Objective Design Method for Lightweight and Modal Optimization of Rear Subframe Structure Based on ML

Abstract

The present invention discloses an object design method for lightweight and modal optimization of the rear subframe structure based on ML, including: (1) constructing a mathematical model corresponding to the simulation of the rear subframe weight and the first-order natural frequency through three-dimensional modeling, static analysis, and modal analysis; (2) generating a population based on the Maximin criterion and Latin hypercube, and establishing a radial basis function machine learning model; (3) using the DPM evolutionary strategy based on the Eplison function driven by Pbest to generate candidate offspring individual vectors; (4) constructing a min-max Pareto front improvement function to screen the true offspring individual vectors; (5) performing simulation evaluation on the true offspring individual vectors, adaptively switching the type of reference vector based on the update of the inverse generation distance, and returning to step (3) until the optimization objective meets the design requirements, and outputting the optimal values of the optimization design parameters. The present invention adaptively adjusts the evolutionary direction according to the population simulation results, and has good optimization design effects for the two objectives of lightweight and modal optimization.

Description

Technical Field

[0001] The present invention relates to the technical fields of artificial intelligence and swarm intelligence. Specifically, it relates to an objective design method for lightweight and modal optimization of the rear subframe structure based on ML. Background Art

[0002] As a key component in vehicle structures, the rear subframe is widely used in various types of transportation vehicles such as cars, trucks, and motorcycles. Its main function is to support the vehicle body and the powertrain, ensuring the stability, safety, and comfort of the vehicle. With the development of the modern automotive industry, consumers have put forward higher requirements for vehicle performance and fuel efficiency. Lightweight design has become one of the core objectives for improving the overall vehicle performance. Lightweighting not only helps reduce the overall vehicle weight, improve fuel economy, but also enhances handling and comfort. At the same time, the rear subframe will vibrate under external forces during vehicle operation, and these vibrations will affect the dynamic response and comfort of the vehicle. Therefore, modal optimization has become another key design task. Modal optimization improves the vibration characteristics of the rear subframe and reduces resonance phenomena by adjusting its geometric shape, material, and structural configuration, thereby enhancing the dynamic performance of the vehicle.

[0003] However, traditional design optimization methods mostly focus on single objectives, usually only considering one aspect of lightweighting or modal optimization, and it is difficult to achieve an effective balance between the two. At the same time, traditional optimization methods often rely on experience and a large amount of experimental data, with high computational costs and long time consumption, which greatly reduces the efficiency and accuracy of the optimization process in high-dimensional design spaces. To address this problem, ML (Machine Learning) optimization methods have emerged. ML can reveal the complex non-linear relationships between design variables through learning a large amount of historical design data. Compared with traditional methods, ML has higher efficiency and accuracy, can quickly find the optimal solution in complex design spaces, significantly reduce computational costs, and improve the automation level of the optimization process. Currently, many optimization technologies have not utilized the capabilities of machine learning and only rely on traditional simulations, which are time-consuming and laborious and cannot meet the optimization requirements in practical engineering. Summary of the Invention

[0004] In view of the limitations of the existing technology or the need for improved technology, the present invention proposes an objective design method for lightweight and modal optimization of the rear subframe structure based on ML. The ML technology is used for the collaborative design of lightweight and modal optimization of the rear subframe. Based on the time-consuming and complex simulation characteristics of the rear subframe structure involving ML, and the design requirements of two objectives of lightweight and modal optimization, a machine learning optimization design method is studied and designed to analyze and adaptively switch the reference vector based on the simulation results to change the optimization direction. The method dynamically adjusts the evolution probability with the number of iterations to achieve targeted optimization of the weight and the first natural frequency. An evolutionary process is constructed based on the radial basis function machine learning model, a min-max Pareto front elevation function is constructed, and the true offspring individual vectors are selected. The static analysis and modal analysis of the population individual vectors are carried out using the lightweight and modal finite element analysis model of the rear subframe to obtain the weight and the first natural frequency, and the optimization direction is adaptively changed according to the inverse generational distance. It can not only effectively balance the two objectives, but also improve the two-objective optimization efficiency of the lightweight and modal optimization of the rear subframe structure, and has broad application prospects. The two-objective design method for lightweight and modal optimization of the rear subframe structure designed by the present invention based on machine learning can adapt to the optimization stagnation caused by complex situations, can achieve higher accuracy and efficiency, and can not only be used for the lightweight and modal optimization design of the rear subframe structure, but also provides a usable algorithm structure for the multi-objective optimization design of other complex structures.

[0005] To achieve the above object, according to one aspect of the present invention, there is provided an objective design method for lightweight and modal optimization of the rear subframe structure based on ML, the method comprising the following steps:

[0006] Step (1): Taking the size parameters of each front cross member, each rear cross member, and each longitudinal beam of the rear subframe as the optimization design parameters, constructing a simulation model of the weight and the first natural frequency of the rear subframe according to the solvers of the three-dimensional modeling software, the finite element analysis pre- and post-processing software, and the structural optimization and multi-physics field simulation software, and deriving a mathematical model for the lightweight and modal optimization design problem of the rear subframe with the weight and the first natural frequency as the two optimization objectives;

[0007] Step (2): Constructing a design space according to the value range of the optimization design parameters, generating a population based on the Maximin criterion and the Latin hypercube, generating a reference vector in the objective space, using the simulation model of the weight and the first natural frequency of the rear subframe to evaluate the population individual vectors by simulation, constructing a database and constructing a radial basis function machine learning model;

[0008] Step (3): Determine the population individual vectors associated with each reference vector based on the perpendicular distance relationship between the population individual vectors and the reference vectors. Using the population individual vectors associated with each reference vector as the benchmark points, adopt a Pbest-driven DPM evolutionary strategy based on the Eplison function to generate candidate offspring individual vectors;

[0009] Step (4): Use a radial basis function machine learning model to predict the weight and first natural frequency of all candidate offspring individual vectors, and construct a min-max Pareto front improvement function to select the true offspring individual vectors from all candidate offspring individual vectors;

[0010] Step (5): Use the rear subframe weight and first natural frequency simulation model to simulate and evaluate the true offspring individual vectors, update the population, database, and radial basis function machine learning model, adaptively switch the reference vector type based on the update situation of the reverse generational distance, and return to Step (3) until the two optimization objectives of weight and first natural frequency meet the design requirements, and output the optimal values of the optimization design parameters.

