Structural simulation optimization method suitable for medium-high dimensional multi-target expensive problem

Abstract

The structure simulation optimization method suitable for the medium-high dimensional multi-target expensive problem comprises the following steps that S1, simulation is conducted in simulation software, and then data are imported into matlab; s2, establishing an initial population from the imported data by using Latin hypercube sampling, and then generating a reference vector by using a simplex lattice method; s3, performing local search on the population by using an RVEA algorithm to obtain an approximate Pareto frontier solution; s4, pre-screening the approximate Pareto frontier solution by using a minimum angle sample filling standard (MDA) to obtain a competitive individual, then importing the individual into simulation software for simulation, finally combining with a real Pareto frontier solution, and adding into a population; s5, selecting a next-generation population; and S6, generating a large number of offspring individuals by using a method for operating repetitive offspring. According to the scheme, the simulation frequency of the complex mechanical structure can be greatly reduced to obtain a good optimization result, and the simulation efficiency of the complex mechanical structure is effectively improved.

Description

Technical Field

[0001] The invention relates to simulation experiment technology, in particular to a structural simulation optimization method suitable for medium- and high-dimensional multi-objective expensive problems. Background Art

[0002] In real life, there are various optimization problems, such as path planning and supply chain optimization. If there are conflicting or competing objectives in the problem to be optimized, such problems are called multi-objective optimization problems, such as the optimization problem between fuel efficiency and overall weight in the aerospace field. Nowadays, there are many methods that can be used to solve such problems. The most common method is to use multi-objective optimization algorithms such as genetic algorithms to solve them. Specifically, a multi-objective optimization algorithm is used to perform several iterative evaluations, and finally a set of new solutions that can be selected by decision makers are obtained. These available solutions are not mutually dominated and are called Pareto frontier solutions. However, with the rapid development of society, traditional multi-objective algorithms are no longer applicable to some optimization problems that consume large computing resources and have long computing time (called expensive multi-objective optimization problems). Taking the simulation design of spacecraft and aircraft as an example, the resource consumption caused by a wind tunnel experiment is extremely huge, and the traditional multi-objective optimization algorithm, which has thousands of evaluations, is obviously not suitable for solving these problems. Therefore, researchers combine the surrogate model technology based on machine learning with the traditional multi-objective algorithm, and use machine learning tools to fit the mapping relationship between the optimization target value and the design variables in the expensive multi-objective optimization problem, and transform it into an optimization problem with low computing resource consumption, thereby replacing the original expensive problem for optimization, thereby greatly reducing the loss of computing resources. The surrogate model technology based on machine learning includes: Kriging surrogate model (Kriging), radial basic function (Radial Basic Function, RBF) and support vector machine (SVM). These surrogate models combined with traditional multi-objective optimization algorithms have derived many algorithms that can be used to solve expensive multi-objective optimization problems, which are called surrogate model-assisted multi-objective optimization algorithms (surrogate assisted evolutionary algorithm, SAEA).

[0003] In the multi-objective optimization algorithm assisted by the surrogate model, how to generate competitive offspring individuals (design parameters, also known as decision variables) is an important issue. There are generally two ways: the first is called the algebraic offspring generation method, which combines the surrogate model with the multi-objective optimization algorithm, uses the surrogate model to replace the original simulation process to calculate the offspring response value, and obtains the final offspring through multiple iterations of the multi-objective algorithm. This seed generation method can effectively greatly improve the convergence speed of the algorithm, but the disadvantage is that it will cause the algorithm to fall into local optimization. The second seed generation method is to generate a large number of offspring by crossover mutation, and also use the surrogate model to replace the original simulation process to calculate the offspring fitness value, which can fully take into account the diversity of the algorithm offspring, but the disadvantage is that the algorithm converges slowly.

[0004] In the multi-objective optimization algorithm assisted by the surrogate model, how to reduce the number of algorithm evaluations is a core issue. For this problem, a method called pre-screening is generally used to select new individuals for evaluation. Take simulation as an example: at the beginning, the simulation data of the mechanical structure is used as the data set, and the surrogate model is used to construct a mathematical mapping relationship between the decision variables (design parameters) and the simulation results to construct an acquisition function. After that, offspring (new design parameters) are generated through crossover mutation and other methods, and the response values ​​of the acquisition function are calculated based on these offspring to select the offspring for simulation, and these offspring are retained for the next generation of the algorithm until the maximum number of evaluations is reached. This method can greatly speed up time-consuming calculations such as simulations.

