Multi-objective optimization method for performance of steel wheel

Abstract

The invention discloses a steel wheel performance multi-objective optimization method, and belongs to the field of optimization algorithms. According to the invention, the agent model and the multi-objective optimization are cooperated to carry out steel wheel performance multi-objective optimization, and the optimization process is carried out by adopting three populations at the same time, so that the convergence and diversity of the whole optimization method are improved. Moreover, the prediction precision of the target function is improved through the iterative updating of the agent model at regular intervals, so that the agent model gradually learns a high-quality solution region in the optimization process, and the optimization search is guided more accurately. According to the method, complex constraint conditions are considered, convergence and diversity of a solution set can be considered in the iteration process, and therefore the wheel design parameters with the optimal comprehensive performance are obtained in the whole reasonable design space of the wheel.

Description

Technical Field

[0001] The present invention belongs to the field of wheel optimization design, and particularly relates to a multi-objective optimization method for the performance of steel wheels. Background Art

[0002] The multi-objective optimization problem of wheels is a typical constrained multi-objective optimization problem, and its core is to simultaneously optimize multiple objective functions while satisfying certain constraint conditions. Generally, the optimization objectives of wheels are to maximize the fatigue life, structural strength, minimize wind noise, and minimize the mass of the wheels while minimizing the manufacturing cost as much as possible. The mass of the wheel is a major factor in the manufacturing cost, and it often conflicts with other objectives. In traditional design, wheel manufacturers usually sacrifice the manufacturing cost of the wheel, that is, increase the amount of wheel material, to ensure other performances such as the fatigue and strength of the wheel. In addition, when designing the wheel, the assembly problem with the tire and the feasibility problem of the production process need to be considered, and the constraint conditions are complex and multi-faceted.

[0003] The genetic algorithm is one of the widely used algorithms in the field of multi-objective optimization, and has the advantages of simplicity, high efficiency, strong versatility, etc. Most of the existing wheel optimization methods also adopt this algorithm. Among them, the second-generation non-dominated sorting genetic algorithm (NSGA-II) is the most widely used. For example, Xu Wenchao used NSGA-II to perform multi-objective optimization on the structure-connection-performance of carbon-aluminum assembled wheels. The core mechanism of NSGA-II is based on non-dominated sorting and crowding distance. When designing, it is mainly aimed at low-dimensional multi-objective optimization problems with 2-3 objectives. However, the multi-objective optimization of wheels is a complex high-dimensional optimization problem. Most of the objective solution sets calculated by using NSGA-II will become non-dominated solutions, resulting in the inability to effectively distinguish the advantages and disadvantages of the solutions. In addition, in the high-dimensional space, the distances between solutions tend to be similar, the crowding distance is prone to failure, the fitness values of population individuals tend to be the same, and no effective selection pressure can be formed. The solution set distribution is unbalanced, and a uniformly distributed Pareto front cannot be obtained. In addition, NSGA-II has limited ability to handle complex constraint problems and insufficient utilization of infeasible solutions. When dealing with constraints, it preferentially selects feasible solutions and simply discards infeasible solutions, resulting in premature convergence to a small feasible region, making it difficult to explore the solution space under complex constraint conditions. The optimal solution of the wheel performance obtained may be a local optimal solution and cannot represent the global optimal solution of the design space. Some scholars have also used the third-generation non-dominated sorting genetic algorithm (NSGA-III) to perform multi-objective optimization on wheels. This algorithm introduces reference point allocation and solves the problem of the uniform distribution of multiple optimization objectives of wheels. However, it still has disadvantages such as a high demand for population size, insufficient ability to handle infeasible solutions, difficulty in maintaining the diversity and convergence of the solution set during the iteration process, etc., which affect the multi-objective optimization effect of wheels and cannot obtain the wheel design with the best performance. Summary of the Invention

[0004] The technical problem to be solved by the present invention is that in the multi-objective optimization method of existing wheels, when performing multi-objective optimization design on the fatigue performance, impact performance, NVH performance and manufacturing cost of wheels, the infeasible solutions are not fully utilized, it is difficult to maintain the diversity and convergence of the solution set, and it is easy to fall into local optimum and difficult to reach the global optimum solution. To solve this technical problem, the present invention intends to provide a multi-objective optimization method for the performance of steel wheels. This method considers complex constraint conditions and can take into account convergence and the diversity of the solution set during the iterative process, so as to obtain the wheel design parameters with the best comprehensive performance within the entire reasonable design space of the wheels.

[0005] The specific technical solution adopted by the present invention is as follows:

[0006] The present invention provides a multi-objective optimization method for the performance of steel wheels, and the steps are as follows:

[0007] S1. For the steel wheel model to be optimized, decompose the optimizable positions in the complete rim and complete spoke, and define the design variables, constraint conditions and objective function of the multi-objective optimization problem of the steel wheel performance;

[0008] The design variables include the rim thickness at different rim positions and the spoke thickness at different spoke positions;

[0009] The constraint conditions include the independent constraints of each design variable itself, the coupling constraints between different design variables and the material property constraints;

[0010] The objective function contains four objectives, namely minimizing the mass of the steel wheel, maximizing the fatigue performance of the steel wheel, maximizing the impact performance of the steel wheel and maximizing the lateral stiffness of the steel wheel;

[0011] S2. Uniformly sample each design variable in the design variable space to form a series of design variable value combinations. Based on each design variable value combination, model and generate a three-dimensional model of the steel wheel, and output each objective value in the objective function through finite element simulation, so as to construct a training data set and train a machine learning model to obtain a unified surrogate model with a set of design variable value combinations as the input and each objective value in the objective function as the output;

[0012] S3. Solve the multi-objective optimization problem based on the multi-population genetic algorithm, and for each group of feasible solutions during the solution process, use the unified surrogate model to predict each objective value in the objective function, and take the obtained optimal solution as the optimization result.