[0011] Further, the said Step (1) specifically includes the following steps:

[0012] First step, according to the geometric structure characteristics of the rear subframe, take the length of each front cross member, the width of each front cross member, the thickness of each front cross member, the length of each rear cross member, the width of each rear cross member, the thickness of each rear cross member, the length of the longitudinal beam, the width of the longitudinal beam, the thickness of the longitudinal beam, the number of front cross members, and the number of rear cross members as the optimization design parameters;

[0013] Second step, according to the selected material of the rear subframe and the structural force conditions, construct a 3D model in 3D modeling software and parameterize the 3D model to obtain a parameterized model;

[0014] Third step, import the parameterized model, the load conditions of the rear subframe, and the key boundary conditions into the finite element analysis pre- and post-processing software for finite element analysis to obtain a finite element analysis model;

[0015] Fourth step, use the solver of the structural optimization and multi-physics field simulation software to perform static analysis and modal analysis on the finite element analysis model to obtain the rear subframe weight and first natural frequency simulation model;

[0016] Fifth step, based on the rear subframe weight and first natural frequency simulation model, derive the mathematical model of the rear subframe lightweight and modal optimization design problem with the rear subframe weight and first natural frequency as the two optimization objectives. The specific expression is as follows:

[0017] ,

[0018] In the above formula, M represents the optimization design parameters of the rear subframe, represents the length of the first front cross member, represents the N Cm length of the represents the width of the first front cross member, represents the N Cm width of the represents the thickness of the first front cross member, represents the N Cm thickness of the N Cm represents the number of front cross members, represents the length of the first rear cross member, represents the N RCm length of the represents the width of the first rear cross member, represents the N RCm width of the represents the thickness of the first rear cross member, represents the N RCm thickness of the N RCm represents the number of rear cross members, represents the length of the longitudinal beam, represents the width of the longitudinal beam, represents the thickness of the longitudinal beam, represents the total weight of the rear subframe, represents the M total mass function of the rear subframe obtained by dividing the rear subframe into Z parts and summing them when optimizing the design parameters, represents the density of the rear subframe, represents the M volume of the rear subframe corresponding to the optimized design parameters at represents the M first natural frequency of the rear subframe corresponding to the optimized design parameters at represents the M first natural frequency function of the rear subframe corresponding to the optimized design parameters at represents the design domain composed of the optimized design parameters of the rear subframe; Find represents finding the optimal solution of the optimized design parameters, Min represents minimizing the weight of the rear subframe, Max represents maximizing the first natural frequency of the rear subframe, and S.t. represents the conditions that the optimized design parameters need to satisfy.

[0019] Further, the step (2) specifically includes the following steps:

[0020] In the first step, considering the material strength and design requirements, determine the value range of the optimization design parameters according to the strength range required by the rear subframe, and construct the design space according to the value range of the optimization design parameters;

[0021] In the second step, use Latin hypercube to generate multiple candidate populations within the design space;

[0022] In the third step, according to the Maximin criterion, calculate the scoring value of each candidate population. The specific steps for calculating the scoring value according to the Maximin criterion are as follows:

[0023] Calculate the Euclidean distance from each individual vector in the candidate population to all other individual vectors;

[0024] Find the minimum Euclidean distance from each individual vector in the candidate population to other individual vectors;

[0025] Take the minimum Euclidean distance of all individual vectors in the candidate population as the scoring value of the candidate population;

[0026] In the fourth step, take the candidate population with the maximum scoring value as the population;

[0027] In the fifth step, use the simplex method to generate uniformly distributed reference vectors in the objective space composed of the two optimization objectives of weight and first-order natural frequency;

[0028] In the sixth step, use the rear subframe weight and first-order natural frequency simulation model to simulate and evaluate the population individual vectors, obtain the rear subframe weight and first-order natural frequency corresponding to the population individual vectors, and store all the population individual vectors and the corresponding rear subframe weight and first-order natural frequency in the database;

[0029] In the seventh step, use all the population individual vectors in the database to establish a radial basis function machine learning model. The expression of the radial basis function machine learning model is as follows:

[0030] ,

[0031] ,

[0032] In the above formula, represents the radial basis function machine learning model constructed for weight, x represents the value of the optimization design parameter, represents the i th reference vector for weight, represents the basis function vector, represents the calculation result of each basis function at the value x of the optimization design parameter, represents the center point of the \(i\)-th basis function, represents the values of the optimized design parameters \(x\) and the center of the basis function the Euclidean distance between represents the radial basis function machine learning model constructed for the first-order natural frequency, represents the i th reference vector for the first-order natural frequency.

[0033] Furthermore, step (3) specifically includes the following steps:

[0034] First step, in the target space, according to the uniform distribution of the reference vectors, calculate the perpendicular distance between each population individual vector and all reference vectors;

[0035] Second step, establish a connection between each population individual vector and the reference vector with the closest distance, and randomly select one from the population individual vectors associated with the reference vector as the reference point according to the uniformly distributed random number;

[0036] Third step, for the reference vectors that have not established a connection with the population individual vectors, select the neighboring reference vector with the closest distance to it, and randomly select one from the population individual vectors associated with the neighboring reference vector as the reference point according to the uniformly distributed random number;

[0037] Fourth step, for the selected reference points, randomly select an optimized design parameter inside the design space to perform the DPM evolutionary strategy based on the Eplison function to obtain an individual vector pool containing optimized design parameter information;

[0038] Fifth step, perform a Pbest-driven selection strategy on the individual vector pool to screen out the first p % of the temporarily selected offspring individual vectors. Among them, the specific steps of the Pbest-driven selection strategy are as follows:

[0039] Use the established radial basis function machine learning model to predict the target values corresponding to all temporarily selected offspring individual vectors, including weight and the first-order natural frequency;

[0040] Construct a Chebyshev aggregation function according to the ideal reference vector and the nadir reference vector, and calculate the fitness values of all temporarily selected offspring individual vectors;

[0041] Perform non-dominated sorting with the target values and fitness values of the temporarily selected offspring individual vectors as two objectives to obtain the non-dominated ranking levels of all temporarily selected offspring individual vectors;

[0042] According to the obtained non-dominated ranking levels, based on the ascending order principle, use the calculated fitness values as the screening index to select the temporarily selected offspring individual vectors;

[0043] According to the fitness value, select the top p % of the temporary candidate offspring individual vectors in ascending order to form a temporary candidate population;

[0044] Among them, the Chebyshev aggregation functions constructed based on the ideal reference vector and the nadir reference vector are as follows:

[0045] ,

[0046] ,

[0047] In the above formula, is the Chebyshev aggregation function constructed based on the ideal reference vector, is the Chebyshev aggregation function constructed based on the nadir reference vector, is the reference vector, is the temporary candidate offspring individual vector, represents the component of the reference vector on the j th objective, is the objective value of the temporary candidate offspring individual vector on the j th objective, represents the minimum value of the ideal reference vector on the j th objective in the database, represents the maximum value of the nadir reference vector on the j th objective in the database, represents the maximum-minimum normalization operation, is the minimum value of the ideal reference vector on the j th objective in the database, is the maximum value of the nadir reference vector on the j th objective in the database, represents the j th objective value in the database, is the i th individual vector in the database, DataBase represents the database, n is the number of objectives;

