Modeling and optimization of laser beam percussion drilling of thin aluminum sheet

Optics & Laser Technology 48 (2013) 461–474

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Modeling and optimization of laser beam percussion drilling of thin aluminum sheet Sanjay Mishra, Vinod Yadava n Motilal Nehru National Institute of Technology, Allahabad, India

a r t i c l e i n f o

abstract

Article history: Received 2 August 2012 Received in revised form 10 September 2012 Accepted 25 October 2012 Available online 20 December 2012

Modeling and optimization of machining processes using coupled methodology has been an area of interest for manufacturing engineers in recent times. The present paper deals with the development of a prediction model for Laser Beam Percussion Drilling (LBPD) using the coupled methodology of Finite Element Method (FEM) and Artificial Neural Network (ANN). First, 2D axisymmetric FEM based thermal models for LBPD have been developed, incorporating the temperature-dependent thermal properties, optical properties, and phase change phenomena of aluminum. The model is validated after comparing the results obtained using the FEM model with self-conducted experimental results in terms of hole taper. Secondly, sufficient input and output data generated using the FEM model is used for the training and testing of the ANN model. Further, Grey Relational Analysis (GRA) coupled with Principal Component Analysis (PCA) has been effectively used for the multi-objective optimization of the LBPD process using data predicted by the trained ANN model. The developed ANN model predicts that hole taper and material removal rates are highly affected by pulse width, whereas the pulse frequency plays the most significant role in determining the extent of HAZ. The optimal process parameter setting shows a reduction of hole taper by 67.5%, increase of material removal rate by 605%, and reduction of extent of HAZ by 3.24%. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Finite element method (FEM) Laser beam percussion drilling (LBPD) Multi-objective optimization

1. Introduction The combined property of monochromaticity, high coherence, and low diffraction facilitate the precise control of laser beam movement with respect to workpiece and helps to focus a large amount of radiant energy on a very small area of irradiated surface to impart high laser energy density. Therefore, it can be advantageously used for material processing of difficult-tomachine advanced engineering materials. Laser Beam Percussion Drilling (LBPD) is a unidirectional thermal energy-based contactfree optical machining process for the creation of miniature holes in the workpiece. In this process a series of laser pulses of specified energy and pulse duration is applied at a single spot to melt or vaporize the material from the workpiece. In LBPD the melt front propagates along the direction of the laser beam to produce a hole; therefore, it is considered as a onedimensional laser machining process. The efficiency of LBPD depends on the thermal and optical properties of the sheet material instead of its mechanical properties, therefore it can be conveniently used for producing small diameter holes in highstrength thermal-resistant materials used in the technologically

n Corresponding author. Tel.: þ91 5322271812; fax: þ 91 5322445101, þ91 5322445077. E-mail address: [email protected] (V. Yadava).

0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.10.035

advanced industries. Holes with diameter ranging from 127 to 1270 mm with depth to diameter ratio of 100:1 can be produced using LBPD in alloys, ceramics, and composites. The primary advantage of using LBPD for fabrication of small diameter holes over the trepanning and helical laser drilling technique is its ability to drill through holes in a thick workpiece (upto 20 mm) in short time (o1 s). It is the most cost effective laser drilling technique for producing a large number of small diameter holes in a very small area. LBPD is extensively used for the economical production of a large number of closely spaced holes in aircraft wings and aircraft engine components, for creating fine cooling holes in nozzle guide vanes, and turbine blades to operate the engine at higher temperature. It is also widely used in the fabrication of miniature components for electronics, automotive, biomedical, Micro Electro Mechanical Systems (MEMS), etc. As the LBPD process requires the temperature of target workpiece to be raised above its melting point, a laser with short pulse length is generally preferred. Moreover, the requirement of small diameter holes with high aspect ratio also necessitates the use of a laser with short wavelength; therefore, a solid state-pulsed Nd:YAG laser is widely used for LBPD. It yields high intensity of laser beam with very good focusing characteristics [1]. Increasingly tighter geometrical tolerances and more stringent hole quality requirements of modern industrial components demand minimum defects in laser-drilled components. In order

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to increase the viability and cost effectiveness of LBPD for fabrication of high aspect ratio holes in industries, the manufacturing engineers should be able to produce geometrically accurate holes with minimum metallurgical distortions in the shortest possible machining time. The ability to predict the hole characteristics with respect to various input parameters will serve the dual purpose of better understanding the interaction mechanisms between the laser beam and workpiece, as well as enhancing the productivity of the process. A hole produced by LBPD suffers with a large number of geometrical inaccuracies in terms of hole taper (Ta) and rough holes with a high degree of roundness error. The Ta reveals the non-cylindrical nature of holes.The walls of the hole exhibit a recast layer and heat-affected zone that can vary in thickness from 2 to 100 mm depending on thermo-physical property of the material, thickness of workpiece, and laserrelated parameters like pulse width, pulse frequency, peak power, etc. The extent of HAZ shows the structural changes due to the dispersion of thermal waves into the workpiece and thus it reveals the metallurgical characteristic of a laser-drilled hole. Since the material removal during LBPD depends on the amount of laser energy entering the workpiece, lasers related parameters i.e., pulse width, pulse frequency, and peak power play a vital role in this process. Moreover, the amount of total laser energy required to produce through holes also depends on the geometrical dimension of the workpiece; therefore, thickness of the workpiece has also been considered as an input process parameter. The productivity of LBPD can be adjudged in terms of materials removed per unit time, better known as material removal rate (MRR). It not only indicates the machinability of materials with respect to the laser but also help us to identify those laser parameters which will provide the required power density for the LBPD process. Considerable research has been carried out to develop a holistic model of the LBPD process to divulge the mechanism involved in laser drilling. Hanon et al. [2] compared the experimental and simulation results of single-pulse laser drillings of 5 and 10.5 mm thick alumina ceramics using a pulsed Nd:YAG laser of 600 W. They found that in single-pulse drilling, peak power and pulse duration can be effectively used for changing the crater depth without any defects. Yan et al. [3] developed 2D axisymmetric Finite Element Method (FEM)-based thermal model to predict the spatter deposition and HAZ during LBPD of 4.4 mm thick alumina ceramics using a CO2 laser of 3.5 kW. It was observed that spatter deposition primarily depends on the peak power and pulse repetition rate of the laser. Ganesh et al. [4] developed a 2D axisymmetric computational model incorporating the effect of vaporization and melt ejection to determine the crater geometry and material removal in single-pulse laser drilling. They found that the effect of vaporization on hole geometry is very small and the rate of material removal is inversely proportional to the square root of the peak laser intensity. Nowak and Prypurniewicz [5] introduced the 3D FDM-based thermal model for the determination of cross-sectional areas of holes produced by LBPD in a partially transparent fired Al2O3 ceramic of 0.75 mm thickness using a solid state-pulsed Nd:YAG laser. For materials having low thermal diffusivity the shape of the laserdrilled hole is found to be strongly dependent on shape and irradiance distribution of laser beam. Cheng et al. [6] proposed a 3D FDM-based thermal model to predict the depth and diameter of holes in thin sheets of metals (aluminum, copper, and mild steel of thickness 2 mm) and carbon fibre-reinforced thermoplastic sheets (1 mm thick) using a pulsed Nd:YAG laser. Yilbas et al. [7] developed 1D analytical thermal model to investigate the drilling efficiency during laser drilling in titanium, nickel, and stainless steel of 0.7 mm and 1 mm thickness. They found that that the drilling efficiency improves when the drilling is

