A Multi-Material Topology Optimization Method for Synchronous Reluctance Motors Based on Variable Density Method

Abstract

This invention belongs to the field of motor topology optimization design, specifically relating to a multi-material topology optimization method for synchronous reluctance motors based on the variable density method. This invention proposes a multi-material topology optimization method for synchronous reluctance motors based on the variable density method to facilitate the automated design process of synchronous reluctance motors. For the stator, a multi-material topology optimization scheme including winding and core materials, as well as current angle, is established to improve the torque output of the synchronous motor. For the rotor, under the same optimization objective, mechanical performance constraints such as structural flexibility and strength are established, enabling synchronous topology optimization of the rotor and stator, improving the completeness and flexibility of the automated design process. Compared with optimizing the rotor or stator separately, synchronous optimization can improve the overall performance of the motor. The method proposed in this invention has been verified on a synchronous motor. The optimization results demonstrate the effectiveness of the proposed optimization method and ensure that the new topology has smooth boundaries and is manufacturable.

Description

Technical Field

[0001] This invention belongs to the field of motor topology optimization design, specifically involving a multi-material topology optimization method for synchronous reluctance motors based on the variable density method. Background Technology

[0002] Synchronous reluctance motors (SRRMs) have been widely researched and applied in industry and academia due to their advantages such as low material cost, small rotor inertia, and good acceleration performance. High-performance motor design requires detailed design optimization. Traditional parameter size and shape optimization typically requires dozens of design variables, significantly increasing computational costs. Furthermore, traditional parameter optimization methods heavily rely on the designer's experience, greatly limiting the freedom of motor design. The development of efficient topology optimization methods provides a highly flexible and automated design approach for the structural design optimization of SRRMs. Among numerous topology optimization methods, the Variable Density Method (SIMP) is more widely used in motor topology optimization due to its simplicity. The general process of traditional SIMP-based motor rotor topology optimization is as follows: creating the initial rotor structure and meshing → interpolating materials for elements in the rotor design domain → establishing an optimization model → finite element analysis → solving the optimization problem using a gradient optimization solver → outputting the rotor optimization results.

[0003] Currently, motor topology optimization focuses almost entirely on the rotor, while stator optimization and stator-rotor synchronization optimization have received less attention due to the complex material distribution of the stator and the magnetic field interactions between the stator and rotor. In synchronous reluctance motors, the rotor and stator are two key components: the rotor is the rotating part of the motor, carrying the magnetic field, while the stator is the stationary part containing the winding arrangement. Optimizing only the rotor without considering the stator may lead to an uncoordinated magnetic field distribution, affecting the overall electromagnetic performance and operating efficiency of the motor. Furthermore, as a stationary component of the motor, the stator's structure and winding design significantly impact the machine's stability, heat dissipation, and mechanical strength. Therefore, neglecting stator optimization may result in potential defects in these critical aspects. Summary of the Invention

[0004] In view of the above-mentioned shortcomings and deficiencies of the prior art, this invention provides a multi-material topology optimization method for synchronous reluctance motors based on the variable density method. First, the design domains of the rotor and stator are determined, and material interpolation models for the electromagnetic and mechanical properties of these two regions are given based on SIMP, namely, permeability, Young's modulus, mass, and current angle. A motor topology optimization model considering multiple physics fields is constructed, and an augmented Lagrange equation framework is built to reduce the dimensionality of the optimization problem and decrease computation time. Finally, sensitivity analysis is performed, and a gradient-based MMA algorithm is used for optimization solution.

[0005] This invention provides a multi-material topology optimization method for synchronous reluctance motors based on the variable density method, the method comprising the following steps:

[0006] Step 1: Establish the geometric model of the motor, define the design domains of the rotor and stator, and mesh the design domains of the rotor and stator.