[0048] Step 6: Select a temporary candidate offspring individual vector from the temporary candidate population as the reference point according to the uniformly distributed random number, and execute the DPM evolutionary strategy based on the Eplison function to generate a candidate offspring individual vector;

[0049] Among them, the construction process of the Eplison function is as follows:

[0050] Determine the initial mutation probability according to the number of optimization design parameters, which can be calculated by the following formula:

[0051] ,

[0052] In the above formula, represents the initial mutation probability, represents the number of optimized design parameters, represents taking the minimum value between and 1;

[0053] Calculate the decay rate calculation parameter that changes with the number of iterations. The calculation formula is as follows:

[0054] ,

[0055] In the above formula, represents the decay rate calculation parameter that changes with the number of iterations, represents the adjustment factor, T c represents the set critical threshold, represents the maximum number of iterations calculation parameter to meet the design requirements, represents the initial mutation probability;

[0056] According to the decay rate calculation parameter that changes with the number of iterations and the initial mutation probability, calculate the mutation probability that changes with the number of iterations. The mutation probability can be calculated by the following formula:

[0057] ,

[0058] In the above formula, represents the mutation probability function that changes with the number of iterations, represents the current number of iterations, represents if the current number of iterations is less than the critical threshold, represents if the current number of iterations is greater than or equal to the critical threshold.

[0059] Furthermore, step (4) specifically includes the following steps:

[0060] First step, use the radial basis function machine learning model to predict the weight and the first natural frequency of all candidate offspring individual vectors, and obtain the predicted values of the weight and the first natural frequency of all candidate offspring individual vectors;

[0061] Second step, taking the predicted values of the weight and the first natural frequency of all candidate offspring individual vectors as the objectives, perform non-dominated sorting on all candidate offspring individual vectors to obtain the non-dominated individual set of candidate offspring individual vectors;

[0062] Third step, construct the min-max Pareto front improvement function, and calculate the minimum improvement value of all candidate offspring individual vectors in the non-dominated individual set relative to the Pareto front. Among them, the min-max Pareto front improvement function is as follows:

[0063] ,

[0064] In the above formula, PF represents the set of objective vectors of the optimal non-dominated individuals obtained in the current iteration in the objective space, that is, the current Pareto front, represents the current candidate offspring individual vector on the current Pareto front PF the minimum improvement value, is the minimum improvement value of the current candidate offspring individual vector relative to the current Pareto front PF represents the sum of the improvement amounts between all objective values and the current Pareto front PF represents the improvement amount between the current objective value and the current Pareto front PF is the value of the j th candidate offspring individual vector in the current Pareto front on the k th objective, is the current candidate offspring individual vector on the k th objective;

[0065] Step 4: According to the ascending order rule, sort the calculated minimum improvement values of all candidate offspring individual vectors relative to the Pareto front in ascending order, and select the first two candidate offspring individual vectors with the minimum improvement values as the true offspring individual vectors.

[0066] Furthermore, the step (5) specifically includes the following steps:

[0067] Step 1: Use the simulation model of the rear subframe weight and the first-order natural frequency to simulate and evaluate the true offspring individual vectors, so as to obtain the rear subframe weight and the first-order natural frequency corresponding to the true offspring individual vectors, store the true offspring individual vectors and the corresponding rear subframe weight and the first-order natural frequency in the database, and update the radial basis function machine learning model according to all the individual vectors in the database;

[0068] Step 2: Perform non-dominated sorting on the set formed by the true offspring individual vectors and the population individual vectors with the weight and the first-order natural frequency as the objectives to obtain the non-dominated ranking levels of all the individual vectors in the set;

[0069] Step 3: According to the non-dominated ranking levels of all the individual vectors in the set, determine the satisfied level value. If the number of all the individual vectors with the top W levels is greater than the population size, and all the individual vectors with the top W-1 ​​​If the number of individual vectors at a certain level is less than the population size, the satisfied level value is equal to W-1 ;

[0070] Fourthly, determine the candidate individual vector set and the remaining individual vector set according to the satisfied level value. Among them, the candidate individual vector set contains all individual vectors from the first level to W-1 level, and the remaining individual vector set contains all individual vectors with W level; if the number of all individual vectors with the highest W level is equal to the population size, the candidate individual vector set contains all individual vectors from the first level to W level, and the remaining individual vector set is reset to be empty;

[0071] Fifthly, for each individual vector in the candidate individual vector set, calculate the Euclidean distance between it and the reference vector, associate the individual vector with the reference vector having the minimum Euclidean distance, remove the individual vector from the candidate individual vector set, and remove the associated reference vector from the reference vector set to obtain the remaining reference vector set;

[0072] Sixthly, if the remaining reference vector set is not empty, calculate the Chebyshev aggregation value of all individual vectors in the remaining individual vector set based on the current reference vector, select the individual vector with the minimum Chebyshev aggregation value and store it in the candidate individual vector set until the remaining reference vector set is an empty set, output the candidate individual vector set, and update the population with the candidate individual vector set;

[0073] Seventhly, calculate the reverse generational distance and compare the current generation's reverse generational distance with the previous generation's reverse generational distance. If the current generation's reverse generational distance is less than or equal to the previous generation's reverse generational distance, it proves that the optimization direction is effective and the reference vector for the optimization process is not switched; if the current generation's reverse generational distance is greater than the previous generation's reverse generational distance, it proves that the optimization direction has poor effect and the reference vector for the optimization process is switched. The formula for calculating the reverse generational distance is as follows:

[0074] ,

[0075] In the above formula, IGD ( PF , PF true ) represents the function for calculating the reverse generational distance, PF true represents multiple uniformly distributed reference vectors, x represents PF true a reference vector on dist (x, PF ) represents the individual vector in the current set and PFthe Euclidean distance between reference vectors in

[0076] Step 8: Determine whether the two optimization objectives, i.e., the weight of the optimal individual vector optimized so far and the first natural frequency value, meet the design requirements. If they meet the design requirements, output the values of the optimal optimized design parameters. If they do not meet the design requirements, return to step (3) and continue to execute all steps from step (3) to step (5) until the design requirements are met.

[0077] In a second aspect, the present invention also provides a computer device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements the steps of the target design method for lightweight and modal optimization of the rear subframe structure based on ML.

[0078] In a third aspect, the present invention also provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, it implements the steps of the target design method for lightweight and modal optimization of the rear subframe structure based on ML.

[0079] Generally speaking, compared with the prior art, the limitations of the target design method for lightweight and modal optimization of the rear subframe structure based on ML provided by the present invention are improved as follows:

[0080] 1. For the focuses in different stages of the optimization process, an Eplison function is designed. This function pays attention to improving the quality of solutions in the early stage of optimization and focuses on maintaining the diversity of solutions in the later stage of optimization, thus effectively balancing the exploration and exploitation processes.

