## ABSTRACT

Today the aluminium matrix composite materials are with the improved mechanical properties replacing many conventional metals and alloys in today’s modern technological applications. Machining of aluminium matrix composites requires the need for a better understanding of cutting processes regarding accuracy and efficiency. This paper focused on the empirical modelling and optimization of kerf width in abrasive water jet machining (AWJM) of Al359/30%SiC composites. The kerf width in AWJM represents the cut accuracy of the machined component. Abrasive water jet pressure, stand-off distance, nozzle traverse rate and abrasive rate are treated as the significant machining variables to measure the kerf width during machining. DOE to reduce the number of experimentations, RSM for quadratic expression for kerf width to input variables, ANOVA for the sufficiency of the model. Finally to the optimal processing parameter values were derived by implementing the evolutionary-based optimization approach called a genetic algorithm. The derived optimal processing variables were analyzed and reported.

*Keywords:*

Abrasive water jet machining, Kerf width, modelling, optimization, response surface methodology, genetic algorithm.

## I. INTRODUCTION

### 1.1 DEVELOPMENT OF MACHINING PROCESS

In the developing world according to the design of a model, there is a requirement of abnormal shapes, material removal of different material and unusual sizes. Conventional machining process could not fulfil the requirement as it is not easy to obtain needed complex structure, further machining process causes damage to the conventional machines and surface finish of the workpiece will have irregularities. To overcome the flaws in the convention process non-conventional process has been developing. The main difference in conventional and non-conventional process is the contact of the workpiece with the cutting tool. Non-convention machining process involves CNC coding and there is no contact between the cutting tool and workpiece. The non-conventional machining process is also known as non-traditional machining process. There are many nonconventional machining processes like Jet Cutting (AJC), Water Jet Cutting (WJC), Abrasive Water Jet Cutting (AWJC) and Laser Beam Cutting (LBC) etc among mentioned Abrasive Water jet cutting (AWJC) and Laser Beam Cutting (LBC) are more productive. Considering Abrasive water jet cutting (AWJC) machining is carried on aluminium MMC material. Jet pressure, standoff distance, transverse rate, abrasive rate as the input parameters, material removal rate, kerf width, time taken for the single cut are output parameters. Design of experiments in this experiment is planned systematically with reference to previous experiments, analysis of data with optimum usage of obtainable resource. Response Surface Methodology (RSM) with Central Composite Design (CCD) is used as the design of experiments (DOE) expertise in this machining processing experiments. As there are many uses of Abrasive water jet cutting (AWJC) in manufacturing process improvement in the cutting process is included.

#### 1.1.1 APPLICATION OF NON-CONVENTIONAL PROCESS

Nowadays most of the application involves metal matrix composites in manufacturing of automotive vehicles, aerospace, automobiles and air crafts. Due to the improved properties lower coefficient of thermal expansion, strength to weight ratio, ability to work at high temperatures and magnificent wear defiance these are classified as advanced materials. There are several manufacturing procedures to produce metal matrix composites such as spray forming, compo casting, powder metallurgy, stir casting, extrusion, Rheocasting and iso-static pressing. Metal matrix composites having improved properties terminately scattered hard ceramics would create tough machining to the industries involving these materials. Due to substantial machining process elaborate in the cutting of metal matrix composites they are confined to particular applications. There is a number of factors involving such as processing rout, chemical compositions, ceramic reinforcement and its distribution which are important factors for machinability of MMC. In the developing world according to the design of a model, there is a requirement of abnormal shapes, material removal of different material and unusual sizes. Conventional machining process could not fulfil the requirement as it is not easy to obtain needed complex structure, further matching process causes damage to the conventional machines and surface finish of the workpiece will have irregularities. To overcome the flaws in the convention process non-conventional process has been developing. The main difference in conventional and non-conventional process is the contact of the workpiece with the cutting tool. Non-convention machining process involves CNC coding and there is no contact between the cutting tool and workpiece.

#### 1.1.2 TYPES OF NON-CONVENTIONAL PROCESS

The non-conventional machining process is also known as non-traditional machining process. There are many nonconventional machining processes like Jet Cutting (AJC), Water Jet Cutting (WJC), Abrasive Water Jet Cutting (AWJC) and Laser Beam Cutting (LBC) etc among mentioned Abrasive Water jet cutting (AWJC) and Laser Beam Cutting (LBC) are more productive. Considering Abrasive water jet cutting (AWJC) machining is carried on aluminium MMC material. Jet pressure, standoff distance, transverse rate, abrasive rate as the input parameters, material removal rate, kerf width, time taken for the single cut are output parameters. Design of experiments in this experiment is planned systematically with reference to previous experiments, analysis of data with optimum usage of obtainable resource. Response Surface Methodology (RSM) with Central Composite Design (CCD) is used as the design of experiments (DOE) expertise in this machining processing experiments. As there are many uses of Abrasive water jet cutting (AWJC) in manufacturing process improvement in the cutting process is included

### 1.2 ABRASIVE WATER JET MACHINING

Abrasive water jet machining is one of the non-conventional machining processes which is the combination of both the abrasive jet machining process and water jet machining process. The abrasive material and the water are subjected to a certain pressure to cut the workpiece. The high-speed water combines with the abrasive material and flows through the nozzle. This jet water which passes through the nozzle will make a neat cut on the workpiece.

