Parametric Optimization in WEDM of WC-Co Composite by Neuro-Genetic Technique

Transcription

1 Parametric Otimization in WEDM of WC-Co Comosite by Neuro-Genetic Technique P. Saha*, P. Saha, and S. K. Pal Abstract The resent work does a multi-objective otimization in wire electro-discharge machining (WEDM) of tungsten carbide-cobalt (WC-Co) comosite. Otimization was done with the hel of a Neuro-Genetic technique which was develoed through a combination of a radial basis function network (RBFN) and non-dominated sorting genetic algorithm (NSGAII). Exeriments were carried out based on Taguchi design of exeriments involving six control factors such as ulse on-time, ulse off-time, eak current, caacitance, ga voltage, and wire feed rate. Cutting seed, surface roughness and kerf width were considered as the measures of erformance of the rocess. The roosed Neuro-Genetic technique was found to be otential in finding alternative otimal inut conditions of the rocess. These otimal solutions may lead to efficient utilization of WEDM in industry. Index Terms NSGA-II, RBFN, WC-Co, WEDM W I. INTRODUCTION ITH the introduction of new hard materials to be machined, suer-hard tool materials were develoed. Among several suer-hard tool materials, cemented carbides, esecially WC-Co comosites were found to be otential materials to making cutting tools, metal-working tools, mining tools and wear resistance comonents. This is because of its greater hardness, strength and wears resistance over a wide range of temeratures. WC-Co comosite is synthesized by the sintering of WC granules that are held together by Co binders. Wire electro-discharge machining rocess was found to be an extremely owerful electro-thermal rocess for machining WC-Co comosites with any intricate shae. This rocess erodes materials by minute electrical discharges between the wire-electrode ( micron diameter) and the work-iece. Although WEDM was found to be otential for machining WC-Co comosites, it has some major limitations articularly while machining this comosite. A large difference in melting and evaoration temeratures of it constituents, makes the rocess unstable [1]. The melting and evaoration temeratures of tungsten Manuscrit received Feb 3, 011; revised March 5, 011. P. Saha* is with the Indian Institute of Technology Patna, Patna , India, Deartment of Mechanical Engineering (corresonding author, hone: ; fax: ; ( saha@iit.ac.in). P. Saha is with Indian Institute of Technology Kharagur, Kharagur- 7130, India, Deartment of Mechanical Engineering ( asha@mech.iitkg.ernet.in). S. K. Pal is with Indian Institute of Technology Kharagur, Kharagur- 7130, India, Deartment of Mechanical Engineering ( skal@mech.iitkg.ernet.in) carbide are C and C, resectively while those for Co are C and C, resectively. Therefore, Co gets melted, evaorated and removed by the discharge energy even before the melting of WC. As a result, in absence of a binder (Co) the WC grains may be released without melting and may lead to unstable machining. Unstable machining causes short circuit, arcing etc. Hence, it is quite difficult to model such rocess by analytical aroach based on the hysics of this rocess. There have not been significant research ublications till today on rocessing of these hard WC-Co comosite materials by WEDM. Thus, there is nonavailability of machining databases for this tye of materials. In absence of a database, an aroriate model, caable of redicting machining behavior for a wide range of oerating conditions, finds immense utility. Few mathematical models develoed so far []-[3] relies uon several assumtions to simlify the rocess and eventually redicts a rocess erformance which is quite far from the reality. In this context, as neural networks are highly flexible modeling tool with the ability to learn the maing between inut and outut without knowing a rior relationshi between them, they could be used for modeling of such random and comlex rocess. In the resent work RBFN was imlemented to develo this model. The authors, in their revious work [4], have already done an extensive arametric study on this material by WEDM. This arametric study reveals that there is not available even a single inut condition which can maximize cutting seed and minimize surface roughness or kerf width simultaneously. Therefore, it is a multi-objective otimization roblem. In multi-objective otimization, user may require several solutions instead of a single one. These solutions are called Pareto-otimal solutions and they are equally imortant. Deending uon the requirement of the user, any one of them can be selected. Say for examle, in rough cutting, a rocess engineer must select such an inut condition from the Pareto-otimal solutions which can rovide