performed above the threshold power. Collins and Gremaud [8] introduced the 1D mathematical model of LBPD to determine the hole radius at a particular depth assuming constant, conical, and parabolic profiles of the laser-drilled hole. Zhang and Faghri [9] developed a 1D analytical thermal model to predict the geometric shape of laser-drilled holes in Hastelloy-X, incorporating the simultaneous transfer of heat in the melt and solid. It was found that loss of heat due to conduction plays a vital role in the determination of hole shape because it controls the thickness of the liquid layer. Salonitis et al. [10] developed a 3D analytical thermal model for the determination of the minimum power required to drill a hole of specified depth in 10 mm thick structural steel (St.37) rods using a CO2 laser, incorporating the effect of pulse repetition rate and power density. Samant et al. [1] proposed a 3D computational thermal model to determine the depth of laser-drilled holes and number of pulses needed to drill a hole of required depth in 2 mm and 3 mm thick silicon carbide plates, incorporating the effect of decomposition, recoil pressure, and surface tension using a pulsed Nd:YAG laser. Recently, several researchers have employed various statistical and numerical approaches for the prediction and optimization of various advance-machining processes. ANN models have been widely used for process modeling due to their non-linear, adaptive, and learning abilities, using the collected data. Biswas et al. [11] developed an ANN-based experimental model of laser microdrilling of a 0.496 mm thick titanium nitride–alumina (TiN–Al2O3) composite to analyze the effects of lamp current, pulse frequency, pulse width, air pressure, and focal length on output responses viz. circularity at the entrance and exit hole taper. Ghoreishi and Nakhjavani [12] developed an ANN-based process model to establish the relation between process parameters (peak power, pulse width, pulse frequency, number of pulses, assist gas pressure, and focal plane position) and output responses (diameter at hole entrance and hole exit, circularity of hole entrance and hole exit, and hole taper) of LBPD using the experimental data. A pulsed Nd:YAG laser was used to drill sheets of a 2.5 mm thick stainless steel. Panda and Yadava [13] developed an ANN model for the die-sinking electrochemical spark machining of silicon nitride using the combined approach of FEM and ANN. Dhara et al. [14] developed an ANN model for the laser micro-machining of a 35 mm thick high speed steel using a pulsed Nd:YAG laser. Dhupal et al. [15] developed a predictive model of laser turning using ANN to predict the depth and width of micro-grooves on a cylindrical aluminum oxide workpiece of diameter 10 mm and length 40 mm. Majumder [16] compared the predictive accuracy of two separate models based on Response Surface Methodology (RSM) and ANN to predict the surface roughness of laser-drilled holes in mild steel of 0.65 mm thickness using a Nd:YAG laser. The authors found that the ANN model is more accurate and robust in predicting the output performance. Similarly, Karazi et al. [17] compared the performance of a model based on ANN and Design of Experiment (DoE) to predict the width and depth of laser machined micro-channels in glass. It was observed that the ANN-based models are more efficient in the accurate prediction of laser parameters than the counterpart. Grey relational analysis (GRA) has been used by few researchers for the multi-objective optimization (MOO) of the laser machining operation. In GRA, the multi quality characteristics are transformed into one index known as the grey relational grade, which represents the correlation between two sequences so that the distance of two factors can be measured discretely. Panda et al. [18] applied GRA to find the optimum process parameters (pulse width, number of pulses, assist gas flow rate, and gas pressure) with multiple quality characteristics (i.e., HAZ, hole circularity, and MRR) during laser drilling of high carbon

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steel. The GRA approach was used by Padhee et al. [19] to obtain optimum cutting parameters for multi quality characteristics during laser drilling of metal-matrix composites (MMC). The effect of pulse width, number of pulses, and concentration of SiCp in MMC has been studied with the objective to minimize the Ta, spatter, and HAZ. Ganguly et al. [20] used GRA to determine the laser micro-drilling parameters (lamp current, pulse frequency, air pressure, and pulse width) for the simultaneous optimization of multi quality characteristics (Ta and width of HAZ) during the laser micro-drilling of 1 mm thick zirconium oxide ceramic. Mondal et al. [21] applied the GRA approach for the MOO of the laser cladding process so that performance characteristics like clad height, clad width, and clad depth can be improved, corresponding to the input process parameters (laser power, scan speed, and powder feed rate). GRA with the entropy measurement method to find the weight of the quality characteristics has been used by a few authors for the simultaneous optimization of the laser cutting process using Nd:YAG lasers [22–24]. Moreover, some authors have illustrated the successful application of the GRA approach independently for the simultaneous optimization of multi-performance characteristics during laser cutting [25–27]. In the LBPD process it is generally desired to achieve the multi performance characteristics which are correlated. When the multiple performance characteristics are correlated, the use of GRA may yield a local optimum. To eliminate this, the Principal Component Analysis (PCA) is used, which transforms a set of correlated variables into a set of uncorrelated principal components. Sibalija et al. [28] applied the combined approach of PCA and GRA for the multi-response optimization of seven quality characteristics (hole diameter at entry and exit, circularity at entry and exit, aspect ratio, taper, and spatter) during pulsed Nd:YAG laser drilling of a 1.2 mm thick nickel-based superalloy sheet. Gauri and Chakraborty [29] applied the PCA approach for the multi-response optimization of wire electrical discharge machining. Dubey and Yadava [30] proposed a hybrid approach of Taguchi methodology (TM) and PCA to optimize the quality characteristics of kerf width, kerf deviation, and kerf taper during the laser cutting of Ni-based superalloy sheets. In this paper, a 2D axisymmetric FEM-based thermal model has been developed, incorporating the effect of temperaturedependent thermal properties, optical properties, and phasechange phenomena to determine the transient temperature distribution in the sheet form of highly reflective, difficult to laser-drill aluminum sheets. The temperature profile so obtained is further utilized to obtain the Ta, MRR, and extent of HAZ. Sufficiently large number of input-output data sets has been generated using the validated FEM-based thermal model which is further used to train the ANN model of LBPD. Attempts have also been made for the multi-objective optimization of the LBPD process using the coupled approaches of GRA and PCA.

2. Thermal modeling and finite element formulation for temperature In LBPD, absorption of laser energy from a high intensity laser beam takes place and it is converted into thermal energy to raise the temperature within the laser-irradiated zone. It is a nonsteady process because, once the material is removed from the workpiece, the hole shape and temperature distribution inside the workpiece change continuously with respect to coordinate axes fixed at the laser beam. The efficiency of LBPD primarily depends on the thermo-optic interaction between the laser beam and the workpiece. Therefore, several aspects like: (i) temperature-dependent thermal properties (ii) temperaturedependent optical properties (absorptivity), and (iii) phase change

463

phenomena have been incorporated in the thermal model to evaluate the temperature distribution during LBPD. Due to the complex nature of the LBPD process, a number of assumptions have been made in deriving the thermal model. 1 The zone of influence of pulsed laser beam in the sheet form of the workpiece is considered to be axisymmetric; i.e., (qT/qy ¼0). 2 The sheet material is homogeneous and isotropic in nature. 3 On-time of pulsed laser is considered to be much shorter than the pulse-off time, and therefore plasma generation does not take place in the laser-drilled hole. 4 The evaporated material is transparent and does not interfere with the incoming laser beam. 5 The metal vapour is optically thin so that its absorption of the high-energy beam is negligible. 6 Gaussian spatial distribution of the laser heat flux is assumed due to smooth drop of irradiance from the beam center towards the radial direction. 7 Since in the LBPD process it is very difficult to track the solid– liquid and liquid–vapour interfaces, it is assumed that all the molten material has been removed from the hole once the melting take place. 8 Multiple reflections of the laser irradiation within the hole are neglected.

