A motor topology optimization method based on conforming mesh and variable density method

Abstract

The application belongs to the field of motor topology optimization design, and particularly relates to a motor topology optimization method based on a skin mesh and a variable density method. The application is based on a traditional variable density method (SIMP), and simultaneously optimizes a rotor structure and a current angle of a synchronous reluctance motor by combining real-time level set segmentation and skin mesh technology. The SIMP is used to realize automatic evolution of the rotor topology by taking advantages of a gradient algorithm and a hole generation capability in solving a minimization problem. Meanwhile, in order to avoid the influence of virtual material, the unit density is converted into a level set value, the level set surface is used for segmentation, and the skin mesh technology is used for accurate subdivision of the segmented area, so that the performance calculation accuracy is ensured. The optimization result obtained by the method in the application is not only optimal in performance, but also obtains a novel structure with clear boundaries, and ensures the performance calculation accuracy.

Description

Technical Field

[0001] This invention belongs to the field of motor topology optimization design, specifically involving a motor topology optimization method that combines body-fitted mesh and variable density method. Background Technology

[0002] The development of efficient topology optimization methods has provided a highly flexible and automated design approach for the structural design optimization of synchronous reluctance motor rotors. The purpose of topology optimization is to solve the optimal layout problem of the structure to achieve the best performance. Various methods for applying topology optimization to motor optimization design have been developed and researched, including the variable density method (SIMP), level set method, on / off method, and methods based on machine learning techniques. Among these, the variable density method is more widely used in motor topology optimization due to its simplicity. The general process of a traditional variable density method-based motor topology optimization method is as follows: creating the initial structure of the motor rotor and meshing it → performing material interpolation on the elements of the design domain → defining the objective function and constraints of the optimization problem → electromagnetic field and mechanical finite element calculations → solving the optimization problem using a gradient optimization solver → outputting the results.

[0003] The core of traditional variable-density methods for motor topology optimization is to transform discrete design variables into continuous design variables and further utilize interpolation models to represent materials, thus accelerating the solution of high-dimensional mathematical programming problems. However, material interpolation schemes result in the presence of intermediate virtual materials in the optimization design, and it is difficult to eliminate the jagged edges of the materials. Post-processing is usually required before manufacturing, such as restoring virtual materials to real materials and smoothing material boundaries. Due to the nonlinearity of electromagnetic systems, these phenomena may lead to a significant difference between the processed motor performance and the performance calculated during optimization, resulting in unreliable optimization designs. Compared to variable-density methods, level set methods can provide smooth and clear boundaries for geometric descriptions. However, most level set methods utilizing virtual materials and fixed elements still face shortcomings such as insufficient hole generation capability and poor compatibility with mathematical programming algorithms. Summary of the Invention

[0004] In view of the above-mentioned shortcomings and deficiencies of the prior art, this invention provides a motor topology optimization method that combines body-fitted meshing with variable density methods. First, the design domain is determined, and a material interpolation model considering multiphysics, i.e., electromagnetic and mechanical properties, including permeability, Young's modulus, mass, and current angle, is given based on SIMP. Then, a motor topology optimization model considering multiphysics is constructed, and real-time level set partitioning and body-fitted meshing techniques are combined to reduce the impact of intermediate density on the design and ensure computational accuracy, achieving automatic evolution of the rotor topology. An augmented Lagrange equation is constructed to reduce the dimensionality of the optimization problem and decrease computation time. Finally, sensitivity analysis is performed, and a gradient-based MMA algorithm is used for optimization.