a larger cutting seed. This selected inut condition may also roduce higher kerf width (higher the kerf width, lower will be the dimensional accuracy). But the rocess engineer is not worried for that as his objective is to maximize cutting seed irresective of any value of kerf width. Similarly in fine cutting, the selected inut condition must be such that it will minimize kerf width irresective of any value of cutting seed. As NSGA-II works with a oulation of oints, it may cature several solutions which would be required for the multi-objective otimization roblem. NSGA-II requires a fitness function which is nothing but a relationshi between inut and outut arameters. As RBFN is making that relationshi, hence, it

2 is couled with NSGA-II, and thus making a Neuro-Genetic technique. Several attemts have been made to model and otimize the rocess [5]-[8]. The modeling and otimization techniques, used by the ast researchers have some limitations. Say for examle, the limitation of regression method is that it may not deict the underlying nonlinear comlex relationshi between the decision variables and resonses. In addition to that it also requires a rior assumtion regarding functional relationshi (such as linear, quadratic, higher order olynomial and exonential) between inut and outut decision variables. This aer attemts to ma inut and outut variables of WEDM by RBFN. The RBFN is suerior over the back-roagation neural network due to the following reasons: it is less comlex, it requires fewer training samles, only one layer of nonlinear elements is sufficient for establishing arbitrary inut-outut maing, it requires less training time, and chance of getting local minimal convergence is less [9]. Similarly in otimization, most of the researchers used either simle weighting method, converting multi-objective roblem into a single objective or constraint otimization method. The drawback of the simle weighting method is that it is very much sensitive to the weight vectors and needs rior knowledge to the roblem. The limitation in constraint otimization technique is that the otimal value of one resonse is obtained while keeing the other one at a desired value and hence, roviding single solution at a time. As in the real world situation user may require different alternatives in decision making; therefore, the above methods need to be solved number of times as the requirement changes. As the wire electro-discharge machining database for WC-Co is not readily available and hence, any attemt to model and otimize the WEDM rocess arameters would be very much useful to the rocess engineers. II. EXPERIMENTATION In the resent research work, a 4-axis CNC WEDM (Make: Electronica Machine Tools Ltd., India, Model: Electra Maxicut) was used for the study. The machine has a transistor controlled RC relaxation circuit and deionized water is used here as a dielectric fluid. This study took WC- Co comosite with some alloying elements as the workiece material. The comosition and the hysical roerties of the work-iece are given in Table I. TABLE I COMPOSITION AND PHYSICAL PROPERTIES OF WC-CO COMPOSITE Comosition WC 76.5 wt%, Co 11.5 wt%, Alloying elements 1 wt% Grain size.4-.8 µm Batch hardness HV30 According to the Taguchi quality design concet a mixed orthogonal array, L 3 was used for the exeriments. A total of six machining arameters such as ulse on-time, ulse off-time, eak current, caacitance, average ga voltage, and wire feed rate were considered as the controlling factors. Among the six inut arameters, one has two levels and the remaining five have four levels. The detailed machining condition is shown in Table II. After comletion of the exeriments the erformances of the WEDM rocess were measured by the methods exlained in the next section. TABLE II MACHINING CONDITION Process arameter Level Unit Wire feed rate (W F ) 4 6 mm/min Pulse on-time (T on ) μs Pulse off-time (T off ) μs Peak current setting (I ) ste Caacitance (C) μf Average ga voltage (V ga ) V A. Measurement Procedure Cutting seed (mm/min): Cutting seed was evaluated under each cutting condition by dividing the cutting length with the required cutting time. Kerf width (mm): A diagram of kerf width is shown in Fig. 1. Fig.1. Work-iece showing kerf width Once the machining was over, each machined surface was cleaned with acetone bath in an ultrasonic cleaner, and kerf width was measured by an otical microscoe attached with a micro-hardness tester (Make LECO, Model: M-400-H1) equied with a digital micrometer. TABLE III EXPERIMENTAL RESULTS Sl Inut rocess arameters CS SR KW No. W F T on T off I C V ga CS = cutting seed, SR = Surface roughness, KW = Kerf width For each cut, total twelve kerf width measurements have been taken at different ositions along the length of the cut

