A Response Surface Methodology to Optimize Multiple Discharge Step Parameters for Sinking Electrical Discharge Machining

Conventional die-sinking electrical discharge machining (EDM) employs a single electrode operating under constant discharge conditions. We explored a two-electrode scenario, in which roughing and nishing were coupled. We developed a multiple discharge step (MDS) method that uses three discharge depths. The discharge current is highest in step 1 and lowest in step 3. Response surface methodology (RSM) was employed to optimize the discharge conditions. Experimentally, the MDS method combined with RSM decreased electrode edge wear and surface roughness compared to the conventional method, with no increase in the average discharge current.


Introduction
Die-sinking electrical discharge machining (EDM) is commonly used to manufacture molds. The workpiece material is removed by electrical discharge between the electrode and workpiece, which are separated by a dielectric uid (Fig. 1). The shape of a soft copper or graphite electrode is stamped onto high-hardness tool steel, but the process time is long. Sinking EDM provides good geometrical accuracy when manufacturing high-aspect-ratio hardened tool steels, but a poorer surface nish than milling because discharge craters are created [1] . A great deal of research has been devoted to the effects of process conditions and gap lled with dielectric uid on electrode wear, machining time, and surface roughness.
The electrical spark not only removes the target metal but also wears the electrode proportionally to the material removal volume. The volumetric relative wear of graphite and copper electrodes with positive polarity and long pulse duration is smaller than that with negative polarity on the nishing EDM [2,3] . The pulse duration is most in uential, but the discharge current is relatively less important on the relative wear of graphite electrode at the roughing stage [4] . The wear rates of copper electrodes increase as the discharge current rises, and decrease with shorter pulse duration [5] . Tungsten-carbide electrodes wear quickly; their high carbide contents and ner grain sizes make them inappropriate for EDM [6] .
The edge wear of the electrode is much faster than the front wear, and very rapid at the beginning because the spark occurs more frequently at the edge, where the local electric intensity is higher than the at surface. The edge and front wear geometries are affected by machining parameters and dielectric ushing conditions [7] . The electrode wear pattern is affected by the electrode geometry and electrode path in multi-axis EDM [8] . A block-divided EDM process involving an electrode with a slender square cross-section was developed to compensate for electrode wear [9] . A cylindrical tool applied at a relief angle, and land thickness, reduced the machining time, radial overcut, and internal surface roughness of machined holes to a greater extent than a process involving a straight cylindrical tool [10] . Electrode wear geometry during die-sinking EDM has been simulated by searching the minimal gap associated with local discharge [11,12] . Few studies have explored the geometry of serial machining using different roughing and nishing graphite electrodes.
established for the three responses (surface roughness, edge wear, and machining time) and simultaneously optimized using the popular desirability function approach [36] .
Three EDM approaches (One-electrode SDS, two-electrode SDS, and two-electrode MDS) are compared in Section 2, and the experimental results presented in Section 3. To optimize the two-electrode MDS approach, RSM is employed in Section 4. The conclusions are summarized in Section 5.

Three Edm Approaches
In this article, we are going to investigate the in uence of the number of electrodes and discharge steps for the sinking EDM. Our MDS method divides sinking EDM into six steps involving the sequential use of two electrodes at three energy discharge levels, depending on the discharge depth. A roughing graphite electrode rapidly removes metal, and the nishing graphite electrode provides a precise workpiece. The discharge energy level is high initially, decreases during the second step, and then decreases further during the nal step, thus improving discharge e ciency.

One-electrode SDS approach
Conventional EDM research used single electrodes and an SDS system, as shown in Fig. 2(a). The use of one electrode for both roughing and nishing is cheaper than a two-electrode approach. A One-electrode SDS approach is appropriate if the removal volume is very small. However, this approach is inappropriate when machining large volumes; use of a single electrode increases the machining errors caused by severe electrode wear. In particular, extensive edge wear causes rounding errors [11,12] .

Two-electrode SDS approach
The sharp convex edge of the roughing electrode becomes worn after extensive machining, causing errors in the concave edges of the workpiece. We employed a new approach using two sequential electrodes. The roughing electrode has high discharge power, thus increasing material removal. This electrode was replaced by a new, sharp-edged nishing electrode to improve the surface in the nishing stage. The approach is shown in Fig. 2(b).

