Literature Reviews of Topology Optimal Design Methods and Applications in Magnetic Devices

Abstract

With the evolution of magnetic devices toward structural innovation and high reliability, the traditional design methods, such as size optimization and shape optimization, are limited by preset structural forms, making it challenging to generate novel structures and topologies. Topology-optimized design methods can achieve an optimal distribution of constituent materials of magnetic devices by optimizing the objective performance subject to certain constraints, and can provide greater freedom for designers. Based on the above background, this paper firstly investigates the principles of deterministic topology optimization methods, and introduces the latest specific applications in magnetic devices. It also demonstrates the advantages of topology optimization technology in enhancing operating performance, fostering structural innovation, and improving material utilization in magnetic devices. To manage uncertainties in design and manufacturing processes of magnetic devices, this paper analyzes uncertainty topology optimization methods, respectively, reliability and robustness-based topology optimization algorithms. To facilitate manufacturing, this paper summarizes the filter strategy for the new structure obtained by topology optimization. Finally, the problems faced by the topology optimization method in the field of magnetic devices are discussed, and a future development direction is projected.

1. Introduction

With the rapid advancement of society, science, and technology, researchers in structural optimization design pay more attention to effectively utilizing materials and arranging them rationally to enhance the stability and reliability of structures. Structural optimization integrates disciplines such as mechanics, mathematics, computer science, and other engineering fields, and is widely applied across aerospace, mechanical, hydraulic, bridge, civil, automotive, energy, and military engineering. It can be categorized into three levels: size optimization, shape optimization, and topology optimization. Size optimization achieves optimal design by adjusting the geometric parameters of the structure [1]. Shape optimization focuses on determining the geometric boundary shapes of continuum structures [2], while topology optimization optimizes material distribution within a given domain [3].

Traditional optimization design methods mainly rely on the engineer’s design experience. However, due to the nonlinear characteristics of electrical materials, empirical calculations can only be used as a reference, making it difficult to meet the continuous innovation demands of electrical equipment, to fully utilize materials, and to control production costs in highly competitive markets. From the perspective of rational material utilization and the generation of novel topological structures, the investigation of topology optimization methods for magnetic devices proves to be highly valuable [4,5].

Through applications in permanent magnet motors, synchronous reluctance motors, and electromagnetic couplers, topology optimization design methods demonstrate significant advantages, including material utilization reduction, innovative structure generation, and engineering cost reductions [6,7,8]. To facilitate production and manufacturing for the complex structures obtained, topology optimization is increasingly integrated with additive manufacturing technology, thereby promoting the development of next-generation magnetic devices [9,10,11].

On the other hand, in practical engineering, uncertain factors such as numerical errors, manufacturing tolerances, and variations in material properties are inevitable. These uncertainties can significantly impact the operational performance of magnetic devices, potentially degrading the performance of optimal design solutions during actual operation or violating constraint conditions. Therefore, it is essential to consider uncertainties in the design process.

The topology optimization in this paper focuses on methods that obtain the optimal structure of magnetic devices through optimizing material distribution in the design space. It provides a comprehensive summary of topology optimization from design philosophy to final structure based on specific applications in magnetic devices. The strategies for managing uncertainties in electrical engineering are combined with topology optimization. The main contributions of this paper are summarized as follows:

(1)

Implementations of existing deterministic topology optimization methods and applications in magnetic devices are reviewed. The characteristics of topology optimization, such as convergence, searching accuracy, and scope of applications, are discussed.

(2)

To deal with uncertainties in electrical engineering, topology optimization, considering reliability and robustness, is analyzed. The problem of difficult manufacturability encountered after topology optimization is also considered, and the filtering strategies are summarized.

(3)

Challenges in the application of magnetic devices are discussed, and predictions are made about future solutions.

2. Topology Optimization

Topology optimization is a structural optimization technique aimed at determining the optimal material distribution within a given design domain to achieve specified performance objectives. The following section presents the theory of deterministic topology optimization and the ways to implement topology optimization.

2.1. Theory of Topology Optimization Method

In this section, the theory of common deterministic topology optimization methods is summarized.

2.1.1. Homogenization Method

Scholar Bendsøe introduced the homogenization method in 1988, which ignited a trend in the development of topology optimization for continuum structures [12]. In this method, each finite element within the discretized design domain is composed of an infinite number of microstructures, typically represented by rectangular elements with rectangular voids [13]. The size (x, y) and rotation angle of these voids serve as the primary design variables, as shown in Figure 1. During optimization, if a void disappears, the corresponding element becomes solid; conversely, it remains void. By iteratively solidifying and hollowing out these elements, the optimal material distribution within the design domain is ultimately achieved.

Subsequently, the homogenization method was applied within the field of magnetic devices [14,15]. This approach modifies only the material properties within the microstructural units of electrical materials, necessitating the calculation of characteristic equations and material constants for each unit, which is computationally intensive and can lead to convergence at local optima during the optimization process.

2.1.2. Variable Density Method

In the variable density method, the design variable is the density ρ of each element in the design domain; ρ continuously changes between 0 and 1, as shown in Figure 2.

The optimal structure topology is determined by the distribution of each design material in the design domain. When transitioning from discrete topology optimization to continuous topology optimization, intermediate density values often arise, complicating the determination of whether material should be removed. Therefore, a penalty model for solid isotropic materials has been proposed to address the issue of intermediate material density. The general formulation for problem minimization is as follows:

$$\begin{array}{l}\mathit{min} f=C\left(u\left({\rho }_k\right),{\rho }_k\right)\\ s.t. g=V-V_0={\displaystyle \sum {\rho }_k-}{V}_0\le 0\\ 0\le {\rho }_k\le 1\end{array}$$

(1)

where C is the total compliance, which is a function of structural displacement u and ρ_k; V is the total volume; V₀ is the target volume; ρ_k is the density of the kth element. To determine properties of electrical materials, relative permeability and magnetoresistance are used as material characteristic variables of density, expressed as

$${\mu }_r=\left({\mu }_{iron}\left(B\right)-{\mu }_{air}\left(B\right)\right){\rho }^p+{\mu }_{air}\left(B\right)$$

(2)

where μ_iron and μ_air are the relative magnetic permeabilities of iron and air, respectively; ρ^P is the penalty factor. Due to varying levels of discretization in the optimized structure, issues such as lattice dependency and checkerboard patterns may arise, posing significant challenges for the practical manufacturing of magnetic devices [16,17,18].

