Multi-objective optimization of concave radial forging process parameters based on response surface methodology and genetic algorithm

In order to improve the forming quality of the forging and reduce the forging cost in the concave radial forging process. In this paper, the influence of process parameters (radial reduction ∆ h , rotation angle β , friction coefficient μ ) on the forging process is investigated by numerical simulation, and the trade-off between the objective functions (strain homogeneity 𝐸 , forging load 𝐹 ) is achieved by a multi-objective optimization method. First, sample points for different combinations of process parameters were obtained by the central composite experimental design. Then the mathematical model between the process parameters and the objective function was established using the response surface method, and the model was subjected to variance analysis and sensitivity analysis. Finally, the optimal process parameter combination was obtained according to the NSGA-II algorithm and the satisfaction function. The optimization results were also verified by finite element simulations. The optimized process combination:


Introduction
With the development of lightweight design ideas for automobiles, the structure of hollow shaft-type parts for vehicles is becoming increasingly complex, which brings great challenges to the manufacturing process of axle parts.In particular, the production of slender shafts with stepped inner holes is particularly difficult.The concave radial forging process is an advanced near-net forming process.It designs a corresponding hammerhead die according to different shaft structures, the shaft parts with multi-step structure can be forged directly by radial forging, and the cross-section size of the billet can be reduced effectively.This process has the advantages of high precision, high manufacturing efficiency and material saving [1].And the fibrous tissue structure after forging can greatly improve the strength of the part.However, the deformation of the material is not absolutely homogeneous during the continuous forging process with multiple work steps.Uneven strain can lead to an increase in residual stresses and uneven microstructure.This affects the stiffness, stability and fatigue resistance of shaft parts [2] [3].At the same time, with the reduction of the cross-sectional size will lead to the forging load creeping up and thus increase the processing cost.Therefore, it is particularly important to control the reasonable process parameters.
Radial forging has two process variants.Feed radial forging (FRF) and concave radial forging (CRF) are classified according to whether the relative axial movement between the dies and the workpiece takes place or not [4].Usually, the feed radial forging is used for drawing length of the workpiece, while the concave radial forging is used for further forging of the shaft with specific structural dimensions.In previous studies, researchers have focused on the process parameters of feed radial forging.Ameli et al. [5] investigated the effect of process parameters such as die geometry, deformation percentage and workpiece motion on residual stresses and pressure on the mandrel.Azari et al. [6] used initial workpiece temperature, die entry angle, feed rate and cross-section reduction as input parameters and radial forging force as output parameters, a sensitivity analysis was performed using two intelligent systems to determine how the radial forging force was affected by the input parameters.Sanjari et al. [7] calculated the effect of process parameters such as different friction coefficients, axial feed rates, back and front thrusts, and die angles on the strain field using the finite element method.The results showed that the deformation inhomogeneity decreased with increasing die angle, decreasing back thrust and decreasing friction coefficient.Afrasiab et al. [8] stated that larger forging loads led to die wear and failure, it was found that an increase in die entry angle increased die life, and the increase of rotational feed rate has a negative effect on die life.Li et al. [9] studied the effect of three process parameters, initial temperature, rotational speed and radial reduction, on strain homogeneity using an orthogonal design method.It was pointed out that the maximum and minimum effective strains of hollow stepped gear shafts occurred in the outermost and innermost regions of different cross sections, respectively.The current concave radial forging process is influenced by the special die structure and is relatively little studied.The selection of process parameters is usually left to the experience of the designer and is an iterative process of trial and error.In addition, this process is generally used as a final forging process to forge the stepped structure of the shaft body.This is more closely related to the final quality of the product.Therefore, there is an urgent need to propose an efficient and accurate method to obtain the optimal combination of concave radial forging process parameters.
The study of radial forging process generally uses two methods: one is the theoretical analysis method such as slab method and upper limit method, and the other is the numerical analysis method represented by the finite element method.Lahoti et al. [10] used the slab method to establish an analytical model for radial forgings of bars and pipes, and studied the influence of various process variables on radial pressure distribution and metal flow in the radial forging process of manufactured pipes [1].The basic assumption was that the deformation in the slab was uniform [11], so the strain nonuniformity could not be studied.In contrast, since the strain is characterized by the velocity field, the upper bound method can be used to study the non-uniformity of the strain.Wu et al. [12] proposed an upper limit method model with continuous velocity field to study the strain nonuniformity in radial forging process.
However, the two theoretical analysis methods mentioned above are limited to two-dimensional mathematical model analysis, ignoring the circumferential inhomogeneity in three-dimensional space, while the finite element method can more accurately and intuitively characterize the changes in metal flow [13], strain and stress distribution [14], thermodynamic problems [15], and microstructure evolution [16] during radial forging and forming of forgings compared with the analytical methods mentioned above.
Researchers usually use the traditional control variable method to analyze the effect of relevant parameters on the objective function.Sun et al. [17] and Li et al. [9] used orthogonal experimental design to study the influence of radial forging process parameters on forging quality, and selected the best combination of process parameters through variance analysis.Sanjari et al. [18] conducted numerical experiments with Taguchi method by selecting the combination of process parameters with orthogonal array.Using the finite element results as input and genetic algorithm as the global optimization program, an artificial neural network model was trained.Then, the optimal process parameters were predicted by this model and compared with the results obtained by Taguchi method.The results of the two groups were found to be in good agreement.In the existing researches, most of the radial forging process parameter combinations are selected by orthogonal experimental design, which ignores the interaction effect between various factors.When multiple optimization objectives cannot be unified, it is difficult for designers to choose the optimal process parameter combinations.Therefore, it is necessary to use multi-objective optimization method to obtain the optimal compromise solution to achieve the optimal global effect, and provide scientific reference and theoretical basis for the selection of process parameter combination.
The combination of response surface method and optimization algorithm can effectively solve the multi-objective optimal solution problem.The response surface method is a method that uses a small number of sample points to develop an approximate model between several influencing factors and an objective function to analyze the sensitivity of each factor to the objective function and the interaction between the factors.Tang et al. [19] proposed a response surface method based on a preform die optimization method.The axisymmetric two-step forging problem was investigated and optimized using a pattern search optimization method.Yang et al. [20] proposed a response surface model with the preform shape geometry parameters as the relationship between the design variables and the response, and verified the effectiveness of the model by optimizing the variables of a typical aero-engine disc prefabricated parts Researchers from our group [21] developed a model for predicting strain homogeneity and material damage in radially forged forgings using the results of finite element analysis and the response surface method.The non-dominated ranking genetic algorithm-II (NSGA-II) was used to obtain the Pareto optimal solution, and the optimization results showed improved forging quality.
The above analysis shows that the process parameters have an important influence on the strain homogeneity and forging load of the forgings.However, the current research mainly considers the influence of a single process parameter on the forming characteristics.In particular, multi-objective process optimization that considers both strain homogeneity and forging load has not been performed.Therefore, this paper addresses the study of the process characteristics of concave radial forging.Based on numerical simulation, the response surface method is combined with genetic algorithm.A more efficient process parameter design method is proposed: firstly, a central composite experimental design method is used to select process parameter combinations and perform finite element simulations to establish a response surface model.Then the accuracy of the model is verified by ANOVA.The sensitivity analysis and interaction analysis are used to explore the influence law of each process parameter on the objective function.Finally, the Pareto front is obtained by using genetic algorithm for global search and the optimal combination of process parameters is obtained by using satisfaction function.
2 Finite element modeling of a concave radial forging process

