A multi-response optimization of the multi-directional forging process for aluminium 7075 alloy using grey-based taguchi method

The multi-directional forging process of aluminium alloy 7075 (AA 7075) is studied using Deform 3D Version 11.0 simulation software. This process results in grain refinement in the bulk material. The 7075 aluminium alloy is used widely in the aerospace and automobile industries. Thermomechanical processing affects the mechanical properties of this alloy. This study focuses on optimising process parameters that affect the multi-directional forging using simulation. In the Taguchi design of experiment, four-factors and five levels are selected. The process input parameters considered are temperature, the strain per pass, the plunger speed, and the friction coefficient (μ). From Taguchi’s orthogonal array, forging simulations are undertaken and analysed. The significance of the process output parameters: material damage, stress and strain are analysed by analysis of variance. The results show that the friction coefficient and strain per pass highly affect the stress/strain distribution. Grey relational analysis is adopted to determine the optimum process parameters. The results show that the optimal combination of parameters is: temperature (200 °C), plunger speed (5 mm/s), friction coefficient (0.6), and strain per pass (0.6). Confirmation of simulation is carried out using the optimum input parameters. From the simulation results, the grey relational grade's optimal parameters have the highest maximum effective strain of 5.57, maximum effective stress of 665 MPa, and maximum damage of 0.416 compared to other simulated results.


Introduction
Multi-directional forging (MDF) is an important severe plastic deformation (SPD) process used in the grains refinement of metals and their alloys [1]. The process is cheap and easy to carry out because it does not need complex equipment when compared to other SPD processes, such as high-pressure torsion (HPT) and accumulative roll bonding (ARB) [2]. Additionally, it is capable of forging large samples. In the MDF process, materials are subjected to very high strains without necessarily resulting in dimensional changes to the samples. The samples are compressed in all the faces, one face after another repeatedly, as shown in Fig. 1. The compression results in a decrease in grain sizes and improved mechanical properties [3]. Noda et al. reported that superplastic behaviour also improved due to accumulated redundant plastic deformation [4].
The MDF is a nonlinear process that can result in undesirable results due to variations in various factors before or during processing. Some of these factors include; the geometry and size of the specimen, temperature changes, plunger speed, number of passes, coefficient of friction, strain per pass, forging equipment errors, strain rate, positional errors and material properties [5][6][7]. Combining any of these uncertainties could result in production losses, product distortion (geometry or/and quality), equipment failure and sometimes accidents. Therefore, identification, control and optimisation of these parameters minimizes the production losses and improves the production process.
During the MDF process, various studies have used input parameters differently. Some researches have been conducted at temperatures ranging from room temperature [8] to extremely high temperatures of up to 500 K [9,10]. Other input process parameters studied include; strain per pass (the strain of every single pass ranging from 0.2 to 0.7), die velocity, friction coefficient, and the geometry of the sample (size and shape) [9]. Most researchers have considered some variables of the process as constants while varying others to examine how they affect the MDF [3,8,[11][12][13]. Although studies on the MDF process have been widely conducted, the literature on process optimisation is scanty.
To carry out the optimisation reliably and at less costly means, the use of Deform™ 3D, a software with Finite Element Method (FEM) is preferred [14]. Deform™ 3D has a well-written algorithm that can mimic the real process effectively [15]. Other simulation softwares available for metal forming operations include; Transvalor Forge™, ANSYS, Simufact.forming™, Hyperextrude, SUPERFORM, ABACUS and MSC.MARC™ [16][17][18]. To get optimised output parameters for the process, several simulations are required. Therefore, the Taguchi optimisation methods crucial in reducing the number of simulations through the design of experiments (DOE) [19]. However, when several output responses are viewed with interdependent goals, the method is unsuitable. Therefore, the Grey-based Taguchi approach was developed in 1982, by Deng Ju-long to determine solutions to multi-response optimisation data [20]. The grey relational analysis helps solve complex interrelationships for multiple responses based on the theory of the grey system [21,22]. After evaluating each response, a grey relational grade is obtained. The multiple responses for optimisation are then transformed into a single optimization grade. The optimum conditions are determined from the grade rank.
In this study, the grey-based Taguchi approach quality tool was applied to optimise the process parameters of the multi-directional forging simulations of AA 7075. The main focus was to minimize damage, maximize effective strain and effective stress. This paper is organised as follows; section one is an introduction, where the general MDF process optimisation details of AA 7075 are discussed. The simulation and optimisation methodologies are outlined and discussed in section two. Section three provides results obtained after simulation and are later discussed in section four. Section five concludes the paper's main findings, research contributions and recommends future potential studies.