3 and finally, the average of those readings were considered. Surface roughness (µm): Average surface roughness (R a ) was measured by SJ-01P Mitutoyo ortable roughness tester. Multile readings were taken for each cut and average was considered. The cut off length and samling length were been ket as 0.8 and 3 mm, resectively. A total thirty-nine exeriments were conducted. Out of that, thirty exerimental data were used to train the network and rests were used for validation of the models. Due to limitation of sace, only fifteen exerimental results are shown in Table III. III. RBFN BASED MODELING OF THE PROCESS A. An Overview of RBFN RBF networks consist of an inut layer, one hidden layer, and an outut layer. The hidden layer contains an array of nonlinear radial basis functions, which are generally Gaussian functions. The outut of the k th hidden node is obtained by the following exression: - ex X C - k k k = (1) n - ex X C - k k=1 k where, X x 1, x, x3,..., x l is the inut vector (also a C k c c c is the center vector of the k th attern), [ 1k, k,..., lk] hidden node having same dimension as that of the inut vector, l is the dimension of inut and center vector, k is the Gaussian width of the k th hidden neuron and n is the number of neurons or centers in the hidden layer. X- C k is the Euclidean distance between inut and k th center. The outcome of the j th unit of the outut layer is obtained through a linear combination of the nonlinear oututs from the hidden layer and is given by the following exression: n s j = kwkj, () k=1 where, w kj is the synatic weight between the k th hidden node and j th outut neuron. It is indeed imortant to indicate that the erformance of the RBFN deends on the centers of the Gaussian functions. Ideally, the centers should reresent the whole inut data sace. Initially the centers are chosen at random. The initial random centers may not artition the whole inut sace effectively, resulting in oor aroximation caability of the RBF network. Therefore, it is very much essential to otimize the initial centers. In the resent work, the centers were otimized by two methods. The first method is k- means clustering and the second one is gradient descent. K-means clustering technique [10] K-means clustering algorithm artitions the whole inut sace into some regions and finds the otimal centers of each region, which can reresent the inut dataset belonging to that region. In this algorithm, P number of l-dimensional inut vectors of the training dataset, X i, i =1,,..., P, are artitioned into n number of clusters denoted by Gk, k = 1,,..., n. It is assumed that at otimal artition, a cost function of dissimilarity measure would be minimized. This cost function is resented by the following exression: n Cost function X k 1 i Ck, (3) i, Xi Gk where, Gk is the k th cluster grou. The otimal center C k that minimizes the cost function is the mean of all inut vectors in grou k. 1 Ck Xi (4) Gk i, Xi Gk where, P Gk uji (5) i 1 where, P is the number of training attern, and the membershi matrix is given by 1, if X for all k j u ji i Cj Xi Ck (6) 0, otherwise Gradient descent technique [11] K-means clustering technique may achieve a local otimum solution which also deends on the initial choice of the cluster centers. The consequence of the local otimality is that it would never reach to the desired value and causes wasting of comutational time. In order to obtain a better result, a gradient descent learning algorithm was imlemented. Irresective of the value of initial centers, the otimal centers could be obtained by this technique. After (+1) th attern resentation, the centers are udated by the following exression: 1 1 c ik c ik c c ik 1 where, c ik is the comonent of k th center vector after (+1) th attern resentation, c is the learning rate for centers,. is given by the following equation: 1 q t s j j (8) j 1 where, q is the number of oututs, 1 t j (7) is the target outut of the (+1) th 1 attern resentation, s j is the redicted outut of the same attern. After finding the otimal centers of the Gaussian functions by the two methods, the forward calculation was done. Based on the difference between redicted outut and target outut, the synatic weights, connecting the neurons of hidden layer and outut layer were udated. This was done by the gradient descent learning algorithm. The synatic weights were modified by the following exression: 1 1 E j w kj w kj w (9) w kj where, 1 w kj is the synatic weight between k th hidden