Research Subjects
In this paper, the stepped shape of the left side of the automobile steering shaft is forged by the concave radial forging process.The steering shaft has the characteristics of variable cross-section and variable wall thickness.The shaft body is divided into forging zone (L1) and non-forging zone (L2).Parts Ⅰ, Ⅱ and Ⅲ of the forging area are defined as extension section, finishing section and sinking section respectively, as shown in Fig. 1.The outer tensioning of the die in extension section is beneficial to the axial flow of the material, thus reducing the balanced axial forging load and reducing the difficulty of forging.The finishing section increases the wall thickness and improves the strength and durability of the material by high frequency forging.The concave fillet in the sinking section is used to reduce the material flow on the cone surface and avoid thinning of the wall thickness at the joint.The advantage of this process is that the forging surface of the die fits perfectly with the final shape of the product, so that shaft parts of various sizes can be forged.

Fig. 1 Schematic diagram and main dimensions of the concave radial forging process
Since the concave radial forging process has little effect on the non-forging zone of the shaft.In order to save the time cost of finite element analysis.In the following study, only the FEM of the forging zone was established.The basic dimensions of the workpiece and dies in the figure as well as the annotations are recorded in Table 1.Considering the dimensional requirements and process redundancy, the forging is required to achieve an exact outer diameter of 14.6 mm and an inner diameter of less than 6 mm.Therefore, the total down stroke is 3 mm and the forging ratio is 27.5%.