Methodology
In this study, MDF simulation was carried out using a finite element software called Deform™ 3D (version 11.0). The optimisation of the process parameters during the MDF simulation was carried out using Taguchi's design of experiments. The obtained results were analyzed using S/N ratios, ANOVA and grey relational analysis to determine optimum parameters.

Design of experiment
The design of experiments by the Taguchi technique is a powerful instrument for analyzing and modelling the influence of input parameters on the output quality. During the analysis, the most significant stage in the design of experiments is the choice and control of the input variables. The Taguchi method helps to efficiently determine the range of input data in the design stage, reducing the cost and time [23]. In the current study, four factors were considered, each having five levels to represent the various factors' data range. The four factors were; temperature, coefficient of friction, strain per pass and plunger speed. These factors were chosen because they have been shown in the literature to be the most significant in forging operations [10]. The respective levels of the factors were chosen based on the published experimental results and the Taguchi designs [2,24]. Table 1 gives the identified process parameters and levels. The interaction of the four process parameters at the five levels was designed using  Table 2.

Simulation model
In this study, workpieces of AA 7075, having dimensions of 50 mm × 50 mm × 50 mm, were used. The material properties of the workpiece are available in the commercial Deform™ 3D software database. Figure 2 shows the simulation set-up with (a) a schematic diagram of the dies and the meshed workpiece and (b) the simulation in progress at step 30. As shown, the MDF simulation set-up consists of a top and bottom die, which are assumed to be rigid, while the workpiece is assumed to be plastic. The bottom die is fixed while the top die moves in a normal direction to the workpiece. Using the primitive modeller of the Deform™ 3D tool, the geometry and dimensions of the rectangular dies and workpiece were developed. The die dimensions were 200 mm × 200 mm × 30 mm. The simulation process was conducted at a room temperature of 25 °C in an isothermal mode. The convectional coefficient was taken as 0.02 N/(s mm °C), whereas the coefficient of heat transfer used during the simulation; between the dies and the workpiece was taken as 5 W/(m 2 K) (g). The simulation type used was Lagrangian incremental with a direct iterative method [25]. The global re-meshing and conjugate-gradient solver were selected. The workpiece was discretised into 20,000 tetrahedral elements. Sensitivity analysis was used to determine the optimum number of elements. The use of tetrahedral elements is preferred as it gives a better and fitting mesh algorithm with only minor distortions. This type of mesh results in more accurate data within a short computation time, compared to the cartesian mesh type, even when the number of elements is high in the simulation model [26].

Optimisation using the grey Taguchi method
The Taguchi method is vital in developing the optimum parameters. The signal-to-noise ratio (S/N) is instrumental in obtaining the means of the output responses. Depending on the desired quality, the following characteristics of the S/N ratios were applied in this study; (i) the lower-thebetter represented by Eq. 1 to minimize damage and (ii) the higher-the-better, represented by Eq. 2 to maximise effective stress and strain. The greater the value of the S/N, the better the attainable quality.

Using the grey relational analysis (GRA) to determine the optimum process parameters
The grey relational analysis (GRA) is used for multi-objective optimisation, and in this study, there were three objectives; (i) to minimize damage, (ii) to maximise effective stress and (iii) to maximise effective strain. The GRG combines different interrelated performance objectives to form a single value called grade. The grade is then used to optimise multi-response problems. The general flow of the process is summarised in Fig. 3. Before computing the grey correlational coefficients, the grey data analysis must be carried out. The S/N ratios are useful for normalization of the results between the range 0 and 1 using Eqs. 3 and 4. Normalizing helps reduce the variability of the results and lowers the impact of the adoption of different units.
Normalization is then carried out after getting Δ (absolute difference) from the y oi and y ij values which can be said to be a loss or a deviation from the target and it can be expressed as; where Z oj is the optimum or ideal value that has been normalized for the jth response and it is equal to 1, Z ij is the value of the jth response after normalization. λ is considered as a distinct coefficient that ranges from 0 to 1.
The grey relation coefficient attempts to show the affiliation between ideal results and the actual normalized results. To compute the grey relational coefficient for the normalized ratios, the expression in Eq. 6 is applied, where Z ij is the next arrangement after the processed data, j = 1, 2, …., n; i = 1, 2, …, n, n is the nth number of the data results from the experiment and GC ij is the grey relational coefficient for experiment number i and the dependent response number j.
The grey relational grade is set after computing the mean value of the grey relational coefficient for each performance characteristic. The total performance characteristic for multi-response processes depends on the determined grey relational grade. To calculate the overall grey relational grade (G i ) for ith experiment and m number of the responses, Eq. 7 is used.
The Principal Component Analysis (PCA) was used in this work to approximate the weight.