[0013] Preferably, in the multi-objective optimization problem of the steel wheel performance, the independent constraint of each design variable itself needs to satisfy that the value of the design variable is within its value space. The coupling constraint between different design variables needs to satisfy that the difference in the rim thickness between two adjacent rim positions should be less than the thickness difference threshold. The material property constraint needs to satisfy that the maximum Mises stress of the steel wheel under the fatigue simulation condition is less than the yield stress of the wheel material.

[0014] Preferably, in the multi-objective optimization problem of the steel wheel performance, the fatigue performance of the steel wheel is the weighted sum of the bending fatigue life, the radial fatigue life, and the biaxial fatigue life.

[0015] Preferably, in the multi-objective optimization problem of the steel wheel performance, the impact performance of the steel wheel is the negative value of the maximum Mises stress during the impact simulation process of the wheel.

[0016] Preferably, in S2, the Latin hypercube sampling method is used to uniformly sample each design variable in the design variable space.

[0017] Preferably, in S2, the machine learning model adopts Gaussian process regression, support vector regression, radial basis function, or neural network.

[0018] Preferably, in S3, the specific method for solving the multi-objective optimization problem based on the multi-population genetic algorithm is as follows:

[0019] S31. Randomly generate three initial populations: the first population P 1 、the second population P 2 、the third population P 3 , and the combination of design variable values x corresponding to each individual in the initialized population needs to be within the design variable space.

[0020] S32. A two-stage update strategy for the early stage and the late stage is used to update the three populations by the genetic algorithm, and the fitness of each feasible solution is obtained after the unified surrogate model predicts each objective value in the objective function.

[0021] In the early stage, the three populations are iteratively optimized synchronously. In each round of iteration, the corresponding offspring populations are updated through genetic operations consisting of selection, crossover, and mutation. However, the selection strategies adopted by the three populations in the selection link are different. The first population P 1 needs to select a part of the feasible solutions with the maximum fitness from the feasible solutions that fully meet the constraint conditions as the offspring population. The second population P 2 needs to select a part of the feasible solutions with the maximum fitness from the feasible solutions that ignore the constraint conditions as the offspring population. The third population P 3It is necessary to select the partial feasible solution with the largest fitness from the feasible solutions that satisfy the constraints after dynamic relaxation of the boundary conditions as the offspring population, and in each round of iteration, the second population P 2 and the third population P 3 of the offspring population need to be passed to the offspring population of the first population P 1 to participate in the next round of iteration of the first population P 1 ;

[0022] In the later stage, stop the iterative update of the second population P 2 while the first population P 1 and the third population P 3 continue to update and iterate the offspring population in the same update manner as in the previous stage until the iteration termination condition is reached, and then obtain the Pareto solution set from the final first population P 1 for the user to determine the optimal solution of the steel wheel design variables according to the performance index preference.

[0023] Preferably, in S32, a default function is set to calculate whether each solution satisfies the constraints. When the constraints are fully satisfied, the default function value is 0. When the constraints are not fully satisfied, the default function value is greater than 0 and the greater the degree of non - satisfaction, the greater the default function value; during the iterative update process, each feasible solution of the first population P 1 must satisfy that the default function value is 0, and each feasible solution of the third population P 3 must satisfy that the default function value is less than a preset positive threshold.

[0024] Preferably, during the iterative optimization process in the previous stage, it is necessary to detect every other generation whether the second population P 2 reaches the unconstrained Pareto front. The detection criterion is that the standard deviation of each objective value in the objective function in the offspring population obtained in the current iteration round is less than the convergence threshold. If this detection criterion is met, it is considered to reach the unconstrained Pareto front, and the iteration in the previous stage is no longer continued but enters the later stage.

[0025] Preferably, during the two - stage update process of the previous stage and the later stage, it is necessary to screen out the individuals with errors exceeding the threshold from the offspring population and add them to the incremental learning data set, and obtain each objective value in the objective function through finite - element simulation as the sample label, and then regularly perform incremental learning optimization on the surrogate model using the incremental learning data set.

[0026] The present invention has the following beneficial effects compared with the prior art:

[0027] The present invention provides a multi-objective optimization method for the performance of steel wheels by collaborating a surrogate model with multi-objective optimization. In the optimization process, three populations are used simultaneously, which improves the convergence and diversity of the entire optimization method; the iterative update of the surrogate model enhances the prediction accuracy of the objective function. The surrogate model gradually learns the high-quality solution region in the optimization process, thereby guiding the optimization search more accurately.

