A multi-objective optimization method for double-layer Y-tube hydroforming process

Abstract

The present invention provides a multi-objective optimization method for a double-layer Y-tube internal high pressure forming process, and relates to the technical field of optimization design of internal high pressure forming process. The method comprises: according to the actual operation requirements of the double-layer Y-tube internal high pressure forming process, using the Latin hypercube sampling method to design N groups of process parameters; constructing a finite element simulation model of the double-layer Y-tube and using the N groups of process parameters to perform finite element simulation respectively, and calculating the performance evaluation index of each group of process parameters according to the simulation results; constructing a PSO-BP neural network prediction model of the double-layer Y-tube and using it as the objective function of NSGA-III to generate a Pareto optimal solution set; using a linear weighted method to establish a multi-factor comprehensive objective function and using the objective function to select the optimal process parameters of the double-layer Y-tube internal high pressure forming process from the Pareto optimal solution set, thereby realizing the coordinated deformation of the inner and outer layers of the tube in the double-layer Y-tube internal high pressure forming process.

Description

Technical Field

[0001] The invention relates to the technical field of optimization design of an internal high pressure forming process, and in particular to a multi-objective optimization method for an internal high pressure forming process of a double-layer Y-shaped tube. Background Art

[0002] In the field of internal high pressure forming, although traditional single-layer material pipes perform well in specific applications, they are often limited in more complex or demanding scenarios. Double-layer tubes have the best properties of both metal materials and can give full play to the excellent properties of both pipes. Therefore, they are more widely used in some fields with special requirements. However, the structure of double-layer tubes is more complicated. Zhang Bing studied a double-layer T-tube internal high pressure forming process and analyzed the reasons for wrinkling and rupture of parts during the forming process. However, the inner and outer double-layer structure of the double-layer tube will make the forming process of the part more uncontrollable. The inner layer may be difficult to form due to the restriction of the outer layer, and it is difficult to control the thickness and uniformity of the material and match the shapes of the inner and outer layers.

[0003] The finite element simulation method can greatly reduce the economic cost of finding the optimal parameters through trial and error in hydroforming production. However, in the actual process of optimizing process parameters, it is impractical to rely on hundreds or thousands of finite element simulation results to obtain the required process parameters. Therefore, there is an urgent need for a mathematical model that can accurately and efficiently predict finite element simulation, thereby greatly reducing time and economic costs.

[0004] The optimization of traditional single-layer tube process parameters is usually determined based on experience and less than 3 objective functions. Tianlun Huang et al. used the second-generation non-dominated sorting genetic algorithm (NSGA-II) to optimize the process parameters of single-layer T-tubes, using contact area evaluation correction degree, maximum thinning rate and convex height as the objective function. However, due to the complexity of the double-layer tube forming process, at least 5 objective functions should be considered in the process of multi-objective optimization of process parameters, including the thinning rate, thickening rate and forming effect of the inner and outer tubes. At present, the multi-objective optimization algorithms commonly used in the field of internal high-pressure forming process optimization can only optimize less than 3 objective functions, and all have certain defects. For example, when dealing with multi-objective optimization problems with more than 3 objectives, the NSGA-II algorithm and the genetic algorithm (GA) have the problem of slow convergence speed and low precision, which cannot achieve a good optimization effect and is not conducive to guiding actual production. Summary of the invention

[0005] In view of the shortcomings of the above-mentioned prior art, the present invention proposes a multi-objective optimization method for the double-layer Y-tube internal high-pressure forming process, aiming to establish a multi-objective optimization model for the process parameter design of the double-layer Y-tube internal high-pressure forming process, so as to achieve the coordinated optimization of the loading path and the main influencing factors of the double-layer Y-tube internal high-pressure forming process, and realize the coordinated deformation of the inner and outer layers of the double-layer Y-tube internal high-pressure forming process.

[0006] The present invention proposes a multi-objective optimization method for a double-layer Y-tube high pressure forming process, the method comprising the following steps:

[0007] Step 1: According to the actual operation requirements of the double-layer Y-tube hydroforming process, the process parameters and performance evaluation indicators are determined, and the Latin hypercube sampling method is used to design N groups of process parameters;

[0008] Step 2: Use finite element simulation software to build a finite element simulation model of the double-layer Y-shaped tube, use N groups of process parameters to perform finite element simulation on the finite element simulation model of the double-layer Y-shaped tube, and calculate the performance evaluation index corresponding to each group of process parameters according to the simulation results;

[0009] Step 3: Based on N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, a PSO-BP neural network prediction model of the double-layer Y-type tube is constructed;

[0010] Step 4: The PSO-BP neural network prediction model of the double-layer Y-type tube is used as the objective function of the non-dominated genetic algorithm NSGA-III based on the reference point, and the Pareto optimal solution set including process parameters and performance evaluation indicators is obtained by using NSGA-III;

[0011] Step 5: A multi-factor comprehensive objective function is established by using a linear weighted method, and the optimal process parameters of the double-layer Y-tube hydroforming process are selected from the Pareto optimal solution set using the multi-factor comprehensive objective function;

[0012] The process parameters in step 1 include: left punch feed amount, right punch feed amount, back punch retreat amount and shaping pressure;

[0013] The performance evaluation indexes include: inner layer wall thickness reduction rate, outer layer wall thickness reduction rate, inner layer wall thickness increase rate, outer layer wall thickness increase rate and branch pipe fillet filling area;

[0014] The method for designing N groups of process parameters using the Latin hypercube sampling method is as follows: for any one process parameter in each group of process parameters, a value range of the process parameter when no failure behavior occurs in the double-layer Y-shaped tube during the forming process is determined, and based on the determined value ranges of all process parameters, N groups of process parameter values ​​are randomly generated using the Latin hypercube sampling method as the designed N groups of process parameters;

[0015] The step 2 further comprises:

[0016] Step 2.1: Use finite element simulation software to build a finite element simulation model of the double-layer Y-shaped tube, including: a rigid die, a left punch, a right punch, a back punch, an inner tube blank and an outer tube blank;

[0017] Step 2.2: Setting the finite element simulation parameters of the finite element simulation model of the double-layer Y-shaped tube, including: analysis step, field variable output, interaction relationship between components in the finite element simulation model of the double-layer Y-shaped tube, load and boundary conditions;