2.1.3. Evolutionary Structural Optimization Method

The evolutionary structural optimization (ESO) method involves the gradual removal of low-stress elements, which progressively decreases the difference between the maximum and minimum stresses, ultimately achieving uniform stress distribution. The flowchart of the method is shown in Figure 3.

The optimization model is written as follows:

$$\begin{array}{l}min W={\displaystyle \sum _{e=1}^N{w}_e}{\beta }_e \\ s.t. {\sigma }_{\max }<{\sigma }^{\ast }\\ {\beta }_e\in \left\{0,1\right\},e=1,2,\dots ,N\end{array}$$

(3)

where W is the total weight of the structure, w_e and β_e (e = 1, 2,..., N) are the weight and design variable of the eth element, respectively; σ_max is the maximum stress of the structure; σ* is the specified upper limit of σ_max. The ESO can only remove materials and cannot add materials, resulting in lower efficiency [19,20]. The subsequently developed bidirectional evolutionary structural optimization (BESO) method can be applied to the electrical field to add or remove materials bidirectionally. This method has a simple concept and efficient implementation but suffers from issues such as numerical instability.

2.1.4. Normalized Gaussian Network Method

The normalized Gaussian network (NGnet) is employed to optimize the topology of electrical materials in [21,22], and can achieve smooth shapes without introducing filtering. Considering the finite element discretization of the computational domain, the material property assigned to each mesh element is determined by function f(x), defined as

$$\begin{array}{l}f\left(x\right)={\displaystyle {\sum }_{i=1}^{N_g}{w}_i{b}_i\left(x\right)}\\ b_i\left(x\right)=G_i\left(x\right)/{\displaystyle {\sum }_{j=1}^{N_g}{G}_j\left(x\right)}\end{array}$$

(4)

where G_i is a Gaussian function centered on the center of the ith element, b_i is the ith normalization function, and w_i is the ith weight. In cases involving two materials (e.g., iron and air), the material of each element g can be decided based on value of f(x) in Equation (5), and the material distribution is shown in Figure 4 [23].

$$g\leftarrow \left\{\begin{array}{l}\mathrm{iron} f\left(x\right)\ge 0 \\ \mathrm{air} f\left(x\right)<0 \end{array}\right.$$

(5)

The application of the NGnet method in magnetic devices can effectively suppress the abnormal topology of electrical materials during the optimization process, with small computational complexity, but its optimization capability remains relatively limited.

2.1.5. Level Set Method

The level set method (LSM) is one boundary-based topology optimization method. The interface formed by the material boundary within its design domain is equivalent to a two-dimension closed curve, as shown in Figure 5.

As shown in Figure 5, in the design area D, the implicit form of the curve enclosing the region Ω is expressed as

$$\Omega =\left\{\left(x,y\right)|\mathit{\Phi }\left(x,y\right)=\mathrm{c}\right\}$$

(6)

where c is a constant, and Φ is a level set equation satisfying

$${\displaystyle \frac{\mathit{\partial }\mathit{\Phi }}{\mathit{\partial }t}}+v\cdot \nabla \mathit{\Phi }=0$$

(7)

where v is the speed that drives the boundary motion, and t is time. When the LSM is applied to topology optimization, electrical materials have clear material boundaries while obtaining new topological structures [24,25,26].

2.1.6. Binary Structure Method

The topology optimization of binary structure (TOBS) method uses sequence approximation to solve optimization problems by linearizing objective and constraint functions [27]. In the kth iteration, the optimization model is

$$\begin{array}{l} \min {\displaystyle \frac{\mathit{\partial }F\left({\rho }^k\right)}{\mathit{\partial }\rho }}\cdot \Delta {\rho }^k\\ \mathrm{s}.\mathrm{t}. {\displaystyle \frac{\mathit{\partial }{G}_i\left({\rho }^k\right)}{\mathit{\partial }\rho }}\cdot \Delta {\rho }^k\le \Delta G_i\left({\rho }^k\right)i\in \left[1,N_g\right]\\ ‖\Delta {\rho }^k‖\le \beta N_d\\ \Delta {\rho }_j^k\in \left\{-{\rho }_j^k,1-{\rho }_j^k\right\}j\in \left[1,N_d\right]\end{array}$$

(8)

where N_g is the number of constraint conditions, and N_d is the number of design variables. The number of state flips from (0,1) to (1,0) in each iteration is controlled by the parameter β ≤ 1, which only allows the β part of the elements to change state. The flowchart for the TOBS method is shown in Figure 6.

2.1.7. ON/OFF Method

The ON/OFF method is based on heuristic criteria to remove or add discrete materials within the design domain [28,29,30,31]. During optimization, the electrical material design domain is discretized into design elements, each possessing a single state, void or solid. As shown in Figure 7, the grey elements represent solid states (referred to as ON), indicating that they are filled with the corresponding material, whereas white elements signify void states (termed OFF), denoting areas filled with air. When determining material assignment, the sensitivity analysis is typically employed as the decision-making basis. If the sensitivity of the objective function with respect to design variables is negative, the element is designated as air; otherwise, the element material is assigned to a solid material. Due to the discrete nature of the material properties, there are no intermediate density materials in the ON/OFF method. However, in post-optimization, electrical materials often exhibit a checkerboard pattern, which poses significant challenges for manufacturing.

2.2. Implementation of Topology Optimization

Normally, the performance analysis of an electromagnetic device can be obtained through finite element analysis (FEA). The FEA can be implemented through direct code programming and commercial software. Therefore, there are mainly two types of implementations for topology optimization. One combines the optimizer with topology optimization methods; the other is the combination of the optimizer with commercial software.