FEM settings
A 3D model of the forging zone section was created in the commercial finite element analysis software Deform, as shown in Fig. 2. The concave radial forging process can be described as follows: four symmetrically structured dies perform radial motion and the workpiece moves rotating around the Y-axis as shown in the figure.Several circumferential feeds are performed between the same radial feed stroke to ensure that the gap between the dies is adequately forged.Then it goes to the next radial feed cycle until the outer diameter size is forged to the target size.Since the gap between the dies has a significant effect on the stress-strain distribution on the workpiece surface, numerical simulations must be performed using a three-dimensional finite element method.

Fig. 2 The finite element model of forging zone
The computational accuracy of the finite element simulation depends heavily on the number of meshes.First, the workpiece was meshed with a four-node quadrilateral mesh.Since almost the entire forging zone underwent large deformation, a uniform division was used.The mesh of the entire forging zone section has a total of 128,781 cells and 28,480 nodes, which can guarantee the accuracy of the simulation.Despite the large deformation of the workpiece during the forging process, Deform software has a powerful adaptive re-mesh capability to maintain mesh quality.
In addition, the shear friction model was chosen for the finite element friction model of the concave radial forging process [22], and the friction coefficient is 0.12.The die deformation in forging is minimal and can be set to rigid material.And this steering shaft forging process is cold forging, so the initial temperature of both the workpiece and the die is set to 20°C.The heat generated in forging is small and the contact time between the die and the product is relatively short.Therefore the effect of heat transfer between the die and the workpiece can be neglected [23].In a series of simulations, all the basic finite element parameter settings were kept constant.

Material property description
The material of the steering shaft is 25CrMo4 alloy steel, and the chemical composition is shown in Table 2.In order to obtain a more accurate material flow behavior during forging.The Johnson-Cook [24] intrinsic model was used to describe the mechanical behavior of the metal.The model includes strain hardening term, strain rate hardening (or softening) term, and thermal softening term.It is widely used in various finite element programs for studying strain and strain rate.A large number of experimental studies show that the Johnson-Cook constitutive model is in good agreement with the experimental results in describing the stress-strain curves of metal materials under dynamic impact loads.The equation of its intrinsic constitutive model is as follows.
where  is the flow stress;  is the equivalent plastic strain; ̇ is the loading strain rate;  0 ̇ is the reference strain rate; ,   and   are the experimental temperature, material melting point and ambient temperature, respectively; and , , ,  and  are the material parameters as shown in Table 3.In order to verify the accuracy of the material model, the stress-strain curve fitted in Deform software was compared with the compression test results of the researcher [25].The stress-strain flow curves at  = 0.001 −1  ̇and  = 20℃ are shown in Fig. 3, and the fitted curves are in better agreement with the experimental data, with an error of no more than 10% under these conditions.Fig. 3 Stress-strain curve of 25CrMo4 steel In addition, it is necessary to assume in the simulation： (1) The volume of the material does not change during deformation due to the force, that is, the volume is incompressible.
(2) The material is homogeneous and satisfies isotropy.
(3) The elastic deformation of the material is ignored during deformation.
(4) The effect of volume forces on forming is not considered in the deformation process.
(5) Both strain rate strengthening and strain strengthening of the deformed material exist during the deformation process.
Fig. 4 Multi-objective optimization process