#### 1.2.1 IMPORTANT COMPONENTS OF THE ABRASIVE WATER JET CUTTING

- RESERVOIR
- PUMP
- INTENSIFIER
- ACCUMULATOR
- CONTROL VALVE
- FLOW REGULATOR
- NOZZLE

The difference in working of abrasive water jet machining and water jet machining is the abrasive material which we used along with the water pressure while machining on the workpiece

#### 1.2.2 WORKING PRINCIPLE ABRASIVE WATER JET MACHINING

The process of abrasive water jet cutting initiates from the reservoir through pump water flow to the intensifier their the water accumulates control valve controls the flow through its regulator to the nozzle. The abrasive feed is added through an abrasive tube in the nozzle and mixture of water and abrasive makes the clean-cut on the workpiece. Schematic Diagram of working of water jet cutting.

### 1.3 PARAMETERS FOCUSED IN ABRASIVE WATER JET MACHINING

INPUT PARAMETERS – Transverse rate, Jet pressure, Abrasive rate, Standoff distance.

OUTPUT PARAMETERS – Kerf width, Time, MRR, surface finish.

#### 1.3.1 MECHANISMS OF MATERIAL REMOVAL

Though the process is commercially used for many years, the details of the material removal mechanism are yet to be fully understood. However, the past works done to understand the process parameters, have thrown light on the possible mechanism of material removal in abrasive water jet machining. The main mechanisms responsible for the material removal in abrasive water jet machining are listed below.

- The direct impact of the abrasive particles on the workpiece.
- Impact of the free moving abrasive particles on the workpiece.
- Erosion of the work surface due to cavitations effect of the abrasive slurry.
- Chemical action associated with the fluid used.

It has been reported that among the above-mentioned mechanisms, the first two are primarily responsible for major stock removal. The part played by erosion has been reported as insignificant for normal materials machined by this process

### 1.4 OPTIMIZATION TECHNIQUES

There are several optimization methods available within the software as the as technology is developing. There are a complicated polynomial equation, the complex calculation in optimization method like Taguchi method, RSM (response surface methodology)etc.. This complicated calculation for optimization techniques is reduced using several available software design expertise, Minitab, Matlab, etc.

## II. LITERATURE REVIEW

1. Momber, Andreas W, Radovan Kovacevic. Principles of abrasive waterjet machining. Springer Science & Business Media, 2012. Classified the water jet into three categories, as low-pressure jets and high-pressure jets, as continuous jets and discontinuous jets and plain water jets, water additive jets and abrasive water jets.

2. Qiang, Zhengrong, et al. CFD research on particle movement and nozzle wear in the abrasive waterjet cutting head. The International Journal of Advanced Manufacturing Technology(2018): 1-10. In the year 1930 Elmo Smith and Leslie Tirrell developed abrasive water jet cutting, the advantage of AWJ cutting over another non-traditional machining process like laser beam machining is, the negligible thermal zone is found during cutting thereby no metallurgical changes in the machined material.

3. Nag, Akash, et al.Influence of Abrasive Water Jet Turning Parameters on Variation of Diameter of Hybrid Metal Matrix Composite. Applications of Fluid Dynamics. Springer, Singapore, 2018. 495-504. Removal of material is due to impact erosion. A359/Al2O3/B4C composite material is used by the researchers and reported that the type of abrasive used has more influence in the deviations of cut diameter than the mass flow rate. Electrically conducting and non – conducting materials can be machined

4. Prabhuswamy, N. R., et al. Machinability Studies of Aluminium 6061 cut by Abrasive WaterJet. Materials Today: Proceedings 5.1 (2018): 2865-2870. Using full factorial experiments 27 experiments were conducted for machining Al6061 and parameters are waterjet pressure, jet traverse speed, abrasive mass flow rate and were considered and

5. Carach, Jan, et al. Surface Roughness of Graphite and Aluminium Alloy After Hydro-abrasive Machining. Advances in Manufacturing. Springer, Cham, 2018. 805-813. Graphite and aluminium alloy were used to compare the surface roughness by considering focusing tube diameters of abrasive waterjet and reported that changes in the shape of the workpiece are also observed and concluded that machined workpiece and the quantity of abrasive particle circulated in water jet mainly affects the final shape.

6. Shanmugavel, Rajesh, et al. Mechanical and Machinability characteristics of Al–NiTi composites reinforced with SiC particulates. Journal of the Australian Ceramic Society 53.1 (2017): 177-185. Experiments are conducted on Al–NiTi–SiC composite by abrasive water jet, kerf angle and surface roughness is measured as the production value. The experimental results were compared with optimal results and reported that the surface roughness value is decreased by 14%, and kerf angle decreased by 10% and concluded the percentage contributions, abrasive waterjet pressure (56.40%), feed rate (17.99%), wt.% of the reinforcement (15.49%), and standoff distance (10.10%) upon which the kerf angle and surface roughness depends on.

