Mathematical Modelling of a Friction Stir Welding Process to Predict the Joint Strength of Two Dissimilar Aluminium Alloys Using Experimental Data and Genetic Programming

Friction stir welding (FSW) is the most popular and efficient method of solid-state joining for similar as well as dissimilar metals and alloys. It is mostly used in applications for aerospace, rail, automotive, and marine industries. Many researchers are currently working with different perspectives on this FSW process for various combinations of materials.The general input process parameters are the thickness of the plate, axial load, rotational speed, welding speed, and tilt angle. The output parameters are joint hardness, % of elongation, and impact and yield strengths. Genetic programming (GP) is a relatively new method of evolutionary computing with the principal advantage of this approach being to evaluate efficacious predictive mathematical models or equations without any prior assumption regarding the possible form of the functional relationship.This paper both defines and illustrates how GP can be applied to the FSW process to derive precise relationships between the output and input parameters in order to obtain a generalized prediction model. A GP model will assist engineers in quantifying the performance of FSW, and the results from this study can then be utilized to estimate future requirements based on the historical data to provide a robust solution. The obtained results from the GP models showed good agreement with experimental and target data at an average prediction error of 0.72%.


Introduction
Some metals such as aluminium and its alloys are known as nonweldable materials using traditional methods of welding and are unable to provide enough strength due to porosity in the fusion zone.However, recent improvements in welding methodology and researchers' studies have led to the development of a new welding technique known as friction stir welding (FSW) [1,2].In 1991, the Welding Institute, UK, invented a solid-state joining technique that was initially applied to aluminium alloys.The FSW basic concept is exceptionally plain in which a nondevourable solid-state heat-treated hard metal tool (with a pin and shoulder) is introduced into the butting ends of sheets or plates to be joined and moved along the line of joint at specific rotational speed, traverse speed, axial force, and tilt angle.This process of welding produces good-quality welding [3].As the rotating tool is in contact with joining materials, it heats up the joint, which further leads to plastic deformation of the joint.This process comes under the solid-state welding category as there is no weld pool formation at the joint; rather a rotary motion of the tool transfers the material to produce the joint.The different tool geometries lead to the development of various crystallization structures of grains, which finally results in a different strength of weld joint according to the type of geometry [4,5].
1.1.Literature Review.The joining of heat-treated aluminium with non-heat-treated aluminium alloys requires using rotational tool speed, traverse speed, and tilt angle as input and hardness, tensile strength, and yield strength as output parameters.The output parameters for the dissimilar metals are difficult to achieve due to distinctive heat coefficients, the base metal chemical composition, and the properties of heat-treated aluminium and non-heat-treated aluminium alloy [6].The study of the optimized effect of the welding parameters on 5 mm thick AA6082 aluminium plates using ANOVA technique is based on L8 orthogonal array showing that the tool's rotational, welding, and plunge speed are the significant parameters in deciding the strengths and percentage elongation [7,8].
FSW is a solid-state joining process, which is suitable for joining aluminium-copper alloy compared to fusion welding processes.The mathematical model with process parameters and tool geometry to predict the responses such as yield strength, tensile strength, and ductility of friction stir welds of the AA2014-T6 aluminium alloy are formulated [6] with empirical relations.Optimization of the process parameters to achieve high tensile strength to find the effect of the process parameters of the FSW joint for similar aluminium alloys H30 helped in developing a practical approach of determining the optimum conditions leading to higher tensile strength [9].
The significant optimal transverse feed is achieved with the help of a square tool while finding the effect of welding parameters on mechanical properties such as the hardness and tensile strengths of the axial type of weld zone [10].ANOVA and Taguchi method were found to be very helpful in the case of the AA6063 alloy for the submerged status of FSW to obtain the optimal welding conditions for maximum hardness strength with welding process parameters such as rotational speed, welding speed, and tool pin profiles (cylindrical, threaded, and tapered) [4,[11][12][13][14].
The tool geometry parameters (shoulder and pin diameter, pin and shoulder shape, and length) played a crucial role in heat generation and material flow in finding the influence of welding parameters on mechanical strengths of similar or dissimilar aluminium alloys [15,16].Tool rotational speed was the most influential factor during the optimal process parameters of FSW of dissimilar Al alloys (AA6262 and AA7075) using grey relational analysis of multiple response characteristics such as tensile strength (UTS) and Vickers hardness (VHN) [17].