2.1. Governing equation Due to the axisymmetric nature of the LBPD process, a small cylindrical portion of the workpiece around the axis of the laser pulse is used as the domain (D). The governing equation for determining the transient temperature distribution within the axisymmetric workpiece domain can be expressed using Eq. (1) [31]. This governing equation is obtained by combining the Fourier law with the principle of conservation of energy and it can be solved with known conditions of temperature or heat at a boundary. The transient temperature distribution inside the sheet due to the interaction of a laser pulse is obtained by solving Eq. (1), subject to the satisfaction of the boundary conditions (Eqs. (2–4)) and the initial condition. " #   1 @ @T ðr,z,t Þ @2 T ðr,z,t Þ kðT Þ þ r @r @r @z2 ¼ rðT ÞC ðT Þ

@T ðr,z,t Þ @t

in domain ABCD

ð1Þ

where T is the transient temperature at (r, z, t), t is the time, k(T) is the temperature-dependent thermal conductivity, r(T) is the temperature-dependent density, C(T) is the temperaturedependent specific heat of the workpiece material in solid state, and r and z are the coordinate axes as shown in Fig. 1. 2.1.1. Boundary and initial conditions At the start of the LBPD process the initial temperature of the entire domain is equal to the ambient room temperature (T0); i.e., T(r, z, 0) ¼T0 in the workpiece domain ABCD at t ¼0. The energy transferred to the workpiece as heat input serves as the thermal boundary condition. On the top surface B1 when the duration of the pulse is lower than the applied pulse width and the radial distance (r) is larger than the laser pulse radius (R), convection will take place, whereas for distances smaller than or equal to the laser radius, heat flux will enter the sheet as shown by Eq. (2). Similarly, when the duration is higher than the pulse width but lower than the total pulse duration convection will take place on boundary B1, as shown in Eq. (3). The boundaries B2 and B3 are considered to be at such a large distance that no heat transfer

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PP ðW Þ ¼

Average laser power ðWÞ  1000 Repetition rate ðHzÞ  pulse width ðmsÞ

ð7Þ

2.1.3. Temperature-dependent absorptivity In LBPD, the optical properties of the workpiece plays an important role in removal of materials. Absorptivity is defined as the fraction of incident laser radiation that is absorbed by the workpiece at normal incidence and it depends on the wavelength of laser beam and the temperature of the workpiece. It is higher for a laser having short wavelength and increases with an increase in workpiece temperature. Based on the electrical properties of the material and the wavelength of the given laser the temperature-dependent absorptivity of a material can be determined by Eq. (8) [33]. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pce0 ½1þ ar ðT ðr,t ÞT 0 ðr,t ÞÞ As ¼ ð8Þ

Fig. 1. Thermal model of pulsed Nd:YAG LBPD.

ls0

takes place across them. The boundary B4 is an axis of symmetry, and therefore the net heat loss or gain is absolutely zero on this surface. Therefore, the boundary conditions are:when 0 ot r t ( hðTðr,z,t ÞT 0 Þ for r 4R on B1 @T ðr,z,t Þ ¼ kðT Þ ð2Þ qin or r r R on B1 @z when t otrtp kðT Þ

@T ðr,z,t Þ ¼ hðT ðr,z,t ÞT 0 Þ @z

on B1

ð3Þ

when t 40 @T ðr,z,t Þ ¼0 @n

on B2 , B3 and B4

where e0 is the permittivity of vacuum, ar is the coefficient of resistance of the workpiece, l is the wavelength of the Nd:YAG laser beam, c (ms  1) is the velocity of light, and s0 is the target conductance at initial temperature. The values of all these parameters for aluminum are given in Table 1. Fig. 2(a–b) illustrates the change in absorptivity with respect to temperature and the variation of heat flux with respect to absorptivity at a constant peak power of laser pulse.

ð4Þ

where t is the pulse width, tp is the total pulse duration (i.e., on and off time ), qin is the amount of heat flux entering the workpiece, R is the laser beam radius, h is the heat transfer coefficient, T0 is the ambient room temperature, and direction n is outward, normal to the boundary surface. In the laser drilling of nickel superalloys the absorption occurs in a very thin layer at the surface of the workpiece and therefore, for many practical cases, it can be considered as a boundary condition to Eq. (1). Secondly, we know that the absorption coefficient is the inverse of penetration depth (also known as absorption depth). The value of this absorption depth relative to the thickness of irradiated workpiece is negligible (for metals), and therefore the energy supplied by the laser source can be modeled as a boundary condition, whereas in case of non-metals like ceramics this value is not negligible and the laser beam must be modeled as a volumetric heat source. 2.1.2. Heat input The laser beam is radially symmetric with a Gaussian heat flux profile. Heat flux at a distance r from the center of the laser beam on the surface of the workpiece is given by the Eq. (5) [32]. The change of laser beam radius with depth can be evaluated using Eq. (6).   2P P 2r 2 qin ¼ As exp ð5Þ pR2 R2     4l zm þ f c d ð6Þ R ¼ 1 þ M2 2 pd2 where As is the absorptivity, PP is the laser peak power, R is the effective beam radius which varies with the depth of the hole due to defocusing, d is the beam diameter (500 mm), fc is the focal length of lens ( 50 mm), zm is the melt depth, and l is the wavelength of the laser beam. The value of PP can be obtained using Eq. (7).

2.1.4. Phase change consideration The finite element simulation of the phase change problem has been done by the capacitance method, in which the mesh remains fixed and the phase change boundary moves through the mesh. In this method the effect of latent heat is accounted for by altering the heat capacity of the material during phase change [34]. 8 < C 0 þ DDHT m from T r T rT m þ DT m m C ðT Þ ¼ ð9Þ : C 0 þ DDHT v from T v r T r T v þ DT v v where C0 is the normal constant specific heat, DHv and DHm are the latent heats of boiling and fusion, respectively, and DTv and DTm are the temperature bands over which the transition occurs. 2.1.5. Finite element equations The Galerkin weighted residual method is used to convert the governing differential equation (Eq. 1) into algebraic equations. To obtain the finite element equation the domain is discretized into nem number of elements. Further, over a typical element, the temperature is approximated using the unknown nodal values and the known interpolation (or shape) function. The following expressions are obtained for the elemental stiffness matrix[K]e, capacitance matrix[C]e, and right side vector {f}e when the Galerkin method is applied to Eqs. (1)–(4). 9 R R ½Ke ¼ De k½Be ½BeT rdrdzþ Bh hfNgb fNgbT rdBh > > > = R ½Ce ¼ De rC fNge fNgeT rdrdz, ð10Þ > b R b R > ; f ¼ T hfNgb rdB þ fNgb fN gbT q rdB > Bh