Principal component analysis (PCA)
Various studies have used PCA to determine the weight (correlation) of different performance characteristics in various processes. Some of these studies include; Dubey & Yadava, who studied how to improve the cutting outcome of Nd:YAG pulsed laser beam cutting (LBC), the results obtained were found to be similar to experimental results [27]. Viswanathan et al. used Taguchi, grey relational analysis and PCA to optimise turning parameters of magnesium alloy. These methods proved to be effective for multi-response optimisation [28]. The procedure of the PCA involves [29]. The output quality performance characteristics are identified then normalization is carried out using Eqs. 8 and 9.
where x * i ,k is the standardized ith experiment and kth is the output data. In this study, the damage was minimized and maximization of effective stress and strain was done. Standardization was carried out to transform the variables into the same scale.
The covariance matrix is then computed using Eq. 10 to determine whether the variables at the input relate to each other; by determining the variation from the mean for each of them.
where Cov. means covariance and σ the standard deviation, i = 1, 2, 3 ….m, and m is the number of experiments, k = 1, 2, 3 …. n, n is the number of outputs measured.
Computation of eigenvalues and eigenvectors are then determined using Eq. 12. The eigenvectors computed are ordered according to their eigenvalues in descending order, to aid in determining the principal components significance order.
Finally, through the eigenvector, the weight factor of each output response of the principal component is calculated using Eq. 13.

Determining the optimum parameters and their level
This Grey relational grade method was used to transform the output responses of the multi-objective optimisation problem into a single optimised solution. The higher the grey relation grade, the closer the outcome to the ideal normalized value and the higher the grade value, the closer the factors in the combination are optimum.

Analysis of variance
The primary objective of the analysis of variance (ANOVA) is to use a statistical system to identify the influence of each factor. The Taguchi scientific design is unable to determine the impact of input variables over the entire process, but when using ANOVA by the percentage contribution, the effects can be determined. The cumulative total of the squared deviations is broken down into two: the cumulative of the squared absolute deviation due to the sum of squared error and each parameter. Each percentage in the total sum of the squared deviations is applied to evaluate the significance of the changes in the performance characteristics in the method parameter. Usually, when the F(variance ratio) value is significant, changes in the process parameters significantly impact performance. The degrees of freedom of an effect should be taken into account, i.e., a mean square effect of the factor will help decide if the effect is weak-the greater the mean square, the stronger the effect.

Results
The simulations were carried out as per the Taguchi design of experiment and corresponding values of the responses were recorded in Table 3. The output responses include; maximum damage, maximum effective strain and stress.
Damage is a characteristic that leads to the cracking or fracture of a material. The extent of the crack is dependent on the induced stresses in the material. The cracks created during deformation enhance fracture when the material is in use [30]. Effective stress is directly related to plastic deformation and higher effective stress is desirable during deformation. A reduction in effective stress results in low material strength. The quantity of effective strain is critical in measuring the degree of deformation within a material [31]. Higher values of effective strain are preferred.
The S/N ratios for the output responses in Table 4 were computed using the Taguchi criteria, represented in Eqs. 1 and 2. Normalization of the S/N values was then computed using Eq. 3 for damage and Eq. 4 maximum effective stress and strain. The absolute difference was computed by deducting each of the normalized values of the S/N ratio from the maximum normalized value according to Eq. 5. Table 4 shows the results of S/N ratios, normalization and absolute difference.
The grey relational coefficient values of the normalized ratios were determined as per Eq. 6. The distinguishing coefficient values were substituted in Eq. 7. Usually, when the outputs have an equal weightage, 0.5 is used [32]. Most researchers widely assumed ξ = 0.5. However, some studies question this practice. Mahmoudi et al. showed that a variation ξ influenced the ranking contrary to the other studies [33]. The distinguishing coefficient ranges ζ ∈ [0, 1]. The maximum value equals one and the minimum equals zero [34,35].
In this study, the PCA was used to determine the distinguishing coefficient. The PCA was analysed in Minitab software, and the results are shown in Fig. 4 and Table 5. According to the PCA results, the damage was found to have more weight, in this study it was assigned as 0.6, followed by effective strain, 0.3 and finally effective stress, 0.1.
From the values of the grey relational coefficient, the grey relational grade was determined according to Eq. 7. The results are shown in Table 6.