[0081] 2. A min-max Pareto front improvement function is constructed for screening, combined with a radial basis function machine learning model for prediction, and individual vectors are selectively picked for simulation of the lightweight and modal finite element analysis model of the rear subframe, improving the efficiency and accuracy of traditional optimization;

[0082] 3. To better judge the direction of the optimization process, based on whether the inverse generational distance obtained from the simulation results of the lightweight and modal finite element analysis model of the rear subframe is updated, the reference vectors in the optimization process are adaptively switched to adjust the evolution direction, enabling better problem-solving ability for complex problems and making the optimization always proceed efficiently;

[0083] The present invention can optimize multi-objective problems involving complex simulations, improve the optimization efficiency of complex simulations, and can optimize multiple objectives simultaneously to achieve the overall optimum, which is beneficial to the optimization application of various complex structures and has practicality. Description of the Drawings

[0084] Figure 1It is a flow diagram of an object design method for lightweight and modal optimization of a rear subframe structure based on ML provided by the present invention. Specific embodiments

[0085] To more clearly elaborate the purpose, technical solution, and advantages of the present invention, it will be described in detail below in conjunction with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit it. In addition, as long as the technical features in the following embodiments do not conflict with each other, they can be combined with each other.

[0086] Please refer to Figure 1 , an object design method for lightweight and modal optimization of a rear subframe structure based on ML provided by the present invention, which is applicable to the two-object design optimization problem of lightweight and modal optimization of the rear subframe structure. Specifically, the method includes steps (1) to (5).

[0087] Step (1): Taking the size parameters of each front crossbeam, each rear crossbeam, and each longitudinal beam of the rear subframe as the optimization design parameters, constructing a simulation model of the weight and the first natural frequency of the rear subframe according to the solvers of 3D modeling software (such as CATIA), finite element analysis pre- and post-processing software (such as HyperMesh), and structural optimization and multi-physics field simulation software (such as OptiStruct), and deriving a mathematical model for the lightweight and modal optimization design problem of the rear subframe with weight and the first natural frequency as the two optimization objectives.

[0088] Step (1) specifically includes the following steps:

[0089] The first step is to take the length, width, thickness of each front crossbeam, the length, width, thickness of each rear crossbeam, the length, width, thickness of the longitudinal beam, the number of front crossbeams, and the number of rear crossbeams as the optimization design parameters according to the geometric structure characteristics of the rear subframe;

[0090] The second step is to construct a 3D model in 3D modeling software and parameterize the 3D model according to the selected material of the rear subframe and the structural stress situation to obtain a parameterized model;

[0091] The third step is to import the parameterized model, the load-bearing situation of the rear subframe, and the key boundary conditions into the finite element analysis pre- and post-processing software for finite element analysis to obtain a finite element analysis model;

[0092] The fourth step is to use the solver of the structural optimization and multi-physics field simulation software to perform static analysis and modal analysis on the finite element analysis model to obtain a simulation model of the weight and the first natural frequency of the rear subframe;

[0093] Step 5: Based on the simulation model of the rear subframe weight and the first-order natural frequency, a mathematical model for the lightweight and modal optimization design of the rear subframe is derived with the rear subframe weight and the first-order natural frequency as two optimization objectives. The specific expression is as follows:

[0094] ,

[0095] In the above formula, M represents the optimization design parameters of the rear subframe, represents the length of the first front crossmember, represents the N Cm th length of the front crossmember, represents the width of the first front crossmember, represents the N Cm th width of the front crossmember, represents the thickness of the first front crossmember, represents the N Cm th thickness of the front crossmember, N Cm represents the number of front crossmembers, represents the length of the first rear crossmember, represents the N RCm th length of the rear crossmember, represents the width of the first rear crossmember, represents the N RCm th width of the rear crossmember, represents the thickness of the first rear crossmember, represents the N RCm th thickness of the rear crossmember, N RCm represents the number of rear crossmembers, represents the length of the longitudinal beam, represents the width of the longitudinal beam, represents the thickness of the longitudinal beam, represents the total weight of the rear subframe, represents the total mass function of the rear subframe obtained by summing up M parts when the optimization design parameters are Z , represents the density of the rear subframe, represents the volume of the rear subframe corresponding to the optimization design parameters M , represents the first-order natural frequency of the rear subframe corresponding to the optimization design parameters M , represents the rear subframe at the optimization design parametersM The first-order natural frequency function corresponding to represents the design domain formed by the optimized design parameters of the rear subframe; Find represents finding the optimal solution of the optimized design parameters, Min represents minimizing the weight of the rear subframe, Max represents maximizing the first-order natural frequency of the rear subframe, and S.t. represents the conditions that the optimized design parameters need to meet.

[0096] Step (2): Construct a design space according to the value range of the optimized design parameters, generate a population based on the Maximin criterion and Latin hypercube, generate a reference vector in the objective space, use the simulation models of the rear subframe weight and the first-order natural frequency to simulate and evaluate the individual vectors of the population, construct a database and construct a radial basis function machine learning model.

[0097] Step (2) specifically includes the following steps:

[0098] The first step is to determine the value range of the optimized design parameters (the length, width, and thickness of each front crossbeam, the length, width, and thickness of each rear crossbeam, the length, width, and thickness of the longitudinal beam, the number of front crossbeams, and the number of rear crossbeams) according to the strength range required by the rear subframe considering material strength and design requirements, and construct a design space according to the value range of the optimized design parameters;

[0099] The second step is to generate multiple candidate populations in the design space using Latin hypercube;

[0100] The third step is to calculate the scoring value of each candidate population according to the Maximin (Maximize the Minimum Distance) criterion. The specific steps for calculating the scoring value using the Maximin criterion are as follows:

[0101] Calculate the Euclidean distance from each individual vector in the candidate population to all other individual vectors;

[0102] Find the minimum Euclidean distance from each individual vector in the candidate population to other individual vectors;

[0103] Take the minimum Euclidean distance of all individual vectors in the candidate population as the scoring value of the candidate population;

[0104] The fourth step is to take the candidate population with the maximum scoring value as the population;

[0105] The fifth step is to generate uniformly distributed reference vectors in the objective space composed of the two optimization objectives of weight and the first-order natural frequency using the simplex method;

[0106] Step 6: Use the simulation model of the rear subframe weight and the first-order natural frequency to simulate and evaluate the individual vectors of the population, obtain the rear subframe weight and the first-order natural frequency corresponding to the individual vectors of the population, and store all the individual vectors of the population and the corresponding rear subframe weight and first-order natural frequency in the database;

[0107] Step 7: Use all the individual vectors of the population in the database to establish a radial basis function machine learning model, where the expression of the radial basis function machine learning model is as follows:

[0108] ,

[0109] ,

[0110] In the above formula, represents the radial basis function machine learning model constructed for the weight, x represents the value of the optimization design parameter, represents the i th reference vector for the weight, represents the basis function vector, represents the calculation result of each basis function at the value x of the optimization design parameter, represents the center point of the ith basis function, represents the Euclidean distance between the value x of the optimization design parameter and the center of the basis function represents the radial basis function machine learning model constructed for the first-order natural frequency, represents the i th reference vector for the first-order natural frequency.