[0028] In the multi-population genetic algorithm provided by the present invention, iterative optimization of three populations is carried out simultaneously at the initial stage of optimization, emphasizing global search. The second population focuses on unconstrained optimization, allowing exploration of infeasible regions, and can quickly find the unconstrained Pareto front. The solutions in the second population are transferred to the first population through information transmission to help the first population break through the constraint boundary and accelerate the finding of the constrained Pareto front; at the later stage of optimization, the iterative optimization of the second population is stopped, and the computing resources are concentrated on the iterative update of the first population and the third population. In the optimization process of the third population, the exploration of the boundary region is strengthened by relaxing the constraint conditions to generate diverse boundary solutions, and the high-quality boundary solutions of the third population are transmitted to the first population through information transmission to expand the search range of the first population, enhance the diversity of the first population, and improve the distribution of the solution set on the Pareto front.

[0029] In addition, in the multi-population genetic algorithm of the present invention, the surrogate model is also iteratively updated based on incremental learning, and the finite element simulation results of high-error data are used to further supervise and optimize the surrogate model, which improves the prediction accuracy of the objective function. The present invention enables the surrogate model to gradually learn the high-quality solution region in the optimization process, thereby guiding the optimization search more accurately. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a schematic diagram of the steps of a multi-objective optimization method for the performance of steel wheels;

[0031] Figure 2 It is a schematic diagram of 6 design variables of the steel wheel in the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0032] In order to make the above objects, features, and advantages of the present invention more obvious and understandable, the following detailed description of the specific embodiments of the present invention will be given in conjunction with the accompanying drawings. Many specific details are set forth in the following description in order to fully understand the present invention. However, the present invention can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without departing from the connotation of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below. The technical features in various embodiments of the present invention can be combined correspondingly without conflict.

[0033] In a preferred embodiment of the present invention, a multi-objective optimization method for the performance of steel wheels is provided, and its main steps are as follows:

[0034] S1. For the steel wheel model to be optimized, decompose the optimizable positions in the complete rim and complete spoke, and define the design variables, constraint conditions, and objective function of the multi-objective optimization problem for the performance of steel wheels.

[0035] In the above multi-objective optimization problem, the design variables include the rim thickness at different rim positions and the spoke thickness at different spoke positions. It should be noted that for the complete rim and complete spoke in the steel wheel, since different positions often require different thicknesses, the optimizable positions in the rim and spoke can be segmented (some special positions may not have optimization conditions and can be excluded from segmentation). The optimal thickness can be designed for the rim segments at different positions and the spoke segments at different positions respectively. Therefore, the rim thicknesses and spoke thicknesses at these different positions can all constitute design variables.

[0036] In the above multi-objective optimization problem, the constraint conditions include the independent constraints of each design variable itself, the coupling constraints between different design variables, and the material property constraints. In the embodiment of the present invention, the independent constraint of each design variable itself needs to satisfy that the value of the design variable is within its value space. The coupling constraint between different design variables needs to satisfy that the difference in rim thickness between two adjacent rim positions should be less than the thickness difference threshold. The material property constraint needs to satisfy that the maximum Mises stress of the steel wheel under the fatigue simulation condition is less than the yield stress of the wheel material.

[0037] In the above multi-objective optimization problem, the objective function includes four objectives, namely minimizing the mass of the steel wheel, maximizing the fatigue performance of the steel wheel, maximizing the impact performance of the steel wheel, and maximizing the lateral stiffness of the steel wheel.

[0038] In the embodiment of the present invention, the fatigue performance of the steel wheel is the weighted sum of the bending fatigue life, radial fatigue life, and biaxial fatigue life, and the impact performance of the steel wheel is the negative value of the maximum Mises stress of the wheel during the impact simulation process.

[0039] S2. Uniformly sample each design variable in the design variable space to form a series of design variable value combinations. Based on each design variable value combination, model and generate a three-dimensional model of the steel wheel, and output the respective objective values in the objective function through finite element simulation, so as to construct a training data set and train a machine learning model to obtain a unified surrogate model with a set of design variable value combinations as the input and the respective objective values in the objective function as the output.

[0040] It should be noted that the above design variable space refers to the value space of all design variables during the optimization process, and each design variable cannot exceed its own value space. The specific sampling method within the design variable space can be selected according to the actual situation. In the embodiment of the present invention, Latin hypercube sampling is used to uniformly sample each design variable within the design variable space. Each design variable samples a value within its own value space, and the sampled values of all design variables together form a set of design variable value combinations. After a series of design variable value combinations are sampled, for each set of design variable value combinations x, a three-dimensional wheel model can be automatically generated through parametric modeling, and then imported into finite element software to perform corresponding wheel mass calculation, wheel fatigue simulation, wheel impact simulation, and wheel lateral stiffness simulation, and output the corresponding objective function value f(x) composed of each objective value, obtaining sample data (x, f(x)). All the sample data (x, f(x)) can constitute a training dataset for training a machine learning model. The input of this machine learning model is a set of design variable value combinations x, and the output of this machine learning model is the objective function value f(x).

[0041] In the embodiment of the present invention, the above machine learning model adopts Gaussian process regression, support vector regression, radial basis function, or neural network, and the specific form is not limited. After training a unified surrogate model based on the sample data obtained from finite element simulation, the true finite element simulation model can be approximated through the unified surrogate model, quickly evaluating the objective function value, reducing the number of calls to the simulation, and reducing the calculation cost.

