have dealt with the design of only iron (i.e. soft ferromagnetic) part. In PMSMs, both shapes of permanent magnet (PM) and iron core affect their performances significantly. Thus, it is evident that the co-design of both materials is required in PMSMs.

Accordingly, the present work aims to propose a multi-material topology optimization scheme for the simultaneous design of PM and back iron in PMSMs. Few studies have been conducted for the multi-material topology optimization in electric motors. In [13,14], PM and iron core shapes of PMSM rotors were co-designed using non-gradient topology optimization based on genetic algorithms. Non-gradient topology optimization is easy to implement, but it is computationally inefficient. Thus, the number of design variables is limited, which result in a low-resolution design result [15]. A gradient-based multi-material topology optimization was applied to the design of electric motors in [16,17]. In [16], copper and iron materials were co-designed in wound field synchronous motors (WFSMs). In [17], PM and iron materials were simultaneously designed in PMSMs. However, a PM design result in [17] has severe oscillations near the air-gap, which results in a scattered layout.

Fig. 1.

Multi-material topology optimization for the rotor design of PMSMs.

In the present work, the three-field density based topology optimization [18] is applied for the co-design of PM and iron materials in the rotor of PMSMs. Figure 1 illustrates the design problem of a PMSM rotor that is addresses in the present work. In the design domain 𝛺_d, the distribution of air, PM and iron materials are determined by the proposed topology optimization approach. Two design variable fields are defined in the design domain, and they are projected into two density fields through a Helmholtz filtering and regularized Heaviside step function [19,20]. Here the Helmholtz filtering aims to avoid oscillating design result, and the Heaviside function is applied to acquire clear borders in the design result (i.e. zero-one solution). Two density fields controls the relative permeability and PM residual magnetic flux density, which determines material states among PM, iron, or air. In addition, the orientation of PM magnetization is also designed during a topology optimization procedure. The optimization goal is set as the maximization of average torque, which is calculated using the Maxwell stress tensor (MST) method [21]. Here, the maximum torque per ampere (MTPA) control is implemented [11] by updating the current angle using the golden section search algorithm. It is noted that studies in [22–24] demonstrates that the curved complex shapes of PM and iron that are obtained by topology optimization can be fabricated by conventional or additive manufacturing techniques.

The remaining paper is organized as follows. Section 2 explains the multi-material interpolation scheme that is proposed for the co-design of PM and iron materials. In Section 3, the torque analysis of PMSM is described. Then, the optimization strategy is explained in Section 4. Here, a design domain is defined, and an optimization problem is formulated. In Section 5, PMSM design results are provided and discussed. Finally, the conclusion is provided in Section 6.