[0018] Step 2.3: Meshing the finite element simulation model of the double-layer Y-shaped tube, inputting N groups of process parameters into the finite element simulation model of the double-layer Y-shaped tube for simulation, and recording the simulation results of the N groups of process parameters; wherein the simulation results of each group of process parameters include: the wall thickness of the double-layer Y-shaped tube and the coordinates of the three outer tube fillets used to calculate the outer tube fillet filling area;

[0019] Step 2.4: Calculate the performance evaluation index corresponding to each group of process parameters using the simulation results of the group of process parameters, and determine the performance evaluation indexes corresponding to N groups of process parameters respectively;

[0020] The step 3 further comprises:

[0021] Step 3.1: Build a BP neural network, including: input layer, initial hidden layer and output layer;

[0022] Step 3.2: Determine the number of neurons in the initial hidden layer of the BP neural network according to the empirical formula;

[0023] Step 3.3: normalize the N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, and divide the normalized process parameters and the performance evaluation indicators corresponding to each group of process parameters into a training set and a validation set according to a preset ratio;

[0024] Step 3.4: Initialize the connection weights and thresholds of each layer in the BP neural network, and set the training parameters of the BP neural network, including: the number of training times, the minimum error of the training target, the learning rate, the activation function, and the training algorithm of the BP neural network;

[0025] Step 3.5: According to the set training parameters, the BP neural network is iteratively trained using the training set to obtain a trained BP neural network. The process parameters in the validation set are then input into the trained BP neural network to obtain the predicted value of the performance evaluation index. The predicted value is compared with the true value of the performance evaluation index in the validation set, and the root mean square error E is used to evaluate the prediction accuracy of the trained BP neural network.

[0026] Step 3.6: Use the particle swarm optimization algorithm PSO to optimize the prediction accuracy of the trained BP neural network and obtain the PSO-BP neural network prediction model of the double-layer Y-type tube;

[0027] Step 3.7: Save the PSO-BP neural network prediction model of the double-layer Y-tube and use it to calculate the determination coefficient R 2 Evaluate the prediction accuracy of the PSO-BP neural network prediction model for the double-layer Y-tube;

[0028] The step 3.6 further comprises:

[0029] Step 3.6.1: Use the connection weights and thresholds of each layer in the trained BP neural network as particles to initialize the initial parameters of the particle swarm optimization algorithm PSO, including: population size, learning factor, inertia weight, number of iterations, maximum particle speed limit, and particle definition area limit;

[0030] Step 3.6.2: Take the root mean square error E as the fitness of the current particle, and compare the fitness of the current particle with the individual best fitness of the particle P best Compare and update. If the current particle's fitness is less than the particle's individual best fitness P best , then the fitness of the current particle is taken as the individual optimal fitness P of the new particle best Otherwise, the individual optimal fitness of the particle is P best constant;

[0031] Step 3.6.3: Update the individual best fitness of the particle P best and the optimal fitness G of the particle swarm best Compare and update, if the individual best fitness of the updated particle P best Less than the optimal fitness G of the particle swarm best , then the individual best fitness of the updated particle P best As the optimal fitness of the new particle swarm G best Otherwise, the optimal fitness of the particle swarm is G best constant;

[0032] Step 3.6.4: According to the individual best fitness P of the updated particle best And the optimal fitness G of the updated particle swarm best Update the particle velocity and particle position of the particle;

[0033] Step 3.6.5: Use the updated particles to recalculate the root mean square error E, repeat steps 3.6.2-3.6.4 for iteration, and terminate the iteration after reaching the preset number of iterations. The optimal particle is obtained and used as the optimal initial weight and threshold after optimization;

[0034] Step 3.6.6: Use the optimized optimal initial weights and thresholds to update the connection weights and thresholds of each layer in the trained BP neural network to obtain the PSO-BP neural network prediction model of the double-layer Y-type tube;

[0035] The step 4 further comprises:

[0036] Step 4.1: Define the objective function set of the multi-objective optimization of the double-layer Y-type tube according to the performance evaluation index, including: the objective function of minimizing the thinning rate of the inner tube, the objective function of minimizing the thinning rate of the outer tube, the objective function of minimizing the thickening rate of the inner tube, the objective function of minimizing the thickening rate of the outer tube, and the objective function of minimizing the filling area of ​​the branch fillet;

[0037] Step 4.2: Set the population size, number of iterations, crossover ratio, and mutation ratio;

[0038] Step 4.3: Design N using Latin hypercube sampling pop The process parameters are input into the PSO-BP neural network prediction model of the double-layer Y-type tube, and N pop A set of process parameters and the performance evaluation index of the set of process parameters are taken as an individual in the initial population, and the generated initial population is taken as the parent population Q t ;

[0039] Step 4.4: Take each objective function in the objective function set of the double-layer Y-tube multi-objective optimization as a target area, set the number of partitions for each target area, and then set a number of reference points according to the set number of partitions;

[0040] Step 4.5: Generate a new generation of population P through selection, crossover and mutation t and the parent population Q t Mix and get population R t , for the population R t Perform non-dominated sorting;

[0041] Step 4.6: After the non-dominated sorting, the population R t The individuals in are adaptively normalized to obtain the adaptively normalized individuals of the population;

[0042] Step 4.7: Establish the mapping relationship between the population individuals after adaptive normalization and each reference point and perform the microhabitat preservation operation, and use the obtained individuals as the individuals on the last Pareto frontier in the Pareto optimal solution;

[0043] Step 4.8: Repeat steps 4.5-4.7 until the preset maximum number of iterations is reached, and a set of Pareto optimal solutions is obtained as the Pareto optimal solution set;

[0044] The objective function for minimizing the fillet area of ​​the branch pipe fillet described in step 4.1 is expressed as:

[0045]

[0046] Where min represents the minimum value; f5 represents the fillet filling area after the tube is formed; D represents the circumscribed circle radius of the outer tube fillet;

[0047] The step 5 further comprises:

[0048] Step 5.1: Use linear normalization to normalize all Pareto optimal solutions in the Pareto optimal solution set to obtain the normalized values ​​of all Pareto optimal solutions;

[0049] Step 5.2: Use linear weighting method to establish multi-factor comprehensive objective function;

[0050] Step 5.3: Set the weighted coefficients of each objective function in the objective function set of the multi-objective optimization of the double-layer Y-tube according to user needs, and use the multi-factor comprehensive objective function to calculate the comprehensive evaluation index of each group of process parameters in the Pareto optimal solution set;

[0051] Step 5.4: All calculated comprehensive evaluation indicators J set Sort in ascending order and select the first comprehensive evaluation index J in the sorting result set The corresponding process parameters are taken as the optimal process parameters for double-layer Y-tube hydroforming;

[0052] The multi-factor comprehensive objective function described in step 5.2 is expressed as:

[0053]

[0054] Among them J set is a comprehensive evaluation index; l is the normalized value of the lth objective function in the Pareto optimal solution, q l It is the weight coefficient of the lth objective function in the objective function set of multi-objective optimization of the double-layer Y-tube; L represents the number of objective functions.

