modelling

CIRP Annals - Manufacturing Technology 62 (2013) 775–797

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CIRP Annals - Manufacturing Technology

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Modelling of ECM and EDM processes

S. Hinduja (1)a,*, M. Kunieda (1)b

a School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, UKb Department of Precision Engineering, The University of Tokyo, Japan

A R T I C L E I N F O

Keywords:

ECM

EDM

Modelling

A B S T R A C T

The modelling of ECM and EDM processes requires not one but several models to simulate the different

phenomena that occur during machining. This paper reviews the models that have been developed to

simulate each of these phenomena, e.g. potential models to calculate the current density distribution in

ECM, thermal models for the plasma arc in EDM, moving boundary models to simulate the anodic

dissolution in ECM and probabilistic models to determine the discharge location in EDM. In addition to

discussing the relative merits of the techniques deployed in these models, the paper describes some

salient applications and concludes with desirable future enhancements to these models.

� 2013 CIRP.

1. Introduction

Because of their ability to machine tough, hard and heat-resistant materials with complicated shapes, electro-chemical andelectrical discharge machining (ECM and EDM) processes werefirst applied for the machining of aerospace alloys and die andmould making in the 1950s. Although both processes arecategorized as electrical machining to differentiate them fromconventional mechanical processes and both have similar machinetool structures and applications, their principle and machiningcharacteristics are significantly different.

The encouraging trend of ECM was not sustained for longbecause of inherent difficulties such as:

(i) c

* C

E

0007

http:

ontrolling/predicting the process due to electro-chemical,hydrodynamic and thermal factors;

(ii) p
redicting the equilibrium shape of the workpiece; and (iii) d etermining the tool shape for a given workpiece geometry.
In the case of EDM, the uncertainty of tool electrode wear wasan early difficulty.

Researchers have tried to overcome these difficulties bydeveloping analytical and numerical models of the processesbut progress has been slow. The reasons for this are as follows.

(i) E
CM and EDM are unlike conventional machining processessuch as turning or milling. In the latter, standard tool shapes areusually used and some of the process parameters such as feedrate and cutting velocity can be changed without affecting theshape of the final workpiece. But in ECM and EDM, the toolshape is unique to the workpiece geometry and a change infeed rate results in a different workpiece shape.
orresponding author.

-mail address: [email protected] (S. Hinduja).

-8506/$ – see front matter � 2013 CIRP.

//dx.doi.org/10.1016/j.cirp.2013.05.011

(ii) C
onventional processes usually require only one or twophysical phenomena to be modelled whereas ECM and EDMrequire several, such as fluid flow, gas evolution, chemicalreactions, heat generation at the electrodes and in theelectrolyte, and mass transport of the species.
In spite of the above complexities, in the last 50 years,empirical, analytical and numerical models have been developed.This paper reviews the development of these models and theirapplication in industry, and discusses some of the computingissues.

2. Overview of modelling the ECM/EDM processes

Both ECM and EDM have several common modelling require-ments. In both, the primary goal is to predict the shape of theworkpiece. EDM is predominantly a thermal process and therefore,in its case, thermal modelling is more important. In the case ofECM, however, determination of the current density distribution isof primary interest. Once the distribution is known, otherparameters such as workpiece dissolution rate can be computed.

ECM has the advantage that there are no residual stresses butthey have to be evaluated in the case of EDM.

In the case of EDM, removal occurs at the discharge locationonly. Even under the same pulse conditions, the thermo-hydrodynamic and electromagnetic behaviours of the anode,cathode, and working fluid materials bring about significantlydifferent results in material removal. The material removal inconsecutive discharges is a cumulative result of single pulsedischarges. However, results of multiple discharges cannot beobtained from a linear superposition of the results of a singledischarge, because the medium in the gap is composed of thedielectric liquid, gas bubbles generated due to discharge and soliddebris particles. The composition, pressure, gap width, andtemperature vary both temporally and spatially, which makesEDM simulation significantly difficult.

http://dx.doi.org/10.1016/j.cirp.2013.05.011

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http://dx.doi.org/10.1016/j.cirp.2013.05.011

S. Hinduja, M. Kunieda / CIRP Annals - Manufacturing Technology 62 (2013) 775–797776

Workpiece shape change, in the case of ECM, is dependentprimarily on the current density on the anode surface. Theelectrolyte’s electrical conductivity depends, amongst otherfactors, on the electrolyte temperature, which varies spatially,and the hydrogen bubbles released at the cathode surface.

In EDM, wear of the tool electrode necessitates the considera-tion of geometrical change of the tool electrode as well. To designthe tool electrode shape, not only the gap width but also the toolelectrode wear has to be considered. In the case of ECM, there is notool wear except when reversed polarity is used to remove anysolid debris adhering to the tool.

In continuous ECM, the tool is fed towards the anode at aconstant value and in pulsed ECM, the cathode or anode is movedinto a fixed position for each cycle. But in EDM, because the toolelectrode is fed by a servo feed control, where the feed rate is basedon the averaged gap voltage, modelling of the feed control issignificantly important to obtain the gap width distribution.Moreover, modelling of the feed control is also necessary toestimate the processing time, which normally results in a muchgreater error when compared with other cutting processes.

3. Modelling the ECM processes

Although the fundamental principles involved in ECM are wellknown [26,104], it is difficult to model the process because thereare several physical and chemical phenomena, some of whichoccur simultaneously. The phenomena that need to be modelledare discussed below and summarized in Table 1.

(i) E

TablMod

Aim

m

Phy

t

Ge

c

Pow

a

Ma

Ad

c

Inv

Inf

r

d

lectro-chemical reactions. The chemical reactions occurringat the anode and cathode cause ions, oxygen and hydrogen tobe released from the electrodes and the electrolyte; this masstransfer as well as the spatial and temporal concentrations ofthe species has to be determined, leading eventually to thecurrent distributions within the electrolyte and on theelectrode surfaces.

(ii) E
lectrolyte flow. The flow of electrolyte through the gap maybe laminar or turbulent. The chemical reactions at the anodeand cathode result in hydrogen bubbles being released at thecathode and oxygen bubbles at the anode. The presence of
e 1elling requirements for ECM and EDM.

ECM EDM

of

odelling

Predict workpiece shape,

cycle times, optimum

process parameters

Predict workpiece shape,

tool wear, gap width,

wire vibration,

temperature distribution

sics of

he process

Electro-chemical

modelling (potential and

current density

distributions, ion mobility,

mass transport) thermal

modelling, fluid flow

modelling

Thermal modelling

(heat-affected zone,

residual stresses) fluid

flow electromagnetic

modelling

ometry

onsiderations

Modelling workpiece

shape (efficiency,

Faraday’s law)

Modelling both tool and

workpiece shapes

er source

nd polarity

Continuous and pulsed

dc; Tool (cathode)

Workpiece (anode)

Pulsed dc and ac

chine control Constant feed rate

(continuous ECM) or

static (pulsed ECM)

Adaptively controlled

aptive

ontrol model

None Pulse conditions, feed,

jumping

erse problem Predict tool shape Predict tool shape

ormation

equired in

atabase

Electrical and thermal

properties of electrolyte,

valency, over-potential

and current efficiency

Discharge delay time,

energy distribution,

plasma diameter,

removal per pulse,

electrical and

thermophysical properties

of electrodes

these bubbles causes the flow of electrolyte to become two-phase; these bubbles affect the electrolyte’s conductivity, thusincreasing the complexity of the process.

(iii) T
hermal effects. The electro-chemical reactions cause heat tobe generated in the double layer and in the bulk of theelectrolyte (Joule heating). This heat energy causes theelectrolyte temperature to increase, resulting in a furtherchange to the electrolyte’s electrical conductivity.
(iv) A
nodic dissolution. The electrochemical reactions occurringcause dissolution of the workpiece, resulting in its shapechanging with time. This temporal change to the workpiece ismodelled as a slow moving boundary problem.
The modelling of these phenomena requires the development ofvarious models (see Fig. 1) which are inter-dependent.

3.1. Multi-ion and potential models

The aim of these models is to determine the distribution of thecurrent density on the electrode surfaces and in the electrolyte.With the multi-ion model, the total current i is given by the net fluxof charged species [117]:

i ¼X

k

zkFNk (1)

where F is Faraday’s constant, zk the charge, and Nk the flux densityof species k is given by:

Nk ¼ �zkukFckrV � Dkrck þ ckv (2)

where c is the molar concentration, D the diffusion co-efficient, u

the mechanical mobility, V the potential, and v the velocity of theelectrolyte. Two other fundamental equations are the electro-neutrality condition

Xk

zkck ¼ 0 (3)

and the conservation of charge.

@ck

@t¼ �r � Nk þ Rk (4)

Rk is the production rate of a species in the bulk of theelectrolyte. Since in ECM the reactions occur on the electrodesurfaces, Rk is zero.

If the electrolyte velocity distribution is determined first, thenEqs. (1)–(4) can be solved to give (i) the voltage distribution in theelectrolyte, and (ii) the concentrations of each species (ck) andhence the total current. Researchers have developed models whichtake all three components of Eq. (2) into account, i.e. migration,diffusion and convection. Qui and Power [137] developed a two-dimensional boundary element model which they applied to aparallel cell reactor and predicted the change in the cathode shape.Bortels et al. developed a more comprehensive model but it wasbased on two-dimensional finite elements and they referred totheir model as a multi-ion transport and reaction (MITReM) model[(Fig._1)TD$FIG]

Fig. 1. Modelling the ECM process (adapted from [125]).

S. Hinduja, M. Kunieda / CIRP Annals - Manufacturing Technology 62 (2013) 775–797 777

[9]. These models are applicable to both electro-chemicalmachining and deposition.

Generally, in ECM the following assumptions can be made.

(i) T
he time-dependent concentration term is zero. (ii) T he electrolyte is continuously refreshed and therefore the
concentration gradient terms can be ignored.

Because of these assumptions, it can be shown that the aboveequations reduce to Laplace’s equation which, in the context ofECM, is often referred to as the potential problem.

ker � ðrVÞ ¼ 0 (5)

where the electrical conductivity ke is given by

F2X

k

z2kukck (6)

One can use the multi-ion mass transport model for ECM tocalculate the concentration of the species and the current densitywithin the electrolyte; this gives a greater insight into the process[30,31] but at the expense of greatly-increased processing time. Onthe other hand, the potential model is popular with researchersbecause of its simplicity and the relative speed with which theresults can be computed.

3.1.1. Empirical models

Over the years, researchers have used various techniques topredict the workpiece (anode) shape. The advent of computers inthe 1970s enabled researchers to use numerical techniques tosolve for the potential distribution in the inter-electrode gap butprior to this, researchers developed empirical and mathematicalmodels.

The earliest empirical equations to predict the overcut and sidegaps in drilling were developed by Koenig and Pahl [75]. Koenigand Degenhardt [73] enhanced these equations for completelyinsulated and partially bare cylindrical tools. They also developednomograms from which one could determine the side gap for agiven tool tip radius, land width and equilibrium gap. Ippolito andFasalio [59], using a multiple regression technique, also developedequations to predict the over-cut. The main disadvantage withthese empirical equations is that they are valid only for the rangeover which the experiments were conducted.

3.1.2. Mathematical models

Of all the analytical techniques, the simplest and effectiveapproach is the cos u method first pioneered by Tipton [159]. Forworkpiece shape prediction, u is the angle between the normal tothe tool surface and the feed direction. Kubeth [85] and Kawafuneet al. [68] also suggested the same method but used thecomplementary angle and referred to it as the sin u method. Inthe cos u method, the tool is subdivided into several straight-linesegments and the corresponding workpiece segment is deter-mined to be at a distance of he=cos u where he is the frontalequilibrium gap. Lawrence suggested something similar butreferred to it as the flat inclined cathode theory [97,98]. However,according to Jain et al. [61], the cos u method gives rise todifficulties when u > 458, and the tool has sharp corners or hasradiused edges because it assumes parallel flux lines and neglectsthe effect of stray currents. If the tool has a complex profile, thenthis method is not to be recommended for obtaining the final

equilibrium workpiece shape; however, most researchers, whenusing an iterative numerical technique to determine the workpieceshape, use the cos u method to determine the starting shape.

