An equal geometry non-uniform magnetization motor optimization method

Abstract

This invention discloses an optimization method for non-uniform magnetization motors with uniform geometry, relating to the field of permanent magnet motor magnetization design. Under the condition that the mechanical pole arc, inter-pole spacing, and rotor outer circle air gap geometry remain unchanged, the radial and tangential magnetization component coefficients of the magnetization partition are used as independent design variables, with the magnetization direction determined by these two variables. Candidate magnetization codes are mapped to permanent magnet subdomain magnetization source terms and interface magnetization source terms, and the air gap magnetic field is solved by combining analytical subdomain boundary continuity conditions. The total harmonic distortion rate of back EMF, torque ripple rate, negative average electromagnetic torque, and magnetization manufacturability index are minimized as objectives. With complete demagnetization parameters, a demagnetization margin evaluation and constraint are added. The final magnetization coding scheme is determined through multi-objective optimization, trade-off selection, and verification. This method allows for the reuse of coefficient matrices and their decomposition results under fixed geometry and equivalent permeability conditions, reducing redundant assembly and decomposition calculations.

Description

Technical Field

[0001] This invention relates to the field of magnetized motor technology, and specifically to an optimization method for a non-uniform magnetized motor with equal geometry. Background Technology

[0002] Surface-mounted permanent magnet synchronous motors (PMSMs) are widely used in servo drives, robot actuators, automotive drives, and precision equipment due to their simple structure, high power density, and good control performance. The air gap magnetic field distribution of these motors directly affects the back EMF waveform, cogging torque, electromagnetic torque pulsation, average torque, and the risk of permanent magnet demagnetization. Therefore, optimizing the air gap magnetic field to improve the overall performance of the motor has always been a key research focus in this field.

[0003] To improve the air gap magnetic field waveform and reduce harmonic content, existing technologies mainly focus on two aspects: magnetic pole geometry and magnetization method. Common methods for optimizing geometry include adjusting the pole arc coefficient, using eccentric or bread-shaped magnetic poles, segmenting the magnets, and creating a non-uniform air gap. For example, some studies have optimized the design of surface-mounted permanent magnet synchronous motors with segmented outer circular arc eccentric structures, reducing air gap magnetic flux density harmonics and suppressing torque ripple by adjusting the eccentricity and pole arc coefficient. Other studies have used bread-shaped or eccentrically clipped pole structures to create a non-uniform air gap, aiming for sinusoidal back electromotive force and low torque ripple. Furthermore, by dividing the rotor magnetic poles into an array of permanent magnets of equal amplitude but unequal width and arranging them according to a modulation method, the air gap magnetic flux density waveform can also be made to approximate a sine wave. While the aforementioned geometric optimization methods can improve magnetic field distribution to some extent, they also have significant limitations: once the mechanical pole arcs, inter-pole spacing, or rotor outer circle air gap geometry change, the prototype structure, tooling fixtures, magnet dimensions, and assembly tolerances often need to be adjusted simultaneously; for motor projects with already determined rotor structures and magnet mechanical dimensions, relying solely on geometric modifications is not always applicable. Furthermore, during the finite element optimization process, changes in geometric boundaries can cause model reconstruction, mesh re-division, and boundary condition resetting, leading to a large amount of repetitive calculations and significantly increasing the optimization cycle and computational cost.

[0004] In terms of magnetization optimization, the Halbach array is a typical technical approach. Halbach magnet arrays can effectively enhance the air gap side magnetic field, weaken the rotor yoke flux, and simultaneously reduce the harmonic content of the air gap magnetic flux density waveform. Building upon this, further improved structures such as non-uniform Halbach arrays and unequal-thickness magnetic poles have emerged. For example, some studies have employed a three-segment non-uniform Halbach magnetized external rotor permanent magnet synchronous motor structure, which improves torque density and reduces harmonic distortion and torque ripple while reducing the amount of permanent magnets used. However, most existing magnetization optimization schemes still rely on adjusting magnet geometric parameters (such as magnet width, thickness, and position). The design variables of the magnetization scheme are coupled with the geometric parameters, lacking a universal expression that can independently describe and adjust the magnetization distribution while keeping the magnet mechanical boundaries completely unchanged.

[0005] In the field of magnetic field calculation methods, the analytical subdomain method is widely used due to its high computational efficiency and good accuracy. Existing studies have divided the cross-section of the motor into subdomains such as permanent magnets, air gaps, slots, and stator slots, and solved the air gap magnetic field by establishing Poisson's equation and Laplace's equation and utilizing boundary continuity conditions. However, most existing analytical subdomain models are established for specific magnetic pole geometries (such as tile-shaped, bread-shaped, and eccentric types). When the magnetization scheme changes, it is often necessary to rebuild or adjust the structure of the analytical model. A unified method that directly incorporates the magnetization distribution as an independent variable into the analytical subdomain solution framework has not yet been formed.

[0006] In summary, existing technologies still have shortcomings in the following aspects: First, geometric optimization methods change the shape of the magnet or the rotor structure, increasing the complexity and cost of prototype manufacturing and assembly, and facing the computational burden of repeated modeling and mesh reconstruction in finite element iterative optimization; Second, magnetization optimization lacks a unified encoding method with radial and tangential components as independent variables; Third, there is a lack of a direct coupling mechanism between the magnetization scheme and the analytical subdomain model, making it difficult to efficiently map changes in magnetization distribution into inputs for analytical solutions; Fourth, existing optimization methods do not adequately consider engineering constraints such as demagnetization risk and magnetization manufacturability, and the optimization results face significant challenges when implemented in actual engineering projects.

[0007] In view of the above, this application is hereby submitted. Summary of the Invention

[0008] This invention provides an optimization method for non-uniform magnetized motors with uniform geometry, which can at least partially improve the above-mentioned problems.

[0009] To achieve the above objectives, the present invention adopts the following technical solution: A method for optimizing a non-uniformly magnetized motor with equal geometry, comprising: Step S1: Obtain the input motor fixed model parameters, evaluation condition parameters, design variable boundary and process feasibility parameters, engineering performance constraint parameters, and optimization auxiliary parameters; when the permanent magnet evaluation temperature and the corresponding demagnetization knee magnetic field strength data are complete, enable demagnetization margin evaluation and demagnetization margin constraint; keep the geometric parameters, material equivalent permeability, winding parameters, and analytical expansion order in the motor fixed model parameters unchanged during the same optimization process; Step S2: Determine the magnetization coding allowable domain based on the design variable boundary and process feasibility parameters, obtain multiple candidate schemes corresponding to the magnetic pole to be optimized based on the magnetization coding allowable domain, and construct the total magnetization coding matrix of each candidate scheme; Step S3: Calculate the magnetization intensity vector field of the candidate scheme based on the total magnetization coding matrix, determine the volume magnetization source term according to the spatial variation of magnetization intensity within each magnetization partition, and determine the interface magnetization source term according to the magnetization intensity jump on both sides of the common interface between adjacent magnetization partitions and between the magnetized region and the adjacent non-magnetized analytical subdomain. Step S4: Project the basis functions within the domain and the basis functions of the interface magnetization source terms into the volume magnetization source terms, respectively, and assemble them into the right-hand term vector of the magnetization source. Based on the fixed analytical subdomain geometry, material equivalent permeability, interface connection relationship and analytical expansion order, establish the linear equation between the coefficient matrix, analytical expansion coefficient vector, right-hand term vector of the magnetization source and right-hand term vector of the winding current. Step S5: For each evaluation sampling point, decompose the coefficient matrix corresponding to the evaluation sampling point; reuse the decomposition result when evaluating different candidate schemes, and update the right-hand term vector of the magnetization source according to the current candidate scheme to obtain the corresponding analytical expansion coefficient vector, and calculate the air gap magnetic flux density and the magnetic field strength inside the permanent magnet from the analytical expansion coefficient vector. Step S6: Calculate the total harmonic distortion rate of the back electromotive force, torque ripple rate, average torque, and magnetization manufacturability index based on the air gap magnetic flux density and magnetization intensity vector field to form the mandatory evaluation vector of the candidate scheme; when the demagnetization margin evaluation is enabled, calculate the demagnetization margin based on the magnetic field strength inside the permanent magnet. Step S7: Repeat steps S3 to S6 to obtain the evaluation vector of each candidate solution. Based on the evaluation vector, establish a multi-objective minimization evaluation function for each candidate solution, and select candidate solutions that meet the preset engineering constraints to generate a Pareto feasible non-dominated solution set. Step S8: Perform normalization compromise evaluation on the candidate schemes in the Pareto feasible non-dominated solution set, and select the candidate scheme with the smallest compromise evaluation value as the coding scheme to be reviewed. The review is carried out using the review method corresponding to the reviewed evaluation quantity, and the coding scheme to be reviewed that meets the preset allowable deviation and still meets the engineering performance constraints of this round is determined as the final magnetization coding scheme.

[0010] In summary, the purpose of this invention is to provide an optimization method for non-uniform magnetization motors with uniform geometry. This method addresses the problems of difficulty in uniformly representing non-uniform magnetization distribution, lack of direct correspondence between magnetization encoding and analytical magnetic field models, and high computational complexity in selecting candidate magnetization schemes, all while maintaining the mechanical geometry of the permanent magnets. This method is applicable to the design of pole magnetization distribution in surface-mounted permanent magnet synchronous motors. It can encode, calculate, filter, and output the radial and tangential magnetization components of each magnetization zone while keeping the mechanical pole arcs, adjacent pole spacing, and rotor outer circumference air gap geometry constant.

[0011] This method is implemented under the condition that the mechanical pole arc of the permanent magnet, the spacing between adjacent magnetic poles, the air gap geometry of the rotor outer circle, and the geometric division of the analytical subdomain remain unchanged. Specifically, a magnetization coding matrix is ​​established according to the magnetic pole number and the magnetization partition number. The matrix elements of the magnetization coding matrix include radial magnetization component coefficients and tangential magnetization component coefficients. The magnetization direction is determined by the two component coefficients and is not used as an independent design variable. The magnetization intensity distribution of each magnetization partition of the permanent magnet is obtained according to the magnetization coding matrix, and the magnetization source contribution inside each magnetization partition and the magnetization vector jump contribution at the common interface of adjacent magnetization partitions are mapped to the right-hand side terms of the volume magnetization source and the interface magnetization source of the analytical subdomain equation, respectively. The rotor core subdomain is then combined with the magnetization source matrix. , permanent magnet subdomain air gap subdomain slot subdomain and the subdomain within the slot Boundary continuity conditions (such as) Figure 3 (As shown), the air gap magnetic field corresponding to the candidate schemes is solved. Under the condition that the geometric boundary, interface connection relationship, analytical expansion order and material equivalent permeability remain unchanged, the coefficient matrix and its decomposition results corresponding to each evaluation sampling point can be reused among different candidate magnetization schemes, and the right-hand side of the magnetization source is updated with the candidate magnetization coding. A multi-objective optimization method is adopted, with the total harmonic distortion rate of back EMF, torque ripple rate, negative average torque and magnetization manufacturability index as four minimization objectives, and the lower limit of average torque and the upper limit of magnetization manufacturability as engineering constraints. When the evaluation temperature of permanent magnet and the magnetic field strength at the demagnetization knee point corresponding to that temperature are obtained, the demagnetization margin is further calculated and the corresponding constraints are enabled. A compromise candidate scheme is selected from the Pareto feasible non-dominated solution set that satisfies the engineering constraints of this round, and the final magnetization coding scheme is determined after verification.