[0055] The beneficial effects of adopting the above technical solution are:

[0056] The method of the present invention applies a novel multi-objective optimization method to the optimization design of the double-layer Y-tube internal high-pressure forming process, uses the data obtained by finite element simulation to build a back propagation (BP) neural network prediction model, and constructs a prediction model for the internal high-pressure forming process design with the inner and outer layer wall thickness thinning rates, the inner and outer layer wall thickness thickening rates and the branch pipe fillet filling area as objective functions. The process parameters of the double-layer Y-tube internal high-pressure forming are used as decision variables, and the prediction model is optimized using a particle swarm optimization algorithm (PSO). In addition, a Latin hypercube sampling method is added to a non-dominated genetic algorithm (NSGA-III) based on a reference point to initialize an initial population composed of decision variables, thereby improving the optimization effect.

[0057] The proposed method establishes a multi-factor comprehensive objective function through a linear weighted method, and selects a set of solutions that best meet the requirements from the Pareto solution set as the optimal process parameters for the final double-layer Y-tube high pressure forming process according to actual needs.

[0058] The method of the present invention combines machine learning, intelligent optimization algorithms and the optimization design of the double-layer Y-tube high-pressure forming process, improves the accuracy and efficiency of the optimization, and provides a new method for achieving the coordinated deformation of the inner and outer layers of the tube during the double-layer Y-tube high-pressure forming process. The method of the present invention can optimize at least five performance evaluation indicators of the double-layer Y-tube high-pressure forming tube, considers the comprehensive influence of multiple process parameters on the inner and outer layers of the tube during the double-layer Y-tube high-pressure forming process, and significantly improves the problem that the traditional multi-objective optimization method has insignificant optimization results when dealing with more than three objectives, and achieves the coordinated deformation of the double-layer Y-tube high-pressure forming process. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Flow chart of a multi-objective optimization method for a double-layer Y-tube high pressure forming process in this embodiment;

[0060] Figure 2 It is an algorithm flow chart of a multi-objective optimization method of a double-layer Y-tube high pressure forming process in this embodiment;

[0061] Figure 3 Schematic diagram of the geometric model of the double-layer Y-shaped tube in this embodiment;

[0062] Figure 4 This is a schematic diagram of points for calculating the fillet area of ​​the branch pipe fillet in this embodiment;

[0063] Figure 5 Schematic diagram of the prediction accuracy of the PSO-BP neural network prediction model of the double-layer Y-type tube in this embodiment; Figure (a) is a comparison diagram of the true value and the predicted value of the maximum thickening rate of the inner layer in the verification set; Figure (b) is a comparison diagram of the true value and the predicted value of the maximum thinning rate of the inner layer in the verification set; Figure (c) is a comparison diagram of the true value and the predicted value of the maximum thickening rate of the outer layer in the verification set; Figure (d) is a comparison diagram of the true value and the predicted value of the maximum thinning rate of the outer layer in the verification set; Figure (e) is a comparison diagram of the true value and the predicted value of the fillet filling area of ​​the branch pipe in the verification set;

[0064] Figure 6 This is a schematic diagram of the results of selecting three representative objectives from the multi-objective optimization results of NSGA-III in this implementation method;

[0065] Figure 7 1 is a comparison diagram of the wall thickness of the inner and outer layers before and after the process parameters are optimized in this embodiment; wherein Figure (a) is a comparison diagram of the wall thickness of the inner and outer layers before the process parameters are optimized; Figure (b) is a comparison diagram of the wall thickness of the inner and outer layers after the process parameters are optimized;

[0066] Figure 8 This is a diagram of formed parts after process parameters are optimized in this embodiment. DETAILED DESCRIPTION

[0067] In order to facilitate the understanding of the present application, the specific embodiments of the present invention are further described in detail below in conjunction with the accompanying drawings and embodiments. The following embodiments are used to illustrate the present invention, but are not intended to limit the scope of the present invention. On the contrary, the purpose of providing these embodiments is to make the disclosure of the present application more thoroughly understood.

[0068] The core idea of ​​the present invention is: using Latin hypercube sampling method to design different process parameters; constructing a finite element simulation model of a double-layer Y-type tube through Abaqus finite element analysis software, and calculating the finite element simulation results according to the designed process parameters; using BP neural network to construct a mathematical model between process parameters and performance evaluation indicators of inner and outer two-layer tubes; using particle swarm optimization algorithm (PSO) to optimize the prediction accuracy of the mathematical model; and using non-dominated sorting genetic algorithm (NSGA-III) based on reference points to construct multi-objective optimization of the double-layer Y-type tube, iteratively searching for the Pareto optimal solution of the process parameters; using linear weighted method to establish a multi-factor comprehensive objective function, and then selecting a set of optimal solutions from the Pareto solution set.

[0069] A multi-objective optimization method for a double-layer Y-tube high pressure forming process in this embodiment is as follows: Figure 1 and Figure 2 As shown, the method comprises the following steps:

[0070] Step 1: According to the actual operation requirements of the double-layer Y-tube high pressure forming process, the process parameters and performance evaluation indicators are determined, and the Latin hypercube sampling method is used to design N groups of process parameters.

[0071] The process parameters include: left punch feed amount, right punch feed amount, back punch retreat amount and shaping pressure.

[0072] In this embodiment, the machine operating parameters set during the internal high pressure forming experiment can be used as the actual operating requirements of the double-layer Y-tube internal high pressure forming process, and the process parameters can be determined according to the specific requirements of the double-layer Y-tube internal high pressure forming, where the unit of the left punch feed amount is mm, the unit of the right punch feed amount is mm, the unit of the back punch retreat amount is mm, and the unit of the shaping pressure is MPa.