[0111] Step (3): Determine the individual vectors of the population associated with each reference vector according to the vertical distance relationship between the individual vectors of the population and the reference vectors. Taking the individual vectors of the population associated with each reference vector as the reference points, adopt the DPM evolutionary strategy based on the Eplison function driven by Pbest to generate candidate offspring individual vectors.

[0112] Step (3) specifically includes the following steps:

[0113] First step: In the target space, calculate the vertical distance between each individual vector of the population and all reference vectors according to the uniform distribution of the reference vectors;

[0114] Second step: Establish a connection between each individual vector of the population and the reference vector with the closest distance, and randomly select one from the individual vectors of the population associated with the reference vector as the reference point according to the uniformly distributed random number;

[0115] In the third step, for the reference vectors that have not established a connection with the population individual vectors, select the neighboring reference vector closest to it, and select one from the population individual vectors associated with the neighboring reference vector as the reference point according to a uniformly distributed random number;

[0116] In the fourth step, for the selected reference point, randomly select an optimization design parameter within the design space to perform the DPM evolutionary strategy based on the Eplison function to obtain an individual vector pool containing the information of the optimization design parameter;

[0117] In the fifth step, perform a Pbest-driven selection strategy on the individual vector pool to screen out the top p % of the temporary candidate offspring individual vectors. The specific steps of the Pbest-driven selection strategy are as follows:

[0118] Use the established radial basis function machine learning model to predict the target values corresponding to all temporary candidate offspring individual vectors, including weight and the first-order natural frequency;

[0119] Construct a Chebyshev aggregation function based on the ideal reference vector and the nadir reference vector, and calculate the fitness values of all temporary candidate offspring individual vectors;

[0120] Perform non-dominated sorting with the target values and fitness values of the temporary candidate offspring individual vectors as two objectives to obtain the non-dominated ranking levels of all temporary candidate offspring individual vectors;

[0121] According to the obtained non-dominated ranking levels, based on the ascending order principle, use the calculated fitness values as the screening index to select the temporary candidate offspring individual vectors;

[0122] According to the fitness values, based on the ascending order principle, select the top p % of the temporary candidate offspring individual vectors to form a temporary candidate population;

[0123] Among them, the Chebyshev aggregation functions constructed based on the ideal reference vector and the nadir reference vector are as follows:

[0124] ,

[0125] ,

[0126] In the above formula, is the Chebyshev aggregation function constructed based on the ideal reference vector, is the Chebyshev aggregation function constructed based on the nadir reference vector, is the reference vector, is the temporary candidate offspring individual vector, represents the component of the reference vector on the j th target, is the objective value of the temporary candidate offspring individual vector on the j th objective, represents the minimum value of the ideal reference vector in the database on the j th objective, represents the maximum value of the nadir reference vector in the database on the j th objective, represents performing the maximum-minimum normalization operation, is based on the minimum value of the ideal reference vector in the database on the j th objective, is the maximum value of the nadir reference vector in the database on the j th objective, represents the j th objective value in the database, is the i th individual vector in the database, DataBase represents the database, n is the number of objectives;

[0127] Step 6: Select a temporary candidate offspring individual vector from the temporary candidate population as the reference point according to the uniformly distributed random number, and execute the DPM evolutionary strategy based on the Eplison function to generate the candidate offspring individual vector;

[0128] The construction process of the Eplison function is as follows:

[0129] Determine the initial mutation probability according to the number of optimization design parameters, which can be calculated by the following formula:

[0130] ,

[0131] In the above formula, represents the initial mutation probability, represents the number of optimization design parameters, represents taking the minimum value between and 1;

[0132] Calculate the decay rate calculation parameter that changes with the number of iterations. The calculation formula is as follows:

[0133] ,

[0134] In the above formula, represents the decay rate calculation parameter that changes with the number of iterations, represents the adjustment factor, T c represents the set critical threshold, represents the maximum number of iterations calculation parameter to meet the design requirements, represents the initial mutation probability;

[0135] Calculate the parameters and the initial mutation probability according to the decay rate varying with the number of iterations, and calculate the mutation probability varying with the number of iterations. The mutation probability can be calculated by the following formula:

[0136] ,

[0137] In the above formula, represents the mutation probability function varying with the number of iterations, represents the current number of iterations, represents that if the current number of iterations is less than the critical threshold, represents that if the current number of iterations is greater than or equal to the critical threshold.

[0138] Step (4): Use the radial basis function machine learning model to predict the weights and first natural frequencies of all candidate offspring individual vectors, and construct the min-max Pareto front improvement function to select the true offspring individual vectors from all candidate offspring individual vectors.

[0139] Step (4) specifically includes the following steps:

[0140] First step, use the radial basis function machine learning model to predict the weights and first natural frequencies of all candidate offspring individual vectors, and obtain the predicted values of the weights and first natural frequencies of all candidate offspring individual vectors;

[0141] Second step, take the predicted values of the weights and first natural frequencies of all candidate offspring individual vectors as the objectives, perform non-dominated sorting on all candidate offspring individual vectors, and obtain the non-dominated individual set of candidate offspring individual vectors;

[0142] Third step, construct the min-max Pareto front improvement function, and calculate the minimum improvement value of all candidate offspring individual vectors in the non-dominated individual set relative to the Pareto front. Among them, the min-max Pareto front improvement function is as follows:

[0143] ,

[0144] In the above formula, PF represents the set of objective vectors of the optimal non-dominated individual set obtained in the current iteration in the objective space, that is, the current Pareto front, represents the current candidate offspring individual vector on the current Pareto front PF the minimum improvement value, is the minimum improvement value of the current candidate offspring individual vector relative to the current Pareto front PF the minimum lift value, represents the sum of the lift amounts between all objective values and the current Pareto front PF between, Indicates the improvement amount between the current target value and the current Pareto front PF , is the value of the j th candidate offspring individual vector in the current Pareto front on the k th objective, and is the objective value of the current candidate offspring individual vector k on the

[0145] Step 4: According to the ascending order rule, sort the minimum improvement values of all calculated candidate offspring individual vectors relative to the Pareto front in ascending order, and select the first two candidate offspring individual vectors with the minimum improvement values as the real offspring individual vectors.