[0012] Compared with existing technologies, the advantages of this method are as follows: 1. It uses radial and tangential magnetization component coefficients to uniformly encode each magnetization zone, and the magnetization direction is determined by the two component coefficients, enabling non-uniform magnetization distribution to be used as a calculable design variable in the optimization process, thus solving the problem of difficult parametric modeling of magnetization distribution. 2. It forms the spatial variation of magnetization intensity within the zone and the abrupt change in magnetization intensity at the common interface into the volume magnetization source term and the known term on the right side of the interface magnetization source, respectively, establishing the correspondence between magnetization encoding and analytical subdomain magnetic field solution, ensuring the physical accuracy of magnetic field calculation under different magnetization schemes. 3. Under the condition that the geometry, material equivalent permeability, interface connection relationship, and analytical expansion order remain unchanged, the coefficient matrix and its decomposition results are reused, and the right-side term of the magnetization source is updated with the candidate magnetization encoding to reduce repeated matrix assembly and decomposition calculations, significantly improving optimization efficiency. 4. Incorporate total harmonic distortion rate of back EMF, torque ripple rate, negative average electromagnetic torque, and magnetization manufacturability into the multi-objective evaluation. When relevant material data is complete, add demagnetization margin evaluation and constraints to ensure that the selection of candidate magnetization schemes considers both electromagnetic performance and process feasibility. 5. Use a verification method corresponding to the evaluated quantities to verify the magnetization coding scheme to be verified, providing an independent verification basis for determining the final magnetization coding scheme and ensuring the reliability of the design results. Attached Figure Description

[0013] Figure 1 This is a flowchart illustrating the optimization method for a non-uniform magnetized motor with equal geometry provided in this embodiment of the invention.

[0014] Figure 2 This is a schematic diagram of the dual-component magnetization encoding matrix and its magnetization partition mapping provided in an embodiment of the present invention.

[0015] Figure 3 This is a non-limiting structural diagram of the magnetization source term analytic subdomain model provided in an embodiment of the present invention.

[0016] Figure 4 This is a flowchart of multi-objective optimization and conditional constraint screening of candidate magnetization codes provided in an embodiment of the present invention.

[0017] Figure 5 This is a flowchart of the compromise solution selection and condition verification provided in the embodiments of the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0019] refer to Figure 1As shown, the first embodiment of the present invention discloses an optimization method for a non-uniform magnetized motor with equal geometry, which can be executed by a non-uniform magnetized motor optimization device (hereinafter referred to as the optimization device), specifically, by one or more processors within the optimization device, to implement the following method: Step S1: Obtain the input motor fixed model parameters, evaluation condition parameters, design variable boundary and process feasibility parameters, engineering performance constraint parameters, and optimization auxiliary parameters; when the permanent magnet evaluation temperature and the corresponding demagnetization knee magnetic field strength data are complete, enable demagnetization margin evaluation and demagnetization margin constraint; keep the geometric parameters, material equivalent permeability, winding parameters, and analytical expansion order in the motor fixed model parameters unchanged during the same optimization process; Specifically, in this embodiment, the fixed model parameters of the motor include basic motor parameters, material parameters, and winding parameters. The basic motor parameters include the geometric dimensions of the rotor core, permanent magnet, air gap, slot and in-slot sub-domain, number of pole pairs, mechanical pole arc of permanent magnet, spacing between adjacent magnetic poles, air gap length, and effective axial length. (For calculations of magnetic flux linkage, back electromotive force, and electromagnetic torque), the material parameters include the relative permeability of each subdomain. Remanence of permanent magnets Reference equivalent magnetization ( (where the permeability is in vacuum), the winding parameters include the number of turns in series per phase. Normalized A-phase winding function ; The evaluation operating condition parameters include electric angular velocity. Winding current corresponding to each evaluation sampling location The total number of sampling points within one electrical cycle Discrete mechanical rotor position (The positions of each discrete mechanical rotor are arranged in the order of the electrical cycle; the following steps use the same set of position numbers to calculate the back EMF waveform and torque waveform); when demagnetization margin evaluation is enabled, the permanent magnet evaluation temperature is also included. and the magnetic field strength at the marked demagnetizing knee point corresponding to that temperature. ; The design variable boundaries and process feasibility parameters include the upper and lower limits of the radial magnetization component coefficient ( ), Upper and lower limits of tangential magnetization component coefficient ( The set of magnetization directions allowed by the magnetization process Maximum permissible deviation in magnetization direction and minimum magnetization coefficient (Amplitude boundaries and magnetization process parameters are used to determine the allowable range of magnetization encoding.) The engineering performance constraints include the lower limit of average torque. Upper limit of manufacturability of magnetization When demagnetization margin evaluation is enabled, the engineering performance constraint parameters also include a minimum demagnetization margin lower limit. ; The optimization auxiliary parameters include a normalized reference value for magnetization amplitude variation. Direction change normalized reference angle , Direction change weight And the weights in the trade-offs between the four minimization objectives. .in, The selection weights of the four minimization objectives in step S8 are denoted as follows: For m = 1, 2, 3, 4, the following conditions are met: The above parameters are all derived from the motor design specifications, permanent magnet material data, and magnetization process requirements, and remain unchanged during the same round of candidate magnetization coding optimization.

[0020] In this embodiment, during the same optimization round, the permanent magnet mechanical pole arc, adjacent magnetic pole spacing, inner and outer radii of the permanent magnet, rotor outer circle, air gap, slot opening, and slot area boundaries remain unchanged; the number of magnetization zones and zone boundaries are determined by step S2 and remain unchanged during the same optimization round. Different candidate schemes only independently change the radial and tangential magnetization component coefficients in the magnetization coding matrix. The magnetization direction is determined by these two component coefficients and is not considered an independent design variable. The aforementioned mechanical geometry and magnetization zone boundaries do not change with the candidate schemes. Figure 3 The diagram is used to illustrate the connection relationships between the various analytical subdomains. The number of slots, poles, number of magnetization zones per pole, and specific slot shapes shown do not constitute a limitation of this invention. Figure 3 The outermost gray annular area represents the fixed stator core, and the central circular area represents the shaft hole; neither has a magnetization code, and their relevant outer boundary conditions are included in the outer boundary and reference condition matrix of step S4 according to the analytical model adopted. The reuse of the coefficient matrix is ​​conditional on the linearity of the material or the use of a fixed equivalent permeability. During implementation, the slot pole combination, the number of magnetization zones per pole, material parameters and their sources, evaluation conditions, the total number of evaluation sampling points, harmonic truncation order, the number of analytical expansion terms in each subdomain, population size, termination conditions, and the type and threshold of the verification deviation are all input from step S1 and form the implementation parameter record. The evaluation calculation and verification call the same implementation parameter record.

[0021] This embodiment uses a unified index notation: i is the magnetic pole number, j is the magnetization partition number, t is the candidate scheme number, c is the evaluation sampling point number, q is the analytic subdomain category, and n is the analytic basis function number. denoted as the row number of the projection equation, a and b represent the adjacent regions on both sides of the common interface, u represents the allowable magnetization direction number, h represents the harmonic order, and m represents the multi-objective number; q∈{r, pm, g, o, s} represent the rotor core subdomain, permanent magnet subdomain, air gap subdomain, slot subdomain, and slot subdomain, respectively. Indicates transpose, superscript This indicates the number of candidates pending review or as final candidates. , and These represent the radial, tangential, and axial unit vectors, respectively. Represents the L2 norm, ² and × represent the Laplace operator and curl operator, respectively; dΩ and ds represent the area element and the interface line element, respectively; and arg min represents the independent variable that minimizes the objective function.

[0022] Please see Figure 2 Step S2: Determine the magnetization coding allowable domain based on the design variable boundary and process feasibility parameters, obtain multiple candidate schemes corresponding to the magnetic pole to be optimized based on the magnetization coding allowable domain, and construct the total magnetization coding matrix of each candidate scheme; Specifically, step S2 further includes: constructing a magnetization coding allowable domain based on the design variable boundaries and process feasibility parameters determined in step S1, and generating multiple candidate schemes corresponding to the magnetic pole to be optimized based on the magnetization coding allowable domain and the preset total number of candidate schemes. The magnetization parameters are uniformly numbered using the pole number i, magnetization zone number j, and candidate scheme number t; where, if the number of motor pole pairs is... Then the number of magnetic poles Each magnetic pole to be optimized is divided into multiple magnetization zones along the circumference (e.g.: In the t-th candidate scheme, the magnetization encoding vector corresponding to the j-th magnetization partition of the i-th magnetic pole is defined as follows: ,in, The radial magnetization component coefficient is... The two component coefficients are tangential magnetization coefficients, and are independent design variables. The magnetization direction offset angle of this magnetization zone relative to the radial direction of the i-th magnetic pole reference is... Where, atan2 is a four-quadrant arctangent function that determines the angular quadrant based on the tangential and radial components; the magnetization direction offset angle is determined by the two magnetization component coefficients and is not considered an independent design variable; and a spatial indication function corresponding to the j-th magnetization partition of the i-th magnetic pole is defined. ,in, This is the spatial indication function corresponding to the j-th magnetization partition of the i-th magnetic pole, used to determine the magnetization encoding vector. The corresponding position within the permanent magnet region, where r is the radial coordinate. Circumferential angular coordinates This represents the j-th magnetization region of the i-th magnetic pole; Arrange the magnetization partition codes of the i-th magnetic pole according to the magnetization partition number to obtain the sub-magnetization coding matrix corresponding to the i-th magnetic pole: ; Further arranging the sub-magnetization coding matrices sequentially according to the magnetic pole number yields the total magnetization coding matrix of the t-th candidate scheme: , , Based on the design variable boundaries and process feasibility parameters determined in step S1, a magnetization coding allowable domain is constructed. For any candidate scheme, the total magnetization coding matrix satisfies: ; in, This represents the total magnetization coding matrix of the t-th candidate scheme. This represents the sub-magnetization code of the i-th magnetic pole and j-th magnetization partition. The total number of motor poles is fixed in the same round of optimization, and is related to... Figure 2 The number of rows in the magnetization coding matrix shown is consistent. The number of magnetization partitions for each magnetic pole fixed in the same round of optimization, and compared with... Figure 2 The number of columns in the magnetization encoding matrices shown is consistent. This indicates the allowable range of magnetization codes determined in advance based on motor design requirements and magnetization process conditions. Indicates the t-th candidate solution. Sub-magnetization coding matrix of each magnetic pole.