7. Bhowmik, Sumit, and Amitava Ray. Prediction and optimization of process parameters of green composites in AWJM process using response surface methodology. The International Journal of Advanced Manufacturing Technology87.5-8 (2016): 1359-1370. Selected optimal parameters using multi-response optimization developed by response surface methodology based optimization for machining green composite.

8. Azmir, M. A., A. K. Ahsan, and A. Rahmah. Effect of abrasive water jet machining parameters on aramid fibre reinforced plastics composite. International Journal of Material Forming2.1 (2009): 37-44. Aramid fibre reinforced plastic – composite is machined by abrasive waterjet and for the conduct of experiments Taguchi’s design of the experiment is used.

9. Santhanakumar, M., R. Adalarasan, and M. Rajmohan. Parameter design for cut surface characteristics in abrasive waterjet cutting of Al/SiC/Al 2 O 3 composite using grey theory-based RSM. Journal of Mechanical Science and Technology30.1 (2016): 371-379. Al/SiC/Al2O3 composite is selected for machining by abrasive waterjet was grey theory-based response surface methodology is used for parameter design for the optimal cutting conditions.

10. Kumar, K. Ravi, V. S. Sreebalaji, and T. Pridhar. “Characterization and optimization of Abrasive Water Jet Machining parameters of aluminium/tungsten carbide composites.” Measurement 117 (2018): 57-66. Experimented on A359/Al2O3/B4C composite material and concluded that turning operations can be easily performed by using abrasive waterjet and it is possible to get required levels of surface finish by the using different abrasive grains

11. Carter C B and Norton M G (2013), Ceramic Materials Science and Engineering, Springer Science, Business Media, New York, 4-31. Use of ceramic materials various fields including structural, semiconductor, micro-electro-mechanical systems, medical, defence, aerospace and electronics, ceramic materials properties like wear resistance, high hardness, brittleness, oxidation resistant and specific strength.

12. Wang J and Liu H (2006), Profile Cutting on Alumina Ceramics by Abrasive Water jet Part 1 & 2: Cutting Performance Models, Proceedings of the Institution of Mechanical Engineers, Vol. 220, 715-725.developed predictive models and estimated kerf angle, depth of cut, and kerf top width in abrasive waterjet machining of alumina ceramic. Response surface methodology with the central composite design is employed to design experiments. Based on a review of the literature, some of the issues and research gaps are identified and discussed in the following section.

13. Yuvaraj, N, Pradeep Kumar, M. Multiresponse Optimization of Abrasive Water Jet Cutting Process Parameters Using TOPSIS Approach. Materials and Manufacturing Processes 2015, 30 (7), 882-889.

14. Ramprasad, U.G, Kamal, H. Optimization MRR of Stainless Steel 403 In Abrasive Water Jet Machining Using ANOVA And Taguchi Method. International Journal of Engineering Research and Applications 2015, 5 (5), 86-91.

Metal Matrix Composites with improved mechanical and physical properties MMCs with silicon carbide as reinforcement have important benefit. MMC mainly modify the most excellent properties of the constituents, namely toughness and ductility of the matrix and high strength and modulus reinforcements.

15. D.B. Miracle, Composites Sci. and Tech.65 (2005) 2526-2540.

Removal of material in waterjet machining is due to a continuous stream of forced water droplet jet impingement, this forced water droplet jet and the abrasives present in the water will make the cut possible.

16. Kumar, Harmesh, Alakesh Manna, and Rajesh Kumar. “Modeling of Process Parameters for Surface Roughness and Analysis of Machined Surface in WEDM of Al/SiC-MMC.” Transactions of the Indian Institute of Metals 71.1 (2018): 231-244.

17. Srivastava, Ashish Kumar, Amit Rai Dixit, and Sandeep Tiwari. “A review on the intensification of metal matrix composites and its nonconventional machining.” Science and Engineering of Composite Materials 25.2 (2018): 213-228. Regarding the metal matrix composite usage in the developing world. The required shape of then can be obtained through non-conventional machining process rather than conventional machining process which has better output. The most efficient kind of MMC of various compositions is made, according to previous studies the coherent composites are suggested basing with the reference of their properties. As manufacturing of the product in present-day involves material requirement with lightweight and strength so non-convention would make better machining of such materials. Categories of the machining process are framed focusing on viridium abrasive water jet machining, LAM, UVC. Discussion is carried on those machining process comparing with machining ability with various MMC’s.

18. Mardi, K. Bimla, et al. “Surface integrity of Mg-based nanocomposite produced by Abrasive Water Jet Machining (AWJM).” Materials and Manufacturing Processes 32.15 (2017): 1707-1714. The varying output parameters with respect to the inputs are mainly focused, considering the output parameters as surface roughness and input parameters transverse rate the alternating values are noted. Basing on the values as surface roughness is more there is an increase in irregularities of surface and increase in transverse speed. surface roughness testing methods are used and the value ranges where the parameters vary are noted

19. Shetty, Nagaraja, et al. “A review on finite element method for machining of composite materials.” Composite Structures 176 (2017): 790-802.

20. Sasikumar, K. S. K., et al. “A study on kerf characteristics of hybrid aluminium 7075 metal matrix composites machined using abrasive water jet machining technology.” Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture (2016): 0954405416654085.

