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

Authors: Mohammed Yunus; Mohammad S. Alsoufi
Journal: Modelling and Simulation in Engineering
 (2018)

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## Abstract

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%.


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## Body

## 1. 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, 11–14].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].


## 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, 11–14].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].


## 2. 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 Table1 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.Table 1
Chemical composition of AA6061 and AA7075.
Elements Mg Mn Fe Si Cu Cr Zn Ti Al Base metal (AA6061) 0.8–1.2 0.0–0.15 0.0–0.70 0.4–0.8 0.15–0.40 0.05–0.35 0.0–0.25 0.15 Remaining Base metal (AA7075) 2.9 0.3 0.5 0.4 2 0.28 6.1 0.2 RemainingTable 2
Important input factors and their levels.
Process parameters Symbol Unit Level 1 Level 2 Level 3 Tilt angle v 0 (°) 3 4 -- Axial load v 1 (KN) 2 2.5 3 Tool rotational speed v 2 (rpm) 600 900 1200 Welding speed v 3 (mm/sec) 70 90 115 Thickness of plate v 4 (mm) 3 4 5Table 3
Parameter setting for genetic programming.
Parameters Value assigned  Population size (P) 600  Number of generations 1000  Maximum depth of tree 6  Maximum generation 60  Functional set Multiply (×), plus (+), minus (−), divide (/)  Terminal set Tilt angle, axial load, tool rotational speed, welding speed, thickness of plate, tensile strength, elongation, and impact strength.  Number of runs 100  Mutation rate 15%  Crossover rate (i) 0.80 for nonhomologous(ii) 0.20 for homologous  Reproduction rate 0.05  Fitnessr2 ∑ k = 1 n e r r o r = P r o g r a m   d a t a - o b s e r v e d   d a t a 2 n  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 (r2) is obtained (when r2 is less than 0.1)  Terminal set T = { P , r a n d o m - c o n s t a n t s }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.


## 3. 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 setF (arithmetic operators {+,-,∗and/}) and set T, 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 (r/R) plays an important role, where the term r represents the linear relationship coefficient between two variables using strength and the path. r varies in -1<r<+1, where (+) and (−) signs indicate positive and negative linear relationships, respectively. When r=+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 r=-1 (a perfect negative fit). If r=0, this indicates no linear relationship and if r approaches zero, it represents a random, nonlinear relation between the two variables. r2 gives the coefficient of determination to predict from a particular model. 