[0023] The magnetization coding allowed field Each magnetization encoding vector satisfies the radial magnetization component coefficient boundary, the tangential magnetization component coefficient boundary, the minimum magnetization intensity constraint, and the magnetization direction process constraint, namely: ,in, and These are the lower and upper limits of the radial magnetization component coefficient, respectively; and These are the lower and upper limits of the tangential magnetization component coefficient, respectively; It is the minimum magnetization coefficient; This is the set of magnetization directions allowed by the magnetization process. For the first in the set of allowed directions One direction; This represents the minimum included angle function that takes into account the periodicity of the angle. This represents the maximum permissible deviation in the magnetization direction; the above parameters remain unchanged during the same optimization process.

[0024] In this embodiment, the allowable domain for magnetization coding is determined based on the upper and lower limits of the radial magnetization component coefficient, the upper and lower limits of the tangential magnetization component coefficient, the minimum magnetization intensity coefficient, the set of magnetization directions allowed by the magnetization process, and the maximum allowable deviation of the magnetization direction input in step S1. This allowable domain defines the two magnetization component coefficients of each magnetization zone and limits the deviation between the magnetization direction determined by the two component coefficients and the allowable direction of the magnetization process, thereby ensuring the feasibility of the subsequently generated candidate magnetization codes in terms of process. By setting the minimum magnetization intensity coefficient, the two magnetization component coefficients are not simultaneously zero, thus giving the magnetization direction a definite definition. Based on this, multiple candidate schemes corresponding to the magnetic pole to be optimized are generated based on the allowable domain for magnetization coding and the total number of preset candidate schemes. The magnetization codes of each candidate scheme are all within the allowable domain, satisfying both the boundary requirements of the design variables and the engineering constraints of the magnetization process.

[0025] For each candidate scheme, each magnetic pole to be optimized is divided into multiple magnetization zones along the circumference. In the t-th candidate scheme, the sub-magnetization code corresponding to the j-th magnetization zone of the i-th magnetic pole includes two independent design variables: radial magnetization component coefficient and tangential magnetization component coefficient. These two component coefficients take values ​​within the allowed domain of the magnetization code, and the magnetization direction is determined by the two component coefficients but is not considered as an independent design variable. To establish the correspondence between the magnetization code and the actual magnetization area of ​​the permanent magnet, a spatial indication function corresponding to each magnetization zone is defined. This function determines whether a point belongs to a specific magnetization zone based on its radial and circumferential angular coordinates: the function takes a value of 1 when the coordinate point is located inside the magnetization zone, and a value of 0 otherwise. Through the above processing, each element of the magnetization code matrix can form a one-to-one correspondence with the actual spatial position on the permanent magnet, providing a clear geometric positioning basis for the subsequent calculation of the magnetization intensity vector field and the solution of the analytical subdomain magnetic field.

[0026] After defining the sub-magnetization codes for all magnetization zones within each magnetic pole, the sub-magnetization code matrices for each magnetic pole are arranged sequentially according to the magnetic pole index to form the total magnetization code matrix for the t-th candidate scheme. The rows of the total magnetization code matrix correspond to the magnetic pole index, the columns correspond to the magnetization zone index for each magnetic pole, and each matrix element is a two-dimensional magnetization code vector for the corresponding magnetization zone. The structure is well-organized, facilitating unified calling and batch updates in subsequent steps.

[0027] By combining the overall magnetization coding matrix with the spatial indicator function, a unified digital description of any non-uniform magnetization scheme is achieved. Each magnetization code in the candidate scheme can be correlated with the actual magnetization region of the permanent magnet, serving as input for subsequent construction of the magnetization intensity vector field and analytical subdomain magnetic field solution. The radial and tangential magnetization component coefficients are taken according to the design variable boundaries determined in step S1. The magnetization direction determined by the two component coefficients simultaneously satisfies the allowable direction of the magnetization process and its maximum allowable deviation requirements. By pre-defining the above conditions within the allowable domain of the magnetization code, coding schemes that do not meet the design variable boundaries or magnetization process requirements can be eliminated before the candidate schemes enter subsequent magnetic field calculations. This avoids a large number of invalid calculations due to process infeasibility during later screening, improving overall optimization efficiency.

[0028] Step S3: Calculate the magnetization intensity vector field of the candidate scheme based on the total magnetization coding matrix, determine the volume magnetization source term according to the spatial variation of magnetization intensity within each magnetization partition, and determine the interface magnetization source term according to the magnetization intensity jump on both sides of the common interface between adjacent magnetization partitions and between the magnetized region and the adjacent non-magnetized analytical subdomain. Specifically, step S3 further includes: in the t-th candidate scheme, let the polarity sign of the i-th magnetic pole be . ,in, Define the reference radial unit vector and reference tangential unit vector of the i-th magnetic pole as follows: According to the radial magnetization component coefficient determined in step S2 and tangential magnetization component coefficient The magnetization vector corresponding to the j-th magnetization region of the i-th magnetic pole is: .in, The reference equivalent magnetization intensity is used; the magnetization direction offset angle of the j-th magnetization region of the i-th magnetic pole relative to its reference radial direction is . The magnetization direction offset angle is determined by the radial magnetization component coefficient and the tangential magnetization component coefficient, and is not used as an independent design variable; each magnetization code satisfies the magnetization code allowable domain determined in step S2; according to the spatial indication function defined in step S2, the magnetization intensity vector corresponding to each magnetization partition is configured to the corresponding region of the permanent magnet, and the magnetization intensity vector field of the t-th candidate scheme in the permanent magnet region is obtained: Different candidate schemes use the same magnetic pole division, number of magnetization zones, and zone boundaries, only changing the radial magnetization component coefficient and tangential magnetization component coefficient corresponding to each magnetization zone, thereby forming different magnetization intensity vector fields.

[0029] The magnetization vector field is decomposed into radial and tangential components: And based on the radial and tangential components, the magnetization space changes within each magnetization zone are used to form the magnetization source corresponding to the t-th candidate scheme, which is then used as the volume magnetization source term: ,in, Let be the vacuum permeability. The above partial derivatives are calculated within each magnetization region and do not include the contribution of abrupt changes at the common interface of the magnetization regions due to the discontinuity of the magnetization vector.

[0030] In the t-th candidate scheme, the common interface between adjacent regions a and b Define the unit normal vector Pointing from region a to region b, and selecting a unit tangent vector according to a preset direction. The equivalent axial magnetization surface current density at the common interface is: When assembling the equations using the continuity condition of the tangential magnetic field strength, the known magnetization term moved to the right-hand side of the equations is: ,in, Used to explain the physical meaning of interface bound current. Used for assembling the right-hand side of the continuous equation for tangential magnetic field strength; when one side of the common interface does not belong to the permanent magnet region, the magnetization vector on that side is zero; and These represent the values ​​of the magnetization vectors on both sides at the common interface. and Let represent the relative permeability of region a and region b, respectively.

[0031] In this embodiment, in the t-th candidate scheme, the magnetization vector of the magnetization partition is determined based on the radial and tangential magnetization component coefficients of the j-th magnetization partition of the i-th magnetic pole, combined with the reference equivalent magnetization intensity. The radial magnetization component coefficient acts on the reference radial magnetization component, and the tangential magnetization component coefficient determines the tangential magnetization component. After obtaining the magnetization vector of each magnetization partition, the magnetization vector of each magnetization partition is multiplied by the corresponding spatial indicator function and summed. That is, the corresponding magnetization vector is assigned to the partition where the spatial indicator function is 1, and no contribution is made where the value is 0, thereby forming the magnetization intensity vector field in the entire permanent magnet region. Through the above processing, the discrete magnetization coding matrix is ​​completely mapped to the magnetization vector distribution in continuous space, realizing the accurate conversion of the magnetization scheme from digital coding to physical field.

[0032] After decomposing the magnetization vector field into radial and tangential components, the spatial variation of magnetization within each magnetization zone forms a volume magnetization source term. Since the magnetization vectors within different magnetization zones may have different amplitudes and directions, the non-uniform spatial distribution of magnetization intensity generates an equivalent volume magnetization source within the permanent magnet subdomain. This source term reflects the contribution of the spatial variation of magnetization intensity itself to the magnetic field equation. Simultaneously, when the magnetization vectors of two adjacent magnetization zones are not identical, the abrupt change in the magnetization vectors at the common interface forms an equivalent interface magnetization current. The magnitude of this interface magnetization current depends on the vector operation result of the difference between the two magnetization vectors and the interface normal direction; its physical essence is the equivalent magnetization current distribution resulting from the discontinuity of magnetization intensity at the interface.

[0033] Through the above processing, the non-uniform magnetization distribution described by the magnetization encoding matrix is ​​transformed into two types of source terms in the analytical subdomain equation: the volume magnetization source term corresponding to the spatial variation of magnetization within the partition. And the equivalent axial magnetization surface current density corresponding to the magnetization jump at the common interface. The known terms on the right-hand side of the continuity equation for tangential magnetic field strength Step S4 involves projecting the volume magnetization source term and the known term on the right side of the interface into the domain basis functions and the interface basis functions, respectively.

[0034] Please see Figure 3 Step S4: Project the known terms of the volume magnetization source and the right-hand side of the interface magnetization source into the domain basis function and the interface basis function respectively, assemble them into the right-hand side term vector of the magnetization source, and establish the linear equation between the coefficient matrix, the analytical expansion coefficient vector, the right-hand side term vector of the magnetization source and the right-hand side term vector of the winding current based on the fixed analytical subdomain geometry, the material equivalent permeability, the interface connection relationship and the analytical expansion order. Specifically, step S4 further includes: projecting the known terms on the right-hand side of the volume magnetization source and the interface magnetization source into the domain basis functions and the interface basis functions, respectively, to form the vector of the right-hand side of the volume magnetization source corresponding to the t-th candidate scheme. and the right-hand term vector of the interface magnetization source ; The right-hand term vectors of the volume magnetization source and the interface magnetization source are assembled according to the corresponding equations to generate the right-hand term vector of the magnetization source for the t-th candidate scheme: The nth component is represented as: The nth component of the right-hand side term of the volume magnetization source is: ,in, For permanent magnet subdomains. Let be the analytic basis function within the nth domain corresponding to the permanent magnet subdomain. For the interface magnetization source generated by the common interface of the magnetization partition, its right-hand side is projected from the interface magnetization source term obtained in step S3 according to the corresponding interface basis function, and its corresponding component can be written as: ;in, The set of common interfaces to be included in the assembly includes common interfaces between adjacent magnetized partitions and between magnetized regions and adjacent non-magnetized parse subdomains. This is the common interface between adjacent magnetization zones a and b. This refers to the known term on the right-hand side of the interface magnetization source formed after the tangential magnetic field strength continuity condition is moved to the right-hand side of the equation in step S3. This is the nth interface basis function corresponding to the common interface; and the coefficient matrix is ​​determined based on the right-hand side term vector of the magnetization source. The analytical expansion coefficient vector corresponding to the t-th candidate solution The calculation formula between them is: , This is the right-hand term formed by the in-slot winding current. (Coefficient matrix) It is determined by fixed geometric parameters, the relative permeability of each subdomain, and the order of analytical expansion.