Two-electrode MDS approach
We also considered another approach based on MDS. During both roughing and nishing, discharges were applied over three steps, to increase die-sinking EDM e ciency. The three steps are shown in Fig. 2(c). High discharge energy during roughing shortens the machining time but increases surface roughness. The MDS approach divides roughing into three steps. The high discharge energy of the rst step increases the material removal rate, and the low energy of the third step reduces surface roughness. The low discharge energy during nishing guarantees a good surface nish but increases the nishing time. In the MDS approach, nishing is also divided into three steps. The high discharge energy of the rst step reduces the nishing time, and the low energy of the third step improves the surface nish.

Comparative Experiments
The three approaches de ned in Section 2 were compared in terms of the wear, surface roughness, and machining time. The die-sinking EDM machine was used to fabricate wedges of HP1 mold steel, using graphite electrodes and our MDS method. The One-electrode SDS method, and the two-electrode SDS and MDS methods, were compared.

Equipment and materials
The MDS experiment used a die-sinking EDM machine (U2610-2H; UNiTECH) [ Fig. 3(a)]. The HP1 mold steel was placed on a magnetic table, and the graphite electrode was bonded to the head of the machine. The dielectric uid jet was ushed in six directions to remove debris in the gap between the electrode and workpiece. The wedge-shaped graphite electrode and fabricated HP1 mold steel are shown in Fig. 3(b).
The HP1 workpiece composition was as follows: 0.

Experimental conditions
The discharge current in the roughing stage was set to 14 A based on the results of Kiyak and Cakır [14] , while the discharge current at the nishing stage was set to 5 A according to Jeong et al. [19] . The other experimental parameters were determined based on the equipment maker's recommendation. The pulse duration was 10 µs/A of peak current; all other conditions were unchanged. The pulse off-time, roughing gap distance, and roughing allowance ( nishing depth) were 40 µs, 0.07 mm, and 0.15 mm, respectively.
The experimental conditions and results of the three approaches are shown in Table 1. With the Oneelectrode SDS approach, roughing and nishing were performed by the same electrode. The two-electrode SDS approach uses one electrode for roughing and the other for nishing. The discharge current in the roughing stage is 14 A, to remove material quickly, and changes to 5 A at the nishing stage for good surface quality. With the two-electrode MDS approach, the discharge currents in the roughing stage were 16 A for the rst step, 14 A for the second step, and 12 A for the third step. The discharge currents in the nishing stage were 6 A for the rst step, 5 A for the second step, and 4 A for the third step. All three approaches were repeated three times to determine the reproducibility of surface roughness, edge wear, and machining time.

Edge wear
To explore the relationship between discharge depth and electrode wear, the workpiece plane was discharged using a wedge-shaped electrode [37] . After EDM, the electrode was divided into cross-sections with 1-mm intervals and the edges were photographed using an optical microscope (Fig. 4). The edge wear of the vertex where two sides and one front face meet is greater than the upper crosssection edge where the single side and one front face meet. Fig. 5(a) presents the edge wears of the Oneelectrode SDS approach. The One-electrode approach results in large edge wear of 0.216 mm because one electrode is used to remove the entire 4 mm depth. Dull electrode edges rounded out the concave edges of the mold steel. The edge wear of the vertex where two sides and one front face meet was greater than that at the upper cross-section edge where a single side and one front face meet. Fig. 5(a) shows the edge wear with the One-electrode SDS approach. The edge wear is substantial (0.216 mm) because one electrode is used to remove the entire 4-mm-deep metal. The dull electrode edges rounded the concave edges of the mold steel. In the two-electrode SDS approach, the precision of the nal shape is determined by the wear of the nishing electrode. The machining allowance a after roughing is removed by the electrode in the nishing stage. The edge wear of the nishing electrode is given by equation (2): In this model, the nal edge wear of the nishing electrode of the two-electrode SDS approach is predicted to be approximately 70% less than that of the single-electrode SDS approach. Fig. 5(b) shows the edge wear of the two-electrode SDS approach. As expected, the edge wear of the nishing electrode was reduced to 0.065 mm. The machined edge became more concave because the rounded edges were removed by the sharp nishing electrode after roughing. The edge wear of the two-electrode MDS approach was the same as that of the two-electrode SDS approach; the number of steps did not affect edge wear. Figure 6 compares the edge wear, surface roughness, and machining time among the approaches. The edge wear of the two-electrode SDS approach was about 70% less than that of the One-electrode SDS approach. The edge wear result of the two-electrode MDS approach was inferior to that of the twoelectrode SDS approach. The average surface roughness (Ra) values were similar among all three approaches. The two-electrode SDS approach was better than the other approaches in terms of machining time. A limitation of this comparison is that the current difference of the two-electrode MDS approach is xed at 2 A in the roughing stage, and at 1 A in the nishing stages, while the roughing gap is also xed at 0.07 mm. In the next section, we use RSM to determine conditions minimizing edge wear, surface roughness, and machining time when the MDS approach is employed.