[0042] S3. Solve the multi-objective optimization problem based on a multi-population genetic algorithm, and for each set of feasible solutions during the solution process, use the unified surrogate model to predict each objective value in the objective function, and take the obtained optimal solution as the optimization result.

[0043] In the embodiment of the present invention, the specific method for solving the multi-objective optimization problem based on a multi-population genetic algorithm is as follows:

[0044] S31. Randomly generate three initial populations: the first population P 1 、the second population P 2 、the third population P 3 , the size of each population is N, and the design variable value combination x corresponding to each individual in the initialized population needs to be within the design variable space;

[0045] S32. Adopt a two-stage update strategy for the early stage and the late stage to update the three populations through the genetic algorithm, and the fitness of each feasible solution is obtained after the unified surrogate model predicts each objective value in the objective function. The specific processes of the two stages are introduced below respectively.

[0046] In the early stage, three populations are iteratively optimized synchronously. In each round of iteration, the corresponding offspring population is obtained through genetic operations consisting of selection, crossover, and mutation; however, the selection strategies adopted by the three populations in the selection link are different. The first population P 1 needs to select a part of the feasible solutions with the largest fitness from the feasible solutions that fully meet the said constraint conditions as the offspring population. The second population P 2 needs to select a part of the feasible solutions with the largest fitness from the feasible solutions that ignore the said constraint conditions as the offspring population. The third population P 3 needs to select a part of the feasible solutions with the largest fitness from the feasible solutions that meet the said constraint conditions after the dynamic relaxation of the boundary conditions as the offspring population. And in each round of iteration, the offspring populations of the second population P 2 and the third population P 3 need to be passed to the offspring population of the first population P 1 to participate in the next round of iteration of the first population P 1 .

[0047] In the later stage, the iterative update of the second population P 2 is stopped, while the first population P 1 and the third population P 3 continue to update and iterate the offspring population in the same update manner as in the said early stage until the iteration termination condition is reached. Then, a Pareto solution set is obtained from the final first population P 1 for the user to determine the optimal solution of the steel wheel design variables according to the performance index preference.

[0048] It should be noted that the solution space for selecting the feasible solution with the largest fitness in the above-mentioned third population P 3 is the feasible solution that meets the said constraint conditions after the dynamic relaxation of the boundary conditions. The said constraint conditions after the dynamic relaxation of the boundary conditions here refer to that it is not necessary to strictly meet the said constraint conditions, and it can slightly not meet the said constraint conditions, so as to expand the search space by dynamically relaxing the constraint boundary and retain the infeasible solutions located at the constraint boundary in the population.

[0049] In the above step S32, a violation function can be set to calculate whether each solution meets the said constraint conditions. When the said constraint conditions are fully met, the value of the violation function is 0. When the said constraint conditions are not fully met, the value of the violation function is greater than 0, and the greater the degree of non - satisfaction, the greater the value of the violation function; during the iterative update process, each feasible solution of the first population P 1 must satisfy that the value of the violation function is 0, and each feasible solution of the third population P 3 must satisfy that the value of the violation function is less than a preset positive threshold.

[0050] In an embodiment of the present invention, a typical multi-objective optimization problem of the performance of a steel wheel can be represented as a combination of the following objective function f(x) and constraint conditions:

[0051] min f(x) = (M(x), -F(x), -I(x), -S(x))

[0052]

[0053] Where: M(x) is the wheel mass, F(x) is the fatigue performance, I(x) is the impact performance, and S(x) is the lateral stiffness; m represents the total number of rim positions to be optimized, and the coupling constraint f 1 (x i , x i+1 ) = |x i+1 - x i | - l, x i , x i+1 represent the rim thicknesses (in mm) of two adjacent rim positions, and l represents the thickness difference threshold that the thickness difference between two adjacent rim positions is not allowed to exceed; and the material property constraint f 2 (x) = σ F max (x) - σ 0 , where σ 0 represents the yield stress σ 0 of the wheel material, and σ F max (x) represents the maximum Mises stress of the wheel under the fatigue simulation condition; Ω is the design variable space. For any i-th design variable x i , it is expressed as x i min ≤ x i ≤ x i max , i = 1, 2,..., 6, x i min and x i max respectively represent the lower and upper limits of the value range of x i .

[0054] Thus, among the above three constraints, the independent constraint x = (x 1 , x 2 ,..., x n ) T ∈ Ω of each design variable itself must be satisfied anyway, that is, the value of each design variable must be within its value range, but the other two constraints can be used to design the penalty function. In an embodiment of the present invention, in the above step S32, the penalty function CV k (x) in the following form can be set to calculate whether each solution satisfies the constraint conditions:

[0055]

[0056] Among them, when k = 1 and 2, CV is satisfied k (x) = 0 indicates that x is a feasible solution that fully satisfies the constraint conditions.

[0057] For the third population, if it is necessary to expand the search space by dynamically relaxing the constraint boundary, a constraint boundary threshold can be set to retain the infeasible solutions with smaller violation function values in the population, avoid prematurely discarding potential high-quality solutions, increase population diversity, and improve the search ability for complex regions. At this time, the violation function value used to screen and retain the feasible solutions can satisfy:

[0058] CV k (x) ≤ γ a

[0059] Among them: a represents the generation number of the current population, and γ a represents the constraint boundary threshold of the current population. When k = 1 and 2, CV is satisfied k (x) ≤ γ a indicates that x is a feasible solution that satisfies the constraint conditions after the dynamic relaxation of the boundary conditions.