[0007] Step 2: Perform Helmholtz filtering and Heaviside projection filtering on the design variables within the stator and rotor design domains; the Helmholtz filtering equation is:

[0008]

[0009] Where i = 0, 1, 2 represents the design variable type, and its design interval is (0, 1], N e The number of elements in the design domain, z i,k Let x be the i-th design variable in the k-th unit. i,k Let M be the centroid coordinates of the k-th unit in the i-th design variable. i,j,k The number of cells within the filter radius R. The design variables are those in the filtered density field; the linear weighting function between the centroids of the two elements can be expressed as:

[0010]

[0011] Where R is the filtration radius;

[0012] Using Heaviside projection filtering on density variables After smoothing, the equation is:

[0013]

[0014] Where β is the steepness of the projection, and η is the threshold of the filter density. It is a physical density field variable;

[0015] Step 3: Construct interpolation models for the multiphysics material properties of the stator and rotor. The rotor's material property used to distinguish between steel and air is permeability, and the interpolation formula is:

[0016]

[0017] Where ν fe and ν air The magnetic permeability of iron and air, respectively, p rThe penalty factor is used. Since only the rotor section's design variables are involved, the subscript i in (4) is equal to 0. The stator section contains copper coils, electric steel, and air. Resistivity and current density are chosen to define the distribution of these materials. Two design variables are involved in the interpolation model. and The interpolation formula is:

[0018]

[0019] Where, ν cu J is the reluctance of the copper coil. cu Current density of copper, and p represents the filtered design variables in the stator design domain. v and p c The penalty factor is used. In the rotor region, considering the mechanical aspects of finite element analysis, the interpolation model for Young's modulus is established as follows:

[0020]

[0021] Where E fe Let be the Young's modulus of the electric steel, and p be the penalty factor. Considering the mechanical load calculation of the rotor, the interpolation model between the design variables and the physical mass is established as follows:

[0022]

[0023] Where, m fe Let be the physical mass of the unit, and ⊙ be the Hadamard product;

[0024] Considering the influence of excitation current on the design, the interpolation model for the current angle is established as follows:

[0025]

[0026] Among them, z γ For design variables, γ is the current angle, and p γ The penalty factor is used; for the static magnetic finite element model, considering the continuous evolution of the winding current, the three-phase current density at different rotation positions is:

[0027]

[0028] Among them, J m Let θ be the amplitude of the current density. k This refers to the rotor's rotational position;

[0029] Step 4: Finite element analysis, solve the governing equations;

[0030] The governing equations for the electromagnetic field of the motor are calculated using the following formula:

[0031] SA = Q (10)

[0032] Where S is the stiffness matrix of the nonlinear material, A is the unknown nodal vector potential, and Q is the excitation source vector related to the current density J.

[0033] The governing equations of elasticity are calculated using the following formula:

[0034] KU=F (11)

[0035] Where K is the mechanical stiffness matrix, U is the unknown nodal displacement vector, and F is the load vector; the expression for the element load is:

[0036]

[0037] Where m e For the unit physical mass, r e Let ω be the centroid radius of each element in the rotor design domain. e Angular velocity;

[0038] Step 5: Establish a synchronous optimization model for the rotor and stator, and determine the objectives and mechanical constraints for the rotor;

[0039] The objective of the optimization model is to maximize the motor output torque. Together with the inequality constraints related to stress and compliance, this forms the optimization model's equation set, which is as follows:

[0040]

[0041] Among them, T i The torque at the i-th rotor position, For the element von Mises stress in the design domain, σ lim N is the allowable stress of the material. e Where C is the total number of elements in the design domain, C is the compliance of the rotor structure, and C0 is a given compliance reference value.

[0042] Step 6: Construct the augmented Lagrange framework. The calculation formula is as follows:

[0043]

[0044] Where k is the iteration step, and N = N e +1 represents the number of constraints, P (k) This is a penalty item;

[0045] Step 7: Perform sensitivity analysis using the adjoint vector method;

[0046] The chain rule is used to derive the sensitivity calculation formula for the augmented Lagrange equation to design variables:

[0047]

[0048] Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (15) can be rewritten as follows:

[0049]

[0050] In the formula, the sensitivity of the penalty term to Young's modulus and physical mass can be expressed as follows:

[0051]

[0052]

[0053] In the formula, To define the penalty term P (k) Equality constraints; μ (k) The quadratic penalty factor is μ, and the update equation is μ. (k+1) =min[αμ (k) ,μ max ], where α>1 is the update parameter; Let be the Lagrange multipliers; using the adjoint variable method to accelerate the sensitivity calculation, we obtain:

[0054]

[0055]

[0056] The accompanying variables in the formula are as follows:

[0057]

[0058]

[0059] In equation (17), the chain rule yields the sensitivity of the objective function to magnetoresistivity and current density, respectively:

[0060]

[0061]

[0062] The sensitivity of torque to reluctance and current density in the two equations is solved using the adjoint variable method:

[0063]

[0064]

[0065] Setting the terms within the brackets in equations (26) and (27) to 0, we obtain the accompanying variables:

[0066]

[0067]

[0068] Therefore, the sensitivity of torque to reluctance and current density is obtained:

[0069]

[0070]

[0071] Finally, combining equations (24)(25) and (30)(31), the sensitivity of the objective function to magnetoresistivity and current density are respectively:

[0072]

[0073]

[0074] Where n is the number of sampling points for calculating the average torque, and I is the row vector;

[0075] Step 8: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle:

[0076] L=f+λ T (SA-Q) (33)

[0077]

[0078] Setting the last term to 0, we obtain the solution for the adjoint variable:

[0079]

[0080] Ultimately, the sensitivity of the objective function to the current angle is:

[0081]

[0082] Step 9: Based on the sensitivity information, the design variables are updated and iterated using the gradient-based MMA optimization algorithm;

[0083] Step 10: Determine convergence. If convergence is not achieved, return to step 3 until the convergence condition is met, and output the optimization result.

[0084] Furthermore, the design variables z in formulas (16) and (17) of step 7 fall into three categories: one category is the reluctance, Young's modulus, and unit physical mass of the electric steel used for interpolation within the rotor design domain; the other two categories are the reluctance and current density of the copper coil used for interpolation within the stator design domain. In formula (17), Young's modulus, mass, and unit reluctance in the rotor design domain are only related to the design variables. Related to the unit current density and unit reluctance in the stator design domain and design variables. and Related, the subscripts i and j represent the unit numbers in the rotor and stator design domains, respectively; in the stator design domain, the reluctance in formula (17) is related to the design variables. The sensitivity is obtained by the direct method:

[0085]

[0086] Similarly, current density affects design variables. The sensitivity is also derived by the direct method:

[0087]

[0088] The beneficial effects of this invention are as follows: This invention proposes a multi-material topology optimization method for synchronous reluctance motors based on the variable density method. The method includes: designing a method for synchronous topology optimization of the stator and rotor. The rotor and stator are inseparable components of the motor, and their performance is closely related. By considering the interdependence and interaction between the rotor and stator, the optimization process can achieve optimal material allocation in both regions, improving the overall performance of the motor compared to optimizing a single region individually. Furthermore, this synchronous optimization improves the integrity and flexibility of the automated design process, facilitating the exploration of new material configurations. Simultaneously, considering structural design and current angle, the limits of the average torque output designed under a given current amplitude are explored. According to mechanical calculations, the rotor structure's flexibility and stress after synchronous optimization with the stator satisfy the constraints of the optimization model, ensuring the rotor's structural stiffness and manufacturability. Attached Figure Description

[0089] The present invention is described with reference to the following figures:

[0090] Figure 1 This is a flowchart of a multi-material topology optimization method for synchronous reluctance motors based on the variable density method.

[0091] Figure 2 Design domain partitioning diagram;

[0092] Figure 3 To optimize the method iteration graph;

[0093] Figure 4 This is a structural configuration diagram during the stator iteration process;

[0094] Figure 5 This is a structural configuration diagram during the rotor iteration process; Detailed Implementation

[0095] To better explain and facilitate understanding of the present invention, a detailed description of the invention is provided below with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention. Furthermore, it should be noted that, unless otherwise specified, the embodiments and features described in the embodiments of the present invention can be combined with each other; for ease of description, only the parts relevant to the invention are shown in the accompanying drawings.