[0073] The performance evaluation indexes include: inner layer wall thickness reduction rate, outer layer wall thickness reduction rate, inner layer wall thickness increase rate, outer layer wall thickness increase rate and branch pipe fillet filling area.

[0074] In this embodiment, the inner layer wall thickness reduction rate, the outer layer wall thickness reduction rate, the inner layer wall thickness increase rate and the outer layer wall thickness increase rate are all expressed in the form of percentage.

[0075] The method for designing N groups of process parameters using the Latin hypercube sampling method is as follows: for any one process parameter in each group of process parameters, a value range of the process parameter is determined when no failure behavior occurs in the double-layer Y-shaped tube during the forming process, and based on the determined value ranges of all process parameters, N groups of process parameter values ​​are randomly generated using the Latin hypercube sampling method as the designed N groups of process parameters.

[0076] In this embodiment, according to the different selected materials, different tube wall thicknesses and different forming processes, in order to save costs, it is generally necessary to determine the range of process parameter values ​​in which no failure behavior occurs during the forming process of the double-layer Y-tube through finite element simulation. If the cost is not considered, this range can be obtained by trial and error on the internal high-pressure forming equipment. When the materials of the tubes are different, the range of process parameter values ​​selected for the double-layer Y-tube in which no failure behaviors such as wrinkling and cracking occur during the forming process is also inconsistent. Therefore, the range of process parameter values ​​selected in this embodiment is a shaping pressure of 65-85MPa, a left punch feed of 22-40mm, a right punch feed of 12-25mm, and a back punch retreat of 18-25mm. MATLAB software is used to generate 80 sets of process parameter values ​​according to the Latin hypercube sampling method.

[0077] Step 2: Use finite element simulation software to construct a finite element simulation model of the double-layer Y-type tube, use N groups of process parameters to perform finite element simulation on the finite element simulation model of the double-layer Y-type tube, and calculate the performance evaluation index corresponding to each group of process parameters according to the simulation results.

[0078] Step 2.1: Use finite element simulation software to build a finite element simulation model of the double-layer Y-tube, including: a rigid die, a left punch, a right punch, a back punch, an inner tube blank and an outer tube blank.

[0079] In this embodiment, if Figure 3 As shown, the "Part" module of the Abaqus finite element simulation software is used to perform three-dimensional modeling on the die of the three-dimensional discrete rigid shell element, the left punch, the right punch, the back punch, the inner tube blank and the outer tube blank of the three-dimensional deformable shell element, and they are assembled in the "Assemble" module to obtain a finite element simulation model of the double-layer Y-type tube, and the left punch, the right punch and the back punch are the same.

[0080] Step 2.2: Set the finite element simulation parameters of the finite element simulation model of the double-layer Y-tube, including: analysis step, field variable output, interaction relationship between components in the finite element simulation model of the double-layer Y-tube, load and boundary conditions.

[0081] In this embodiment, the analysis step and field variable output are set in the "step" module; the total time length of the analysis step is 0.01, and there are three steps, which are 0.001, 0.007, and 0.002 respectively; the field variable outputs the thickness STH of the cross section; in the "interaction" module, the constraint relationship between the left and right punches and the inner and outer tubes is set to binding, and the friction between the inner and outer tubes and the friction between the outer tube and the mold are both set to 0.01; in the "load" module, the determined process parameters are input to set the load and boundary conditions, including the shaping pressure, the feed speed of the left punch, the right punch, and the back punch.

[0082] Step 2.3: Mesh the finite element simulation model of the double-layer Y-tube, input N groups of process parameters into the finite element simulation model of the double-layer Y-tube for simulation, and record the simulation results of the N groups of process parameters; the simulation results of each group of process parameters include: the wall thickness of the double-layer Y-tube and the coordinates of the three outer tube fillets used to calculate the filling area of ​​the outer tube fillets.

[0083] In this embodiment, the double-layer Y-shaped three-way pipe is meshed, and 80 sets of process parameter values ​​and their corresponding simulation results are recorded.

[0084] Step 2.4: Calculate the performance evaluation index corresponding to each set of process parameters using the simulation results of the set of process parameters, and determine the performance evaluation indexes corresponding to N sets of process parameters respectively.

[0085] In this embodiment, the wall thickness of the double-layer Y-shaped tube is used to calculate the wall thickness reduction rate and the wall thickness increase rate. Figure 4 As shown, three points (p, q), (u, v) and (r, s) are taken at the fillet of the outer tube respectively, and the circumscribed circle radius of the fillet of the outer tube is calculated according to the coordinates of the three points, and then the fillet filling area of ​​the branch tube of the outer tube is calculated.

[0086] Step 3: Based on N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, a PSO-BP neural network prediction model for the double-layer Y-type tube is constructed.

[0087] Step 3.1: Build a BP neural network, including: input layer, initial hidden layer and output layer.

[0088] In this embodiment, there are an input layer and an output layer in the BP neural network, and there are usually several hidden layers between the input layer and the output layer to complete the mapping from any input dimension to the output dimension. The number of neurons in the input layer and the output layer is fixed, while the number of neurons in the hidden layer is uncertain, so it is necessary to determine the number of neurons in the initial hidden layer in the constructed BP neural network according to an empirical formula. The left punch feed amount, the right punch feed amount, the back punch retreat amount and the shaping pressure are used as the input of the BP neural network, and the inner and outer layer wall thickness thinning rate, the inner and outer layer wall thickness thickening rate, and the fillet filling area of ​​the branch pipe are used as the output of the BP neural network.

[0089] Step 3.2: Determine the number of neurons in the initial hidden layer of the BP neural network according to the empirical formula.

[0090] The empirical formula is:

[0091]

[0092] Where n is the number of neurons in the initial hidden layer; n1 is the number of neurons in the input layer; n2 is the number of neurons in the output layer; and a is an arbitrary constant in [1, 10].

[0093] In this embodiment, the number of neurons in the initial hidden layer of the network is determined to be 9 according to an empirical formula.

[0094] Step 3.3: Normalize the N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, and divide the normalized process parameters and the performance evaluation indicators corresponding to each group of process parameters into a training set and a validation set according to a preset ratio.