[0146] Step (5): Use the simulation model of the rear subframe weight and the first-order natural frequency to simulate and evaluate the real offspring individual vectors, update the population, database, and radial basis function machine learning model, adaptively switch the reference vector type based on the update situation of the inverse generational distance, and return to Step (3) until the two optimization objectives of weight and first-order natural frequency meet the design requirements, and output the optimal values of the optimization design parameters.

[0147] Step (5) specifically includes the following steps:

[0148] First step: Use the simulation model of the rear subframe weight and the first-order natural frequency to simulate and evaluate the real offspring individual vectors, so as to obtain the rear subframe weight and the first-order natural frequency corresponding to the real offspring individual vectors, store the real offspring individual vectors and the corresponding rear subframe weight and first-order natural frequency in the database, and update the radial basis function machine learning model according to all individual vectors in the database;

[0149] Second step: Perform non-dominated sorting on the set formed by the real offspring individual vectors and the population individual vectors with weight and first-order natural frequency as the objectives to obtain the non-dominated ranking levels of all individual vectors in the set;

[0150] Third step: According to the non-dominated ranking levels of all individual vectors in the set, determine the satisfied level value. If the number of all individual vectors with the first W levels is greater than the population size (the number of individual vectors in the population), and the number of all individual vectors with the first W-1 levels is less than the population size, then the satisfied level value is equal to W-1 ;

[0151] Fourth step: Determine the candidate individual vector set and the remaining individual vector set according to the satisfied level value. Among them, the candidate individual vector set contains all individual vectors from the first level to the W-1 th level, and the remaining individual vector set contains all individual vectors with WAll individual vectors of the level; if the number of individual vectors with the highest W level is equal to the population size, the candidate individual vector set contains all individual vectors from the first level to W level, and the remaining individual vector set is reset to empty;

[0152] Step 5: For each individual vector in the candidate individual vector set, calculate the Euclidean distance between it and the reference vector, associate the individual vector with the reference vector having the minimum Euclidean distance, remove the individual vector from the candidate individual vector set, and remove the associated reference vector from the reference vector set to obtain the remaining reference vector set;

[0153] Step 6: If the remaining reference vector set is not empty, calculate the Chebyshev aggregation value of all individual vectors in the remaining individual vector set based on the current reference vector, select the individual vector with the minimum Chebyshev aggregation value and store it in the candidate individual vector set until the remaining reference vector set is an empty set, output the candidate individual vector set, and update the population with the candidate individual vector set;

[0154] Step 7: Calculate the reverse generation distance and compare the current generation reverse generation distance with the previous generation reverse generation distance. If the current generation reverse generation distance is less than or equal to the previous generation reverse generation distance, it proves that the optimization direction is effective and the reference vector for the optimization process is not switched; if the current generation reverse generation distance is greater than the previous generation reverse generation distance, it proves that the optimization direction has poor effect and the reference vector for the optimization process is switched. The formula for calculating the reverse generation distance is as follows:

[0155] ,

[0156] In the above formula, IGD ( PF , PF true ) represents the function for calculating the reverse generation distance, PF true represents multiple uniformly distributed reference vectors, x represents PF true a reference vector on, dist (x, PF ) represents the Euclidean distance between the individual vector in the current set and the PF reference vector in;

[0157] Step 8: Judge whether the two optimization objectives of the weight and the first-order natural frequency value of the optimal individual vector optimized so far meet the design requirements. If they meet the design requirements, output the optimal optimized design parameter values. If they do not meet the design requirements, return to step (3) and continue to execute all steps from step (3) to step (5) until the design requirements are met.

[0158] Example 1

[0159] In this example, the benchmark function ZDT2 is used to illustrate the optimization performance of a proposed ML-based objective design method for lightweight and modal optimization of the rear subframe structure. The expression of the benchmark function ZDT2 with two objectives is as follows:

[0160] ,

[0161] ,

[0162] In the above formula, and are the first objective function and the second objective function respectively, is the auxiliary construction function, represents the first optimization design parameter, are the remaining optimization design parameters other than the first optimization design parameter in the multi-dimensional design space, is the i th optimization design parameter, n is the number of optimization design parameters.

[0163] To elaborate on the superiority of the method in this example, a ML-based objective design method for lightweight and modal optimization of the rear subframe structure in this example is compared with another classic and excellent Kriging-assisted reference vector guided evolutionary algorithm, and the maximum number of simulation evaluations in this example is set to 300 times, and the number of optimization design parameters is set to 30. The average inverted generational distance (abbreviated as average IGD) of 25 rounds of optimization is compared. The results are shown in Table 1. The results show that under the same number of simulation evaluations, the method in this example is significantly better than the Kriging-assisted reference vector guided evolutionary algorithm. It can be considered that the method in this example can excellently solve the two-objective design optimization problem of lightweight and modal optimization of the rear subframe structure.

[0164] Table 1: Comparison table of optimization results of different methods

[0165]

[0166] An object design method for lightweight and modal optimization of the rear subframe structure based on ML provided by the present invention realizes targeted optimization of weight and the first natural frequency by dynamically adjusting the evolution probability with the number of iterations. An evolution process is constructed based on a radial basis function machine learning model, a min-max Pareto front elevation function is constructed, and the true offspring individual vectors are screened out. The static analysis and modal analysis of the population individual vectors are carried out using the lightweight and modal finite element analysis model of the rear subframe to obtain the weight and first natural frequency information, and the optimization direction is adaptively changed according to the inverse generational distance, improving the two-objective optimization efficiency and accuracy of the lightweight and modal optimization of the rear subframe structure and providing a systematic solution for the two-objective optimization design of the lightweight and modal optimization of the rear subframe structure.

[0167] Example 2. An embodiment of the present invention provides a computer device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of an object design method for lightweight and modal optimization of the rear subframe structure based on ML in the foregoing embodiment are implemented.

[0168] Example 3. An embodiment of the present invention provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of an object design method for lightweight and modal optimization of the rear subframe structure based on ML in the foregoing embodiment are implemented.

[0169] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present invention and is not used to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A target design method for lightweight and modal optimization of rear subframe structure based on ML, characterized in that: The method comprises: Step (1): Taking the size parameters of each front cross beam, each rear cross beam, and each longitudinal beam of the rear subframe as optimization design parameters, constructing a weight and first-order natural frequency simulation model of the rear subframe according to the solver of the 3D modeling software, the finite element analysis pre- and post-processing software, and the structural optimization and multi-physics field simulation software, and deriving a mathematical model of the lightweight and modal optimization design problem of the rear subframe with weight and first-order natural frequency as two optimization targets; Step (2): construct a design space based on the value range of the optimization design parameters, generate a population based on the Maximin criterion and Latin hypercube, generate a reference vector in the target space, use the rear subframe weight and first-order natural frequency simulation model to simulate and evaluate the individual vectors of the population, build a database and construct a radial basis function machine learning model; Step (3): Determine the population individual vectors associated with each reference vector based on the vertical distance relationship between the population individual vector and the reference vector, take the population individual vectors associated with each reference vector as the reference point, and adopt the Pbest-driven DPM evolution strategy based on the Eplison function to generate candidate offspring individual vectors; Step (4): Use the radial basis function machine learning model to predict the weight and first-order natural frequency of all candidate offspring individual vectors, and construct the minimum maximum Pareto frontier lifting function to select the true offspring individual vector from all candidate offspring individual vectors; Step (5): Use the rear subframe weight and first-order natural frequency simulation model to simulate and evaluate the real offspring individual vectors, update the population, database and radial basis function machine learning model, adaptively switch the reference vector type based on the reverse generation distance update situation, return to step (3), until the two optimization goals of weight and first-order natural frequency meet the design requirements, and output the optimal optimization design parameter values.