[0060] In the embodiments of the present invention, the constraint boundary threshold can change dynamically according to the following formula:

[0061]

[0062] Among them: γ 0 is the violation function value of the initial population, A represents the maximum generation number of the population, and r is the fixed relaxation coefficient of the constraint boundary. The initial constraint boundary threshold is large, allowing the population to explore the infeasible region, and gradually decreases in the later stage, converging to the feasible region near the constraint boundary. Thus, the diverse solutions in the third population can be transferred to the first population, providing diverse solutions near the constraint boundary, improving the coverage and uniformity of the Pareto front, and avoiding the premature convergence of the first population to the local optimal solution. However, for the second population, the constraint conditions can be completely ignored, that is, the violation function CV k (x) can be used as a feasible solution regardless of any value, and only the offspring individuals need to be screened according to the fitness.

[0063] In addition, during the iterative optimization process in the above-mentioned early stage, it is necessary to detect every other generation whether the second population P 2 reaches the unconstrained Pareto front. The detection criterion is that the standard deviation of each objective value in the objective function of the offspring population obtained in the current iteration round is less than the convergence threshold. If this detection criterion is met, it is considered to have reached the unconstrained Pareto front, and the iteration in the early stage is no longer continued but enters the later stage.

[0064] In an embodiment of the present invention, during the two-stage update process of the pre-stage and the post-stage, it is necessary to screen out individuals with errors exceeding the threshold from the offspring population and add them to the incremental learning dataset, and obtain each target value in the objective function as a sample label through finite element simulation. Then, the surrogate model is incrementally learned and optimized using the incremental learning dataset at regular intervals.

[0065] It should be noted that in S3, the final Pareto solution set needs to be obtained from the first population after the final iteration, and the optimal solution of the steel wheel design variables needs to be further selected from this first population. Since there are multiple optimization objectives in the present invention, the specific selection criteria can be determined by the user according to the performance index preferences. Whichever optimization objective is given priority can be used as the index to select the optimal solution with the best performance of that optimization objective.

[0066] The following is based on a specific embodiment to illustrate the specific implementation manner of the above-mentioned multi-objective optimization method for the performance of steel wheels.

[0067] Embodiment

[0068] In this embodiment, taking the Figure 1 shown steel wheel as an example, the specific implementation steps of the multi-objective optimization method for the performance of the steel wheel are as follows:

[0069] Step 1: For a steel wheel model with a given structure, define the multi-objective optimization problem for the performance of the steel wheel.

[0070] Define design variables: According to the user's requirements for the steel wheel structure, define n = 6 wheel design variables x i , i = 1, 2,..., n, which constitute the design space x, namely the first rim thickness x 1 , the second rim thickness x 2 , the third rim thickness x 3 , the fourth rim thickness x 4 , the first spoke thickness x 5 , the second spoke thickness x 6 . The wheel positions represented by each design variable are as Figure 1 shown.

[0071] Define constraint conditions: Each design variable has independent constraints according to user requirements, rim / tire assembly requirements, and production process requirements, that is, it must take values within a certain range, x i min ≤x i ≤x i max , i = 1, 2,..., 6. In addition, there are coupling constraints, that is, the thickness difference between adjacent rim parts is less than 1 mm, that is, f(x i , xi+1 ) = |x i+1 -x i | - 1 ≤ 0, i = 1, 2, 3. There are also specific material property constraints, that is, the maximum Mises stress of the wheel under fatigue simulation conditions should be less than the yield stress σ 0 : σ F max (x) ≤ σ 0

[0072] Define the objective function of the multi-objective optimization of the steel wheel as follows:

[0073] (1) Minimize the wheel mass M(x), and the individual mass under different combinations of design variables is calculated by the following formula:

[0074] M(x) = ρ·V(x)

[0075] Among them, ρ is the density of the wheel material, and V(x) is the total volume of the wheel, which is measured by finite element software.

[0076] (2) Maximize the fatigue performance F(x), and the fatigue performance is the weighted sum of the bending fatigue life F 1 (x), the radial fatigue life F 2 (x), and the biaxial fatigue life F 3 (x) as follows:

[0077] F(x) = λ 1 F 1 (x) + λ 2 F 2 (x) + λ 3 F 3 (x)

[0078] Among them, λ 1 , λ 2 , λ 3 are weight coefficients, which are defined according to user requirements. Generally speaking, in simple designs, only the bending fatigue life and the radial fatigue life can be considered (set λ 3 = 0), and the finite element calculation cost of these two is relatively low; when the fatigue performance requirements for the wheel are relatively strict, it is recommended to consider the biaxial fatigue life of the wheel (set λ 3 ≠0), because the biaxial fatigue condition more realistically describes the load history of the wheel during actual service on the road and can more accurately reflect the actual service life of the wheel.

[0079] (3) Maximize the impact performance I(x), and the impact performance of the wheel is measured by the maximum Mises stress σ I max during the impact simulation of the wheel (I(x) = -σ Imax ), σ I max The smaller it is, the smaller the local deformation of the wheel when resisting impact, and the better the impact performance.