[0095] In this embodiment, 80 groups of process parameters and their corresponding performance rating indicators are normalized, and the first 85% of the normalized data are used as a training set, that is, the first 65 groups of process parameters and their corresponding performance evaluation indicators after normalization are used as a training set; the last 15% of the normalized data are used as a validation set, that is, the last 15 groups of process parameters and their corresponding performance evaluation indicators obtained after normalization are used as a validation set.

[0096] Step 3.4: Initialize the connection weights and thresholds of each layer in the BP neural network, and set the training parameters of the BP neural network, including: number of training times, minimum error of training target, learning rate, activation function and training algorithm of the BP neural network.

[0097] In this embodiment, the number of training times is set to 1000, and the minimum error of the training target is 1*10 -5 , the learning rate is 0.01, the activation function uses sigmoid and purelin, and the training algorithm uses LM optimization algorithm (trainlm).

[0098] Step 3.5: According to the set training parameters, the BP neural network is iteratively trained using the training set to obtain a trained BP neural network. The process parameters in the verification set are then input into the trained BP neural network to obtain the predicted value of the performance evaluation index. The predicted value is compared with the true value of the performance evaluation index in the verification set, and the root mean square error E is used to evaluate the prediction accuracy of the trained BP neural network.

[0099] The root mean square error is expressed as:

[0100]

[0101] Where E is the root mean square error, which is used to evaluate the prediction accuracy of the BP neural network; e q Represents the deviation between the predicted value of the BP neural network and the true value in the validation set; K is the number of data sets in the validation set.

[0102] In this embodiment, the prediction ability of the BP neural network is trained using the training set, the process parameters in the training set are input into the BP neural network, the output of the BP neural network is used as the predicted value of the performance evaluation index corresponding to the input process parameters, the predicted value is compared with the performance evaluation index in the training set and the error is calculated, the BP neural network is optimized using the error, and the above process is repeated until the preset number of training times is met or the preset training target minimum error is met, the training is terminated, and a trained BP neural network is obtained. The validation set is input into the trained BP neural network to predict the performance evaluation index, and after the prediction is completed, the root mean square error E is obtained by using the true value of the performance evaluation index in the validation set and the obtained predicted value to evaluate the prediction accuracy of the BP neural network.

[0103] Step 3.6: Use the particle swarm optimization algorithm PSO to optimize the prediction accuracy of the trained BP neural network and obtain the PSO-BP neural network prediction model of the double-layer Y-type tube.

[0104] Step 3.6.1: Use the connection weights and thresholds of each layer in the trained BP neural network as particles to initialize the initial parameters of the particle swarm optimization algorithm PSO, including: population size, learning factor, inertia weight, number of iterations, maximum particle speed limit, and particle definition area limit.

[0105] In this embodiment, the velocity and position of each particle are initialized, the learning factors c1=2 and c2=2 are set, the population size is 30, the inertia weight is [0.4, 0.9], and the number of iterations is 50.

[0106] Step 3.6.2: Take the root mean square error E as the fitness of the current particle, and compare the fitness of the current particle with the individual best fitness of the particle P best Compare and update. If the current particle's fitness is less than the particle's individual best fitness P best , then the fitness of the current particle is taken as the individual optimal fitness P of the new particle best Otherwise, the individual optimal fitness of the particle is P best constant.

[0107] Step 3.6.3: Update the individual best fitness of the particle P best and the optimal fitness G of the particle swarm best Compare and update, if the individual best fitness of the updated particle P best Less than the optimal fitness G of the particle swarm best , then the individual best fitness of the updated particle P best As the optimal fitness of the new particle swarm G best Otherwise, the optimal fitness of the particle swarm is G best constant.

[0108] Step 3.6.4: According to the individual best fitness P of the updated particle best And the optimal fitness G of the updated particle swarm best Updates the particle's particle velocity and particle position.

[0109] The updating formula of the particle speed is: For the i-th particle in the particle group, the updating formula of the speed of the d-th generation of the particle is:

[0110]

[0111] in represents the velocity of the dth generation of the i-th particle; w represents the inertia weight of the velocity; represents the speed of the d-1th generation of the i-th particle; c1 and c2 are both learning factors; represents the best position of the ith particle up to the dth generation; represents the position of the dth generation of the ith particle; G best t d Represents the best position of all particles in the particle swarm up to the dth generation; r1 and r2 are both random numbers on [0.1].

[0112] The updating formula of the particle position is: For the i-th particle in the particle group, the updating formula of the position of the d-th generation of the particle is:

[0113]

[0114] in represents the position of the ith particle in the d+1th generation.

[0115] In this embodiment, the optimal particle is found by updating the speed and position of the particle. In the process of updating the speed and position of the particle, if the speed of the particle exceeds the maximum speed limit, the speed of the particle is adjusted; if the position of the particle exceeds the defined area, the position of the particle is adjusted, so that the learning factor is continuously adjusted and improved.

[0116] Step 3.6.5: Use the updated particles to recalculate the root mean square error E, repeat steps 3.6.2-3.6.4 for iteration, and terminate the iteration after reaching the preset number of iterations. The optimal particle is obtained and used as the optimal initial weight and threshold after optimization.

[0117] Step 3.6.6: Use the optimized optimal initial weights and thresholds to update the connection weights and thresholds of each layer in the trained BP neural network to obtain the PSO-BP neural network prediction model of the double-layer Y-tube.

[0118] Step 3.7: Save the PSO-BP neural network prediction model of the double-layer Y-tube and use it to calculate the determination coefficient R 2 Evaluate the prediction accuracy of the PSO-BP neural network prediction model of the double-layer Y-tube.

[0119]

[0120] Where R 2 is the determination coefficient, which is used to evaluate the prediction accuracy of the PSO-BP neural network prediction model for double-layer Y-type tubes. 2 The closer it is to 1, the higher the prediction accuracy; is the average of the true values ​​of the bth performance evaluation index in the validation set;b is the true value of the bth performance evaluation indicator in the validation set, is the predicted value of the bth performance evaluation indicator in the validation set.

[0121] In this embodiment, if Figure 5 As shown, the R of the inner layer maximum thickening rate test set data 2 The R of the inner layer maximum thinning rate test set data is 0.99. 2 is 0.97, and the maximum thickness increase rate of the outer layer is R 2 is 0.97, the maximum thinning rate of the outer layer is R 2 is 0.98, the R of the fillet area 2 It is 0.98.