2. The method according to claim 1, characterized in that The step (1) specifically comprises the following steps: In the first step, according to the geometric structure characteristics of the rear subframe, the length of each front cross beam, the width of each front cross beam, the thickness of each front cross beam, the length of each rear cross beam, the width of each rear cross beam, the thickness of each rear cross beam, the length of the longitudinal beam, the width of the longitudinal beam, the thickness of the longitudinal beam, the number of front cross beams and the number of rear cross beams are used as optimization design parameters; The second step is to construct a 3D model in a 3D modeling software and parameterize the 3D model according to the selected material and structural stress of the rear subframe to obtain a parametric model; The third step is to import the parameterized model, the load condition of the rear subframe and the key boundary conditions into the finite element analysis pre- and post-processing software to perform finite element analysis and obtain the finite element analysis model; The fourth step is to use the solver of the structural optimization and multi-physics simulation software to perform static analysis and modal analysis on the finite element analysis model to obtain the rear subframe weight and first-order natural frequency simulation model; The fifth step is to derive the mathematical model of the rear subframe lightweight and modal optimization design problem based on the rear subframe weight and first-order natural frequency simulation model, taking the rear subframe weight and first-order natural frequency as the two optimization targets. The specific expression is as follows: ， In the above formula, M represents the optimized design parameters of the rear subframe, represents the length of the first front crossbar, Indicates N Cm The length of the front crossbeam, Indicates the width of the first front crossbar, Indicates N Cm The width of the front crossbeam, represents the thickness of the first front crossbeam, Indicates N Cm The thickness of the front crossbeam, N Cm Indicates the number of front beams, Indicates the length of the first rear crossbeam, Indicates N RCm The length of the rear crossbeam, Indicates the width of the first rear cross member, Indicates N RCm The width of the rear crossbeam, Indicates the thickness of the first rear cross member, Indicates N RCm The thickness of the rear crossbeam, N RCm Indicates the number of rear crossbeams, represents the length of the longitudinal beam, Indicates the width of the longitudinal beam, Indicates the thickness of the longitudinal beam, Indicates the total weight of the rear subframe, Indicates that the optimal design parameters M The rear subframe is divided into Z The total mass function of the rear subframe obtained by summing up the parts is: represents the density of the rear subframe, Indicates that the optimal design parameters M The volume of the rear subframe corresponding to Indicates that the rear subframe is optimized in design parameters M The first-order natural frequency corresponding to Indicates that the rear subframe is optimized in design parameters M The first-order natural frequency function corresponding to It represents the design domain formed by the optimization design parameters of the rear subframe; Find represents the optimal solution of the optimization design parameters, Min represents minimizing the weight of the rear subframe, Max represents maximizing the first-order natural frequency of the rear subframe, and St represents the conditions that the optimization design parameters need to meet.

3. The method according to claim 1, characterized in that The step (2) specifically includes the following steps: The first step is to determine the value range of the optimization design parameters according to the required strength range of the rear subframe, taking into account the material strength and design requirements, and construct the design space according to the value range of the optimization design parameters; In the second step, Latin hypercube is used to generate multiple candidate populations in the design space; The third step is to calculate the score of each candidate population according to the Maximin criterion. The specific steps of calculating the score according to the Maximin criterion are as follows: Calculate the Euclidean distance from each individual vector in the candidate population to all other individual vectors; Find the minimum Euclidean distance from each individual vector to other individual vectors in the candidate population; The minimum Euclidean distance of all individual vectors in the candidate population is used as the score of the candidate population; The fourth step is to select the candidate population with the maximum score value as the population; The fifth step is to use the simplex method to generate uniformly distributed reference vectors in the target space composed of the two optimization objectives of weight and first-order natural frequency; Step 6: Use the rear subframe weight and first-order natural frequency simulation model to simulate and evaluate the population individual vectors, obtain the rear subframe weight and first-order natural frequency corresponding to the population individual vectors, and store all population individual vectors and the corresponding rear subframe weight and first-order natural frequency in the database; The seventh step is to use all population individual vectors in the database to establish a radial basis function machine learning model, where the radial basis function machine learning model expression is as follows: ， ， In the above formula, represents the radial basis function machine learning model built for weight, x represents the value of the optimized design parameter, Indicates the weight i A reference vector, represents the basis function vector, represents the calculation result of each basis function at the value x of the optimized design parameter, represents the center point of the 𝑖th basis function, represents the value x of the optimization design parameter and the center of the basis function The Euclidean distance between represents the radial basis function machine learning model built for the first-order natural frequency, represents the first order natural frequency i A reference vector.