[0035] Specifically, the coefficient matrix The analytical expansion coefficient vector corresponding to the t-th candidate solution The reasoning process for the calculation formula between them is as follows: For the axial components of the magnetic vector potential of the q-th subdomain, the projection corresponding to the source term is used. Expansion of analytic basis functions of order 1: In the above equation, Φ represents the analytic basis function, n represents the basis function index, and q represents the subdomain category. Let be the number of analytical expansion terms for the q-th subdomain, and x represent the corresponding analytical expansion coefficients. The expansion coefficients of each analytical subdomain are arranged according to a fixed subdomain order and basis function indices to form the analytical expansion coefficient vector corresponding to the t-th candidate magnetization code. Since the geometric boundaries are fixed, and the material parameters, analytical expansion order, and interface connectivity remain unchanged in the same round of optimization, the analytical basis functions and their spatial derivatives are independent of the candidate indices t.

[0036] First, substituting the above magnetic vector potential expansion into the governing equations of each analytic subdomain, we obtain: , , ,in, The volume magnetization source term obtained in step S3, The axial current density in the subdomain s within the slot is calculated from the three-phase winding current group at the c-th evaluation sampling point based on the winding connection relationship and the conductor distribution within the slot. The relative permeability of the subdomain s within the slot. The governing equations within the above domain are projected term by term using the corresponding domain basis functions. Let... Indicates the row number of the projection equation; for the q-th subdomain, The value range is 1 to Then the corresponding components of the domain coefficient matrix, the right-hand side of the bulk magnetization source, and the right-hand side of the winding current are respectively: , , Therefore, after assembling according to the unified order of the analytical expansion coefficient columns and the order of the projection equations within the domain, it can be represented as follows: .

[0037] Secondly, based on the relationship between magnetic flux density and magnetic vector potential, substituting the expansion of the magnetic vector potential into the formulas for calculating radial and tangential magnetic flux density, we obtain: , At the common interface of adjacent parse subdomains a and b Above, set Let be the unit normal vector pointing from subdomain a to subdomain b, and let the normal magnetic flux density satisfy the continuity condition: Substituting the magnetic vector potential expansion into the above condition for continuity of normal magnetic flux density, we get: ,in, and These represent the magnetic flux density values ​​of regions a and b on the common interface, respectively. and These are the nth analytical expansion coefficients corresponding to the analytical subdomains a and b on either side of the common interface. All terms on the left-hand side of this equation are related to the analytical expansion coefficients. Projecting the above normal magnetic flux density continuity equation according to the corresponding interface basis functions and arranging the coefficient terms according to a unified analytical expansion coefficient sequence, we form the normal magnetic flux density continuity equation, i.e.: It should be noted that substituting the magnetic flux density expression into the continuity condition for the normal magnetic flux density and extracting the corresponding terms of the analytical expansion coefficients forms the coefficient matrix of the continuity equation for the normal magnetic flux density. Substituting the material relations into the continuity condition of the tangential magnetic field strength, the coefficient matrix of the continuity equation of the tangential magnetic field strength is formed after projection and assembly. In this context, the zero on the right-hand side of the continuity equation for normal magnetic flux density is the zero vector, which has the same block dimension as the equation.

[0038] Furthermore, regarding the common interface between adjacent parse subdomains a and b... Under the condition that there is no free surface current at the interface, the tangential magnetic field strength satisfies the continuity condition: ,in, and These represent the magnetic field strength values ​​for regions a and b at their common interface. For public interface The unit tangent vector, according to the material constitutive relation: We can obtain: Substituting the constitutive relation of the material into the continuity condition of the tangential magnetic field strength, and moving the known terms related to the magnetization to the right-hand side of the equation, we obtain: ,in, and These are the magnetization vectors on either side of the common interface; when one side of the common interface does not belong to the permanent magnet subdomain, the magnetization on that side is zero. Projecting the above tangential magnetic field continuity condition according to the corresponding interface basis functions, and assembling them according to a unified analytical expansion coefficient column order and equation row order, we obtain: ,in, Let be the coefficient matrix of the continuity equation for the tangential magnetic field strength. The corresponding components of the right-hand side term of the interface magnetization source can be expressed as: .

[0039] Finally, the domain governing equations, the normal magnetic flux density continuity equation, and the tangential magnetic field strength continuity equation are assembled according to a unified analytical expansion coefficient column order and equation row order to form the domain governing equation coefficient matrix. Normal magnetic flux density continuity equation coefficient matrix and the coefficient matrix of the continuity equation for tangential magnetic field strength The coefficient matrix is ​​obtained by assembling the above three matrix blocks: Then, the right-hand terms of the bulk magnetization source, the interface magnetization source, and the winding current are arranged according to their corresponding equation rows, and zeros are added to the non-corresponding equation rows before assembly. The analytical subdomain equation corresponding to the t-th candidate scheme can then be expressed as: .

[0040] In this embodiment, the volume magnetization source terms are projected according to the basis functions used in the analytical subdomain model to form the volume magnetization source right-hand term vector; simultaneously, the interface magnetization source terms are projected according to the interface basis functions used in the analytical subdomain model to form the interface magnetization source right-hand term vector. Through the above basis function projection processing, the volume magnetization sources and interface magnetization sources distributed in continuous space are transformed into discrete right-hand term vector forms that can be directly used to solve the analytical subdomain equations, so that the source terms generated by non-uniform magnetization can form an accurate mathematical match with the basis function space of the analytical subdomain model. The two types of right-hand terms are arranged according to the corresponding equation rows, and zeros are padded in the non-corresponding equation rows before assembly to obtain the corresponding magnetization source right-hand term vector, which contains the contribution of the non-uniform magnetization distribution described by the magnetization encoding matrix to the magnetic field equation. Based on this, combined with the right-hand terms formed by the winding current in the slot, a calculation formula is established between the coefficient matrix and the analytical expansion coefficient vector corresponding to the t-th candidate scheme. In this formula, the coefficient matrix on the left side of the equation is multiplied by the analytical expansion coefficient vector to be solved, while the right side is the total right-hand side, composed of the magnetization source right-hand side vector and the slot winding current right-hand side. It is worth noting that in this calculation formula, the coefficient matrix is ​​determined only by fixed geometric parameters, the relative permeability of each subdomain, and the analytical expansion order, independent of the specific candidate magnetization codes. This means that regardless of how the candidate magnetization codes change, as long as the geometric boundaries and material properties remain unchanged, the structure and values ​​of the coefficient matrix do not need to be altered. Each iteration only requires updating the magnetization source right-hand side vector according to the current magnetization code to construct the complete solution equation. This characteristic lays a crucial mathematical foundation for the rapid batch solution of different candidate schemes in subsequent steps.

[0041] Step S5: For each evaluation sampling point, decompose the coefficient matrix corresponding to the evaluation sampling point; reuse the decomposition result when evaluating different candidate schemes, and update the right-hand term vector of the magnetization source according to the current candidate scheme to obtain the corresponding analytical expansion coefficient vector, and calculate the air gap magnetic flux density and the magnetic field strength inside the permanent magnet from the analytical expansion coefficient vector. Specifically, step S5 further includes: for the c-th evaluation sampling point in the t-th candidate scheme (each evaluation sampling point corresponds to a determined rotor position and winding current condition), based on the calculation formula between the coefficient matrix and the analytical expansion coefficient vector, calculating its corresponding analytical subdomain linear equation: ,in, Let c be the coefficient matrix corresponding to the c-th evaluation sampling point. Let be the analytical expansion coefficient vector of the t-th candidate solution at the c-th evaluation sampling point. Let c be the magnetization source right-hand term vector corresponding to the c-th evaluation sampling point. The right-hand term is formed by the slot winding current corresponding to the c-th evaluation sampling point; The analytical expansion coefficient vector of the t-th candidate solution at all evaluation sampling points is denoted as: , This is the analytical expansion coefficient vector corresponding to the first evaluation sampling point. For the first The analytical expansion coefficient vector corresponding to each evaluation sampling point, where... The total number of evaluation sampling points used within one electrical cycle; Under the same evaluation sampling point c, It is determined solely by fixed geometric parameters, material equivalent permeability, interface connectivity, boundary conditions, and the order of analytical expansion, and is independent of the candidate magnetization coding index t. The formula for performing a LU decomposition with row pivoting is: And when evaluating different candidate magnetization codes, reuse Update only Obtained by previous generation and back generation The value of , where, Let c be the permutation matrix corresponding to the c-th evaluation sampling point. Let c be the lower triangular matrix corresponding to the c-th evaluation sampling point. Let be the upper triangular matrix corresponding to the c-th evaluation sampling point; when any fixed condition of the influence coefficient matrix changes, reassemble and decompose the corresponding matrix. If the magnetization vector field is defined in the rotor coordinate system, then the magnetization vector field at the c-th evaluation sampling position in the stator coordinate system is calculated according to... Obtain, among which, This is a two-dimensional rotation matrix from the rotor coordinate system to the stator coordinate system; based on this, generate... .

[0042] according to Reconstruct the magnetic vector potentials of the permanent magnet subdomain and the air gap subdomain, and calculate the air gap magnetic flux density. and the internal magnetic field strength of permanent magnets Where pm is the permanent magnet subdomain and g is the air gap subdomain. The relative permeability of a permanent magnet. Let be the magnetization vector field corresponding to the t-th candidate scheme at the c-th evaluation sampling position.

[0043] In this embodiment, for the c-th evaluation sampling point in the t-th candidate scheme, the evaluation sampling point corresponds to the determined rotor position and winding current condition. The right-hand term vector of the magnetization source formed by the projection of the magnetization source is added to the right-hand term vector formed by the winding current in the slot under the evaluation sampling point to construct the analytical subdomain linear equation under the sampling point. In this equation, the analytical expansion coefficient vector corresponding to the sampling point is to be determined. Since the rotor position is different for different evaluation sampling points, the coefficient matrix corresponding to each sampling point may be different. However, for the same sampling point, the coefficient matrix is ​​determined only by fixed geometric parameters, material parameters, interface connection relationship and analytical expansion order, and is independent of the candidate magnetization code number.