A response surface methodology (RSM) model was developed to investigate the effects of tool shoulder and probe geometries on the Al FSW with respect to weld strength, cross-sectional area, and grain size to define what was appropriate for the FSW characteristics [5].Data variance analysis showed that the rotational speed and diameter of a tool are the major and the minor significant factors, respectively.Similarly, this is seen in studying the effect of welding factors on the microstructure and mechanical strength of heterogeneous tailored welded blank sheets (5083-H12 and 6061-T6 Al alloys) [18].Optimization of FSW parameters was established in different conditions of base material and the microstructures of the welded condition compared with the postweld heat-treated microstructures welded in annealed and T6 condition [19].Work was also extended on the FSW using aluminium alloys and the influence of tool geometry on microstructure, defect type, and mechanical properties of a joint.For a higher shoulder and pin diameter, the resultant defects were least compared to smaller dimensions of them [20].The proper adjustment of welding parameters is presented in such a way that the welding interface is heated to a plastic state at the place of welding.The functional behavior of the weldments is determined by the tensile strength, metallurgical behavior, surface roughness, weld hardness, and microhardness.Various works on the welding parameters and tool pin profile in defining weld quality under the influence of rotational and travel speed on friction stir processed zone formation in AA 5083 aluminium alloy plates are carried out.The welded samples were examined by using optical microscopy to define the different joined zones and to identify possible defects [21].
1.2.Genetic Programming.Genetic programming (GP) is a computerized process of building programs closely based on the Darwinian principle of natural selection to solve specific problems carefully and is considered as an expansion of genetic/evolutionary algorithms which uses them effectively.GP evolves a program that forecasts the actual output from an experimental data file of inputs and outputs [22,23].The GP Discipulus software uses Java, C/C ++ , and assembly interpreter [24] to write a program that maps inputs onto output data.In general, GP primarily generates considerable numbers of random computer programs, where each program is executed and rated according to a fitness metric defined by the developer.The best problem-solver program is selected in each generation and breeds it [25].Spontaneously, if two computer programs are even slightly useful in solving a problem, then some of their parts probably have some merit.By combining temporary chosen merit parts of effective programs, we may produce new computer programs that may be even more well fitted to solving the problem.After the application of genetic methods to the developed current population (the group of programs), the population of the new generation programs supersedes the old ones.Each in the population of computer programs is then measured for fitness, and the process is repeated over many generations.This process will produce populations of computer programs which, over many generations, tend to exhibit increasing average competence in dealing with their environment.
The best individual that appeared in any generation of a run (the best-so-far individual) is designated as the result produced by genetic programming.The grading level of the computer programs that are produced is the essential feature of GP.The results of GP are thus inherently hierarchical.
The Discipulus software uses Automatic Induction of Machine Code technology to deal with the speed problem of machine learning, which allows the user to conduct a greater number of runs to investigate relationships between inputs and output and to uncover bad data or outliers, assess time, and so on [26].The generated models are intended to develop process prediction mathematical models.Hence, the GP method is selected for the present work.GP solutions are computer-based programs that are easy to inspect, document, evaluate, and test.The GP solutions help in understanding the type of the derived correlation between input and output data, which was previously unknown.GP evolves both the structure and the constants of the solution simultaneously [27].
Discipulus GP actively discriminates between relevant input data and inputs that have no bearing on a solution [28].In other words, Discipulus performs input variable selection as a by-product of its learning algorithm.GP is successfully used to solve problems in a vast number of fields such as systems, data, time series prediction and economic modelling, curve fitting, symbolic regression, industrial process control, financial trading, optimization and scheduling, and bioinformatics [29].
There are numerous techniques available, such as linear regression, statistical and artificial neural network, fuzzy logic ANFIS, and response surface methodology, to derive correlations but they have shown limited accuracy.This study aims at deriving a mathematical model showing empirical correlations between inputs and output for predicting the mechanical properties of joint strength of FSW using a genetic programming approach that can be used in forecasting to an accuracy level of 99.2 to 99.6%.This new model would be more straightforward by eliminating the large computations involved in any conventional equation of state applications.It would likewise be very helpful in developing and examining the developed correlations without conducting experiments for the validation of data in the future [30].