[0080] (4) Maximize the lateral stiffness S(x). The greater the lateral stiffness of the wheel, the greater the first-order resonance frequency of the wheel, and more external disturbances can be isolated, and the better the NVH performance of the wheel.

[0081] The multi-objective optimization problem of the steel wheel performance in this embodiment can be expressed as follows:

[0082] min f(x) = (M(x), -F(x), -I(x), -S(x))

[0083]

[0084] In the formula: i = 1, 2, 3 respectively correspond to 3 rim positions to be optimized, and the coupling constraint f 1 (x i , x i+1 ) = |x i+1 - x i | - 1, where x i , x i+1 represent the rim thicknesses (in mm) of the i-th and the (i + 1)-th rim positions, and 1 means that the difference in rim thickness between two adjacent rim positions is not allowed to exceed 1 mm; and the material property constraint f 2 (x) = σ F max (x) - σ 0 , where σ 0 represents the yield stress σ 0 of the wheel material, and σ F max (x) represents the maximum Mises stress of the wheel under the fatigue simulation condition; Ω is the design variable space. For any i-th design variable x i , it is expressed as x i min ≤ x i ≤ x i max , i = 1, 2,..., 6, and x i min and x i max respectively represent the lower limit and the upper limit of the value range of x i .

[0085] Step 2: Train the unified surrogate model

[0086] Uniformly sample within the design variable space by Latin hypercube sampling, and the sampling capacity is N 0, the sampling capacity is determined by comprehensively considering the accuracy requirements and the performance of computer equipment. For each input design variable x, first, a three-dimensional wheel model is automatically generated through parametric modeling, and then it is imported into the finite element software for corresponding wheel mass calculation, wheel fatigue simulation, wheel impact simulation, and wheel lateral stiffness simulation, and the corresponding objective function value f(x) is output to obtain the sample data (x, f(x)). Use machine learning methods (Gaussian process regression, support vector regression, radial basis function, neural network) to train a unified surrogate model based on the sample data. Approximate the real simulation model through the surrogate model, quickly evaluate the objective function value, reduce the number of simulation calls, and reduce the calculation cost.

[0087] Step 3: Multi-objective optimization process based on multi-population genetic algorithm

[0088] This process is specifically divided into steps 3.1) to 3.3), which are described separately as follows:

[0089] 3.1) Initialize the population:

[0090] In the initial stage, three initial populations are randomly generated: the first population P 1 , the second population P 2 , the third population P 3 , the size of each population is N, and the design variables x of each individual in the initialized population are within the specified value range. Predict the objective function value for the individuals in the initial population through the unified surrogate model. Evaluate the constraint conditions and calculate the violation function CV k (x):

[0091]

[0092] For k = 1 and 2, when CV k (x) = 0, it means that x is a feasible solution that fully satisfies the constraint conditions.

[0093] 3.2) Iteratively update the population with multiple populations

[0094] This link is divided into the early stage of a) and the later stage of step b), and different strategies are used for iterative update in two stages.

[0095] a) In the early stage, for the first population P 1 , the second population P 2 , the third population P 3 , genetic operations (selection, crossover, mutation) are used to generate the corresponding offspring O 1 , O 2 , O 3 , and the populations P are jointly iteratively optimized through environmental selection (evaluating the fitness of each individual using the non-dominated sorting method of the widely used multi-objective genetic algorithm and selecting the superior individuals)1 , P 2 , P 3 。Meanwhile, update and optimize the surrogate model. Specifically, pause the update of the population every D generations (D is a hyperparameter), and perform an update of the unified surrogate model once.

[0096] In this embodiment, the main purposes of setting three populations are described as follows:

[0097] The first population: responsible for exploring the Pareto front composed of feasible solutions with constraints.

[0098] The second population: directly optimize the objective function predicted by the surrogate model by ignoring the constraints, and quickly reach the unconstrained Pareto front. Through information transmission, the second population can help the first population break through the constraint boundary in the early stage and search the infeasible region. As the second population iteratively updates, the offspring gradually transfer to a more favorable feasible region, and these high-quality offspring are provided to the first population, guiding the first population to explore the constraint boundary region faster and accelerating convergence.

[0099] The third population: expand the search space by dynamically relaxing the constraint boundary, retain the infeasible solutions with smaller violation function values in the population, avoid prematurely discarding potential high-quality solutions, increase population diversity, and improve the search ability for complex regions.

[0100] For the third population, the specific constraint form for retaining feasible solutions is:

[0101] CV k (x) ≤ γ a

[0102] In the formula: the current population generation is represented as a, and γ a represents the constraint boundary threshold of the current population, and this constraint boundary changes dynamically according to the following formula:

[0103]

[0104] In the formula: γ 0 is the violation function value of the initial population, A represents the maximum population generation, and r is the fixed relaxation coefficient of the constraint boundary. The initial constraint boundary threshold is relatively large, allowing the population to explore the infeasible region. It gradually decreases in the later stage and converges to the feasible region near the constraint boundary, transferring the diverse solutions in the third population to the first population, providing diverse solutions near the constraint boundary, improving the coverage and uniformity of the Pareto front, and avoiding the first population prematurely converging to the local optimal solution.

