Target design method for lightweight and modal optimization of rear subframe structure based on ML

Abstract

The invention discloses an ML-based rear subframe structure lightweight and modal optimization target design method, which comprises the following steps: (1) through three-dimensional modeling, statics analysis and modal analysis, constructing a mathematical model corresponding to the weight of a rear subframe and first-order inherent frequency simulation; (2) generating a population based on a Maximin criterion and a Latin hypercube, and establishing a radial basis function machine learning model; (3) generating candidate offspring individual vectors by adopting a Pbest-driven DPM evolutionary strategy based on an Eplison function; (4) constructing a minimum and maximum Pareto front lifting function to screen real offspring individual vectors; and (5) performing simulation evaluation on a real offspring individual vector, adaptively switching the reference vector type based on a reverse generation distance updating condition, returning to the step (3) until an optimization target meets a design requirement, and outputting an optimal optimization design parameter value. According to the method, the evolution direction is adaptively adjusted according to the population simulation result, and the optimization design effect on the two targets of lightweight and modal optimization is good.

Description

Technical Field

[0001] The present invention relates to the field of artificial intelligence and swarm intelligence technology, and in particular to a target design method for lightweight and modal optimization of a rear subframe structure based on ML. Background Art

[0002] As a key component in the vehicle structure, the rear subframe is widely used in various types of vehicles such as cars, trucks, and motorcycles. Its main function is to support the body and transmission system to ensure the stability, safety, and comfort of the vehicle. With the development of the modern automobile industry, consumers have put forward higher requirements for vehicle performance and fuel efficiency. Lightweight design has become one of the core goals to improve vehicle performance. Lightweighting not only helps to reduce the weight of the vehicle and improve fuel economy, but also improves handling and comfort. At the same time, the rear subframe will vibrate due to external forces during the driving process of the vehicle. These vibrations will affect the dynamic response and comfort of the vehicle. Therefore, modal optimization becomes another key design task. Modal optimization improves the vibration characteristics and reduces resonance by adjusting the geometry, material, and structural configuration of the rear subframe, thereby improving the dynamic performance of the vehicle.

[0003] However, traditional design optimization methods mostly focus on a single goal, usually only considering one aspect of lightweight 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 space. To address this problem, ML (Machine Learning) optimization methods have emerged. ML can reveal the complex nonlinear relationship between design variables by learning from a large amount of historical design data. Compared with traditional methods, ML has higher efficiency and accuracy, can quickly find the optimal solution in a complex design space, significantly reduce computational costs, and improve the automation level of the optimization process. At present, many optimization technologies have not yet used the capabilities of machine learning, and only use traditional simulation, which is time-consuming and labor-intensive, and cannot meet the optimization needs in actual engineering. Summary of the invention

[0004] In view of the limitations of the prior art or the need for improved technology, the present invention proposes a target design method for lightweight and modal optimization of the rear subframe structure based on ML, and adopts ML technology to perform collaborative design of lightweight and modal optimization of the rear subframe. The rear subframe structure based on ML involves time-consuming and complex simulation characteristics, as well as the design requirements of the two goals of lightweight and modal optimization. A machine learning optimization design method for changing the optimization direction by adaptively switching reference vectors based on the analysis of simulation results is studied and designed. The method dynamically adjusts the evolution probability with the number of iterations to achieve targeted optimization of weight and first-order natural frequency, constructs the evolution process based on the radial basis function machine learning model, constructs the minimum and maximum Pareto front lifting function, selects the real offspring individual vector, uses the rear subframe lightweight and modal finite element analysis model to perform static analysis and modal analysis on the population individual vector, obtains the weight and the first-order natural frequency, and adaptively changes the optimization direction according to the reverse generation distance, which can not only effectively balance the two goals, but also improve the optimization efficiency of the two goals of 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 based on machine learning designed in the present invention can adapt to the optimization stagnation caused by complex situations and can achieve higher accuracy and efficiency. It can not only be used for 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, a target design method for lightweight and modal optimization of a rear subframe structure based on ML is provided, the method comprising the following steps: 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.

[0006] Furthermore, 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 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, NCm Indicates the number of front beams, represents 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.

[0007] Furthermore, 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.

[0008] Furthermore, 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.

[0009] Furthermore, 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.

[0010] Furthermore, 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.

[0011] In a second aspect, the present invention further provides a computer device, comprising a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the steps of a target design method for lightweighting and modal optimization of a rear subframe structure based on ML are implemented.

[0012] In a third aspect, the present invention further provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of a target design method for lightweighting and modal optimization of a rear subframe structure based on ML.

[0013] In summary, compared with the prior art, the target design method for lightweight and modal optimization of the rear subframe structure based on ML provided by the present invention has the following improvements over the limitations of the prior art: 1. Aiming at the emphasis of different stages in the optimization process, an Eplison function is designed. This function focuses on improving the quality of solutions in the early stage of optimization and on maintaining the diversity of solutions in the later stage of optimization, thereby effectively balancing the exploration and development process.