0

h

Bq

in

q

Table 1 Electrical property of aluminum. Definition

Values

Permittivity of vacuum (C2 N  1 m  2) Coefficient of resistance(K  1) Conductance(O  1 m  1)

8.85  10  12 0.004210 3.77  107

S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474

0.060

6.0

0.055

5.5

465

Heat flux (W) x 109

Peak power= 10 kW

Absorptivity

0.050 0.045 0.040 0.035 0.030

5.0 4.5 4.0 3.5 3.0

0.025 200

300

400

500 600 700 Temperature (K)

800

900

1000

2.5 0.025

0.030

0.035

0.040 0.045 Absorptivity

0.050 0.055

0.060

Fig. 2. (a) Temperature dependence of absorptivity and (b) change of heat flux with absorptivity at constant peak power.

where [B]e is the matrix relating temperature derivatives with its nodal values, Bh is the convective boundary, Bq is boundary of the input heat flux, {N}b is the shape function vector for the boundary element, {N}e is the nodal shape function vector of area element, and De is domain of a typical area element. The integrals in the elemental equations are evaluated using the Gauss quadrature rule for numerical integration. Assembly of elemental equations is carried out using the assembly rule and the resulting assembled equations are no ½GC T þ ½GKfT g ¼ fGF g

ð11Þ

where

½GC ¼

nem X e¼1

½Ce , ½GK ¼

nem X

½Ke , fGF g ¼

e¼1

nbm X

½f e

b¼1

where nem is the number of area elements in the mesh, nbm is the number of line boundary elements in the mesh, [GC] is the global heat capacitance matrix, [GK] is the global stiffness matrix, [GF] is the global right side force vector, {T} is the global nodal tempera ture, and fT g is the time derivative of {T}. Eq. (11) represents the set of ordinary differential equations in the variable {T} as a function of time t. These equations are converted to a set of algebraic equations by the application of implicit FDM. Here, the solution marches in time, in steps of Dt until the desired final time is reached. In the present model, Dt is divided into two-time steps of Dt1 and Dt2, where Dt1 is the pulse width and Dt2 the pulse-off time of the laser pulse.

2.2. Determination of temperature distribution For the determination of temperature distribution, initial workpiece domains of thicknesses 700 mm, 1000 mm, and 1300 mm with a radius of 2500 mm are respectively used for workpiece thicknesses of 0.7 mm, 1.0 mm, and 1.3 mm. All the domains were discretized into eight noded quadratic serendipity elements. The nodal coordinates and connectivity matrices of the elements were obtained using ANSYS 10. Based on the results of the convergence test (h-convergence) an element size of 50  50 mm was used for this work. For the convenience of computation, the radius of the laser beam is kept as a whole number multiple of the element length used in the finite element analysis. The initial temperature of the workpiece is taken as 298 K and the model takes into account the phase change effects by taking into consideration the latent heat of fusion or latent heat of vaporization.

2.3. Estimations of Ta, MRR, and extent of HAZ When a laser pulse strikes the workpiece, the temperature begins to rise and at the end of the pulse the volume of material having temperature more than the melting temperature (933 K) of aluminum is removed from the workpiece to create a blind hole. The melt-isotherm is obtained from the result of temperature distribution. In order to find the melt-isotherm, the melt, unmelt, and partially melt elements were identified in the workpiece domain. The unmelt elements remain unaffected and do not affect the hole profile, whereas in the partially melt elements some of the nodes have been melted and some remain unaffected, thus, to obtain the actual shape and size of the drilled hole, the locus of all points in the partially melt elements which lie within the temperature range of TM70.1 (TM is the melting temperature of selected workpiece) were obtained using the developed code. Due to the removal of material the domain correspondingly changes for each consecutive strike of laser pulse with the workpiece, therefore re-meshing of the remaining domain (using ANSYS) and the corresponding prediction of new melt-isotherms were repetitively performed until a through hole was obtained. The nodal temperature obtained for each preceding step is used as an initial temperature of the workpiece to determine temperature distribution due to successive laser pulses. After the formation of a through hole the diameter at the entry and exit side can be finally obtained from the final melt-isotherm and the Ta is calculated using Eq. (12) [35].   dentry dexit 180 ð12Þ T a ðdegreeÞ ¼ tan1 2t h p where th is the material thickness. HAZ is determined to identify the region upto which the microstructure properties of the workpiece has been altered due to the thermal energy of the laser beam. Therefore, to estimate the extent of HAZ, the isothermal line corresponding to the recrystallization temperature of aluminum (473 K) is obtained from the temperature distribution as described earlier for the case of melt-isotherm. The extent of HAZ is obtained by determining the width of melting point isotherm and recrystallization isotherm as shown in Eq. (13). Extent of HAZ ¼ Rrecrystallization Rmelt

ð13Þ

where Rrecrystallization ¼Radius predicted by isothermal line at the recrystallization temperature and Rmelt ¼Radius predicted by isothermal line at the melting temperature. The volume of material removed by each pulse of laser is determined from the respective melt-isotherm. To determine the

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Fig. 3. Schematic to determine volume of material removed by (a) theoretical model and (b) experimental method.

Table 2 Percentage chemical composition of aluminum-alloy (wt %). Cu

Si

Fe

Mg

Ni

Mn

Zn

Ti

Pb

Sn

Cr

Al

0.042

0.618

0.752

0.059

0.008

0.026

0.029

0.012

0.011

0.013

0.005

98.42

material removed by each laser pulse, the region bounded by the melt-isotherm is divided into (c  1) circular strips, where c is the total number of coordinate points lying on the actual meltisotherm obtained from the developed code. The mean radius for each strip from the axisymmetric axis between two consecutive z-coordinates is obtained by averaging two radii values as shown in Fig. 3(a). The total volume of material removed is obtained using Eq. (14). VT ¼

n X

VP

where VT is the total volume of material removed, n is the number of laser pulses, and VP is the volume of material removed by each pulse, given by r þ r 2   iþ1 Vi ¼ p i zi þ 1 zi 2 c1 X

Vi

i¼0

where r and z are the coordinates and c is the number of coordinate points on the melt-isotherm. The MRR (mm3/sec) can be predicted from the developed model using Eq. (15). V t  1000 MRRth ¼ Pn i ¼ 1 tp

T(K)

300

400

500

600

800

1000

1200

k (W/m K) Cp (J/kg K) r (kg/m3)

237 902 2701

240 949 2681

– 997 2661

231 1042 2639

218 1134 2591

– 921 2365

– 921 2305

ð14Þ

p¼1

VP ¼

Table 3 Temperature-dependent material properties of aluminum [37].

Table 4 Average thermo-physical properties of aluminum [37]. Definition value Melting point(Tm) Density(r) Specific heat(Cp) Conductivity(k) Latent heat fusion(Hm) Latent heat boiling(Hv) Vaporization temperature(Tv) Thermal diffusivity Density(rl) (liquid phase) Specific heat(Cpl) (liquid phase) Conductivity (kl) (liquid phase)

933 K 2702 kg/m3 903 J/kg K 237 W/m K 360,000 J/kg 10,900,000 J/kg 2793 K 0.000097 m2/s 2385 kg/m3 1080 J/kg K 100 W/m K

ð15Þ

where MRRth is the MRR predicted by the developed model, Vt (mm3) is the predicted total volume of material removed, tp (ms) is the total pulse duration, and n is the number of pulses used to make the through hole. 2.4. Experimental comparison In order to verify the proposed model, experiments were conducted on a flat aluminum sheet of thickness 0.7 mm with composition as mentioned in Table 2. The temperaturedependent thermal properties and other thermo-physical properties of aluminum are given in Tables 3 and 4, respectively.

