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

)e 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 formachining 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.)eir 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. )is 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.


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 B 4 C, SiC, and Al 2 O 3 as reinforcement (long fibers, short whiskers, or particulates in irregular or spherical shapes). e properties of MMCs are judged by matrix, reinforcement, and interface between them [2]. ey 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.

Laser Drilling.
e 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

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. e 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 (v 0 ), pulse frequency (v 1 ), assistance of gas pressure (v 2 ), and pulse width (v 3 ) at different levels have been selected. e 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. e 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]. e input parameters of different levels can improve the quality of drilled holes. e 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. e 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].

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.
e accumulated data were randomized using notitia Discipulus TM software and were provided to the software in three groups, viz., training, validation, and applied [17]. e 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]. e 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].

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.

Correlation Coefficient, r/R.
e 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].
e 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. e square of the correlation coefficient provides you with the coefficient of determination, r 2 , to find the proportion of the variance of output that is predictable from the inputs. is helps us to determine how certain one can be in making predictions from a defined model. r 2 defined from the ratio of the illustrated variation to the total variation in the range of 0 < r 2 < 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 r 2 � 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].

Factors Involved in GP Modeling.
e 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.
Architecture-altering operations: choose an architecture-altering operation from the available repertoire of such operations, and create one new offspring program for the new population by applying the chosen architecture-altering operation to one selected program Create new individual program(s) for the population by applying the following genetic operations with specified probabilities by reproduction, cross over, and mutation Execution of each program in the population or selects one or two individual program(s) from the population to ascertain its fitness (explicitly or implicitly) using the problem's fitness measure Iterations following substeps (called a generation) on the population carried out until the termination criterion is satisfied Random generation of an initial population (called generation zero) of individual computer programs consisting of the available functions and terminals  e 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}

Results and Discussion
e 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. e 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]. ese analyses require high value of digit numbers not lower and decimal numbers for output values compared to values of inputs to accomplish accuracy. ese methods are suitable for small number of inputs; otherwise, erroneous empirical relation would be generated.
e 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. e 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. e 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%. is 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%). e 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. erefore GP-based solutions maintain an accuracy level beyond statistical analysis irrespective of number of outputs and inputs [25]. e model used experimental measurements collected by converting into three independent datasets such as training, validation, and applied. Independent input variables used are pulse power (v 0 ), pulse frequency (v 1 ), assistance of gas pressure (v 2 ), and pulse width (v 3 ) 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) taper 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.
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. e 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. e analysis of the mathematical expression of the GP Modelling and Simulation in Engineering 5 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. e 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.

Conclusions
In this present work, new models of the circularity, spatter, heat affected zone, and taper of MMCs drilling properties at      Modelling and Simulation in Engineering more information was supplied through experimental measurements. erefore, 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. e 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.

Conflicts of Interest
e 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.