[0044] For each evaluation sampling point, a fixed geometric coefficient matrix is ​​subjected to a single LU decomposition with row pivot selection. Under constant conditions, different candidate magnetization codes reuse the corresponding decomposition results, updating only the right-hand side of the magnetization source based on the current candidate magnetization code, and obtaining the analytical expansion coefficient vector through forward and backward substitution. This approach fully leverages the characteristic that the coefficient matrix is ​​independent of candidate schemes under equal geometric conditions, avoiding repeated matrix assembly and decomposition operations due to geometric changes. In multi-objective optimization iterations involving numerous candidate schemes and multiple evaluation sampling points, this strategy significantly reduces redundant computations, making the magnetic field solution process, which originally required substantial computational resources, highly efficient and feasible, thus significantly improving optimization efficiency.

[0045] After obtaining the analytical expansion coefficient vector at each evaluation sampling point, the magnetic vector potential distribution of the permanent magnet subdomain and the air gap subdomain is reconstructed based on this coefficient vector, and the air gap magnetic flux density is calculated. When demagnetization margin evaluation is enabled, the magnetic field strength inside the permanent magnet is also calculated. Through the above method, the magnetic field response of each candidate scheme at each evaluation sampling point can be obtained quickly while ensuring calculation accuracy, providing a reliable data foundation for the comprehensive evaluation of subsequent multi-dimensional performance indicators.

[0046] Step S6: Calculate the total harmonic distortion rate of the back electromotive force, torque ripple rate, average torque, and magnetization manufacturability index based on the air gap magnetic flux density and magnetization intensity vector field to form the mandatory evaluation vector of the candidate scheme; when the demagnetization margin evaluation is enabled, calculate the demagnetization margin based on the magnetic field strength inside the permanent magnet. Specifically, step S6 further includes: in the t-th candidate scheme, calculating the A-phase flux linkage and back electromotive force based on the radial component of the air gap magnetic flux density corresponding to the c-th evaluation sampling point, with the following formula: , The electrical angle corresponding to the c-th evaluation sampling point is ,in, The number of turns in series per phase. This is the effective axial length of the motor. The radius selected within the air gap region is used to calculate the field quantities. This radius is then used to extract the radial and tangential components of the air gap magnetic flux density, and further to calculate the magnetic flux linkage and electromagnetic torque. To normalize the A-phase winding function, Electric angular velocity, This represents the number of pole pairs of the motor. The position of the mechanical rotor corresponding to the c-th evaluation sampling point is given; Fourier decomposition is performed on the back EMF sequence obtained within one electrical cycle to obtain the fundamental effective value. and the effective value of the h-th harmonic ;when At that time, according to Calculate the total harmonic distortion rate of the back electromotive force, where the highest retained harmonic order satisfies ;when When this happens, the corresponding candidate solutions are determined to be infeasible. To retain the highest harmonic order, it is predetermined based on the total number of evaluation sampling points and harmonic evaluation requirements, and is kept consistent within the same round of optimization.

[0047] Based on the radial and tangential components of the air gap magnetic flux density corresponding to each evaluation sampling point, the electromagnetic torque at the c-th evaluation sampling point is calculated using the Maxwell stress method. According to one electrical cycle The electromagnetic torque at each evaluation sampling point is used to calculate the average electromagnetic torque. ;when The torque ripple rate is expressed as: ,in, and These represent the maximum and minimum values ​​of the electromagnetic torque at each evaluation sampling point within one electrical cycle; when The corresponding candidate solutions are determined to be infeasible.

[0048] When the evaluation temperature of the permanent magnet has been obtained in step S1 and the magnetic field strength at the demagnetizing knee point corresponding to that temperature. At that time, demagnetization margin evaluation is activated. This applies to the spatial location of the c-th evaluation sampling point and within the permanent magnet region. The component of the magnetic field strength inside the permanent magnet along the local magnetization direction corresponding to the c-th evaluation sampling position is: And calculate the minimum demagnetization margin based on this component. ,in, The signed demagnetizing knee magnetic field strength is defined along the local magnetization direction, i.e. , This indicates the spatial location within the permanent magnet region. Under this definition, This indicates that the evaluated location has not yet exceeded the demagnetization knee point at the corresponding temperature.

[0049] Based on the magnetization vector field, the manufacturability index of adjacent magnetization zones is determined. ,in, Let represent the local manufacturability quantity between the j-th magnetization zone and the (j+1)-th magnetization zone within the i-th magnetic pole, and max represent the maximum value within the index range. This represents the smallest included angle considering the periodicity of the angle. , and The normalized parameters and weights determined in step S1 and kept constant in the same round of optimization; The smaller the value, the smoother the changes in magnetization amplitude and direction between adjacent magnetization zones within the same magnetic pole; and based on the total harmonic distortion rate of the back EMF, torque ripple rate, average electromagnetic torque, and magnetization manufacturability index, an evaluation vector for the t-th candidate scheme is formed. , For transposition; when demagnetization margin assessment is enabled, additional assessment quantities are added in addition to the four mandatory assessment quantities. As a condition evaluation metric, it is used for subsequent engineering constraint judgment and scheme review.

[0050] In this embodiment, for the t-th candidate scheme, the A-phase flux linkage is first calculated based on the radial component of the air gap magnetic flux density at each evaluation sampling point, combined with the number of series turns per phase, effective axial length, the field calculation radius selected at the air gap, and the A-phase winding function. Then, the A-phase back EMF at each sampling point is obtained by taking the derivative of the flux linkage with respect to the electrical angle. Fourier decomposition is performed on the back EMF waveform within one electrical cycle to obtain the fundamental effective value and the effective values ​​of each harmonic. Based on this, the total harmonic distortion rate of the back EMF is calculated. This index directly reflects the harmonic content of the motor's back EMF waveform and its deviation from a sine wave, and is one of the indicators for evaluating the electromagnetic performance of the motor.

[0051] In electromagnetic torque evaluation, the electromagnetic torque at each sampling point is calculated using the Maxwell stress method based on the radial and tangential components of the air gap magnetic flux density. After obtaining the instantaneous electromagnetic torque at all sampling points, the average electromagnetic torque is statistically calculated. Simultaneously, the torque ripple rate is calculated based on the maximum, minimum, and average values ​​of the electromagnetic torque waveform. The average electromagnetic torque reflects the torque output level of the candidate scheme, while the torque ripple rate reflects the degree of fluctuation of the electromagnetic torque with rotor position; together, they constitute the core dimensions of electromagnetic performance evaluation.

[0052] For the evaluation of demagnetization performance, the evaluation temperature of the permanent magnet has already been obtained in step S1. and the magnetic field strength at the demagnetizing knee point corresponding to that temperature. When activated, based on the internal magnetic field strength and local magnetization direction of the permanent magnet at each evaluation sampling point, the component of the magnetic field strength along the local magnetization direction is calculated. The minimum demagnetization margin is obtained by taking the minimum margin relative to the demagnetization knee point across all evaluation sampling points and the permanent magnet region. This value reflects how close the permanent magnet is to the demagnetization boundary under operating conditions. A positive value indicates that the evaluated location has not yet crossed the demagnetization knee point at the corresponding temperature; a larger value indicates stronger demagnetization resistance. Using the demagnetization margin as one of the evaluation indicators ensures that the reliability of the motor is considered while pursuing improved electromagnetic performance.

[0053] For the evaluation of magnetization manufacturability, based on the magnetization intensity vector field of each magnetization zone, the changes in magnetization vector amplitude and magnetization direction between adjacent magnetization zones within the same magnetic pole are considered. The amplitude change is normalized according to a preset reference value, and the direction change is normalized according to a preset reference angle after considering the angular periodicity and weighted accordingly. The maximum weighted value among all combinations of adjacent zones is taken as the magnetization manufacturability index. The smaller this index, the smoother the change in magnetization state between adjacent zones, which is more conducive to the realization of the magnetization process. By incorporating this index into the evaluation system, the optimization process not only focuses on electromagnetic performance but also takes into account the manufacturability of the magnetization scheme, avoiding optimization results that deviate from the actual process.

[0054] Finally, the above indicators are combined to form the basic evaluation vector for the t-th candidate scheme, including the total harmonic distortion rate of the back EMF, torque ripple rate, average electromagnetic torque, and magnetization manufacturability index. When the demagnetization margin evaluation is enabled, a minimum demagnetization margin is added to the basic evaluation vector as a conditional evaluation quantity. This evaluation vector comprehensively covers multiple dimensions such as electromagnetic performance, reliability, and process feasibility, providing a complete data foundation for subsequent multi-objective optimization and engineering constraint screening, enabling the final output magnetization code to be evaluated with trade-offs among multiple performance indicators.

[0055] Please see Figure 4 Repeat steps S3 to S6 to obtain the evaluation vector of each candidate solution. Based on the evaluation vector, establish a multi-objective minimization evaluation function for each candidate solution, and select candidate solutions that meet the preset engineering constraints to generate a Pareto feasible non-dominated solution set. Specifically, step S7 further includes: based on the evaluation vector, establishing a multi-objective minimization evaluation function for each candidate solution, the formula of which is: , Let be the multi-objective minimization evaluation function for the t-th candidate solution, where , and Smaller is better; by adopting The objective of minimizing the average electromagnetic torque is transformed into a unified objective of maximizing the average electromagnetic torque.

[0056] For the t-th candidate solution, the engineering constraints include: When the demagnetization margin evaluation and demagnetization margin constraint in step S1 are enabled, the following conditions are also met: ,in, This is the lower limit of the average torque. This is the upper limit of magnetization manufacturability. This is the minimum demagnetization margin lower limit. The above threshold is determined by the design task and magnetization process requirements in step S1. The allowable deviation of the magnetization direction relative to the set of allowable directions for the magnetization process is already included in the candidate coding allowable domain and will not be recalculated as an independent engineering constraint.