0<r2<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 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.”


## 4. 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). (1)Tensile  strength=381.45I2-1-G+cos⁡v2,(2)A=-1.51-20.011v2+2.116v0-1.51+0.011v2+2.116v01.51v2-1.6412,B=1.51+0.011v2+2.116v01.51v2-1.64121.51v2-1.641,C=21.51v2-1.641+7.6v2B+AB2,D=-15.2v2B+AB2+1.51v2-1.641,E=D+C-v1/1.51/v2-1.6411.253+3v1/1.51/v2-1.641-2D-2√C,F=1.51v2-1.641-1.253+3∗v11.51/v2-1.641-2∗D-2√C2,G=1.253+3v1/1.51/v2-1.641-2D-2C21.51/v2-1.641-1.253+3v11.51/v2–1.641-2D-2C2,H=FG+G2-25.04+2v025.04+2v0-v32,I=Hv0-3.153H/v0-3.153–v0-3.153/v2v0-3.153v1v2,J=√H/v0-3.153-v0-3.153v2v0-3.153v1v2E-2I2+v1,K=1.383I-J2v32.5085J4v3-1.54997I-J+G-1+0.8077/GH6+0.8077/GHJ,(3)elongation=0.1-KG-1+0.8077/GH6+0.8077/GH/J22-1.953K+v2/I-J+I-J22.22v2-1.953K/I-J+I-J+1.54v0+2.4,(4)A=2∗-1.1+v0v1-0.964+1.1,B=-v2+1.1v3∗-1.1+v0v1,C=-v2+1.1v3∗-1.1+v0v1--1.1+v0v1,D=-1.1+B+0.26∗B1.1+B-1.38∗-0.26∗v12∗B1.1+B∗C42,E=5.52∗-0.26∗v12∗B1.1+B∗C422,F=C--0.26∗v12∗B1.1+B∗C42-5∗E,G=2∗1.1+-0.1∗2.64-v1+F-v2∗F2+D-E/FFv31,H=D-E+2∗-6.6∗v3∗G∗D-E2F∗D-E2,I=H+v4∗-9.33∗v32∗G∗D-E2F∗D-EF10.51F∗v3,J=1.26-0.04∗1.43∗I/F–H∗v2/F+v3/v1v3,(5)impact  strength=F∗H∗Jv1,(6)A=-cos⁡v3+cos⁡381.45cos⁡v3+v3+v3,B=-cos⁡381.45cos⁡v3+v3+v3v2A2,C=381.454∗-1.175+v1v0+v3,D=A-cos⁡B-1.365-cos⁡2-v2cos⁡381.45C-v3,E=cos⁡-2.02D+0.742D+0.743v3-383.45v3-C+0.74-v2,F=sin⁡sin⁡cos⁡E-v2+v2+1383.45v3-C+1.175+v1,G=2D+0.74-cos⁡E-v2cos⁡381.45381.454F+v30.25,H=21-G+v02-G-v4383.45v3-C-381.454F+v3,I=381.458cos⁡v22-12+0.7+v1-G-cos⁡v2+v3+G-cos⁡v2-v3.Comparison between the experimental and predicted outputs using a mathematical model from GP is shown in Tables4, 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.Table 4
Comparison between experimental and predicted values of tensile strength atR2= 99.5%.
V0(°) V1(KN) V2(rpm) V3(mm/s) V4(mm) Tensile strength (MPa) Error Linear regression Error Experimental GP 3 2 600 75 4 201.4173 201.4149 0.002396 199.4963 1.92105 3 2 600 90 4 207.5238 206.971 0.552811 199.6827 7.8411 3 2 900 115 4 200.4701 200.8218 −0.35167 204.3825 −3.91235 3 2 1200 115 4 210.9491 211.2845 −0.33536 208.7715 2.17765 3 2.5 600 75 4 212.8087 212.9875 −0.17885 204.0413 8.76745 3 2.5 600 90 4 214.7545 214.4605 0.294006 204.2277 10.5268 3 2.5 900 90 4 226.1082 226.7305 −0.62225 208.6167 17.4915 3 2.5 1200 115 4 221.0869 221.4523 −0.36539 213.3165 7.77045 3 3 600 75 4 208.9469 208.806 0.140945 208.5863 0.36065 3 3 600 115 4 217.343 217.9036 −0.56067 209.0835 8.25955 3 3 900 75 4 229.8519 229.2028 0.649048 212.9753 16.87665 3 3 900 90 4 233.5722 233.628 −0.0558 213.1617 20.4105 3 3 900 90 5 233.9161 233.6219 0.29422 213.2857 20.6304 3 3 900 115 4 242.6929 242.4103 0.282562 213.4725 29.22045 3 3 1200 75 3 227.4726 227.1649 0.307632 217.2403 10.23235 3 3 1200 75 4 227.1322 227.1745 −0.04233 217.3643 9.76795 3 3 1200 75 5 226.8476 227.184 −0.33641 217.4883 9.35935 3 3 1200 115 3 227.9172 227.7119 0.205277 217.7375 10.17975 3 3 1200 115 4 228.1085 227.7122 0.396347 217.8615 10.24705 3 3 1200 115 5 228.2549 227.7125 0.54248 217.9855 10.26945 4 2 600 75 4 197.3443 196.8444 0.499908 198.4663 −1.12195 4 2 600 75 5 197.1878 196.8481 0.339767 198.5903 −1.40245 4 2 600 90 3 212.6613 212.6237 0.037552 198.5287 14.1326 4 2 600 90 4 213.2592 212.7725 0.486725 198.6527 14.6065 4 2 900 75 3 213.7823 213.4791 0.303131 202.7313 11.05105 4 2 900 75 4 213.8493 214.1454 −0.29617 202.8553 10.99405 4 2 900 75 5 214.0453 214.1446 −0.09932 202.9793 11.06605 4 2 900 90 4 227.3222 227.8775 −0.55531 203.0417 24.2805 4 2 900 90 5 227.7843 227.8655 −0.08118 203.1657 24.6186 4 2 900 115 3 226.5605 227.1803 −0.61986 203.2285 23.33205 4 2 900 115 4 226.9803 227.1879 −0.20764 203.3525 23.62785 4 2 900 115 5 227.3916 227.191 0.200684 203.4765 23.91515 4 2 1200 90 3 216.5077 217.1303 −0.62257 207.3067 9.201 4 2 1200 90 4 217.0157 217.0639 −0.04823 207.4307 9.585 4 2 1200 90 5 217.5187 217.2333 0.2854 207.5547 9.964 4 2 1200 115 3 215.5096 215.5605 −0.05092 207.6175 7.89215 4 2 1200 115 4 215.2814 215.5672 −0.28584 207.7415 7.53995 4 2 1200 115 5 215.202 215.5741 −0.37207 207.8655 7.33655 4 2.5 600 75 3 210.9097 211.1155 −0.20586 202.8873 8.02245 4 2.5 600 75 4 210.9576 210.9654 −0.00786 203.0113 7.94635 4 2.5 600 75 5 210.8665 210.847 0.019516 203.1353 7.73125 4 2.5 600 90 4 205.5315 205.905 −0.37355 203.1977 2.3338 4 2.5 600 90 5 206.0258 206.0515 −0.02574 203.3217 2.7041 4 2.5 600 115 3 209.6681 209.8436 −0.17548 203.3845 6.28365 4 2.5 900 75 4 225.0363 224.4874 0.548965 207.4003 17.63605 4 2.5 900 90 4 220.4077 220.5082 −0.10057 207.5867 12.821 4 2.5 900 115 4 227.377 227.1711 0.205856 207.8975 19.47955 4 2.5 1200 75 4 216.1377 216.0029 0.134796 211.7893 4.34845 4 2.5 1200 90 4 215.9696 215.655 0.314621 211.9757 3.9939 4 2.5 1200 115 4 220.2623 219.8801 0.382263 212.2865 7.97585 4 3 600 75 4 204.118 204.1865 −0.06847 207.5563 −3.43825 4 3 600 115 4 217.8454 218.1242 −0.27884 208.0535 9.79195 4 3 900 115 4 238.2042 238.2691 −0.06488 212.4425 25.76175 4 3 1200 75 4 218.2925 218.8536 −0.56111 216.3343 1.95825 3 2 1200 115 5 211.2005 211.2984 −0.09787 208.8955 