Modelling and Simulation in Engineering
Volume 2019 (2019), Article ID 1024365, 15 pages
https://doi.org/10.1155/2019/1024365

Research Article

Mathematical Modeling of Multiple Quality Characteristics of a Laser Microdrilling Process Used in Al7075/SiCp Metal Matrix Composite Using Genetic Programming

Mohammed Yunus and Mohammad S. Alsoufi

Department of Mechanical Engineering, College of Engineering and Islamic Architecture, Umm Al-Qura University, Al-Abdiah, Makkah 24231, Saudi Arabia

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

Received 16 August 2018; Accepted 12 November 2018; Published 2 January 2019

Academic Editor: Aiguo Song

Copyright © 2019 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.

The conventional method for machining metal matrix composites (MMCs) is difficult on account of their excellent characteristics compared with those of their source materials. Modern laser machining technology is a suitable noncontact method for machining operations of advanced engineering materials due to its novel advantages such as higher productivity, ease of adaptation to automation, minimum heat affected zone (HAZ), green manufacturing, decreased processing costs, improved quality, reduced wastage, removal of finishing operations, and so on. Their application includes hole drilling in an aircraft engine components such as combustion chambers, nozzle guide vanes, and turbine blades made up of MMCs which meet quality standards that determine their suitability for service use. This paper presents a derived mathematical model based on evolutionary computation methods using multivariate regression fitting for the prediction of multiple characteristics (circularity, taper, spatter, and HAZ) of neodymium: yttrium aluminum garnet laser drilling of aluminum matrix/silicon carbide particulate (Al/SiCp) MMCs using genetic programming. Laser drilling input factors such as laser power, pulse frequency, gas pressure, and pulse width are utilized. From a training dataset, different genetic models for multiple quality characteristics were obtained with great accuracy during simulated evolution to provide a more accurate prediction compared to empirical correlations.

1. Introduction

Metal matrix composites (MMCs) are substances which blend a tough metallic matrix with a hard ceramic reinforcement possessing excellent features such as high strength to wear ratio, high modulus, and wear and corrosion resistance [1]. MMCs are broadly used in the fields of the aerospace, automotive, electronics, and metallic industries. MMCs comprise a metal as a base material (matrix) and hard ceramic particles such as B₄C, SiC, and Al₂O₃ as reinforcement (long fibers, short whiskers, or particulates in irregular or spherical shapes). The properties of MMCs are judged by matrix, reinforcement, and interface between them [2]. They are a material which is difficult to machine due to the presence of hard ceramic particles [3]. Most of the research in the machining of Al/SiCp MMCs has focused on turning and milling, whereas drilling has been given less attention.

1.1. Laser Drilling

The laser drilling system is growing exponentially to suit the alternative program for achieving the major demands of aerospace, automobile, metallic, and electric industrial potential, especially microhole drilling in different components such as watches and turbine blades, fuselages, printed circuit boards, and so on [3]. Pulsed Nd:YAG laser microhole drilling has gained popularity in recent years to be used as an indispensable tool for microhole drilling of components for technologically advanced industries. Laser microhole drilling processes are successfully employed to both conductive and nonconductive materials to remove by evaporation, and the amount of molten material removed depends on the penetration of laser energy generated from a sequence of laser pulses at the same place [4]. Laser drilling in the production industry has been associated with the various advantages of a high rate of machining, noncontact method; hence, there is no damage to tool or tool wear, increased product quality, and less wastage. Highly reflective materials require a low amount of laser power with a short wavelength of Nd:YAG (compared with CO₂ lasers) [5]. While it possesses several advantages and is widely approved by advanced industries, it also has some defects such as taperness, noncircular holes, HAZ, recast layer, and so on [6].