[0096] To overcome the shortcomings of current technologies that only perform topology optimization on the rotor portion of the motor, such as design incompleteness and neglect of the interaction between the stator and rotor, this invention proposes a multi-material topology optimization method for synchronous reluctance motors based on the variable density method. The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0097] Example 1

[0098] Figure 1 This is a schematic diagram of a multi-material topology optimization method for a synchronous reluctance motor based on the variable density method, as shown in another embodiment of the present invention. Figure 1 As shown, the method includes:

[0099] Step 1: Establish the geometric model of the motor, define the design domains of the rotor and stator, and mesh the design domains of the rotor and stator. Figure 2 This is a mesh partitioning diagram of the stator and rotor design domain in another embodiment of the present invention. Typically, the size of the design domain, i.e., the number of elements within it, significantly impacts computational speed. To alleviate the computational burden of complex finite element analysis, this study employs a fractional geometric model. Figure 2 The diagram shows a quarter-finite element model of the synchronous reluctance motor, along with the design domains for the rotor and stator, highlighted in blue and green, respectively. The stator design domain is half the length of a slot, with the centerline of the slot spacing aligned with the x-axis. A mirrored copy of the design domain achieves the complete pitch shape design of the stator winding slots. Therefore, for the stator, the design domain is a 1 / 48 model. For the optimized design of the double-layer winding, to reduce optimization difficulty, the two winding structures are also separated along the axis of symmetry. The number of coil turns in the winding is negligible; the current excitation is reflected through the current in the design domain. For the rotor, half of the analysis domain is set as the design domain, i.e., a 1 / 8 model. Meanwhile, the periodic boundary conditions are also guaranteed.

[0100] Using the same symmetric replication strategy, all information, including the stator and rotor mesh information and other physical properties, can be mirrored from the design domain to the other half of the model in each iteration, and then mirrored to other areas of the model. Therefore, the number of design variables related to the number of elements is significantly reduced.

[0101] Step 2: Perform Helmholtz filtering and Heaviside projection filtering on the design variables within the stator and rotor design domains;

[0102] In SIMP-based motor structure topology optimization, the final numerical results often exhibit a checkerboard pattern, which is detrimental to the feasibility and manufacturability of the design. Therefore, before performing material interpolation, this invention employs a three-field filtering strategy to ensure mesh independence and clear black-and-white optimization results, thereby mapping design variables from the original design field to the physical density field.

[0103] First, Helmholtz filtering is performed. The filtering equation is:

[0104]

[0105] Where i = 0, 1, 2 represents the design variable type, and its design interval is (0, 1], N e The number of elements in the design domain, z i,k Let x be the i-th design variable in the k-th unit. i,k Let M be the centroid coordinates of the k-th unit in the i-th design variable. i,j,k The number of cells within the filter radius R. These are the design variables in the filtered density field; these design variables belong to both the rotor and stator design domains. The linear weighting function between the centroids of the two elements can be expressed as:

[0106] w(x i,k )=R-|x i,k -x i,j | (40)

[0107] Where R is the filtration radius;

[0108] Using Heaviside projection filtering on density variables After smoothing, the equation is:

[0109]

[0110] Where β is the steepness of the projection, and η is the threshold of the filter density. These are physical density field variables; the value of β increases regularly with the number of iterations. Finally, through two filtering steps, the original design variables are mapped to the physical density field.

[0111] Step 3: Construct interpolation models for the multiphysics material properties of the stator and rotor, respectively;

[0112] This invention involves the simultaneous optimization of two different regions, and material interpolation will be performed separately for each. For the rotor domain, the materials include steel and air. To distinguish between these two materials in a static electromagnetic field, a robust and stable electromagnetic performance analysis based on magnetoresistance is chosen. Magnetic permeability is selected to differentiate the material properties of the steel and air in the rotor section, and the interpolation formula is:

[0113]

[0114] Where ν fe and ν air The magnetic permeability of iron and air, respectively, p r The penalty factor is used. Since the interpolation formula only involves the design variables of the rotor, the subscript i in (4) is equal to 0.

[0115] The stator domain mainly consists of three materials: copper coils, electrical steel, and air; therefore, two design variables are defined. and To optimize the synchronization of the electric steel and coil, magnetoresistivity and current density are selected to define the distribution of these materials, and the interpolation formula is as follows:

[0116]

[0117] Where, ν cu J is the reluctance of the copper coil. cu Current density of copper and p represents the filtered design variables in the stator design domain. v and p c This is a penalty factor.