[0105] The three populations cooperate and optimize. By transferring the high-quality solution sets in the second population and the third population to the first population, the optimization convergence performance is improved while the diversity of the solution set is not lost.

[0106] In addition, during the multi-objective optimization process, the algorithm generates and evaluates a large number of design variable combinations (i.e., candidate solutions). These candidate solutions are used to quickly predict the objective function values (such as wheel mass, fatigue life, impact performance, and NVH performance) through a surrogate model. When selecting the population, first obtain the function value of each individual according to the surrogate model, then calculate the violation function value, and then perform environmental selection and iterative update. Since the surrogate model is only an approximation of the real simulation model, there may be large errors in its prediction results at some design points. Therefore, the surrogate model is updated and optimized every D generations during the collaborative optimization of the three populations. The specific method for updating the surrogate model is as follows:

[0107] (1) Screen high-error design points

[0108] During each generation of the evolutionary process, the algorithm generates new candidate solutions (design variable combinations) through operations such as selection, crossover, and mutation. These new solutions first quickly predict the objective function values through the surrogate model, and high-error design points are screened based on the uncertainty inside the surrogate model (such as the variance in Gaussian process regression). The screening sample source is the population of D generations.

[0109] (2) Select high-error design points for high-precision simulation

[0110] For the high-error design points screened from the population of D generations, re-call a high-precision simulation tool (such as finite element analysis software) for real evaluation to obtain accurate objective function values. Add these new real data to the training dataset.

[0111] (3) Update the surrogate model

[0112] Retrain the unified surrogate model using the new dataset to improve the prediction accuracy of the surrogate model for the objective function. The surrogate model gradually learns the high-quality solution region during the optimization process, thus guiding the optimization search more accurately.

[0113] (4) After updating the surrogate model, perform iterative optimization of the population again

[0114] b) Detect the second population P every other generation during the optimization process 2 Whether it reaches the unconstrained Pareto front, which determines whether the algorithm enters the later stage. The specific method is as follows:

[0115] Define the evolutionary interval G and the convergence threshold The current population is denoted as a, and SD i is the standard deviation of the objective function values of the i-th generation population. If for the past G generations of populations, the standard deviation of the objective function values of each generation of populations is less than the convergence threshold Flag a state variable Γ = 1, indicating that the second population P 2It has converged to the unconstrained Pareto front, stops optimizing it, and the algorithm enters the later stage; otherwise, Γ = 0, and continue to optimize the second population. The collaborative optimization of the second population can accelerate the convergence speed of the overall optimization process.

[0116] The above state variables can be expressed by the formula:

[0117]

[0118] After entering the later stage, the second population P 2 stops iterative update, and the third population P 3 generates offspring O through genetic operators 3 , and adds the feasible solutions in the offspring O 3 and the infeasible solutions with smaller violation function values to the first population P 1 through information transfer, generates new offspring O through genetic operators 1 , and adds the feasible solutions in the offspring O 1 back to the first population P 1 , and then perform fitness-based environmental selection on P 3 , O 1 , O 3 for co-iterative optimization (using the non-dominated sorting method of the widely used multi-objective genetic algorithm to evaluate the fitness of each individual and select the best individuals).

[0119] It should be noted that during the optimization process, the collaborative optimization of the surrogate model is also carried out simultaneously. The surrogate model gradually learns the high-quality solution region in the optimization process, so as to more accurately guide the optimization search.

[0120] In summary, in the early stage of optimization, the present invention simultaneously performs iterative optimization of three populations, emphasizing global search. The second population focuses on unconstrained optimization, allowing exploration of the infeasible region, and can quickly find the unconstrained Pareto front. The solutions in the second population are transferred to the first population through information transfer, helping the first population break through the constraint boundary and accelerate the finding of the constrained Pareto front. At the same time, the optimization of the surrogate model is carried out to improve the prediction accuracy of the surrogate model for the objective function and enhance the evaluation accuracy of the population; in the later stage of optimization, the iterative optimization of the second population is stopped, and the computing resources are concentrated on the iterative update of the first population and the third population. The optimization process of the third population strengthens the exploration of the boundary region by relaxing the constraint conditions, generates diverse boundary solutions, and transfers the high-quality boundary solutions of the third population to the first population through information transfer, expanding the search range of the first population, enhancing the diversity of the first population, and improving the distribution of the solution set on the Pareto front. During this process, the optimization of the surrogate model is also carried out simultaneously. The surrogate model gradually learns the high-quality solution region in the optimization process, so as to more accurately guide the optimization search.

[0121] 3.3) Convergence Judgment and Output

[0122] After each iteration, it is necessary to determine whether the number of iterations reaches the set maximum number of generations A or meets other convergence conditions. If so, output the constrained Pareto solution set in the first population. If not, continue to execute the collaborative optimization of the surrogate model and multi-objective optimization. After obtaining the Pareto solution set, since this invention is a multi-objective optimization, the user can determine the optimal solution of the steel wheel design variables according to the performance index preference. In this embodiment, the goal of minimizing the wheel mass is given priority. Therefore, the solution set with the minimum wheel mass can be selected as the compromise solution, and thus the design of the steel wheel with the best performance is completed.