Under the non-conventional machining process, there are many parameters involving, considering kerf width, surface roughness, kerf top angle and kerf angles the experiment is carried out on the hybrid aluminium 7075 MMC notifying the values. Undertaking the analysis of those values and applying the optimization methods

21. Pramanik, Alokesh. “Developments in the non-traditional machining of particle reinforced metal matrix composites.” International Journal of Machine Tools and Manufacture 86 (2014): 44-61. Improvements are made in the non-convectional processes to rectify certain flaws involved in it. Basing on the surface finish of material after the machining process the better output can be estimated. Hence making the study of different machining non-convectional process comparison is done. Abrasive water jet machining, electro discharge and laser beam machining would be better suggested than EDMs

22. Arul Kumar, B., and G. Kumaresan. “Abrasive water jet machining of aluminum-silicon carbide particulate metal matrix composites.” Materials Science Forum. Vol. 830. Trans Tech Publications, 2015. Most of the composites are difficult to be machined using a conventional machining process, due to the high tool wear. This paper gives us the complete results when SiC particles reinforced aluminium matrix composites are machined with the abrasive water jet machining process.

23. Bhowmik, Sumit, and Amitava Ray. “Abrasive Water Jet Machining of Composite Materials.” Advanced Manufacturing Technologies. Springer, Cham, 2017. 77-97.

The required of lightweight and hard materials is the reason for the creation of composite materials and replacing the metals by them. It is very hard to machine composite materials with the conventional machining process. This paper deals with the problems involved in the machining process, including corrosion and gives solutions to those problems.

24. Markopoulos, Angelos P., et al. “Machining and Machining Modeling of Metal Matrix Composites—A Review.” Modern Manufacturing Engineering. Springer, Cham, 2015. 99-141.

This paper is regarding overall review of machining processes with various composite materials. Machining processes involves turning milling drilling, tapping etc the machinability of non-conventional machining and errors involved in it the rectifying measures used to overcome the errors or alternative machining process involved.

25. Kavya, J. T., R. Keshavamurthy, and GS Pradeep Kumar. “Studies on parametric optimization for abrasive water jet machining of Al7075-TiB2 in-situ composite.” IOP Conference Series: Materials Science and Engineering. Vol. 149. No. 1. IOP Publishing, 2016. The study of variation in the parameters like a stand of distance, cutting speed is taken on the material Al7075-TiB2 in-situ composite and machined with the water jet machining process. The optimization method TAGUCHI is used and analysis is made. The influence of variation in parameters is observed.

26. Kandpal, Bhaskar Chandra, and Hari Singh. “Machining of aluminium metal matrix composites with electrical discharge machining-a review.” Materials Today: Proceedings 2.4-5 (2015): 1665-1671. Electrical discharge machining (EDM) is the machining process of electrically conductive materials which controllers sparks that occur between electrode and a workpiece in the presence of dielectric fluid. The metal matrix composites(MMC) which have more applications in automobile, aircraft, and railway sectors. Aluminium metal matrix composites are one of the MMC types and their advanced properties like the highest strength and lightweight.

27. Markopoulos, Angelos P., et al. “Modelling and optimization of machining with the use of statistical methods and soft computing.” Design of Experiments in Production Engineering. Springer, Cham, 2016. 39-88.

28. Lauro, Carlos H., et al. “Design of Experiments—Statistical and artificial intelligence analysis for the improvement of machining processes: A review.” Design of Experiments in Production Engineering. Springer, Cham, 2016. 89-107. The Response Surface Methodology (RSM) is a method to optimise and model a problem to define the relationship between parameters and the responses with the desired criteria. the factorial DoE can forecast surface roughness using a small number of experiments. The RSM will have at least three levels for each factor to avoid questions for the estimated values for the combinations of not tested factor.

29. Mishra, Srimant Kumar, Sandhyarani Biswas, and Alok Satapathy. “A study on processing, characterization and erosion wear behaviour of silicon carbide particle-filled ZA-27 metal matrix composites.” Materials & Design 55 (2014): 958-965. the simple liquid metallurgy technique called stir casting technique it is used for the fabrication of composite materials. abrasion trials are made as per the experimental design and The results indicate the abrasion wear rate have more impact velocity. It also shows good filler characteristics of SiC particles as the wear rate decreases with increase in filler in the matrix material.

30. Shukla, Rajkamal, and Dinesh Singh. “Experimentation investigation of abrasive water jet machining parameters using Taguchi and Evolutionary optimization techniques.” Swarm and Evolutionary Computation 32 (2017): 167-183. Abrasive water jet machining (AWJM) is used in hard material for removal of metals.it has high-quality cutting and good surface finish. AWJM process uses a mixture of water and abrasive to dissolve material from the target surface.

31. Mardi, K. Bimla, et al. “Surface integrity of Mg-based nanocomposite produced by Abrasive Water Jet Machining (AWJM).” Materials and Manufacturing Processes 32.15 (2017):

32. Gilles, Patrick, et al. “A new cutting depth model with rapid calibration in abrasive water jet machining of titanium alloy.” The International Journal of Advanced Manufacturing Technology 93.5-8 (2017): 1499-1512.