2.30505 3 2.5 600 75 5 213.5635 213.1032 0.460327 204.1653 9.39825 3 2.5 900 75 5 224.5181 225.1232 −0.60516 208.5543 15.96385 3 2.5 1200 115 5 220.9586 221.4597 −0.50114 213.4405 7.51815 3 3 600 75 5 209.4358 208.8087 0.627182 208.7103 0.72555 3 3 900 75 5 230.5879 230.988 −0.4001 213.0993 17.48865 3 3 900 90 5 233.9161 233.6219 0.29422 213.2857 20.6304 3 3 1200 75 5 226.8476 227.184 −0.33641 217.4883 9.35935 3 3 1200 115 5 228.2549 227.7125 0.54248 217.9855 10.26945 4 2 600 75 5 197.1878 196.8481 0.339767 198.5903 −1.40245 4 2 900 75 5 214.0453 214.1446 −0.09932 202.9793 11.06605 4 2 900 90 5 227.7843 227.8655 −0.08118 203.1657 24.6186 4 2 900 115 5 227.3916 227.191 0.200684 203.4765 23.91515 4 2 1200 90 5 217.5187 217.2333 0.2854 207.5547 9.964 4 2 1200 115 5 215.202 215.5741 −0.37207 207.8655 7.33655 4 2.5 600 75 5 210.8665 210.847 0.019516 203.1353 7.73125 4 2.5 600 90 5 206.0258 206.0515 −0.02574 203.3217 2.7041 4 2.5 900 75 5 225.7942 225.9323 −0.13811 207.5243 18.26995 4 2.5 900 90 5 220.4536 220.4198 0.033768 207.7107 12.7429 4 2.5 900 115 5 228.0017 228.3694 −0.36765 208.0215 19.98025 4 2.5 1200 75 5 216.5734 216.9923 −0.41885 211.9133 4.66015 4 2.5 1200 90 5 216.3825 215.9084 0.474014 212.0997 4.2828 4 2.5 1200 115 5 220.4091 219.8816 0.527542 212.4105 7.99865 4 3 600 75 5 204.4071 204.1895 0.217575 207.6803 −3.27315 4 3 600 115 5 217.7037 218.1192 −0.4155 208.1775 9.52625 4 3 900 90 5 226.0705 226.6372 −0.56667 212.2557 13.8148 4 3 1200 75 5 219.0428 219.1528 −0.11002 216.4583 2.58455 4 3 1200 90 5 219.528 219.6197 −0.09166 216.6447 2.8833Table 5
Comparison between experimental and predicted values of % elongation atR2= 99.3.
V0(°) V1(KN) V2(rpm) V3(mm/s) V4(mm) Elongation (%) Error Linear regression Error Experimental GP 3 2 600 75 4 10.6 10.58972 −0.01028 13.42925 −2.82925 3 2 600 90 4 9.8 9.935991 0.135991 13.4663 −3.6663 3 2 600 115 4 12.3 12.31535 0.015351 13.52805 −1.22805 3 2 900 90 4 8.2 8.212519 0.012519 15.4433 −7.2433 3 2 900 115 4 13 12.61499 −0.38501 15.50505 −2.50505 3 2 1200 90 4 7.2 7.499517 0.299517 17.4203 −10.2203 3 2.5 600 75 4 7.99 7.825613 −0.16439 13.73675 −5.74675 3 2.5 600 90 4 8.8 8.686644 −0.11336 13.7738 −4.9738 3 2.5 600 115 4 8.4 8.531275 0.131275 13.83555 −5.43555 3 2.5 900 75 4 6.8 6.794509 −0.00549 15.71375 −8.91375 3 2.5 900 90 4 7.9 7.832968 −0.06703 15.7508 −7.8508 3 2.5 900 115 4 8.1 7.955772 −0.14423 15.81255 −7.71255 3 2.5 1200 75 4 7.6 7.602399 0.002399 17.69075 −10.0908 3 2.5 1200 90 4 7.5 7.640862 0.140862 17.7278 −10.2278 3 2.5 1200 115 4 8.1 8.265376 0.165376 17.78955 −9.68955 3 3 600 75 4 9 9.003488 0.003488 14.04425 −5.04425 3 3 600 