A mathematical technique, which can predict the workpieceshape albeit under idealized machining conditions, is that due toCollett et al. [17]. They used a conformal mapping technique andwere able to analytically determine the workpiece shape for plane-faced tools with and without insulation on the vertical face.Hewson-Browne [50] extended the work of Collett et al. byconsidering a tool with rounded corners and partial insulation. This

method can only be applied to a limited number of geometries thatcan be transformed to the complex plane; moreover it is restrictedto two-dimensional problems and transient workpiece shapescannot be predicted. However, it can provide exact workpieceshapes for certain tool geometries and these shapes can be, andhave been, used by researchers to benchmark their solutions.

A more recent analytical-cum-computational method has beensuggested by Hocheng et al. [51] to predict the shape of an ECdrilled hole. At each time step, they consider a point on theworkpiece surface as being influenced by all the point sources onthe tool and the amount of dissolution at an anodic point isobtained by integrating over the entire width of the tool.

3.1.3. Numerical models

As processing power began to become readily available in thelate 1970s and early 1980s, researchers started to use numericalmethods such as the finite difference and finite element methodsto determine the potential distribution within the inter-electrodegap. The boundary element method (BEM) remained in the shadowof the other two methods and it was not until the early 1980s that itwas used to model the ECM process.

The finite difference method (FDM) was the first of thenumerical methods to be used; Tipton [159] is generally creditedfor having pioneered the application of this method to ECM. Tiptontried different methods of moving the workpiece; instead ofmoving a point along a direction normal to the surface, heconsidered only the vertical component of the cut vector. With thismethod the final equilibrium shape is not affected but theintermediate transient shapes do not correspond to reality.Klingert et al. [71] determined the primary and secondary currentdistributions with the activation over-potential modelled as alinear/logarithmic current density-dependent function. Lawrence[98] predicted the shape he would obtain with a semi-cylindricaltool and verified his predictions with both experimental andanalytical results. In their FDM model, Koenig and Humbs [74]made the current efficiency dependent on the current density andtemperature whereas Dabrowski and Kozak [19] combined theirFDM model with the analogue method. Subsequently Kozakdeveloped FDM-based models suitable not only for EC drilling butalso for die sinking [79]. This model considered the change in theelectrolyte’s electrical conductivity due to temperature increaseand void fraction. Like Klingert et al. [71], Prentice and Tobias [135]also developed their FDM model to include primary, secondary andtertiary current distributions but they applied their model toelectroplating.

It is generally accepted that the standard FDM cannot dealaccurately with curved boundaries; it assumes a linear variation ofthe governing variable. Both these factors make it necessary todeploy thousands of grid-points in the inter-electrode gap,although Nanayakkara [112] and Nanayakkara and Larsson[113] tried to reduce the number of grid-points by assuming aquadratic variation for each grid-point. These disadvantages can beovercome by using the finite element method (FEM) becausehigher-order iso-parametric elements can be used, the sides ofwhich can be curved. Therefore, no approximations have to bemade when representing curved tool and workpiece shapes; also,these elements allow a quadratic or cubic variation of the potential,thus obviating the need for a large number of elements.

The FEM was first used in ECM by Pandey and Jain; in [63] theymodelled the inter-electrode gap with a very coarse lineartriangular mesh and in [62] they repeated the same but this timethey obtained the temperature distribution within the gap. Thegeometry of the reactor cell was limited to two parallel plates.Alkire et al. [3] highlighted the difficulties in obtaining accuratedistributions of the potentials in the vicinity of a singularity.Assuming ideal machining conditions, Hardisty et al. predicted theworkpiece shape generated by a parabolic tool and verified it withan analytical solution [44].

Although the use of finite elements is a considerable improve-ment over finite differences, it has one serious drawback when

Table 2Comparison of results for a rectangular tool [115].

Tool with no insulation Tool with insulation

hc=hg ho=hg hc=hg ho=hg h1=hg

Collett [17] 0.80 1.159 0.80 1.159

Hewson-Browne [50] 1.0

Pera [132] 1.0 1.7 1.0 1.7

BEM [115] 0.8043 1.1586 0.7409 0.7548 1.079

FE [11] 0.83 1.191 0.775 0.875 1.25

FDM [97] – 1.3 – –


modelling the ECM process. Because the workpiece shape changesafter every time step, one has to re-mesh the entire domain afterevery few time steps. In the 1980s, when the availability ofprocessing power was still limited, Brookes [11] found the re-meshing of the inter-electrode gap very time-consuming. Instead,he used the same mesh but after every time step, he deleted someelements, modified others and improved the shape of the elementsclose to the workpiece boundary. Unfortunately, he found thatmodifying the mesh was almost as time-consuming as re-meshingthe inter-electrode gap.

Another disadvantage of the FEM is that the values of thevoltage gradient, denoted by @V=@n or q, at nodes on theworkpiece surface are not directly calculated. They are subse-quently determined from the computed values of potentials at thenodes in an element and this gives rise to discontinuity of @V=@n

between adjacent elements.The BEM overcomes these problems to some extent. Although

Christiansen and Rasmussen [15] were the first to apply theintegral equation method to modelling ECM, it was Hansen whoemployed it more generally for different axi-symmetric config-urations [42]. Narayanan et al. also developed a general 2-D modelbut were able to demonstrate the accuracy of their model bycomparing it with known analytic solutions [115]. Since the BEMreduces the dimensionality of a problem by one, it means that a 3-D model of the ECM process requires the surfaces of the tool andworkpiece to be discretised with 2-D triangular elements. Fig. 2shows a boundary element model consisting of a workpiece (greysurfaces) in which a slot is to be machined by a tool (goldensurfaces) with a rectangular cross-section. The figure shows thetool about to start machining. Virtual surfaces (green) areintroduced to form a closed shell.

It is interesting to compare the accuracy that is obtainable withthe different numerical methods. Narayanan et al. using a plane-faced rectangular tool, with and without insulation, did one suchcomparison [115]. They compared the results obtained from FD, FEand BE models with those obtained by Collet et al. [17] andHewson-Browne [50] for a rectangular tool (Fig. 3) under idealmachining conditions (Table 2). Assuming that the results from themathematical methods are exact, it is clear that the resultsobtained by PERA [132] and Lawrence [97] using finite differencesare considerably in error whereas Brookes [11] using finite

[(Fig._2)TD$FIG]

Fig. 2. A BE model for a milled component [129].[(Fig._3)TD$FIG]

Fig. 3. Equilibrium frontal and side gaps [115].

elements, predicted the ratios hc/hg and ho/hg to be 0.83 and1.191, respectively, for the tool with no insulation (see Fig. 3). TheBEM, using quadratic elements, calculated these ratios as 0.8043and 1.1586 respectively. The FE value for h1/hg is considerably faraway from the theoretical value of 1.0 as calculated by Hewson-Browne; the BE value of 1.079 is considerably closer, demonstrat-ing the superiority of the BEM. It should be noted that these resultswere obtained in the mid-1980s and the meshes deployed thenwere not as fine as those one would use today. However, theexample serves to illustrate the relative accuracy that can beobtained with the different methods. In summary, the accuracydepends upon the type of iso-parametric element used (higher-order elements give more accurate results) the time step (if a largetime step is used, oscillations are induced in the workpiece shapewhich become difficult to suppress) and the mesh density.

The BEM has one major disadvantage. The electrolyte’selectrical conductivity (ke) varies in the inter-electrode gapbecause of Joule heating and the release of hydrogen bubbles.This variation can be modelled to some extent by subdividing thegap into several zones and assuming a constant value of ke for eachzone. Subdivision into zones makes the use of the BEM lessappealing. If one can assume that ke is constant in the entire gap,then the BEM is the ideal choice; otherwise, if a more accuratemodel of the process is required wherein the velocity, temperatureand ke variations are taken into account, then the FEM should beused.

3.2. Workpiece shape change model

ECM is a slow-moving boundary problem and for modellingpurposes, the total machining time is divided into several timesteps. The rate of dissolution at a point on the workpiece isgoverned by Faraday’s law

dh

dt¼ hM

rzFJ (7)

where h is the current efficiency, M the atomic mass, h the inter-electrode gap, and J the normal current density which is given by

J ¼ keq (8)

Eq. (7) is usually solved using the marker method althoughmore recently the level set method (LSM) has been deployed. Withthe former, using the explicit Euler integration scheme, the gap atnode i on the workpiece surface, after the kth time step, is given by

hkþ1i ¼ hk

i þDt � hM

rzFJ � f cosu

� �(9)

where Dt is the time increment and f the feed rate. Eq. (9) requiresnot only the value of J but also its direction. At a node lying on theworkpiece boundary, one of two cases may arise. In the first case,the elements meeting at this node may share a common tangent.For example, in two-dimensional models, if the workpiece surfaceis modelled as a B-spline curve using higher-order elements, thenslope continuity is ensured. Alternatively quadratic elements canbe used and slope continuity enforced [115]; in such cases, a singlevalue of the voltage gradient q is computed by the BE or FEprogram, and its direction is also uniquely defined, i.e. normal to


the tangent. In the second case, there is a geometric discontinuity(e.g. sharp corner) at the node and the voltage gradient q becomesmulti-valued. One solution to this problem, which is popular withresearchers, is the method suggested by Alacron et al. [2]. Accordingto this method, a single value of q is forced on the solution and itsdirection is determined by assuming a linear variation of thepotential over the element edges meeting at the node.

In 3-D BE models, linear triangular elements are used resultingin the workpiece surface becoming faceted. Therefore, there maybe several elements meeting at a node, with each element havingits own normal ðneÞ. In such cases, the unit normal (~nnd) at thatnode is given by the average of the element normal vectors [135],

~nnd ¼1

Ne

Pe¼1Ne

ne!

Pe¼1Ne

ne!��

264

375 (10)

where Ne is the number of elements meeting at node i. In somecases, the above does not yield a valid normal direction. Accordingto Purcar [136], for a normal to be valid, it should be visible from allthe faces. He deployed a method first suggested by Kallinderis andWard [67] wherein the two faces subtending the most acute angleat a node is determined, followed by the plane bisecting them. Thenormal is forced to lie on this plane and its direction is chosen sothat it makes equal angles with the remaining faces meeting atnode i.

Another approach to solve the sharp corner problem with BEmodels is by the use of discontinuous or partially discontinuouselements [131]. With these elements the nodes are not located atthe ends of an edge but along the edges or even within the element.Yet another technique to cope with the corner problem withoutchanging the element type is to use the double node techniqueintroduced by Brebbia [10]. There are disadvantages with the useof double nodes. The elements meeting at the double node aredisplaced in different directions, creating a virtual gap between thenodes which have to be bridged somehow. Also, the use of doublenodes increases the number of equations, and hence thecomputing time. However, in the case of ECM, double nodes areideal to represent sharp corners on the tool surface [129] (e.g. atthe junction of the end and side faces) because there is no relativemovement between nodes on the tool surface (at the end of everytime step, all the nodes on the tool undergo the same rigid bodymovement).

Another modelling problem occurs when the cathode(or anode) isadjacent to an insulating surface and the angle between the twosurfaces is greater than p/2 (see Fig. 4(a)). For example, during then + 1th time step the cathode/anode will be moved due to deposition/dissolution resulting in the two surfaces becoming disconnected(Fig. 4(a)). In such cases, Deconinck suggests either an additionalvirtual element to connect the end of the current electrode profilewith the insulated surface (Fig. 4(b)) or extending the electrodeprofile until it meets the insulated surface (Fig. 4(c)) [28].

The accuracy of the computed workpiece shape depends also onthe magnitude of the time step. For a simple cell geometryconsisting of two parallel plates, Hardisty et al. predicted the[(Fig._4)TD$FIG]

Fig. 4. Junction between electrode and insulated surface [27].

number of time steps that would be required to reach equilibrium;they tested their improved algorithm for small and large initialstarting gaps [43]. Narayanan [114] showed that too large a timestep induced oscillations in the computed workpiece surface thatwere difficult to suppress. Deconinck [27] showed that the errordepends upon Wagner’s number W and the magnitude of the inter-electrode gap h. Based on a one-dimensional analysis, Purcar [136]suggested that the time step is given by

Dt ¼ffiffiffiffiffiffiffiffiffiffie

2mb

r� Jav

Jmax(11)

where Jav and Jmax are the average and maximum values of thecurrent density respectively, e the permissible error, m the numberof time steps and b = 1/(2(W + hmin)3) where hmin is the minimumgap.