Building response surface models
The purpose of the response surface model is to relate the response indicators of concave radial forging to the process parameters.In this way the proposed model can easily predict the forging results for different combinations of process parameters.It can also be used to find the critical factors that have a significant influence on the response.In addition, the developed response surface model is used for subsequent genetic algorithm optimization to obtain the solution of the population during the iterative process.A representative response surface model includes experimental design, empirical model building and statistical analysis of the constructed model [26].
For a concave radial forging process.Linear, square and interaction terms are usually considered to describe the relationship between the variable  and the response .This is expressed as follows.
where  is the response target,  0 is a constant,   and   are the linear and quadratic coefficients of the model, respectively,   is the mixture coefficient of the model, and  is the residual between the approximation and the actual value。 The choice of design variables and response is the key to the design of the experiment.According to previous studies, the larger the radial reduction (∆ℎ ) per forging stroke the higher the forging efficiency and the better the forging penetration.However, it will increase the forging load and cause die wear and other effects.The mismatch of circumferential rotation angle () tends to cause uneven material flow.It leads to inner circle depression and flying edge phenomenon between die gaps.And the concave radial forging die has a large contact area range with the workpiece.Friction coefficient () is another important process parameter.
Therefore, the radial radial reduction ∆ℎ, the circumferential rotation angle  and the friction coefficient  of the concave radial forging process will be selected as design variables in this paper, and the upper and lower bounds of the process range will be explored by numerical simulation.In the test scheme, the range of process parameters should be as large as possible, in the test scheme, the range of process parameters should be as large as possible to allow for optimization in a larger design space.In addition, the range of variation should ensure that no failures of fracture, folding, or flying edges occur during the forging process.Referring to the literature [27] [28] with the results of the finite element analysis, the range of process parameters was determined to be ∆ℎ∈[0.1,0.5]mm, ∈ [15,25]°, ∈[0.05,0.15].The values of each design variable are given at three levels, as shown in Table 4.The response is used as a criterion for judging the reasonableness of the process parameters.Obviously, the forging load has to be considered.This is because the forging load directly determines the choice of equipment tonnage.By optimizing the process to reduce the forging load, a smaller tonnage of equipment can be selected for manufacturing.A smaller forging load also reduces the wear [29] and impact fatigue of the equipment and dies, extending the life of the equipment and resulting in a corresponding saving in the production cost of the product.The response  is used to represent the peak load as an evaluation function of the forging load.
Where,   is the maximum peak load during forging.
Strain homogeneity reflects the level of discrete distribution of effective strain on the workpiece.When the forging deformation is more uniform, the local difference is reduced, and the internal stress will be correspondingly reduced, thus reducing the possibility of cracks caused by the uneven deformation of the structure.The uniform deformation of forging product is defined by the mean standard deviation of the equivalent strain of all nodes of forging workpiece and the average equivalent strain of the whole final forming part, the strain heterogeneity evaluation function is represented by the response E.
Where   is the effective strain of cell  in the final workpiece,  is the total number of cells used in the numerical analysis, and   is the average effective strain.
The intermediate process of variables and response was determined, and the combination scheme of different process parameters was designed according to the level of design variables and factors, as shown in Table 5.And the numerical simulation experiments are performed in turn, and the obtained response values are recorded in the table.From the results, it can be seen that the forging load  varies from 541.25KN to 722.007 KN and the strain homogeneity  varies from 0.228162 to 0.319764.The range of variation of the responses are both large, which means that the optimization of the process parameters in the design space is very necessary.

NSGA-II optimization algorithm
The multi-objective optimization algorithm used in this study is NSGA-II proposed by Deb et al. [30].It is characterized by the introduction of an elite strategy in the fast non-dominated ranking algorithm, which allows the retention of the best individuals.The selection based on non-dominated rank values and crowding ensures that a set of non-dominated solutions can be identified and analyzed [31].These features allow the NSGA-II algorithm not only to achieve convergence of the set of Pareto frontier solutions, but also to improve the speed of the algorithm and to maintain the diversity of results.The basic idea of the algorithm is as follows: the initial population is first generated randomly.The first generation is obtained by selection, crossover and mutation of the genetic algorithm.Then, the offspring population is combined with the parent population for non-dominance ranking.The crowding distance of each individual is calculated.Based on the ranking and crowding level, we select the best individuals to form a new parent population.Finally, the next generation population is generated again by the basic operations of the genetic algorithm.The optimization process ends when a certain number of evolutionary generations is reached or the algorithm converges.
According to the previous research, different concave radial forging process parameters have different effects on the evaluation function, which involves the problem of deformation non-uniform and forging load decision-making at the same time, that is, when the minimum response E and F are sought as far as possible in the design space, it is a multiobjective optimization problem.The specific mathematical model can be expressed as follows: 4 Results and Discussion

Response surface model analysis
Based on the data in Table 5, the regression analysis of the response surface model was performed using the commercial statistical software Minitab.The coefficients in Eq (3) were obtained.The regression equations for responses  and  are specifically expressed as follows.

