A simplified mathematical model development for the

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ISSN: 1091-0344 (Print) 1532-2483 (Online) Journal homepage: http://www.tandfonline.com/loi/lmst20

A simplified mathematical model development for the design of free-form cathode surface in electrochemical machining Hasan Demirtas, Oguzhan Yilmaz & Bahattin Kanber To cite this article: Hasan Demirtas, Oguzhan Yilmaz & Bahattin Kanber (2017) A simplified mathematical model development for the design of free-form cathode surface in electrochemical machining, Machining Science and Technology, 21:1, 157-173, DOI: 10.1080/10910344.2016.1275192 To link to this article: http://dx.doi.org/10.1080/10910344.2016.1275192

Published online: 31 Jan 2017.

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Date: 27 July 2017, At: 05:57

MACHINING SCIENCE AND TECHNOLOGY , VOL. , NO. , – http://dx.doi.org/./..

A simplified mathematical model development for the design of free-form cathode surface in electrochemical machining Hasan Demirtasa , Oguzhan Yilmazb , and Bahattin Kanberc a Vocational School, Kilis  Aralık University, Kilis, Turkey; b Department of Mechanical Engineering, Gazi University, Ankara, Turkey; c Department of Mechanical Engineering, Bursa Technical University, Bursa, Turkey

ABSTRACT

KEYWORDS

High-performance machining of free-form surfaces is highly critical in automotive, aerospace, and die–mold manufacturing industries. Therefore, electrochemical machining (ECM) process has been used in such cases in that sense. The most important challenges of using ECM process are the lack of accuracy and difficulty in designing proper machining tool (cathode) surfaces. In this article, a simplified mathematical model is presented to obtain a cathode surface for ECM of free-form surfaces which have high curvatures. In this theoretical approach, the finite-element method (FEM) is used to solve the 3-D Laplace equation and to determine the potential distribution between the anode (workpiece) and cathode (tool) surfaces. A compact and simple program was developed to obtain a proper cathode surface that only requires some nodal coordinates on the anode surface and boundary conditions. In this work, a trial cathode surface is constructed for a given gap distance. For the determined ECM parameters, cathode shape that satisfies the boundary conditions is obtained for the 45th layer. The results are compared with the literature and ANSYS Workbench for verification. The developed theoretical approach benefits simpler and faster FEM solutions, accurate cathode surface, and consequently correct form of machined surface.

Cathode design; ECM; free-form surface; surface quality

Introduction Electrochemical machining (ECM) is a nontraditional machining process which is theoretically based on Faraday’s and electrolysis laws. According to these laws, the electron flow from one electrode to another (connected to a direct current power supply) is performed with the help of an electrical conductive liquid. In the ECM process, workpiece and tool (electrode) are charged as positive and negative, respectively. The tool is called a cathode and the workpiece is called an anode. In the machining mechanism, the material is removed from the anode without a contact between the anode and cathode. The electrolyte is pumped through the machining CONTACT Hasan Demirtas [email protected] Pasa Bul., No , , Kilis, Turkey. ©  Taylor & Francis Group, LLC

Kilis  Aralik University, Mehmet Sanlı Mah., Dogan Gures

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gap between the anode and cathode, while direct current is passed through the gap at a low voltage to dissolve metal from the anode. ECM is more advantageous than conventional machining, since it can be applicable regardless of material hardness, tool wear, and cutting forces in some cases. In fact, ECM is very useful for producing a bright surface finish, machining difficult-to-cut materials, and manufacturing complex geometry components (Rajurkar et al., 1999). The application of the ECM process is sometimes limited due to the troubles encountered with peripheral factors, cathode design, and process monitoring. Among them, the cathode design is the most important challenging factor that must be overcome. The critical point in the cathode design is to obtain an optimum cathode surface geometry for a specified anode surface geometry under the condition of changes in surface normalities, electrolyte flow rate, type, current density, and material. Various methods have been applied for the cathode design in ECM process, such as the boundary-element method (BEM), finite-difference method, and finiteelement method (FEM). FEM is applied more frequently for free-form surfaces than the others (Mount et al., 2003; Zhitnikov et al., 2004; Sun et al., 2006). Hocheng et al. (2003) used the iteration integral method for a 2-D cathode design. This model is based on electric field theory. The errors induced by the model make it inappropriate for short machining times, but the errors decrease with time. Li and Ji (2010) investigated the model error difference between the 2-D and 3-D Cos θ method and showed that the 3-D Cos θ method induced fewer errors than the 2-D Cos θ method. However, this model is based on geometric rules and does not indicate the ECM parameters (except the cathode feed rate). Kozak et al. (1998) used the electrochemical shaping theory to make a computer simulation for electrochemical shaping. The model was developed with considerations of some ECM parameters, such as electrochemical machinability, applied voltage, electrical conductivity, and so forth, but the important parameters, such as initial gap distance and cathode feed rate, were not taken into account. Some studies have utilized BEM to design computer simulations for ECM (Purcar et al., 2004, 2008). These models, however, were generally suitable only for simple surface forms. Pattavanitch et al. (2010) offered an ECM process simulation by BEM, and Li and Ji (2009) developed a model to machine aeroengine blades through the ECM process. The latter model is based on a BP neural network in which the inputs are the applied voltage, initial machining gap, feed rate of the cathode, pressure difference between the inlet and outlet of the electrolyte and electrolyte temperature. ECM cathode designs have also been made by FEM (Sun et al., 2006; Li and Niu, 2007; Wang and Zhu, 2009). Sun et al. (2006) used a nonuniform rotational B-spline surface (NURBS) model to explain the workpiece surface. In this model, the cathode points are found using electrochemical law. The potential distribution from anode to cathode is described by a Laplace equation. The Laplace equation is solved using FEM to find the gap which will be used to obtain the cathode coordinates. But the procedure of the cathode design is too long and unclear for practical usage. In this article, a simplified mathematical model is presented for the designing of a free-form cathode surface in ECM process using FEM. A dedicated FEM algorithm was developed and implemented in Mathematica . The developed mathematical