[0005] Among them, the method of constructing the acquisition function generally takes into account the mining and exploration of the algorithm at the same time. The acquisition function based on the Kriging proxy model can balance the above characteristics because it has the prediction mean and prediction error provided by the Kriging model. However, the acquisition function based on the Kriging proxy model will cause the algorithm to tend to search for some highly nonlinear or multimodal uncertainty areas during the search process, which will cause the obtained Pareto front solution to fail to take into account diversity. In addition, previous algorithms often only use one proxy model, which makes the algorithm often focus on only one characteristic, resulting in the algorithm balancing the common characteristics.

[0006] In addition, as the dimension of decision variables increases, the efficiency of general methods will become extremely low. Summary of the invention

[0007] In view of the above-mentioned defects, the purpose of the present invention is to propose a structural simulation optimization method suitable for medium and high dimensional multi-objective expensive problems. This scheme can significantly reduce the number of simulations for complex mechanical structures to obtain good optimization results, and effectively improve the simulation efficiency of complex mechanical structures.

[0008] To achieve this object, the present invention adopts the following technical solutions:

[0009] One of the above technical solutions includes the following beneficial effects: 1. Compared with the previous multi-objective optimization algorithm assisted by the proxy model, a local search method with the RVEA algorithm as the main framework is proposed for the first time: the local search based on the RBF proxy model and the RVEA algorithm as the framework can greatly improve the convergence speed of the algorithm in the search process. 2. Compared with the previous algorithm based on the improved RVEA, the present invention proposes for the first time to use two proxy models (RBF proxy model and Kriging proxy model) to balance the various characteristics of the algorithm. 3. In view of the lack of diversity of Pareto front solutions caused by EIM pre-screening, the present invention proposes for the first time a filling standard pre-screening method based on an angle penalty mechanism guided by a reference vector to enhance the diversity of Pareto front solutions after EIM pre-screening. 4. Most of the previous multi-objective optimization algorithms assisted by the proxy model are only applicable to low dimensions. Once the dimension of the decision variable becomes high, the efficiency of the algorithm will drop sharply. The present invention proposes a multi-objective optimization algorithm assisted by a proxy model applicable to medium and high dimensions, and the test results prove that it is extremely efficient in dealing with medium and high dimensional problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 It is a schematic diagram of the overall process of an embodiment of the present invention;

[0011] Figure 2 It is a schematic diagram of a method for generating offspring by repeating operator operation according to an embodiment of the present invention;

[0012] Figure 3 Schematic diagram of selecting reference vectors for individuals not assigned the true Pareto frontier according to an embodiment of the present invention. DETAILED DESCRIPTION

[0013] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and cannot be understood as limiting the present invention.

[0014] like Figure 1 As shown in FIG. 1 , the structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems includes the following steps:

[0015] S1, simulate in simulation software, and then import the data into MATLAB;

[0016] S2, using Latin hypercube sampling to build the initial population from the imported data, and then using the simplex lattice method to generate the reference vector;

[0017] S3, use RVEA algorithm to perform local search on the population to obtain the approximate Pareto front solution;

[0018] S4, using the minimum angle sampling filling standard (MDA) to pre-screen the approximate Pareto front solution to obtain competitive individuals, and then import the individuals into the simulation software for simulation, and finally combine them with the real Pareto front solution and add them to the population;

[0019] S5, select the next generation population;

[0020] S6. Use the operation repeated progeny method to produce a large number of offspring individuals;

[0021] S7, use the expected improvement matrix filling standard (EIM) to pre-screen the offspring individuals to obtain competitive individuals, then import the individuals into the simulation software for simulation, and finally combine them with the real Pareto front solution, add them to the species, and then select the next generation population;

[0022] S8. Use the repeated offspring method to generate a large number of offspring, and then use the filling standard (APD) based on the angle penalty mechanism guided by the reference vector to pre-screen competitive offspring, and then import the individual into the simulation software for simulation, and finally combine it with the real Pareto front solution, and add the seed to subsequently select the next generation population.

[0023] The benefits of doing this are: 1. Compared with the previous multi-objective optimization algorithms assisted by proxy models, this is the first time that a local search method with the RVEA algorithm as the main framework is proposed: using a local search based on the RBF proxy model and the RVEA algorithm as the framework can greatly improve the convergence speed of the algorithm during the search process.