Signal to noise ratio analysis
Main effect plots for S/N are useful in determining the optimal parameters. Figure 5 shows the main effects plot of S/N ratios for maximum damage, maximum effective strain, maximum effective stress and grey relational grade. It was observed that when using the smaller is better quality characteristic for the damage, the factors are optimum at the following conditions; temperature-400℃, plunger speed-10 mm/s, friction coefficient-0.6 and strain per pass-0.6. The optimum values for all other responses are as recorded in Table 7. The temperature showed a decreasing increasing trend, implying that its influence on the outputs was not linear. A change in strain per pass would affect the output differently depending on the strains.

Analysis of variances
The analysis of variance was carried out as shown in Table 8 to determine the significance levels and the contribution of the factors. Additionally, the error contribution is included to ascertain if the results arise from residual errors or were they from the factors' contribution. The error contributions were 5.45%, 0.84%, 1.90% and 0.91% for damage, effective strain and stress and grey relational grade respectively. These error values indicate the percentage of the data that is due to error [36]. The error values were found to be very low to be considered. Twenty per cent should be the maximum value of the percentage error for good statistical data [37]. The strain per pass was the most significant factor that affected damage with a percentage contribution of 72.15% and a p value of 0. The outcome showed that damage would be highly affected by the slightest change in strain per pass. A 95% confidence interval was used in this study. The other factor affecting damage was the friction coefficient with a percentage contribution of 20.1% and a p value of 0.009. It also meant that friction between dies and the workpieces played an important in determining the deformation. The plunger speed and temperature had p values of 0.771 and 0.807, respectively and had an insignificant contribution to damage of 1.22% and 1.08% respectively. Therefore, strain per and friction coefficient should be carefully minimised to reduce damage. For maximum effective strain, it was observed that strain per pass was the most significant having a P-value of 0 and a percentage contribution of 87.68%. The friction coefficient had a percentage contribution of 7.95% and a P-value of 0. The other parameters had little contribution to the effective strain. They indicated that strain per pass and friction coefficient are essential in inducing effective strain with strain per pass having the dominant effect. For maximum effective stress, strain per pass was the only significant factor with a contribution of 88.93% and a P-value of 0. The contribution results of the strain per pass imply that the strain per pass should be significantly considered during deformation.
In the grey relational grade, ANOVA results showed that; the most significant parameters were; friction coefficient with a percentage contribution of 7.94% and a P-value of 0.001. The strain per pass had a percentage contribution of 88.93% and a P-value of 0. This implied that both friction coefficient and strain per pass highly affected the process's output. The temperature and the plunger speed least affect the MDF process.

Confirmatory simulation
The confirmatory simulations were carried out at the optimum results. The results obtained were recorded in Table 9. From the table, the least damage was 0.111, recorded when a simulation was carried out at S/N optimised result for damage. In comparison, the most excellent effective strain was 5.57, observed from optimised grey relational grade data. Figure 6 shows the contours of damage and effective strain for sliced samples at optimum conditions. The damage distribution results demonstrated that damage was more on the sample's surface than in the innermost parts of the material. The edges and corners experienced the most damage. The damage resulted from the increased strains due to the high loads and abrasive wear of the surfaces. The other factor that caused the damage was friction. The strain contour showed that along the edges, the strains increased. The central region experienced low strains. Although the multi-directional forging achieves a certain level of homogeneity, there is still a need to enhance the distribution of the stresses and the strain.