Experiments were performed using a 200 W pulsed Nd:YAG laser beam system (SLP-200) with CNC work table supplied by Suresh Indu Laser, Pune (India). In the existing laser beam the pulse width can be varied from 0.1 to 20 ms with a maximum pulse frequency of 50 Hz. All the experiments were performed without assist gas in the percussion drilling mode so that that the independent effect of various laser parameters on the hole characteristics can be determined without being affected by the assist gas. Holes were drilled normal to the surface. Table 5 shows the values of process variables used for experimentation. The diameter of laser-drilled holes was measured using an Optical Measuring Microscope (Model SDM-TR-MSU, Sipcon Instrument Industries, India) at 10  magnifications. Based on the diameter

S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474 Table 5 Experimental parameters used in the experiment.

14

Value/range

12

Average power Pulse width Pulse frequency Nozzle standoff distance

200 W 0.5–4 ms 10–25 Hz 1 mm

10

Ta (degree)

Parameter

obtained at the entry side (dentry) and exit side (dexit) of the hole, Ta was calculated using Eq. (12). Due to hole taper the final geometry of the hole can be considered as a frustum of a cone (Fig. 3(b)). The volume of material removed (Vexp) during LBPD was calculated using Eq. (16). pt  ð16Þ V exp ¼ h r 2entry þ r 2exit þ r entry r exit 3

6

0 0.5

1.0

2.0

5.0

2.5

3.0

3.5

4.0

Predicted values(with phase change) predicted values(without phase change) Experimental values

4.5

4.0

3.5

3.0

2.5 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5

Pulse frequency (Hz)

220

Predicted values Experimental values

200 180

τ = 1.5 ms

160

3

MRR(mm /s)

3. Modeling of LBPD using ANN coupled with the FEM model From the above discussion it has been observed that even though the FEM-based thermal model is capable of predicting the performance characteristics of LBPD satisfactorily for different sets of input process parameters, the repetitive re-meshing technique (using ANSYS) adopted here makes it very time consuming and tedious. A need thus exists to develop a model which can predict the performance characteristics of the LBPD process at a faster rate with more precision. The use of ANN for process modeling is gaining popularity in the manufacturing arena due to its generalization capabilities from imprecise and noisy data and their ability to approximate non-linear and complex relationships between the process parameters and output responses. A very important feature of ANNs is their adaptive nature, where ‘learning by example’ replaces ‘programming’ in solving the problems, which makes them more suitable for applications in which the complete understanding of process behavior is lacking but the training data are available [12]. In the present study, a correlation between the input parameters and output responses is to be established, therefore the backpropagation neural network (BPNN) is appropriate due to its universal approximation capabilities [36]. The BPNN consists of an input layer, output layers, and some hidden layers between the input and output layers. The network needs to be trained before employing it for predicting the performance characteristics, therefore the architecture of the network, i.e., the number of hidden layer(s) and number of neurons in each layer, should be known. During training the free

1.5

Pulse width (ms)

Ta (degree)

The results predicted by the FEM model in terms of Ta (at different pulse width and pulse frequency) as well as MRR at different pulse frequencies has been compared with the respective experimental values (Fig. 4). The Ta and MRR predicted by the developed model are in good agreement with the experimental results. The difference observed between the experimental and predicted values could be due to the non-inclusion of certain aspects in the developed model, like the formation of plasma and multiple reflections of the laser beam inside the hole cavity. Fig. 5(a) illustrate the hole profile obtained after simulation (at a pulse width of 2 ms and pulse frequency of 20 Hz), whereas Fig. 5(b–c) shows the hole obtained at the entrance and exit sides of the laser-drilled hole.

8

2

V exp  1000 Pulse frequency  time of drilling operation ðsÞ  pulse width ðmsÞ

ð17Þ

Predicted values( without phase change) Predicted values( with phase change) Experimental values

4

where rentry and rexit are the radii at the entry side and exit sides of the hole respectively. The calculated value of Vexp is further used for the calculation of MRR using Eq. (17). MRR ¼

467

140 120 100 80 60 40 4

6

8

10

12

14

16

18

20

22

Pulse frequency (Hz) Fig. 4. Comparison of experimental and predicted results.

parameters or connection weights are determined and the mapping between the input and output is accomplished. In BPNN the weights are initialized randomly and outputs are calculated based on these weights. The calculated output is compared with the actual outputs by the network and the error is transferred to the previous layer for weight correction. These two phases are iterated till the weight factors stabilize their values. 3.1. Network training and testing data In this work, the data generated by the FEM-based computational thermal model to predict the Ta, MRR and extent of HAZ in

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Fig. 5. (a) Laser hole profile obtained after simulation (at pulse frequency of 10 Hz and pulse width of 2 ms), (b) micrograph of hole at top surface (598 mm) and (c) micrograph of hole at bottom surface (258 mm).

the LBPD process has been used for training and testing of the ANN model. The selected input process parameters, i.e., pulse width, pulse frequency, peak power, and thickness of workpiece at three different levels have been considered to perform 81 (34) numerical experiments based on the full factorial approach, and correspondingly the performance characteristics have been predicted. A subset of 9 experiments based of Taguchi’s L9 orthogonal array has been separated from the total data set for the testing of the network and the remaining 72 data set is utilized for training of the network. The input and output data set are normalized to lie between 0.1 and 0.9 so that training of the network can take place at a faster rate and the chances of occurrence of local optima can be prohibited. Eq. (18) is used to normalize the training and testing datasets. xn ¼

0:8ðxxmin Þ þ 0:1 xmax xmin

t

APE

Extent of HAZ

Workpiece thickness

Input layer

Hidden layer

Output layer

Fig. 6. Configuration of the proposed ANN model. 70

ð20Þ

Averge mean prediction error Maximum prediction error

60 50

Error (%)

As there are four input process parameters and three performance characteristics, the number of neurons selected for the input and output layer are 4 and 3 respectively. The number of neurons in the hidden layer may lead to underfitting or overfitting of the training points; therefore, selection of the number of hidden layers and number of neurons in the hidden layers plays a very critical role in deciding the optimal neural network architecture. In this study, the scaled conjugate gradient (SCG) algorithm is used for training of the network to avoid the timeconsuming line search. This algorithm combines the model-trust region approach used in the Levenberg–Marquardt (LM) algorithm, with the conjugate gradient approach. The number of neurons in the hidden layer in which the total mean prediction error becomes minimum has been selected. The mean prediction error (MPE) of the testing datasets for each of the output performance parameters (Ta, MRR, and extent of HAZ) is determined from the average prediction error (APE). The APE and MPE is defined as follows: 



FEMresult ANNpredicted result

 100 APEð%Þ ¼

ð19Þ

FEMresult i¼1

MRR

ð18Þ

3.2. Network topology, training and testing

Pt

Ta

Pulse frequency Peak power

where xn is the normalized value of variable x, and xmax and xmin are the maximum and minimum values of x in total datasets respectively.