[0024] 2. Compared with the previous algorithms based on improved RVEA, the present invention proposes for the first time to use two proxy models (RBF proxy model and Kriging proxy model) simultaneously to balance various characteristics of the algorithm.

[0025] 3. In response to the lack of diversity in Pareto front solutions caused by EIM pre-screening, the present invention proposes for the first time a reference vector-guided filling standard pre-screening method based on an angle penalty mechanism to enhance the diversity of Pareto front solutions after EIM pre-screening.

[0026] 4. Most of the previous proxy model-assisted multi-objective optimization algorithms are only applicable to low dimensions. Once the dimension of the decision variable becomes high, the efficiency of the algorithm will drop sharply. The present invention proposes a proxy model-assisted multi-objective optimization algorithm suitable for medium and high dimensions, and test results show that it is extremely efficient in dealing with medium and high dimensional problems.

[0027] Preferably, in step S1, the mechanical structure is first put into the simulation software for simulation, and the simulation data is imported into the matlab software;

[0028] The decision variables and optimization objectives in the simulation data are transformed into:

[0029]

[0030] Among them, x is the decision variable, F(x) is the target to be optimized, m is the number of targets, and D is the number of decision variables.

[0031] Preferably, in step S2, a Latin hypercube sampling method is used to uniformly select some data from the simulation data as the initial population, denoted as P 0 , the population size is N;

[0032] And use P 0 All the data in are used to fit the Kriging model and the RBF proxy model, and a database DB for fitting the proxy model is established. 0 All data are entered into DB, and the number of simulations FE increases by N times;

[0033] And select P through the non-dominated sorting algorithm 0 The Pareto front solution in is denoted as APF 0 ; Then, a series of uniformly distributed reference vectors are generated by the simple grid method, denoted as V 0 , the simple lattice method is as follows:

[0034]

[0035] Where i = 1, 2, ..., N is the number of uniformly distributed points, k is the number of targets, and H is a positive integer that can be adjusted independently and is used for simple grid design;

[0036]

[0037] Among them, ||p i || is p i The module, v i is the generated reference vector.

[0038] Preferably, in step S3, after fitting the proxy model, the proposed local search phase assisted by the proxy model based on the RVEA algorithm is entered, specifically:

[0039] First, use the DB data to fit the RBF proxy model, and then use the simple lattice method to generate a series of reference vectors, denoted as V 1 , the number is N, and then a new population P is generated through Latin hypercube sampling1 , the population size is N, and then P 1 With APF 0 Combined, denoted as P 1 ', and then produce offspring O through crossover mutation 1 , the number is N, and then use the RBF proxy model to calculate O 1 The response value of P 1 'With O 1 Combine, and select the minimum value Zmin of each target, then subtract the combined individual response value from Zmin, and then calculate the combined individual and V 1 The reference vector individuals in are bound by the maximum cosine value, specifically:

[0040]

[0041] Among them, f i，i＝1,...N P 1 'Individual, v j,j＝1,…,N V 1 vector, by calculating the maximum cosθ for each P 1 'Individual allocation reference vector;

[0042] After allocating a reference vector to each, the next generation of population is selected through the angle penalty mechanism, and the ideal point of each generation of population needs to be compared with the ideal point of the previous generation, and the smaller response value after comparison is selected as the new ideal point Zmin'. The angle penalty mechanism is as follows;

[0043] d i,j =(1+P(θ i,j )·||f i ||)

[0044]

[0045] Among them, ||f i || represents the Euclidean distance between the i-th individual and the origin, M is the number of targets, and g max is the predefined maximum number of iterations, a is the control P(θ i,j ) parameter of the rate of change, θ i,j is the angle between the i-th individual and its corresponding reference vector, and γ is the minimum angle between all reference vectors;

[0046] At iteration number g, g is reached max Finally, the final virtual Pareto front solution PF' is obtained through the non-dominated sorting algorithm.

[0047] The benefit of doing this is that after a local search based on the RVEA algorithm, an approximate Pareto front solution with relatively good convergence will be obtained, which greatly speeds up the algorithm's search to the area where the true Pareto front solution is located.