Analysis of variance
ANOVA is used to determine the accuracy of the coefficients in the regression model and the significance of the design variables.Before ANOVA can be performed, its necessity needs to be tested.The residual plots of strain homogeneity  and forging load  are shown in Fig. 5 and Fig. 6, respectively, it can be seen from the figures, the overall normal probability plots for  and  are straight, indicating that the residuals at the sample points are normally distributed.In addition, it can be seen from the residual fitting diagram that the maximum residual of E and F are 0.005 and 14.2, respectively, and the residuals roughly form a horizontal band near the zero line, indicating that the squares of the error terms are roughly equal.the histograms of the residuals of  and  show a smaller skewed distribution.However, they tend to be symmetric in general.From the plot with the order, it can be seen that the residuals of  and  are randomly distributed below and above the zero line, indicating that the residuals are not correlated with each other.Therefore, it can be shown that the model satisfies the necessary requirements for ANOVA.The ANOVA results for strain homogeneity  and forging load  are shown in Tables 6 and 7.In the table, DF represents degrees of freedom.SS indicates the change of various factors, Adj.SS indicates the adjusted SS.MS is the sum of squares divided by the number of degrees of freedom, Adj.MS represents the adjusted MS, and the f-value is the ratio of the mean inter-group variance to the mean intra-group variance.The larger the f-value, the more significant the influence of the model item.p-value is a measure of the difference between the control and experimental groups, and a p-value <0.05 indicates a significant difference between the two groups. 2 is the coefficient of determination, and the closer the value is to 1, the better the model fits.Engineering generally requires  2 >0.9.  2 represents the adjusted coefficient of determination, and   2 represents the model's predictive effect on the new observations.As can be seen from the table, the overall f-values of the model are 114.79 and 112.55, respectively, and the p-value is zero, indicating that the model is statistically significant.Based on the magnitude of f-value, it can be judged that the significant levels affecting  from low to high are linear, squared and interaction terms, while the significant levels affecting  from low to high are linear, interaction and squared terms.And the most important factors affecting both  and  are the radial reduction ∆ℎ.Further, the significant model terms with p-value < 0.05 for response  include: ∆ℎ、、、∆ℎ 2 、 2 、 ∆ℎ * , and the significant model terms with p-value < 0.05 for response  include: ∆ℎ、、、∆ℎ * .The  2 is the coefficient of determination, the closer the value is to 1, the better the model fit is.The  2 for response  and  are 0.9904 and 0.9902, which is very close to 1, indicating that the developed models fully satisfy the accuracy requirements.the difference between   2 and   2 is less than 10%, indicating that the models have high predictive performance in the design space.The fitted relationship between the predicted and actual values of the regression model is shown in Fig. 7. From the figure, it can be seen that there is a basic agreement between the predicted and actual values without any deviation from the larger dispersion.It further proves that the mathematical model can represent the relationship between the response and the variables.

Sensitivity analysis
The purpose of sensitivity analysis is to quantitatively describe the degree of influence of different design variables on the response.In this paper, the behavior of the variation in the prediction model is observed by the following equation.The influence of three process parameters on the optimization objective is quantitatively described.
Where X and Y are the sample standard operating variances.S and SX and SY are the total sample variances for each dependency term in the independent interval, respectively.
The sensitivity analysis histogram of the process parameters is shown in Fig. 8.In the figure, it is clear that the amount of radial reduction ∆ℎ has the greatest effect on the responses.This is consistent with the findings in the ANOVA.Among them，the effect for the strain homogeneity target  is as high as 87.82% with a negative correlation.The effect for the forging load  reaches 78.84% with a positive correlation.The effect of circumferential rotation angle β is negatively correlated for both targets and has a greater effect on strain homogeneity 24.47%.The friction coefficient μ, on the other hand, is another important factor that cannot be neglected for the forging load target, with an effect of 39.56% and a positive correlation.Therefore, in the recessed forging process, the amount of radial reduction is the most important consideration for a reasonable process.The circumferential rotation angle and friction coefficient have non-negligible effects on strain homogeneity and forging load, respectively.

Interaction analysis
A surface plot of the effect of different process parameters on the objective function is given in Fig. 9.It visualizes the interaction between the process parameters.
As shown in Fig. 9a, it can be seen that a larger ∆ℎ leads to an increase in .A larger  favors a lower .And it can be found that ∆ℎ has a more significant effect than .A larger  favors a lower .It can be found that ∆ℎ has a more significant effect than .This is because a larger ∆ℎ means that the die bites deeper into the workpiece, making the radial force increase significantly.As can be seen in Fig. 9b, the  decreases as ∆ℎ increases and the rate of decrease decreases.This is because the increase of the radial reduction increases the forging permeability, which helps to improve the homogeneity of the strain between the inner and outer surfaces of the forging.And the contour line in the figure is elliptical, which means that there is a strong interaction effect between ∆ℎ and β on the response .
As shown in Fig. 9c and d, an increase in  leads to an increase in both E and .This is consistent with reality.This is because the larger friction coefficient makes the flow of the workpiece axially and circumferentially obstructed, leading to an increase in wall thickness.This reduces the strain homogeneity as well as the increase in forging load.It can also be seen from the figure that the contours in figure 9c show parallel curves, indicating a relatively stable interaction between the variables.In contrast, the contour lines in Fig. d are excessive from elliptical to parallel.This indicates that the interaction between  and ∆ℎ is significant at larger ∆ℎ .But at smaller ∆ℎ , there is almost no interaction of  with ∆ℎ.
This can be seen in Fig. 9e and f.The effect of  is more significant than  for the forging load  and the contour plot is uniformly distributed, indicating that  does not interact with .While for , the effect of  is rather less than  And the contour plot also indicates that there is a strong interaction effect between  and .When the value of  is small, the main influence on  is technological parameter As the value of  increases, its influence becomes more and more significant.