2. Multi-material interpolation scheme

This section explains a material interpolation scheme for multi-material layout design. In this scheme, a material state at position $\vec{x}$ is determined by a design variable vector fields $\vec{{\phi}}(\vec{x})$ , which is defined as $\begin{eqnarray}\vec{{\phi}}(\vec{x})=\left[\begin{array}{@{}c@{}}{\phi}_{1}(\vec{x})\\ {\phi}_{2}(\vec{x})\end{array}\right],\end{eqnarray}$ (1) with bounds ${\phi}_{1}(\vec{x})\in [-1,1]$ and ${\phi}_{2}(\vec{x})\in [-1,1]$ . A raw design variable vector may suffer from grayscale and checkerboard problems, which result in unclear borders and scattered layout in a design result. To prevent these problems, the three-field density formulation [18] is applied in the present work. In this formulation, a Helmholtz filtering [19,20] is first applied to a design variable vector field $\vec{{\phi}}(\vec{x})$ for smooth material layouts: $\begin{eqnarray}-{R_{{\phi}}}^{2}\left[\begin{array}{@{}c@{}}{\nabla}^{2}\tilde{{\phi}_{1}}(\vec{x})\\ {\nabla}^{2}\tilde{{\phi}_{2}}(\vec{x})\end{array}\right]+\left[\begin{array}{@{}c@{}}\tilde{{\phi}_{1}}(\vec{x})\\ \tilde{{\phi}_{2}}(\vec{x})\end{array}\right]=\left[\begin{array}{@{}c@{}}{\phi}_{1}(\vec{x})\\ {\phi}_{2}(\vec{x})\end{array}\right],\end{eqnarray}$ (2) where R _𝜙 is a parameter for filter radius. Next, a material density vector field, $\vec{{\rho}}(\vec{x})$ , is defined as $\begin{eqnarray}\vec{{\rho}}(\vec{x})=\left[\begin{array}{@{}c@{}}{\rho}_{1}(\vec{x})\\ {\rho}_{2}(\vec{x})\end{array}\right]=\left[\begin{array}{@{}c@{}}H_{r}(\tilde{{\phi}_{1}}(\vec{x}))\\ H_{r}(\tilde{{\phi}_{2}}(\vec{x}))\end{array}\right],\end{eqnarray}$ (3) where H _r is a regularized Heaviside step function. This function aims to obtain a zero-one design result with clear border, and defined as $\begin{eqnarray}H_{r}({\tau})=\left\{\begin{array}{@{}ll@{}}0\quad & ({\tau}<-h)\\ \displaystyle \frac{1}{2}+\frac{15}{16}\left(\frac{{\tau}}{h}\right)-\frac{5}{8}\left(\frac{{\tau}}{h}\right)^{3}+\frac{3}{16}\left(\frac{{\tau}}{h}\right)^{5}\quad & (-h\leq {\tau}\leq h)\\[8.0pt] 1\quad & (h<{\tau})\end{array}\right.,\end{eqnarray}$ (4) where h is a bandwidth parameter. Finally, a material state is determined by the material density vector field $\vec{{\rho}}(\vec{x})$ . For this, material properties (i.e., the relative magnetic permeability μ_r, and the residual magnetic flux density $\vec{B}_{r}$ ) are interpolated as $\begin{eqnarray}\displaystyle {\mu}_{r} & = & \displaystyle {\rho}_{1}(\vec{x})^{p}({\rho}_{2}(\vec{x})^{p}{\mu}_{iron}+(1-{\rho}_{2}(\vec{x})^{p}){\mu}_{\mathit{PM}})+(1-{\rho}_{1}(\vec{x})^{p}){\mu}_{air},\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \vec{B}_{r} & = & \displaystyle \vec{B}_{\mathit{PM}}{\rho}_{1}(\vec{x})^{p}(1-{\rho}_{2}(\vec{x})^{p}),\end{eqnarray}$ (6) where p is a density penalty parameter, which is set to 3 in the present work. In (5), μ_iron, μ_PM, and μ_air are the relative magnetic permeability of iron, PM, and air respectively. In (6), $\vec{B}_{PM}$ is the residual magnetic flux density of PM. It is noted that the saturation effect of soft ferromagnetic materials is considered in the present work, and thus μ_iron is set as a nonlinear function with respect to the magnetic flux density.

Through material property interpolations in (5)–(6), a material density vector variable $\vec{{\rho}}(\vec{x})$ determines three material states (i.e. Iron, PM, air) at position $\vec{x}$ . Table 1 summarizes material properties and corresponding material states when the material density variables 𝜌₁ and 𝜌₂ are either 0 or 1.

Table 1

Material states defined by design variables

𝜌₁	𝜌₂	μ_r	$\vec{B}_{r}$	Material state
1	1	μ_iron	0	Iron
1	0	μ_PM	$\vec{B}_{PM}$	PM
0	1	μ_air	0	Air
0	0