[0014] 2. Construct the minimum and maximum Pareto front lifting function for screening, combine it with the radial basis function machine learning model for prediction, and select individual vectors for rear subframe lightweighting and modal finite element analysis model simulation in a targeted manner, thereby improving the efficiency and accuracy of traditional optimization; 3. In order to better judge the direction of the optimization process, based on whether the reverse generation distance obtained by the simulation results of the rear subframe lightweight and modal finite element analysis model is updated, the reference vector in the optimization process is adaptively switched to adjust the evolution direction, so that it has better solution capabilities for complex problems and the optimization is always carried out efficiently; 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 overall optimization, which is beneficial to the optimization application of various complex structures and has practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 A simplified flow chart of a target design method for lightweighting and modal optimization of a rear subframe structure based on ML provided by the present invention. DETAILED DESCRIPTION

[0016] In order to more clearly explain the purpose, technical solutions and advantages of the present invention, it will be described in detail below in conjunction with the accompanying drawings and examples. 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.

[0017] See also Figure 1 The present invention provides a target design method for lightweight and modal optimization of a rear subframe structure based on ML, which is applicable to the two-target design optimization problem of lightweight and modal optimization of a rear subframe structure. Specifically, the method includes steps (1) to (5).

[0018] Step (1): Taking the size parameters of the front cross beams, rear cross beams and longitudinal beams of the rear subframe as the optimization design parameters, a rear subframe weight and first-order natural frequency simulation model is constructed based on the solver 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). Taking weight and first-order natural frequency as the two optimization objectives, a mathematical model for the lightweight and modal optimization design problem of the rear subframe is derived.

[0019] Step (1) specifically includes 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 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, represents 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 parametersM 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.

[0020] Step (2): Construct a design space based on the range of optimal design parameter values, 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.

[0021] Step (2) specifically includes the following steps: The first step is to determine the value range of the optimization design parameters (length of each front cross beam, width of each front cross beam, thickness of each front cross beam, length of each rear cross beam, width of each rear cross beam, thickness of each rear cross beam, length of longitudinal beam, width of longitudinal beam, thickness of longitudinal beam, number of front cross beams and number of rear cross beams) according to the strength range required by 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 (Maximize the Minimum Distance) 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.

[0022] 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.

[0023] 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.

[0024] Step (4): Use the radial basis function machine learning model to predict the weights and first-order natural frequencies of all candidate offspring individual vectors, and construct the minimum maximum Pareto frontier lifting function to select the true offspring individual vectors from all candidate offspring individual vectors.

[0025] 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.

[0026] 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.

[0027] 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 the level is greater than the population size (the number of individual vectors in the population), and all 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.

[0028] Example 1 This embodiment uses the benchmark test function ZDT2 to illustrate the optimization performance of the proposed ML-based target design method for lightweight and modal optimization of the rear subframe structure. The expression of the benchmark test function ZDT2 with two objectives is as follows: , , In the above formula, and are the first objective function and the second objective function respectively, is an auxiliary constructor, represents the first optimization design parameter, are the remaining optimization design parameters in the multidimensional design space except the first optimization design parameter. It is i Optimized design parameters, n is the number of optimized design parameters.

[0029] In order to further illustrate the superiority of the method of the present embodiment, a target design method for lightweight and modal optimization of the rear subframe structure based on ML in the present embodiment is compared with another classic and excellent Kriging-assisted reference vector guided evolutionary algorithm, and the maximum number of simulation evaluations in the present embodiment is set to 300 times, and the number of optimized design parameters is set to 30. The average inverse generation distance (abbreviated as average IGD) of 25 rounds of optimization is compared. The results are shown in Table 1. The results show that when the number of simulation evaluations is the same, the method of the present embodiment is significantly better than the Kriging-assisted reference vector guided evolutionary algorithm. It can be considered that the method of the present embodiment can excellently solve the two-objective design optimization problem of lightweight and modal optimization of the rear subframe structure.

[0030] Table 1: Comparison of optimization results of different methods

[0031] The present invention provides a target design method for lightweight and modal optimization of a rear subframe structure based on ML. The method achieves targeted optimization of weight and first-order natural frequency by dynamically adjusting the evolution probability with the number of iterations, constructs an evolution process based on a radial basis function machine learning model, constructs a minimum-maximum Pareto front lifting function, screens out real offspring individual vectors, uses a rear subframe lightweight and modal finite element analysis model to perform static analysis and modal analysis on the population individual vectors, obtains weight and first-order natural frequency information, and adaptively changes the optimization direction according to the reverse generation distance, thereby improving the two-target optimization efficiency and accuracy of lightweight and modal optimization of the rear subframe structure, and providing a systematic solution for the two-target optimization design of lightweight and modal optimization of the rear subframe structure.

[0032] Embodiment 2, the 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, it implements the steps of a target design method for lightweight and modal optimization of a rear subframe structure based on ML in the aforementioned embodiment.

[0033] Embodiment 3, the 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 a target design method for lightweight and modal optimization of a rear subframe structure based on ML of the aforementioned embodiment are implemented.

[0034] It will be easily understood by those skilled in the art that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in 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 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; A Chebyshev aggregation function is constructed 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: Determine the initial mutation probability according to the number of optimized design parameters, which 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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