[0123] The above-described embodiments are only some of the preferred implementation solutions of the present invention, but are not intended to limit the present invention. Those of ordinary skill in the relevant technical field can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, all technical solutions obtained by means of equivalent replacement or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A multi-objective optimization method for steel wheel performance, characterized in that: Here are the steps: S1. For the steel wheel model to be optimized, the optimizable positions in the complete rim and the complete spoke are decomposed, and the design variables, constraints and objective functions of the multi-objective optimization problem of steel wheel performance are defined; The design variables include rim thickness at different rim positions and spoke thickness at different spoke positions; The constraints include independent constraints of each design variable, coupling constraints between different design variables, and material performance constraints; The objective function includes four objectives, namely, minimizing the mass of the steel wheel, maximizing the fatigue performance of the steel wheel, maximizing the impact performance of the steel wheel and maximizing the lateral stiffness of the steel wheel; S2. uniformly sampling each design variable in the design variable space to form a series of design variable value combinations, modeling and generating a three-dimensional model of a steel wheel based on each design variable value combination, and outputting each target value in the objective function through finite element simulation, thereby constructing a training data set and training a machine learning model to obtain a unified proxy model with a set of design variable value combinations as input and each target value in the objective function as output; S3. Solving the multi-objective optimization problem based on a multi-population genetic algorithm, and using the unified agent model to predict each target value in the target function for each feasible solution in the solving process, and obtaining the optimal solution as the optimization result.

2. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In the multi-objective optimization problem of steel wheel performance, the independent constraint of each design variable itself needs to satisfy that the value of the design variable is within its value space, the coupling constraint between different design variables needs to satisfy that the difference in rim thickness between two adjacent rim positions should be less than the thickness difference threshold, and the material property constraint needs to satisfy that the maximum Mises stress of the steel wheel under fatigue simulation conditions is less than the yield stress of the wheel material.

3. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In the multi-objective optimization problem of steel wheel performance, the fatigue performance of the steel wheel is the weighted sum of bending fatigue life, radial fatigue life and biaxial fatigue life.

4. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In the multi-objective optimization problem of steel wheel performance, the impact performance of the steel wheel is the negative value of the maximum Mises stress of the wheel during the impact simulation process.

5. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In S2, each design variable is uniformly sampled in the design variable space using Latin hypercube sampling.

6. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In S2, the machine learning model adopts Gaussian process regression, support vector regression, radial basis function or neural network.

7. The multi-objective optimization method for steel wheel performance according to claim 1, characterized in that: In S3, the specific method for solving the multi-objective optimization problem based on a multi-population genetic algorithm is as follows: S31. Randomly generate three initial populations: the first population, the second population, and the third population. The design variable value combination corresponding to each individual in the initialization population must be within the design variable space. S32, a two-stage update strategy of an early stage and a late stage, wherein the three populations are updated by a genetic algorithm, and the fitness of each feasible solution is obtained by predicting each target value in the objective function by the unified agent model; In the early stage, the three populations are iteratively optimized synchronously. In each round of iteration, the corresponding offspring population is updated through genetic operations consisting of selection, crossover, and mutation. However, the three populations use different selection strategies in the selection process. The first population needs to select the feasible solutions with the largest fitness as the offspring population from the feasible solutions that fully meet the constraints. The second population needs to select the feasible solutions with the largest fitness as the offspring population from the feasible solutions that ignore the constraints. The third population needs to select the feasible solutions with the largest fitness as the offspring population from the feasible solutions that meet the constraints after the boundary conditions are dynamically relaxed. In each round of iteration, the offspring populations of the second and third populations need to be passed to the offspring population of the first population to participate in the next round of iteration of the first population. In the later stage, the iterative update of the second population is stopped, while the first and third populations continue to iterate the update of the sub-populations in the same updating manner as in the previous stage until the iteration termination condition is reached. A Pareto solution set is obtained from the final first population for the user to determine the optimal solution of the steel wheel design variables according to the performance indicator preference.

8. The multi-objective optimization method for steel wheel performance according to claim 7, characterized in that: In S32, a default function is set to calculate whether each solution satisfies the constraint condition. When the constraint condition is fully satisfied, the default function value is 0. When the constraint condition is not fully satisfied, the default function value is greater than 0, and the greater the degree of non-satisfaction, the greater the default function value. In the iterative update process, each feasible solution of the first population must satisfy that the default function value is 0, and each feasible solution of the third population must satisfy that the default function value is less than a preset positive threshold.

9. The multi-objective optimization method for steel wheel performance according to claim 7, characterized in that: During the iterative optimization process in the early stage, it is necessary to detect whether the second population has reached the unconstrained Pareto front every other generation. The detection standard is that the standard deviation of each target value in the objective function of the offspring population obtained in the current iteration round is less than the convergence threshold. If the detection standard is met, it is regarded as reaching the unconstrained Pareto frontier, and the iteration of the early stage will no longer continue but enter the later stage.

10. The multi-objective optimization method for steel wheel performance according to claim 7, characterized in that: In the two-stage update process of executing the early stage and the late stage, it is necessary to screen out individuals with errors exceeding the threshold from the offspring population and add them to the incremental learning data set, and obtain the target values ​​in the objective function as sample labels through finite element simulation, and then regularly use the incremental learning data set to perform incremental learning optimization on the proxy model.

Images