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

Authors: Mohammed Yunus; Mohammad S. Alsoufi
Journal: Modelling and Simulation in Engineering
 (2019)

---

## 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.


---

## Body

## 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 (v0), pulse frequency (v1), assistance of gas pressure (v2), and pulse width (v3) [11].


## 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 (v0), pulse frequency (v1), assistance of gas pressure (v2), and pulse width (v3) [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 40–50μm 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 (v0), pulse frequency (v1), assistance of gas pressure (v2), and pulse width (v3) 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 180μm [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 Figure1. 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 Table1. 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.


## 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 Table1. 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 setF 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 (v0), pulse frequency (v1), assistance of gas pressure (v2), and pulse width (v3) 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)(1)circularity=2GF2−v0v0−1.745v0+0.904515,(2)spatter=1+N2K+L−4L−1.9/K+L+24L−1.9/K+L0.5−0.552−1.1524,(3)taper=VP−N/N−M+L/M+V/M−N3+OV/M−N3/M−N3−VF+G+QF+H2Q2−10.25Q2+2H2Q2+4/3Q−3−2F−2G−2F+H2Q2−10.25Q2+2H2Q2+HQ+5−2v3+0.73/3Q−3−2F−2G−2F+H2Q2−10.25Q2+2H2Q2+HQ+5−H2Q2−1/M−N3P−N/N−M+L/M+V/M−N3+OV/M−N3+1.05,(4)heataffectedzone=1.22L/I−0.682v1v3−0.079−1.22FG−0.5356Gv0+1.22L/I−0.682v1.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 Tables2–5 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 atR2 = 99.68.
No.
PP (W) (v0)
PF (Hz) (v1)
AGP (kg/cm2) (v2)
PW (ms) (v3)
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 atR2 = 99.88.
No.
PP (W) (v0)
PF (Hz) (v1)
AGP (kg/cm2) (v2)
PW (ms) (v3)
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
−0.0001Table 4
Comparison between experimental and predicted values of taper atR2 = 99.43.
No.
PP (W) (v0)
PF (Hz) (v1)
AGP (kg/cm2) (v2)
PW (ms) (v3)
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
0.020539Table 5
Comparison between experimental and predicted values of spatter atR2 = 99.3.
No.
PP (W) (v0)
PF (Hz) (v1)
AGP (kg/cm2) (v2)
PW (ms) (v3)
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
0.000556Furthermore, Figure2 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.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.
(a)
(b)
(c)
(d)


## 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.



---

*Source: 1024365-2019-01-02.xml*