[0118] To further ensure the structural strength of the rotor, finite element analysis considering mechanics is performed in the rotor region, using Young's modulus for interpolation. The model is as follows:

[0119]

[0120] Where E fe Let be the Young's modulus of the electric steel, and p be the penalty factor. Considering the mechanical load calculation of the rotor, and given the non-uniformity of the triangular element, an interpolation model between the design variables and the physical mass is established as follows:

[0121]

[0122] Where, m fe Let be the physical mass of the unit, and ⊙ be the Hadamard product;

[0123] Furthermore, to further maximize the motor's output performance under MTPA conditions, this invention investigates the influence of the current angle on the output torque. Therefore, the current angle, which is related to current density, is optimized along with other design variables. Considering the influence of the excitation current on the design, an interpolation model for the current angle is established as follows:

[0124]

[0125] Among them, z γ For design variables, γ is the current angle, and p γ The penalty factor is used; for the static magnetic finite element model, considering the continuous evolution of the winding current, the three-phase current density at different rotation positions is:

[0126]

[0127] Among them, J m Let θ be the amplitude of the current density. k This refers to the rotor's rotational position;

[0128] Step 4: Finite element analysis, solve the governing equations;

[0129] The governing equations for the electromagnetic field of the motor are calculated using the following formula:

[0130] SA = Q (48)

[0131] Where S is the stiffness matrix of the nonlinear material, A is the unknown nodal vector potential, and Q is the excitation source vector related to the current density J.

[0132] The governing equations of elasticity are calculated using the following formula:

[0133] KU=F (49)

[0134] Where K is the mechanical stiffness matrix, U is the unknown nodal displacement vector, and F is the load vector; the expression for the element load is:

[0135] F e =m e r e ω e 2 (50)

[0136] Where m e For the unit physical mass, r e Let ω be the centroid radius of each element in the rotor design domain. e Angular velocity;

[0137] Step 5: Establish a synchronous optimization model for the rotor and stator, and determine the objectives and mechanical constraints for the rotor;

[0138] The purpose of the optimization model in this invention is to improve the output performance of the motor while ensuring the mechanical performance of the rotor. Therefore, the objective of the optimization model is defined as maximizing the output torque of the motor. This objective, together with the inequality constraints of stress and compliance, constitutes the system of equations for the optimization model. The system of equations is as follows:

[0139]

[0140] Among them, T i The torque at the i-th rotor position is For the element von Mises stress in the design domain, σ lim N is the allowable stress of the material. e Where C is the total number of elements in the design domain, C is the compliance of the rotor structure, and C0 is a given compliance reference value.

[0141] Step 6: Construct the augmented Lagrange framework;

[0142] As can be seen from equation (13), the number of constraints is related to the number of elements in the finite element mesh. In order to reduce the computation time, this invention adopts the augmented Lagrangian (AL) method to solve large-scale optimization problems. The calculation formula is as follows:

[0143]

[0144] Where k is the iteration step, and N = N e +1 represents the number of constraints, P (k) This is a penalty item;

[0145] Step 7: Perform sensitivity analysis using the adjoint vector method;

[0146] Since the method proposed in this invention employs a gradient-based algorithm, the derivation of the sensitivity of motor performance and characteristics to design variables is of great significance. Because the Adjoint Variable Method (AVM) is computationally efficient and easily implemented in finite element models, this paper uses AVM to calculate the sensitivity. The chain rule is used to derive the sensitivity calculation formula for the augmented Lagrange equation to design variables:

[0147]

[0148] Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (15) can be rewritten as follows:

[0149]

[0150] In the formula, the sensitivity of the penalty term to Young's modulus and physical mass can be expressed as follows:

[0151]

[0152]

[0153] In the formula, To define the penalty term P (k) Equality constraints; μ (k) The quadratic penalty factor is μ, and the update equation is μ. (k+1) =min[αμ (k) ,μ max ], where α>1 is the update parameter; Let be the Lagrange multipliers; using the adjoint variable method to accelerate the sensitivity calculation, we obtain:

[0154]

[0155]

[0156] The accompanying variables in the formula are as follows:

[0157]

[0158]

[0159] In equation (17), the chain rule yields the sensitivity of the objective function to magnetoresistivity and current density, respectively:

[0160]

[0161]

[0162] The sensitivity of torque to reluctance and current density in the two equations is solved using the adjoint variable method:

[0163]

[0164]