When the explicit scheme is used, Volgin and Lubiynov foundthat, irrespective of which of the four variations (right, left, centraland upwind) of the explicit difference method is used, numericalinstability sets in when sharp corners are encountered [168]. Asharp corner results in the normal being defined ambiguously andsubsequent iterations cause a swallow tail or self-intersection inthe workpiece profile to be formed. Special topological routineshave to be developed to detect and eliminate them. Fig. 5 shows aworkpiece containing a recess and as it grows, its shape self-intersects [136].

This problem can be avoided by using the level set method, whichwas first pioneered by Sethian [144] and later described in detail inRef. [145]. This method has been applied to other dynamic problemssuch as flame propagation and waves. In this method, a scalarimplicit function ? is used to represent the moving front and also itsevolution. For example, the following implicit function would beused to represent the evolution of the workpiece front in ECM

@?

@tþ vr? ¼ 0 (12)

where v is the velocity with which the dissolution of the workpieceoccurs. The LSM was first applied to electro-chemical machining byVolgin and Lubiynov [168] who used the stationary formulation ofthe LSM. This method is more computing intensive than themarker method, although the computing time can be reduced tosome extent if the velocities are calculated only for a band of nodeson either side of the workpiece surface. It has been applied in 2-Dmodels but has yet to be applied to 3D ECM models.

3.3. Thermal models

Most of the early potential models assumed a constant value ofconductivity when computing the current density distribution. Inreality, the increased temperature of the electrolyte and thepresence of hydrogen gas in the inter-electrode gap affect ke. Thereare two empirical equations for ke and the one suggested by Thorpeand Zerkle [158] is given by:

ke ¼ ke0ð1� aÞm½1þ gðT � T0Þ� (13a)

where a is the void fraction, T the electrolyte’s currenttemperature, g the conductivity constant of the electrolyte and
[(Fig._5)TD$FIG]
Fig. 5. Self-intersection of the work-piece profile [136].

[(Fig._6)TD$FIG]

Fig. 6. Variation of current density with gap size in PECM [83].


m a constant varying between 1.5 and 2 for the heterogeneousmixture of liquid and gas. Suffix 0 refers to the initial properties ofthe electrolyte at the time of entering the gap. The secondempirical equation is due to Riggs [139] and Riggs et al. [140]:

ke ¼ k18e0:024ðT�18Þð1� aÞ1:5 (13b)

where k18 is the conductivity at 18 8C. Needless to say, use of theabove equations requires knowledge of the temperature distribu-tion within the electrolyte which can be determined by solving thefollowing convection-diffusion equation,

rC p@T

@tþ rc pvrT ¼ r � ðktrTÞ þ Pbulk (14)

where r is the density, cp the specific heat, v the velocity, kt thethermal conductivity and Pbulk the heat load. Researchers avoidusing this equation in its entirety because of the followingdifficulties:

(i) a
priori knowledge of the velocity distribution of the electrolytewithin the gap is required; and
(ii) t

[(Fig._7)TD$FIG]

Fig. 7. Time-temperature evolution in PECM for a point located on the anode and at

the gap exit (Sh = 0.006, pulse period = 0.1 s) [148].

he thermal conductivity of the electrolyte in the equation isthe sum of the molecular and turbulent conductivities; thelatter is determined from the k – v Reynolds averagedturbulence model.

The heat load usually consists of Pbulk, the Joule heat generatedin the electrolyte, and Pdl, the heat generated in the double layer.The latter can be neglected as most of it goes to heat the anode andcathode surfaces.

Most of the early researchers used a one-dimensional approachto calculate the increase in electrolyte temperature. In thisapproach, the temperature at a section along the gap is obtainedby performing a simple energy balance. Using this one-dimen-sional approach, Clark and McGeough [16] compared theirpredictions with experimentally measured temperature values.Although a good correlation was found between the two sets oftemperatures at the exit of the gap, there was considerablediscrepancy between the two sets at intermediate sections. Themeasured temperatures of the electrolyte were much higher. Thisis understandable because the predicted temperatures did not takeinto the account the additional resistivity caused by the hydrogenbubbles. Loutrel and Cook [100] observed through a microscopethat the average void fraction of the flowing bubble layer is 0.5;they, as in [16], used a one-dimensional model but took intoaccount the increased resistivity due to the bubbles and were ableto make more accurate predictions of the workpiece shape.

Instead of having to assume a value for the void fraction, manyresearchers have developed a two-phase numerical model for theelectrolyte flow which they solved using finite differences. Some ofthese two-phase models incorporate one-dimensional flow byaveraging the variables across the width of the gap at each section[37,52]. Using a two-dimensional flow model, Hourng and Changwere able to compute the temperature increase along the flowstream under equilibrium conditions taking into account the voidand other process parameters [53].

Jain and Pandey were one of the first to use a two-dimensionalFE model to predict the current densities and hence calculate theincrease in temperature of the electrolyte [62]. Their FE resultsconsistently underestimated the temperature increase, probablybecause of the very coarse mesh used.

With the advent of pulsed electro-chemical machining (PECM)in the late 1980s, the interest in electro-chemical machining hasbeen revived because PECM offers, by virtue of smaller gap sizes(10–100 mm), better accuracy and surface finish. The determina-tion of the increase in electrolyte temperature depends largely onthe Strouhal number (Sh), which during the off-time is given by

Sh ¼L

vto

where L is the length of the electrode and to is the pulse-off time.When Sh�1, there is sufficient time for the electrolyte to bereplenished and by-products to be flushed away; hence the systemcan be considered to have completely recovered during the restperiod. Under these circumstances, it is sufficient to model thePECM process by considering the effect of just one pulse (or groupof pulses) applied during the on-time. Kozak, Rajurkar and Rossadopted this approach and they derived the characteristic PECMequations using a simple energy balance and then solved themusing finite differences [82]. In a more refined mathematical modelfor PECM, Kozak, Rajurkar and Wei suggest a critical upper limit forthe pulse-on time for different process parameter settings [83].Fig. 6 shows the variation of the charge density with gap size, fordifferent pulse-on times ðt pÞ. One can operate below this criticaltime limit by ensuring that for a given current density, thecorresponding pulse time and gap size are to the left of the dottedline.

Smets et al. [148] developed an analytical model that was basedon the complete convection-diffusion equation; they studied thetemperature evolution in PECM in a rectangular gap taking theelectrodes into consideration. They showed that the temperaturetransient curve during the off-period comprises two parts (seeFig. 7); in the first part which shows a steep fall in temperature,most of the heat is transported away by the electrolyte due toconvection. In the second part, the fall in temperature is moregradual as heat flows from the electrode across the thermalboundary layer and into the electrolyte. The second part hastherefore a much higher time constant than the first.

Even when the Strouhal number is small, the temperatureevolution is transient in nature taking several cycles to reachsteady state. Since the time scale for the pulse-on time is farsmaller than the time required for the temperature to reach steadystate, a complete transient may become computationally veryexpensive as several thousand time steps will be required. Smetset al. overcame these difficulties by developing several analyticaland numerical models to calculate the averaged and pulsedtemperature history [149,150]. Their quasi-steady state short-cut(QSSSC) model is capable of calculating the average temperatureuntil a certain time period and then switching over to smaller timesteps to determine the cyclic temperature variation over the nextfew cycles.

[(Fig._8)TD$FIG]

Fig. 8. Variation of polarization voltage with current [4].

[(Fig._9)TD$FIG]

Fig. 9. Workpiece shapes for different over-potentials [114].


3.4. Over-potential

The over-potential of a cell has three components, i.e.activation, concentration and resistance over-potentials. Whilstthese over-potentials are dependent, amongst other factors, on thecurrent density and flow, it is difficult to characterize each of themseparately. Usually, for the sake of simplicity, the three over-potentials are combined together and represented by a single over-potential (Vpol) and only its relationship with the current density istaken into consideration either by a Tafel-like logarithmicrelationship

V pol ¼ aþ b ln J (15a)

or by a simple linear relationship

V pol ¼ aþ bJ (15b)

where a and b are constants or by using the Butler–Volmerequation [117]

J ¼ Jo½eaaFV pol=RT � eacFV pol=RT � (15c)

where R is the gas constant, aa and ac are the transfer co-efficients,and Jo the exchange current density. The values of Jo, aa and ac haveto be experimentally determined.

Assuming a linear relationship and a simplified analysis, Altenashowed that the current density is given by [4]:

J ¼ ðV � aÞke

ðhþ bkeÞ(16)

where V is the applied voltage. The product bke expresses theeffect of concentration on the current density and becomesimportant when the gap sizes are small as is the case in pulsedECM. In the case of continuous ECM where the gap sizes can be asmuch as 0.5 mm, h is much greater than bke. Hence, serious errorsare not introduced if Vpol is assumed to be independent of J.However, when the inter-electrode gap is in the range of 100 mmor less, the effect of the concentration over-potential becomesmore pronounced and neglecting the term b�ke will introduceerrors.

Consideration of over-potential using the logarithmic or linearrelationship presents an awkward problem because both Vpol and J

are not known. It causes the problem to become non-linear, thusrequiring an iterative solution at every time-step. Danson et al. [20]and Adey [1] used the following iterative method.

Vkþ1pol ¼ Vk�1

pol þ CðVkpol � Vk�1

pol Þ (17)

where C is a damping factor and k the current iteration. Prentice[134,135] also used a similar iterative technique but made thedamping factor a variable dependent on the normalized changes inVpol and J, and also on Wagner’s number. When the iterativetechnique of Danson et al. was tried for a stepped tool with linearand logarithmic over-potential relationships, it took more thantwelve iterations per time step to converge. A much faster methodbased on Newton-Raphson’s method was developed by Narayananand who was able to achieve convergence within three iterations[114].

One of the difficulties that modellers face in taking polarizationinto account is the lack of experimental data for the constants inEq. (15). Altena [4] experimentally determined the polarizationvoltage data for different amounts of concentration of NaNO3

electrolyte; one set of his results is shown in Fig. 8 which clearlyexhibits a linear relationship between Vpol and J.

To investigate the effect of over-potential, Narayanan assumedthe following normalized linear over-potential relationships forthe tool and workpiece surfaces and an equivalent logarithmicrelationship [114].

V pol�cathode ¼ 0:2 J (19a)

V pol�anode ¼ 1:0� 0:2J (19b)

The computed workpiece shapes are shown in Fig. 9, fromwhich it is clear that polarization decreases the equilibrium gap.Also the computed workpiece shapes with linear and logarithmicover-potential relationships are different only in the verticalsection of the workpiece, a section where the current densityvalues are relatively small.

3.5. Current efficiency

One of the parameters required for modelling is the currentefficiency (h); its value depends upon whether the electrolyte ispassivating (e.g. NaNO3) or not (e.g. NaCl). If it is a non-passivatingelectrolyte, then h can be assumed to be a constant. Otherwise, it isa function of the current density, pulse time and electrolyteconcentration. It is influenced to a lesser extent by the build up ofthe anions on the anode surface [23].

Instead of h, Kozak et al. advocate the use of the electro-chemical machinability coefficient (kv) which is given by

kv ¼h:kc

r(20)

where kc is the electrochemical equivalent of the workpiecematerial [81]. Kozak et al. argue that kv should be determinedexperimentally because, during alloy dissolution, the electroche-mical equivalent of each constituent is different from that whenthe constituent materials are dissolved individually. Instead ofhaving to determine values for two parameters (i.e. h and kc)experimentally, they showed that only one is necessary, i.e. kv

which is given by f=Jo where Jo is the mean current density; theythen went on to determine experimentally the value of kv forpassivating and activating electrolytes when machining differentalloys. For example, using a 13% water solution of NaNO3 andmachining an alloy NC10, they obtained the relationship kv = 1.64–2.13e�0.034 J which, of course, can be easily programmed into asystem.