[0057] In this embodiment, the non-dominated sorting genetic algorithm NSGA-II is used to perform multi-objective optimization of candidate magnetization codes. An initial population is constructed based on the magnetization code allowable domain determined in step S2. Steps S3 to S6 are sequentially executed on each candidate magnetization code in the population to obtain its four-objective evaluation results and engineering constraint evaluation results. Offspring are then generated through selection, crossover, and mutation. For offspring exceeding the allowable domain of candidate codes, the nearest feasible code within the allowable domain is projected. If projection is not possible, resampling is performed. The parent and offspring candidate codes are compared using constraint dominance rules: when only one of the two candidate schemes satisfies all engineering constraints enabled in this round, the feasible scheme is retained first; when neither is feasible, the candidate with the smaller constraint violation is retained first; when both are feasible, the priority order is determined based on Pareto dominance and crowding distance. After merging the parent and offspring, non-dominated sorting is performed, and the next generation population is selected in order of non-dominated level from best to worst (layer number from smallest to largest) and crowding distance from largest to smallest. Iteration stops when the preset maximum number of iterations or the convergence condition is reached. A Pareto feasible non-dominated solution set that simultaneously satisfies the above engineering constraints is output and passed to step S8 for compromise solution selection and verification. Solutions that still cannot be feasiblely encoded after resampling are discarded.

[0058] In this embodiment, for each candidate magnetization coding scheme generated in step S2, it is mapped to a magnetization intensity vector field in step S3, and the volume magnetization source term and interface magnetization source term are calculated. In step S4, the magnetization source term is projected into a right-hand term vector of the magnetization source and a solution equation is established. In step S5, the magnetic field response is solved at each evaluation sampling point to obtain the air gap magnetic flux density and the magnetic field strength inside the permanent magnet. Then, in step S6, the total harmonic distortion rate of the back electromotive force, torque ripple rate, average electromagnetic torque, and magnetization manufacturability index of the scheme are calculated and an evaluation vector is formed. When the demagnetization margin evaluation is enabled, the minimum demagnetization margin is calculated simultaneously. Through the above batch processing, the electromagnetic performance, reliability, and process feasibility of each candidate scheme are comprehensively and quantitatively evaluated.

[0059] Based on this, a multi-objective minimization evaluation function is established based on the evaluation vectors of each candidate scheme. This multi-objective evaluation function simultaneously incorporates four objectives: total harmonic distortion rate of back EMF, torque ripple rate, negative average electromagnetic torque, and magnetization manufacturability index. Specifically, the smaller values ​​of the total harmonic distortion rate of back EMF, torque ripple rate, and magnetization manufacturability index are preferred. For the average electromagnetic torque, its negative value is used as the optimization objective, transforming the "the larger the average electromagnetic torque, the better" approach into a minimization form. This unifies the mathematical expression of the four objectives, facilitating multi-objective optimization. These four objectives reflect motor performance from different dimensions. Single-objective optimization can easily lead to the deterioration of other performance aspects, while multi-objective optimization can seek a reasonable balance among the objectives.

[0060] After constructing the multi-objective evaluation function, the candidate schemes are further screened using engineering constraints. Specifically, for the t-th candidate scheme, its average electromagnetic torque is required to be no less than a preset lower limit of average torque to ensure that the motor's output capacity meets the design requirements; its magnetization manufacturability index is required to be no more than a preset upper limit of magnetization manufacturability to ensure that the magnetization scheme is feasible under existing magnetization process conditions; when the demagnetization margin evaluation and corresponding constraints are enabled, its minimum demagnetization margin is required to be no less than a preset lower limit of minimum demagnetization margin to ensure that the permanent magnet has sufficient anti-demagnetization safety margin in the working state. Candidate schemes that meet all enabled engineering constraints in this round are determined to be feasible schemes. The above engineering constraints fully consider the actual needs of motor design and organically unify performance indicators with engineering feasibility.

[0061] After dual screening using multi-objective evaluation functions and engineering constraints, a Pareto feasible non-dominated solution set that satisfies all engineering constraints enabled in this round is output from all candidate solutions. Each candidate solution in this solution set satisfies its corresponding engineering constraints, and for any given candidate solution, no other feasible candidate solution is superior to it in all optimization objectives or at least in one objective. This constitutes the Pareto candidate solution set under given constraints. Through this process, the output solutions satisfy the engineering constraints set in this round, and a sufficient set of high-quality candidates is provided for subsequent selection of the final solution that meets design preferences. This avoids ineffective calculations on infeasible solutions and improves overall optimization efficiency.

[0062] Please see Figure 5 Step S8: Perform normalization compromise evaluation on the candidate schemes in the Pareto feasible non-dominated solution set, and select the candidate scheme with the smallest compromise evaluation value as the coding scheme to be reviewed. Use the review method corresponding to the reviewed evaluation quantity to review the scheme, and determine the coding scheme to be reviewed that meets the preset allowable deviation and still meets the performance constraints of the current round of activation engineering as the final magnetization coding scheme.

[0063] Specifically, step S8 further includes: for the Pareto feasible non-dominated solution set For each candidate scheme in the list, to eliminate the influence of different objective dimensions and numerical ranges, for the t-th candidate scheme, the four minimization objectives (labeled sequentially as follows) are: ), for the first Each target is normalized. ,in, and The first One goal is The minimum and maximum values; the weights selected based on the trade-offs predetermined in step S1. Calculate the compromise evaluation value of the t-th candidate solution. ,satisfy: Candidate solutions in the Pareto feasible non-dominated solution set are sorted in ascending order of their compromise evaluation values.

[0064] Select the candidate scheme with the smallest compromise evaluation value as the coding scheme to be reviewed: ,in, The magnetization code to be verified is used; a preset verification method (e.g., at least one of finite element calculation, air gap magnetic flux density measurement, back EMF test, or torque test) is used to verify the code scheme (the motor geometric parameters, material parameters, speed, current, and temperature conditions used for verification should be consistent with the evaluation conditions set in step S1), and the verification results are obtained. The corresponding performance evaluation results include the total harmonic distortion rate of the back EMF. Torque ripple rate Average electromagnetic torque and magnetization manufacturability indicators When demagnetization margin evaluation is enabled, the minimum demagnetization margin is also obtained. .

[0065] When it is determined that the verification result is within the preset allowable deviation range and still meets the lower limit of average torque and the upper limit of magnetization manufacturability, and when the demagnetization margin constraint is enabled, it also meets the lower limit of minimum demagnetization margin, the coding scheme to be verified is determined as the final magnetization coding scheme and output. When it is determined that the verification result is not within the preset allowable deviation range or does not meet the engineering constraints, the magnetization coding scheme to be verified is removed from the current Pareto feasible non-dominated solution set, and the candidate scheme with the smallest compromise evaluation value is selected from the updated Pareto feasible non-dominated solution set as the new coding scheme to be verified. When it is determined that there is no candidate solution that meets the verification requirements in the Pareto feasible non-dominated solution set, return to step S7, reinitialize the population or update the optimization parameters, generate a new Pareto feasible non-dominated solution set, and continue to execute the trade-off and verification process in step S8. In this embodiment, for each candidate scheme in the Pareto feasible non-dominated solution set, its four minimization objectives are the total harmonic distortion rate of the back EMF, the torque ripple rate, the negative average torque, and the magnetization manufacturability index. Since these four objectives have different dimensions and numerical ranges, direct weighted comparison is unreasonable. Therefore, each objective is first normalized to eliminate the influence of differences in dimensions and orders of magnitude. Based on this, according to the pre-set trade-off weights for the four objectives according to the design task, the trade-off evaluation value of each candidate scheme is calculated. This trade-off evaluation value comprehensively reflects the overall performance of the candidate scheme on multiple optimization objectives; the smaller the evaluation value, the more the scheme conforms to the design preference. The Pareto feasible non-dominated solution set is arranged in ascending order of trade-off evaluation values, and the candidate scheme with the smallest trade-off evaluation value is selected as the coding scheme to be reviewed. This scheme represents the magnetization coding scheme with the smallest trade-off evaluation value under the pre-set weights, while satisfying all engineering constraints. Through the above normalization and weighting processes, the final scheme selection process has a clear and quantifiable decision-making basis, making the scheme selection criteria clear and repeatable.

[0066] Subsequently, the coding scheme to be reviewed is reviewed using a preset review method to obtain the review results. The review method includes at least one of finite element numerical calculation, air gap magnetic flux density measurement, back EMF testing, or torque testing. The motor geometric parameters, material parameters, speed, current, and temperature conditions used in the review should be consistent with the evaluation conditions set in step S1. Through the above review verification, the calculation results of the analytical model can be independently verified, ensuring the reliability of the optimization results. When the review results are within the preset allowable deviation range and still meet the lower limit of average torque and the upper limit of magnetization manufacturability, and when the demagnetization margin constraint is enabled, also meet the lower limit of minimum demagnetization margin, the coding scheme to be reviewed is determined as the final magnetization coding scheme and output. This review step provides an independent verification channel for the analytical optimization results, enabling the design results to withstand the test of subsequent engineering stages, enhancing the credibility and engineering persuasiveness of the method.

[0067] When the verification result is not within the preset allowable deviation range, or although it is within the deviation range but no longer meets the preset engineering constraints, the coding scheme to be verified is removed from the Pareto feasible non-dominated solution set. The candidate scheme with the smallest compromise evaluation value is then selected from the updated Pareto feasible non-dominated solution set as the new coding scheme to be verified, and the verification process is repeated. Through this removal and replacement mechanism, even if the preferred scheme fails the verification, alternative schemes can still be found in the original high-quality candidate solution set, fully utilizing the effective solution set resources obtained from the previous multi-objective optimization. When there are no candidate schemes that meet the verification requirements in the updated Pareto feasible non-dominated solution set, it means that all schemes in the current Pareto feasible non-dominated solution set cannot pass independent verification. At this time, the process returns to step S7 to re-perform candidate coding optimization and generate a new Pareto feasible non-dominated solution set. This closed-loop processing mechanism ensures that even in extreme cases, the entire optimization process still has the ability to self-correct and iterate, further improving the robustness and engineering practicality of the method. Through the aforementioned verification and branching mechanisms, a complete closed loop is formed, from theoretical design and numerical optimization to engineering verification. Schemes that pass verification are confirmed and output, while those that fail are replaced or re-optimized. This ensures that the final magnetization coding scheme not only meets the multi-objective optimization requirements in theory but also withstands independent verification in engineering practice. Thus, it provides an efficient, reliable, and practically applicable technical solution for the pole design of high-performance permanent magnet synchronous motors.

[0068] In simple terms, this method uses a magnetization coding matrix to uniformly describe non-uniform magnetization schemes, allowing the radial and tangential magnetization component coefficients to enter the optimization process as independent design variables, with the magnetization direction determined by these two component coefficients. By mapping the magnetization coding matrix to the magnetization intensity vector of each magnetization zone of the permanent magnet, and converting the magnetization spatial variation within the magnetization zone and the magnetization jump at the common interface of adjacent magnetization zones into the right-hand side terms of the volume magnetization source and the interface magnetization source of the analytical subdomain equation, a correspondence is established between the candidate magnetization scheme and the analytical subdomain magnetic field model. By reusing the coefficient matrix and its decomposition results under equal geometric conditions, the repetitive matrix assembly and decomposition calculations during the solution of different candidate schemes can be reduced. Through multi-objective optimization consisting of back EMF total harmonic distortion rate, torque ripple rate, average electromagnetic torque, and magnetization manufacturability index, and combined with engineering constraint screening using the lower limit of average torque, the upper limit of magnetization manufacturability, and the minimum lower limit of demagnetization margin when demagnetization margin evaluation is enabled, the output magnetization coding scheme can be made more suitable for engineering implementation. Through the above steps, the present invention can incorporate non-uniform magnetization design variables, analytical subdomain magnetic field calculations, and engineering constraint screening into the same optimization process without changing the magnetic pole mechanical geometry. This makes the final magnetization code output have a clear analytical calculation source and is also convenient for subsequent finite element verification, process evaluation, or prototype testing.