The application of numerical methods enables topology optimization within the CST/ANSYS environment, which has become a research hotspot in the field of electromagnetic design in recent years. By utilizing the physical field adaptability and computational efficiency advantages of combining commercial software and an optimizer to solve the multi-physics field coupling, high-precision modeling and large-scale computing requirements in complex engineering problems are met. The commonly used combination is MATLAB version R2020a and ANSYS version 18.2. MATLAB scripts are coupled to ANSYS software, FEA is performed using ANSYS, and functional evaluations are conducted in MATLAB. For example, the structural topology optimization problem of 2-layer and 3-layer permanent magnet hybrid synchronous reluctance motors was analyzed, with the aim of reducing the required amount of permanent magnet material while minimizing the risk of demagnetization, in [32]. In the optimization process, the thickness of the center column and bridge of the structure needs to be combined with machine analysis to generate reliable designs. The simulation results demonstrate the performance advantages of optimizing the topology. A parallel computing algorithm was developed in MATLAB, which uses FEA, topology optimization methods, and parallel cloud computing to design radial flow rotors in centrifugal pumps [33]. The objective was to minimize energy dissipation and vorticity within half of the rotor circumference, and the topology optimization objective was validated using ANSYS. The results indicate that numerical analysis software can help engineering designers achieve non-intuitive designs and enhance the results. FEA simulation was used to guide the electromagnetic shielding and filtering design of unmanned aerial vehicles in [34]. A decoupled electromagnetic integrated structure for symmetrical LCL-EMI filters (The manufacturer of LCL-EMI filters is Harbin institute of Technology, Harbin, China.) used in grid-connected converters for V2G applications was proposed to guide the topology selection of integrated filters in [35]. FEA simulation calculations were conducted using ANSYS, and prototype design and related performance testing were carried out to verify the effectiveness of the proposed method. However, this method to achieve topology optimization based on commercial software has some limitations, such as requiring a simplified model and the increasing complexity of engineering problems. Future breakthroughs depend on algorithm innovation, hardware acceleration, and technological progress.

From the perspective of the performance analysis of engineering problems, topology optimization can be achieved through commercial software or proxy models. The above content has already explained the use of commercial software; the implementation of topology optimization with the use of deep learning (DL) will be investigated next. Traditional topology optimization methods depend on manually designed parameterized models and mathematical formulas, which face bottlenecks when dealing with complex boundary conditions, multi-physics field coupling, or large-scale high-dimensional problems. The introduction of DL provides a novel methodology for topology optimization [36,37], in which the core is to assist the optimization process through data modeling or algorithm learning. The optimization flowchart of DL is shown in Figure 8.

The DL directly learns the mapping relationship between input conditions, such as loads and boundary constraints, and output topology through neural networks, without the need for manually defining mathematical expressions for material distribution. The DL avoids the accumulation of modeling errors in traditional methods by using multi-input neural networks, and captures the potential correlations of high-dimensional design variables through multi-layer nonlinear transformations. It compresses millions of parameters into a low-dimensional hidden space for optimization and reduces computational complexity. In addition, the DL utilizes the generative capabilities of generative adversarial networks to overcome the shape limitations in traditional parametric methods. Therefore, DL provides a transformation from “manual trial and error” to “intelligent generation” for topology optimization, especially in cutting-edge fields such as electrical equipment additive manufacturing, intelligent new structures, and extreme environmental engineering, which have irreplaceable value. For example, multi-physics DL was employed to predict the average torque, torque ripple, maximum stress, and iron loss characteristics of permanent magnet motors for topology optimization in [38]. The use of deep neural networks as proxy models accelerates topology optimization. A comprehensive analysis reveals that the two-stage optimization algorithm based on neural networks reduces computational costs while ensuring accuracy. A research study was carried out on the multi-objective optimization of shape generation using an autoencoder (AE) based on convolutional neural networks in [7]. The AE constructed low-dimensional features of compressed input data and optimized the SynRM design shape of the magnetic flux barrier by associating the component values of the intermediate layer with the values of the objective function, surpassing the optimized shape structure of traditional topology optimization methods.

Regarding optimizers, topology optimization normally applies swarm intelligence algorithms. Commonly used algorithms include the genetic algorithm (GA), differential evolution (DE) algorithm, ant colony (ACO) algorithm, hummingbird algorithm (HA), and grey wolf (GWO) algorithm. The GA and DE algorithm achieve global search by iteratively optimizing gene sequences [39,40]. The ACO algorithm performs optimization through simulating the path selection behavior of ants during foraging [41], which simulates the behavior of ants releasing and sensing pheromones in a topological structure to find the optimal solution. The HA combines global Levy flight exploration and local neighborhood development, and updates positions through competition among nectar sources, balancing strong exploration and development capabilities [42]. The GWO algorithm achieves unidirectional information transmission within a rigid hierarchical structure, and has a distinguishable convergence direction [43]. In topology optimization, continuous structural design domains can be divided into discrete mesh elements, with each mesh element corresponding to a gene; each gene is represented by 0/1, indicating the presence or absence of material, and the material property of each mesh element is determined. The optimization flowchart of a swarm intelligent optimization algorithm is shown in Figure 9.

Swarm intelligent optimization algorithms easily handle multi-objective and multi-constraint problems, and their challenges lie in computational efficiency in high-dimensional discrete spaces. Numerous topology optimization methods exist, each with its own set of advantages and disadvantages.

Table 1. Comparison of topology optimization methods.

Methods	Accuracy	Convergence	Complexity	Constraints	Speed	Applications	References
Homogenization method	Moderate	Low	High	Volume constraint	Low	Truss/suspended beam structure, etc.	[12,13,14,15]
Variable density method	High	Stable	Moderate	Volume constraint	Fast	Motors/sensors/thermal management device, etc.	[16,17,18]
ESO	Moderate	Moderate	Low	Stress/displacement constraints	Moderate	Motor/sensor/electrical equipment structural components, etc.	[19,20]
NGnet	Higher	Moderate	High	Network weight constraint	Low	Motor/magnetic resonance system, etc.	[21,22,23]
LSM	Higher	Moderate	High	Volume/displacement constraints	Low	Microelectromechanical systems/sensors, etc.	[24,25,26]
TOBS	High	Unstable	Moderate	Geometric/topological constraints	Moderate	Motor/antenna design, etc.	[27]
ON/OFF	Moderate	Fast	Low	Material/geometric/topological constraints	Fast	Microelectromechanical systems/electromagnetic components, etc.	[28,29,30,31]

summarizes the problems and the applicability of different topology optimization methods.

3. Applications of Deterministic Topology Optimization in Magnetic Device

The deterministic topology optimization (DTO) optimizes the distribution of materials within a given domain based on predefined constraints and performance metrics to achieve maximum resource utilization. This process is to ensure that the structure achieves optimal performance while satisfying all necessary conditions. Through extensive exploration and research by scientists, topology optimization methods have been widely adopted to address complex deterministic optimization problems in magnetic devices [44,45,46]. The following is a brief overview of specific application cases of deterministic topology optimization for magnetic devices.