1.2. Genetic Programming

Genetic programming (GP) is the most effective tool used to predict the behaviour of various processes and for the formation of empirical modeling. Normally, GP supports the process of evolution in nature—Darwin’s theory of “survival of the fittest”—to find the best solution to an assigned problem. GP is known as the generalized form of the genetic algorithm (GA) and has been extensively studied [7–9]. In GP, the model is represented by terminals and functions. A well-known implementation of GP is in symbolic regression and is used to determine the mathematical expression for a given set of variables and functions. The function is generated using a Boolean operator (AND, OR, NOT, or AND NOT), nonlinear operators (sin, cos, tan, exp, tanh, and log), and from basic mathematical operators (+, −, /, and ×). The fitness function is calculated as the error between the actual value and the predicted value of the symbolic expressions. In GP, individual terms are randomly initialized, and the population is progressed to find out the optimal solutions through various operations such as reproduction, crossover, and mutation. The reproduction process produces children as an input to the next generation by replicating a fraction of the parent selected by the current generation. Individuals having highest fitness values in the population are selected as the parent and used for reproduction. Normally, the crossover operation produces children by exchanging some parts of their selected parents. The crossover operation is divided into two types (subtree and node crossover). However, the subtree crossover has shown more significant effect than the node crossover.

A variety of methods are available to develop relationships between inputs and outputs for evaluating outputs under varying input conditions without experimental work. These forecasting output data are derived in the form of equations using artificial neural network, statistical regression methods like linear regression, response surface, ANOVA, etc., with limited accuracy. To obtain higher accuracy at about 99.3 to 99.8% for the mathematical model derived, the effective method available is genetic programming which is an application of machine learning or artificial intelligence using inbuilt algorithms. The use of the GP approach with experimental data to develop a mathematical model involving laser microhole drilling input and output parameters is presented in this work. These mathematical models can be used to study the production of high-quality microdrills of the pulsed Nd:YAG laser used in Al7075/SiCp MMCs by minimizing the microhole drilling defects, such as the degree of taperness of a hole, spatter, and heat affected zone width, and to maximize a hole circularity [10]. The various drilling input parameters involved are pulse power (), pulse frequency (), assistance of gas pressure (), and pulse width () [11].

2. Materials and Methods

In the present work, using the stir casting technique, an MMC consisting of aluminum alloy 7075 as a base metal reinforced with particulates of silicon carbide having a size of 40–50 μm for 10% volume fractions is produced. The microholes were drilled onto MMC plates using the pulsed Nd:YAG laser beam system (Model: JK300D), and the tests were carried out at its maximum power capacity of 16 kW. Various input parameters such as laser input power (), pulse frequency (), assistance of gas pressure (), and pulse width () at different levels have been selected. The experimental results for various levels of input factors of laser microhole drilling of MMC (Al7075/10%SiCp) plate of 2 mm thick are shown in Tables 2–5. For a smaller diameter hole, laser microdrilling is preferred, especially when the materials are very hard, extra thin, or made of glass and composites. The quality of these holes mainly depends on heat affected zone (HAZ) of hole walls, taperness formed when the hole is enlarged, circularity to maintain the uniform dimension of the circular hole, and spatter occurring at the ends of a hole during resolidification of material [12, 13].

The input parameters of different levels can improve the quality of drilled holes. The quality of holes was determined using optical measuring microscope OLYMPUS STM6 on a cut sectioned hole sample by measuring circularity, spatter, HAZ, and taper characteristics, and in each experimental run, the laser was drilled at a spot size of 180 μm [10]. Various studies have been conducted to investigate the effects of input control parameters on the defects developed by the laser microdrilling process [14, 15]. Due to an internal focusing of the issue of a laser drilling process, hole taperness and noncircularity affect the quality of holes [16]. Normally, there is spatter accumulation due to an incomplete suspension of removed MMC at the drilling zone which resolidifies and adheres around the whole circumference. Hence, it is advisable to produce high-quality circular microdrilled holes with the minimum amount of taper, spatter, and HAZ width. The above-said qualities are to be determined by the level of input parameters which requires a mathematical model to study the process condition capable of producing the desired quality of product. However, this model is to be derived in such a way that all the characteristics measuring qualities are quickly measurable simultaneously [11].