Materials and Methods
In this study, two dissimilar aluminium alloys AA7075 and AA6061 having different thicknesses of 3, 4, and 5 mm whose chemical compositions are listed in Table 1 were examined.Butt joint welding configuration using a vertical milling machine with a special purpose tool (cylindrical taper profile with pin diameter of 6 mm, 10 ∘ taper, pin length of 1.8 mm, and shoulder diameter of 20 mm) is carried out.To derive a mathematical model, we require an enormous amount of data from experimentation for which various process parameters at different levels having more influence on the mechanical properties of the joint formed by FSW are derived from the literature.They are axial load, the rotational speed of the tool, welding speed, tilt angle, and plate thickness and their levels are shown in Table 2. Butt welded samples are prepared with a rig consisting of MS backing plate to support base metal along the rolling direction.The various mechanical properties of AA6061 and AA7075, namely, hardness, elongation, and tensile and impact strengths, are noted in Table 3. Welding is carried out by placing plates on the advancing side and retreating side alternately and clamping these firmly to prevent the butt faces from being forced apart from proper stirring and mixing of material and finishing by rotation and translation movement of the tool.The tilt of the tool towards a trailing direction ensured that the shoulder of the tool held the stirred material by a threaded pin and moved material efficiently from the front to the back of the pin.The amount of penetration of a pin depth and tool shoulder radius contacting the workpiece is decided by the pin length.Pin length criteria are helpful in producing welds with the inner channel, surface groove, excessive flash, or local thinning of the weld plates and so on.
Each butt joint was cut to the pieces of specimens for yield and impact strength tests based on the ASTM standard.A universal testing machine was used to conduct the tensile tests and their loading and elongation record of specimens was also taken to calculate tensile strength and elongation rate on the bases of their fracture loading of specimens.The Charpy type of impact test is used to evaluate the impact energy of FSW joint as per ASTM standards.

Genetic Models
GP is the most common approach among evolutionary computation methods in which the programs adapt hierarchically classified programs obtained dynamically by changing the size and form evolved during simulation.The random program was developed using available function and terminal genes of symbolic regression from set  (arithmetic operators {+, −, * and /}) and set , respectively, producing an initial population to adapt to the environment of the solution.The program structures keep being modified subject to adaptability based on guidelines using fitness measurement function.Fitness measurement shows how far the predicted data by the GP coincides with the experimental data.Genetic modification of program contents starts, and it follows that when the calculation of fitness measurement is applied to an initial population and becomes repetitive, it is called evolution.Once termination criteria are met, the evolution process stops to get the number of generations of the solution.Data sets collected from the various measurements of mechanical properties of FSW joint strength were taken for this analysis.The tensile strength, impact strength, and elongation are our targeted output.Axial load, welding speed, tool rotational speed, tilt angle, and thickness of plate are taken as inputs.The data samples were randomized manually using Notitia software.Feeding the randomized data set into the software is initially done by splitting them into three categories, namely, training, validation, and applied testing [2,24].Discipulus always accepts the last column as the proposed output.Many experimental runs help generate the optimal solution in finding the best parameters within the least time using arbitrary settings as shown in Table 3.Another term called the correlation coefficient (/) plays an important role, where the term  represents the linear relationship coefficient between two variables using strength and the path. varies in −1 <  < +1, where (+) and (−) signs indicate positive and negative linear relationships, respectively.When  = +1, the inputs and output have a perfect positive correlation; it can be seen that as input values increase, the output values also increase and vice versa when  = −1 (a perfect negative fit).If  = 0, this indicates no linear relationship and if  approaches zero, it represents a random, nonlinear relation between the two variables. 2 gives the coefficient of determination to predict from a particular model.0 <  2 < 1 denotes the strength of the linear correlation inputs and outputs or represents the percent of the data closest to the line of best fit.In the crossover rate, homologous crossover acts as a modification to imitate natural evolution more precisely than the conventional crossover where the two new emerging programs are set to form a subsequent and most suitable program by switching groups of neighbouring instruction.Blocks between the two emerging programs to deliver the upcoming group are such that they are of the same length and above the position from both emerging programs.Nonhomologous crossover occurs when instruction blocks are swapped between two evolving programs with no reference to the size and location of the two