4 neuron and j th weights and outut neuron, w is the learning rate for E 1 k 1 1 t j s j k W kj. The training will be stoed when mean square error (MSE) for training will reach to a threshold value. The MSE is calculated by the following exression: 1 P q MSE t s P j j 1 j 1 (10) B. Develoment and Validation of Models The dataset as listed in Table III was used to train the networks. In order to do a fare comarison between the models, a same value of MSE for training was used for both the models. Here it is Once the otimal center locations were obtained, the synatic weights were udated by gradient descent learning algorithm. In case of RBFN with k-means, the network arameters such as n, and w were obtained by trial and error method. This was carried out by varying the MSE for testing against these arameters. It was found that a architecture yields better results while and w were ket as 0.3 and 0.5, resectively. On the other hand, in case of RBFN with gradient descent technique, the otimal center locations along with other network arameters were obtained by gradient descent method. Here also a architecture yields better rediction while c, (learning rate for Gaussian width) and w were ket as 0.5, 0.6 and 0.3, resectively. The models were validated against the exerimental results. The detailed statistical analysis for the rediction error for cutting seed, surface roughness and kerf width was given in Table IV. From the table, it is clear that excet surface roughness, the absolute rediction error for cutting seed and kerf width were within the accetable limit. RBFN with gradient descent technique yields better result than with k-means technique. TABLE IV MODEL VALIDATION RESULTS Outut Statistical M V Range Max Min measures Model Cutting K-means seed Gradient descent Surface K-means roughness Gradient descent Kerf K-means width Gradient descent M= mean, V= variance IV. OPTIMIZATION BY NEURO-GENETIC TECHNIQUE Although three outut arameters were considered in the resent study, multi-objective otimization was done only on two arameters. They are cutting seed and kerf width. The objective of a rocess engineer is to maximize the cutting seed and minimize the kerf width simultaneously. These two objectives are conflicting in nature as already discussed in section I. Therefore, this roblem is a multiobjectives otimization roblem. In case of multi-objective otimization roblem, a single solution which is the best with resect to all the objectives may not exist. The solutions for such situation are known as Pareto-otimal solutions or non-dominated solutions. Since NSGA-II works with a oulation of oints, a number of Pareto-otimal solutions may be obtained using this technique. In the resent work, RBFN based NSGA-II (Neuro-Genetic technique) was used to obtain Pareto-otimal solutions. The next sections will rovide a brief idea of the NSGA-II algorithm [6] and otimization results. A. Develoment of RBFN based NSGA-II The goals of the NSGA-II algorithm are to find a set of solutions as close as ossible to the Pareto-otimal front and simultaneously as diverse as ossible. Excet for the fitness assignment method, the basic structure of NSGA-II is similar to that of GA. The stes involved in this algorithm are briefly exlained. Ste 1: Random binary oulation initialization: A binary oulation P of size N is generated randomly. A oulation consists of a set of inut rocess arameters and is also used in making a solution which will roduce oututs from the inut-outut relationshi. The binary coded arameters are then converted into real value by a linear maing, considering their uer and lower limits. The binary string of each arameter is converted into real value by the following exression: x U L i x i x L D i x i x i (11) ls i 1 where, x i is the real value of the i th inut arameter, L x i and xi U are the lower and uer limits of the i th inut arameters, resectively; ls i is the string length of the i th inut arameter, and x D i is the decoded value of the i th arameter. Before feeding these real values to RBFN model, these raw values are required to be normalized. After normalizing these real values in the range 0.1 to 0.9, the fitnesses of the objective functions are obtained by the best RBFN model. Here RBFN network makes the hidden relationshi between inut and outut variables, and thus formulates the objective functions. Ste : Fast non-dominated sorting: The oulation is sorted based on their non domination levels. The detailed exlanation of this technique is described else-where [1]. In this technique two entities are calculated, first one is the domination count ( n i ) that reresents the number of solutions which dominate the solution i and the second one is S i that reresents the number of solutions which are dominated by the solution i. This is accomlished by comaring each solution with every other solution and checked whether the solution under consideration satisfies the rules given below.