[0005] This invention provides a method for motor topology optimization that combines body-fitted meshes with a variable density method. The method includes the following steps:

[0006] Step 1: Establish the motor geometric model and mesh the rotor design domain;

[0007] Step 2: Preprocess the design variables, i.e., perform Helmholtz filtering and Heaviside projection filtering on the design variables; the Helmholtz filtering equation is:

[0008]

[0009] Among them, z j Let x be the design variable for element j, and its design interval be (0,1], x j Let N be the centroid coordinates of element j. i,j The number of elements within the filtering radius R; the weight function between the centroids of two elements is expressed as:

[0010] w(x j )=R-|x j -x i | (2)

[0011] Where R is the filtration radius;

[0012] For intermediate variables Perform Heaviside projection filtering, the equation is:

[0013]

[0014] Where β is the steepness of the projection and η is the threshold of the filter density;

[0015] Step 3: Construct a multiphysics interpolation model, including the electromagnetic and mechanical properties of the material. The material properties used to distinguish between electrical steel and air are permeability, Young's modulus, and mass. Considering electromagnetic properties, calculate the differences between ferromagnetic materials and air in the rotor, define the distribution of ferromagnetic materials and air, and establish the interpolation model between design variables and electromagnetic properties as follows:

[0016]

[0017] Where the subscripts fe and air represent ferromagnetic materials and air, respectively, ν is the magnetic permeability, and p is the penalty factor;

[0018] Considering mechanical properties, the Young's modulus and mass characteristics of the unit are calculated, and an interpolation model between the design variables and mechanical properties is established as follows:

[0019]

[0020]

[0021] Where E is Young's modulus, m is physical mass, and ⊙ is the Hadamard product;

[0022] Step 4: Establish the interpolation model for the current angle. Considering the influence of the excitation current on the design, the interpolation model for the current angle is established as follows:

[0023] γ=πx γ (7)

[0024] Where, x γ γ is the design variable, and γ is the current angle;

[0025] For the static magnetic finite element model, the current depends on the current angle at different rotational positions, and the phase current is expressed as:

[0026]

[0027] Among them, I m Let θ be the current amplitude. k I represents the rotor's rotational position. A I B and I C These are the currents of phases A, B, and C, respectively.

[0028] Step 5: Establish the initial hypothetical structure for rotor optimization design;

[0029] Step 6: Initial finite element analysis, including level set splitting and body mesh generation;

[0030] The unit design variables are mapped to level set values, and the mapping equation is as follows:

[0031]

[0032] Where, n i z is the number of units sharing node k. max and z min These are the maximum and minimum values ​​of the design variables, respectively.

[0033] After converting the element design variables into node-level level set values, the design domain is divided using the zero level set surface. The division rule is as follows:

[0034]

[0035] Wherein, Ω+ represents the ferromagnetic material domain, and Ω- represents the air domain. For the boundary;

[0036] The design area was re-divided using body-fitting mesh technology;

[0037] Step 7: Finite element analysis. Calculate the governing equations for the electromagnetic field of the motor. The calculation formula is as follows:

[0038] SA = q (11)

[0039] Where S is the stiffness matrix of the nonlinear material, A is the unknown nodal vector potential, and q is the excitation source vector related to the current density J.

[0040] The governing equations of elasticity are calculated using the following formula:

[0041] KU=f (12)

[0042] Where K is the mechanical stiffness matrix, U is the unknown displacement vector, and f is the load vector;

[0043] Step 8: Construct an optimization model and determine the objectives and constraints;

[0044] The optimization model is represented by a set of control equations with the motor output torque as the objective and stress and compliance as constraints. The set of equations is as follows:

[0045]

[0046] Among them, T i The torque at the i-th rotor position, The element von Mises stress in the design domain is given by subscript j, which is the element number, and σ is the element stress. lim N is the allowable stress of the material. e Where C is the total number of elements in the design domain, C is the compliance of the rotor structure, and C0 is a given compliance reference value.

[0047] Step 9: Construct the augmented Lagrange framework. The calculation formula is as follows:

[0048]

[0049] Where k is the iteration step, and N = N e +2 represents the number of constraints. As a penalty item, To define the penalty term P (k) Equality constraints; μ (k) The quadratic penalty factor is μ, and the update equation is μ. (k+1) =min[αμ (k) ,μ max ], where α > 1 is the update parameter; For Lagrange multipliers;

[0050] Step 10: Perform sensitivity analysis using the adjoint vector method;

[0051] The chain rule is used to derive the sensitivity calculation formula for the augmented Lagrange equation to design variables:

[0052]

[0053] Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (15) can be rewritten as follows:

[0054]