Termination
Individual program emerges when the sum of absolute errors is zero (is less than specified) (a) the required number of runs is accomplished (10000) or (b) the required correlation coefficient ( 2 ) is obtained (when  2 is less than 0.1) Terminal set  = {, random-constants} sets of instruction blocks.The homologous crossover parameter sets the percentage of crossover events that are "homologous" as opposed to the percentage that is "nonhomologous."

Results and Discussion
The evolutionary method using precise instructions and terminal genes from a set of function library and termination library types builds an organism (known as a mathematical model) fit for the estimation of results behaving similarly to the nature of computer programs that differ in form and size [24,26].Various models for outputs are developed by GP using the training data [29].The best mathematical models obtained from GP simulation are given from (1) to (5).
Comparison between the experimental and predicted outputs using a mathematical model from GP is shown in Tables 4, 5, and 6 for elongation and tensile and impact strengths of a FSW joint, respectively.Errors in determining predicted outputs are negligible, and the percentage of error is less than ±0.6, which shows that results obtained from a GP mathematical model are highly acceptable.
Figures 1(a), 2(a), and 3(a) show the percentage deviation between the best model regarding individual generation and experimental results during regression fit using the set of function genes with a normal distribution behavior.It is evident that in early generations the best models are not as precise as the models generated subsequently.The GP model analysis suggests specific outputs of mechanical properties of FSW joint for the experimental scheme under various input factors that can be associated with the performance.The GP models generated allow representation of the experimental data without thorough awareness of the occurrence.Besides, their study allows us to obtain a deeper insight into the relevant input conditions in describing the output phenomenon by showing changes in any of the inputs observed in properties of the FSW joint produced which are linked to the changes in the performance.
Figures 1(b), 2(b), and 3(b) show the normal Q-Q plot for tensile strength, elongation, and impact strength with a 95% level of confidence.The blue circles in this Q-Q plot start out on one side of the line and then are almost entirely on the other side for a long stretch and then move to the other side of the reference line again.This behavior indicates some degree of skewing.In Figure 1(a), the mean and standard deviation of the normal Q-Q distribution plot of tensile strength for the error variation between the experimental and predicted values was −0 ± 0.4 MPa.In Figure 2(b), the mean and standard deviation of the normal Q-Q distribution plot of elongation for the error variation between the experimental and predicted values was −0±0.2%.However, in Figure 3(b), the mean and standard deviation of the normal Q-Q distribution plot of impact strength for the error variation between the experimental and predicted values was −08 − 4 ± 0 J/m 2 .

Linear Regression Analysis.
In order to compare the GP results with the other types of regression analysis, an attempt has been made to perform the linear regression analysis of experimental results.The linear regression analysis of elongation and tensile and impact strengths of a FSW joint involving V0 to V4 input variables has been carried out using Minitab5 V18 statistical software with the experimental data of Tables 4, 5, and 6.The corresponding linear regression models are presented in (7), (8), and (9) and error obtained in comparison with experimental data is also tabulated in Tables 4, 5, and 6.Various residual plots of outputs are shown in Figures 4, 5, and 6.The amount of error is very high during building up of empirical relation with elongation and impact strength compared to tensile strength as tensile strength has higher experimental values to 3 digit numbers.Linear regression requires higher digit number for output values in comparison with input parameters (V0-V4) to achieve higher level of accuracy.It is more suitable for two inputs and more number of inputs leads to erroneous empirical relation derived.
Figure 7 shows that the boxplots of various outputs of FSW joint give a clear indication of no error or error at a fourth decimal place, which is negligible.There are two boxes representing statistical data on a plot consisting of a rectangle with a vertical line inside to indicate the median value.The quartile is defined as the middle number between the smallest number, highest value, and the median of the data set and median of data.The higher and lower quartiles are shown by horizontal lines on either side of the rectangle with the box on the left for experimental and the box on the right for GP.From the boxplots, it is seen that there is absolutely no variation between both boxes of experimental and GP results.Therefore, GP results are capable of producing results without experimentation at a higher accuracy level.Thus, boxplots are very effective in comparing two or more results of different methods.By comparing the errors of experimental data with GP and linear regression, it is seen that the equations derived by GP gave higher accuracy as regards results with least or no error, whereas the linear regression models generate output with higher error value, which may lead to an erroneous or wrong prediction of outputs.Linear regression also increases the uncertainty in the models, where the precision has to be maintained but GP outputs are very helpful.