33. Klich, Jiri, et al. “Influence of Variously Modified Surface of Aluminium Alloy on the Effect of Pulsating Water Jet.” Strojniški Vestnik-Journal of Mechanical Engineering 63.10 (2017): 577-582.

34. Klich, Jiří, Dagmar Klichová, and Petr Hlaváček. “Effects of the pulsating water jet on aluminium alloy with the variously modified surface.” Tehnički vjesnik 24.2 (2017): 341-345.

35. Ushasta Aicha,*, Simul Banerjeea, Asish Bandyopadhyaya, Probal Kumar Dasb “Abrasive Water Jet Cutting of Borosilicate Glass” Borosilicate glass being a brittle material convectional machining process would put the limitations. Depth of cut is calculated with various machine input parameter settings water pressure, abrasive flow rate, transverse speed and standoff distance. The output parameters give a depth of cut gives a scheme of the impact of various parameters on machining of amorphous borosilicate glass.

36. Vishal Guptaa, P.M. Pandeya, Mohinder Pal Garg*b, Rajesh Khannab, N.K.Batrab “Minimization of kerf taper angle and kerf width using Taguchi’s method in abrasive water jet machining of marble”

This paper gives information regarding abrasive water jet cutting is a traditional machining method provides productive replacement of conventional techniques. According to Taguchi design of experiments were conducted. Three various process parameters were considered water pressure nozzle, transverse speed and abrasive flow rate machining on the marble.

## III. RESPONSE SURFACE METHODOLOGY

### 3.1 RESPONSE SURFACE METHODOLOGY

Response surface methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building. By careful design of experiments, the objective is to optimize a response (output variable) which is influenced by several independent variables (input variables). An experiment is a series of tests, called runs, in which changes are made in the input variables in order to identify the reasons for changes in the output response. Originally, RSM was developed to model experimental responses (Box and Draper, 1987), and then migrated into the modelling of numerical experiments. The difference is in the type of error generated by the response. In physical experiments, inaccuracy can be due, for example, to measurement errors while, in computer experiments, numerical noise is a result of incomplete convergence of iterative processes, round-off errors or the discrete representation of continuous physical phenomena (Giunta et al., 1996; van Campen et al., 1990, Toropov et al., 1996). In RSM, the errors are assumed to be random. Response surface methodology 16 The application of RSM to design optimization is aimed at reducing the cost of expensive analysis methods (e.g. finite element method or CFD analysis) and their associated numerical noise. The problem can be approximated as described in Chapter 2 with smooth functions that improve the convergence of the optimization process because they reduce the effects of noise and they allow for the use of derivative-based algorithms. Venter et al. (1996) have discussed the advantages of using RSM for design optimization applications. The engineer wants to find the levels of temperature (x1) and time (x2) that maximize the early age strength (y) of the cement. The early age strength is a function of the levels of temperature and time, as follows:

y = f (x_{1}, x_{2}) + ε (3.1)

Where ε represents the noise or error observed in the response y. The surface represented by f(x1, x2) is called a response surface. The response can be represented graphically, either in the three-dimensional space or as contour plots that help visualize the shape of the response surface. Contours are curves of constant response drawn in the xi , xj plane keeping all other variables fixed. Each contour corresponds to a particular height of the response surface.

This chapter reviews the two basic concepts in RSM, first choice of the approximate model and, second, the plan of experiments where the response has to be evaluated.

### 3.2 APPROXIMATE MODEL FUNCTION

The structure of the relationship between the response and the independent variables is unknown. The first step in RSM is to find a suitable approximation to the true relationship. The most common forms are low-order polynomials (first or second-order). In this thesis, a new approach using genetic programming is suggested. The advantage is that the structure of the approximation is not assumed in advance, but is given as part of the solution, thus leading to a function structure of the best possible quality. In addition, the complexity of the function is not limited to a polynomial but can be generalised with the inclusion of any mathematical operator (e.g. Response surface methodology trigonometric functions), depending on the engineering understanding of the problem. The regression coefficients included in the approximation model are called the tuning parameters and are estimated by minimizing the sum of squares of the errors (Box and Draper, 1987):

where wp is a weight coefficient that characterizes the relative contribution of the information of the original function at the point p, p=1,…,P. The construction of response surface models is an iterative process. Once an approximate model is obtained, the goodness-of-fit determines if the solution is satisfactory. If this is not the case, the approximation process is restarted and further experiments are made or the GP model is evolved with different parameters. To reduce the number of analyses in computer simulations, sensitivity data may 17 be used in the model fitting, although this information is not always available at low cost. If in addition to the values of the original function Fp = F(xp) their first-order derivatives at point

are known, the problem (3.2) is replaced by the following one (Toropov et al., 1993

where γ >0 is the parameter characterizing a degree of inequality of the contribution of the response and the sensitive data. In this thesis, γ is taken as 0.5, following recommendations by Toropov et al. (1993). Van Keulen et al. (2000) have presented a methodology for the construction of responses using both function values and derivatives on a weighted least-squares formulation. The authors conclude that the use of derivatives provides better accuracy and requires a reduced number of data.