[0055] The objective function f is independent of Young's modulus E and mass m, therefore the corresponding partial fraction is 0, and the formula can be rewritten as:

[0056]

[0057] In the formula, the sensitivity of the penalty term to the three unit parameters can be expressed as follows:

[0058]

[0059]

[0060]

[0061] Using the adjoint variable method to accelerate the sensitivity calculation, we obtain:

[0062]

[0063]

[0064]

[0065] The accompanying variables in the formula are as follows:

[0066]

[0067]

[0068]

[0069] Step 11: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle:

[0070] L=f+λ T (SA-q) (27)

[0071]

[0072] Setting the last term to 0, we obtain the solution for the adjoint variable:

[0073]

[0074] Ultimately, the sensitivity of the objective function to the current angle is:

[0075]

[0076] Step 12: Based on the sensitivity information, the design variables are updated and iterated using the gradient-based MMA optimization algorithm;

[0077] Step 13: Determine convergence. If convergence is not achieved, return to step 3 until the convergence condition is met, and output the optimization result.

[0078] Furthermore, the design steps for the initial conjectured structure in step 5 are as follows:

[0079] (1) Mesh the design domain and mirror the mesh to the entire computational domain;

[0080] (2) Use the SIMP method to determine the design variables and interpolation formulas;

[0081] (3) Define a square with a side length equal to the rotor diameter;

[0082] (4) Define the number of columns and rows of the circle, as well as the number and diameter of the circles along the side length of the square;

[0083] (5) Estimate the position of the element relative to the circle based on the element center point, i.e., inside or outside the circle;

[0084] (6) Define the initial values ​​of the design variables for the inner and outer elements of the circle, respectively;

[0085] (7) Map the initial variables of the design domain to the entire rotor domain.

[0086] Furthermore, in step 6, the body-fitted mesh is generated using a mesh adaptive method based on level sets.

[0087] Furthermore, the electromagnetic finite element equation SA=q in step 7 is solved using the Newton-Raphson iteration method.

[0088] Furthermore, the solution equation for the element load in the mechanical finite element equation KU=f in step 7 is:

[0089]

[0090] Where f e The centripetal force for each unit, m e For the unit mass, r e Let ω be the radius of the unit centroid on the rotor. e ω is the angular velocity.

[0091] Furthermore, in step 10, the objective function in formula (17) has a partial fraction of the element permeability. The solution is obtained using the adjoint variable method. The specific steps are as follows:

[0092] First, using the chain rule, we obtain:

[0093]

[0094] The torque at each sample point is obtained using the Maxwell tensor method:

[0095]

[0096] Among them, T i Let r be the torque at sampling point i, μ0 be the free permeability, and r be the torque at sampling point i. r and r s These are the rotor and stator radii, S, respectively. ag Let B be the air gap perimeter, r be the distance from the origin to the air gap integral path, and B be the distance from the origin to the air gap integral path. r and B t These are the normal and tangential components of the magnetic flux density, respectively; using the adjoint variable method, we obtain:

[0097] G = T i +λ i T (S i A i -q) (34)

[0098]

[0099] Simplified, we get:

[0100]

[0101] make:

[0102]

[0103] Differentiate the element torque with respect to permeability sensitivity The expression:

[0104]

[0105] Wherein, the accompanying variable λ i Confirmed by formula (37); combining formulas (13) and (38), we get:

[0106]

[0107] Where n is the number of sampling points for calculating the average torque, and I is the row vector.

[0108] The beneficial effects of this invention are as follows: This invention proposes a motor topology optimization method that combines body-fitted meshing with a variable density method. The method includes: introducing level set partitioning and body-fitted meshing to the traditional variable density method-based motor topology optimization method. This ensures that the optimized rotor not only has smooth boundaries in the new topology structure but also effectively avoids optimization errors caused by intermediate densities, making performance calculations more accurate and allowing for manufacturing without further post-processing. Simultaneously, considering structural design and current angle, the limits of the average torque output designed under a given current amplitude are explored. Furthermore, the method in this invention starts from an initial hypothetical structure with different numbers of holes, effectively generating a rotor structure with multiple magnetic barriers, reducing the influence of the initial design on the optimization results. According to mechanical calculations, the optimized rotor structure's flexibility meets the requirements of the optimization model, ensuring the integrity of the ferromagnetic barriers, i.e., the stiffness of the generated geometry; after calculating the performance of all elements, the maximum equivalent stress is less than the allowable stress, ensuring mechanical strength. Attached Figure Description