[0069] In summary, this method first inputs the geometric parameters, material parameters, winding parameters, and operating condition parameters of the motor to be optimized. The geometric parameters include at least the rotor core radius, the inner and outer radii of the permanent magnet, the mechanical pole arc of the permanent magnet, the spacing between adjacent magnetic poles, the air gap size, the slot size, and the parameters of the slot area. The material parameters include at least the remanence of the permanent magnet or the reference equivalent magnetization, relevant magnetic parameters, and the evaluation temperature and corresponding demagnetization knee point data required when demagnetization margin evaluation is enabled. The winding parameters and evaluation operating condition parameters are used for subsequent calculations of back EMF, electromagnetic torque, and corresponding evaluation quantities.

[0070] During the optimization process, the permanent magnet mechanical pole arc, adjacent pole spacing, rotor outer circle, air gap, and magnetization zone boundary remain unchanged within the same optimization round. A magnetization coding matrix is ​​established according to the pole number i and magnetization zone number j. The rows of the magnetization coding matrix correspond to the pole numbers, and the columns correspond to the magnetization zone numbers. Matrix elements include radial and tangential magnetization component coefficients. These two component coefficients are independent design variables, while the magnetization direction is determined by them but not considered as independent design variables. For the j-th magnetization zone of the i-th pole, its magnetization intensity vector can be represented as being composed of both radial and tangential magnetization components. Their relative values ​​are used to adjust the magnetization intensity modulus and magnetization direction, describing the magnetization state of the non-uniform magnetization scheme in different magnetization zones. Therefore, different candidate magnetization coding matrices can form different magnetization intensity distributions under the same mechanical boundary and magnetization zone conditions.

[0071] The magnetization encoding matrix is ​​mapped to magnetization source terms in the analytical subdomain magnetic field equations. Specifically, the magnetization intensity distribution of each magnetization zone within the permanent magnet subdomain is determined based on the magnetization encoding matrix, and this magnetization intensity distribution is expanded into source term coefficients that can be used to solve the analytical subdomain. For the permanent magnet subdomain, a vector magnetic potential Poisson equation containing magnetization source terms is established; for the air gap subdomain, a Laplace equation satisfying the source-free condition is established; for the rotor core subdomain, slot subdomain, and slot subdomain, vector magnetic potential expressions matching the corresponding domain control equations and boundary conditions are established, where the slot winding current is used as a known right-hand side term in the equations of the corresponding domain. When the candidate magnetization encoding changes, the corresponding magnetization intensity distribution and magnetization source right-hand side terms are updated accordingly; the coefficient matrix remains unchanged under the condition that the geometric boundaries of the analytical subdomain, the material equivalent permeability, the interface connection relationship, and the analytical expansion order remain unchanged.

[0072] The air gap magnetic field is solved by simultaneously applying boundary conditions to five subdomains: rotor core, permanent magnet, air gap, slot opening, and slot interior. A system of linear equations is formed at the interfaces of adjacent subdomains, applying continuous conditions for normal magnetic flux density and tangential magnetic field strength, to solve for the analytical expansion coefficients. Since the geometric boundaries, material equivalent permeability, interface connectivity, and analytical expansion order remain unchanged during iteration, the coefficient matrix can be kept consistent. Each iteration primarily updates the magnetization source term vector based on the current magnetization encoding matrix and solves for the corresponding air gap magnetic field distribution.

[0073] Based on the obtained air gap magnetic field, internal magnetic field strength of the permanent magnet, and magnetization vector field, the performance indicators of the candidate magnetization codes are further calculated. These performance indicators include the total harmonic distortion rate of the back EMF, torque ripple rate, average electromagnetic torque, and magnetization manufacturability index. When the demagnetization margin evaluation conditions are met, the minimum demagnetization margin is also calculated. The total harmonic distortion rate of the back EMF can be calculated from the harmonic components and the fundamental component of the back EMF; the torque ripple rate can be calculated from the maximum electromagnetic torque, minimum electromagnetic torque, and average electromagnetic torque; the minimum demagnetization margin is used to evaluate the safety of the permanent magnet's operating point relative to the demagnetization risk boundary under a specified current or operating condition; the magnetization manufacturability index is used to constrain the changes in magnetization amplitude and magnetization direction between adjacent magnetization zones.

[0074] Candidate magnetization coding matrices can be generated by the NSGA-II multi-objective optimization algorithm or by other multi-objective optimization algorithms capable of forming Pareto feasible non-dominated solution sets. During the screening process, candidate magnetization codes are subject to engineering constraints based on the lower limit of average electromagnetic torque and the upper limit of magnetization manufacturability; when demagnetization margin constraints are enabled, a preset minimum demagnetization margin lower limit must also be met. The deviation of the magnetization direction from the set of allowable directions for the magnetization process is limited within the allowable domain of the magnetization code. For candidate schemes that satisfy all enabled engineering constraints in this round, based on... , - and A multi-objective evaluation is performed to form a Pareto feasible non-dominated solution set, and the final magnetization coding scheme is determined through compromise and verification. The output results include the final magnetization coding matrix and its corresponding radial magnetization component coefficients, tangential magnetization component coefficients, the magnetization direction determined by the two, and the corresponding performance evaluation results.

[0075] Compared with existing technologies, the advantages of this method are as follows: First, by defining a magnetization coding matrix composed of radial magnetization component coefficients and tangential magnetization component coefficients, a unified parameterized description of different non-uniform magnetization schemes is achieved, transforming the magnetization design problem into an coded and iterative optimization variable, fundamentally solving the problem of the difficulty in parameterized modeling of non-uniform magnetization distribution.

[0076] Second, by mapping the magnetization code to the magnetization intensity vector of each magnetization zone of the permanent magnet, and determining the volume magnetization source term and the interface magnetization source term according to the spatial variation of the magnetization intensity within the zone and the magnetization intensity jump at the common interface of adjacent zones, the two types of magnetization sources are then projected into the corresponding right-hand side terms of the analytical subdomain equations. This establishes the computational relationship between the magnetization code and the analytical subdomain magnetic field solution, enabling the magnetic field effect caused by non-uniform magnetization to be rigorously and completely mathematically described, thus ensuring the physical accuracy and theoretical rigor of the magnetic field calculation.

[0077] Third, by making full use of the conditions that the geometric boundaries of the analytical subdomains, the equivalent permeability of the material, the interface connection relationship, and the order of the analytical expansion remain unchanged, the coefficient matrix and its LU decomposition results can be reused among different candidate magnetization schemes. When the candidate magnetization encoding changes, the right-hand side of the corresponding magnetization source is mainly updated, and the analytical expansion coefficients are obtained through previous and back substitutions. This reduces the repeated matrix assembly and decomposition calculations in the candidate scheme evaluation process, greatly improves the efficiency of multi-objective optimization, and makes it possible to evaluate the full performance of large-scale candidate schemes within an acceptable engineering time.

[0078] Fourth, a multi-dimensional comprehensive evaluation system was constructed, including total harmonic distortion rate of back EMF, torque ripple rate, negative average electromagnetic torque, and magnetization manufacturability index. Engineering constraints were screened by combining the lower limit of average electromagnetic torque and the upper limit of magnetization manufacturability. When the evaluation temperature and corresponding demagnetization knee point data for permanent magnets are complete and demagnetization margin constraints are enabled, the minimum demagnetization margin is further used for evaluation and constraint. Through multi-objective optimization, a Pareto feasible non-dominated solution set satisfying all enabled engineering constraints in this round is obtained. A compromise evaluation is then conducted to select a magnetization coding scheme to be reviewed, ensuring that the final output magnetization code balances electromagnetic performance and process feasibility within the set evaluation index range. Furthermore, it can constrain demagnetization risk when demagnetization parameters are complete, preventing the optimization results from deviating from engineering practice.

[0079] Fifth, the magnetization coding scheme to be verified is verified by at least one of the following methods: finite element calculation, air gap magnetic flux density measurement, back electromotive force test, or torque test. The final magnetization coding scheme is determined when the verification results meet the preset allowable deviation range and the engineering constraints of this round. When the verification results do not meet the requirements, other schemes are selected from the remaining Pareto feasible non-dominated candidate schemes for verification. If necessary, the multi-objective optimization step is returned to obtain the candidate solution set again. This forms a complete closed loop from theoretical design, numerical optimization to experimental verification, ensuring the reliability and feasibility of the design results.

[0080] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A method for optimizing a non-uniformly magnetized motor with uniform geometry, characterized in that, include: Step S1: Obtain the input motor fixed model parameters, evaluation condition parameters, design variable boundary and process feasibility parameters, engineering performance constraint parameters and optimization auxiliary parameters. When the permanent magnet evaluation temperature and the corresponding demagnetization knee magnetic field strength data are complete, enable demagnetization margin evaluation and demagnetization margin constraint. In the same round of optimization, keep the geometric parameters, material equivalent permeability, winding parameters and analytical expansion order in the motor fixed model parameters unchanged. Step S2: Determine the magnetization coding allowable domain based on the design variable boundary and process feasibility parameters, obtain multiple candidate schemes corresponding to the magnetic pole to be optimized based on the magnetization coding allowable domain, and construct the total magnetization coding matrix of each candidate scheme; Step S3: Calculate the magnetization intensity vector field of the candidate scheme based on the total magnetization coding matrix, determine the volume magnetization source term according to the spatial variation of magnetization intensity within each magnetization partition, and determine the interface magnetization source term according to the magnetization intensity jump on both sides of the common interface between adjacent magnetization partitions and between the magnetized region and the adjacent non-magnetized analytical subdomain. Step S4: Project the basis functions within the domain and the basis functions of the interface magnetization source terms into the volume magnetization source terms, respectively, and assemble them into the right-hand term vector of the magnetization source. Based on the fixed analytical subdomain geometry, material equivalent permeability, interface connection relationship and analytical expansion order, establish the linear equation between the coefficient matrix, analytical expansion coefficient vector, right-hand term vector of the magnetization source and right-hand term vector of the winding current. Step S5: For each evaluation sampling point, decompose the coefficient matrix corresponding to the evaluation sampling point, reuse the decomposition result when evaluating different candidate schemes, update the right-hand term vector of the magnetization source according to the current candidate scheme, obtain the corresponding analytical expansion coefficient vector, and calculate the air gap magnetic flux density and the magnetic field strength inside the permanent magnet from the analytical expansion coefficient vector. Step S6: Calculate the total harmonic distortion rate of the back electromotive force, torque ripple rate, average torque and magnetization manufacturability index based on the air gap magnetic flux density and magnetization intensity vector field to form the mandatory evaluation vector of the candidate scheme. When the demagnetization margin evaluation is enabled, the demagnetization margin is additionally calculated based on the magnetic field strength inside the permanent magnet. Step S7: Repeat steps S3 to S6 to obtain the evaluation vector of each candidate solution. Based on the evaluation vector, establish a multi-objective minimization evaluation function for each candidate solution, and select candidate solutions that meet the preset engineering constraints to generate a Pareto feasible non-dominated solution set. Step S8: Perform normalization compromise evaluation on the candidate schemes in the Pareto feasible non-dominated solution set, and select the candidate scheme with the smallest compromise evaluation value as the coding scheme to be reviewed. The review is carried out using the review method corresponding to the reviewed evaluation quantity, and the coding scheme to be reviewed that meets the preset allowable deviation and still meets the engineering performance constraints of this round is determined as the final magnetization coding scheme.

2. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 1, characterized in that, The fixed model parameters of the motor include basic motor parameters, material parameters, and winding parameters. The basic motor parameters include the geometric dimensions of the rotor core, permanent magnet, air gap, slot and sub-domains within the slot, number of pole pairs, mechanical pole arc of the permanent magnet, spacing between adjacent magnetic poles, air gap length, and effective axial length. The material parameters include the relative permeability of each sub-domain, remanence of the permanent magnet, and reference equivalent magnetization. When demagnetization margin evaluation is enabled, the material parameters also include the evaluation temperature of the permanent magnet and the magnetic field strength at the demagnetization knee point corresponding to that temperature. The winding parameters include the number of turns in series per phase and the normalized A-phase winding function. The evaluation operating condition parameters include operating parameters and evaluation sampling parameters. The operating parameters include electrical angular velocity and winding current corresponding to each evaluation sampling position. The evaluation sampling parameters include the total number of sampling points in one electrical cycle and the discrete mechanical rotor position. When demagnetization margin evaluation is enabled, the evaluation operating condition parameters also include permanent magnet evaluation temperature. The design variable boundaries and process feasibility parameters include the upper and lower limits of the radial magnetization component coefficient, the upper and lower limits of the tangential magnetization component coefficient, the set of magnetization directions allowed by the magnetization process, the maximum allowable deviation of the magnetization direction, and the minimum magnetization intensity coefficient. The engineering performance constraints include the lower limit of average torque and the upper limit of magnetization manufacturability. When demagnetization margin evaluation is enabled, the engineering performance constraint parameters also include a minimum demagnetization margin lower limit; The optimization auxiliary parameters include a normalized reference value for magnetization amplitude change, a normalized reference angle for direction change, a weight for direction change, and a trade-off weight for the four minimization objectives.

3. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 2, characterized in that, Step S2 is as follows: The magnetization coding allowable domain is determined based on the design variable boundaries and process feasibility parameters, and a preset number of candidate magnetization coding schemes are generated within the magnetization coding allowable domain. Each magnetic pole to be optimized is divided into multiple magnetization partitions. For the t-th candidate scheme, the magnetization code of the j-th magnetization partition of the i-th magnetic pole is represented as follows: ,in, The radial magnetization component coefficient is... The tangential magnetization component coefficient; the magnetization direction offset angle of the magnetization partition relative to the reference radial direction is... Each magnetization code satisfies the magnetization code allowable field; Arrange the magnetization codes according to the magnetic pole number and magnetization partition number to obtain the total magnetization code matrix of the t-th candidate scheme. ,in, , , This represents the total number of magnetic poles. This represents the number of magnetization zones for each magnetic pole.

4. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 3, characterized in that, Step S3 is as follows: In the t-th candidate scheme, the magnetization vector is determined based on the radial and tangential magnetization component coefficients corresponding to each magnetization zone. ,in, As the reference equivalent magnetization, and These are the reference radial unit vector and the reference tangential unit vector of the i-th magnetic pole, respectively; The magnetization vectors are combined according to the spatial location of each magnetization zone to form the magnetization vector field of the candidate scheme, and the volume magnetization source term is determined according to the spatial variation of the magnetization intensity within each magnetization zone; for the common interface of adjacent regions a and b , Let represent the unit normal vector pointing from region a to region b. This represents the unit tangent vector selected according to a preset direction, and the equivalent axial magnetization surface current density. , and These represent the values ​​of the magnetization vectors on both sides at the common interface. It is a unit vector along the axial direction; in the assembly of the continuity equation for the tangential magnetic field strength, the known term on the right-hand side of the interface magnetization source is... The two are used separately. and Let represent the relative permeability of region a and region b, respectively.

5. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 4, characterized in that, Step S4 is as follows: The known terms on the right-hand side of the volume magnetization source and the interface magnetization source are respectively projected using the in-domain basis functions and the interface basis functions to form the vector of the right-hand side of the volume magnetization source. and the right-hand term vector of the interface magnetization source ; Based on the right-hand term vectors of the volume magnetization source and the interface magnetization source, the right-hand term vector of the magnetization source is assembled. A coefficient matrix is ​​established based on the domain governing equations, the continuity condition of the normal magnetic flux density, and the continuity condition of the tangential magnetic field strength. And determine its analytical expansion coefficient vector corresponding to the t-th candidate scheme. The calculation formula between them is: ,in, This is the right-hand term formed by the winding current in the slot.

6. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 5, characterized in that, Step S5 is as follows: For the c-th evaluation sampling point in the t-th candidate scheme, the corresponding analytic subdomain linear equation is calculated based on the calculation formula between the coefficient matrix and the analytic expansion coefficient vector: ,in, Let c be the coefficient matrix corresponding to the c-th evaluation sampling point. Let c be the analytical expansion coefficient vector corresponding to the c-th evaluation sampling point. Let c be the magnetization source right-hand term vector corresponding to the c-th evaluation sampling point. The right-hand term is formed by the slot winding current corresponding to the c-th evaluation sampling point; Perform LU decomposition with row pivoting on the coefficient matrix: , Let c be the permutation matrix corresponding to the c-th evaluation sampling point. Let c be the lower triangular matrix corresponding to the c-th evaluation sampling point. For the c-th evaluation sampling point, the upper triangular matrix is ​​used. For the same evaluation sampling point, when the geometric boundary of the analytical subdomain, the equivalent permeability of the material, the interface connection relationship and the analytical expansion order remain unchanged, the result of the LU decomposition is reused among different candidate schemes, and the right-hand term of the magnetization source is updated according to the current candidate magnetization encoding. The analytical expansion coefficient vector is obtained by previous and back substitution. The magnetic vector potentials of the air gap subdomain and the permanent magnet subdomain are reconstructed based on the analytical expansion coefficient vector, and the air gap magnetic flux density and the magnetic field strength inside the permanent magnet are calculated.

7. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 6, characterized in that, Step S6 is as follows: In the t-th candidate scheme, the back electromotive force is calculated based on the air gap magnetic flux density corresponding to each evaluation sampling point, and harmonic analysis is performed on the back electromotive force to obtain the total harmonic distortion rate of the back electromotive force. ; The electromagnetic torque at each evaluation sampling point is calculated based on the radial and tangential components of the air gap magnetic flux density, and the average electromagnetic torque is obtained. and torque ripple rate ; Magnetization manufacturability indices are calculated based on the changes in magnetization amplitude and magnetization direction between adjacent magnetization zones. ; The required evaluation vector for the t-th candidate scheme is formed by the total harmonic distortion rate of the back EMF, the torque ripple rate, the average electromagnetic torque, and the magnetization manufacturability index. , For transpose; When demagnetization margin evaluation is enabled, the minimum demagnetization margin is calculated based on the component of the magnetic field strength inside the permanent magnet along the local magnetization direction. And use it as a conditional evaluation metric; Determine the component of the magnetic field strength inside the permanent magnet along the local magnetization direction, and calculate the minimum demagnetization margin based on this component. .

8. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 7, characterized in that, Step S7 is as follows: A multi-objective minimization evaluation function is established for the t-th candidate solution based on four mandatory evaluation parameters. , Let be the multi-objective minimization evaluation function for the t-th candidate scheme, where the objective of minimizing the average electromagnetic torque is transformed into minimizing the objective by taking the negative of the average electromagnetic torque. Filtering satisfies The candidate schemes; when the demagnetization margin constraint is enabled, they also satisfy... For candidate solutions that satisfy all engineering constraints in this round, multi-objective optimization and non-dominated screening are performed to generate a Pareto feasible non-dominated solution set. This is the lower limit of the average torque. This is the upper limit of magnetization manufacturability. This is the lower limit of the minimum demagnetization margin.

9. The optimization method for a non-uniform magnetized motor with equal geometry according to claim 8, characterized in that, Step S8 is as follows: For each candidate solution in the Pareto feasible non-dominated solution set, the four minimization objectives in the candidate solution are normalized, and the compromise evaluation value of the candidate solution is calculated based on the normalization result of the four minimization objectives and the preset compromise selection weights. The solutions are then arranged in ascending order of the compromise evaluation values. The candidate scheme with the smallest compromise evaluation value is selected as the coding scheme to be reviewed, and the review is carried out by finite element calculation or air gap magnetic flux density measurement, back electromotive force test or torque test corresponding to the evaluated quantity being reviewed; the magnetization manufacturability is rechecked according to the candidate code and the allowable domain of the magnetization code; when the demagnetization margin constraint is enabled, the minimum demagnetization margin is verified by a review model that can obtain the magnetic field strength inside the permanent magnet or the corresponding test method. When the absolute or relative deviation between the verification evaluation quantity and the corresponding analytical calculation result is within the preset allowable range, and the coding scheme to be verified meets the lower limit of average electromagnetic torque and the upper limit of magnetization manufacturability, and meets the lower limit of minimum demagnetization margin when the demagnetization margin constraint is enabled, the coding scheme to be verified is determined as the final magnetization coding scheme. If the verification result does not meet the above conditions, the coding scheme to be verified is removed from the Pareto feasible non-dominated solution set, and the next candidate scheme is selected for verification according to the compromise evaluation value order; when there is no candidate scheme that can be verified in the updated Pareto feasible non-dominated solution set, the candidate population is re-initialized or the optimization parameters are updated and the process returns to step S7 to generate a new Pareto feasible non-dominated solution set.

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