5 Objective1i Objective1j and Objectivei Objective j or (1) Objective1i Objective1j and Objectivei Objective j where, Objective1 i and Objective1 j are the fitness of objective1 for the i th and j th solution, resectively. Similarly, Objective i and Objective j are the fitness of objective for the i th and j th solution, resectively. If the rules are satisfied, then the solution j is dominated else non-dominated. Thus the whole oulation is divided into different ranks. Ranks are defined as the several fronts generated from the fast non-dominated sorting technique such that Rank1 solutions are better than the Rank solutions and so on. This technique drastically reduces the comutational time. Ste 3: Crowding distance: Once the oulations are sorted, crowding distance is assigned to each individual belonging to each rank. This is because the individuals of the next generation are selected based on the rank and the crowding distance. This crowding distance ensures a better sread among the solutions. A better sread means a better diversity among the solutions. In order to calculate crowding distance, fitness of the objective functions for the solutions belonging to a articular rank were sorted in descending order with resect to each objective. An infinite distance is assigned to the boundary solutions i.e., for the first and n th solution, if n number of solutions belong to a articular rank. This ensures that the individuals in the boundary will always be selected and hence, result in better sread among the solutions. For other solutions belonging to that rank, the crowding distances are initially assigned to zero. For r to n 1solutions, this is calculated by the following formula: ( 1) ( 1) () () f m r Irm Irm f m r (13) f max f min m m where, I () rm is the crowding distance of the of the r th individual for m th objective, where, m =1 to. fm( r 1) is the value of the m th objective for ( r 1 ) th individual, f max m and f min m are the maximum and minimum values of the m th objective, resectively. Ste 4: Crowded tournament selection: A crowded comarison oerator comares two solutions and returns the winner of the tournament. A solution i wins a tournament with another solution j if any of the following conditions are true: (i) If solution i has a better rank than j (ii)if they have the same rank but solution i has larger crowding distance than solution j. Ste 5: Recombination and selection: The offsring and current oulation are combined and selection is done in order to obtain the oulation of the next generation. The offsring are generated by -oint cross over and bitwise mutation. The elitism is ensured, as the best oulation from the offsring and arent solutions are selected for the next generation. The N solutions are then sorted based on their non-domination; and crowding distances are calculated for all the individuals belonging to a rank. In order to form the oulation of the current generation, the individuals are taken from the fronts subsequently unless it reaches to the desired oulation number (N). The filling starts with the best non-dominated front (Rank1 solutions), with the solutions of the second non-dominated front, followed by the third non-dominated front, and so on. If by adding all individuals in a front the oulation exceeds N, then individuals are selected based on their crowding distance. The stes are reeated until maximum generation number is reached. B. Otimization Results The objective of the resent study is to maximize the cutting seed and minimize the kerf width. In order to convert the second objective into maximization, it is suitably modified. The objectives then take the following form: Objective 1 Cutting seed 1 (14) Objective Kerf width As the range and the ste length of the rocess variables were different, hence, different bit size was assigned to individual arameters. Initially bit lengths for individual arameters were taken arbitrarily. Finally otimal bit lengths were obtained by trial and error method. The otimal bit lengths for W F, T on, T off, I P, C and V ga were found to be 5, 10, 15, 5, 5, and 15, resectively. And hence, the string length of each chromosome (oulation) became 55. NSGA-II starts with 100 initial oulations (N) which were generated randomly. Any other value of N could also be considered. Here a solution indicates a set of combination of inut rocess arameters. Each set of arameters is roducing a cutting seed and kerf width data through RBFN model. NSGA-II ranks the individuals based on dominance. The fast non-dominated sorting rocedure allows us to find out the non-domination frontiers (Ranks) where individuals of the frontier set are non-dominated by any solution. By using the non-dominated sorting rocedure, 100 scattered initial solutions were making four frontiers after 13 generation. Therefore, the whole initial scattered solutions are now groued into four ranks. This is shown in Fig.. It is obvious that Rank1 solutions are better than Rank solutions and so on.