[0165] Setting the terms within the brackets in equations (26) and (27) to 0, we obtain the accompanying variables:

[0166]

[0167]

[0168] Therefore, the sensitivity of torque to reluctance and current density is obtained:

[0169]

[0170]

[0171] Finally, combining equations (24)(25) and (30)(31), the sensitivity of the objective function to magnetoresistivity and current density are respectively:

[0172]

[0173]

[0174] Where n is the number of sampling points for calculating the average torque, and I is the row vector;

[0175] Step 8: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle:

[0176] L=f+λ T (SA-Q) (71)

[0177]

[0178] Setting the last term to 0, we obtain the solution for the adjoint variable:

[0179]

[0180] Ultimately, the sensitivity of the objective function to the current angle is:

[0181]

[0182] Step 9: Based on the sensitivity information, the design variables are updated and iterated using the gradient-based MMA optimization algorithm;

[0183] Step 10: Determine convergence. If convergence is not achieved, return to step 3 until the convergence condition is met, and output the optimization result.

[0184] Figure 3 The diagram shows the iteration history of the optimization method, where curves marked with "*" represent the iteration curves for average torque, and curves marked with "o" represent the optimization iteration curves for lead angle.

[0185] Figure 4 The complete topological configuration of different iteration steps is obtained by mirror copying the rotor design domain after optimization. Its initial structure is a conjectured porous structure. Figure 5 The 1 / 4 topology configurations for different iteration steps are obtained after optimizing the stator design domain. Figure 4 and Figure 5 The topology optimization iterative process shows that the overall configuration of the rotor and stator structures will emerge in fewer iterations, and subsequent iterations will further refine the details of the configuration, demonstrating the computational efficiency of the solution method and the close relationship between the two.

[0186] It should be noted that any reference numerals placed between parentheses in the claims should not be construed as limiting the claims. The word "comprising" does not exclude the presence of components or steps not listed in the claims. The words "a" or "an" preceding a component do not exclude the presence of a plurality of such components. The invention can be implemented by means of hardware comprising several different components and by means of a suitably programmed computer.

[0187] Furthermore, it should be noted that in the description of this specification, the terms "one embodiment," "some embodiments," "embodiment," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Furthermore, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0188] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the claims should be interpreted to include both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0189] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, then this invention should also include these modifications and variations.