### 3.3 DESIGN OF EXPERIMENTS

An important aspect of RSM is the design of experiments (Box and Draper, 1987), usually abbreviated as DoE. These strategies were originally developed for the model fitting of physical experiments, but can also be applied to numerical experiments. The objective of DoE is the selection of the points where the response should be evaluated. Most of the criteria for the optimal design of experiments are associated with the mathematical model of the process. Generally, these mathematical models are polynomials with an unknown structure, so the corresponding experiments are designed only for every particular problem. The choice of the design of experiments can have a large influence on the accuracy of the approximation and the cost of constructing the response surface. In a traditional DoE, screening experiments are performed in the early stages of the process, when it is likely that many of the design variables initially considered have little or no effect on the response. The purpose is to identify the design variables that have large effects for further investigation. Genetic Programming has shown good screening properties (Gilbert et al., 1998), as will be demonstrated in Section 6.2, which suggests that both the selection of the relevant design variables and the identification of the model can be carried out at the same time. A detailed description of the design of experiments theory can be found in Box and Draper (1987), Myers and Montgomery (1995) and Montgomery (1997), among many others. Schoofs (1987) has reviewed the application of experimental design to structural optimization, Unal et al. (1996) discussed the use of several designs for response surface methodology and multidisciplinary design optimization and Simpson et al. (1997) presented a complete review of the use of statistics in design. As introduced in Section 3.1, a particular combination of runs defines an experimental design. The possible settings of each independent variable in the N-dimensional space are called levels. A comparison of different methodologies is given in the next section.

#### 3.3.1 FULL FACTORIAL DESIGN

To construct an approximation model that can capture interactions between N design variables, a full factorial approach (Montgomery, 1997) may be necessary to investigate all possible combinations. A factorial experiment is an experimental strategy in which design variables are varied together, instead of one at a time. The lower and upper bounds of each of N design variables in the optimization problem needs to be defined. The allowable range is then discretized at different levels. If each of the variables is defined at only the lower and upper bounds (two levels), the experimental design is called 2N full factorial. Similarly, if the midpoints are included, the design is called 3N full factorial and shown in Figure 3.2.

Factorial designs can be used for fitting second-order models. A second-order model can significantly improve the optimization process when a first-order model suffers a lack of fit due to interaction between variables and surface curvature. A general second-order model is defined as

where xi and xj are the design variables and are the tuning parameters. The construction of a quadratic response surface model in N variables requires the study at three levels so that the tuning parameters can be estimated. Therefore, at least (N+1) (N+2) / 2 function evaluations are necessary. Generally, for a large number of variables, the number of experiments grows exponentially (3 N for a full factorial) and becomes impractical. A full factorial design typically is used for five or fewer variables. If the number of design variables becomes large, a fraction of a full factorial design can be used at the cost of estimating only a few combinations between variables. This is called fractional factorial design and is usually used for screening important design variables. For a 3 N factorial design, a fraction can be constructed, resulting in 3N-p points. For example, for p=1 in a 3 3 design, the result is a one-third fraction, often called 33-1 design, as shown in Figure 3.3 (Montgomery, 1997).

#### 3.3.2 CENTRAL COMPOSITE DESIGN

A second-order model can be constructed efficiently with central composite designs (CCD) (Montgomery, 1997). CCD are first-order (2 N ) designs augmented by the additional centre and axial points to allow estimation of the tuning parameters of a second-order model. Figure 3.4 shows a CCD for 3 design variable

In Figure 3.4, the design involves 2 N factorial points, 2N axial points and 1 central point. CCD presents an alternative to 3N designs in the construction of second-order models because the number of experiments is reduced as compared to a full factorial design (15 in the case of CCD compared to 27 for a full-factorial design). CCD has been used by Eschenauer and Mistree (1997) for the multiobjective design of a flywheel. In the case of problems with a large number of designs variables, the experiments may be timeconsuming even with the use of CCD. 3.3.3 D-optimal designs The D-optimality criterion enables more efficient construction of a quadratic model (Myers and Montgomery, 1995). The objective is to select P design points from a larger set of candidate points. Equation (3.4) can be expressed in matrix notation as:

*Y = X*B+C*

where Y is a vector of observations, e is a vector of errors, X is the matrix of the values of the design variables at plan points and B is the vector of tuning parameters. B can be estimated using the least-squares method as:

B = (X^{t}+X)^{-1} X^{T} Y

The D-optimality criterion states that the best set of points in the experiment maximizes the determinant | X T X |. “D” stands for the determinant of the X T X matrix associated with the model. From a statistical point of view, a D-optimal design leads to response surface models for which the maximum variance of the predicted responses is minimized. This means that the points of the experiment will minimize the error in the estimated coefficients of the response model. The advantages of this method are the possibility to use irregular shapes and the possibility to include extra design points. Generally, D-optimality is one of the most used criteria in the computer-generated design of experiments. Several applications are described in Giunta et al. (1996) for the wing design of a high-speed civil transport and Unal et. al. (1996) for a multidisciplinary design optimization study of a launch vehicle. Haftka and Scott (1996) have reviewed the use of D-optimality criteria for the optimization of experimental designs.