[0109] The present invention is described with reference to the following figures:

[0110] Figure 1 A flowchart of a motor topology optimization method that combines body-fitted mesh and variable density method;

[0111] Figure 2 Design domain partitioning diagram;

[0112] Figure 3 A graph showing the changes in the initial design variables;

[0113] Figure 4 Example diagram of level set segmentation and body-fitted mesh;

[0114] Figure 5 Flowchart for solving the augmented Lagrange equation for MMA;

[0115] Figure 6 To optimize the method iteration graph;

[0116] Figure 7 For the structural configuration diagram during the iteration process Detailed Implementation

[0117] To better explain and facilitate understanding of the present invention, a detailed description of the invention is provided below with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention. Furthermore, it should be noted that, unless otherwise specified, the embodiments and features described in the embodiments of the present invention can be combined with each other; for ease of description, only the parts relevant to the invention are shown in the accompanying drawings.

[0118] To overcome the shortcomings of traditional variable-density topology optimization methods, such as unsmooth boundaries, unreliable post-processing optimization design, and insufficient hole generation capability of level set topology optimization methods, this invention proposes a motor topology optimization method that combines body-fitted meshes with variable-density methods. The invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0119] Example 1

[0120] Figure 1 This is a schematic diagram of a motor topology optimization method that combines body-fitted mesh and variable density method in another embodiment of the present invention, as shown below. Figure 1 As shown, the method includes:

[0121] Step 1: Establish the motor geometric model and mesh the rotor design domain;

[0122] Figure 2 This is a schematic diagram of the design domain mesh generation in another embodiment of the present invention. First, a mesh for 1 / 8 of the model is generated, and then the mesh for the other half of the model is copied according to mirror symmetry. Changes in design variables and material properties caused by updates to design variables during the optimization process will be mapped to the entire computational domain.

[0123] Step 2: Preprocess the design variables;

[0124] Helmholtz filtering and Heaviside projection filtering are applied to the design variables to reduce local features and improve design fluency. To ensure the existence of solutions and avoid checkerboard patterns, Helmholtz filtering is first performed to transform the design variable field z into the intermediate density field. The transformation and filtering equation is:

[0125]

[0126] Among them, z j Let x be the design variable for element j, and its design interval be (0,1], x j Let N be the centroid coordinates of element j. i,jLet R be the number of elements within the filtering radius R. The weight function between the centroids of two elements can be expressed as:

[0127] w(x j )=R-|x j -x i | (41)

[0128] Where R is the filtration radius;

[0129] By utilizing Heaviside projection, grayscale phenomena in the optimization design are reduced, and intermediate density fields are completed. To physical density field Conversion:

[0130]

[0131] Where β is the steepness of the projection and η is the threshold of the filter density.

[0132] Step 3: Establish a multiphysics interpolation model;

[0133] Accordingly, in order to perform finite element analysis and optimization, it is necessary to establish an interpolation scheme between continuous design variables and material physical properties. The structural optimization of the motor rotor is reflected in the layout of ferromagnetic materials in the design domain. Since this invention mainly considers the electromagnetic and mechanical properties of materials, the material properties used to distinguish electrical steel from air are permeability, Young's modulus, and mass.

[0134] Considering electromagnetic performance, the differences between ferromagnetic materials and air in the rotor are calculated, the distribution of ferromagnetic materials and air is defined, and an interpolation model between design variables and electromagnetic characteristics is established as follows:

[0135]

[0136] In this context, the subscripts fe and air represent ferromagnetic materials and air, respectively, ν is the magnetic permeability, and p is the penalty factor, which can reduce the influence of units with intermediate density (grey units) on the optimization results.