The Pareto front by NSGA-II
For multi-objective optimization problems, it means that two or more conflicting objectives need to be optimized simultaneously.The difficulty is that the improvement of one objective function will inevitably lead to a negative change in another objective function.Therefore, multi-objective optimization results in a set of solutions that are not worse for all objectives, the well-known Pareto non-inferiority solution [32].In this study, the NSGA-II algorithm is used to solve the multi-objective optimization problem of Eq (6).The parameter settings of NSGA-II are shown in Table 8.Among them, the parameter values of NSGA-II algorithm are referred to [33] [34].Finally, 32 points were selected as the Pareto fronts for the strain homogeneity E and forging load F objective functions.Each frontier solution represents a trade-off between two objective functions, with Pareto frontier points being the most advantageous.As shown in Fig. 12, where A and B are the optimal solutions of each objective function, respectively.

Optimal combination of process parameters
In order to determine the final process parameter solution from the many Pareto non-inferiority solutions.The satisfaction function   was used to evaluate the response values of the target in a comprehensive manner [35].The calculation method is shown in Eq (10) below.The values of the satisfaction function for the Pareto solution set are shown in Table 9.
Where  1 and  2 denote the response values for each scenario  and ，respectively. 1 ,  2 ,  1 , and  2 are the maximum and minimum values of  1 and  2 in the Pareto non-inferiority solution.As can be seen from Table 9, the solution NO.14 is the optimal compromise solution, which has the smallest value of the satisfaction function (  =0.8729).The strain homogeneity  and forging load  of the compromise solution are 0.241367 and 577.029KN, respectively.The corresponding combinations of process parameters are ∆ℎ =0.25 mm,  =21.68° and  =0.05.In order to show the validity of the compromise solution.In this paper, the compromise solution is also compared with the optimal solution of each objective function.Overall, the compromise solution balances the conflicting relationships among the objective functions well.The details are shown in Table 10.Plan A has the smallest forging load.It reduces 9.36% compared to the initial process, but the strain homogeneity increases by 4.29%.Plan B has the lowest strain homogeneity.It is 21.04% lower compared to the initial process, but the forging load increases by 0.17%.While the compromise solution reduces the strain homogeneity by 14.25%, the forging load also decreases by -1.76%.In conclusion, the multi-objective optimization method based on the response surface method and NSGA-II algorithm in this paper is very effective and provides a reliable way to save time and effort for the development of the concave radial forging process.

Analysis of optimization results
First, to verify the accuracy of the optimal solution.Finite element simulations were performed for the combination of process parameters outputted by the NSGA-II algorithm.And the simulated values were compared with the output values of NSGA-II algorithm for error analysis, and the specific results are shown in Table 11 below.From the table, it can be seen that the simulated values and the output values of the NSGA-II algorithm are basically consistent.The output values of the objective function  are large compared with the FEA results, and the errors are 2.55%.The output values of the objective function  are smaller than the FEA results, with errors of -2.70%.However, the errors of both objective functions are within 5%.This indicates that the above response surface model with NSGA-II algorithm is reliable for the prediction of the concave radial forging process.To further verify the validity of the compromise solution.Finite element simulations of the process parameter combinations of the compromise solution were carried out and compared with the initial process.The details are as follows.