Altena investigated the effect of current density, pulse time andconcentration on the current efficiency. Fig. 10 shows one set of

[(Fig._10)TD$FIG]

Fig. 10. Current efficiency for different current densities and pulse-on times for

concentration of 250 g NaNO3/l. [4].


results that he obtained using a water solution containing 250 g/lof NaNO3 [4]. These results were captured by Altena into a singlehyperbolic tangent equation:

h ¼ aðtanhðbt p þ cÞÞ � J þ ðd � t p þ eÞ þ f (21a)

where a–f are constants and t p is the pulse on-time. In the case ofcontinuous ECM, the relationship is much simpler, i.e.

h ¼ a � ðtanhðb � J þ cÞÞ (21b)

Instead of having to experimentally determine the value of h,Van Damme et al., in a seminal work, developed a FE model topredict the current efficiency when machining steel using asolution of NaNO3 as the electrolyte [165,166]. They predicted thecurrent efficiency by calculating three variables at each node, i.e.the potential, and ion concentrations cMe

z+ and cOH�. Thesevariables were obtained by solving three differential equations, thefirst of which was the potential equation. Eq. (2) (but without themigration term) was solved twice, once for each of the two ionicspecies. The other highlight of this work is the introduction of awater depletion function which was used as a weighting factor forthe assumed current density function. Their results for differentpulse-on times and for a particular concentration are shown inFig. 11 along with the experimental results obtained by Altena [4].Although there is some discrepancy between the two sets ofresults, the proposed procedure is very encouraging and the wayforward for modelling current efficiency.

3.6. Flow models

Flow is an important process parameter because in addition toforming an electrical bridge between the two electrodes, ittransports away most of the heat energy (and solid debris)generated during machining. How effectively it does this dependsupon the flow patterns generated, whether there are regionswithin the gap which are starved of the electrolyte and/or regionswhere the electrolyte forms eddies.[(Fig._11)TD$FIG]

Fig. 11. Predicted current efficiency for different current densities and pulse-on

times [166].

Single phase flow is obtained by solving the incompressibleNavier–Stokes equations.

rv ¼ 0 (22a)

r@v

@tþ v � rv

� �¼ �r pþmDv (22b)

where p is the pressure, v is the velocity and m is the dynamicviscosity.

Jain et al. were among the early researchers to develop anumerical model which accounted for the void fraction, andtemperature and pressure changes in the inter-electrode gap [64].However, the model was very specific in its application as itrequired the anode and cathode to be cylindrical which made itpossible to solve some of the equations analytically. Hourng andChang [52] initially generalized the problem by developing anintegrated model in which the bubbly two-phase flow was firstcomputed using Navier Stokes equations using a one-dimensionalmodel. The computed velocities were then fed into a thermalenergy model to calculate the temperature distribution in theelectrolyte. The temperatures were used to update the values ofthe electrolyte’s electrical conductivity, followed by a re-computa-tion of current distribution in the gap. This iterative process wasrepeated until convergence of the anode shape was obtained.Hourng and Chang were able to predict the spatial variation of theprocess variables within the inter-electrode gap.

However, one-dimensional flow models have limited useespecially when the anode and cathode shapes contain sharpbends because they cannot identify any eddies or separation of theelectrolyte. In a subsequent paper, Hourng and Chang [53]enhanced their model by considering the flow to be two-dimensional. They were able to demonstrate that better workpieceaccuracy was achievable with this model than that with a similar1D model. A very convincing example is shown in Fig. 12; it clearlydemonstrates that flow, even if it is single phase, should bemodelled at least in two dimensions [29]. This figure clearly showsthat re-circulations are formed, on the upstream and downstreamsides, near the two concave vertices of the cathode. These re-circulations reduce the efficiency of heat removal and actuallyresult in an unsymmetrical workpiece shape (see Section 5.4). Thisexample also shows that a visualization of the flow in the gap couldbe of invaluable assistance in the design of the cathode shape.

But flow in ECM should be modelled as two-phase becausesufficient quantities of hydrogen are released at the cathode.Oxygen is also released at the anode but the quantity is notsignificant. Since a considerable portion of the inter-electrode gap,especially the downstream part, is occupied by hydrogen gas,considering flow as single phase would be an approximation.

Because modelling two-phase flow in two-dimensions is notstraightforward, early researchers simplified the problem byconsidering the two-phase flow only as a one-dimensional model.This made it possible for researchers, notably Thorpe and Zerkel[158] to study the dynamics of the system of the processanalytically. Loutrel and Cook [100] were among the first to builda two-phase 1D numerical model of the process; in this model theyassumed a linear increase of the void fraction from gap entrance toexit. This causes the gap to decrease towards the exit. The use of a[(Fig._12)TD$FIG]

Fig. 12. Electrolyte flow in the inter-electrode gap [29].


two-phase two-dimensional model enabled Chang and Hourng[14] not to make this assumption. Instead the rate of dissolution ofhydrogen gas was incorporated into the model using Faraday’slaws of electrolysis and by so doing they were able to obtain amuch better correlation with experimental results.

3.7. Tool shape prediction model

Predicting the tool (cathode) shape for a given workpiece shapeis often referred to as the inverse problem. The inverse problem ismore important than the direct problem because, in practice,several trial tools have to be manufactured resulting in highdevelopment costs. Yet tool design has received far less attentionfrom researchers probably because the problem is ill-posed, i.e. insome cases, for a given workpiece shape, there is no correspondingtool shape and sometimes for a given workpiece shape, there areseveral solutions. The inverse problem is also a free boundaryproblem because one of the boundaries, i.e. tool boundary, isundefined. There are three main approaches to solve this inverseproblem, i.e. analytical, transformation to the complex plane andembedded. The analytical method is due to Tipton who suggestedthat the tool surface distance is proportional to cos u, where u, inthis particular case, is the angle between the normal to the anodeand the tool feed direction [160]. Since this method gives rise todifficulties, it is now normally used by researchers as a startingapproximation to the required cathode shape.

As early as 1968, Krylov [84], followed by Nilson and Tsuei[118,119], showed that it is possible to calculate the cathode shapedirectly if:

(i) t
he workpiece shape is defined using an analytic function, e.g.Fourier series; and
(ii) t

Fig. 13. Determining the tool shape [116].

he shape can be transformed from the physical plane to thecomplex.

Lacey [94] demonstrated this transformation technique bycalculating the cathode surface for a concave hyperbolic anodesurface and for different values of keðV= f Þ.

However, there are difficulties with this approach. Only thesimplest shapes can be transformed to the complex plane; also acorner on the anode surface gives rise to a singularity, theconsequence of which is that the direct solution can be obtainedonly for small values of keðV= f Þ i.e. high feed rates, low anodevoltage or low electrolyte conductivity.

Hunt transformed the potential equation to a second-orderdifferential equation which he integrated and then, using verticallines or a multi-grid, searched for points at which Laplace’sequation was satisfied [54]. The curve joining these pointsdefined the cathode shape. Hunt also suggested the embeddedtechnique [55] which removes the ill-posedness associated withthe inverse problem. He did this by searching through severaldirect solutions obtained by successive modifications to thecathode shape until a shape was found which matched, or wasclose to, the required anode shape. This embedded approach, insome form or another, is still the most popular techniquedeployed by researchers.

Narayanan et al. using the BEM utilized the fact that theconditions on the anode surface are over-specified [116]. Given aworkpiece shape and the feed rate, the required value of thevoltage gradient qr is equal to the dissolution rate, i.e. f cos u=M

where M is the dissolution rate. They considered each flux lineindependently and suggested three different formulations tocalculate the geometrical error at the termination point of theflux line on the cathode surface. For example, in the thirdformulation, the most promising of the three, the geometrical errorat the end point of a flux line is given by

Derror ¼ l2ðqr � qwÞðVw � VtÞ þ lðqr � qwÞ

where l is the length of the flux line, qw the calculated voltagegradient on the workpiece surface, and Vw and Vt the voltages onthe tool and workpiece surfaces respectively. They repeatedlymodified the cathode shape until the error was acceptable. In manyways, this is similar to Hunt’s embedded technique [55] but it doesnot suffer from the disadvantage that analytical functions have tobe used for representing the anode and cathode surfaces. Fig. 13shows that even after 50 iterations, the converged calculatedcathode shape does not have the sharp corner that is present in the‘‘exact’’ tool shape, i.e. the tool which was used to generate theworkpiece shape in the first place. But this does not matter becausethe calculated converged cathode shape yielded the same work-piece shape as the exact tool. Bhattacharya et al. [6] used finiteelements to determine the cathode shape iteratively but used amuch simpler criterion: the difference between the required andcalculated anode shapes. Das and Mitra [22] viewed it as a non-linear optimization problem in which they minimized

XN

i¼1

ðqr � qwÞ2

where N is the number of nodes on the anode surface. Zhouand Derby also considered it as an optimization problem butthey minimized the difference between the required andcalculated shape with respect to the coefficients in the analyticalexpression defining the cathode shape [179]. Like otherresearchers, they showed that the number of co-efficients inthe analytical expression affects the accuracy to which thesolution converges.

Chang and Hourng developed a comprehensive model forpredicting the tool shape [14]. Using Hunt’s embedded technique[55] and representing the tool and workpiece shapes as analyticalfunctions, they computed the cathode shape considering thetemperature increase due to Joule heating, void fraction and flow intwo dimensions.

The only work to-date in predicting the inverse problem in 3-Dis by Sun et al. [155] who claim to have obtained the tool shape inone iteration by calculating the theoretical value of the equilibriumgap using the cos u method. They were probably able to do thisbecause the anode surface, in their case, was a turbine blade with avery gentle curvature.[(Fig._13)TD$FIG]

4. Modelling of EDM processes

4.1. Generalized model for EDM processes

4.1.1. Discharge location

The generalized model for EDM processes is shown in Fig. 14. InEDM, removal occurs at the discharge spot, where a tiny crater isgenerated. For each pulse, discharge occurs only at a single locationwhere the dielectric strength is lowest. Hence, the EDM simulation

[(Fig._14)TD$FIG]

Simula�on of flow field in gap: bubbles, debris, dielectric liquid, and forces ac�ng on electrodes

Simula�on of removal due to single discharge

Simula�on of geometry

Heat input, diameter of heat source, pressure, flow velocity

Simula�on of EDM arc plasma

Discharge delay �me

Machine control

Temperature distribu�on

4.2.4, 4.2.5, 4.2.6

4.2.7, 4.2.8, 4.2.9, 4.2.10

4.3.1, 4.3.2, 4.3.3, 4.3.4

4.2.1, 4.2.2, 4.2.3

4.4

Discharge loca�on

Fig. 14. Generalized model for EDM processes.

[(Fig._16)TD$FIG]

FeedWire

Tool electrode

Unit area


starts from the determination of discharge location. If dischargealways occurs where the gap width is narrowest, the tool electrodeshape can be replicated on the workpiece with a high accuracy.However, discharge does not occur deterministically at thenarrowest gap. As shown in Fig. 15, Schumacher [143] indicatedthat the gap space is not filled with the dielectric liquid, but a largefraction of the gap is occupied by bubbles and debris particles inconsecutive pulse discharges. Hence, it is necessary to determinethe discharge location taking into account the influence of debrisparticles and bubbles.

4.1.2. Simulation of EDM arc plasma

Simulation of arc plasma is a necessary pre-requisite fordetermining the temperature distribution and material removal inelectrodes as it provides the necessary boundary conditions, i.e.heat input; diameter of heat source; pressure, and flow velocity.The heat input, which is the energy received from the arc plasma bythe electrode, and diameter of the arc plasma exert a significantinfluence on the temperature distribution in the electrode. Asshown in Fig. 15, discharge occurs in a narrow gap filled withdielectric liquid between parallel plane electrodes within a shortduration of nano to micro seconds order. Plasma constituents arevariable due to mass transfer from the evaporated dielectric andelectrode materials. The conservation equations of mass, momen-tum and energy, Ohm’s law and Maxwell’s equations have to besolved considering moving boundaries. Moreover, considering thatan analysis of arc plasma needs the temperature distribution andtopography of both the anode and cathode surfaces as boundaryconditions, the arc plasma simulation must be performediteratively together with simulations of temperature distributionand material removal. Thus, a precise analysis of the EDM plasma isdifficult.
[(Fig._15)TD$FIG]
Fig. 15. Schematic view of discharge gap due to multiple discharge.