[0137] Considering mechanical properties, the Young's modulus and mass characteristics of the unit are calculated, and an interpolation model between the design variables and mechanical properties is established as follows:

[0138]

[0139]

[0140] Where E is Young's modulus, m is physical mass, and ⊙ is the Hadamard product.

[0141] Step 4: Establish an interpolation model for the current angle;

[0142] To account for the impact of excitation current on the design, the current angle and structural design variables are optimized simultaneously to obtain the maximum output performance under the condition of maximum torque per ampere (MTPA). The interpolation model for the current angle is established as follows:

[0143] γ=πx γ (46)

[0144] Where, x γ γ is the design variable, and γ is the current angle;

[0145] For the static magnetic finite element model, the current depends on the current angle at different rotational positions, and the phase current is expressed as:

[0146]

[0147] Among them, I m Let θ be the current amplitude. k I represents the rotor's rotational position. A I B and I C These are the currents of phases A, B, and C, respectively.

[0148] Step 5: Establish the initial conjecture structure;

[0149] Topology optimization of electric motors is an ultra-high-dimensional, highly nonlinear problem. The initial design can affect the optimization results. Therefore, this invention proposes a method for establishing an initial hypothetical structure. The initial design is defined as follows: Define a square with a side length equal to the rotor diameter; define the number of columns and rows of circles, as well as the number and diameter of circles arranged along the side length of the square; estimate the position of the element relative to the circle based on the element center point, i.e., inside or outside the circle; define the initial values ​​of the design variables for elements inside and outside the circle respectively; map the initial variables of the design domain to the entire rotor domain. Figure 3 This describes the changes in the initial design variable z, where... Figure 3 (b) and Figure 3 (c) are the filtered results respectively. After Heaviside projection The distribution of .

[0150] Step 6: Initial finite element analysis, including level set partitioning;

[0151] The unit design variables are mapped to level set values, and the mapping equation is as follows:

[0152]

[0153] Where, n i z is the number of units sharing node k. max and z min These are the maximum and minimum values ​​of the design variables, respectively.

[0154] Even with filtering and projection before material interpolation, elements with virtual material properties, such as a virtual density of 0.6, cannot be avoided due to intermediate values ​​in the range of 0 to 1. Because of the nonlinearity in motor topology optimization, using electromagnetic finite element performance calculations with virtual material properties as the criterion for iterative loops will affect the output of the optimal solution. To avoid using virtual material properties in analysis and to perform finite element analysis with clear material boundaries, this invention proposes a boundary generation method based on the characteristics of virtual materials. This mapping method converts element design variables into node-level level set values ​​and uses the zero level set surface to partition the design domain. The partitioning rule is as follows:

[0155]

[0156] Wherein, Ω+ represents the ferromagnetic material domain, and Ω- represents the air domain. For the boundary.

[0157] In order to clearly delineate materials within the design domain, the geometric boundary is determined by the boundary between the level set and the design domain. Figure 4 In the diagram, (a) and (b) represent the generated level set and the region partitioning based on the boundary between the level set and the design domain. Using the zero-level set contour values ​​obtained on a fixed mesh, the design region is re-partitioned using body-fitting meshing techniques. The goal is to partition the design domain based on clear boundaries, and then transform the material properties into distinguishable materials with clear boundaries. Figure 4 (c) is the body-fitted mesh generated by the mesh adaptation method based on level sets.

[0158] Step 7: Finite element analysis. Calculate the governing equations for the electromagnetic field of the motor. The calculation formula is as follows:

[0159] SA = q (50)

[0160] Where S is the stiffness matrix of the nonlinear material, A is the unknown nodal vector potential, and q is the excitation source vector related to the current density J.

[0161] The governing equations of elasticity are calculated using the following formula:

[0162] KU=f (51)

[0163] Where K is the mechanical stiffness matrix, U is the unknown displacement vector, and f is the load vector; the solution equation for the element load is:

[0164]

[0165] Where f e The centripetal force for each unit, m e For the unit mass, r e Let ω be the radius of the unit centroid on the rotor.e ω is the angular velocity.