3. Genetic Programming Methodology

Various stages of GP involved in the flowchart are as follows:

For the analysis of multiple characteristics of neodymium: yttrium aluminum garnet laser drilling of aluminum matrix/silicon carbide particulate MMCs such as circularity, taper, spatter, and HAZ, data accumulations of experiments were carried out as per stages involved in Figure 1. The accumulated data were randomized using notitia DiscipulusTM software and were provided to the software in three groups, viz., training, validation, and applied [17]. The three sets of data are generated by experimental results provided. Higher numbers of experiment running around more than 100 are required for best solution. Test runs were conducted to determine desirable parameters generating an optimal solution in the minimum possible time. Initially, the tests were carried out at the default parameter settings, such as population size, crossover rate, DSS subset size, and so on, and later varied to find optimum values [18]. The different parameters involved in achieving the final mathematical model satisfying the results are tabulated in Table 1. It shows the flow of set required to achieve the final model which could provide us a mathematical model satisfying the above conditions of the quantity involved [19].

Figure 1: Stages involved in genetic programming.

Table 1: Parameter setting for genetic programming.

Parameters
Value assigned

Population size (P)
500

Number of generations
1000

Maximum depth of tree
6

Maximum generation
50

Functional set
Multiply, plus, minus, divide

Terminal set
(v1, v2, v3, v4, v5, v6, −10, 10)

Number of runs
110

Mutation rate
0.10

Crossover rate
75% nonhomologous

25% homologous

Reproduction rate
0.05

Fitness, r²
The square root of the sum of the square of absolute value of the differences (errors), between the program’s output and the observed data.

Termination
An individual emerges whose sum of absolute errors is less than specified: (a) required number of runs be completed or (b) required correlation coefficient is obtained

Terminal set
T = {P, random-constants}

3.1. Regression and Fitness Measurement

Regression analysis is a stochastic method in which symbolic regression finds both the working model of output (or target) function and its inputs (or fixed coefficients), or at least an approximation (error measurement fit by linear or square). Fitness measurement indicates how far the output value predicted by the GP concurs with the experimental value.

3.2. Correlation Coefficient, r/R

The linear correlation coefficient refers to measuring the strength and direction of a linear relationship held between two variables (output, M, and input, N), where r value lies such that −1 < r < +1 [8]. The signs indicate a strong positive linear correlation between M and N or perfect fit if r is close to +1 with an increase of M values; N values also increase, whereas r is close to −1 if M and N have a strong negative linear correlation such that with an increase of M values, N values decrease. For r = 0, there is no linear/weak correlation, and a value approaching zero represents a random, nonlinear relationship between the M and N. The square of the correlation coefficient provides you with the coefficient of determination, r², to find the proportion of the variance of output that is predictable from the inputs. This helps us to determine how certain one can be in making predictions from a defined model. r² defined from the ratio of the illustrated variation to the total variation in the range of 0 < r² < 1 signifies the strength of the linear correlation between P and Q or represents the percentage of the data which is closest to the line of best fit [20]. If r = 0.977, then r² = 0.994, which means that 99.4% of the total variation in Q can be explained by the linear relationship between P and Q and the remaining 0.6% of the variation in Q continues unexplained [21].

3.3. Factors Involved in GP Modeling

The various parameters involved in modeling GP are tabulated in Table 1. It shows the flow of set required to achieve the final model which could provide you with a mathematical model satisfying the above conditions of the quantity involved.