[0048] Further, in step S4, after obtaining PF', competitive offspring are selected for pre-screening by the minimum distance angle sampling filling standard, specifically: first, the kmeans clustering method is used to divide PF' into n clusters, and then n offspring individuals are selected in each cluster by the minimum distance angle mechanism, and then the decision variables of the n offspring individuals are substituted into the simulation software for simulation, and the number of simulations FE is increased by n times, and then the n offspring individuals are placed in the database DB of the fitting agent model and the current true Pareto front solution PF, and then the PF is updated by the non-dominated sorting algorithm, and the minimum distance angle mechanism is:

[0049]

[0050] in, It indicates the i-th individual of the n-th cluster of PF' obtained after local search, y j is the jth individual in the current real PF, Z' is the ideal point formed by the minimum value of the j objectives of all individuals in PF', and represents the Euclidean distance of the ideal point of the ith individual of the nth cluster in PF', is the minimum angle between the individual in the true PF and the individual after local search, and ε is a parameter to avoid the score being zero.

[0051] The advantage of doing this is that by using MDA pre-screening on the approximate Pareto front solution after local search, individuals with better convergence and diversity can be selected.

[0052] Furthermore, in step S5, after the local search, similar to the population selection method in step three, after binding a reference vector to each individual, the next generation population is selected using the APD mechanism.

[0053] like Figure 2 As shown, in step S6, after the local search, although the convergence speed of the algorithm is improved, it may fall into local optimization, so it is necessary to jump out of the local optimization, specifically: use the operation repetition offspring generation method to generate a large number of candidate individuals, and the operation repetition operator offspring generation method is specifically to generate N*S candidate offspring from the parent population by simulating binary crossover and polynomial mutation;

[0054] Simulate a binary crossover operation:

[0055]

[0056] in, There are two parent individuals, and are two offspring individuals, rand is a random number, and η is a custom parameter;

[0057] Polynomial mutation operation:

[0058] v k '=v k +δ×(u k -l k )

[0059]

[0060] Among them, δ 1 =(ν k -l k ) / (u k -l k ), δ 2 =(u k -v k ) / (u k -l k ), u is a random number in the interval [0,1], v k is the individual of the previous generation, v k ' is the newly generated population individual, u k represents the upper bound of the decision variable, l k represents the lower bound of the decision variable, η m A custom distribution index.

[0061] The advantage of doing this is that the repeated operator offspring generation method can generate a large number of candidate offspring, and certain characteristics can be fully taken into account in the selection of candidate offspring, so that more competitive individuals can be obtained.

[0062] Further, in step S7, after generating n groups of candidate offspring, the Kriging proxy model is fitted with the database DB, and then the EIM pre-screening is performed. The EIM pre-screening is specifically as follows: the generated n groups of candidate offspring are divided into S groups by the kmeans clustering method. 1 clusters, and then select the candidate individuals with the maximum MOEI value from each cluster. The specific MOEI value is:

[0063]

[0064] Among them, m is the target number, n is the number of individuals in the current true Pareto frontier solution, is the expected improvement value of the jth individual in the i-th objective of the candidate individual x for the current true Pareto front solution, is the i-th target value of the j-th individual in the current true Pareto front solution, With s i (x) are the predicted mean and variance of candidate individual x, Φ and φ are the standard normal density function and distribution function, respectively;

[0065] After selecting the candidate individual with the largest MOEI value for each cluster, the decision variables of these n individuals are substituted into the simulation software for simulation, the number of simulations FE increases by n times, and the data of the n individuals are entered into the database DB of the fitting agent model, and then combined with the current true Pareto front solution PF, and then the PF is updated using the non-dominated sorting algorithm; in addition, these n individuals are combined with the previous generation population, and similar to step three, after binding a reference vector for each individual, the next generation population is selected using the APD mechanism.

[0066] The benefits of doing this are: This can screen out individuals with the largest MOEI value, and the MOEI value combines the predicted response value with uncertainty (variance), so it can be used for global optimization to avoid the algorithm falling into local optimality.

[0067] like Figure 3 As shown, because the EIM pre-screening will cause the algorithm to tend to search for some highly nonlinear or multimodal uncertainty areas, resulting in the lack of diversity in PF, the present invention proposes a filling standard pre-screening method based on an angle penalty mechanism guided by a reference vector, specifically using a simple grid method to generate N reference vectors uniformly distributed in the decision space, and then binding the individuals of the current true Pareto front solution to the generated reference vectors according to the maximum cosine value. The method is consistent with the individual binding reference vector in step three. After binding the individuals in the current true Pareto front solution to the vectors uniformly distributed in the decision space, n reference vectors v' without bound individuals are selected. u ;

[0068] Select v' u After that, the target value of individuals in the true Pareto front solution is converted, specifically:

[0069]

[0070] Among them, f i,j (i=1,2,…,N,j=1,2,…,M) is the target value of the jth target of the i-th solution of the current true Pareto frontier, The ideal point consisting of the minimum target value of each target in the current true Pareto front solution;

[0071] Then use the DB data to fit the RBF proxy model, and then use the repetitive operation operator generation method in step 6 to generate a large number of candidate offspring. For these candidate offspring, use the fitted RBF proxy model to calculate the candidate individuals and v' u APD value, and select v'u There are n candidate individuals with the minimum APD value corresponding to each reference vector in . The specific APD value is:

[0072]

[0073] P(θ i,g )=theta·θ i,g

[0074] Where d is the value of each candidate offspring and v' u The corresponding APD value, v' u,g is the selected reference vector v' u The g-th reference vector, is the fitness value of the mth target of the ith candidate offspring calculated by the RBF proxy model, and theta is a parameter set by the user to control P(θ i,g ) rate of change, θ i,g is the angle between the i-th candidate offspring and the g-th selected reference vector;

[0075] After pre-screening n candidate offspring, substitute their decision variables into the simulation software for simulation, increase the number of simulations FE by n times, and 2 The data of the candidate offspring are entered into the database DB and the current true Pareto front solution PF, and then the PF is updated using the non-dominated sorting algorithm, and then combined with the previous generation population. Similar to step S3, after binding a reference vector for each individual, the APD mechanism is used to select the next generation population and return to step S3 for the next iteration until the number of simulations FE reaches the preset upper limit.

[0076] The benefits of doing this are: EIM pre-screening will cause the algorithm to tend to some areas with more complex calculations, resulting in a lack of diversity in the Pareto front solution. The reference vector is used to guide the identification of areas with lack of diversity, and then the APD fitness value is used to select promising individuals in these areas, thereby supplementing the lack of diversity in the Pareto front solution.

[0077] In order to show the advantages of the present invention over other methods, this experiment uses the inverted generational distance (IGD) to test various test problems with a target of 3 to evaluate the advantages and disadvantages of various methods. The specific IGD values ​​are:

[0078]

[0079] Among them, P represents the true Pareto front solution, Q represents the solved population (the solved Pareto front solution), d(v,Q) represents the minimum Euclidean distance from individual v in P to population Q, and |P| represents the number of solutions distributed in the Pareto front.

[0080] Experimental parameters:

[0081] The population size N is 100, the target number M is 3, the number of evaluations (simulations) M is 300, the decision variables D are 10, 20, 30, and the number of individuals pre-screened each time is 3 times.

[0082] Experimental data:

[0083]

[0084] Table 1: Comparison of IGD values ​​between the present invention and other methods

[0085] Table 1 shows the comparison of IGD values ​​of DTLZ test functions of three targets of the overall algorithm and other advanced methods. The parameters are 300 evaluations of the true function and 20 independent tests of each test function. Where M represents the number of test function targets, D represents the decision variable dimension, the value outside the brackets represents the mean of the IGD value, and the value inside the brackets represents the variance of the IGD value. "+" indicates that HS-RVEA (the method of the present invention) is better than other methods in the test function of this dimension, "-" indicates that HS-RVEA is inferior to other methods in the test function of this dimension, and "≈" indicates that HS-RVEA performs similarly to other methods.

[0086] The technical principle of the present invention is described above in conjunction with specific embodiments. These descriptions are only for explaining the principle of the present invention and cannot be interpreted as limiting the scope of protection of the present invention in any way. Based on the explanations herein, those skilled in the art can associate other specific implementations of the present invention without paying creative labor, and these methods will fall within the scope of protection of the present invention.

Claims

1. A structural simulation optimization method suitable for medium- and high-dimensional multi-objective expensive problems, characterized by: The steps include: S1, simulate in simulation software, and then import the data into MATLAB; S2, using Latin hypercube sampling to build the initial population from the imported data, and then using the simplex lattice method to generate the reference vector; S3, use RVEA algorithm to perform local search on the population to obtain the approximate Pareto front solution; S4, using the minimum angle sampling filling standard (MDA) to pre-screen the approximate Pareto front solution to obtain competitive individuals, and then import the individuals into the simulation software for simulation, and finally combine them with the real Pareto front solution and add them to the population; S5, select the next generation population; S6, using the operation repetition operator generation method to generate offspring individuals; S7, use the expected improvement matrix filling standard (EIM) to pre-screen the offspring individuals to obtain competitive individuals, then import the individuals into the simulation software for simulation, and finally combine them with the real Pareto front solution, add them to the species, and then select the next generation population; S8. Use the repeated offspring method to generate a large number of offspring, and then use the filling standard (APD) based on the angle penalty mechanism guided by the reference vector to pre-screen competitive offspring, and then import the individual into the simulation software for simulation, and finally combine it with the real Pareto front solution, and add the seed to subsequently select the next generation population.

2. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 1 is characterized in that: In step S1, the mechanical structure is first put into the simulation software for simulation, and the simulation data is imported into the matlab software; The decision variables and optimization objectives in the simulation data are transformed into: Among them, x is the decision variable, F(x) is the target to be optimized, m is the number of targets, and D is the number of decision variables.

3. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 2 is characterized in that: In step S2, a Latin hypercube sampling method is used to uniformly select some data from the simulation data as the initial population, denoted as P0, and the number of populations is N; And use all the data in P0 to fit the Kriging model and the RBF proxy model, establish a database DB for fitting the proxy model, enter all the data in P0 into DB, and increase the number of simulations FE by N times; And the Pareto front solution in P0 is selected by the non-dominated sorting algorithm, denoted as APF0; then a series of uniformly distributed reference vectors are generated by the simplex lattice method, denoted as V0. The simplex lattice method is as follows: Where i = 1, 2, ..., N is the number of uniformly distributed points, k is the number of targets, and H is a positive integer that can be adjusted independently and is used for simple grid design; Among them, ||p i || is p i The module, v i is the generated reference vector.

4. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 3 is characterized in that: In step S3, after fitting the proxy model, the proposed local search phase assisted by the proxy model based on the RVEA algorithm is entered, specifically: First, use the DB data to fit the RBF proxy model, and then use the simple lattice method to generate a series of reference vectors, denoted as V1, with a number of N. Then use Latin hypercube sampling to generate a new population P1 with a population size of N. Then combine P1 with APF0, denoted as P1', and then generate offspring O1 with a number of N through crossover mutation. Then use the RBF proxy model to calculate the response value of O1, then combine P1' with O1, and select the minimum value Zmin of each target, then subtract the combined individual response value from Zmin, and then calculate the combined individual and the reference vector individual in V1 through the maximum cosine value to bind, specifically: Among them, f i，i＝1,...N is the individual of P1', v j,j＝1,…,N is the vector of V1, and a reference vector is assigned to each individual in P1' by calculating the maximum cosθ; After allocating a reference vector to each, the next generation of population is selected through the angle penalty mechanism, and the ideal point of each generation of population needs to be compared with the ideal point of the previous generation, and the smaller response value after comparison is selected as the new ideal point Zmin'. The angle penalty mechanism is as follows; d i，j =(1+P(θ i，j )·||f i ||) Among them, ||f i || represents the Euclidean distance between the i-th individual and the origin, M is the number of targets, and g max is the predefined maximum number of iterations, a is the control P(θ i,j ) parameter of the rate of change, θ i,j is the angle between the i-th individual and its corresponding reference vector, and γ is the minimum angle between all reference vectors; At iteration number g, g is reached max Finally, the final virtual Pareto front solution PF' is obtained through the non-dominated sorting algorithm.

5. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 4 is characterized in that: In step S4, after obtaining PF', competitive offspring are selected for pre-screening by the minimum distance angle sampling filling standard. Specifically, PF' is first divided into n clusters by the kmeans clustering method, and then n offspring individuals are selected in each cluster by the minimum distance angle mechanism. After that, the decision variables of the n offspring individuals are substituted into the simulation software for simulation, and the number of simulations FE increases by n times. Then, the n offspring individuals are placed in the database DB of the fitting agent model and the current real Pareto front solution PF, and then the PF is updated by the non-dominated sorting algorithm. The minimum distance angle mechanism is: in, It indicates the i-th individual of the n-th cluster of PF' obtained after local search, y j is the jth individual in the current real PF, Z' is the ideal point formed by the minimum value of the j objectives of all individuals in PF', and represents the Euclidean distance of the ideal point of the ith individual of the nth cluster in PF', is the minimum angle between the individual in the true PF and the individual after local search, and ε is a parameter to avoid the score being zero.

6. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 5 is characterized in that: In step S5, after the local search, similar to the population selection method in step 3, after binding a reference vector to each individual, the next generation population is selected using the APD mechanism.

7. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 6 is characterized in that: In step S6, after the local search, although the convergence speed of the algorithm is improved, it may fall into local optimization, so it is necessary to jump out of the local optimization, specifically: use the operation repetition operator offspring generation method to generate a large number of candidate individuals, and the operation repetition operator offspring generation method is specifically to generate N*S candidate offspring from the parent population by simulating binary crossover and polynomial mutation; Simulate a binary crossover operation: in, There are two parent individuals, and are two offspring individuals, rand is a random number, and η is a custom parameter; Polynomial mutation operation: v k ′=v k +δ×(u k -l k ) Among them, δ1=(v k -l k ) / (u k -l k ),δ2=(u k -v k ) / (u k -l k ), u is a random number in the interval [0,1], v k is the individual of the previous generation, v k ' is the newly generated population individual, u k represents the upper bound of the decision variable, l k represents the lower bound of the decision variable, η m A custom distribution index.

8. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 7 is characterized in that: In step S7, after generating n groups of candidate offspring, the Kriging proxy model is fitted with the database DB, and then the EIM pre-screening is performed. The EIM pre-screening is as follows: the generated n groups of candidate offspring are divided into S1 clusters by the kmeans clustering method, and then the candidate individuals with the maximum MOEI value are selected from each cluster. The MOEI value is specifically: Among them, m is the number of targets, n is the number of current real Pareto front solutions, is the expected improvement value of the jth individual in the i-th objective of the candidate individual x for the current true Pareto front solution, is the i-th target value of the j-th individual in the current true Pareto front solution, With s i (x) are the predicted mean and variance of candidate individual x, Φ and φ are the standard normal density function and distribution function, respectively; After selecting the candidate individual with the largest MOEI value for each cluster, the decision variables of these n individuals are substituted into the simulation software for simulation, the number of simulations FE increases by n times, and the data of the n individuals are entered into the database DB of the fitting agent model, and then combined with the current true Pareto front solution PF, and then the PF is updated using the non-dominated sorting algorithm; in addition, these n individuals are combined with the previous generation population, and similar to step three, after binding a reference vector for each individual, the next generation population is selected using the APD mechanism.

9. The structural simulation optimization method applicable to medium- and high-dimensional multi-objective expensive problems according to claim 8, characterized in that: Because the EIM pre-screening will cause the algorithm to tend to search for some highly nonlinear or multimodal uncertainty areas, resulting in a lack of diversity in PF, the present invention proposes a filling standard pre-screening method based on an angle penalty mechanism guided by a reference vector, specifically using a simple grid method to generate N reference vectors uniformly distributed in the decision space, and then binding the individuals of the current true Pareto front solution to the generated reference vectors according to the maximum cosine value. The method is consistent with the individual binding reference vector in step three. After binding the individuals in the current true Pareto front solution to the vectors uniformly distributed in the decision space, n reference vectors v' without bound individuals are selected. u ; Select v' u After that, the target value of individuals in the true Pareto front solution is converted, specifically: Among them, f i,j (i=1,2,…,N,j=1,2,…,M) is the target value of the jth target of the i-th solution of the current true Pareto frontier, The ideal point consisting of the minimum target value of each target in the current true Pareto front solution; Then use the DB data to fit the RBF proxy model, and then use the operation of step 6 to repeat the operator generation method to generate a large number of candidate offspring. For these candidate offspring, use the fitted RBF proxy model to calculate the candidate individuals and v' u APD value, and select v' u There are n candidate individuals with the minimum APD value corresponding to each reference vector in . The specific APD value is: Where d is the value of each candidate offspring and v' u The corresponding APD value, v' u,g is the selected reference vector v' u The g-th reference vector, is the fitness value of the mth target of the ith candidate offspring calculated by the RBF proxy model, and theta is a parameter set by the user to control P(θ i,g ) rate of change, θ i,g is the angle between the i-th candidate offspring and the g-th selected reference vector; After pre-screening n candidate offspring, their decision variables are substituted into the simulation software for simulation, the number of simulations FE increases by n times, and the data of S2 candidate offspring are entered into the database DB and the current true Pareto front solution PF, and then the PF is updated using the non-dominated sorting algorithm, and then combined with the previous generation population. Similar to step S3, after binding a reference vector for each individual, the APD mechanism is used to select the next generation population and return to step S3 for the next iteration until the number of simulations FE reaches the preset upper limit.

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