Strain homogeneity of workpiece
Figure 13 shows the strain nephograms of the outer surface of the workpiece after concave radial forging.In Figure (a), the maximum effective strain of the initial process becomes 2.224, and in Figure (b), the maximum effective strain of the optimized process becomes 1.648.The maximum effective strain of both processes is concentrated in the right side of the sizing zone, and the maximum effective strain of the optimized process is reduced by 25.9%.Comparing the strain nephograms on the outer surface, it can be seen that the optimized strains in the forging zone are mainly concentrated in the range of 0.6-0.8.In contrast, the strain range in the forging zone under the initial process is mainly in the range of 0.6-1.0, and even some intervals of 1.0 to 1.2 appear.The compromise solution has a smaller strain range, which indicates a better strain homogeneity of the forging as a whole, which is conducive to improving the roughness of the outer surface of the workpiece and enhancing the forging accuracy.In the axial profile nephogram Fig. 14, the strain in the radial direction of the workpiece first decreases and then increases from the outside to the inside.The strain range on the outer and inner surfaces is higher than the central region, which indicates the occurrence of work hardening.However, it can be seen from the figure that the optimized strain range (0.4~0.6) in the sizing zone (II) covers a larger wall thickness and the depth of the hardened layer on the outer surface is significantly reduced.This helps to reduce the difficulty of subsequent processing of the workpiece.In the sinking section (III), the strain nephogram under the initial process shows a sawtooth distribution, which is prone to fracture failure of the workpiece.After optimization, the strain transition is smoother, indicating that the strain homogeneity of the workpiece in the radial direction has been significantly improved.This will make the metal tissue flow of the forgings more consistent and improve the strength and service life of the workpiece.In order to illustrate the variation of effective strain with forging time, the point tracking function in Deform was used to select three points on the outer surface, middle layer and inner surface of the middle section of the sizing zone (II) for point tracking.They are labeled P1, P2, and P3, respectively.The effective strains at each location are shown in Fig. 15.In general, with the increase of forging times, the cross section is constantly reduced, and the effective strain of the tracking point is gradually increased in a step-like manner.On the other hand, P1 and P3 in the figure represent the strains on the inner and outer surfaces, respectively.It can be seen that the strain magnitudes on the inner and outer surfaces remain the same under the optimized process combination, which indicates that the forging penetration is better under this process combination.Under the initial process, the strains at the tracking points at each location showed obvious stratification.In particular, the effective strain value of the outer surface reached 0.71297, which was obviously process hardened and unfavorable for subsequent processing.The extreme value of the effective strain for both also shrinks from 0.17515 to 0.11992, indicating a more uniform degree of overall material deformation in the radial direction.It further indicates that the workpiece under the compromise solution has better strain homogeneity after machining.The circumferential strain nephograms of the initial process and compromise solutions are also given in Fig. 16 and 17.The two sections are taken from the intermediate sections of the sizing zone (II) and the sinking section (III), respectively.It is obvious from Fig. 16 and 17.At the initial process circumferential rotation angle  of 15°, a periodic irregular variation of strain occurs.which leads to a decrease in the homogeneity of strain.The circumferential rotation angle  of the optimized compromise solution is 21.68°.It can be seen that there is a circular strain order distribution within the cross section.This indicates a more homogeneous circumferential forging penetration effect.Since the concave radial forging is a mandrelless forging, the inner wall is formed under natural flow.It can also be seen in Fig. 17 that the uneven circumferential strain leads to significant fluctuations in the radius of the inner circle.It tends to cause defects such as denting and folding of the inner circle.This creates a safety hazard for the shaft parts during use.In contrast, the optimized inner circle is well rounded.It further shows that the combination of process parameters under the compromise solution has a superior performance.

Forging load
Figure 18 records the forging load variation for the concave radial forging process in successive multi-steps.In the Deform finite element analysis every 0.01 mm is a calculation step.And three circumferential feeds were performed at the same radial reduction amount.From the figure, it can be seen that the initial process has a radial reduction ∆ℎ of 0.2mm, and the forging needs to go through 15 circumferential cycles for a total of 45 forges to reach the target size.The compromise solution has a downfeed ∆ℎ of 0.25 mm and requires only 12 circumferential cycles for a total of 36 forgings to reach the target size.Obviously, the compromise solution has better processing efficiency, which helps to reduce the production cost.The peak load increases gradually as the section size decreases.The highest loads are found in the first forging of the last cycle.The maximum loads of the initial process and the compromise solution are 587.362KN and 577.029KN, respectively.the maximum load of the compromise solution is reduced by 10.333 KN, which further reduces the machining cost.Although the increase in the radial reduction will lead to an increase in the forging load, the maximum forging load is still reduced due to the lower friction coefficient of the compromise solution.It can also be found from the load variation diagram that in each cycle pass, the peak forging load of three forging is not the same, which is mainly due to the different contact area between the hammer and the forging.Fig. 19 shows the hammer head contact with the forging during the first cycle pass before and after optimization.Dark blue indicates that the distance is zero, indicating that the hammer head is in contact with the forging.It can be seen that in the first forging process, due to the small pressing amount in the initial process, the small arc area of the hammer head did not actually forge the billet, and the contact area of the optimized process in the first forging almost covers the entire hammer head, indicating that under this pressing amount, full forging was achieved from the first cycle pass.In a cycle pass, the last two forging are mainly responsible for rounding, and the process contact area after optimization is reduced, while the process contact area before optimization is almost the same three times.Therefore, in the load travel diagram, the optimized process usually has a smaller peak load for the third forging in each cycle, which also means that the outer roundness is higher after the first two forging.It shows that the optimized circumferential rotation Angle is beneficial to reduce the load in the process of circular correction and improve the forging precision.