4. Results and Discussion

The selection of precise instructions from set F and available terminal genes from set f(0) play a vital role in GP modeling, and evolutionary process will build a mathematical model (i.e., organism) such that it is as fit as for the prediction of results. The model consists of both instructions and function genes behaving similarly to the character of computer programs differing in appearance and dimensions [20, 22]. Previously, extensive work on developing mathematical models was also carried out using linear regression, second order, and higher order equations using response surface methods, ANOVA, box, artificial neural network, and fuzzy logic method where the models of these methods develop the amount of error is very high compared to experimental values upto 3 digit numbers [24]. These analyses require high value of digit numbers not lower and decimal numbers for output values compared to values of inputs to accomplish accuracy. These methods are suitable for small number of inputs; otherwise, erroneous empirical relation would be generated. The above nonGP methods may use limited data according design matrix and orthogonal array of tables, and the equation will be developed by using optimal input values [24].

Using Grey–Taguchi method, the overall performance characteristic in Nd:YAG laser microdrilling of alumina was calculated using grey relational grade (GRG) to find a optimal parameter set for getting highest GRG. The optimal value of GRG was 0.9172 which indicates the effectiveness of the proposed approach and the highest value of GRG of 0.8989 for confirmation experiment. The overall quality feature is improved by 2.03% at optimal condition with HAZ width by 8.78%, and the hole taper worsened by 2.14%. This shows that, in multilevel optimization of performance, characteristics cannot reach simultaneously optimum value, instead they have to compromise between various performance characteristics to accomplish the optimum value pertaining to overall performance. Both the performance characteristics improve at optimum level with GRG of 0.1521 (19.88%). The taperness (from 0.0491 to 0.0476 rad) and HAZ width (from 0.2180 to 0.1683 mm) are decreased at the same time. From the results, we can see that optimum results can provide correlations at the accuracy level of 85% if outputs are limited to two numbers. By increasing output quality characteristics as well as input parameters and number of experiments, it is very difficult to arrive at optimal set and corresponding correlations obtained by regression analysis. Therefore GP-based solutions maintain an accuracy level beyond statistical analysis irrespective of number of outputs and inputs [25].

The model used experimental measurements collected by converting into three independent datasets such as training, validation, and applied. Independent input variables used are pulse power (), pulse frequency (), assistance of gas pressure (), and pulse width () and circularity, spatter, heat affected zone, and taper as the dependent output variable. Various models for outputs are developed by GP using the training dataset [23]. Best mathematical models obtained from GP simulation are given by equations (1)–(4)

Appendixes A, B, C, and D show further details related to the derivation of the circularity, spatter, taper, and heat affected zone, respectively.

Comparison of the experimental and predicted outputs using the mathematical model obtained from GP is shown in Tables 2–5 for circularity, taper, spatter, and HAZ of pulsed Nd:YAG laser microhole drilling of MMC components. Errors in determining predicted outputs are very few, and the percentage of error is less than ±1% which shows that results obtained from a GP mathematical model are highly acceptable.

Table 2: Comparison between experimental and predicted values of circularity at R² = 99.68.