The residual magnetic flux density of PM, $\vec{B}_{PM}$ in (6) can be represented as $\begin{eqnarray}\vec{B}_{\mathit{PM}}=\bar{B}_{\mathit{PM}}\left[\begin{array}{@{}c@{}}\cos ({\theta}_{\mathit{PM}})\\ \sin ({\theta}_{\mathit{PM}})\end{array}\right],\end{eqnarray}$ (7) where $\bar{B}_{PM}$ is the magnetization strength of PM. For the design of PM magnetization orientation, a design variable θ ∈ [−1,1] is defined. In the present work, θ_PM in the design domain is confined as [π,3π∕4] considering the symmetry of PMSM. Thus, the PM magnetization angle, θ_PM, is defined as $\begin{eqnarray}{\theta}_{\mathit{PM}}={\pi}-\frac{{\pi}}{8}({\theta}+1).\end{eqnarray}$ (8)

3. Torque analysis

This section describes the torque analysis used to calculate the average torque of PMSM. The first step is the calculation of magnetic field distribution that can be obtained by solving the governing equation. The governing equation of a magnetostatic analysis including PM and external current is derived from the Maxwell’s equation as $\begin{eqnarray}{\nabla}\times \left(\frac{1}{{\mu}_{0}{\mu}_{r}}{\nabla}\times \vec{A}\right)=\vec{J}+{\nabla}\times \left(\frac{1}{{\mu}_{0}{\mu}_{r}}\vec{B_{r}}\right),\end{eqnarray}$ (9) where μ₀ is the magnetic permeability of air, $\vec{A}$ is the magnetic vector potential, and $\vec{J}$ is the external current density. By solving (9), the magnetic flux density $\vec{B}$ is obtained as $\begin{eqnarray}\vec{B}={\nabla}\times \vec{A}.\end{eqnarray}$ (10) Next, the torque of the PMSM is calculated by the Maxwell’s stress tensor (MST) method [21] as $\begin{eqnarray}T=\frac{L_{Fe}}{{\mu}_{0}}p\oint rB_{r}B_{{\vartheta}}dS,\end{eqnarray}$ (11) where L _Fe is the active length of the core, p is the number of pole pairs, and r is the radius of integration path. B _r and B _𝜗 are the radial and azimuthal components of the magnetic flux density, $\vec{B}$ obtained from (9) and (10). The average torque T _avg is then obtained from torque values calculated at evenly distributed rotor positions: $\begin{eqnarray}T_{avg}=\frac{1}{N}\mathop{\sum }_{i=1}^{N}T_{i},\end{eqnarray}$ (12) where T _i is the torque at i-th rotor position, respectively. In (12), N is total number of rotor positions, which is set to 8 in the present work. Therefore, the average torque T _avg (i.e. objective function) is calculated by performing the finite element analysis 8 times.

The external current density $\vec{J}$ in (9) is determined from input currents of three phase windings, I _A, I _B, and I _C, which are set as: $\begin{eqnarray}\displaystyle I_{A}({\psi}_{i},{\gamma}) & = & \displaystyle I_{m}\cos ({\psi}_{i}+{\gamma}),\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle I_{B}({\psi}_{i},{\gamma}) & = & \displaystyle I_{m}\cos \left({\psi}_{i}+{\gamma}-\frac{2}{3}{\pi}\right),\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle I_{C}({\psi}_{i},{\gamma}) & = & \displaystyle I_{m}\cos \left({\psi}_{i}+{\gamma}-\frac{4}{3}{\pi}\right),\end{eqnarray}$ (15) where I _m is the amplitude of input current, 𝜓_i is an electrical angle of rotor position, and 𝛾 is a current angle. Here, the current angle 𝛾 means the relative angle between the magnetic field generated by a rotor PM, and the magnetic field by a stator input current. Figure 2 shows an example plot of average torque with respect to current angle 𝛾. As shown in Fig. 2, a maximum average torque is generally located where current angle 𝛾 is between 45° and 90° [25]. Based on the maximum torque per ampere (MTPA) control, the current angle 𝛾 is updated during the optimization iterations.

Fig. 2.

Example plot of average torque T _avg with respect to current angle 𝛾.