[0122] Step 4: The PSO-BP neural network prediction model of the double-layer Y-tube is used as the objective function of the reference point-based non-dominated genetic algorithm NSGA-III, and NSGA-III is used to obtain the Pareto optimal solution set including process parameters and performance evaluation indicators.

[0123] In this implementation, the number of groups of Pareto optimal solutions in the Pareto optimal solution set is set according to user needs. The more groups there are, the richer the sample diversity is, but the higher the calculation cost is.

[0124] Step 4.1: Define the objective function set of the multi-objective optimization of the double-layer Y-type tube according to the performance evaluation index, including: the objective function of minimizing the thinning rate of the inner tube, the objective function of minimizing the thinning rate of the outer tube, the objective function of minimizing the thickening rate of the inner tube, the objective function of minimizing the thickening rate of the outer tube, and the objective function of minimizing the filling area of ​​the branch fillet.

[0125] The objective function of minimizing the thinning rate of the inner layer tube or the objective function of minimizing the thinning rate of the outer layer tube is expressed as:

[0126]

[0127] Where min means taking the minimum value; f 1,2 Indicates the maximum thinning rate of the inner layer or the outer layer; t is the original thickness of the tube; t min It is the minimum thickness of the pipe after deformation.

[0128] The objective function of minimizing the thickness increase rate of the inner layer pipe or the objective function of minimizing the thickness increase rate of the outer layer pipe is expressed as:

[0129]

[0130] where f 3,4 Indicates the maximum thickness increase rate of the inner layer pipe or the maximum thickness increase rate of the outer layer pipe; t maxIt is the maximum thickness of the pipe after deformation.

[0131] The objective function of minimizing the fillet area of ​​the branch pipe fillet is expressed as:

[0132]

[0133] Where f5 is the fillet filling area after the tube is formed; D represents the circumscribed circle radius of the outer tube fillet.

[0134] In this embodiment, if Figure 4 As shown, three points (p, q), (u, v) and (r, s) are taken at the fillet of the outer tube respectively, and the circumscribed circle radius of the fillet of the outer tube is calculated according to the coordinates of the three points, and then the fillet filling area of ​​the branch tube of the outer tube is calculated.

[0135] Step 4.2: Set the population size, number of iterations, crossover ratio, and mutation ratio.

[0136] In this embodiment, the population size is set to 80, the number of iterations is set to 100, the crossover ratio is set to 0.5, and the mutation ratio is set to 0.02.

[0137] Step 4.3: Design N using Latin hypercube sampling pop The process parameters are input into the PSO-BP neural network prediction model of the double-layer Y-type tube, and N pop A set of process parameters and the performance evaluation index of the set of process parameters are taken as an individual in the initial population, and the generated initial population is taken as the parent population Q t .

[0138] In this embodiment, the Latin hypercube sampling method is used to redesign N in the same way as the method for designing process parameters in step 1. pop Set process parameters, set N pop =80. The PSO-BP neural network prediction model of the double-layer Y-tube is used to obtain N pop A performance evaluation index of a group of process parameters is obtained, and a group of process parameters and the performance evaluation index of the group of process parameters are combined as an individual in the initial population P, and the initial population P is used as the parent population Qt.

[0139] Step 4.4: Take each objective function in the objective function set of the double-layer Y-tube multi-objective optimization as a target area, set the number of partitions for each target area, and then set a number of reference points according to the set number of partitions.

[0140] The method for setting the reference point is:

[0141]

[0142] Where refCount indicates the number of reference points set; M indicates the number of target areas; and H indicates the number of partitions.

[0143] In this embodiment, the number of partitions is set to 10 and the number of target areas is set to 5, and 1001 reference points are finally generated according to the calculation formula for the number of reference points.

[0144] Step 4.5: Generate a new generation of population P through selection, crossover and mutation t and the parent population Q t Mix and get population R t , for the population R t Perform non-dominated sorting.

[0145] In this embodiment, a simulated binary crossover algorithm and a polynomial mutation method are used to generate a new generation of population P. t , where the simulation of the binary crossover algorithm requires first selecting the parent population Q t Select individuals P1 and P2, then select a random number τ between [0,1), and calculate the new generation of individuals using formulas (10)-(12).

[0146]

[0147] Where β represents the propagation factor; τ represents a random number in the interval [0,1); k represents the propagation factor distribution index; z1 is the new individual corresponding to individual P1; z2 is the new individual corresponding to individual P2.

[0148] Polynomial mutation refers to generating a mutation value through polynomial distribution, and then adding the mutation value to the gene on the chromosome to achieve chromosome mutation. Its mutation form is expressed as:

[0149] v′ k =v k +δ·(u k -l k ) (13)

[0150]

[0151] δ1=(v k -l k ) / (u k -l k ) (15)

[0152] δ2=(u k -v k ) / (u k -l k ) (16)

[0153] where v′k Represents the new generation of population after mutation; v k represents the previous generation population; u k Indicates the upper bound of the variable; l k Indicates the lower bound of the variable; μ m is the random distribution index selected by the user; δ represents the perturbation factor; δ1 represents perturbation factor 1; δ2 represents perturbation factor 2.

[0154] Step 4.6: After the non-dominated sorting, the population R t The individuals in are adaptively normalized to obtain the adaptively normalized individuals in the population.

[0155] In this embodiment, de-dimensionalization is achieved through an adaptive normalization operation of the population individuals, thereby eliminating the influence of different dimensions on the optimization process.

[0156] Step 4.7: Establish the mapping relationship between the population individuals and each reference point after adaptive normalization and perform the microhabitat preservation operation, and use the obtained individuals as the individuals on the last Pareto frontier in the Pareto optimal solution.

[0157] Step 4.8: Repeat steps 4.5-4.7 until the preset maximum number of iterations is reached, and a set of Pareto optimal solutions is obtained as the Pareto optimal solution set.

[0158] In this embodiment, if Figure 6 As shown in the figure, three representative objectives are selected from the Pareto optimal solution set, such as the maximum thinning rate of the outer layer, the maximum thickening rate of the inner layer and the filling area of ​​the branch fillet, and a scatter plot is drawn to represent the results of multi-objective optimization.