No.
PP (W) ()
PF (Hz) ()
AGP (kg/cm²) ()
PW (ms) ()
Circularity (mm)
Error %

Experimental
GP

1
250
210
12
0.4
0.908
0.908066
−6.6E − 05

2
210
210
8
0.2
0.9032
0.903074
0.000126

3
240
220
12
0.4
0.911
0.911254
−0.00025

4
210
210
8
0.6
0.953
0.953124
−0.00012

5
250
220
8
0.6
0.9425
0.941842
0.000658

6
210
250
12
0.6
0.978
0.978011
−1.1E − 05

7
210
230
10
0.4
0.948
0.945836
0.002164

8
230
210
10
0.6
0.957
0.958292
−0.00129

9
210
230
10
0.3
0.923
0.922506
0.000494

10
240
210
11
0.3
0.892
0.89308
−0.00108

11
220
210
9
0.3
0.926
0.926227
−0.00023

12
230
230
12
0.3
0.92
0.920843
−0.00084

13
240
220
12
0.3
0.9205
0.920919
−0.00042

14
220
250
8
0.3
0.896
0.895994
6.11E − 06

15
230
210
10
0.5
0.947
0.94698
1.96E − 05

16
240
230
8
0.4
0.899
0.898971
2.94E − 05

17
240
210
11
0.2
0.892
0.893078
−0.00108

18
220
240
12
0.6
0.974
0.973441
0.000559

19
240
250
10
0.6
0.954
0.955555
−0.00155

20
250
250
11
0.4
0.92
0.919789
0.000211

21
250
240
10
0.2
0.894
0.893536
0.000464

22
230
250
9
0.5
0.953
0.949875
0.003125

23
210
250
12
0.5
0.958
0.963079
−0.00508

24
250
240
10
0.3
0.892
0.893537
−0.00154

25
250
220
8
0.5
0.938
0.933247
0.004753

26
250
250
11
0.3
0.895
0.903726
−0.00873

27
230
230
12
0.2
0.893
0.892581
0.000419

28
230
220
11
0.2
0.89
0.892581
−0.00258

29
240
240
9
0.6
0.95
0.949358
0.000642

30
220
240
12
0.2
0.893
0.892039
0.000961

31
250
230
9
0.2
0.896
0.896958
−0.00096

32
220
220
10
0.5
0.951
0.950144
0.000856

33
240
240
9
0.5
0.95
0.943263
0.006737

34
210
220
9
0.2
0.9
0.896232
0.003768

35
220
250
8
0.2
0.902
0.896907
0.005093

36
250
210
12
0.5
0.948
0.94813
−0.00013

37
220
220
10
0.4
0.936
0.923214
0.012786

38
230
240
8
0.3
0.901
0.90405
−0.00305

39
250
230
9
0.6
0.9475
0.947726
−0.00023

40
230
250
9
0.4
0.911
0.928516
−0.01752

41
240
250
10
0.2
0.895
0.895507
−0.00051

42
220
210
9
0.4
0.91
0.90617
0.00383

43
240
230
8
0.5
0.94
0.936347
0.003653

44
210
240
11
0.5
0.951
0.959169
−0.00817

45
220
230
11
0.5
0.95
0.954643
−0.00464

46
220
230
11
0.6
0.97
0.967844
0.002156

47
230
240
8
0.4
0.901
0.906892
−0.00589

48
230
220
11
0.6
0.966
0.964493
0.001507

49
210
240
11
0.4
0.93
0.943397
−0.0134

50
210
220
9
0.3
0.922
0.918336
0.003664

PP: pulse power (W), PF: pulse frequency (Hz), PW: pulse width (ms), AGP: assist gas pressure (Kg/cm³).

Table 3: Comparison between experimental and predicted values of HAZ at R² = 99.88.