3.1. Permanent Magnet Synchronous Motor

Currently, researchers focus on multi-material topology optimization of motors, including permanent magnets, flux barriers, and iron cores, to enhance the utilization of magnetic materials and improve motor structure and operational characteristics. By employing the LSM to optimize the motor winding, the optimal coil structure in stator slots could be obtained to maximize motor torque [47]. Subsequently, the LSM has been further improved to optimize the rotor magnetic poles of a permanent magnet synchronous motor (PMSM) to reduce cogging torque [48]. However, the LSM faces challenges in accurately estimating iron losses. To address this limitation, researchers have introduced the NGnet to the topology optimization of the PMSM to improve torque performance without increasing iron loss [49]. Despite these advancements, the optimized structural models often encounter practicality and manufacturability issues. To resolve the compatibility issues between complex shapes and manufacturing technologies, a multi-objective topology optimization method based on the immune algorithm was proposed, which was applied in the design of a PMSM rotor to promote the generation of meaningful shapes in practical engineering through filtering strategies [50]. However, the introduction of filtering operators often results in the topology structure being overly dependent on the filtering process and its associated parameters. To address this problem, an improved NGnet topology optimization method was proposed, which could directly obtain a smooth structure without filtering steps. Furthermore, the influence of the Gaussian number on the optimization results was investigated during the optimization process [51]. As illustrated in Figure 10, the analysis revealed that increasing the number of Gaussian numbers significantly enhances the shape representation capability of the Gaussian network. A two-step multi-material topology optimization method was proposed in [26]. The first step was optimization to determine the rough shape, and the second step was to restrict the design domain to the material boundary of the structure obtained in the first step and optimize the design domain based on the ON/OFF method. To evaluate the effectiveness of this method, it was applied to a PMSM and compared with traditional multi-material topology optimization methods. The results indicated that the optimized structure of the proposed method has a higher average torque.

3.2. Magnetic Shielding System

A magnetic shielding system is essential for ensuring the proper operation of high-precision magnetic measuring instruments by eliminating interference magnetic fields generated by nearby ferromagnetic objects and electrical control equipment. Topology optimization of magnetic shielding systems based on the ON/OFF method and the LSM avoided the checkerboard phenomenon and promoted the generation of manufacturable shapes in practical engineering [52].

3.3. Magnetic Recording Head

To reduce the leakage magnetic field in specific regions while enhancing the field strength in the recording area, the shape of the magnetic head can be determined using topology optimization techniques. A topological structure of a magnetic recording head was studied using the density method, with optimization focused on the coil and magnetic yoke shapes, resulting in an increase in magnetic flux in the magnetic recording domain and a decrease in leakage flux in adjacent areas [53,54]. An improved ON/OFF method was used to optimize a pointed field monopole magnetic head with magnetic shielding. As the recording field increased, the leakage flux in adjacent positions and lines decreased, resulting in a recording field of 1.7 T and a field gradient of 0.0314 T/nm. A low stray field of 0.1 T was observed at the center of adjacent tracks, effectively reducing the leakage flux in lines affected by crosstalk by 50% [55].

3.4. Brushless DC Motor

The core goal of topology optimization for brushless DC motors is to achieve comprehensive performance improvements such as light weight and high efficiency by optimizing the internal structure layout of the motor while meeting constraints such as performance, cost, and process. However, post-optimization designs often feature void holes, complicating direct manufacturing. To address the checkerboard problem caused by these holes, a micro-genetic algorithm has been proposed for optimizing the stator teeth of brushless DC motors [56]. This topology optimization eliminated clusters by threshold values, reduced the population size, reduced the number of computations, and increased the continuity of materials.

3.5. Synchronous Reluctance Motors

Synchronous reluctance motors (SynRMs) are widely used due to their simple structure and low manufacturing cost. However, these motors typically exhibit lower maximum torque and efficiency compared to PMSMs. To address these problems, as illustrated in Figure 11a,b, the NGnet was used for topology optimization of SynRM rotor silicon steel (The SynRM rotor is established by the technical committee of IEE of Sapporo, Japan), which can increase torque without increasing iron loss [57]. The optimal distribution of air, iron, and magnets for the SynRM was determined through an interpolation method, resulting in a reduction in magnet usage while simultaneously improving performance. Compared to the initial design, the number of magnets used was reduced by 30% [58]. The flux barrier of the SynRM rotor was optimized by applying initial random hollow circles, resulting in a lower torque ripple and higher average torque [59]. Subsequently, a two-step topology optimization method was proposed to optimize the SynRM rotor by using filtering and penalty strategies to enhance average torque. Experiments were conducted on two topology-optimized SynRM rotor structures of 4000 r/min and 8000 r/min, as shown in Figure 11c,d. The average torque obtained from the simulation and experiments was compared to verify the effectiveness of the method [60]. By utilizing the multi-objective genetic optimization algorithm based on finite element analysis for multi-material topology optimization of the SynRM, three-layer and five-layer SynRM rotors were fabricated (Figure 11e,f). Figure 11g,h show that the experimentally measured torque is slightly lower than the calculated torque. The difference between experiments and calculations is mainly caused by manufacturing and assembly tolerances. In addition, mechanical and thermal stresses during manufacturing may also alter magnetic properties [61].

3.6. Electromagnetic Actuator

Electromagnetic actuators generate magnetic current through coils in the magnetic circuit of the iron yoke, inducing force on the armature and causing it to move. By optimizing the topology of the electromagnetic brake, the magnetic force could be increased [28,62]. Specifically, an improved genetic algorithm was utilized to optimize the magnetic yokes of two types of magnetic brakes. Verification results confirm that the algorithm exhibits strong global search performance, accelerates convergence speed, and maximizes the attractive force borne by the armature, achieving 1649.5 N·m. The two optimized actuator structures are shown in Figure 12 [28]. Furthermore, the variable density method has been employed to perform topology optimization on the electromagnetic actuator, maximizing the average magnetic force acting on the plunger while ensuring the optimal structural shape. The average magnetic force of the optimized structure is 41.9% higher than that of the initial structure [62].

3.7. Electromagnetic Interference Filter

Passive filters for electromagnetic interference (EMI) filtering choke coils and capacitors are generally used in power electronics to solve electromagnetic interference, and topology design and characterization/modeling are very important in EMI filtering components. EMI filtering choke coils account for most of the weight and size of passive filters and can reduce significant EMI noise [63]. Structural optimization design has been extremely promising in enhancing the anti-interference ability of magnetic devices to resist external electromagnetic attacks [64]. To address electromagnetic compatibility issues of electric vehicle charging systems, the layout and wiring rules of a three-phase EMI filter was analyzed using FEM, and parasitic inductance mutual cancellation technology was proposed to optimize the three-phase EMI filters [65].