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Figure . Mapping of a NURBS surface. Note: NURBS, nonuniform rotational B-spline surface.

model is able to apply the symbolic and numerical computation as a hybrid method to solve the 3-D Laplace equation with high precision and less computing time. Furthermore, to reduce the CPU time, nodal coordinates were transferred from local coordinates to a natural coordinate system. To increase the precision of the FEM results, mesh quality factors, such as skewness ratio and aspect ratio, were considered. The developed method deals with the control points of free-form surfaces modeled through NURBS, which are obtained with a 3-D computer-aided design (CAD) program, and these points are imported into Mathematica . The developed method simply uses these control points to trail the cathode shape, with consideration of the electric potential distribution between the anode and cathode surfaces, which are modeled using a 3-D Laplace equation and an FEM solution.

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Mathematical modeling of the cathode surface In CAD and graphics, NURBS is the most popular tool for defining curves and surfaces. It can be used for many applications from manufacturing industry to film industry (Rogers, 2001). Due to capabilities of NURBS modeling for free-form surfaces, NURBS surfaces have been widely adapted in CAD/CAM software and graphics applications (Piegl and Tiller, 1997). A NURBS surface includes an intrinsic rational piecewise polynomial mapping from the 3-D surface to the 2-D parameter domain as shown in Figure 1. Therefore, NURBS modeling is preferred for describing the anode and cathode surfaces in this work. Based on this, anode surface is constructed and NURBS control points are obtained with using a 3-D surface modeling software. To construct the trail cathode geometry in Mathematica , these NURBS control points are used. To obtain the optimum cathode geometry that satisfies the Faraday and Ohm’s laws, a model is developed using FEM. To decrease analysis time, natural coordinate system is used during construction of shape functions.

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Modeling of workpiece (anode) surface

Free-form surfaces have been widely used in aerospace, automotive, and the die/mold industries. They are generally used to improve the functional requirements. The importance of them is increasing with advancing technology and thus NURBS is the best way to describe these surfaces. Therefore, for describing the

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anode surface, NURBS modeling can be expressed as follows: m n i=0 j=0 wi j Pi j Ti,k (u) T j,l (v ) , S (u, v ) = m n i=0 j=0 wi j Ti,k (u) T j,l (v )

(1)

where Pi j are the three-dimensional control net vertices, wi j is the weight of the Pi j , u and v are the biparametric directions, Ti,k (u) and T j,l (v ) are the nonrotational basis functions in the biparametric u and v directions, and k and l are orders of B-Spline basis functions. Using the surface control points, NURBS surface can be plotted in Mathematica , but surface function S(u, v ) cannot be obtained directly. Therefore, all the NURBS surface function components must be defined clearly. wi j describe the point weight and for a point that is on the surface, it is equal to 1. Ti,k (u) and T j,l (v ) are the complex functions and changes with the knots and degrees of the NURBS surface curves in u and v directions. For obtaining the knots and curve degrees, a command that is called “BSplineFunction” was used. It represents a B-spline function for a curve defined by the control points. After defining the knots and curve degrees, to obtain the nonrotational basis functions Ti,k (u) and T j,l (v ), “PiecewiseExpand” and “BSplineBasis” commands are used. “PiecewiseExpand” expands nested piecewise functions in expression to give a single piecewise function, and “BSplineBasis” gives the kth nonuniform B-spline basis function of degree d with knots at positions ui . “PiecewiseExpand” can be used to expand symbolic BSplineBasis functions into explicit piecewise polynomials. Due to NURBS surface function S(u, v ), the unit normal theorem can be used to obtain the function of normal vector for the free-form surface as follows;