Conclusion
Mechanical properties are the critical measurements of technological quality of a FSW joint.They exert considerable influence over both production costs and productivity.The proposed work adopts a potential and novel approach using GP for such significant measurements in deriving empirical modelling of mechanical properties.The proposed GP    approach does not depend on any standard mathematical decree or any past awareness of the type of solution.GP employs evolving theories to develop automatically mathematical models that fit the given experimental historical data without making any assumptions about the shape, size, and complexity of the problem under any number of input parameters.In this study, new models of the elongation and tensile and impact strength at different input conditions are formulated using Discipulus GP software and C programming.Without further conducting any experimental runs, joint properties are determined quickly by substitution of input conditions.The accuracy of new GP models in comparison to the test results is greater by about ±0.06 to 0.001.New GP models are used as an alternative method to estimate the joint properties where the experimental results or correlations are not available.However, the precision of solutions achieved by GP depends on allied evolutionary inputs, the amount of test data, and their level of accuracy.GP are time-consuming processes since they involve a reasonably large number of iterations but remain worth adopting as a method by considering the prediction accuracy of the model developed.The capability of the genetic models for predicting the responses of a process is advantageous as they have a higher degree of accuracy compared to the performance of other analytical methods such as ANN, DFA, ANFIS, and RSM.The proposed mathematical model verifies the test results to the reliability of about 99.4 to 99.6%.The GP approach, therefore, proved itself to be a highly expert and valuable means for recognizing correlations in data.

Future Scope of the Work
The work can be extended further by applying the GP on FSW employing other aluminium alloys such as 5XXX and 7XXX used in the automotive industry to increase its use in the automotive industry.Further implementation of GP on the forces generated during single as well as multiple passes for different alloys, working conditions, and process parameters would possibly be very beneficial.GP could also be implemented in fine-grained FSW aluminium alloys that exhibit improved strength and ductility.
Also, further GP studies may be conducted, considering most of the welding parameters, on a wider range of values.Fatigue analysis and shear tests can be handled.Higher thickness aluminium plates can be welded by employing double-sided FSW.Finally, GP application of FSW is extended by employing on other materials such as copper, titanium, and magnesium for future studies.

Figure 1 :
Figure 1: (a) Comparison between experimental and predicted values of tensile strength and (b) normal Q-Q plot of error between experimental and GP data.

Figure 2 :
Figure 2: (a) Comparison between experimental and predicted values of elongation and (b) normal Q-Q plot of error between experimental and GP data.

Figure 3 :
Figure 3: (a) Comparison between experimental and predicted values of impact strength and (b) normal Q-Q plot of error between experimental and GP data.

Figure 4 :
Figure 4: Residual plots of tensile strength using linear regression method.

Figure 5 :
Figure 5: Residual plots of elongation using linear regression method.

Figure 6 :
Figure 6: Residual plots of impact strength using linear regression method.

Figure 7 :
Figure 7: The quality distribution of GP models and experimental work of different outputs of FSW joint.

Table 2 :
Important input factors and their levels.

Table 3 :
Parameter setting for genetic programming.

Table 4 :
Comparison between experimental and predicted values of tensile strength at R 2 = 99.5%.

Table 5 :
Comparison between experimental and predicted values of % elongation at R

Table 6 :
Comparison between experimental and predicted values of impact strength at R 2 = 99.4.