90 4 9.3 9.507143 0.207143 14.0813 −4.7813 3 3 600 115 4 9.1 9.102601 0.002601 14.14305 −5.04305 3 3 900 90 4 11 10.72836 −0.27164 16.0583 −5.0583 3 3 900 90 5 11.6 11.36213 −0.23787 16.2593 −4.6593 3 3 900 115 3 10.8 11.06383 0.263827 15.91905 −5.11905 3 3 900 115 4 11.1 11.01522 −0.08478 16.12005 −5.02005 3 3 900 115 5 11.8 11.60359 −0.19641 16.32105 −4.52105 3 3 1200 75 4 9.5 9.406263 −0.09374 17.99825 −8.49825 3 3 1200 90 3 9.6 9.381887 −0.21811 17.8343 −8.2343 3 3 1200 90 4 9.7 9.373203 −0.3268 18.0353 −8.3353 3 3 1200 115 3 8.9 8.98249 0.08249 17.89605 −8.99605 3 3 1200 115 4 9 8.98249 −0.01751 18.09705 −9.09705 3 3 1200 115 5 9.2 9.224294 0.024294 18.29805 −9.09805 4 2 600 75 3 12.6 12.564 −0.036 15.06025 −2.46025 4 2 600 75 4 12.3 12.50683 0.206832 15.26125 −2.96125 4 2 600 75 5 12.9 12.62934 −0.27066 15.46225 −2.56225 4 2 600 90 3 10.3 10.46052 0.160518 15.0973 −4.7973 4 2 600 90 4 10.8 10.66317 −0.13683 15.2983 −4.4983 4 2 600 90 5 11 10.83043 −0.16957 15.4993 −4.4993 4 2 600 115 3 8.2 8.279408 0.079408 15.15905 −6.95905 4 2 600 115 4 8.4 8.279408 −0.12059 15.36005 −6.96005 4 2 600 115 5 8.7 8.663003 −0.037 15.56105 −6.86105 4 2 900 75 3 8.3 8.192486 −0.10751 17.03725 −8.73725 4 2 900 75 4 8.1 8.220214 0.120214 17.23825 −9.13825 4 2 900 75 5 8.5 8.29337 −0.20663 17.43925 −8.93925 4 2 900 90 3 7.8 8.055097 0.255097 17.0743 −9.2743 4 2 900 90 5 8.5 8.278371 −0.22163 17.4763 −8.9763 4 2 900 115 3 9.6 10.02004 0.420042 17.13605 −7.53605 4 2 900 115 4 9.8 10.03759 0.237595 17.33705 −7.53705 4 2 1200 75 3 13.1 12.78323 −0.31677 19.01425 −5.91425 4 2 1200 75 4 12.7 12.71508 0.015078 19.21525 −6.51525 4 2 1200 90 3 11.2 11.36919 0.16919 19.0513 −7.8513 4 2 1200 90 5 11.3 11.42188 0.121882 19.4533 −8.1533 4 2 1200 115 3 12.2 12.17725 −0.02275 19.11305 −6.91305 4 2 1200 115 4 12 12.14144 0.141442 19.31405 −7.31405 4 2 1200 115 5 12.6 12.38238 −0.21762 19.51505 −6.91505 4 2.5 600 75 3 12.7 12.72953 0.029527 15.36775 −2.66775 4 2.5 600 75 4 12.25 12.60696 0.356962 15.56875 −3.31875 4 2.5 600 75 5 13 12.73714 −0.26286 15.76975 −2.76975 4 2.5 600 90 3 12.4 12.3729 −0.0271 15.4048 −3.0048 4 2.5 600 90 4 11.8 12.19971 0.399713 15.6058 −3.8058 4 2.5 600 90 5 12.6 12.23691 −0.36309 15.8068 −3.2068 4 2.5 900 90 4 9.25 9.1138 −0.1362 17.5828 −8.3328 4 2.5 900 115 4 10.1 10.11175 0.011749 17.64455 −7.54455 4 2.5 1200 75 4 11 11.33745 0.337448 19.52275 −8.52275 4 2.5 1200 90 4 11.4 11.42875 0.028755 19.5598 −8.1598 4 2.5 1200 115 4 12.2 11.88495 −0.31505 19.62155 −7.42155 4 3 600 75 4 12.7 12.6509 −0.0491 15.87625 −3.17625 4 3 600 90 4 12.3 12.44309 0.143089 15.9133 −3.6133 4 3 600 115 4 11.8 11.48007 −0.31993 15.97505 −4.17505 4 3 900 75 4 10.1 9.74895 −0.35105 17.85325 −7.75325 4 3 900 90 4 9.3 9.357395 0.057395 17.8903 −8.5903 4 3 900 115 4 9.35 9.391239 0.041239 17.95205 −8.60205 4 3 1200 75 4 12 12.40683 0.406835 19.83025 −7.83025 4 3 1200 90 4 12.2 12.08855 −0.11145 19.8673 −7.6673 3 2 600 75 3 10.5 10.54728 0.047283 13.22825 −2.72825 3 2 600 90 3 10.1 10.04854 −0.05146 13.2653 −3.1653 3 2 600 115 3 11.8 11.50068 −0.29932 13.32705 −1.52705 3 2 900 90 3 7.9 8.212519 0.312519 15.2423 −7.3423 3 2 900 115 3 11.6 11.83127 0.231271 15.30405 −3.70405 3 2 1200 90 3 7.8 7.551078 −0.24892 17.2193 −9.4193 3 2.5 600 75 3 8 7.825613 −0.17439 13.53575 −5.53575 3 2.5 600 90 3 8.4 8.406551 0.006551 13.5728 −5.1728 3 2.5 600 115 3 8.9 8.74754 −0.15246 13.63455 −4.73455 3 2.5 900 75 3 7.1 7.337808 0.237808 15.51275 −8.41275 3 2.5 900 90 3 7.6 7.651297 0.051297 15.5498 −7.9498 3 2.5 1200 75 3 7.2 7.602399 0.402399 17.48975 −10.2898 3 2.5 1200 90 3 7.1 6.999386 −0.10061 17.5268 −10.4268 3 2.5 1200 115 3 7.9 7.564971 −0.33503 17.58855 −9.68855 3 3 600 75 3 9.1 9.036617 −0.06338 13.84325 −4.74325 3 3 600 90 3 9.3 9.520501 0.220501 13.8803 −4.5803 3 3 600 115 3 8.7 9.102601 0.402601 13.94205 −5.24205 3 3 900 90 3 11 10.80059 −0.19941 15.8573 −4.8573 3 3 900 115 3 11.1 11.06383 −0.03617 15.91905 −4.81905 3 3 1200 115 3 8.9 8.98249 0.08249 17.89605 −8.99605 4 2 600 75 3 12.7 12.564 −0.136 15.06025 −2.36025 4 2 600 90 3 10.7 10.68751 −0.01249 15.0973 −4.3973 4 2 600 115 3 8.3 8.279408 −0.02059 15.15905 −6.85905 4 2 900 75 3 8.5 8.192486 −0.30751 17.03725 −8.53725 4 2 900 90 3 7.8 8.055097 0.255097 17.0743 −9.2743 4 2 1200 75 3 13 12.78323 −0.21677 19.01425 −6.01425 4 2 1200 90 3 11.2 11.36919 0.16919 19.0513 −7.8513 4 2 1200 115 3 12.2 12.17725 −0.02275 19.11305 −6.91305 4 2.5 600 75 3 12.7 12.72953 0.029527 15.36775 −2.66775 4 2.5 600 90 3 12.4 12.3729 −0.0271 15.4048 −3.0048 4 2.5 900 75 3 9 9.157757 0.157757 17.34475 −8.34475 4 2.5 900 90 3 8.7 9.142856 0.442856 17.3818 −8.6818 4 2.5 1200 75 3 11.2 11.39373 0.193726 19.32175 −8.12175 4 2.5 1200 90 3 11.6 11.58502 −0.01498 19.3588 −7.7588 4 3 600 75 3 12.9 12.57058 −0.32942 15.67525 −2.77525 4 3 600 90 3 12.7 12.49178 −0.20822 15.7123 −3.0123 4 3 600 115 3 11.5 11.57182 0.071815 15.77405 −4.27405 4 3 900 75 3 10.3 9.788599 −0.5114 17.65225 −7.35225 4 3 900 90 3 9.5 9.395347 −0.10465 17.6893 −8.1893 4 3 900 115 3 9.3 9.485898 0.185898 17.75105 −8.45105 4 3 1200 75 3 12.5 12.42801 −0.07199 19.62925 −7.12925 4 3 1200 115 3 11.8 12.07808 0.278084 19.72805 −7.92805Table 6
Comparison between experimental and predicted values of impact strength atR2= 99.4.