No.
PP (W) ()
PF (Hz) ()
AGP (kg/cm²) ()
PW (ms) ()
HAZ (mm)
Error %

Experimental
GP

1
240
230
8
0.4
0.04
4.19E − 02
−0.00188

2
230
210
10
0.5
0.112
0.107232
0.004768

3
230
230
12
0.3
0.068
7.13E − 02
−0.00328

4
210
230
10
0.3
0.068
7.17E − 02
−0.00368

5
220
240
12
0.2
0.081
8.02E − 02
0.000789

6
240
220
12
0.3
0.078
7.82E − 02
−0.00023

7
220
210
9
0.4
0.088
8.75E − 02
0.000453

8
250
220
8
0.6
0.104
0.103178
0.000822

9
210
210
8
0.2
0.065
6.80E − 02
−0.00305

10
250
210
12
0.4
0.1
9.97E − 02
0.000311

11
210
230
10
0.4
0.069
7.04E − 02
−0.00139

12
240
220
12
0.4
0.073
7.20E − 02
0.00103

13
230
220
11
0.6
0.085
8.46E − 02
0.000417

14
220
250
8
0.3
0.102
0.101978
2.21E − 05

15
220
230
11
0.6
0.089
8.67E − 02
0.002272

16
220
220
10
0.5
0.094
9.50E − 02
−0.00098

17
240
240
9
0.6
0.105
0.102995
0.002005

18
240
210
11
0.2
0.085
8.12E − 02
0.003847

19
220
250
8
0.2
0.0747
7.28E − 02
0.001893

20
230
250
9
0.5
0.1
0.103908
−0.00391

21
210
220
9
0.3
0.082
8.05E − 02
0.001512

22
210
250
12
0.5
0.0909
9.14E − 02
−0.00051

23
210
250
12
0.6
0.0921
9.28E − 02
−0.00071

24
220
210
9
0.3
0.101
0.101721
−0.00072

25
240
240
9
0.5
0.1
0.10225
−0.00225

26
230
250
9
0.4
0.094
9.11E − 02
0.002885

27
240
250
10
0.6
0.096
9.56E − 02
0.000406

28
250
210
12
0.5
0.097
9.83E − 02
−0.00131

29
230
230
12
0.2
0.076
7.74E − 02
−0.0014

30
220
230
11
0.5
0.092
9.06E − 02
0.001379

31
210
240
11
0.4
0.072
7.25E − 02
−0.00049

32
230
210
10
0.6
0.095
9.25E − 02
0.002548

33
250
250
11
0.3
0.078
7.72E − 02
0.000826

34
250
220
8
0.5
0.105
0.104177
0.000823

35
210
220
9
0.2
0.084
7.68E − 02
0.007168

36
220
240
12
0.6
0.095
9.20E − 02
0.003025

37
230
240
8
0.4
0.055
4.67E − 02
0.008281

38
210
210
8
0.6
0.1
9.79E − 02
0.002054

39
250
230
9
0.2
0.07
7.45E − 02
−0.00448

40
240
250
10
0.2
0.095
8.29E − 02
0.012098

41
240
210
11
0.3
0.08
7.90E − 02
0.001025

42
230
240
8
0.3
0.09
8.87E − 02
0.001309

43
210
240
11
0.5
0.084
9.04E − 02
−0.00638

44
250
240
10
0.3
0.065
7.74E − 02
−0.01241

45
250
230
9
0.6
0.095
9.90E − 02
−0.00402

46
250
250
11
0.4
0.08
0.081854
−0.00185

47
230
220
11
0.2
0.077
0.0783
−0.0013

48
220
220
10
0.4
0.075
0.07292
0.00208

49
250
240
10
0.2
0.091
8.82E − 02
0.002776

50
240
230
8
0.5
0.09
0.090104
−0.0001

Table 4: Comparison between experimental and predicted values of taper at R² = 99.43.