[0166] Step 8: Construct an optimization model and determine the objectives and constraints;

[0167] In this invention, the optimization aims to simultaneously consider the current angle and rotor structure design to find an optimal rotor structure layout that generates the maximum output torque, while ensuring the mechanical rigidity and strength in actual production and use. Therefore, the optimization model is represented as a set of control equations with the motor output torque as the objective and stress and flexibility as constraints. The set of equations is as follows:

[0168]

[0169] Among them, T i The torque at the i-th rotor position, The element von Mises stress in the design domain is given by subscript j, which is the element number, and σ is the element stress. lim N is the allowable stress of the material. e Where C is the total number of elements in the design domain, C is the compliance of the rotor structure, and C0 is a given compliance reference value.

[0170] Step 9: Construct the augmented Lagrange framework;

[0171] Because stress constraints are considered, the number of constraints is as large as the number of elements in the design domain. To overcome the computational burden caused by high-dimensional constraints, this invention employs the augmented Lagrangian method. The calculation formula is as follows:

[0172]

[0173] Where k is the iteration step, and N = N e +2 represents the number of constraints. As a penalty item, To define the penalty term P (k) Equality constraints; μ (k) The quadratic penalty factor is μ, and the update equation is μ. (k+1) =min[αμ (k) ,μ max ], where α > 1 is the update parameter; These are Lagrange multipliers. The penalty term is relative to the number of constraints N = N. e +2 is normalized to ensure the stability of the numerical solution.

[0174] Step 10: Perform sensitivity analysis using the adjoint vector method;

[0175] The optimization model is solved based on the MMA gradient optimization algorithm. Therefore, it is necessary to calculate the sensitivity of the augmented Lagrange equation to the design variables. The chain rule is used to list the calculation formula:

[0176]

[0177] Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (55) can be rewritten as follows:

[0178]

[0179] The objective function f is independent of Young's modulus E and mass m, therefore the corresponding partial fraction is 0, and the formula can be rewritten as:

[0180]

[0181] In the formula, the sensitivity of the penalty term to the three unit parameters can be expressed as follows:

[0182]

[0183]

[0184]

[0185] Using the adjoint variable method to accelerate the sensitivity calculation, we obtain:

[0186]

[0187]

[0188]

[0189] The accompanying variables in the formula are as follows:

[0190]

[0191]

[0192]

[0193] The partial component of the objective function with respect to the element permeability in the formula To solve this using the adjoint variable method, firstly, we obtain the following using the chain rule:

[0194]

[0195] The torque at each sample point was calculated using the Maxwell tensor method.

[0196]

[0197] Among them, T i Let r be the torque at sampling point i, μ0 be the free permeability, and r be the torque at sampling point i. r and r sThese are the rotor and stator radii, S, respectively. ag Let B be the air gap perimeter, r be the distance from the origin to the air gap integral path, and B be the distance from the origin to the air gap integral path. r and B t These are the normal and tangential components of the magnetic flux density, respectively. Using the adjoint variable method, we obtain:

[0198] G = T i +λ i T (S i A i -q) (69)

[0199]

[0200] Simplified, we get:

[0201]

[0202] make:

[0203]

[0204] get The expression:

[0205]

[0206] Wherein, the accompanying variable λ i This is confirmed by formula (72). Combining formulas (13) and (73), we obtain:

[0207]

[0208] Where n is the number of sampling points for calculating the average torque, and I is the row vector.

[0209] Step 11: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle:

[0210] L=f+λ T (SA-q) (75)

[0211]

[0212] Setting the last term to 0, we obtain the solution for the adjoint variable:

[0213]

[0214] Ultimately, the sensitivity of the objective function to the current angle is:

[0215]

[0216] Step 12: Solve for the minimum value of the Lagrange equation using the gradient optimization solver MMA:

[0217] Based on the sensitivity solution, the moving asymptote method (MMA) is used to search for the approximate minimum of the augmented Lagrange equation, that is, to find the approximate solution to the Lagrange equation minimization problem. Figure 5 This is a flowchart for solving the Lagrange equation using MMA. First, input the data related to the finite element problem and the optimization solver, and initialize the Lagrange multipliers and penalty factors; then, use MMA to solve for an approximate solution to the Lagrange equation and update... and μ (k) Repeat the above steps until the convergence condition is met.