V0(°) V1(KN) V2(rpm) V3(mm/s) V4(mm) Impact strength (J/m2) Error Linear regression Error Experimental GP 3 2 600 115 3 0.62 0.627465 −0.007465129 0.130185 0.489815 3 2.5 600 115 3 0.56 0.56651 −0.006510201 −0.05781 0.617815 3 3 900 75 3 0.89 0.888732 0.001268461 −0.21518 1.105175 3 3 900 90 3 0.95 0.956007 −0.006007421 −0.21339 1.16339 3 3 1200 75 3 0.9 0.90866 −0.008660352 −0.17978 1.079775 3 3 1200 90 3 0.89 0.890057 - 5.67889 E - 05 −0.17799 1.06799 4 2 600 115 3 0.89 0.890585 −0.000584826 0.133285 0.756715 4 2 900 75 3 0.88 0.876978 0.003022139 0.163925 0.716075 4 2 900 115 3 0.9 0.898867 0.001133347 0.168685 0.731315 4 2 1200 90 3 0.81 0.810876 −0.000875952 0.20111 0.60889 4 2.5 900 75 3 0.95 0.956236 −0.006236064 −0.02408 0.974075 4 2.5 900 115 3 0.89 0.880833 0.009166732 −0.01932 0.909315 4 2.5 1200 90 3 0.91 0.912273 −0.002273049 0.01311 0.89689 4 3 600 90 3 0.718 0.716218 0.001782065 −0.24569 0.96369 4 3 900 75 3 0.73 0.724777 0.005222838 −0.21208 0.942075 4 3 900 90 3 0.745 0.739808 0.005191798 −0.21029 0.95529 4 3 1200 75 3 0.72 0.715923 0.004076512 −0.17668 0.896675 4 3 1200 90 3 0.71 0.713943 −0.003943005 −0.17489 0.88489 4 3 1200 115 3 0.72 0.727521 −0.007521241 −0.17192 0.891915 3 2 600 90 4 0.8 0.799481 0.000518906 0.08501 0.71499 3 2.5 600 90 4 0.63 0.630611 −0.00061142 −0.10299 0.73299 3 2.5 600 115 4 0.58 0.575422 0.00457819 −0.10002 0.680015 3 3 900 90 4 0.91 0.918671 −0.008671191 −0.25559 1.16559 3 3 1200 75 3 0.9 0.90866 −0.008660352 −0.17978 1.079775 3 3 1200 90 3 0.89 0.890057 - 5.67889 E - 05 −0.17799 1.06799 4 2 600 75 4 0.65 0.651921 −0.001920617 0.086325 0.563675 4 2 600 115 3 0.89 0.890585 −0.000584826 0.133285 0.756715 4 2 600 115 5 0.82 0.813068 0.006931789 0.048885 0.771115 4 2 900 75 3 0.88 0.876978 0.003022139 0.163925 0.716075 4 2 900 115 3 0.9 0.898867 0.001133347 0.168685 0.731315 4 2 900 115 5 0.865 0.865083 - 8.29792 E - 05 0.084285 0.780715 4 2 1200 75 5 0.71 0.712427 −0.002426603 0.114925 0.595075 4 2 1200 90 3 0.81 0.810876 −0.000875952 0.20111 0.60889 4 2 1200 90 5 0.77 0.777844 −0.007843952 0.11671 0.65329 4 2 1200 115 4 0.78 0.78315 −0.003149838 0.161885 0.618115 4 2.5 600 90 4 0.86 0.860959 −0.000958576 −0.09989 0.95989 4 2.5 1200 75 4 0.8 0.808779 −0.008778763 −0.03088 0.830875 4 2.5 1200 90 4 0.93 0.931005 −0.001005299 −0.02909 0.95909 4 3 600 75 4 0.72 0.717328 0.002671809 −0.28968 1.009675 4 3 900 75 4 0.74 0.737732 0.002268126 −0.25428 0.994275 4 3 1200 75 4 0.725 0.715911 0.009089208 −0.21888 0.943875 4 3 1200 90 4 0.718 0.713928 0.004072254 −0.21709 0.93509 3 2 600 90 4 0.8 0.799481 0.000518906 0.08501 0.71499 3 2.5 600 90 4 0.63 0.630611 −0.00061142 −0.10299 0.73299 3 2.5 600 115 4 0.58 0.575422 0.00457819 −0.10002 0.680015 3 3 900 90 4 0.91 0.918671 −0.008671191 −0.25559 1.16559 3 3 1200 75 3 0.9 0.90866 −0.008660352 −0.17978 1.079775 3 3 1200 90 3 0.89 0.890057 - 5.67889 E - 05 −0.17799 1.06799 4 2 600 75 4 0.65 0.651921 −0.001920617 0.086325 0.563675 4 2 600 115 3 0.89 0.890585 −0.000584826 0.133285 0.756715 4 2 600 115 5 0.82 0.813068 0.006931789 0.048885 0.771115 4 2 900 75 3 