4. Optimization strategy

This section presents an optimization strategy for co-design of iron and PM in a PMSM rotor. Figures 3 and 4 show the design domain and finite elements of the design model in the present work, respectively. A PMSM is composed of 4 poles and 12 slots. To reduce a computational cost for torque analysis, a half analysis model is built, and the design domain is set to a 1/8 region of whole rotor area. The design variable vector field $\vec{{\phi}}(\vec{x})$ and the PM orientation design variable θ are assigned at the design domain. Then, they are reflected and copied to the remaining part of rotor area. It is noted that the direction of the PM magnetization (i.e., outward and inward) is reversed at every 1/4 region of the rotor since the PMSM has 4 poles. The total number of elements and degrees of freedom are 9,382 and 17,591 in the design model, respectively.

Fig. 3.

Design domain for the PMSM.

Fig. 4.

Finite element mesh in the design model.

The relative permeability of the PM material, μ_PM, is set to 1.05. The nonlinear BH relationship of the silicon steel NGO 50PN 470 is used in the iron material. For this, the simultaneous exponential extrapolation (SEE) approach is applied [26]. The BH curve can be easily estimated using interpolation on existing data region. However, it is important to extrapolate in over-magnetic flux region for nonlinear magnetic problem. The BH curve beyond the existing data is expressed as $\begin{eqnarray}B(H)=B_{s}(1-ae^{-bH})+{\mu}_{0}H,\end{eqnarray}$ (16) where a and b are coefficients of SEE and B _s is the saturation magnetic flux density. They are estimated from the last several data of the sample data. Figure 6 shows the BH curve graph used in the present work.

This work aims to design a PMSM rotor that maximizes average torque T _avg under MTPA control. For this, an optimization problem is formulated as $\begin{eqnarray}\displaystyle \text{Find}\text{} & \text{} & \displaystyle \quad \vec{{\phi}}(\vec{x}),∼\text{and}∼{\theta}\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{Maximize}\text{} & \text{} & \displaystyle \quad T_{avg}(\vec{{\phi}}(\vec{x}),{\theta},{\gamma})\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{Subject to}\text{} & \text{} & \displaystyle \quad \vec{K}(\vec{{\phi}}(\vec{x}))\vec{A}=\vec{f}(\vec{{\phi}}(\vec{x}),{\theta},{\gamma})\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \quad V_{\mathit{PM}}(\vec{{\phi}}(\vec{x}))\leq V_{\mathit{PM}}^{\ast }.\end{eqnarray}$ (20) The objective function (18) is set as the average torque T _avg. The system of equation in (19) is composed of the stiffness matrix $\vec{K}$ and the force vector $\vec{f}$ . In (20), the volume of PM V _PM is restricted less than $V_{PM}^{\ast }$ that is set to 30% of the design domain. The PM volume, V _PM, is calculated from two material density variables 𝜌₁ and 𝜌₂: $\begin{eqnarray}V_{\mathit{PM}}(\vec{{\phi}}(\vec{x}))=\int _{{\Omega}_{D}}{\rho}_{1}(\vec{x})(1-{\rho}_{2}(\vec{x}))dV.\end{eqnarray}$ (21)

The optimization problem is solved using the Globally Convergent Method of Moving Asymptotes (GCMMA) [27,28], implemented in MATLAB with COMSOL V5.5. In the GCMMA, the maximum number of inner iterations is set as 5. Based on the MTPA control, the current angle 𝛾 is updated for every 3 optimization iterations using the golden section search algorithm [29]. A continuation scheme is applied for the bandwidth parameter h in (4). For relaxation, the parameter h is fixed as 1 until the convergence criterion is satisfied. In the present work, the convergence criterion is defined as relative errors between objective function values in last two iterations, which set to 0.001. After then, The parameter h decreases by 0.05 for every 3 optimization iterations. By this work, the material density field converge into a zero-one design value for clear borders. The optimization iteration stops when the parameter h decreases by less than 0.35, that is empirically determined value. The flow chart of proposed topology optimization process is summarized in Fig. 5.

Fig. 5.

Flow chart of proposed multi-material topology optimization procedure.

Fig. 6.

BH curve of the silicon steel NGO 50PN470.

Fig. 7.