#### 3.3.3 TAGUCHI ‘S CONTRIBUTION TO EXPERIMENTAL DESIGN

Taguchi’s methods (Montgomery, 1997) study the parameter space based on the fractional factorial arrays from DoE, called orthogonal arrays. Taguchi argues that it is not necessary to consider the interaction between two design variables explicitly, so he developed a system of tabulated designs which reduce the number of experiments as compared to a full factorial design. An advantage is the ability to handle discrete variables. A disadvantage is that Taguchi ignores parameter interactions.

#### 3.3.4 LATIN HYPERCUBE DESIGN

Latin hypercube design (McKay et al., 1979) can be viewed as an N-dimensional extension of the traditional Latin square design (Montgomery, 1997). On each level of every design variable, only one point is placed. There is the same number of levels as runs and the levels are assigned randomly to runs. This method ensures that every variable is represented, no matter if the response is dominated by only a few ones. Another advantage is that the number of points to be analyzed can be directly defined. An example of the use of such plans can be found in Schoofs et al. (1997). 3.3.5 AUDZE-EGLAIS’ APPROACH Audze and Eglais (1977) suggested a non-traditional criterion for the elaboration of plans of experiments which, similar to the Latin hypercube design, is not dependent on the mathematical model of the problem under consideration. The input data for the elaboration of the plan only include the number of factors N (number of design variables) and the number of experiments K. The main principles in this approach are as follows:

- The number of levels of factors (same for each factor) is equal to the number of experiments and for each level, there is only one experiment. This is similar to the Latin hypercube design.
- The points of experiments are distributed as uniformly as possible in the domain of variables. There is a physical analogy with the minimum of the potential energy of repulsive forces for a set of points of unit mass if the magnitude of these repulsive forces is inversely proportional to the distance squared between the points

where Lpq is the distance between the points having numbers p and q (p≠q). The elaboration of the plans is time-consuming, so each plan of the experiment is elaborated only once and stored in a matrix characterized by the levels of factors for each of the P experiments. For example, for a number of factors (design variables) N = 2 and P = 10, the matrix is

The plan (3.8) is represented in Figure 3.5 and compared with a CCD for two design variables with 9 runs.

#### 3.3.6 VAN KEULEN’S APPROACH

In the course of an iterative optimization process modelled by approximations, new points must be generated in specified domains of the design variable space. A new scheme for the design of experiments (Van Keulen and Toropov, 1999) has been formulated with the following characteristics:

- The scheme works efficiently even if only a single additional design point is generated to the existing plan. For a number of new design points, the algorithm is used several times.
- The scheme remains effective if different types of functions are used within the same optimization task to approximate the objective function and the constraints. The approach distributes points as homogeneously as possible in the subdomains of interest. This is done by the introduction of the following cost function:

which is minimized with respect to the location of the new point d. Symbols denoted refer to coordinates which are normalized in the sub-domain of interest. The first term in the expression attempts to maximize the distance between points, and the second term promotes a homogeneous distribution along the coordinate axes. The third and fourth terms ensure that points do not belong to the boundary of the subdomain. The last term prevents points from aligning along the diagonal of the search sub-region when only a few points are available.

### 3.4 CONCLUSION

The response surface methodology analysis has been reviewed. RSM can be used for the approximation of both experimental and numerical responses. Two steps are necessary, the definition of an approximation function and the design of the plan of experiments. A review of different designs for fitting response surfaces has been given. A desirable design of experiments should provide a distribution of points throughout the region of interest, which means to provide as much information as possible on the problem. This “space-filling” property is a characteristic of three plans: Latin hypercube sampling, Audze-Eglais and van Keulen. All three plans are independent of the mathematical model of the approximation. However, Latin hypercube sampling distributes the points randomly in the space, while Audze-Eglais uses a distribution based on the maximum separation between points. The Audze-Eglais plan has been chosen in this thesis. It should be noted that if the model building is to be repeated within an iterative scheme (e.g. with mid-range approximations), van Keulen’s plan would become an attractive alternative as it adds points to an existing plan. This thesis is primarily focused on building global approximations.

## IV. EXPERIMENTAL DETAILS

### 4.1 MATERIAL USED FOR EXPERIMENT

Aluminium MMC is used to conduct the experiment. Aluminium Sic is the metal matrix composite we used. The properties are:

Typical Properties:

- Density, g/cm3 2.95 – 3.00
- Thermal Conductivity, W/m•K 170 – 200
- Thermal Expansion Coefficient, E-6 K-1 tailor-made from 6.5 to 9.5
- Electrical Resistivity, mW•cm 30 – 50
- Bending Strength, MPa 350 – 500

### 4.2 DIMENSIONS OF THE WORK PIECE:

The work piece of certain dimension is considered and single cut length is 20mm 120 mm × 45 mm and cut is considered in this format.

### 4.3 PARAMETERS

The parameters involved in this experiment are :

- MATERIAL REMOVAL RATE: The amount of material volume removed at instant input values given to the machining.
- STAND-OFF-DISTANCE (SOD): Stand-off-distance is defined as the distance between the face of the nozzle and the working surface of the work
- JET PRESSURE: The pressure of water to remove the material is called jet pressure.
- TRANSVERSE RATE: Material removal from the workpiece through particular time and transverse to the other cut is known as transverse rate
- ABRASIVE RATE: The abrasive material used for a cut along with fluid while machining according to the time is called abrasive rate
- TIME: Time taken for a single cut is noted

4.4 PROCEDURE

- DESIGN OF EXPERIMENTS
- EXPERIMENTATION
- OPTIMIZATION METHOD
- PREDICTING THE OPTIMISTIC PARAMETERS

**LINE DIAGRAM OF PROCEDURE**

#### 4.4.1 DESIGN OF EXPERIMENTS:

In this stage the, according to machining process we choose the input parameters basing on machine specification and later stage we extract the output. machine specification and later stage we extract the output.