[0218] Step 13: Determine convergence. If convergence is not achieved, return to step 2 until convergence is achieved, obtain the optimal topology, and output the result.

[0219] Figure 6 The iteration history of the optimization method. Figure 7 This refers to the complete topological configuration of different iteration steps obtained by mirroring the optimized design domain. Figure 6 and Figure 7 The topology optimization iterative process shows that the overall configuration of the rotor structure emerges in fewer iterations, and subsequent iterations further refine the details of the configuration, demonstrating the computational efficiency of the solution method. Furthermore, after level set partitioning and body-fitting mesh processing, Figure 7 The topological configurations in the model all have smooth and clear edges, proving the reliability of the optimization method.

[0220] It should be noted that any reference numerals placed between parentheses in the claims should not be construed as limiting the claims. The word "comprising" does not exclude the presence of components or steps not listed in the claims. The words "a" or "an" preceding a component do not exclude the presence of a plurality of such components. The invention can be implemented by means of hardware comprising several different components and by means of a suitably programmed computer.

[0221] Furthermore, it should be noted that in the description of this specification, the terms "one embodiment," "some embodiments," "embodiment," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Furthermore, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0222] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the claims should be interpreted to include both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0223] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, then this invention should also include these modifications and variations.

Claims

1. A method for optimizing motor topology using a combination of body-fitted mesh and variable density method, characterized in that, The method steps are as follows: Step 1: Establish the motor geometric model and mesh the rotor design domain; Step 2: Preprocess the design variables, i.e., perform Helmholtz filtering and Heaviside projection filtering on the design variables; the Helmholtz filtering equation is: （1） in, For unit The design variables are defined within the design interval (0,1]. Let J be the centroid coordinates of element j. The number of elements within the filtering radius R; the weight function between the centroids of two elements can be expressed as: （2） Where R is the filtration radius; For intermediate variables Perform Heaviside projection filtering, the equation is: （3） in, Let be the steepness of the projection. The threshold for filter density; Step 3: Construct a multiphysics interpolation model, including the electromagnetic and mechanical properties of the materials. The material properties used to distinguish between electrical steel and air are permeability, Young's modulus, and mass. Considering electromagnetic properties, calculate the differences between ferromagnetic materials and air in the rotor, define the distribution of ferromagnetic materials and air, and establish the interpolation model between design variables and electromagnetic properties as follows: （4） In this context, the subscripts fe and air represent ferromagnetic materials and air, respectively. Permeability, As a penalty factor; Considering mechanical properties, the Young's modulus and mass characteristics of the unit are calculated, and an interpolation model between the design variables and mechanical properties is established as follows: （5） （6） in, For Young's modulus, For physical mass, For Hadamah accumulation; Step 4: Establish the interpolation model for the current angle. Considering the influence of the excitation current on the design, the interpolation model for the current angle is established as follows: （7） in, For design variables, It is the current angle; For the static magnetic finite element model, the current depends on the current angle at different rotational positions, and the phase current is expressed as: （8） in, The current amplitude, The rotor's rotational position, , and These are the currents of phases A, B, and C, respectively. Step 5: Establish the initial hypothetical structure for rotor optimization design; Step 6: Initial finite element analysis, including level set splitting and body mesh generation; The unit design variables are mapped to level set values, and the mapping equation is as follows: （9） in, The number of units sharing node k. and These are the maximum and minimum values ​​of the design variables, respectively. After converting the element design variables into node-level level set values, the design domain is divided using the zero level set surface. The division rule is as follows: （10） in, For ferromagnetic materials, For air domain, For the boundary; The design area was re-divided using body-fitting mesh technology; Step 7: Finite element analysis. Calculate the governing equations for the electromagnetic field of the motor. The calculation formula is as follows: （11） in The stiffness matrix of the nonlinear material. For unknown node potential vectors. To match current density Relevant stimulus source vectors; The governing equations of elasticity are calculated using the following formula: （12） in The mechanical stiffness matrix, The displacement vector is unknown. For load vectors; Step 8: Construct an optimization model and determine the objectives and constraints; The optimization model is represented by a set of control equations with the motor output torque as the objective and stress and compliance as constraints. The set of equations is as follows: （13） in, The torque at the i-th rotor position, The element von Mises stress in the design domain is given by the subscript j, which represents the element number. The allowable stress of the material, The total number of elements in the design domain. For the flexibility of the rotor structure, For the given compliance reference value; Step 9: Construct the augmented Lagrange framework. The calculation formula is as follows: （14） Where k is the iteration step, To constrain the quantity, As a penalty item, To define penalty items Equality constraints; Given a quadratic penalty factor, the update equation is: ,in To update parameters; For Lagrange multipliers; Step 10: Perform sensitivity analysis using the adjoint vector method; The chain rule is used to derive the sensitivity calculation formula for the augmented Lagrange equation to design variables: （15） Since the augmented Lagrange equation consists of an objective function term and a penalty term, formula (15) can be rewritten as: （16） Wherein, objective function With Young's modulus and quality Since it is irrelevant, the corresponding partial fraction is 0, and the formula is rewritten as: （17） In the formula, the sensitivity of the penalty term to the three unit parameters can be expressed as follows: （18） （19） （20） Using the adjoint variable method to accelerate the sensitivity calculation, we obtain: （21） （22） （23） The accompanying variables in the formula are as follows: （24） （25） （26） Step 11: Use the adjoint variable method to solve for the sensitivity of the objective function to the current angle: （27） （28） Setting the last term to 0, we obtain the solution for the adjoint variable: （29） Ultimately, the sensitivity of the objective function to the current angle is: （30） Step 12: Based on the sensitivity information, the design variables are updated and iterated using the gradient-based MMA optimization algorithm; Step 13: Determine convergence. If convergence is not achieved, return to step 3 until the convergence condition is met, and output the optimization result.