Response Surface Methodology For Mds
As shown in Section 3, two-electrode EDM performed better in terms of edge wear than one-electrode EDM. However, for the two-electrode cases, the MDS approach was no better the SDS approach in terms of edge wear, surface roughness, or machining time. We used a Box-Behnken design (one of the most popular second-order RSM) to optimize the roughing and nishing currents of the MDS and the roughing gap.

Experimental plan
The process parameters and their respective levels are shown in Table 2. We considered three parameters: the discharge current differences in the roughing and nishing stages, and the roughing gap.
In the MDS approach, a high discharge energy during the rst step saves machining time, and a low discharge energy during the third step ensures a good surface nish. Therefore, in the roughing stage, the discharge current is increased in the rst step by an amount equivalent the roughing current difference (RCD), xed to 14 A during the second step and decreased during the third step by an amount equivalent to the RCD. The RCD was set to 0, 2, or 4 A. The minimum RCD was 0 A (equivalent to that of the SCD approach), the intermediate RCD was 2 A (equivalent to the MDS approach before RSM optimization) and the maximum RCD was double the intermediate value. During nishing, the discharge current was increased, xed, and then decreased in the rst, second, and third steps, respectively. The nishing current difference (FCD) was set to 0, 1, or 2 A, using a method similar to that applied to derive the RCDs. The roughing gap distance W is the clearance between the workpiece and electrode. The gap was set to 0.04, 0.07, or 0.10 mm. The intermediate level is the same as that of the SDS and MDS before RSM; i.e., 0.03 mm away from both the minimum and maximum levels.  Table 3 shows the BBD matrix and measurement data, where the edge wear is the wear of the electrode after the nishing stage, and the machining time is the sum of the roughing and nishing times. As the roughing stage is followed by the nishing stage, the Ra should be unaffected by the RCD. ANOVA revealed that the R 2 of the tted model was 62.51%, and the adjusted R 2 was 52.28. Thus, 62.51% of the surface roughness variation is attributable to linear effects of the FCD, the gap, and their interaction. The smaller R 2 value may re ect surface roughness variations among the milled electrodes, and errors when measuring inclined surfaces. Fig. 7 shows the estimated Ra, with respect to FCD ΔI f and gap W when RCD is xed at 2 A. The FCD had a greater in uence on Ra. When the FCD increased, Ra tended to decrease to a minimum. The graph also shows that the gap had less effect on Ra when the FCD increased by up to 2 A. Figure 8 shows the surface nish improvement of the MDS approach using FCD 2 A compared to the conventional SDS approach using a single discharge current when the average discharge current is set to 5 A. The average surface roughness Ra was improved to 3.2 µm by increasing the discharge current at the rst step and decreasing it at the third step, compared with 4.6 µm by the conventional approach.

Analysis of edge wear
The edge wear EW of the nishing electrode is in uenced by three parameters. Equation (4)

Analysis of machining time
Machining time (MT) is in uenced by three parameters, as shown in Equation (5)

Optimal conditions
Page 11/27 The desirability function approach is implemented to optimize the three response variables affected by the three process parameters [36] . The desirability function approach is most often employed to optimize multiple responses simultaneously [35] . This approach searches for parameter settings that jointly optimize multiple responses by satisfying the requirements for each response under consideration. In this approach, the estimated response values of each response are transformed to scale-free desirability between 0 and 1. The individual desirability (d) for each response to be minimized is obtained by specifying the target value and upper bound required for the response. If the response is larger than the upper bound, d is set at 0. If the response is smaller than the upper bound, d increases from 0 to 1 as the response variable comes closer to the target value. If the response is smaller than the target value, d is determined to be 1. A weight factor, which determines the desirability function shape for each response, is then assigned to each response. Weight can be given as a value between 0.1 and 10. When the weight is 1, the desirability function is linear. When the response needs to be smaller than the upper bound, a weight less than 1 is determined. If the response should be close to the target value, weight is set at a value greater than 1. In general, if the weight factor is not mentioned, it is set at 1 [35] .
The individual d are combined into an overall desirability D, which is the geometric mean of the individual d. When the response variables vary in terms of importance, D is the weighted geometric mean of the individual d. The relative importance of response variables are re ected by the 'importance values'. The optimal compromise among multiple responses is achieved by maximizing D [36] . The desirability function approach was employed to simultaneously minimize the average surface roughness Ra, the edge wear of the nishing electrode, and the machining time simultaneously. The target values and upper bounds were 3 and 6 µm for Ra, 0.05 and 0.1 mm for edge wear, and 10 and 15 min for the machining time, respectively. As it is more important that the surface roughness and machining time are lower than their upper bounds than that they reach the importance target values, their weights were set to 0.5.
Moreover, edge wear was considered to be twice as important as surface roughness and machining time, and was thus assigned an importance value of 2.
Using the response optimizer in Minitab, the optimal parameter combination was shown to be (RCD, FCD, Gap) = (0.580, 2.0, 0.04) (Fig. 11). These conditions optimize the three responses simultaneously. The estimated Ra was 3.50 µm, the edge wear of the nished electrode was 0.052 mm, and the machining time was 10.78 min. As the RCD is controlled in integral increments, we performed additional experiments at RCDs of 0 A and 1 A.