4. The method according to claim 1, characterized in that The step (3) specifically includes the following steps: The first step is to calculate the vertical distance between each population individual vector and all reference vectors in the target space according to the uniform distribution of the reference vectors; The second step is to establish a connection between each population individual vector and the nearest reference vector, and select one of the population individual vectors associated with the reference vector as a reference point according to a uniformly distributed random number; The third step is to select the nearest neighboring reference vector to the reference vector that has no connection with the population individual vector, and select one from the population individual vectors associated with the neighboring reference vector as the reference point according to the uniformly distributed random number; The fourth step is to randomly select an optimal design parameter within the design space for the selected benchmark point and perform the DPM evolution strategy based on the Eplison function to obtain an individual vector pool containing the optimal design parameter information; The fifth step is to use the Pbest-driven selection strategy for the individual vector pool to screen out the top p % temporary candidate offspring individual vectors, where the specific steps of the Pbest-driven selection strategy are as follows: The established radial basis function machine learning model is used to predict the target values ​​corresponding to all temporary candidate offspring individual vectors, including weight and first-order natural frequency; Construct a Chebyshev aggregation function based on the ideal reference vector and the nadir reference vector to calculate the fitness values ​​of all temporary candidate offspring individual vectors; Perform non-dominated sorting with the target value and fitness value of the temporary candidate offspring individual vector as two goals, and obtain the non-dominated ranking level of all temporary candidate offspring individual vectors; According to the obtained non-dominated ranking level, in accordance with the ascending principle, the calculated fitness value is used as the screening index to select the temporary candidate offspring individual vector; According to the fitness value, select the top p % temporary candidate offspring individual vectors constitute the temporary candidate population; Among them, the Chebyshev aggregation functions constructed based on the ideal reference vector and the nadir reference vector are as follows: ， ， In the above formula, is a Chebyshev aggregation function based on an ideal reference vector. is the Chebyshev aggregation function constructed based on the nadir reference vector, is the reference vector, is the temporary candidate offspring individual vector, Indicates the reference vector j The weight of the target, is the temporary candidate offspring individual vector in j The target value on the target, Indicates that the ideal reference vector in the database is j The minimum value on the target, Indicates that the nadir reference vector is in the database j The maximum value on the target, Indicates the maximum and minimum value normalization operation. is based on the ideal reference vector in the database. j The minimum value of the target, is the nadir reference vector in the database j The maximum value of the target Indicates the first j target value, The first i Individual vectors, DataBase Represents a database, n is the target number; Step 6: Select a temporary candidate offspring individual vector from the temporary candidate population as a reference point according to a uniformly distributed random number, and execute the DPM evolution strategy based on the Eplison function to generate a candidate offspring individual vector; The Eplison function construction process is as follows: The initial mutation probability is determined according to the number of optimized design parameters and can be calculated by the following formula: ， In the above formula, represents the initial mutation probability, represents the number of optimized design parameters, Indicated in Take the minimum value among 1 and 1; Calculate the decay rate calculation parameters that change with the number of iterations. The calculation formula is as follows: ， In the above formula, represents the calculation parameter of the decay rate that changes with the number of iterations, represents the adjustment factor, T c Indicates the critical threshold value set. Indicates the maximum number of iterations required to achieve the design requirement. represents the initial mutation probability; According to the decay rate calculation parameters that change with the number of iterations and the initial mutation probability, the mutation probability that changes with the number of iterations is calculated. The mutation probability can be calculated by the following formula: ， In the above formula, represents the mutation probability function that changes with the number of iterations, Indicates the current iteration number, Indicates that if the current number of iterations is less than the critical threshold, Indicates if the current number of iterations is greater than or equal to the critical threshold.

5. The method according to claim 1, characterized in that The step (4) specifically includes the following steps: The first step is to use the radial basis function machine learning model to predict the weights and first-order natural frequencies of all candidate offspring individual vectors, and obtain the predicted values ​​of the weights and first-order natural frequencies of all candidate offspring individual vectors; The second step is to perform non-dominated sorting on all candidate offspring individual vectors with the weights and the first-order natural frequency prediction values ​​of all candidate offspring individual vectors as the target, and obtain the non-dominated individual set of candidate offspring individual vectors; The third step is to construct the minimum maximum Pareto frontier improvement function and calculate the minimum improvement value of all candidate offspring individual vectors in the non-dominated individual set relative to the Pareto frontier. The minimum maximum Pareto frontier improvement function is as follows: ， In the above formula, PF It represents the target vector set of the optimal non-dominated individual set obtained in the current iteration in the target space, that is, the current Pareto frontier. Represents the current candidate offspring individual vector On the current Pareto frontier PF The minimum improvement on is the current candidate offspring individual vector relative to the current Pareto frontier PF The minimum improvement value of Represents all target values ​​and the current Pareto frontier PF The sum of the lifts between Represents the current target value and the current Pareto frontier PF The amount of improvement, is the first j The candidate offspring individual vector is in k The value on the target, is the current candidate offspring individual vector In the k Target value on each target; The fourth step is to sort all the calculated candidate offspring individual vectors in ascending order according to the ascending rule, with respect to the minimum improvement value of the Pareto frontier, and select the first two candidate offspring individual vectors with the minimum improvement value as the true offspring individual vectors.

6. The method according to claim 1, characterized in that The step (5) specifically includes the following steps: In the first step, the real offspring individual vector is simulated and evaluated using the rear subframe weight and first-order natural frequency simulation model, so as to obtain the rear subframe weight and first-order natural frequency corresponding to the real offspring individual vector, store the real offspring individual vector and the corresponding rear subframe weight and first-order natural frequency in the database, and update the radial basis function machine learning model according to all individual vectors in the database; The second step is to perform non-dominated sorting on the set formed by the real offspring individual vector and the population individual vector with weight and first-order natural frequency as the target, and obtain the non-dominated ranking level of all individual vectors in the set; The third step is to determine the level value that satisfies according to the non-dominated ranking level of all individual vectors in the set. W The number of individual vectors of level 1 is greater than the population size, and all vectors with the previous W-1 The number of individual vectors of level is less than the population size, then the level value satisfied is equal to W-1 ; The fourth step is to determine the candidate individual vector set and the remaining individual vector set according to the level value satisfied, where the candidate individual vector set includes the vectors from the first level to W-1 All individual vectors of level , the remaining individual vector set contains all W All individual vectors of level; if all have the highest W When the number of individual vectors at the first level is equal to the population size, the candidate individual vector set contains W All individual vectors of level , the remaining individual vector sets are reset to empty; Step 5: For each individual vector in the candidate individual vector set, calculate the Euclidean distance between it and the reference vector, and associate the individual vector with the reference vector with the minimum Euclidean distance, remove the individual vector from the candidate individual vector set, and remove the associated reference vector from the reference vector set to obtain the remaining reference vector set; Step 6: If the remaining reference vector set is not empty, then the Chebyshev aggregation values ​​of all individual vectors in the remaining individual vector set are calculated based on the current reference vector, and the individual vector with the smallest Chebyshev aggregation value is selected and stored in the candidate individual vector set until the remaining reference vector set is an empty set, and the candidate individual vector set is output, and the population is updated with the candidate individual vector set; The seventh step is to calculate the reverse generation distance and compare the current generation reverse generation distance with the previous generation reverse generation distance. If the current generation reverse generation distance is less than or equal to the previous generation reverse generation distance, it proves that the optimization direction is effective and the optimization process reference vector is not switched; if the current generation reverse generation distance is greater than the previous generation reverse generation distance, it proves that the optimization direction is not effective and the optimization process reference vector is switched. The formula for calculating the reverse generation distance is as follows: ， In the above formula, IGD ( PF , PF true ) represents the function for calculating the reverse generation distance, PF true represents multiple uniformly distributed reference vectors, x represents PF true A reference vector on dist (x, PF ) represents the individual vectors in the current set and PF The Euclidean distance between the reference vectors in ; The eighth step is to determine whether the two optimization objectives, the weight of the optimal individual vector and the first-order natural frequency value, meet the design requirements. If the design requirements are met, the optimal optimization design parameter value is output. If the design requirements are not met, return to step (3) and continue to execute all steps from step (3) to step (5) until the design requirements are met.

7. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 6 are implemented.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

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