3.8. Induction Motor

Induction motors are widely used in various industrial machinery applications due to their robustness, low cost, and self-starting characteristics. To improve the efficiency of induction motors, the design optimization method combining electromagnetic field analysis with topology optimization is effective in implementing magnetic structures. However, the effect of the actual voltage waveform on the topology optimization results cannot be considered due to the current input being set as a magnetic source. Therefore, according to the strong coupling analysis of the magnetic field and three-phase circuit, a sensitivity-based PWM inverter voltage-driven induction motor time-domain topology optimization method was proposed in [8]. In addition, the discrete cosine transform was applied to reduce the significant memory consumption required for storing state variables in the time-domain attendant variable method.

3.9. Other Applications

Topology optimization has other applications in the field of magnetic devices. The nuclear magnetic resonance instrument requires the main magnet to generate a uniform magnetic field within the resonance zone. Therefore, the topology of the nuclear magnetic resonance magnet was optimized; the optimized structure significantly enhances magnetic field uniformity compared to the initial structure [66]. The topology optimization design method of electromagnetic cloaks constructed from isotropic materials around objects has been studied [67]. The LSM was used to optimize the design, achieving the optimal distribution of ferrite and air, and a ferrite structure with an electromagnetic invisibility cloak function was found. The wound synchronous motors designed in [68] can achieve the distribution of multi-materials in the rotor through a magnetic structure topology optimization method that integrates solid isotropy with a material penalty. The copper loss of the rotor before optimization was 1060 W; after optimization, the rotor copper loss was 486.8 W, which was reduced by 54.1%. The electromagnetic coupler can achieve innovative structural designs through topology optimization, and the requirements for their power consumption and output force are gradually becoming stringent. The variable density method was applied to the design of an electromagnetic coupler to obtain the optimal shape by maximizing the output force in a specified direction under the constraint of power [69]. The optimal converter topology for photovoltaic and wind energy systems coupled to the grid was determined in [70].

The applications of topology optimization in magnetic devices are listed in

Table 2. The applications of topology optimization in magnetic devices.

Devices	Method	Optimized Region	Objective	References
PMSM	LSM/improved LSM/NGnet/multi-objective topology optimization method based on the immune algorithm/improved NGnet/a two-step multi-material topology optimization method	Motor winding/rotor magnetic poles/rotor/rotor/rotor/rotor	Maximize motor torque/reduce cogging torque/improve torque performance without increasing iron loss/manufacturable with filtering/manufacturable without filtering/maximize motor average torque	[26,47,48,49,50,51]
Magnetic shielding system	ON/OFF and LSM	Magnetic shielding system	Facilitated practical engineering of manufacturable shapes	[52]
Magnetic recording head	Variable density method/an improved ON/OFF method	The coil and magnetic yoke shapes/a pointed field monopole magnetic head with magnetic shielding	An increase in magnetic flux in the magnetic recording domain and a decrease in leakage flux in adjacent areas/the leakage flux in adjacent positions and lines decreased	[53,54,55]
Brushless DC motor	A micro-genetic algorithm	The stator teeth of brushless DC motor	Reduced the number of computations, increased the continuity of materials	[56]
SynRM	NGnet/an interpolation method/initial random hollow circles/a two-step topology optimization method/multi-objective genetic optimization algorithm based on FEA	Rotor silicon steel/the optimal distribution of air, iron, and magnets for SynRM/the flux barrier of rotor/rotor/rotor	Increase torque without increasing iron loss/reduction of magnet usage while simultaneously improving performance/lower torque ripple and higher average torque/enhance average torque/enhance torque	[57,58,59,60,61]
Electromagnetic actuator	Variable density method/improved genetic algorithms	Electromagnetic actuator/electromagnetic actuator	Maximizing the average magnetic force acting on the plunger while ensuring the optimal structural shape/maximizing the average magnetic force acting on the plunger	[28,62]
EMI filter	FEA	The layout and wiring rules of a three-phase EMI filter	Address electromagnetic compatibility issues of electric vehicle charging systems	[65]
Induction motor	A sensitivity-based PWM inverter voltage-driven induction motor time-domain topology optimization method	Induction motor	Consider the actual voltage waveform	[8]
Nuclear magnetic resonance magnet	FEA and particle swarm algorithm	Coils and magnets	Enhanced magnetic field uniformity	[66]
Electromagnetic cloaks	LSM	Electromagnetic cloaks	Achieving the optimal distribution of ferrite and air, finding a ferrite structure with electromagnetic invisibility cloak function	[67]
Wound synchronous motor	Integrates solid isotropy with material penalty	Rotor	Achieve the distribution of multi-materials in the rotor	[68]
Electromagnetic coupler	Variable density method	Electromagnetic coupler	Obtain optimal shape by maximizing the output force in a specified direction under constraint of power	[69]

.

It is evident that the application of topology optimization in magnetic devices has the potential to explore innovative electromagnetic structures and enhance operating performance. However, research on topology optimization in the field of magnetic devices remains in its nascent stages, with relatively few reports addressing multi-material systems, magnetic composites, three-dimensional problems, and the integration of multi-physical fields. Further investigation is needed to evaluate the rationality of optimized structures and feasibility for practical engineering applications.

4. Topology Optimization Considering Uncertainties

In the design and manufacturing processes of magnetic devices, uncertain factors such as manufacturing tolerance and variations in material properties are inevitable. These uncertainties can cause performance fluctuations of magnetic devices and may disrupt predefined constraint conditions. Consequently, it is essential to account for uncertainty. In topology optimization, methods that consider uncertain factors are generally divided into the reliability-based topology optimization (RBTO) method and the robust topology optimization (RTO) method. The RBTO method ensures that the working performance of the magnetic device achieves a predetermined level of reliability. The RTO minimizes fluctuations in the objective function or constraint conditions, thereby ensuring the stability of operational performance.

4.1. Reliability-Based Topology Optimization

The RBTO method integrates reliability analysis with topology optimization to ensure that the reliability of key operational performance considerations of electrical equipment meets predetermined requirements under the influence of uncertain factors.