No.
PP (W) ()
PF (Hz) ()
AGP (kg/cm²) ()
PW (ms) ()
Taper (deg)
Error %

Experimental
GP

1
230
250
9
0.4
3.5
3.48E + 00
0.020539

2
230
240
8
0.4
3.702
3.68349
0.01851

3
210
220
9
0.3
3.815
3.816499
−0.0015

4
230
210
10
0.5
3.75
3.72E + 00
0.034738

5
240
250
10
0.2
4.135
4.12E + 00
0.017045

6
210
240
11
0.4
3.45
3.43E + 00
0.023249

7
230
230
12
0.3
3.925
3.86E + 00
0.064339

8
220
240
12
0.6
3.5
3.487621
0.012379

9
220
240
12
0.2
4.12
4.13E + 00
−0.00979

10
240
220
12
0.3
3.8
3.81E + 00
−0.00709

11
230
240
8
0.3
3.8
3.79E + 00
0.010771

12
220
250
8
0.3
3.962
3.96E + 00
0.006659

13
230
220
11
0.2
4.16
4.16E + 00
−0.00037

14
240
220
12
0.4
3.775
3.757363
0.017637

15
250
230
9
0.6
3.8
3.770205
0.029795

16
210
210
8
0.6
4.2
4.200192
−0.00019

17
230
220
11
0.6
3.75
3.718309
0.031691

18
220
250
8
0.2
4.05
4.10E + 00
−0.04831

19
250
210
12
0.4
4.1
3.96E + 00
0.138649

20
250
220
8
0.5
3.5
3.522696
−0.0227

21
240
240
9
0.5
3.4
3.41E + 00
−0.01021

22
210
210
8
0.2
4.1
4.10E + 00
0.00162

23
240
230
8
0.5
3.505
3.523799
−0.0188

24
210
230
10
0.4
3.58
3.633331
−0.05333

25
220
210
9
0.3
3.6
3.774172
−0.17417

26
250
250
11
0.3
3.8
3.846722
−0.04672

27
210
240
11
0.5
3.307
3.441274
−0.13427

28
250
240
10
0.2
4.1
4.105488
−0.00549

29
220
230
11
0.5
3.3
3.441037
−0.14104

30
240
250
10
0.6
3.8
3.81E + 00
−0.00654

31
230
210
10
0.6
3.305
3.638603
−0.3336

32
230
250
9
0.5
3.38
3.41E + 00
−0.03093

33
230
230
12
0.2
4.05
4.110147
−0.06015

34
210
220
9
0.2
4
4.125376
−0.12538

35
250
230
9
0.2
4.15
4.123817
0.026183

36
240
240
9
0.6
3.75
3.748872
0.001128

37
240
210
11
0.3
3.9
3.86E + 00
0.043294

38
250
210
12
0.5
3.65
3.501807
0.148193

39
210
250
12
0.6
3.25
3.47E + 00
−0.21634

40
210
250
12
0.5
3.45
3.49592
−0.04592

41
220
220
10
0.5
3.6
3.572536
0.027464

42
240
210
11
0.2
3.95
4.11E + 00
−0.16276

43
250
220
8
0.6
3.6
4.114297
−0.5143

44
220
220
10
0.4
3.8
3.694675
0.105325

45
250
250
11
0.4
3.9
3.655343
0.244657

46
240
230
8
0.4
3.6
3.615975
−0.01597

47
220
230
11
0.6
3.4
3.677536
−0.27754

48
220
210
9
0.4
3.5
3.48E + 00
0.023193

49
250
240
10
0.3
3.8
3.80E + 00
0.004566

50
230
250
9
0.4
3.5
3.48E + 00
0.020539

Table 5: Comparison between experimental and predicted values of spatter at R² = 99.3.