MPEð%Þ ¼

Pulse width

40 30 20 10 0 3

4

5

6

7

8

9

10

11

Number of neuron in the hidden layer Fig. 7. Plot to determine the number of neurons in the hidden layer.

where t is the number of datasets used for testing. In this study t¼9. The average mean prediction error (AMPE) of all the output performance parameters is then evaluated using Eq. (21). This AMPE is further used to select the optimal configuration of network. AMPEð%Þ ¼

MPEhole taper þ MPEextent of 3

HAZ þ

MPEMRR

ð21Þ

AMPE of all the output performance parameters is evaluated, which is further used for selecting the optimal network configuration. The ANN-based MATLAB code was used to select the optimal neural network architecture. Fig. 6 illustrates the ANN

S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474

architecture used for this study. Simulation was performed by varying the number of neurons in the hidden layer from four to 11. It can be seen from Fig. 7 that the AMPE as well as the maximum prediction error is minimum when the number of neurons in the hidden layer is seven. The network is trained well by 3000 epochs (Fig. 8). Table 6 shows the values of absolute prediction errors of 4-7-3 BPNN architecture for the desired performance characteristics. It is observed that the values of absolute prediction error for Ta, MRR, and extent of HAZ are

469

6.88%, 6.32%, and 5.06% respectively. So, 4-7-3 is the most suitable network for this particular study.Fig. 9 3.3. Prediction of performance characteristics using the developed ANN model In LBPD the performance characteristics are functions of laser input parameters which control the amount of thermal energy imparted to the workpiece surface, i.e., pulse width, pulse frequency, peak power, and the thickness of workpiece required to be drilled. In the subsequent sections the effect of laser drilling parameters on the process performance has been discussed using the developed BPNN model. The input process parameters to the neural network are entirely different from the FEM data set, so the generalization capabilities of the BPNN model in predicting process behavior under different sets of process parameters can be effectively utilized. 3.3.1. Effect of pulse width The effect of pulse width (at a constant pulse frequency of 15 Hz and peak power of 30 kW) on Ta, MRR, and extent of HAZ at different workpiece thicknesses is shown in Fig. 10. Fig. 10(a) illustrates that, for a given thickness, the Ta decreases with increase in pulse width. Secondly, with increase of workpiece thickness the Ta reduces. The pulse width determines the duration for which the laser beam interacts with the workpiece. Therefore, with an increase of pulse width the temperature along the radial direction as well as across the depth increases. In case of aluminum, most of the laser beam is reflected back during the initial period. Therefore, the significant change in entrance hole diameter does not take place due to increase of pulse width. But, once the crater formation takes place and

Fig. 8. Training of 4-7-3 BPNN.

Table 6 Comparison of developed model with FEM results and the errors in prediction. Trial

1 2 3 4 5 6 7 8 9

Machining conditions

Ta (degrees)

MRR (mm3/s)

Extent of HAZ (mm)

Absolute prediction error (%)

t (ms)

f (Hz)

PP (kW)

th (mm)

FEM

ANN

FEM

ANN

FEM

ANN

Ta

MRR

HAZ

0.5 0.5 0.5 1.5 1.5 1.5 2.5 2.5 2.5

10 14 18 10 14 18 10 14 18

10 20 40 20 40 10 40 10 20

0.7 1 1.3 1.3 0.7 1 1 1.3 0.7

12 9 5.5 7.2 7.1 7.8 7 6.1 6.8

11.9 8.7 5.6 7.2 7.6 7.7 6.7 6.5 6.4

44 145 260 196 178 140 226 185 190

41.4 143.1 259.7 202.8 170.7 140.2 226.3 186.3 178.0

145 120 105 115 130 172 118 175 195

148.5 121.4 108.6 110.3 125.5 180.7 118.4 170.4 191.3

1.23 3.19 1.38 0.04 6.88 1.35 3.71 6.40 5.69

6.02 1.33 0.12 3.46 4.08 0.12 0.14 0.68 6.32

2.43 1.16 3.40 4.07 3.49 5.06 0.32 2.64 1.88

Fig. 9. Linear regression analysis between neural network outputs and FEM values for Ta, MRR, and extent of HAZ.

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S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474

9.5

0.8 mm 0.9 mm 1.1 mm 1.2 mm

9.0

Ta(degree)

8.5 8.0 7.5 7.0 6.5 6.0 5.5 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

with the aluminum sheet increases. This increases the melt quantity of metal to increase the MRR. It is also observed that at the same pulse width higher MRR is obtained for a thick aluminum sheet. Thick sheets require a large number of laser pulses and thus the total laser energy increases. High thermal energy possessed by the molten metal gets diffused to the surrounding region and begins to accumulate to yield higher MRR. Fig. 10(c) shows the variation of extent of HAZ on the top surface at different pulse widths. With the increase of pulse width at constant peak power the extent of HAZ increases due to the diffusion of more thermal energy on the surface of the workpiece. This is responsible for the change in material microstructure. As the thickness increases the value of this performance characteristic decreases because the irradiated energy is not sufficient to change the microstructures.

Pulse width (ms) 240

0.8 mm 0.9 mm 1.1 mm 1.2 mm

200

3

MRR (mm /s)

220

180

160

140

120 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Pulse width (ms) 155

0.8 mm 0.9 mm 1.1 mm 1.2 mm

Extent of HAZ(μm)

150 145 140 135

3.3.2. Effect of pulse frequency The effect of pulse frequency on different performance characteristics of laser-drilled holes has been analyzed by varying the pulse frequency at a constant pulse width of 1.5 ms and peak power of 30 kW. Ta decreases with an increase in pulse frequency, as shown in Fig. 11(a). At high pulse frequency the heat loss due to convection is very small, therefore the exit diameter increases once the through hole penetrates the surface. This, in turn, decreases the Ta at high pulse frequency. With a rise in workpiece thickness the Ta decreases. It can be observed that when the pulse frequency increases at constant pulse width and peak power, the heat flux remains the same, but the cooling time between the successive laser pulses decreases due to which the amount of heat energy entering the workpiece per unit time also increases to enhance the MRR (Fig. 11(b)). It has been observed that with increase in workpiece thickness the MRR also increases due to increase in total laser energy. Fig. 11(c) demonstrates the effect of pulse frequency on the extent of HAZ. It increases with the increase of pulse frequency because as the time-lapse between the laser pulses is small, high thermal diffusion takes place in a radial direction to create a large HAZ area. At the same pulse frequency the extent of HAZ decreases with an increase in workpiece thickness because the change of microstructure depends on the temperature rise, which is directly proportional to the amount of irradiated energy available at the top surface.