Con rmation experiment
The optimal conditions (subsection 4.5) were (RCD, FCD, Gap) = (0.58 A, 2 A, 0.04 mm). As the current can be controlled only in integral units, we tested two conditions [(0 A, 2 A, 0.04 mm) and (1 A, 2 A, 0.04 mm)] three times, and compared the data ( Table 4). The three responses were optimized when the RCD was 1 A.

MDS optimization results
The comparisons in Section 3 showed that the two-electrode MDS approach was somewhat inferior to the two-electrode SDS approach (Fig. 6). In this section, we used an RSM to optimize the RCD, FCD, and gap in terms of edge wear, surface roughness, and machining time. Fig. 12 shows the average values of the three responses for the two-electrode SDS and two-electrode MDS approaches, before and after RSM.
RSM for the Two-electrode MDS approach contributed to the improvement of edge wear and surface roughness. Through RSM optimization for the MDS approach, the edge wear of the nishing electrode was improved from 0.072 mm to 0.052 mm, and the average surface roughness was reduced from 4.01 to 3.27. The RSM optimized MDS approach has reduced edge wear by 20% and average surface roughness by 24% compared to the SDS approach. However, the machining time of the MDS approach has increased by 15% compared to the SDS approach, where the discharge current is not changed. The machining time of the MDS is longer than that of the SDS because the time saved from the high discharge current in the rst step is smaller than the time increased from the low discharge current in the third step. Overall, the Two-electrode MDS approach with RSM shows better performance than the Twoelectrode SDS approach.
The RSM of the two-electrode MDS approach improved edge wear and surface roughness. RSM optimization improved the edge wear of the nishing electrode from 0.072 to 0.052 mm, and reduced the Ra from 4.01 to 3.27. The RSM-optimized MDS approach reduced edge wear by 20%, and the Ra by 24%, compared to the SDS approach. However, the machining time of the MDS approach increased by 15% compared to the that of SDS approach (where the discharge current does not change). The MDS machining time is longer than that of SDS because the time saved by using a high discharge current in the rst step is less than the extra time required by the low discharge current in the third step. Overall, the two-electrode RSM-optimized MDS approach performed better than the two-electrode SDS approach.

Conclusion
Conventional die-sinking EDM studies use one electrode operation under constant discharge conditions.
In this article, we employed two electrodes for roughing and nishing, an MDS approach and three discharge steps to improve edge wear, surface roughness, and machining time. Compared to a Oneelectrode SDS, the edge wear of a two-electrode SDS was reduced from 0.216 to 0.065 mm, i.e., by more than three-fold. However, for the two-electrode case, application of the MDS approach before RSM led to poorer performance than the SDS approach.
A Box-Behnken design was used to investigate the effects of RCD, FCD, and gap on edge wear, surface roughness, and machining time. The desirability function approach was employed to minimize the three responses. The optimal conditions were RCD = 0.58 A, FCD = 0.58 A and gap = 0.04 mm. As the current can be controlled only in integral units, RCD was tested at 0 and 1 A, and then set to 1 A (which gave better performance). Three con rmation experiments performed under optimal conditions showed that the RSM-optimized MDS approach reduced edge wear by 20%, and improved the surface roughness by 24%, compared to the SDS approach.
The MDS approach combined with RSM can be bene cial to EDM practitioners who want to optimize process parameters to improve the edge wear, surface roughness, and machining time.

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