Design result when the input current amplitude, I _m, is 20 A and PM strength, $\bar{B}_{PM}$ , is 0.8 T: (a) Full geometry, and (b) 1/4 rotor geometry. In (a), white-colored lines represent the equipotential lines of magnetic field. In both (a) and (b), colored region represent permanent magnet blocks with different magnetization directions that is indicated by white-colored arrows.

5. Design result

Figure 7 presents a PMSM rotor design result that is obtained from the proposed multi-material topology optimization approach. This result is attained by setting the amplitude of input current, I _m, as 20 A, and the strength of PM magnetization, $\bar{B}_{PM}$ , as 0.8 T. Figure 7(a) and (b) show the full geometry and 1/4 rotor part of the design result. The white contour lines in Fig. 7(a) represent the equipotential lines of the magnetic flux density when the electrical angle 𝜓_i = 0. In Fig. 7, white-colored arrows indicate the optimized orientation of PM magnetization, θ_PM. In the design result, the PM segments and iron core are simultaneously designed so that a magnetic field can interact well without leakage between the rotor and stator. A clear and smooth border of PM and iron materials is successfully determined through the proposed multi-material topology optimization scheme. Figure 8(a) and (b) show the history of the objective and constraint functions during optimization iterations. The total numbers of external and inner iterations in the GCMMA optimizer are 60 and 177, respectively. Thus, the objective function is called a total of 237 times during the optimization process. The objective function of the final design is calculated as 3.52 Nm, and PM volume is well constrained less than 30% of the design domain (i.e. 0.3). In the final design, the orientation of PM magnetization, θ_PM, and the current angle 𝛾 under MTPA control are determined as 17.96° and 69.4° respectively.

Fig. 8.

Convergence history of (a) objective function T _avg and (b) PM volume V _PM.

In a PMSM, the net torque consists of PM torque component and reluctance torque component. PM torque component is the interaction between PMs and the stator current. In general, the PMs are placed near air-gap like a surface mounted PMSM for increasing PM torque component. In contrast, reluctance torque is the interaction between the iron structure and the stator current. The iron structure with salience, like a synchronous reluctance motor, has higher reluctance torque than a smooth circular structure. Therefore, observing the geometry of the rotor core provides clues to predict the dominant torque component.

To investigate the effect of input current amplitude, I _m, and PM strength, $\bar{B}_{PM}$ on the design, the geometry of rotor core is observed. Figure 9 show the rotor design results obtained at various setting of I _m and $\bar{B}_{PM}$ . For fair comparison, all parameters are set the same except the filter parameter R _𝜙 in (2). The parameter R _𝜙 is set differently to consider manufacturability in each design. Figure 9(a) and (d) are the design results when the stator input current is relatively stronger than rotor PM strength. In these cases, iron cores are designed to achieve high salience ratio, which results in high reluctance torque component. PMs are designed as V-shape around iron core. On the other hand, Fig. 9(b) and (c) show the rotor design results when rotor PM strength is much stronger than stator input current. In these cases, PMs are designed near the air-gap like surface PMSMs, which maximizes a PM torque component. Iron core is located below PM for guiding magnetic field generated by PM.

Table 2

Comparison of design results obtained at various settings of I _m and $\bar{B}_{PM}$

	Fig. 9(a)	Fig. 9(b)	Fig. 9(c)	Fig. 9(d)
Input current I _m	20 A	20 A	10 A	30 A
Strength of PM $\bar{B}_{PM}$	0.2 T	1.4 T	0.8 T	0.8 T
Orientation of PM θ_PM	23.17°	0°	0°	23.93°
Average torque T _avg	1.81 Nm	4.84 Nm	1.55 Nm	5.33 Nm
Current angle 𝛾	61.3°	85.9°	89.7°	61.3°

Fig. 9.

Rotor design results obtained with (a) I _m = 20 A, $\bar{B}_{PM}=∼0.2$ T, (b) I _m = 20 A, $\bar{B}_{PM}=∼1.4$ T, (c) I _m = 10 A, $\bar{B}_{PM}=$ 0.8 T, (d) I _m = 30 A, $\bar{B}_{PM}=∼0.8$ T.