Experimental verification
According to the results of multi-objective optimization, the peak load in the concave radial forging process is about 577KN, and the equipment with smaller tonnage should be selected as far as possible to meet the forging force, so as to reduce the production cost.Finally, HA-40 radial forging equipment was selected, and its main technical indicators are shown in Table 12.The maximum forging force is 610KN, which meets the requirements, and other key indicators also meet the size requirements of the sample.According to the combination of optimized process parameters, the sample was trial-produced, and the processing sample of hollow steering shaft stage concave radial forging was obtained.The comparison between the sample and the numerical simulation results is shown in Fig. 20.As can be seen from the figure, the sample finishing section formed by concave radial forging is smooth and round, which lays a good material foundation for subsequent extruding splines and turning threads.The transition between the sinking section and the finishing section is uniform, and the 30° convex rounded corner is similar to the numerical simulation results.The large equivalent strain produces work hardening, which is conducive to improving the hardness of the transition surface.At the end of the extension section, due to the difference in axial flow velocity between the inner and outer surfaces of the material, a slight flaring phenomenon appears in both numerical simulation and test.In terms of dimensional accuracy, the outer surface of the hollow steering shaft stage is well fitted to the mold, and the diameter of the finishing section is only 0.01mm deviation.Due to the absence of mandrel limitation in the inner hole and the error in the friction coefficient, the numerical simulation material carries out more axial flow, among which the length error is 1.8% and the diameter error of the inner hole is 6.78%.On the whole, the test samples meet the requirements of steering shaft products, which further demonstrates the scientificity and reliability of the process parameters obtained by the multi-objective optimization method.

Conclusion
In this paper, a mathematical model between the concave radial forging process and forging strain homogeneity and forging load is developed based on response surface methodology.ANOVA was used to evaluate the accuracy requirements of the RSM model and to determine the significance of the influence of each process parameter on the target.Finally, the multi-objective optimization solution problem was solved using genetic algorithm.The Pareto front between forging strain uniformity and forging load is obtained, and an optimal compromise solution is selected from the solution set by the satisfaction function.Specifically, the following conclusions were drawn.
(1) The RSM-NSGA integrated method is proposed to realize the modeling and optimization of the parameters of the concave radial forging process.The results of ANOVA and finite element simulation validate the prediction accuracy of RSM model and the optimization results of genetic algorithm.
(2) The results of ANOVA and sensitivity analysis showed that the most significant factor affecting the strain homogeneity  and forging load  was the amount of radial reduction ∆ℎ.In addition, the effect of  was negatively correlated for both targets ,  and had a greater effect on strain homogeneity -14.47%.In contrast,  was the second influencing factor for , with an effect of 39.56% and a positive correlation.
(3) The compromise solutions have  and  of 0.241367 and 577.029KN, respectively, corresponding to the combination of process parameters ∆ℎ = 0.25 ,  = 21.68°, and  = 0.05.Compared with the initial plan, F decreases by 9.36% but E increases by 4.29% for plan A. Plan B has a 21.04% decrease in , but a 0.17% increase in .The compromise solution, on the other hand, reduces  by 14.25% while  decreases by -1.76%, which better balances the conflicting relationship between the objective functions.
(4) The analysis of the strain nephogram and forging load of the forgings shows that the increase of the ∆ℎ is beneficial to improve the forging penetration and thus the strain homogeneity in the radial direction, and to improve the machining efficiency.And a reasonable  can improve the strain homogeneity in the circumferential direction.

Fig. 9
Fig. 9 Response surface diagram of interaction between process parameters

Fig. 13
Fig.13 Strain nephograms on the outer surface of the workpiece before and after optimization

Fig. 14
Fig. 14 Strain nephogram of axial section of workpiece before and after optimization (a) Initial process point tracking diagram (b) Optimized process point tracking diagram Fig. 15 Variation pattern of effective strain at different locations on the radial direction before and after optimization

Fig. 16
Fig. 16 circumferential strain nephogram in the middle section of the sizing zone (II) (a) Initial process forging load (b) Compromise solution of forging load Fig. 18 Variation of forging load under continuous multi-step

Fig. 19
Fig. 19 Contact between die and forging in the first circumferential cycle

Fig. 20
Fig. 20 Comparison between test results and numerical simulation results under compromise solution process parameters

Table 1
Basic structural parameters of the three-dimensional model

Table 2
Alloy steel 25CrMo4 Chemical composition

Table 3
Parameters of the Johnson-Cook model

Table 4
Design variables and levels.

Table 5
Central composite design and corresponding results.

Table 6
The results of ANOVA for .

Table 9
Satisfaction function values for Pareto's non-inferior solutions NO.

Table 10
Comparison of the performance of the objective function of each scheme

Table 11
Error analysis between NSGA-II optimized value and FEM simulated value

Table 12
Technical specifications of the radial forging equipment of HA-40