No.
PP (W) ()
PF (Hz) ()
AGP (kg/cm²) ()
PW (ms) ()
Spatter (mm)
Error %

Experimental
GP

1
220
220
10
0.4
0.055
5.49E − 02
0.00013

2
230
220
11
0.6
0.044
4.42E − 02
−0.00018

3
210
250
12
0.6
0.044
4.34E − 02
0.000648

4
240
220
12
0.3
0.043
4.32E − 02
−0.00022

5
230
240
8
0.3
0.04
4.36E − 02
−0.00365

6
210
230
10
0.3
0.042
4.21E − 02
−7.3E − 05

7
250
240
10
0.2
0.065
6.50E − 02
3.18E − 05

8
230
230
12
0.2
0.043
4.31E − 02
−7.8E − 05

9
240
240
9
0.6
0.072
7.16E − 02
0.000406

10
230
220
11
0.2
0.045
4.50E − 02
−1.6E − 05

11
250
230
9
0.2
0.068
6.82E − 02
−0.00024

12
220
240
12
0.6
0.052
5.23E − 02
−0.00028

13
210
250
12
0.5
0.043
4.29E − 02
6.36E − 05

14
220
210
9
0.4
0.059
5.94E − 02
−0.00045

15
210
220
9
0.2
0.0445
0.044101
0.000399

16
220
240
12
0.2
0.05
5.02E − 02
−0.00019

17
230
210
10
0.5
0.072
7.03E − 02
0.001661

18
230
240
8
0.4
0.071
6.97E − 02
0.001265

19
240
210
11
0.2
0.056
5.54E − 02
0.000566

20
230
250
9
0.5
0.074
7.25E − 02
0.001524

21
240
230
8
0.4
0.075
0.064522
0.010478

22
220
250
8
0.3
0.074
0.071546
0.002454

23
250
210
12
0.5
0.046
5.18E − 02
−0.00581

24
210
240
11
0.4
0.043
4.20E − 02
0.000993

25
230
210
10
0.6
0.049
5.21E − 02
−0.00307

26
250
250
11
0.3
0.065
5.50E − 02
0.010003

27
210
230
10
0.4
0.04
4.28E − 02
−0.00277

28
210
210
8
0.6
0.043
0.042277
0.000723

29
210
240
11
0.5
0.045
0.044648
0.000352

30
220
230
11
0.6
0.057
5.40E − 02
0.003045

31
240
220
12
0.4
0.042
4.29E − 02
−0.00093

32
240
250
10
0.2
0.066
6.64E − 02
−0.00042

33
230
230
12
0.3
0.041
4.27E − 02
−0.00171

34
230
250
9
0.4
0.069
6.84E − 02
0.000556

Furthermore, Figure 2 shows the regression fit for the percentage of microdrilling process parameters of the MMCs. Figure 3 shows the experimental-predicted relationship of circularity, HAZ, taper, and spatter with the normal distribution behaviour. The models generated by the genetic programming perform better based on statistical terms and the historical dataset (training, validation, and applied data) to exhibit a better predictive capacity on the experimental dataset. The analysis of the mathematical expression of the GP Model suggests specific laser output quality characteristics for the experimental system under various control factors that can be associated with the performance of a laser microdrilled hole in MMCs. The models generated by genetic programming allow representation of the experimental data without a detailed knowledge of the phenomenon. In addition, their study allows us to obtain a deeper insight into the relevant factors in describing the quality phenomenon; for instance, changes in any of the factors observed in quality of the microhole produced that might be associated with changes in the performance of hole production in MMC.

(a)

(b)

(c)

(d)

Figure 2: Regression fit for the percentage of microdrilling process parameters of the MMCs. (a) Experimental-predicted value of circularity. (b) Experimental-predicted value of HAZ. (c) Experimental-predicted value of taper. (d) Experimental-predicted value of spatter.

(a)

(b)

(c)

(d)

Figure 3: Experimental-predicted relationship of (a) circularity, (b) HAZ, (c) taper, and (d) spatter.

5. Conclusions

In this present work, new models of the circularity, spatter, heat affected zone, and taper of MMCs drilling properties at different laser operating parameters are generated using Discipulus GP software and C programming. Using GP-based mathematical models, quality of laser drilled hole properties for MMCs involving various laser input parameters is determined quickly by substitution of laser input conditions, and without conducting any experiments, the actual results can be predicted. The comparison between the new GP-based model and the experimental results indicated that the new model is more accurately close to ± 0.006 to 0.009. Therefore, the new model can be considered an alternative method to estimate the mechanical properties when the experimental measurements or correlations are not available. The correctness of solutions achieved by GP depends on correlated evolutionary parameters, the number of experimental results, and their level of accuracy. To improve the structure of the model during evolution, more information was supplied through experimental measurements. Therefore, the proposed mathematical model has verified its results are adaptable up to reliability of 99.3 to 99.8% to forecast experimental results. In the testing stage, the GP model gives the same result as was found out during the experiment with the reliability of cent percent. The GP approach has thus proved to be a highly skilled and advantageous tool for recognizing correlations in data when no proper theoretical or other methods are possible or available.

Appendix

A. Circularity

B. Spatter

C. Taper

D. Heat Affected Zone (HAZ)

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors have no conflicts of interest.

Authors’ Contributions

MY and MSA participated in the conceptualization of the research, prepared the data for experimentation and the analysis and interpretation of the data. MY used Genetic program Discipulus software for the analysis, including writing equations. MY drafted the manuscript, and figures were sketched by MSA. MY and MSA revised the manuscript for intellectual content and approved the final manuscript.

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