The reliability optimization design considers the impact of uncertain factors while achieving performance optimization [71,72]. The most widely adopted reliability analysis methodologies in the electrical field include Monte Carlo simulation, the reliability index method, and fault tree analysis [73,74,75]. Reliability optimization was used to enhance the output torque of brushless DC motors while simultaneously reducing the failure rate in [76]. Reliability optimization was used to reduce the cogging torque, with considerations of stator magnetic flux density, in [77]. Reliability optimization was used to minimize back electromotive force and improve cogging torque reliability [78]. Reliability optimization was performed on the motor to reduce cogging torque and the failure rate of output torque [79]. Uncertainty was calculated using the Monte Carlo simulation method in [80], and reliability optimization was performed to minimize the mass of the rocket engine. The reliability of centrifugal compressors was enhanced through reliability optimization in [81]. The nonlinear uncertain system method was adopted to optimize trough solar power plants and reduced energy costs [82]. There are inevitably uncertain factors during the design stage and actual operating conditions of magnetic devices. The above research endeavors to enhance the operational performance of magnetic devices through reliability optimization design. To ensure structure reliability, the following sections elaborate on the methods and applications of considering reliability analysis in the topology optimization design.

In deterministic topology optimization, researchers tend to ignore uncertain factors such as material properties, geometric shapes, and loading conditions, so the reliability of the obtained structural topology cannot be guaranteed. The incorporation of reliability criteria into deterministic topology optimization serves to enhance the reliability of structures. The RBTO method considers uncertainty by introducing reliability criteria in the topology optimization process. The fundamental process of RBTO is shown in Figure 13.

According to the different ways of handling reliability constraints during the optimization process, the optimization strategies for RBTO problems can be classified into nested optimization, decoupling, single-loop, and reliability safety factor methods.

(1)

Nested Optimization Method

The nested optimization method consists of an outer loop and an inner loop. The outer loop performs the topology searching. The inner loop performs reliability analysis on each design scheme during each iteration process. According to different reliability expressions, nested optimization methods can be divided into the reliability index approach (RIA) and the performance measure approach (PMA). In the RIA, the optimization model for minimizing the objective function f(x) subject to constraint g(x) ≤ 0 is formulated as

$$\begin{array}{l}\min f\left(x\right)\\ \mathrm{s}.\mathrm{t}. g\left(x\right)\le 0\end{array}$$

(9)

where the design variable x is classified into deterministic variable d and uncertain variable x_p~N(μ_p,σ_p). The uncertain variable follows a certain probability distribution. Under the influence of uncertain factors, the probability of the constraint condition is the reliability index, which serves as a constraint condition in the optimization process:

$$\begin{array}{l}\underset{d}{\min } f\left(d,{\mu }_p\right)\\ \mathrm{s}.\mathrm{t}. \beta \le {\beta }_t\end{array}$$

(10)

where the reliability index β is the minimum distance from the origin to the approximate limit state equation G(u) = 0, expressed as

$$\begin{array}{l}\min \beta =‖u‖=\sqrt{u^{\mathrm{T}}u}\\ \mathrm{s}.\mathrm{t}. G\left(u\right)=0\end{array}$$

(11)

The PMA is the reverse of the RIA. Under the condition that the distance from the most probable point (MPP) to the origin point equals β, the minimum value of the state equation is determined. Then, by comparing whether this minimum value exceeds zero, it is determined whether the structure satisfies the reliability constraint.

Nested optimization satisfies the reliability constraint to some extent. However, due to the necessity of invoking inner loops during each iteration to address the reliability constraint, the computational complexity increases significantly.

(2)

Decoupling Method

The decoupling method separates topology optimization and reliability analysis processes. This method consists of two components, deterministic optimization and reliability evaluation, as shown in Figure 14. However, due to the necessity of conducting repeated reliability analyses following each deterministic optimization step, the computational cost remains relatively high.

(3)

Single-loop Method

Both nested optimization and decoupling methods necessitate the construction of an optimization model to find MPP points, whereas the single-loop method avoids the process of calculating MPP points. It eliminates redundant reliability loops but typically offers lower accuracy compared to the other methods.

(4)

Reliability Safety Factor Method

The reliability safety factor method introduces the reliability factor K_Oβ into the traditional safety factor design, and its mathematical model is

$$\begin{array}{l}\underset{d}{\min } f\left(d\right)={\displaystyle \sum _{i=1}^n{d}_i{v}_i}\\ \mathrm{s}.\mathrm{t}. K_{O\beta }{\mu }_C-{\mu }_{C_0}\le 0\\ 0\le d_i\le 1\\ K_{O\beta }={\displaystyle \frac{\left[1+{\beta }_{C_0}^{\ast }{\left({\gamma }_{C_0}^2+{\gamma }_C^2-{\beta }_{C_0}^{\ast }{\gamma }_{C_0}^2{\gamma }_{C_0}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]}{1-{\beta }_{C_0}^{\ast 2}{\gamma }_{C_0}^2}}\end{array}$$

(12)

where f(d) is total volume of structure, d_i is the design variable, and ${\beta }_{C_0}^{\ast }$ is the given reliability indicators. Although the reliability safety factor method simplifies the optimization model, there is no universal expression for the reliability factor.

The concept of reliability was introduced into structural topology optimization in mechanical engineering in [83]. A preliminary study on the RBTO of a C-type magnetic actuator was carried out in [84], considering one random variable (magnetic permeability of the magnetic material) and two random variables (magnetic permeability and current density). Based on the decoupling RBTO method, in [85], topology optimization and reliability analysis were separated. An RBTO model, using the BESO method and response surface methodology, was developed in [86]. An RBTO method for continuous domains was studied in [87] under conditions of uncertain material properties. The inverse optimal safety factor was proposed in [88] with the aim of reducing the structural mass while improving the structural reliability. An RBTO method based on one independent method for the motor was proposed in [89], and the results showed that this method improved the final properties of the structure. RBTO design was performed on the electromagnetic system [84]. To reduce computational costs, parallel response surface methodology was employed for an RBTO design of microelectromechanical systems [90]. The RBTO design was carried out for clamped T-shaped structures, clamped sleeper beam structures, and three-dimensional corner beam structures under multi-dimensional non-probabilistic uncertainties [91].

Overall, research on RBTO remains in its early stages, with most studies primarily focused on the mechanical field, with limited practical engineering applications in the electrical field. Further in-depth research is needed to accurately simulate uncertain factors based on limited samples, evaluate reliability indicators, and validate RBTO design in engineering problems.

4.2. Robustness-Based Topology Optimization

RTO combines robustness with topology optimization to minimize variations in the objective function or constraints under uncertain factors, thereby ensuring the robustness of electrical equipment performance. The robust optimization design determines the robust optimal solution under the influence of uncertain factors, as shown in Figure 15. The commonly used robust optimization methods include robust linear programming, robust quadratic programming, robust semi-definite programming, and distributed robust optimization, etc. [92,93,94,95].