[0159] Step 5: A multi-factor comprehensive objective function is established using the linear weighted method, and the optimal process parameters of the double-layer Y-tube high pressure forming process are selected from the Pareto optimal solution set using the multi-factor comprehensive objective function.

[0160] Step 5.1: Use linear normalization to normalize all Pareto optimal solutions in the Pareto optimal solution set to obtain the normalized values ​​of all Pareto optimal solutions.

[0161]

[0162] in Represents the normalized value of the lth objective function of the mth Pareto optimal solution in the Pareto optimal solution set; represents the lth objective function value of the mth Pareto optimal solution in the Pareto optimal solution set; min(x l) represents the minimum value of the lth objective function in the Pareto optimal solution set; max(x l ) represents the maximum value of the lth objective function in the Pareto optimal solution set.

[0163] In this implementation, linear normalization is used to process the obtained Pareto optimal solution to eliminate the influence of different dimensions.

[0164] Step 5.2: Use the linear weighting method to establish a multi-factor comprehensive objective function.

[0165] The multi-factor comprehensive objective function is expressed as:

[0166]

[0167] Among them J set is a comprehensive evaluation index; l is the normalized value of the lth objective function in the Pareto optimal solution, q l It is the weight coefficient of the lth objective function in the objective function set of multi-objective optimization of the double-layer Y-tube; L represents the number of objective functions.

[0168] In this embodiment, L is the same as the value of M in formula (9). l It is the normalized value of the inner and outer wall thickness reduction rate (%), inner and outer wall thickness increase rate (%) or branch fillet filling area in the Pareto optimal solution.

[0169] Step 5.3: Set the weighted coefficients of each objective function in the objective function set of the double-layer Y-tube multi-objective optimization according to user needs, and use the multi-factor comprehensive objective function to calculate the comprehensive evaluation index of each group of process parameters in the Pareto optimal solution set.

[0170] In this embodiment, the weighted coefficients of the maximum outer layer thinning rate, the maximum outer layer thickening rate, the maximum inner layer thickening rate, the maximum inner layer thinning rate, and the fillet filling area are set to [0.3, 0.2, 0.3, 0.2, 0.1] respectively, and are substituted into the comprehensive objective function in formula (18), and the obtained N pop Group comprehensive evaluation index J set , that is, 80 groups of comprehensive evaluation indicators J set .

[0171] Step 5.4: All calculated comprehensive evaluation indicators J set Sort in ascending order and select the first comprehensive evaluation index J in the sorting result set The corresponding process parameters are taken as the optimal process parameters for high pressure forming of double-layer Y-tube.

[0172] In this embodiment, J setThe minimum process parameters are taken as the optimal process parameters for double-layer Y-type tube hydroforming. The optimized parameters are used to carry out double-layer Y-type tube hydroforming experiments, and the wall thickness distribution before and after optimization is obtained as shown in the figure. Figure 7 As shown in the figure, it shows that the deformation of the optimized parts is more coordinated, and the formed parts are as follows Figure 8 shown.

[0173] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some or all of the technical features therein. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A multi-objective optimization method for a double-layer Y-tube hydroforming process, characterized in that: The method comprises the following steps: Step 1: According to the actual operation requirements of the double-layer Y-tube hydroforming process, the process parameters and performance evaluation indicators are determined, and the Latin hypercube sampling method is used to design N groups of process parameters; Step 2: Use finite element simulation software to build a finite element simulation model of the double-layer Y-shaped tube, use N groups of process parameters to perform finite element simulation on the finite element simulation model of the double-layer Y-shaped tube, and calculate the performance evaluation index corresponding to each group of process parameters according to the simulation results; Step 3: Based on N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, a PSO-BP neural network prediction model of the double-layer Y-type tube is constructed; Step 3.1: Build a BP neural network, including: input layer, initial hidden layer and output layer; Step 3.2: Determine the number of neurons in the initial hidden layer of the BP neural network according to the empirical formula; Step 3.3: normalize the N groups of process parameters and the performance evaluation indicators corresponding to each group of process parameters, and divide the normalized process parameters and the performance evaluation indicators corresponding to each group of process parameters into a training set and a validation set according to a preset ratio; Step 3.4: Initialize the connection weights and thresholds of each layer in the BP neural network, and set the training parameters of the BP neural network, including: the number of training times, the minimum error of the training target, the learning rate, the activation function, and the training algorithm of the BP neural network; Step 3.5: According to the set training parameters, the BP neural network is iteratively trained using the training set to obtain a trained BP neural network. The process parameters in the validation set are then input into the trained BP neural network to obtain the predicted value of the performance evaluation index. The predicted value is compared with the true value of the performance evaluation index in the validation set, and the root mean square error E is used to evaluate the prediction accuracy of the trained BP neural network. Step 3.6: Use the particle swarm optimization algorithm PSO to optimize the prediction accuracy of the trained BP neural network and obtain the PSO-BP neural network prediction model of the double-layer Y-type tube; Step 3.7: Save the PSO-BP neural network prediction model of the double-layer Y-tube and use it to calculate the determination coefficient R 2 Evaluate the prediction accuracy of the PSO-BP neural network prediction model for the double-layer Y-tube; Step 4: The PSO-BP neural network prediction model of the double-layer Y-type tube is used as the objective function of the non-dominated genetic algorithm NSGA-III based on the reference point, and the Pareto optimal solution set including process parameters and performance evaluation indicators is obtained by using NSGA-III; Step 4.1: Define the objective function set of the multi-objective optimization of the double-layer Y-type tube according to the performance evaluation index, including: the objective function of minimizing the thinning rate of the inner tube, the objective function of minimizing the thinning rate of the outer tube, the objective function of minimizing the thickening rate of the inner tube, the objective function of minimizing the thickening rate of the outer tube, and the objective function of minimizing the filling area of ​​the branch fillet; The objective function of minimizing the fillet area of ​​the branch pipe fillet is expressed as: Where min represents the minimum value; f5 represents the fillet filling area after the tube is formed; D represents the circumscribed circle radius of the outer tube fillet; Step 4.2: Set the population size, number of iterations, crossover ratio, and mutation ratio; Step 4.3: Design N using Latin hypercube sampling pop The process parameters are input into the PSO-BP neural network prediction model of the double-layer Y-type tube, and N pop A set of process parameters and the performance evaluation index of the set of process parameters are taken as an individual in the initial population, and the generated initial population is taken as the parent population Q t ; Step 4.4: Take each objective function in the objective function set of the double-layer Y-tube multi-objective optimization as a target area, set the number of partitions for each target area, and then set a number of reference points according to the set number of partitions; Step 4.5: Generate a new generation of population P through selection, crossover, and mutation t and the parent population Q t Mix and get population R t , for the population R t Perform non-dominated sorting; Step 4.6: After the non-dominated sorting, the population R t The individuals in are adaptively normalized to obtain the adaptively normalized individuals of the population; Step 4.7: Establish the mapping relationship between the population individuals after adaptive normalization and each reference point and perform the microhabitat preservation operation, and use the obtained individuals as the individuals on the last Pareto frontier in the Pareto optimal solution; Step 4.8: Repeat steps 4.5-4.7 until the preset maximum number of iterations is reached, and a set of Pareto optimal solutions is obtained as the Pareto optimal solution set; Step 5: A multi-factor comprehensive objective function is established using the linear weighted method, and the optimal process parameters of the double-layer Y-tube high pressure forming process are selected from the Pareto optimal solution set using the multi-factor comprehensive objective function.