Also, the current angle of MTPA 𝛾 is an important measure for indirectly finding torque component ratio. In general, PM and reluctance torque are maximized at 90^° and 45^°, respectively. Thus, the closer the current angle 𝛾 is to 90^°, the higher the PM torque component ratio. Of course, the closer the current angle 𝛾 is to 45^°, the higher the reluctance torque component ratio.

Table 2 compares the PM orientation θ_PM, average torque, T _avg, and current angle, 𝛾, of the design results at various settings of I _m and $\bar{B}_{PM}$ . The current angle 𝛾 of Fig. 9(a) and (d) is calculated around 60°, which means that both reluctance and PM torque components are utilized like V-shape interior PMSMs. On the other hand, the current angle 𝛾 of Fig. 9(b) and (c) is calculated around 90°, which means that only PM torque component is utilized like surface PMSMs.

6. Conclusion

The present work aims to propose a multi-material topology optimization approach for the simultaneous design of PM and iron core in PMSMs. The proposed approach can determine structural shapes and configurations of PM and iron materials together with the orientation of PM magnetization. To prevent oscillating design result, Helmholtz filtering is applied to density design variables. In addition, a regularized Heaviside step function with a continuation scheme is utilized to achieve clear border without intermediate densities. The average torque (i.e. objective function) is calculated using the MST method. To implement the MTPA control, the current angle of the input current is determined by the golden section search algorithm. The rotor of a PMSM with 4 poles and 12 slots is designed by the proposed topology optimization approach. As expected, PM and iron core are successfully designed into shapes with clear and smooth borders. In addition, the effect of input current amplitude and PM strength on design result is investigated. When the stator input current is relatively stronger than the rotor PM strength, rotor PM and iron core are designed for utilizing both PM and reluctance torque components like V-shape interior PMSMs. When the rotor PM strength is much stronger than stator input current, PM is designed near the air-gap, which utilizes only PM torque component like surface PMSMs. Future work will focus on the fabrication and experimental validation of the design result.

Footnotes

Acknowledgement

This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2019R1A2C1002808), and Korea Institute for Advancement of Technology(KIAT) grant funded by the Korea Government(MOTIE) (P0008763, The Competency Development Program for Industry Specialist).

References

Bends{\o}e

M.P.

and Kikuchi

, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng. 71(2) (1988), 197–224.

Dede

E.M.

Nomura

and Lee

, Multiphysics Simulation: Electromechanical System Applications and Optimization, Springer, Berlin, 2014.

Lee

Seo

J.H.

and Kikuchi

, Topology optimization of switched reluctance motors for the desired torque profile, Struct. Multidisciplinary Optim. 42(5) (2010).

Sato

and Igarashi

, Topology optimization of synchronous reluctance motor using normalized gaussian network, IEEE Trans. Magn. 51(3) (2015), 1–4, Art no. 8200904.

Okamoto

Hoshino

Wakao

and Tsuburaya

, Improvement of torque characteristics for a synchronous reluctance motor using MMA-based topology optimization method, IEEE Trans. Magn. 54(3) (2018), 1–4, Art no. 7203104.

Yamashita

and Okamoto

, Design optimization of synchronous reluctance motor for reducing iron loss and improving torque characteristics using topology optimization based on the level-set method, IEEE Trans. Magn. 56(3) (2020), 1–4, Art no. 7510704.

Guo

and Brown

I.P.

, Simultaneous magnetic and structural topology optimization of synchronous reluctance machine rotors, IEEE Trans. Magn. 56(10) (2020), 1–12.

Credo

Fabri

Villani

and Popescu

, Adopting the topology optimization in the design of high-speed synchronous reluctance motors for electric vehicles, IEEE Trans. Ind. Electron. 56(5) (2020), 5429–5438.

Choi

J.S.