For the applications in the field of energy systems and costs, a long-term robust collaborative optimization model for uncertain electricity and gas systems was proposed in [96]. This model simultaneously optimized the costs associated with generators, transmission lines, natural gas pipelines, etc. A novel industrial microgrid configuration was proposed in [97], with the objective of reducing the operational expenses of industrial microgrids while simultaneously leveraging the advantages offered by electric vehicles. To address the uncertainties associated with renewable energy generation and industrial load variations, robust optimization was implemented. Furthermore, the impact of various uncertain scenarios on the stable operation of industrial microgrids was analyzed. An RTO scheduling model based on real-time electricity price uncertainty was developed and applied to community energy hub systems [98].

For the applications in the field of wind power systems, the robust optimization algorithm for wind power systems based on interval uncertainty was proposed in [99]. A two-stage distributed robust optimization model was proposed for wind farms in [100], which minimized the total system cost while ensuring safe operation constraints.

For the applications in the field of magnetic devices, to estimate the impact of parameter disturbances in the design, a robust optimization algorithm combining six sigma design with a multi-level genetic algorithm was investigated in [101] and applied to the PMSM, which significantly enhanced the robustness of the PMSM’s performance. A robust optimization method was proposed to enhance the instability in back electromotive force and cogging torque caused by static eccentricity [102]. A multi-objective sequential robust optimization method based on orthogonal design and a hyper-volume index was proposed to improve the performance of permanent magnet motors while reducing the computational cost [103].

RTO improves the robustness of the structure by introducing robustness into the topology optimization process, taking uncertainty into account. RTO is generally used to minimize expected compliance or optimize worst-case performance, which can be divided into two types: random probability and non-probability.

(1)

Random Probability RTO Method

For the random probability RTO method, the distribution of the objective function when disturbed is defined by the probability density distribution function, and the optimization objective is typically defined as the expected value of the objective function, that is, the robust expected fitness function f_exp(x). However, in practical magnetic device research, there is no exact analytical expression for the objective function f_exp(x); approximate methods are generally used to calculate f_exp(x):

$$f_{\exp }\left(x\right)=1/N{\displaystyle {\sum }_{i=1}^{N}f\left(x_i\right)}$$

(13)

where x_i is the random variable x after being disturbed, and N is the number of sample points around point x. The standard deviation σ(x) defines the degree of deviation in the objective function under disturbance:

$$\sigma \left(x\right)=\sqrt{1/\left(N-1\right){\displaystyle {\sum }_{i=1}^N{\left[f\left(x_i\right)-f_{\exp }\left(x\right)\right]}^2}}$$

(14)

Therefore, by employing f_exp(x) and σ(x) as objectives in multi-objective optimization, the expected value and standard deviation can be considered simultaneously.

(2)

Non-probability RTO Method

The non-probabilistic RTO method reflects the robustness of the corresponding points by analyzing the worst-case value of the objective function within an uncertain interval. Current research applied to engineering problems primarily includes the variance model, gradient index model, and worst-case model.

• Variance Model

$$\begin{array}{l}\min {\sigma }_f\\ \mathrm{s}.\mathrm{t}. g_i(x)\le 0\\ x^{\mathrm{L}}\le x\le x^{\mathrm{U}} i=1,2,\dots ,m\end{array}$$

(15)

• Gradient Index Model

$$\begin{array}{l}\min GI\left(x\right)=\underset{1\le j\le n}{\max } \left|{\displaystyle \frac{\mathit{\partial }f\left(x\right)}{\mathit{\partial }{x}_j}}\right|\\ \mathrm{s}.\mathrm{t}. g_i(x)\le 0\\ x^{\mathrm{L}}\le x\le x^{\mathrm{U}} i=1,2,\dots ,m\end{array}$$

(16)

• Worst-case Model

$$\begin{array}{l}\min f_w\left(x\right)=\underset{\xi \in U\left(\xi \right)}{\max } f\left(\xi \right)\\ \mathrm{s}.\mathrm{t}. g_i(x)\le 0\\ x^{\mathrm{L}}\le x\le x^{\mathrm{U}} i=1,2,\dots ,m\end{array}$$

(17)

The variance model, gradient index model, and worst-case model ensure the robustness of the design by minimizing the standard deviation σ_f of the objective function, minimizing the maximum value, GI, of the partial derivative of the objective function, and minimizing the worst case of the objective function in the uncertain interval, respectively.

A nonlinear coupled electromechanical system RTO model applied to Coulomb force actuation was proposed [104], ensuring the robustness of the system. The RTO method for optimizing electrical equipment was introduced, and the results demonstrated better performance in terms of search efficiency and robustness [105]. In the RTO method [106], a simplified model of local damage within a continuous structure was adopted, considering the worst-case local fault situation of the inverter to enhance the robustness of local material removal. New mutation and crossover operators were introduced to the RTO algorithm [107], resulting in the more robust optimized topology structure that achieved similarity robustness 50% faster than existing RTO methods. The RTO method based on the moving asymptote method in [108] optimized the rotor of a SynRM to improve its torque characteristics and robustness. A mathematical model for the RTO of DC motors was proposed in [109], and the findings indicated significant improvements in motor robustness. Starting from random field theory, RTO was applied to permanent magnet motors to reduce cogging torque by considering material and manufacturing tolerance uncertainties [110]. Convex models and interval uncertainty RTO methods were studied using random field theory [111]. The mixed uncertainty RTO method with probability and interval uncertainty were studied [112,113]. Overall, research on RTO in the magnetic device field remains in its early stages, with practical applications necessitating further in-depth investigation.

5. Filtering Strategy

Topology optimization is not limited by traditional design experience and has the capability to generate novel structures. To mitigate the difficulties of traditional manufacturing techniques, filtering strategies are commonly employed to smooth the optimized topology structures.

5.1. Density Filtering Method

The density filtering method directly processes the density of the element [114,115]. The expression is as follows:

$${\tilde{\rho }}_e={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^{N_e}W(x_j){\rho }_j}}{{\displaystyle {\sum }_{j=1}^{N_e}W(x_j)}}}$$

(18)

where ρ is the optimized element density, $\tilde{\rho }$ is element density for density filtering, N_e is the neighborhood of e, and W is the weight function. Bruns combined density filtering with their proposed element removal method, which has certain advantages in terms of smoothness [116]. To obtain a smoother weighting function, a smoothed Gaussian function was proposed. However, Wang demonstrated that the smoothed Gaussian function does not show superiority over the original linear function [117].