Claims

1. A multi-material topology optimization method for synchronous reluctance motors based on the variable density method, characterized in that, The method steps are as follows: Step 1: Establish the geometric model of the motor, define the design domains of the rotor and stator, and mesh the design domains of the rotor and stator. Step 2: Perform Helmholtz filtering and Heaviside projection filtering on the design variables within the stator and rotor design domains; the Helmholtz filtering equation is: Where i = 0, 1, 2 represents the design variable type, and its design interval is (0, 1], N e The number of elements in the design domain, z i,k Let x be the i-th design variable in the k-th unit. i,k Let M be the centroid coordinates of the k-th unit in the i-th design variable. i,j,k The number of cells within the filter radius R. The design variables are those in the filtered density field; the linear weighting function between the centroids of the two elements can be expressed as: w(x i,k )=R-|x i,k -x i,j | (2) Where R is the filtration radius; Using Heaviside projection filtering on density variables After smoothing, the equation is: Where β is the steepness of the projection, and η is the threshold of the filter density. It is a physical density field variable; Step 3: Construct interpolation models for the multiphysics material properties of the stator and rotor; the material property used to distinguish steel from air in the rotor is permeability, and the interpolation formula is: Where ν fe and ν air The magnetic permeability of iron and air, respectively, p r The penalty factor; since only the rotor part is involved in the design variables, the subscript i in (4) is equal to 0; the stator part contains copper coils, electric steel and air, and the reluctance and current density are chosen to define the distribution of these materials. Two design variables are involved in the interpolation model. and The interpolation formula is: Where, ν cu J is the reluctance of the copper coil. cu Current density of copper and p represents the filtered design variables in the stator design domain. v and p c As a penalty factor; in the rotor region, considering mechanical finite element analysis, the interpolation model of Young's modulus is established as follows: Where E fe Let p be the Young's modulus of the electric steel, and p be the penalty factor; considering the mechanical load calculation of the rotor, the interpolation model between the design variables and the physical mass is established as follows: Where, m fe Let be the physical mass of the unit, and ⊙ be the Hadamard product; Considering the influence of excitation current on the design, the interpolation model for the current angle is established as follows: Among them, z γ For design variables, γ is the current angle, and p γ The penalty factor is used; for the static magnetic finite element model, considering the continuous evolution of the winding current, the three-phase current density at different rotation positions is: Among them, J m Let θ be the amplitude of the current density. k This refers to the rotor's rotational position; Step 4: Finite element analysis, solve the governing equations; The governing equations for the electromagnetic field of the motor are calculated using the following formula: SA = Q (10) Where S is the stiffness matrix of the nonlinear material, A is the unknown nodal vector potential, and Q is the excitation source vector related to the current density J. The governing equations of elasticity are calculated using the following formula: KU=F (11) Where K is the mechanical stiffness matrix, U is the unknown nodal displacement vector, and F is the load vector; the expression for the element load is: Where m e For the unit physical mass, r e Let ω be the centroid radius of each element in the rotor design domain. e Angular velocity; Step 5: Establish a synchronous optimization model for the rotor and stator, and determine the objectives and mechanical constraints for the rotor; The objective of the optimization model is to maximize the motor output torque. Together with the inequality constraints related to stress and compliance, this forms the optimization model's equation set, which is as follows: Among them, T i The torque at the i-th rotor position is For the element von Mises stress in the design domain, σ lim N is the allowable stress of the material. e Where C is the total number of elements in the design domain, C is the compliance of the rotor structure, and C0 is a given compliance reference value. Step 6: Construct the augmented Lagrange framework. The calculation formula is as follows: Where k is the iteration step, and N = N e +1 represents the number of constraints, P (k) This is a penalty item; Step 7: Perform sensitivity analysis using the adjoint vector method; The chain rule is used to derive the sensitivity calculation formula for the augmented Lagrange equation to design variables: Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (15) can be rewritten as follows: In the formula, the sensitivity of the penalty term to Young's modulus and physical mass can be expressed as follows: In the formula, To define the penalty term P (k) Equality constraints; μ (k) The quadratic penalty factor is μ, and the update equation is μ. (k+1) =min[αμ (k) ,μ max ], where α>1 is the update parameter; Let be the Lagrange multipliers; using the adjoint variable method to accelerate the sensitivity calculation, we obtain: The accompanying variables in the formula are as follows: In equation (17), the chain rule yields the sensitivity of the objective function to magnetoresistivity and current density, respectively: The sensitivity of torque to reluctance and current density in the two equations is solved using the adjoint variable method: Setting the terms within the brackets in equations (26) and (27) to 0, we obtain the accompanying variables: Therefore, the sensitivity of torque to reluctance and current density is obtained: Finally, combining equations (24)(25) and (30)(31), the sensitivity of the objective function to magnetoresistivity and current density are respectively: Where n is the number of sampling points for calculating the average torque, and I is the row vector; Step 8: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle: L=f+λ T (SA-Q) (33) Setting the last term to 0, we obtain the solution for the adjoint variable: Ultimately, the sensitivity of the objective function to the current angle is: Step 9: Based on the sensitivity information, the design variables are updated and iterated using the gradient-based MMA optimization algorithm; Step 10: Determine convergence. If convergence is not achieved, return to step 3 until the convergence condition is met, and output the optimization result.

2. The multi-material topology optimization method for synchronous reluctance motors based on the variable density method according to claim 1, characterized in that: The design variables z in formulas (16) and (17) in step 7 fall into three categories: one category is the reluctance, Young's modulus, and unit physical mass of the electric steel used for interpolation within the rotor design domain; the other two categories are the reluctance and current density of the copper coil used for interpolation within the stator design domain. In formula (17), Young's modulus, mass, and unit reluctance in the rotor design domain are only related to the design variables. Related to the unit current density and unit reluctance in the stator design domain and design variables. and Related, the subscripts i and j represent the unit numbers in the rotor and stator design domains, respectively; in the stator design domain, the reluctance in formula (17) is related to the design variables. The sensitivity is obtained by the direct method: Similarly, current density affects design variables. The sensitivity is also derived by the direct method: .

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