Izui

Nishiwaki

Kawamoto

and Nomura

, Topology Optimization of the stator for minimizing cogging torque of IPM motors, IEEE Trans. Magn. 47(10) (2011), 3024–3027.

10.

Sasaki

and Igarashi

, Topology optimization accelerated by deep learning, IEEE Trans. Magn. 55(6) (2019), 1–5, Art no. 7401305.

11.

Hong

S.G.

and Park

I.H.

, Continuum-sensitivity-based optimization of interior permanent magnet synchronous motor with shape constraint for permanent magnet, IEEE Trans. Magn. 56(2) (2020), 1–4, Art no. 7505504.

12.

Zheng

Lei

Zhu

Jin

and Guo

, Topology optimization of ferromagnetic components in electrical machines, IEEE Trans. Energy Convers. 35(2) (2020), 786–798.

13.

Ishikawa

Nakayama

Kurita

and Dawson

F.P.

, Optimization of rotor topology in PM synchronous motors by genetic algorithm considering cluster of materials and cleaning procedure, IEEE Trans. Magn. 50(2) (2014), 637–640.

14.

Sato

Watanabe

and Igarashi

, Multimaterial topology optimization of electric machines based on normalized Gaussian network, IEEE Trans. Magn. 51(3) (2015), 1–4, Art no. 7202604.

15.

Sigmund

, On the usefulness of non-gradient approaches in topology optimization, Struct. Multidisciplinary Optim. 43(5) (2011), 589–596.

16.

Guo

Salameh

Krishnamurthy

and Brown

I.P.

, Multimaterial magneto-structural topology optimization of wound field synchronous machine rotors, IEEE Trans. Ind Appl. 56(4) (2020), 3656–3667.

17.

Guo

and Brown

I.P.

, Magneto-structural combined dimensional and topology optimization of interior permanent magnet synchronous machine rotors, in: Proc. IEEE Energy Convers. Congr. Expo., 2020, pp. 1399–1406.

18.

Sigmund

and Maute

, Topology optimization approaches, Struct. Multidisciplinary Optim. 48(6) (2013), 1031–1055.

19.

Kawamoto

Matsumori

Yamasaki

Nomura

Kondoh

and Nishiwaki

, Heaviside projection based topology optimization by a PDE-filtered scalar function, Struct. Multidisciplinary Optim. 44(1) (2011), 19–24.

20.

Lazarov

B.S.

and Sigmund

, Filters in topology optimization based on Helmholtz-type differential equations, Int. J. Numer. Methods. Eng. 86(6) (2011), 765–781.

21.

Meessen

K.J.

Paulides

J.J.H.

and Lomonova

E.A.

, Force calculations in 3-D cylindrical structures using fourier analysis and the maxwell stress tensor, IEEE Trans. Magn. 49(1) (2013), 536–545.

22.

Jung

Lee

and Lee

, Design and fabrication of magnetic system using multi-material topology optimization, IEEE Access 9 (2021), 8649–8658.

23.

Wrobel

and Mecrow

, A comprehensive review of additive manufacturing in construction of electrical machines, IEEE Trans. Energy Convers. 35(2) (2020), 1054–1064.

24.

Pham

Kwon

and Foster

, Additive manufacturing and topology optimization of magnetic materials for electrical machines—A review, Energies 14(2) (2021), 283.

25.

Liu

Zhao

Chen

and Du

, Principle of Torque-Angle approaching in a hybrid rotor permanent-magnet motor, IEEE Trans. Ind. Electron. 66(4) (2019), 2580–2591.

26.

Rao

D.K.

and Kuptsov

, Effective use of magnetization data in the design of electric machines with overfluxed regions, IEEE Trans. Magn. 51(7) (2015), 1–9, Art no. 6100709.

27.

Svanberg

, The method of moving asymptotes a new method for structural optimization, Int. J. Numer. Methods Eng. 24(2) (1987), 359–373.

28.

Svanberg

, A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim. 12(2) (2002), 555–573.

29.

Nocedal

and Wright

, Numerical Optimization, Springer-Verlag, New York, 2006.