2. The motor topology optimization method based on body-fitted mesh and variable density method according to claim 1, characterized in that: The design steps for the initial conjectured structure in step 5 are as follows: (1) Mesh the design domain and mirror the mesh to the entire computational domain; (2) Use the SIMP method to determine the design variables and interpolation formulas; (3) Define a square with a side length equal to the rotor diameter; (4) Define the number of columns and rows of the circle, as well as the number and diameter of the circles along the side length of the square; (5) Estimate the position of the element relative to the circle based on the element center point, i.e., inside or outside the circle; (6) Define the initial values ​​of the design variables for the inner and outer elements of the circle, respectively; (7) Map the initial variables of the design domain to the entire rotor domain.

3. The motor topology optimization method based on body-fitted mesh and variable density method according to claim 2, characterized in that: In step 6, the body-fitted mesh is generated using a mesh adaptive method based on level sets.

4. The motor topology optimization method based on body-fitted mesh and variable density method according to claim 3, characterized in that: Electromagnetic finite element equations in step 7 The solution is obtained using the Newton-Raphson iterative method.

5. The motor topology optimization method based on body-fitted mesh and variable density method according to claim 4, characterized in that: The mechanical finite element equations in step 7 The solution equation for the load of the medium element is: （31） in The centripetal force for each unit, For unit mass, The radius of the centroid of a single unit on the rotor. ω is the angular velocity.

6. The motor topology optimization method based on body-fitted mesh and variable density method according to claim 5, characterized in that: The partial fraction of the objective function with respect to the element permeability in formula (17) in step 10 The solution is obtained using the adjoint variable method. The specific steps are as follows: First, using the chain rule, we obtain: （32） The torque at each sample point is obtained using the Maxwell tensor method: （33） in, Sampling points Torque on, The permeability of free space, and These are the rotor and stator radii, respectively. For the air gap perimeter, This represents the distance from the origin to the integral path of the air gap. and These are the normal and tangential components of the magnetic flux density, respectively; using the adjoint variable method, we obtain: （34） （35） Simplified, we get: （36） make: （37） Differentiate the element torque with respect to permeability sensitivity The expression: （38） Among them, the accompanying variables Confirmed by formula (37); combining formulas (13) and (38), we get: （39） in, The number of sampling points used to calculate the average torque. It is a row vector.

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