Discussion
In Fig. 5, the strain per pass has a very steep slope which shows that a slight change in the strain per pass highly affects the MDF process. Strain per pass above 0.5 influences the optimal results for damage, effective strain and stress. The high dependence of the output parameters to the strain per pass is attributed to its contribution to the evolution and refinement of grains in materials: the smaller the grains' size, the greater the effective strain and stress. However, a rapid formation of the new small grains results in misorientation causing microstructural damages [38]. During the MDF, deformation bands and micro shear bands are formed in different directions, intersecting each other, resulting in continuous grain fragmentation. Further deformations increase the fragmentations and misorientation at the borders, and nearly refined material with equiaxed grains are produced at high strains [39]. As uniaxial compression is applied to the material continuously the misorientation at the boundaries intensifies and the spacing between the grains decreases. The accumulation of strains causes microstructural non-uniformities to evolve gradually during each compression pass [9].
When the MDF was conducted for strain per pass above 0.3, the effective stress increased rapidly until the maximum value of 665 Mpa was attained, as shown in Table 4. The maximum effective stress indicates the dislocation density increased and the ultrafine grains were formed [40]. However, when using strain per pass of 0.3, the temperatures must be high and the friction coefficient above 0.4 to get the effective stress above 660 MPa. Temperature highly influences the product's final quality [9,41,42]. Zhao et al. reported that when an aluminium alloy is heated, the grain structure does not change until the temperature gets to about 350 °C. At this temperature, the grains at the boundaries start to form new grains rapidly until a new undistorted grain structure is developed, replacing the old one, a process called recrystallization. The recrystallized aluminium develops different mechanical properties from the former; for example, hardness for such materials decrease, resulting in more damage. Additionally, the tensile strength decreases with an increase in temperature while the ductility and the effective strain increase [43].
The effect of the die speed and friction coefficient can be observed in Table 3. An increase in the plunger speed resulted in more damage. This can be attributed to the reduced cooling during the process, resulting in increased internal temperature and increased internal stress in the workpiece [44]. Considering the friction coefficient, an increase in friction coefficient resulted in increased damage but better effective strain. The friction coefficient is the ratio of the resisting force to the normal force. Temperature and deformation speed are other factors that affect the friction at the interface during a forging process [45]. The MDF utilizes compressive forces from the high loads and speeds, resulting in abrasive wear. When surfaces are excessively abrased, failure initiation is likely to start at the worn-out points. However, friction during the forging process improves the strain and stress distribution and the material flow, thus determining the product quality [46].
In Fig. 6a, b, an inhomogeneous distribution of damage and the effective strain was observed; this was due to the inhomogeneous distribution of nanoparticles [47]. The inhomogeneities of particles and material properties result from the inhomogeneous distribution of strain in the sample during compression. Previous studies indicated that strains and temperature immensely affected the material's structure homogeneity [48,49]. However, the change of axis of deformation during MDF and higher strain per pass led to the development of uniform distribution of strain, stress and damage [50,51], observed in Fig. 6b, c.

Conclusion
In this study, the multi-directional forging of AA 7075 was studied. The principal objective was to optimise the MDF process based on the significant process parameters. The DEFORM™ 3D was used for simulation. The effects of varying the process input parameters (plunger speed, temperature, friction coefficient and strain per pass) on maximum damage, maximum effective stress and maximum effective strain were studied. The signal-to-noise ratio identified the optimum conditions for each response to be as follows; damage, the optimum temperature was 25 °C, the plunger speed, 20 mm/s, the friction coefficient, 0.2 and the strain per pass, 0.2. These conditions resulted in the least damage of 0.111. For maximum effective strain, the optimal values identified were; temperature, 400 °C, plunger speed, 5 mm/s, friction coefficient, 0.6 and the strain per pass at a point of 0.6. Finally, for maximum stress-effective, the values were; temperature, 200 °C, plunger speed, 10 mm/s, the friction coefficient, 0.5 and the strain per pass, 0.5. From the ANOVA, it was noted that the strain per pass is the most crucial input that affects damage, effective strain and stress. Optimisation using the grey relation analysis The current study was limited to simulation. However, this study brings a better understanding of the MDF process and the optimisation of the process parameters when processing AA 7075. The results will be significant in guiding the choice of MDF parameters not only for AA 7075 but also for other metal alloys. The results will go a long way in assisting manufacturers and researchers in developing AA 7075 with better mechanical properties. The AA 7075 with improved mechanical properties can be applied in the automobile, aerospace and construction industries. This research, therefore, contributes immense knowledge to the quality production of high-strength AA 7075.
In future work, further optimisation results of AA 7075 can be obtained if the studies focus on analyzing the alloy's chemical variation, its spatial variations of the stresses and strains in different metal atoms, grain structure, and boundaries dislocation distribution. Further research should also be carried out on other hard-to-machine metals.

Conflict of interest
The authors declare that they have no conflict of interest.
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