4.1.3. Simulation of temperature distribution and material removal

due to single discharge

Most of the simulations of temperature distribution andremoval were performed assuming that the heat input anddiameter of arc plasma are known. Recent developments of highspeed video cameras have made it possible to observe and obtainthe diameter of the plasma. The inverse problem solution based onthe analysis of electrode temperature distribution combined withtemperature measurement gives the heat input from the arcplasma. However, it is still difficult to calculate the removalamount because not all the molten region is removed[126,167,178]. Furthermore, since the removal under the heatflux affects the temperature distribution in the discharge crater,simulations of the temperature distribution and material removalshould be solved as a coupled problem.

4.1.4. Simulation of flow field in gap

To determine the discharge location and to obtain the gap widthdistribution, the flow of debris particles and bubbles should becalculated. These flows are generated by the explosion of thebubble at the discharge spot, flushing flow of dielectric liquid, andthe jumping motion of the tool electrode.

4.1.5. Simulation of geometry

The load of removal, which is the volume of workpiece removedper unit area of the tool, is not uniform on the tool electrode surfacebecause it depends upon the depth of cut [18] and the surfaceinclination as shown in Fig. 16(a). Hence, wear does not occuruniformly over the working surface. In addition, the load ofremoval depends also on the curvature of the tool electrodesurface. A convex/concave unit area faces a greater/smallerworkpiece surface than a flat unit area. This non-uniform loadremoval can easily be reproduced by repeating the removal of bothanode and cathode at the discharge location for each discharge,provided the location is determined correctly as described later.Provided the volume of removal per discharge in the simulation iscorrect, the obtained geometries should agree with the machiningresults. In reality however, the simulation accuracy is notsatisfactorily high. This is because the bubbles existing prior tothe discharge result in different plasma growth and dischargecrater formation. Debris particles decrease the dielectric break-down strength of the gap, thereby increasing the gap width, whichaffects the plasma diameter, varying the removal amount[126,143].

(a) Sinking EDM (b) Wire EDM

Gap widthRemoval volume per unit area

Depth of cut due to feed

Fig. 16. Non-uniform load removal and gap width due to probabilistic nature of

discharge.

4.1.6. Machine control

In EDM, discharge ignition is delayed with increasing gapwidth. On the other hand, if the gap width is too small, dischargeoccurs immediately. If the feed rate is too high, the tool electrodemakes contact with the workpiece, leading to a short circuit.Hence, the tool electrode is fed based on the servo feed control asshown in Fig. 17 [151]. It is therefore necessary to simulate thedischarge delay time, which varies depending on factors such asthe gap width, debris particle concentration and working surfacearea.

[(Fig._17)TD$FIG]

Fig. 17. Principle of servo feed control.

[(Fig._20)TD$FIG]


4.1.7. Simplified and partial models

The idea that EDM can be simulated accurately by repeating allthe steps in the generalized model is not realistic or possible. It israther useful to simulate a part of the generalized model forunderstanding the gap phenomena, optimizing the machinecontrol, evaluating the machining accuracy and for processplanning.

The simulation method of time-sequential repetition of theflowchart which models a single pulse discharge needs a longcalculation time, longer than the actual machining. Hence,Tricarico et al. [164] repeatedly calculated the thickness of theremoval layer on the tool electrode and workpiece for every smallfeed step taking into account the dependence of the materialremoval rate on the local gap width using surface models for thetool electrode and workpiece geometries as shown in Fig. 18. Inthis simulation however, it is difficult to reproduce the gapphenomena for each discharge.

4.1.8. Geometry considerations

Geometrical simulation in EDM needs consideration of toolelectrode wear. In the case of surface models shown in Fig. 18,distance between adjacent nodal points decreases on the toolelectrode due to wear, whilst nodal points on the lateral surface ofthe machine hole move further apart. Calculation of the directionof the normal at each nodal point is susceptible to significant error.Hence, the simulation must be interrupted frequently for re-meshing to avoid topological instability [108]. In contrast, the useof voxel models as shown in Fig. 19, makes re-meshingunnecessary; also the singularity problem at sharp edges can beeliminated [109].
[(Fig._18)TD$FIG]
Fig. 18. Surface model for geometrical simulation [164].

[(Fig._19)TD$FIG]

Fig. 19. Voxel models for sinking EDM [109].

4.2. Modelling of a single discharge

4.2.1. Probability of discharge

To ignite an electric discharge in clean oil at an open voltage of100 V, the gap width must be less than several microns. In the caseof EDM however, since the dielectric liquid is contaminated withelectrically conductive debris particles, whose average diameter iseven more than one third of the gap width [176], discharge canoccur at gap widths of tens of micrometres or more[8,102,143,154]. This fact sometimes reduces the replicatingaccuracy due to the uneven distribution of the debris particles.On the other hand, the extended gap width is favourable for easygap control because it is difficult to keep the gap width constant atseveral microns.

In order to study the influence of debris particles, a debrisparticle with a diameter of 5 mm was placed in a gap of 20 mm asshown in Fig. 20 [89]. If it were true that the discharge occurs at apoint where the gap is shortest, then the discharge should occur atthe point where the debris particle is placed. However, experi-mental results showed that in most cases, the discharge crater wasnot generated at the point where the debris particle was placed.This is because the probability of discharge in a certain area isobtained from the product of the probability of discharge per unitarea and its surface area (area effect). Since the projected area ofthe debris particle in the direction normal to the electrode surfaceis negligible compared with the area of the electrodes(50 mm � 50 mm), the probability that discharge does not occuron the debris is greater than the probability that the dischargeoccurs on the debris. This result indicates that the dischargelocation is determined in a probabilistic way and not deterministic.

Another example which indicates the importance of theprobabilistic model is already shown in Fig. 16. In sinking EDM,when the tool electrode is fed by a distance during a certain time,the duty of removal volume per unit area on the tool electrode isgreater at a curved surface than that at a flat surface. The duty on aninclined surface is lighter than that on a horizontal surface.Therefore, the discharge frequency per unit area must be higher ona convex surface, and lower on an inclined surface compared withthat on flat and horizontal surfaces, respectively. To satisfy theseconditions, the gap width must be smaller on a convex surface, andgreater on an inclined surface than the corresponding gaps on ahorizontal surface. In wire EDM, the gap width distribution aroundthe wire is not uniform as shown in Fig. 16(b), which would nothave happened if the discharge location were to be determineddeterministically from the gap width distribution.

Fig. 20. Probabilistic nature of discharge location.

4.2.2. Discharge delay time

The discharge delay time depends upon the local values of thegap width and concentration of debris particles at the dischargelocation. Therefore, the discharge delay time can be used as ameasure to determine the discharge location. Moreover in reality,the tool electrode feed is controlled based on the discharge delaytime. Hence, modelling of discharge delay time is the key to EDMsimulation.

Morimoto and Kunieda [109] quantified empirically theaverage discharge delay time td,ave [ms] as a function of the gapwidth, gap [mm], concentration of debris particles, conc [mm3/


mm3], machining area, area [mm2], and debris diameter, r [mm]under the following machining conditions: copper anode, carbonsteel cathode, discharge current of 3 A, discharge duration of300 ms and open circuit voltage of 120 V.

td;ave ¼ 8:2� 1012 � ga p8:8 � r2:9

area1:2 � conc1:6

� �(25)

The machining area is a factor because the probability ofdischarge within a specific area is proportional to the ratio of itssurface area to the whole discharge area as described in theprevious section. td,ave was obtained experimentally based on thefact that the discharge delay time td conforms to the exponentialdistribution obtained from the Laue plot method [5,8]. The Laueplot shows the percentage of electric insulation that does not breakdown until time t after the supply of a pulse voltage [58]. Suppose asingle pulse discharge was generated N times and the number ofdischarges which did not occur until time t was n, then n as apercentage of N can be expressed by:

n

N¼ exp � t

td;ave

� �(26)

where td,ave the average value of td, is given by [107]:

td;ave ¼Std

N(27)

The Laue plot enables easy evaluation of the discharge delaytime, which normally has a large scatter, from the slope td,ave asshown in Fig. 21.[(Fig._21)TD$FIG]

Fig. 21. Evaluation of discharge delay time using Laue plot [109].

[(Fig._22)TD$FIG]

Fig. 22. Magnetohydrodynamics analysis of EDM arc coupled with heat conduction

analysis in electrodes [46].

4.2.3. Theory of determination of discharge location

To determine the discharge location, the discharge delay time ofeach voxel of the tool electrode is calculated probabilistically usingEqs. (25) and (26). In Eq. (25), area is set to the area of each voxel.conc and gap are the concentration of debris particles and gapwidth at each voxel whose values are recalculated for everydischarge cycle. conc is defined as the volume of debris particlesdivided by the volume of the gap voxels close to the voxel of thetool electrode. The last variable r, debris diameter, can be obtainedfrom machining experiments. Then, using Eq. (26), the Laue plotcan be drawn for every voxel as shown by the green line in Fig. 21.Note that the slope of the Laue plot for each voxel is significantlysmall when compared to that obtained from actual machining,because the area of each voxel is nearly equal to the dischargecrater size which is much smaller in size than the tool electrodesurface. Thus td,ave calculated for each voxel is much longer thanthat in actual machining. It must be emphasized that td of eachvoxel should be determined probabilistically. Substituting arandom number from 1 to 100% for n/N along the vertical axisin Fig. 21, td can be calculated based on the Laue plot of each voxelas indicated by the blue arrow. This operation is verified by thetheory of statistics which shows that, to generate a value of arandom variable X having a distribution function U = F(X), it

suffices to generate a value of a random variable U that is uniformlydistributed [107]. Since the number of voxels existing over themachining surface is several thousands to several tens ofthousands, td may even be as small as a few micro-seconds.Therefore, if the voxel with the minimum td is determined as thedischarge location, the same orders of td as measured in actualmachining processes can be reproduced. Consequently, the greaterthe machining area, the shorter will be the discharge delay time.Thus, the area effect is successfully reproduced in the simulation.

4.2.4. EDM arc plasma

Since erosion in the EDM process is made by accumulation ofremoval due to single discharges, the simulation model for heatconduction in electrodes is important. With the advent of powerfulcomputers and numerical analysis methods, it is nowadays notdifficult to take into account the time-dependent radius of thecircular heat source and time-dependent heat flux, or temperaturedependence of thermo-physical properties of electrode materials.However, boundary conditions such as arc plasma diameter, whichis time-dependent, and ratios of power distributed into the anodeand cathode have not yet been obtained theoretically, althoughthey exert a significant influence on the calculation results. This isbecause simulation of EDM arc plasma is extremely difficult.

Hayakawa et al. [46,47] first conducted magneto-hydrody-namic analysis of a DC arc in air between parallel plane electrodesunder the gap conditions used in EDM. They assumed that thespecies in high-temperature air which includes copper electrodevapour are N2, O2, NO, N, O, Cu, NO+, N+, O+, Cu+, N2+, O2+, Cu2+ andelectrons. Considering the temperature dependence of the thermo-physical properties of the plasma, the electromagnetic field,temperature, pressure and velocity distributions were calculatedfor the regions including both the electrodes and discharge gap.The conservation equations of mass, momentum and energy,Ohm’s law and Maxwell’s equations were solved. The energyequation included Joule heating, conduction, convection, andradiation terms. Fig. 22 shows an example of the temperaturefields. It was found that most of the discharge power is distributedin the electrodes, and heat transfer due to convection and radiationis negligible. Although this result is in agreement with theexperimental results of a DC arc [48], the arc which they analysedwas not in dielectric liquid but in air. The arc was not in transientbut in steady state, and removal of the electrodes was not takeninto consideration. Furthermore, the arc was assumed to be inthermo-equilibrium, and the equations of motion of the threespecies: electrons; ions and neutral particles, were not solvedseparately. Thus, the gap phenomena were symmetrical betweenthe anode and cathode thereby making the fractions of energydistributed to the anode and cathode equal. Consequently, it wasdifficult to obtain the difference in the energy distribution betweenthe anode and cathode.