5.2. Sensitivity Filtering Method

The sensitivity filtering method addresses the checkerboard phenomenon and mesh dependency issues in topology optimization. The formulation expression is as follows:

$${\displaystyle \frac{\widetilde{\mathit{\partial }C}}{\mathit{\partial }{x}_e}}={\displaystyle \frac{1}{({\displaystyle {\sum }_{f=1}^{N_e}W(x_f)})}}{\displaystyle \sum _{f=1}^{N_e}W}(x_f){\displaystyle \frac{\mathit{\partial }C}{\mathit{\partial }{x}_f}}$$

(19)

$$W(x_f)=r_{\min }-\mathit{dist}(e,f)$$

(20)

where $\widetilde{\mathit{\partial }C}$/$\mathit{\partial }{x}_e$ is the sensitivity of element e after filtering, $\mathit{\partial }C$/$\mathit{\partial }{x}_f$ is the sensitivity of element f before filtering, x_e and x_f are densities of element e and element f, respectively, N_e is the neighborhood of e, W is the weight function, and dist(e, f) is the distance between elements e and f. The sensitivity filtering method was studied [118], and has attracted significant attention in the academic community.

5.3. Morphology-Based Filtering Method

In image analysis, morphological operators including erosion and dilation are used to quantify holes and perform automatic inspection of image data [119]. The erosion operator removes holes smaller than the defined neighborhood in the initial image, while the dilation operator fills any hole smaller than the neighborhood. The erosion and dilation operators do not preserve volume. The open and close operators are defined as the erosion followed by dilation and dilation followed by erosion, respectively, where the volume remains unchanged.

In the actual manufacturing processes of engineering, topology optimization technology has overcome the structural innovation challenges posed by traditional manufacturing techniques. However, the resulting optimized structures are often not directly manufacturable, highlighting the advantages of filtering strategies. This method smooths the optimized structure, eliminates islands and irregularities within the topology, ensures compliance with the requirements of traditional manufacturing processes, enhances the manufacturability of the structure, reduces manufacturing difficulty and costs, and improves production efficiency. Therefore, filtering strategies significantly enhance the compatibility between topology optimization and additive manufacturing processes, promoting the development of innovative designs for practical engineering applications.

6. Challenges and Future Developments

Currently, topology optimization methods face numerous challenges in the application of electrical equipment. The subsequent section will present a comprehensive overview of the prevailing challenges, while providing a description of future developments.

6.1. The Complexity of Multi-Physics Field Coupling

A magnetic device is essentially an electromagnetic thermal mechanical multi-physics coupling system, with strong nonlinear interactions between fields. The electromagnetic field generates joule heating, leading to temperature rise, and a high temperature changes the conductivity/magnetic properties of materials. Structural deformation affects the distribution of electromagnetic fields. However, traditional topology optimization is mostly based on the assumption of a single physical field, and the mathematical difficulty of coupling field sensitivity analysis is high. In the future, a unified multi-physics field topology optimization platform can be developed to integrate the coupling relationships of electromagnetic fields, temperature fields, and structural fields through weak or strong coupling algorithms. The Pareto front optimization strategy can be introduced to handle multi-objective conflicts (such as the trade-off between efficiency, temperature rise, and volume) through weighted summation or constraint methods, combined with adaptive weight adjustment algorithms to dynamically balance optimization objectives.

6.2. Manufacturing Process Constraints

Topology optimization filling materials within a given design domain may generate “hollow” or “sharp edge” structures that are difficult to manufacture, exceeding traditional machining capabilities. For example, wire cutting cannot achieve complex surfaces, and when used, it is difficult to accurately match process constraints. The future development of topology optimization algorithms will be combined with process constraints, which can be introduced into manufacturability scoring functions. In addition, future development can also explore hybrid manufacturing processes, such as additive manufacturing, which can achieve complex topology manufacturing and break the limitations of single processes.

6.3. Material Nonlinearity

The magnetic permeability of actual materials such as silicon steel and soft magnetic composite materials varies with the strength of the magnetic field, and there is a hysteresis effect, which leads to the nonlinearity of the electromagnetic field control equation. Topology optimization is often based on linear assumptions, which can lead to optimization results deviating from true performance. The introduction of nonlinear material interpolation models based on physical mechanisms can address this problem. Alternatively, future developments can introduce machine learning algorithms such as neural networks to construct material constitutive relationships, which can reduce nonlinear computational costs.

6.4. Computing Resource Bottleneck

Large-scale FEA modeling requires a fine mesh, resulting in millions of degrees of freedom. Traditional single-core calculations are difficult to converge in a reasonable time and cannot meet the requirements of engineering design cycles. Currently, large-scale FEA relies on high-performance computing or reduced order models, such as Kriging surrogate models, but the balance between accuracy and efficiency still needs to be found. The introduction of machine learning into topology optimization can improve the efficiency of optimization; for example, DL is used to predict the mapping relationship between material distribution and performance. In addition, the introduction of topology optimization through parallel computing and GPU acceleration technology can improve the efficiency of solving large-scale problems. Future developments can create open-source topology optimization frameworks that integrate the strengths of mathematics, computer science, and electrical engineering to develop more efficient algorithms.

6.5. Uncertainty

The current uncertainties include reliability issues related to multi-physics field coupling, stochastic effects of manufacturing processes, prediction bias of high-frequency electromagnetic responses, time-varying characteristics of material properties, and adaptability to extreme working conditions. The future development of uncertain topology optimization can be achieved by establishing a probabilistic topology optimization model, treating material parameters and operating conditions as random variables, and using polynomial chaos expansion or FEA methods to calculate the probability density function of performance indicators, generating a probabilistic Pareto front for the solution. Alternatively, the data-driven, reliable optimization strategies can be introduced to train generative adversarial networks using historical manufacturing data, and simulate the impact of manufacturing deviations on topology structures.

7. Conclusions

The application of structural optimization design in magnetic devices has increased significantly over the past decade, attracting substantial attention from scholars. In contrast to the limitations of traditional size and shape optimization design methods in the innovation of electrical equipment structures, topology optimization, as an optimization design method that breaks traditional empirical constraints, represents an inevitable trend in the evolution of optimization methodologies. This paper divides topology optimization methods into deterministic topology optimization and uncertain topology optimization for research, and introduces their latest specific application cases in magnetic devices. Filtering strategies employed after topology optimization are summarized. Furthermore, the challenges associated with applying topology optimization methods to magnetic devices are discussed, and potential future research directions are outlined.