Modelling and Simulation in Engineering
Volume 2018 (2018), Article ID 4183816, 18 pages
https://doi.org/10.1155/2018/4183816

Research Article

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

Mohammed Yunus and Mohammad S. Alsoufi

Mechanical Engineering Department, Umm Al-Qura University, Makkah, Saudi Arabia

Correspondence should be addressed to Mohammed Yunus; yunus.mohammed@rediffmail.com

Received 2 December 2017; Revised 8 February 2018; Accepted 20 February 2018; Published 3 April 2018

Academic Editor: Dimitrios E. Manolakos

Copyright © 2018 Mohammed Yunus and Mohammad S. Alsoufi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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%.

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].

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 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.

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
Remaining

Table 2: Important input factors and their levels.

Process parameters
Symbol
Unit
Level 1
Level 2
Level 3

Tilt angle

(°)
3
4
- -

Axial load

(KN)
2
2.5
3

Tool rotational speed

(rpm)
600
900
1200

Welding speed

(mm/sec)
70
90
115

Thickness of plate

(mm)
3
4
5

Table 3: Parameter setting for genetic programming.

Parameters
Value assigned

Population size ()
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

Fitness

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 () is obtained (when is less than 0.1)

Terminal set

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 set (arithmetic operators ) 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 , where (+) and (−) signs indicate positive and negative linear relationships, respectively. When , 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 (a perfect negative fit). If , this indicates no linear relationship and if approaches zero, it represents a random, nonlinear relation between the two variables. gives the coefficient of determination to predict from a particular model. 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).

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.

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

0
(°)
1
(KN)
2
(rpm)
3
(mm/s)
4
(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
2.8833

Table 5: Comparison between experimental and predicted values of % elongation at = 99.3.

0
(°)
1
(KN)
2
(rpm)
3
(mm/s)
4
(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
−7.92805

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

0
(°)
1
(KN)
2
(rpm)
3
(mm/s)
4
(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

−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

−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

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

−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

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
0.93509

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.

(a)

(b)

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.

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  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 %. 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  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 involving to 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.

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 () 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.

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.

Conflicts of Interest

The authors have no conflicts of interest.

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