4.2.5. Energy distribution

When copper was used for both the anode and cathode, theanode removal amount was greater than that at the cathode for a

[(Fig._23)TD$FIG]

Fig. 23. Method to obtain energy distributed into electrode in single pulse discharge

[170].

[(Fig._25)TD$FIG]

Fig. 25. High-speed video frames of EDM arc plasma (ie: 23 A, te: 80 ms, ui: 280 V)

[78].


discharge duration shorter than 20 ms, while it was smaller with adischarge duration longer than 20 ms [170]. Motoki and Hashi-guchi [110], and Van Dijck [167] explained that such aphenomenon is caused by the variation of the energies distributedinto the anode and cathode with the discharge duration, on thebasis of the T–F electron emission theory [99]. Other paperssuggest that this is due to the small mass of electrons compared toions, resulting in quicker impingement of electrons into the anodesurface than the slow ions, which arrive at the cathode surface later[24]. However, there is no scientific evidence that the delay of ionscan be in the order of nanoseconds or more.

On the other hand, Koenig et al. [77] measured the energydistribution by measuring the temperature of the electrodes anddielectric fluid in consecutive pulse discharges. Xia et al. [170]measured the energy distribution in a single discharge bycomparing the measured temperatures of the foil electrodes withthe calculated results obtained under the assumed ratio of theenergy distributed in electrodes, using a FD model and theexperimental setup shown in Fig. 23. When the calculated andmeasured temperatures were in agreement, they found that theenergy distributed to the anode and cathode is about 40–48% and25–34% respectively. They reported that the energy distributed tothe anode is always greater than that to the cathode and isindependent of the discharge duration. Recently, Zahiruddin et al.[178] measured the energy distribution in micro EDM. Fig. 24shows the overall results of energy distribution ratio versusdischarge durations. The energy distribution into the anode isconsistently greater than that into the cathode regardless of thedischarge duration.

Thus, it is obvious that the difference in the volumes removedbetween the anode and cathode cannot be fully explained by theenergy distribution. Motoki et al. [111], Ikai et al. [56] and Mohriet al. [105] explained that the amount of electrode wear is reduceddue to the protective effects of carbon layer which is formed on theanode surface. Since a thicker carbon layer is generated when thedischarge duration is long, tool (anode) wear is smaller than thaton the workpiece (cathode) even though the energy distribution tothe anode is more.[(Fig._24)TD$FIG]

Fig. 24. Energy distribution into anode and cathode [178].

4.2.6. Plasma diameter

Another important boundary condition is the plasma diameter.There are many papers in which a point heat source is assumed atthe cathode spot based on the researches on gas and vacuumdischarges ignited with gap widths which are significantly greaterthan the plasma diameter. DiBitonto et al. [34] and Patel et al. [128]obtained the energy distribution ratios into the anode and cathodeas 8% and 18%, respectively, assuming the point cathode sourcemodel. However, there have been no photographs taken to verifythe point source model in the narrow EDM gap yet. Anothersimplified model, which is widely used without evidence, assumesthat the plasma diameter is equal to the diameter of dischargecraters [180].

To obtain an evidence-based diameter, Snoeys and Van Dijck[152,153,167] systematically analysed the temperature distribu-tion assuming a circular heat source with time-dependent radiusand time-dependent heat flux on a semi-infinite cylinder. The heatsource growth function was obtained from an iterative calculationof the heat conduction equation by assuming that the temperatureat the centre of the heat source corresponds to the metal boilingtemperature at a pressure equal to an average pressure in the gasbubble calculated from the thermo-dynamical model. Goodagreement was found between the calculated melting pointisothermals and those measured from pictures of the cross-sections of the craters cut perpendicular to the surface.

Recent developments of high-speed video cameras havefacilitated measurement of the arc plasma diameter in EDM[33,78]. The radial temperature distribution in the arc plasma wasmeasured by optical emission spectroscopy and the plasma areawhere discharge current flows was determined based on the factthat the degree of ionization depends on the temperature [78]. Themeasured diameter of arc plasma generated in air was 0.5 mm, fivetimes larger than the crater diameter as shown in Fig. 25. Theplasma completed expanding within 2 ms after dielectric break-down, and thereafter, its diameter remained constant duringdischarge. Although the expansion rate was lower when the arcplasma was generated in a dielectric liquid, its diameter was stillgreater than the crater diameter.

4.2.7. Simulation of removal

4.2.7.1. Temperature rise in electrode. Given the energy distributionratio, and time-dependent radius of the circular heat source asboundary conditions, the following heat diffusion equation can besolved to obtain the temperature distribution in the electrodes.

@@x

k@T

@x

� �þ @

@yk@T

@y

� �þ @

@zk@T

@z

� �þ q ¼ rc p

@T

@t(28)

where q is the rate at which energy is generated per unit volume ofthe electrode due to Joule heating. By integrating the solution for apoint source which is liberated instantaneously at a given pointand time with regard to appropriate space and time variables, onecan obtain solutions for instantaneous and continuous sources ofany spatial configuration [12]. The analytical solutions of thetemperature distributions at time t produced by stationaryGaussian energy distribution sources in a semi-infinite solid aregiven by Pittaway [133].

[(Fig._27)TD$FIG]

Fig. 27. Side view of bubble oscillation (ie: 20 A, te: 100 ms, gap width: 0.1 mm,

anode: Cu f5 mm, cathode: Cu f5 mm).


Nowadays, with the development of powerful computers andnumerical methods, it is no longer difficult to take into account thetemperature dependence of thermophysical properties of elec-trode materials and latent heat of melting and vaporization.However, it should be emphasized that correct temperaturedistributions can be obtained only if correct boundary conditionsare used, i.e. energy distribution and plasma diameter. Althoughthere are many papers dealing with temperature analysis, in mostcases there is insufficient discussion about correct boundaryconditions.

Zingerman [181] states that Joule heating does not play asubstantial role in the machining of metallic materials, becauseelectrical potential drops in metallic workpieces are negligiblysmall. According to the calculated results of Rich [138], however,for high-resistivity metals, e.g., Hg, Sb, Fe, and Bi, Joule heating iscomparable to the energy input from the arc plasma. Saeki et al.[141] found that the removal of high-electric-resistivity material(Si3N4–30wt%SiC) in a single discharge can be greater than that ofsteel due to Joule heating.

4.2.7.2. Mechanism of removal. Calculation of the temperaturedistribution in the electrode only is insufficient to obtain thevolume of removal due to a single discharge. Simulations of craterformation and eruption of molten pool are indispensable. There isno doubt that the material can be ejected when its temperatureexceeds the boiling point. However, there are few papers dealingwith the removal model of the melting area. Zolotykh [182], Snoeysand Van Dijck [152,153,167], and Tao et al. [157] reported thatmost of the metal removal occurs due to boiling of the superheatedmolten mass in the crater at the end of the discharge becauseboiling of that superheated metal is prevented by the bubblepressure during the discharge. Van Dijck calculated the volume ofthe region inside the normal boiling point isothermal surface, atthe end of the discharge. The calculated volume agreed well withthe measured material removed per pulse. It was also found thatthe material removal efficiency, which was defined as the ratio ofthe ejected to melted volume, was only 1–10%. However,observation of the gap of single pulse discharge in dielectric liquidusing X-ray showed that 85% of material removal occurs during thedischarge duration [38]. High-speed camera images of flying debrisscattered from discharge points showed that material removal alsooccurs during the discharge [45]. Furthermore, Yoshida et al. [176]found that the metal removed per pulse in air is almost equal tothat in liquid when the discharge duration is longer than 100 ms,indicating that metal removal can occur without a sharp drop inbubble pressure. Thus, development of computation models whichcan simulate the material removal phenomena is eagerly awaited.

4.2.7.3. Advanced analysis methods for simulation of crater formation

and material removal. In some cases, molecular dynamics (MD)simulation can be very helpful to analyse the phenomena whichare difficult to model using conventional thermo-hydrodynamicssimulations. Shimada et al. [146] simulated the formation of a thinelectrode in a single discharge. Yang et al. [173,174] analysed theforming mechanism of a discharge crater ignited in vacuum asshown in Fig. 26. Both the space and time domains which can be
[(Fig._26)TD$FIG]
Fig. 26. Molecular dynamics simulation of material removal [173].

handled are unrealistically small compared with those used inEDM. Nevertheless, Fig. 26 clearly demonstrates the evidence ofthe superheating theory, indicating that removal can occur at anytime when the pressure inside the molten pool exceeds thepressure outside [174]. Tao et al. [157] simulated realistic cratermorphology, such as build-up crest and bulged bottom, usingFLUENT, a commercial computational fluid dynamics (CFD)software. To incorporate both molten and solid materials togetherin the same analysis domain, the volume of fraction (VOF) methodwas used, where both liquid and solid phases are modelled usingliquid type cells.

4.2.8. Bubble

Electrode materials and dielectric liquid evaporate, moleculesare dissociated, and atoms ionized, resulting in a rapid expansionof a bubble. Fig. 27 shows side views of a bubble generated by asingle discharge in deionized water and EDM oil. The bubble anddielectric liquid are analogous to the spring and mass oscillationsystem respectively [36,57]. Starting from the initial condition,where the bubble is compressed in a small volume, the dielectricliquid is accelerated radially. At the moment the pressure insidethe bubble equals the atmospheric pressure, the kinetic energypeaks. Hence, the bubble continues expanding. The diameter of thebubble peaks when all the kinetic energy is transferred to thepotential energy of the bubble. The diameter of the bubble reachesseveral millimetres, several tens of times greater than the gapwidth. Thereafter the bubble starts contracting until it iscompressed to its initial diameter. In reality, the viscosity of theliquid causes damping in the oscillation. After discharge, ions andelectrons are recombined, and the evaporated atoms andmolecules are solidified or condensed to form debris particles ordielectric liquid, but gases such as hydrogen and methane whichare generated by the dissociation of the working oil are left to forma bubble.

4.2.9. Reaction force acting on electrodes

Fig. 28 shows a model to calculate the bubble oscillation in thegap between parallel plane electrodes [36]. The reaction forceacting on the tool electrode can be calculated by integrating boththe pressure in the bubble and that in the dielectric liquid over theworking surface [86]. The reaction force in Fig. 29 was measured byKunieda et al. [91] using the Split Hopkinson Bar method [172]. Atthe initial state in which the bubble is compressed, the forcereaches its peak value. With the expansion of the bubble, thebubble pressure decreases and so does the force. The force evenbecomes negative because the bubble continues to expand evenafter the bubble pressure falls below the atmospheric pressure dueto the inertia of the dielectric liquid. Since the natural frequenciesof EDM machine structures are much lower than the frequencycomponents included in the reaction force waveform, the gapwidth cannot respond to the change in force. Furthermore, thereaction force in a series of pulse discharges decreases with timewhile the working gap is filled with bubbles generated in

[(Fig._29)TD$FIG]

Fig. 29. Calculated force acting on electrodes due to bubble oscillation (ie: 45 A, te:

150 ms, gap width: 150 mm) [91].

[(Fig._30)TD$FIG]

Fig. 30. Bubbles in gap after 300 times of discharge (110 ms after start of machining,

ie: 19 A, te: 250 ms, to: 50 ms, diameter of copper electrode: 30 mm) [156].

[(Fig._31)TD$FIG]

Fig. 31. Calculated equipotential lines generated by negatively charged particle

[92].

[(Fig._28)TD$FIG]

Fig. 28. Pressure distribution in bubble generated by single pulse discharge in gap

filled with dielectric liquid.


consecutive discharges [86,91]. Hence influence of the reactionforce caused by bubble pressure is negligibly small in sinking EDM.However, it causes vibration and deflection of electrodes in thecase of wire EDM [106,123] and micro EDM [72].

4.2.10. Residual stress

The temperature rise due to discharge results in a compressivethermal stress on the electrode surface until the stress is releasedby material yielding at a high temperature. The stress becomesnearly zero when the material becomes molten. After discharge,due to the temperature decrease, the molten material re-solidifiesand starts shrinking, generating a tensile residual stress on thesurface. This model was analysed using FEM [21,175] and MDsimulation [174].