6 Kerf width (mm) Fig.. Solutions are making four frontiers after thirteen generation. After finding the frontiers, the crowding distance was calculated for each individual by alying equation 13. The Crowding distance selection oerator hels NSGA-II in distributing the solution uniformly to the frontier rather than bunching u at several good oints. This was done by maintaining diversity in the oulation. The diversity in the oulation gives a wide range of choices to the users. Subsequently, the solutions of four frontiers were converged into only single front at the end of 50 generations. This was achieved by alying NSGA-II (ste 1 to ste 5) on those 100 initial solutions. Hence, at the end of 50 generations the scattered initial solutions are lying in the Pareto front leading to the final set of solutions. This is shown in Fig. 3. This grah indicates that the Pareto front contains 100 non-dominated solutions. So far as the NSGA- II arameters were concerned, -oint cross over with cross over robability 0.9 and bitwise mutation with mutation robability 0.1 were used. The tuning of these arameters was done by trial and error method Cutting seed (mm/min) Fig.3. Pareto-otimal solutions after 50 generation The Pareto-otimal solutions using NSGA-II with RBFN technique is listed in Table V. In this table, out of hundred data, only first fifteen are resented. TABLE V PARETO-OPTIMAL SOLUTIONS Sl Inut rocess arameters CS KW No. W F T on T off I C V ga CS = cutting seed, KW = Kerf width As none of the solutions in the Pareto front are better than other, any one of them is an accetable solution. The choice of one solution over other exclusively deends uon the requirement of the rocess engineers. If a higher cutting seed or a lower kerf width is required, a suitable combination of rocess variables could be selected from Table V. It could be shown that by using the non-dominated sorting genetic algorithm, the erformance of the WEDM rocess could significantly be imroved by selecting roer inut rocess arameters. For an instance, it could be observed from the exerimental results as tabulated in Table V, that the arametric combination for exeriment number 1 roduces cutting seed values of mm/min and kerf width of 0.39 mm. By otimizing using RBFN based NSGA-II, the cutting seed could be increased by 34.9 % for the same kerf width (Sl. no. 11). This has been achieved by selecting roer inut condition of the rocess. V. CONCLUSION The roosed Neuro-Genetic technique will be useful in finding several otimum solutions in multi-objective otimization roblem. Here only two objectives were considered. In future, Pareto solutions can be obtained by considering three objectives simultaneously. REFERENCES [1] R.A. Mahdavinejad and A. Mahdavinejad, ED machining of WC- Co, Journal of Material Processing Technology, vol , , 005. [] K. P. Rajurkar and W. M. Wang, Thermal modeling and on-line monitoring of wire-edm, Journal of Material Processing Technology, vol. 38, , [3] S. Saha, M. Pachon, A. Ghoshal and M. J. Schulz, Finite element modeling and otimization to revent wire breakage in electrodischarge machining, Mechanics Research Communications, vol. 31, , 004. [4] P. Saha, A. Singha, S.K. Pal and P. Saha, Soft comuting models based rediction of cutting seed and surface roughness in wire electro-discharge machining of tungsten carbide cobalt comosite, International Journal of Advanced Manufacturing Technology, vol. 39,.74-84, 008. [5] Y.S. Trang, S.C. Ma and L.K. Chung, Determination of otimal cutting arameters in wire electrical discharge machining, International Journal of Machine Tools and Manufacturing, vol. 35, , [6] S. Sarkar, S. Mitra and B. Bhattacharyya, Parametric otimization of wire electrical discharge machining of titanium aluminide alloy through an artificial neural network model, International Journal of Advanced Manufacturing Technology, vol. 7, , 006. [7] T.A. Sedding and Z.Q. Wang, Parametric otimization and surface characterization of wire electrical discharge machining rocess, Precision Engineering, vol. 0,. 5-15, [8] D. Scott, S. Boyina and K.P. Rajurkar, Analysis and otimization of arameter combination in wire electrical discharge machining, International Journal of Production Research, vol. 9, , [9] S. Chen and B. M. Mulgrew, Adative Bayesian feedback equalizer based on a radial basis function networks, IEEE International Conference on Communications, Chicago, vol. 3, , 199. [10] J.S.R Jang, C.T Sun and E. Mizutani, Neuro-Fuzzy and Soft Comuting, Prentice Hall of India Private Limited, New Delhi, 005. [11] M. Taghi, V. Baghmisheh and N. Pavesic, Training RBF networks with selective backroagation, Neurocomuting, vol. 6, , 004. [1] K. Deb, A. Prata, S. Agarwal and T. A. Meyarivan, Fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions, Evolutionary Comutation, vol. 6, no., , 00.