0.88 0.876978 0.003022139 0.163925 0.716075 4 2 900 115 3 0.9 0.898867 0.001133347 0.168685 0.731315 4 2 900 115 5 0.865 0.865083 - 8.29792 E - 05 0.084285 0.780715 4 2 1200 75 5 0.71 0.712427 −0.002426603 0.114925 0.595075 4 2 1200 90 3 0.81 0.810876 −0.000875952 0.20111 0.60889 4 2 1200 90 5 0.77 0.777844 −0.007843952 0.11671 0.65329 4 2 1200 115 4 0.78 0.78315 −0.003149838 0.161885 0.618115 4 2.5 600 90 4 0.86 0.860959 −0.000958576 −0.09989 0.95989 4 2.5 1200 75 4 0.8 0.808779 −0.008778763 −0.03088 0.830875 4 2.5 1200 90 4 0.93 0.931005 −0.001005299 −0.02909 0.95909 4 3 600 75 4 0.72 0.717328 0.002671809 −0.28968 1.009675 4 3 900 75 4 0.74 0.737732 0.002268126 −0.25428 0.994275 4 3 1200 90 4 0.718 0.713928 0.004072254 −0.21709 0.93509Figures1(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.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.
(a) (b)Figure 2
(a) Comparison between experimental and predicted values of elongation and (b) normal Q-Q plot of error between experimental and GP data.
(a) (b)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.
(a) (b)Figures1(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 -08E-4±0 J/m2.


### 4.1. 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 involvingv0 to v4 input variables has been carried out using Minitab® 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.(7)Tensile  strength=174.2-1.03V0+9.09V1+0.01463V2+0.1243V3+0.12V4,(8)elongation=1.76+1.832V0+0.615V1-0.000659V2+0.00247V3+0.201V4,(9)impact  strength=0.915+0.0031V0-0.0376V1+0.000118V2+0.000119V3-0.0422V4.Figure 4
Residual plots of tensile strength using linear regression method.Figure 5
Residual plots of elongation using linear regression method.Figure 6
Residual plots of impact strength using linear regression method.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.Figure7 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.Figure 7
The quality distribution of GP models and experimental work of different outputs of FSW joint.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.


## 4.1. 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 involvingv0 to v4 input variables has been carried out using Minitab® 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.(7)Tensile  strength=174.2-1.03V0+9.09V1+0.01463V2+0.1243V3+0.12V4,(8)elongation=1.76+1.832V0+0.615V1-0.000659V2+0.00247V3+0.201V4,(9)impact  strength=0.915+0.0031V0-0.0376V1+0.000118V2+0.000119V3-0.0422V4.Figure 4
Residual plots of tensile strength using linear regression method.Figure 5
Residual plots of elongation using linear regression method.Figure 6
Residual plots of impact strength using linear regression method.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.Figure7 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.Figure 7
The quality distribution of GP models and experimental work of different outputs of FSW joint.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.


## 5. 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.



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*Source: 4183816-2018-04-03.xml*