130 125 120 115 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Pulse width (ms) Fig. 10. Effect of pulse width at constant pulse frequency and peak power on (a) Ta ,(b) MRR, and (c) extent of HAZ.

phase change occurs, the thermal conductivity of aluminum tends to achieve a constant value and the specific heat also does not change remarkably; therefore, with an increase of pulse width more molten material is produced at the bottom and thus the Ta gets reduce. It is observed that as the workpiece thickness increases the Ta decreases due to reduction in the diameter of the hole at the entrance owing to the dissemination of thermal energy to a larger volume. Fig. 10(b) shows that the MRR increases with increase of pulse width because the energy per pulse (Ep) of laser (product of pulse width and peak power) as well as the laser interaction period

3.3.3. Effect of peak power For analyzing the effect of peak power on the three specified performance characteristics the values of pulse width and pulse frequency are kept constant at 1 ms and 15 Hz respectively. Fig. 12(a) demonstrates the effect of peak power on Ta. The increase of peak power increases the radial diffusion of heat as well as the amount of material that can be removed per pulse. Moreover, once the laser beam breaks through the material, the effect of increase of peak power is more on the hole exit resulting in formation of a hole with less taper. As the thickness of workpiece increases the proportional availability of thermal energy at the top surface reduces, thereby decreasing the entrance hole diameter. The amount of heat flux entering the workpiece is governed by the peak power of the laser. Fig. 12(b) illustrates that when peak power is increased at constant pulse width and pulse frequency, the amount of heat flux entering the workpiece also increases and subsequently the melt depth also increases, thereby increasing the MRR. Fig. 12(c) demonstrates that the extent of HAZ reduces with the increase in peak power. At high peak power the energy content of the pulse is high, thus imparting high thermal energy to the surface of the workpiece, but due to very fast temperature decay caused by the high thermal conductivity of aluminum, the

S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474

9.5

471

0.8 mm 0.9 mm 1.1 mm 1.2 mm

9.0

9.0 8.5

Ta(degree)

Ta(degree)

8.5 8.0 7.5 7.0

0.8 mm 0.9 mm 1.1 mm 1.2 mm

6.5 6.0

8.0

7.5

7.0

6.5

5.5 10

11

12

13

14

15

16

17

12

18

15

18

Pulse frequency (Hz) 0.8 mm 0.9 mm 1.1 mm 1.2 mm

27

30

33

36

200 190

3

210

210

MRR (mm /s)

225

3

195 180 165

180

0.8 mm 0.9 mm 1.1 mm 1.2 mm

170 160

150

150

135

140

120

130 120

105 10

11

12

13

14

15

16

17

12

18

15

18

21

24

27

30

33

36

Pulse Peak power (kW)

Pulse frequency (Hz) 140

160

0.8 mm 0.9 mm 1.1 mm 1.2 mm

135

Extent of HAZ(μm)

150

Extent of HAZ (μm)

24

220

240

MRR (mm /s)

21

Pulse Peak power (kW)

140

130

130

125

0.8 mm 0.9 mm 1.1 mm 1.2 mm

120

120 115

110 10

11

12

13

14

15

16

17

18

Pulse frequency (Hz) Fig. 11. Effect of pulse frequency at constant pulse width and peak power on (a) Ta, (b) MRR, and (c) extent of HAZ.

structural change will not occur. With a rise in workpiece thickness at constant peak power the extent of HAZ reduces due to the dissemination of heat flux to a larger volume of material, which will further reduce the value of temperature obtained at the surface of the workpiece.

12

15

18

21

24

27

30

33

36

Pulse Peak power (kW) Fig. 12. Effect of peak power at constant pulse width and pulse frequency on (a) Ta, (b) MRR, and (c) extent of HAZ.

to enhance the suitability of LBPD for industrial application. Therefore, multi-objective optimization becomes increasingly important in the LBPD process. In this study, data sets for the MOO is obtained from the responses predicted by the ANN model. 4.1. Grey relational analysis

4. Multi-objective optimization of LBPD using GRA coupled with PCA In laser-drilled holes several aspects of the quality like geometrical accuracy, minimum metallurgical distortion, along with a high material removal rate need to be considered simultaneously

GRA is advantageous to use in situations in which the relationships among the factors are uncertain. It measures the absolute value of the data difference between two sequences. The data predicted by the ANN model is required to be pre-processed due to difference between the range and unit of the predicted input data. Therefore, ANN results are normalized in the range of zero

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and one, which is known as grey relational generating. Depending on the nature of data sequence various methods may be applied for the normalization. If the target value of data sequence is ‘‘higher the better type’’ the original sequence can be normalized using Eq. (22) xni ðpÞ ¼

xp ðpÞminxi ðpÞ maxxi ðpÞminxi ðpÞ

ð22Þ

When ‘‘lower is better’’ is characteristic of the original sequence, then the original sequence should be normalized as follows: xni ðpÞ ¼

maxxp ðpÞxi ðpÞ maxxi ðpÞminxi ðpÞ

ð23Þ

here, i¼1, 2, 3,yyy.,t, p¼1, 2, 3,yy..z, xni ðpÞ is the normalized value of p-th element in the i-th sequence, max xpi ðpÞis the largest value of xpi ðpÞ, min xpi ðpÞis the smallest value of xpi ðpÞ, t is the number of trials, and z is the number of quality characteristics. After data processing a grey relational coefficient is calculated with the preprocessed sequences using the following expression:

Dminþ zDmax , x0,i ðpÞ ¼ D0,i ðpÞ þ zDmax

p

where z is the distinguishing or identification coefficient and its value lies between zero and one. z is set to 0.5 in this study. A weighting method is then used to integrate the grey relational coefficient of each experiment into the grey relational grade. The overall evaluation of the multiple performance characteristics is based on grey relational grade, i.e., z 1X g0,i ¼ wp x0,i ðpÞ zp¼1

where Cov(xi(j),xi(l) is the covariance of sequences xi(j) and xi(l),sxi ðjÞis the standard deviation of sequence xi(j), and sxi ðlÞis the standard of sequence xi(l). 3. To determine the eigenvalues and eigenvectors from the correlation coefficient array using Eq. (29). ðRlk Im ÞV ik ¼ 0

ð26Þ

where g0,i is the grey relational grade, wp represents the weighting value of the p-th performance characteristics, and Pz p ¼ 1 wp ¼ 1. Since the importance of each quality characteristic is different, the weighting value corresponding to each performance characteristic should reflect its relative importance properly and objectively. 4.2. Principal component analysis PCA is a multivariate statistical method to identify the patterns in the correlated data utilizing the data to explain the similarities and differences between them. In PCA, the original data sets are converted into principal components which are uncorrelated with each other. The steps involved in finding the principal components are:

Y mk ¼

n X

xm ðiÞV ik

ð30Þ

i¼1

where Ymk is the principal components. The principal components are aligned in descending order with respect to variance and therefore the first principal component accounts for most variance in the data. The components with eigenvalues greater than one are chosen to replace the original responses for further analysis.

4.3. Optimized results using coupled method of GRA and PCA Holes drilled by laser should have lower values of Ta and higher values of MRR, with low extent of HAZ. The normalized values of each performance characteristic for different trials as Table 7 Sequence after data preprocessing. Trial no.

Normalized data

Ideal sequence

Ta 1

MRR 1

Extent of HAZ 1

1 2 3 4 5 6 7 8 9

0 0.5079 1 0.7460 0.6825 0.6667 0.8254 0.8571 0.8730

0 0.4659 1 0.7393 0.5923 0.4526 0.8470 0.6638 0.6257

0.5157 0.8452 1 0.9794 0.7956 0.1282 0.8815 0.2527 0

Table 8 Calculated grey relational coefficient and grey relational grade. Trial no.