4.3. Modelling of gap phenomena due to multiple discharges

4.3.1. Difference from single discharge

Generation of bubbles causes fluid flow as described in the nextsection. In addition, pressure in the bubble generated by adischarge is significantly influenced by bubbles which are alreadypresent [91], resulting in different plasma growth and dischargecrater formation. Observation of gap phenomena through atransparent electrode clearly showed that discharge occurs evenin bubbles, and that craters formed in bubbles are different fromthose in the liquid [70]. Since debris particles weaken the dielectricbreakdown strength of the gap, the gap width is increased. Thus,the plasma diameter is increased, thereby varying the removalamount. Therefore, diameters of craters and debris particles inconsecutive discharges are not equal for each discharge and showdifferent intermediate values between single pulse discharges inthe gap filled with gas and liquid [156].

Furthermore, the uneven concentration of debris particlesresults in a non-uniform gap width, deteriorating the machiningaccuracy. The cooling effect of the liquid may also be affected bythe bubbles. Concentration of discharge locations may result inlocally elevated temperature and accumulation of debris particles,leading to unstable discharge. Consequently, results of multiple

discharges are not equal to those obtained from a superposition ofthe results obtained from a single pulse discharge [126]. Therefore,simulation of gap phenomena considering multiple discharges isnecessary.

4.3.2. Bubbles in consecutive discharges

The working gap is mostly occupied by bubbles although bothelectrodes are submerged in dielectric liquid [70]. The flow field inthe gap was calculated by superimposing the dielectric flow causedby the bubble generated at a discharge point and the radial flow ofdielectric flushing supplied from the centre hole [156]. Assumingthat discharge occurs at random on the electrode surface, ananimation of the dielectric flow field was made as shown in Fig. 30.

4.3.3. Debris movement

Fig. 31 shows an electrostatic field distorted by a negativelycharged debris particle [92]. The measured gap width isoccasionally larger than 100 mm, whereas the measured averagediameter of debris particles is 25 mm or less. This result can beexplained in terms of the electrophoresis movement of debrisparticles in the direction perpendicular to the electrode surfaces,which was observed by Suda et al. [154], Bommeli et al. [8], andSchumacher [143]. They also observed that some particles arelinked in series to form chains parallel to the electric fields. Thespeed of the electrophoresis-induced motion was calculated to be0.136 m/s, high enough for a particle to change its position duringthe discharge delay time in the narrow discharge gap [92].

4.3.4. Flushing

To maintain stable machining and to obtain narrower and moreuniform gap width, it is critical to flush bubbles and debris particlesand cool the working gap in order to prevent the localization andconcentration of discharge locations. Pressure or suction flushingthrough holes in the electrode or workpiece remains one of themost efficient flushing methods especially if these holes have to beprovided anyway or do not harm the workpiece. Both in pressureand suction flushing, one can observe lower electrode wear andlarger gap width at the outlet point in comparison with the inletpoint [76]. Normally, the tool electrode is lifted periodically toreplace the contaminated dielectric fluid in the gap with fresh fluid.

[(Fig._32)TD$FIG]

Fig. 32. Simulation of fliud flow in wire EDMed kerf [124].

[(Fig._33)TD$FIG]

Fig. 33. Material removal rate Vw and volumetric tool wear rate VE versus gap width

[164].

[(Fig._34)TD$FIG]

Fig. 34. Pocket with a human-shaped protrusion [129].


Cetin et al. [13] calculated the three-dimensional fluid flow in thegap considering the suspended debris particles and obtained therelationship between the tool lifting velocity and height and theflushing capability. The working gap can be flushed by a freshdielectric fluid jetted from nozzles placed adjacent to the dischargegap. Masuzawa et al. [103] however demonstrated throughhydrodynamic analysis that jetting of dielectric fluid merely fromone direction causes increased density of debris particles in thedownstream region, resulting in uneven distribution of gap widthdeteriorating the machining accuracy.

Okada et al. [124] analysed the three-dimensional fluid flowaround the wire electrode when the workpiece is being cut byWEDM as shown in Fig. 32. They investigated the influence of theposition of the dielectric jet nozzles relative to the upper and lowersurfaces of the workpiece and flushing flow rate on the ability toflush the debris particles out of the gap.

From these fluid dynamics analyses, debris concentration canbe calculated quantitatively. However, the relationship betweenthe debris concentration and gap width cannot be determinedunambiguously. The gap width cannot be defined uniquely by thedebris concentration because the gap width is time dependent. Thegap width increases with the passage of time even if the toolelectrode is not fed, because discharge can still be ignited even ifthe gap width is increased, although the probability of dischargedecreases. Hence, a probabilistic approach should be used to obtainthe gap width distribution from the debris concentration.

4.4. Machine control

Given the shape of the tool electrode, the workpiece shape canbe obtained precisely, only if the gap width distribution is known.Since the gap width is determined by the servo feed control, aprecise geometrical simulation cannot be performed without anaccurate model which can reproduce the feed control of the toolelectrode. Since td can be obtained for each pulse discharge usingthe method described in Section 4.2.2, the moving average of thegap voltage can be calculated from the td values sampled in acertain period of time. Thus the tool electrode can be fed forward orbackward depending on the difference between the average gapvoltage and servo reference voltage as in actual servo-control.

4.5. Empirical data required

Although the EDM process can be modelled as described in theprevious sections, certain data which can only be obtainedempirically is required. For example, a pre-requisite for thedetermination of the discharge location is the evaluation of theaverage discharge delay time, as expressed by Eq. (25). Thisequation has to be determined experimentally for every machiningcondition: open voltage; dielectric liquid; and electrode materials.

Similarly, the simulation of arc plasma is not possible andtherefore the energy distribution to the electrodes and plasmadiameter are not known. Usually researchers assume values forthese two parameters; instead they should be obtained from

experiments and then used as boundary conditions for determin-ing the temperature distribution in electrodes.

Another importation parameter which has to be empiricallydetermined is the removal efficiency. Its value has an importantbearing, during the geometric simulation, on the removal volumesof anode and cathode per single pulse discharge. Even if the pulseconditions, materials used for the electrodes, and dielectric liquidare the same, the removal efficiency will be different depending onthe working surface area and gap width [164] as shown in Fig. 33.Other data which must be supplied are the volume of the bubblegenerated by a single discharge, debris particle size, and thermo-physical properties of the electrode materials.

5. Modelling examples in ECM and EDM

5.1. EC milling

The advantage of EC milling is that the required shape isobtained by moving a tool of simple shape along the threeorthogonal axes. However, calculating the current densitydistribution becomes more difficult as the inter-electrode gap isnow three-dimensional. Kozak et al. used an analytical method inwhich they assumed a linear variation of the current through thethickness of the inter-electrode gap [80]. This assumption helpedthem to calculate the current density and simulate the machiningof a free-form surface with a spherical tool. However, their modeldid not consider stray machining which is a very importantconsideration especially in EC milling. Domain-based methods likefinite elements do not lend themselves readily because the meshwould have to be re-generated with 3-D elements after each timestep.

Pattavanitch simulated and actually milled several componentsvarying in complexity from a simple slot [130] to the diamond-shaped pocket with a human-shaped protrusion shown in Fig. 34[129]. He subsequently modelled the machining of thesecomponents with the BEM. Whilst the accuracy obtained fromthe models was reasonably good, he faced several difficulties inpreparing data, two of which are mentioned below.

(i) S
ince the pocket was machined in several axial passes, theshape of the human figure had to be expanded and theboundary of the pocket contracted by the cumulative overcut.Commercial CAM programs do not have this facility and specialroutines had to be developed.
(ii) A
preliminary investigation showed that the time step shouldnot exceed 0.1 s. Since the workpiece was machined with a feed

[(Fig._36)TD$FIG]

Fig. 36. BE model for a thin ring [130].

[(Fig._37)TD$FIG]


rate of 9 mm/min, it meant that in one time step, only0.015 mm of the tool path could be considered. The tool pathwas 608 mm long and therefore it had to be sub-divided intowell over 40,000 segments, with every few segments requiringa mesh regeneration. Tool paths were generated usingMASTERCAM, a commercial software normally used forgenerating tool paths for turned and milled components.

5.2. Graphical and mathematical simulation

The accuracy of FE and BE models depends, in addition to themesh density and quality, element type and time step, on the inputvalues such as polarization voltage and current efficiency. Unlessthe latter are accurately specified, there will always be discrepancybetween the experimental and theoretically predicted results. Toensure that potential models generate accurate results, one needsto know, for different electrolyte concentrations and pulse times,the relationships between:

Fig. 37. Temperature distribution using multi-ion model [32].
(i) t
[(Fig._35)TD$FIG]

he polarization voltage and current density, and
(ii) t he current efficiency and current density.
Altena derived the above relationships experimentally [4]which have now been used by several modellers. In addition to theabove, Altena determined other relationships such as that betweenthe process voltage and frontal gap. With the help of theserelationships, accurate graphical and mathematical models for theprocess were built by DeSilva et al. [35]. These models were usednot to predict the current density but instead to determine the tooldimensions and the side gaps so that the slots were not undulytapered. For example, to machine the 300 mm wide slots in theshaver head in Fig. 35, the width of each of the 90 radially-orientedlamellae was determined as 120 mm and it was found that theactual width of the machined slots was measured as 285 mm [125].

Fig. 35. Machining slots in the Philips shaver head [125].

[(Fig._38)TD$FIG]

Fig. 38. Active and passive nodes [129].

5.3. EC turning

Although researchers have EC machined axi-symmetric com-ponents, very few have attempted to model the process. Ma et al.[101] proposed a mathematical model based on Faraday’s andOhm’s laws to calculate the amount of material removed in pulsedEC-turning. Their model is a function of the cutting velocity, widthof the tool and pulse-time. However, their model is of limited useas it is one-dimensional in nature. Pattavanitch et al. [130] haveshown that a 3D model of EC turning is feasible; they reported thedevelopment of a BE model to simulate the machining of the discshown in Fig. 36(a). Fig. 36(b) shows part of the BE model - it doesnot show the bounding virtual surfaces. Since the length of the ringwas equal to the depth of the tool, it was not necessary to feed thetool in the longitudinal direction. Also no radial feed was involved.The purpose of modelling was to determine the time required toreduce the diameter of the thin disc by 51 mm which wasdetermined as 90 s and subsequently verified experimentally.

5.4. Multi-ion and transport model for ECM

The importance of modelling the electrolyte flow and thermaleffects was recently demonstrated by Deconinck et al. [30,31].

Fig. 37 shows the anode shape (black curve) obtained using theirmulti-ion transport and reaction model which takes into accountthe electrolyte flow and heat energy generated in the electrolyteand double layer. The shape of the anode curve is asymmetrical(black curve) and Deconinck et al. [32] attributed this asymmetryto the increased temperature of the electrolyte in the upstreamregion. The temperature increase was due to the formation ofeddies, preventing the heat being efficiently being transportedaway. Deconinck et al. also suggested that water depletion may beanother factor causing the asymmetry. Water depletion makesoxygen evolution difficult, causing a greater portion of the currentto be spent on metal dissolution, thus increasing the currentefficiency. It demonstrates that modelling of flow and internalenergy has a significant effect on the computed anode shape.

5.5. Computing time in ECM

No review on modelling would be complete without a fewwords about computing time which can become prohibitiveespecially since several coupled models are involved. Thecomputing time depends, amongst other factors, on the efficiencyof the algorithm to solve the resulting simultaneous equations. Forthis, the FAST Multipole Method suggested by Greengard andRohklin [39] should be used to make the stiffness matrix sparseand then a GMRES technique used to solve the resulting equations[25].

In the case of EC milling, nodes lying on the workpiece surfaceshould be classified as being either ‘‘active’’ or ‘‘passive’’ [129]. Thecurrent density on the workpiece surface has positive values onlyin and around the projected area of the tool (yellow nodes) and alittle distance away from the tool, its magnitude becomes zero. InFig. 38, the flux density at the black nodes (which are on theworkpiece surface but are far away from the projected area), isvirtually zero; such nodes are said to be ‘‘passive’’ since both thepotential and current density are known. Such nodes should notbe considered and this will decrease the computing timesubstantially.