2. According to claim 1, a multi-objective optimization method for a double-layer Y-tube hydroforming process is characterized in that: The process parameters in step 1 include: left punch feed amount, right punch feed amount, back punch retreat amount and shaping pressure; The performance evaluation indexes include: inner layer wall thickness reduction rate, outer layer wall thickness reduction rate, inner layer wall thickness increase rate, outer layer wall thickness increase rate and branch pipe fillet filling area; The method for designing N groups of process parameters using the Latin hypercube sampling method is as follows: for any one process parameter in each group of process parameters, a value range of the process parameter is determined when no failure behavior occurs in the double-layer Y-shaped tube during the forming process, and based on the determined value ranges of all process parameters, N groups of process parameter values ​​are randomly generated using the Latin hypercube sampling method as the designed N groups of process parameters.

3. According to claim 2, a multi-objective optimization method for a double-layer Y-shaped tube high pressure forming process is characterized in that: The step 2 further comprises: Step 2.1: Use finite element simulation software to build a finite element simulation model of the double-layer Y-shaped tube, including: a rigid die, a left punch, a right punch, a back punch, an inner tube blank and an outer tube blank; Step 2.2: Setting the finite element simulation parameters of the finite element simulation model of the double-layer Y-shaped tube, including: analysis step, field variable output, interaction relationship between components in the finite element simulation model of the double-layer Y-shaped tube, load and boundary conditions; Step 2.3: Meshing the finite element simulation model of the double-layer Y-shaped tube, inputting N groups of process parameters into the finite element simulation model of the double-layer Y-shaped tube for simulation, and recording the simulation results of the N groups of process parameters; wherein the simulation results of each group of process parameters include: the wall thickness of the double-layer Y-shaped tube and the coordinates of the three outer tube fillets used to calculate the outer tube fillet filling area; Step 2.4: Calculate the performance evaluation index corresponding to each set of process parameters using the simulation results of the set of process parameters, and determine the performance evaluation indexes corresponding to N sets of process parameters respectively.

4. According to claim 3, a multi-objective optimization method for a double-layer Y-tube hydroforming process is characterized in that: The step 3.6 further comprises: Step 3.6.1: Use the connection weights and thresholds of each layer in the trained BP neural network as particles to initialize the initial parameters of the particle swarm optimization algorithm PSO, including: population size, learning factor, inertia weight, number of iterations, maximum particle speed limit, and particle definition area limit; Step 3.6.2: Take the root mean square error E as the fitness of the current particle, and compare the fitness of the current particle with the individual best fitness of the particle P best Compare and update. If the current particle's fitness is less than the particle's individual best fitness P best , then the fitness of the current particle is taken as the individual optimal fitness P of the new particle best Otherwise, the individual optimal fitness of the particle is P best constant; Step 3.6.3: Update the individual best fitness of the particle P best and the optimal fitness G of the particle swarm best Compare and update, if the individual best fitness of the updated particle P best Less than the optimal fitness G of the particle swarm best , then the individual best fitness of the updated particle P best As the optimal fitness of the new particle swarm G best Otherwise, the optimal fitness of the particle swarm is G best constant; Step 3.6.4: According to the individual best fitness P of the updated particle best And the optimal fitness G of the updated particle swarm best Update the particle velocity and particle position of the particle; Step 3.6.5: Use the updated particles to recalculate the root mean square error E, repeat steps 3.6.2-3.6.4 for iteration, and terminate the iteration after reaching the preset number of iterations. The optimal particle is obtained and used as the optimal initial weight and threshold after optimization; Step 3.6.6: Use the optimized optimal initial weights and thresholds to update the connection weights and thresholds of each layer in the trained BP neural network to obtain the PSO-BP neural network prediction model of the double-layer Y-tube.

5. According to claim 4, a multi-objective optimization method for a double-layer Y-tube hydroforming process is characterized in that: The step 5 further comprises: Step 5.1: Use linear normalization to normalize all Pareto optimal solutions in the Pareto optimal solution set to obtain the normalized values ​​of all Pareto optimal solutions; Step 5.2: Use linear weighting method to establish multi-factor comprehensive objective function; Step 5.3: Set the weighted coefficients of each objective function in the objective function set of the multi-objective optimization of the double-layer Y-tube according to user needs, and use the multi-factor comprehensive objective function to calculate the comprehensive evaluation index of each group of process parameters in the Pareto optimal solution set; Step 5.4: All calculated comprehensive evaluation indicators J set Sort in ascending order and select the first comprehensive evaluation index J in the sorting result set The corresponding process parameters are taken as the optimal process parameters for high pressure forming of double-layer Y-tube.

6. The multi-objective optimization method for a double-layer Y-shaped tube high pressure forming process according to claim 5, characterized in that: The multi-factor comprehensive objective function described in step 5.2 is expressed as: Among them J set is a comprehensive evaluation index; l is the normalized value of the lth objective function in the Pareto optimal solution, q l is the weighted coefficient of the lth objective function in the objective function set of multi-objective optimization of the double-layer Y-tube; n represents the number of objective functions.

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