Now according to this input parameter specification, we need to select the ranges of three value for each parameter for applying analysis method

Four values each parameter range is set and must obtain the output parameters.

### DESIGN OF EXPERIMENTS

#### 4.4.2 EXPERIMENTATION:

The work piece cutting has been done according to the design of experiment values. After the machining the work obtains thirty slots with respect to the input values as shown in figure 120 mm × 45 mm and cut is considered in this format

The gap in the slot of the cut is known as kerf width. Kerf width is calculated using a binocular microscope. Taking the average of three values of each slot is calculated.

BINOCULARS MICROSCOPE

Least count of the microscope 1mm= 50 division 1 division = 0.02 mm

## V. EXPERIMENTAL RESULTS

### 5.1 OPTIMIZATION

Using response surface methodology the values of the experiment were optimized. Using design expertise response surface methodology was applied and obtained results

### 5.2 POSTULATION MODEL

The experimental value at each reading is used to supplement the mathematical model according to the response surface methodology. This model involves the consideration of kerf width with different control variable setting during machining of Al7075/SiCp. Machining performance of Abrasive water jet machining completely significant with the interaction of the control variables. Due to this reason, the output response is coded to be in equation of second-order polynomial models. The postulated model is represented as a regression n coefficient as shown below equation. Kerf Width = +19969.61+96.21 A+42.27 B+195.75 C+14493.82D+0.082 AB+0.23AC+ 35.03AD+0.17BC+15.33B D+71.24C D +0.028 A2+4.937E-003 B2+0.26 C2+2629.96 D2 The decoded values A,B,C and D represents the logarithmic transformation of the control factors jet pressure, standoff distance. Traverse rate, abrasive rate respectively.

### 5.3 TESTS FOR ADEQUACY OF THE MODEL

The supplement empirical model is firm for its acceptability using the tests. Inspection of variance (ANOVA) is done for the quadratic response surface model and the readings are given table 5. From table % it was noticed that the value of “prob>F” for the model is less than 0.005 which shows model is crucial. The multiple regression coefficient R2 is calculated to verify whether the model is fitted according to the described data. The readings of R2 is defined as the ratio of the variability described by the model to the total variability in the real experimental data and is utilized as a measure of righteousness of fit. if R2 is near to the unity then the experimental data confines with this model better.

The “Model F-value” of 0.90 implies the model is not significant relative to the noise. There is a 57.68 % chance that a “Model F-value” this large could occur due to noise.

Values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case, there are no significant model terms. Values greater than 0.1000 indicate the model terms are not significant. If there are many insignificant model terms (not counting those required to support hierarchy), model reduction may improve your model.

The “Lack of Fit F-value” of 1.58 implies the Lack of Fit is not significant relative to the pure error. There is a 29.71% chance that a “Lack of Fit F-value” this large could occur due to noise. Non-significant lack of fit is good — we want the model to fit.

A negative “Pred R-Squared” implies that the overall mean is a better predictor of your response than the current model.

“Adeq Precision” measures the signal to noise ratio. A ratio greater than 4 is desirable. Your ratio of 4.038 indicates an adequate signal. This model can be used to navigate the design space. The variables A, B, C, D indicate parameters jet pressure, standoff distance, transverse rate, abrasive rate respectively.

## CONCLUSION

In this project, we have worked on the specification of abrasive water jet machining. The output parameters how they vary with respect to the input parameters for the optimal working of the machine. An attempt is made for machining aluminium metal matrix composite with silicon carbide using water jet machining and considers the input parameters kerf width is selected as output parameter that is maximization of material removal rate and minimization of the kerf.

*References *

*Nag, Akash, et al. “Influence of Abrasive Water Jet Turning Parameters on Variation of Diameter of Hybrid Metal Matrix Composite.” Applications of Fluid Dynamics. Springer, Singapore, 2018. 495-504.**Gnanavelbabu, A., Kaliyamoorthy Rajkumar, and P. Saravanan. “Investigation on the cutting quality characteristics of abrasive water jet machining of AA6061-B4C-hBN hybrid metal matrix composites.” Materials and Manufacturing Processes (2018): 1-11.**Kumar, K. Ravi, V. S. Sreebalaji, and T. Pridhar. “Characterization and optimization of Abrasive Water Jet Machining parameters of aluminium/tungsten carbide composites.” Measurement 117 (2018): 57-66.**Srivastava, Madhulika, et al. “Journal of Manufacturing Processes.” (2018).455–468**Mardi, K. Bimla, et al. “Surface integrity of Mg-based nanocomposite produced by Abrasive Water Jet Machining (AWJM).” Materials and Manufacturing Processes 32.15 (2017): 1707-1714.*

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