[(Fig._39)TD$FIG]

Start

Determination of discharge location

Removal of tool electrode and workpiece

Servo feed of tool electrode

End of machining

End

Generation and re-location of debris particles

No

Yes

Tool electrode voxels

Debris particlesWorkpiece voxels

Discharge location

Fig. 39. Discharge location searching algorithm [109].

[(Fig._42)TD$FIG]

Fig. 42. Reverse simulation.


5.6. Sinking EDM

Kunieda et al. [88,109] developed a sinking EDM simulationusing the discharge location search algorithm shown in Fig. 39which was obtained by simplifying the generalized model (Fig. 14).In the first step, the discharge location was determined probabil-istically as described in Section 4.2.3. In the second step, voxels ofboth the tool electrode and workpiece at the discharge locationwere converted to gap voxels, reproducing the material removal.Since the simulations of arc plasma, temperature distribution, andremoval were not performed, the removal volumes of toolelectrode and workpiece per discharge were obtained fromexperiments. In the third step, debris particles newly generatedby the discharge were located evenly on the periphery of thebubble generated at the same time, and debris particles presentprior to the discharge were re-located due to the dielectric liquidflow caused by the bubble expansion. In the servo feeding step, theaverage gap voltage was calculated from the moving average of thedischarge delay time of consecutive pulse discharges. Then thefeeding speed of tool electrode was determined depending on thedifference between the average gap voltage and servo referencevoltage as is done on actual EDM machines. These steps wererepeated until the machining depth reached the pre-set value.

A cylindrical steel workpiece was machined using a copper toolelectrode with the same diameter. Dielectric liquid was suppliedinto the gap from the centre of the tool electrode. Fig. 40 showsdistribution of debris particles simulated for two different flushingflow rates. Fig. 41 compares the calculated gap width distributionswith experimental results. The gap width increases in the radialdirection, because debris particles are transferred with the flushing

[(Fig._40)TD$FIG]

Fig. 40. Simulation of debris movement under flushing flow [109].

[(Fig._41)TD$FIG]

Fig. 41. Gap width distribution in radial position.

flow of dielectric liquid. The discharge location search algorithmcombined with heat transfer analysis realizes the simulation ofsurface temperature distribution in consecutive discharges[60,66,171].

5.7. Reverse simulation in EDM

Contrary to the forward simulation where the tool electrode isgiven, it is practically more important to develop a reversesimulation for obtaining the tool electrode shape with which thetarget workpiece shape can be machined precisely [87] (seeFig. 42). The reverse simulation was conducted using the samealgorithm as that developed for the forward simulation in Fig. 39assuming that the tool electrode is machined by the workpiecehaving the same initial shape as the target workpiece shape. Thedata of the removal volumes per discharge obtained fromexperiments were switched between tool electrode and workpiecein the reverse simulation.

5.8. Milling EDM

The machining of a groove by EDM was simulated consideringwear of the tool electrode [177]. In-process measurement of thetool electrode wear enabled the adaptive compensation of thedepth of cut to machine stepped grooves to a high accuracy [7].Using the discharge location search algorithm, a three-dimensionalgeometric simulation of micro-EDM milling was developed asshown in Fig. 43 [49].

[(Fig._43)TD$FIG]

Fig. 43. Simulation of micro EDM milling [49].

5.9. Wire EDM

5.9.1. Forces acting on wire electrode

In WEDM, there are four kinds of forces acting on the wireelectrode [32,127]: discharge reaction force; electrostatic force;hydrodynamic force and electromagnetic force (see Fig. 44). Thedischarge reaction force is caused by the rapid expansion of abubble at the discharge spot during the discharge. The electrostaticforce acts mostly when an open voltage is applied between thewire and workpiece during ignition delay time. The electromag-netic force acts on the wire during the discharge and canbe calculated from the area integral of the vector product betweenthe current density and magnetic flux density in the wire. The

[(Fig._44)TD$FIG]

Explosion force Electrostatic Hydrodynamic Electromagnetic

Workpiece

Wire

Discharge

Potential

difference

Fluid flow

Current

Discharge duration Discharge delay time

Discharge durationAnytime

Fluid flow

Reac�on force

Fig. 44. Forces acting on wire electrode.

[(Fig._46)TD$FIG]

Workpiece

node(x,y,z)

Lower guide

Upper guideWire

x

yz

Spark

Workpiece

Wire

Lower guide

Upper guide

Node(x,y,z)

Cell

Workpiece

node(x,y,z)

Lower guide

Upper guideWire

x

yz

Spark

Workpiece

Wire Upper guide

Node(x,y,z)

Cell

Determining dischargelocation

Removing workpiece

Analyzing wirevibration

Feeding wire electrode


Removing workpiece

Analyzing wirevibration


Lower guideDischarge spot

x

yz

Workpiece

Wire Upper guide

Cell

Node (x, y, z)


Removing workpiece

Analyzing wire vibration


Fig. 46. Geometrical simulation of wire EDM.

[(Fig._47)TD$FIG]

Fig. 47. Simulation of temperature distribution along wire electrode (discharge

current: 90 A, discharge frequency: 18 kHz, wire diameter: 0.25 mm, workpiece


hydrodynamic force is the drag force generated by the flow ofdielectric fluid. These forces cause vibration and deflection of thewire, thereby lowering the machining accuracy, speed, andstability [32,69].

Obara et al. [123] and Mohri et al. [106] obtained the reactionforce from solutions of the inverse problem in which wirevibrations calculated using assumed values of the force werecompared with measured ones. They used thin workpieces toexclude the influences of the electrostatic and electromagneticforces. Obara et al. [121] measured the change in the resultantforce involving the reaction force, electromagnetic force, andelectrostatic force with various discharge frequencies. They thenobtained the electrostatic force by extrapolating the resultantforce to the limit of zero discharge frequency, because both thereaction force and electromagnetic force are zero when thedischarge frequency is zero. As for the hydrodynamic force,Kuriyama et al. [93] conducted a CFD analysis of the force andinvestigated the influence of the jet flushing conditions on thewire deflection.

Regarding the electromagnetic force, Tomura and Kunieda[161] developed a FE model to calculate the electromagnetic fieldand the electromagnetic forces as shown in Fig. 45. Tomura andKunieda [162] found that, using a workpiece 40 mm thick, thereaction force is larger than the electrostatic force with largedischarge energy, while their magnitudes are reversed withdecreased discharge energy. The influence of the electromagneticforce on the wire vibration is not negligibly small under roughcutting conditions, especially with higher discharge frequenciesand larger workpiece thicknesses.

[(Fig._45)TD$FIG]

Fig. 45. Simulation of electromagnetic field in wire EDM to obtain electromagnetic

force acting on wire [161].

thickness: 100 mm [90].

5.9.2. Simulation of wire vibration and deflection

Obara et al. [122], Han et al. [41], and Tomura et al. [163]developed programs for WEDM simulation. The simulation shownin Fig. 46 is based on the repetition of the following routine:calculation of wire vibration considering the forces applied to thewire, determination of the discharge location considering the gapwidth between the wire and workpiece, and removal of workpieceat the discharge location. The electromagnetic force and hydro-dynamic force were ignored, and the influence of debris particleswas not considered. The geometrical simulation error was less than1.5 mm.

5.9.3. Wire breakage

Fig. 47 shows the temperature distribution along the wireelectrode obtained from a heat transfer analysis using FDM [65,90].Zone 3 indicates the part of the wire electrode where dischargeoccurs. Zones 2 and 4 show the part from the upper and lowerfeeding points to the upper and lower surfaces of the workpiece,respectively. Although the average temperature is around theboiling point of water, which is used as the working fluid, thetemperature at the point where the preceding discharge occurredis significantly high so that the tensile strength of the wire weakensat this point. Consideration of this has led to the development ofadaptive control systems in which the pulse energy is reduced orstopped based on the distribution of discharge locations measuredin process [95,120,147].

5.9.4. Optimization of wire electrode composition

Fine wire electrodes with diameter of 30 mm or less arebecoming popular. To resist the tension force, high tensile strengthmaterials such as tungsten or molybdenum are used. However,since these materials are rare metals, steel wires coated with brassor zinc are being developed [142]. Another requirement of the wireelectrode is low impedance at the frequency components involvedin the discharge current waveform. Thus, the electromagnetic fieldshown in Fig. 45 was analysed to investigate the influence of thewire electrode and workpiece materials on the discharge current[40].

6. Conclusions and future work

This paper has described the models to simulate the ECM andEDM processes. In the case of ECM, commercial systems based onthe potential model have become available. In the case of EDM, thepicture of the process is not completely clear probably because thephysics of the process is yet to be completely understood.Therefore, in EDM, the complete machining of a workpiece cannotbe modelled. Instead individual parts of the process such asestimating the ease of machining or durability of a tool, from awear viewpoint can be modelled. Much work still needs to be doneboth in modelling ECM and EDM and this is discussed in thefollowing sections.


6.1. Future work in ECM

To make further progress in modelling the EC milling process,there is a need for geometric pre-processing modules which willshrink/expand the external/internal boundaries of the feature to bemachined. Also, a special CAM system is required which cansubdivide the tool path into segments, the length of each segmentdepending on the time step and traverse feed rate.

To save computing time, for each time step, a node should beautomatically classified as either active or passive.

Modellers would benefit from more reliable experimental dataespecially that for current efficiency and polarization voltage.Whilst some data are available for dilute solutions of NaNO3, verylittle data are available for solutions of H2SO4 and HNO3. Anotherproblem with acidic electrolyte solutions is that their electricalconductivity depends on the amount of metal ion concentration. Itmust be borne in mind that the concentration keeps increasingwith usage of these electrolyte solution and very few modellerstake this into account.

At the moment, the presence of hydrogen bubbles is accountedfor by using empirical equations. The accuracy of an ECM modelcould be improved by developing more realistic bubbly two-phasemodels. So far, modelling of two-phase flow in two- or three-dimensions has not been reported. Modelling of flow is essentialbecause it has a pronounced effect on the quality of the workpiece,especially if it is a small and deep hole.

6.2. Future work in EDM

There are few EDM simulation programs which have beencommercialized. Some machine tools are equipped with empiricalsimulation tools which can suggest appropriate pulse conditionsdepending on the workpiece material and working surface area.The machining time can be estimated based on the removal stock[96,169]. Considering that it takes many days for machine toolbuilders to obtain such empirical data each time when a newmachine is developed, more effort should be devoted to improvingthe simulation accuracy. To achieve this, the following problemsshould be solved.

First of all, it is necessary to obtain correct boundaryconditions, especially the energy distribution and arc plasmadiameter. Since the simulation of arc plasma is difficult, theyshould be obtained from experiments. Given the correct boundaryconditions, powerful multi-physics simulations will realizeprecise simulation of removal. Since the material removalefficiency, the ratio of the ejected to melted volume, can beobtained, the database of pulse conditions could be made off-line.However, since pyrolytic carbon generated on the electrodesurface exerts a significant effect on tool wear reduction, asmentioned in Section 4.2.5, quantitative investigation of itsprotective effect is indispensable. Moreover, to complete thegeneralized EDM model, integration of the simulation of gapphenomena with the pulse generator control and servo feedcontrol is essential.

6.3. Common future work

For practical use, development of inverse modelling is eagerlyanticipated both in ECM and EDM. Preparation of tool electrode is acornerstone to obtaining high accuracy and productivity. Integra-tion of machine tool controllers and process modelling is anotherchallenging issue. Machine tool builders can design and developnew machine structures and pulse generators using the integratedsimulator. The accuracy of adaptive control strategies will beincreased by combining both machine and gap simulations.Meanwhile, simulations of hybrid processes between ECM andEDM and synergistic interaction between electrochemical andelectrical discharge phenomena in the normal ECM and EDMprocesses using water-based working fluid are other challengingworks.

Acknowledgements

The authors would like to express their sincere thanks to J.Atkinson, H. Altena, C. Diver, A. Klink, B. Lauwers, A. Malshe, V.Pattavanitch and M. Zeis for their assistance in the preparation ofthis paper. We would also like to thank G. Levy, J.P. Kruth, G. Levy,J.A. McGeough, A. De Silva, K.P. Rajurkar, X.D. Yang and othermembers of the STC-E group for their input.

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