1. To develop the original multiple quality characteristic array using the grey relational coefficient of each performance characteristics. xi(j), i¼ 1, 2y, t, j¼ 1, 2y, z 2 3 x1 ð1Þ x1 ð2Þ       x1 ðzÞ 6 x ð1Þ x ð2Þ       x ðzÞ 7 2 21 6 2 7 6 7 7 ^ ^       ^ X¼6 ð27Þ 6 7 6 ^ ^   ^ 7 4 5 xt ð1Þ xt ð2Þ       xt ðzÞ

ð29Þ

where lk is the eigenvalue and Vik is the eigenvector corresponding to the eigenvalue. 4. To formulate the uncorrelated principal component using Eq. (30)

ð24Þ

where x0,i(p)is the relative difference of the p-th element between comparative sequence xi and reference sequence x0. D0,i(p)is the deviation sequence of the reference sequence and the comparability sequence. 9 D0,i ðpÞ ¼ jxn0 ðpÞxni ðpÞj, > > = n n Dmax ¼ max maxjx0 ðpÞxi ðpÞj ð25Þ p i > ; Dmin ¼ min min jxn0 ðpÞxni ðpÞj > i

where t is the number of the experiment and z is the number of performance characteristic. x is the grey relational coefficient of each performance characteristic. Here, t ¼9 and z ¼3. 2. To evaluate the correlation coefficient array as follows:   Covðxi ðjÞ,xi ðlÞÞ Rjl ¼ , j ¼ 1,2,. . .,z, l ¼ 1,2,. . .,z ð28Þ sxi ðjÞ  sxi ðlÞ

1 2 3 4 5 6 7 8 9

Grey relational coefficient Ta

MRR

Extent of HAZ

0.333333 0.503981 1 0.66313 0.611621 0.600024 0.74118 0.777726 0.797448

0.333333 0.483512 1 0.657289 0.550843 0.477373 0.765697 0.597943 0.571886

0.508906 0.763592 1 0.96043 0.709824 0.364485 0.808407 0.400866 0.333333

Grey relational grade

Order

0.3719 0.552 1 0.7259 0.6064 0.4941 0.7668 0.6154 0.5958

9 7 1 3 5 8 2 4 6

S. Mishra, V. Yadava / Optics & Laser Technology 48 (2013) 461–474

shown in Table 7 are obtained using Eqs. (22) and (23). The elements of the array for multi-performance characteristics shown in Table 8, representing the grey relational coefficient of each quality characteristic. These data are further used to evaluate the correlation coefficient matrix and determine the corresponding eigenvalues from Eq. (29), shown in Table 9. The eigenvectors corresponding to each eigenvalue are listed in Table 10. The square of eigenvalue matrix represents the contribution of respective performance characteristic to the principal component. The variance contribution for the first principal component (Table 9) characterizing the whole original variables, i.e., the three

Table 9 Eigenvalues and explained variation for principal components. Principal component

Eigenvalue

Explained variation (%)

First Second Third

2.2287 0.7531 0.0183

74.29 25.10 0.61

Table 10 Eigen vectors for principal components. Quality characteristics

Ta MRR Extent of HAZ

Eigenvector First principal component

Second principal component

Third principal component

0.5817 0.6646 0.4690

0.5640 0.0858  0.8213

0.5861  0.7423 0.3249

473

performance characteristics, is 74.29%; therefore, in this study the square of the corresponding eigenvectors are selected as the weighting values of the related performance characteristics as shown in Table 11. Based on Eq. (26) and data listed in Table 8, the grey relational grade is calculated. Higher values of grey relational grade correspond to better performance regardless of the category of the performance characteristics. The response table of the Taguchi method has been used here to calculate the average grey relational grade for each parameter level. Table 12 shows that the optimum parameter level corresponding to maximum average grey relational grade is A3B3C3D3. In order to adjudge the parameter which significantly affects the performance characteristic of LBPD, analysis of variance (ANOVA) was applied. ANOVA is a computational technique for the quantitative estimation of the relative significance (F-ratio) and also the percentage contribution of each process parameter. The results of ANOVA for the grey relational grade are shown in Table 13. It shows that the peak power is the most significant LBPD process parameter followed by workpiece thickness for affecting the multi-performance characteristics. After obtaining the optimal level of process parameters it is essential to verify the improvement of the performance characteristics using this optimal combination. Table 14 compares the results of confirmations experiments using the optimal process parameters and the initial process parameters. A reduction in Ta by 67.5%, increase in MRR by 605%, and decrease in extent of HAZ by 3.2% is observed while machining at optimum parameter setting. Therefore, the combined approach of GRA and PCA can be effectively used for the multi-performance optimization of the LBPD process.

5. Conclusions Table 11 Contribution of each individual quality characteristic for the principal component. Quality characteristics

Contribution

Ta MRR Extent of HAZ

0.3384 0.4417 0.2199

In the present work, a model of LBPD process has been developed using the coupled methodology FEM and ANN. The predicted response of the ANN model is further used for the multi-objective optimization of the LBPD process using the coupled approach of GRA and PCA. The following conclusions have been derived from the present study: 1. The Ta decreases with increase of pulse width, pulse frequency, peak power, and workpiece thickness. The reduction of Ta with

Table 12 Response table for the grey relational grade. Symbol

Input parameter

Level 1

Level 2

Level 3

Max–min

A B C D

Pulse width Pulse frequency Peak power Workpiece thickness

0.6413 0.6215 0.4938 0.5247

0.6088 0.5913 0.6246 0.6043

0.66n 0.6966n 0.7911n 0.7804n

0.0187 0.1053 0.2973 0.2557

n

signify the optimum value.

Table 14 Comparison between initial and optimal level. Best combination Initial design A1B1C1D1 Optimal design A3B3C3D3 Final improvement

Ta

MRR

Extent of HAZ

12.0 3.9 67.5%

44.0 310.31 605%

145.0 140.3 3.24%

Table 13 Results of analysis of variance. Symbol

Input parameter

Degree of freedom

Sum of square

Mean square

F value

Contribution (%)

A B C D Error Total (Error)

Pulse width Pulse frequency Peak power Workpiece thickness

2 2 2 2 0 8 (4)

0.003935n 0.017658n 0.133190 0.102759 0 0.257642 (0.021593)

0.001968 0.008829 0.066595 0.051379 – – (0.00539825)

– – 12.34 9.52 – –

1.53 6.86 51.72 39.89 –

n

Indicates the sum of squares added together to estimate the pooled error sum of squares indicated by parentheses.

474

2.

3.

4.

5.

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increase of pulse width is more significant for workpieces having higher thickness. At a given thickness, the MRR increases with an increase of laser parameters, i.e., pulse width, pulse frequency, and peak power. The effects of pulse width and pulse frequency are more prominent than those of peak power in enhancing the MRR. As the thickness of workpiece increases the MRR also increases. The extent of HAZ increases with increase in pulse width and pulse frequency but it decreases with increase in peak power and workpiece thickness. The pulse frequency has the most significant effect on the extent of HAZ. A feed-forward back-propagation ANN of type 4-7-3 having one hidden layer provides the best prediction with an average prediction error of 2.81% and maximum prediction error of 6.7%. The PCA used to determine the corresponding weights of each performance characteristic while applying the GRA for multiobjective optimization is capable of objectively reflecting the relative importance of each performance characteristic. Based on ANOVA, the major controllable parameters significantly affecting the multi-performance characteristics of the LBPD process are peak power (51.72%), workpiece thickness (39.89%), pulse frequency (6.86%), and pulse width (1.53%). The optimal combination of process parameters obtained from the proposed method is A3B3C3D3.The corresponding confirmation test shows that Ta decreases by 67.5, MRR increases by 605%, and the extent of HAZ decreases by 3.24%.

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