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n (u, v ) =

Su (u, v ) xSv (u, v ) , |Su (u, v ) xSv (u, v )|

(2)

where Su (u, v ) and Sv (u, v ) are the derivatives of the S(u, v ) along u and v directions. As can be seen from Eq. (2), the normal vector n is a function of u and v and to obtain its numerical values at all points, u and v should have a value between one and zero due to the biparametric coordinates of the model as can be seen in Figure 2, (0 ≤ u ≤ 1, 0 ≤ v ≤ 1). The aim of this is to describe the n function for every point due to point coordinates in biparametric coordinate system. According to the electrochemical law, the trail the coordinates of the cathode surface control points can be obtained due to anode surface control points is as follows: xc = xa − b. sin θ . sin α

(3)

yc = ya − b. sin θ . cos α

(4)

zc = za + b

(5)

where b describes the distance along z direction between the anode and cathode, xa , ya , za are the coordinates of the anode surface control point, θ is the angle between the feeding direction of the cathode and normal to the anode, and α is the

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Figure . Representation of the biparametric coordinates for  ×  points along u and v directions.

Figure . The angles due to normal vector.

angle between normal vector’s project on x − y plane and y direction, as shown in Figure 3. As can be seen from Figure 3, θ and α angles associated with the components of the normal vector can be described as follows: nx α = cos−1 (6) ny  2 2 −1 nx + ny θ = tan (7) nz Finite-element modeling of cathode surface

Flow between anode and cathode was considered as a solid layer for meshing and numbering as shown in Figure 4. To create such a solid layer, the flow within the

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Figure . Solid model of the electrolyte between the anode and cathode.

gap had to be distributed in a uniform manner, and this is only possible in laminar flow. Re =

V.LC .ρ μ

(8)

Equation 8 describes the Reynolds number that is used to characterize the different flow regimes within similar fluids, such as laminar and turbulent flow, where Re is the Reynolds number, V is the upstream velocity, LC is the characteristic length of the geometry, ρ is the density, and μ is the dynamic viscosity. For the same electrolyte velocity of the fluid (V), the flow rate defines the flow regimes and therefore the flow rate of the electrolyte is considered to be constant. To avoid the effects of over potential and to make Faraday’s law applicable for obtaining the material removal rate, some assumptions must be made. These are (i) reaction rates of the electrode are very fast, (ii) to prevent the temperature gradient and concentration within the electrode gap, the electrolyte solution is completely mixed, (iii) machining is made by the current at the anode surface without any ECM (Zhou and Derby, 1995). In the most ECM processes, the cathode is fed at constant speed toward the anode so that the anode surface maintains its speed as constant. In this situation, the distance between the anode and cathode can be accepted as a time-independent value. Therefore, ECM system is acceptable as a quasisteady-state and free boundary problem (Hardisty et al., 1993; Domanowski and Kozak, 2001; Sun et al., 2006; Li and Niu, 2007; Wang and Zhu, 2009). According to aforementioned assumptions, the electric potential distribution inside the gap domain can be expressed by Laplace’s equation; ∂ 2ϕ ∂ 2ϕ ∂ 2ϕ + 2 + 2 =0 ∂x2 ∂y ∂z

(9)

Due to Faraday’s and Ohm’s laws, anode and cathode boundaries are shown as follows:  Vf ∂ϕ = ∗ Cos θ On Anode (10) ∂n kv ∗ kc ϕA = U } On Cathode

(11)

ϕC = 0}

(12)

On Anode

where ϕ describes the electric potential at each point on the anode, cathode surfaces and layers of the gap. U is the applied voltage, kc is the electrical conductivity of the electrolyte, which is directly affected by the electrolyte temperature and electrolyte

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Figure . Effect of the temperature on the electrical resistivity of the NaCl electrolyte for different concentrations (ASM Handbook, ).

concentration. The conductivity of an electrolyte changes substantially with temperature. Figure 5 shows the influence of temperature on the electrical resistivity of the NaCl electrolyte for different concentrations (ASM Metal Handbook, 1989). Any modification of the electrolyte temperature changes the electrolyte resistivity or conductivity. Based on Eq. (10), electrolyte conductivity was considered to be constant, and therefore the electrolyte temperature remained constant at room temperature. Vf is the feed rate of cathode along z direction, kv is the electrochemical machinability of the anode material, and θ is the angel between the movement direction of the cathode and normal to the anode. To satisfy the boundary conditions for the anode surface, Laplace equation can be minimize based on variation principle and can be explained as follows:     2  2  2  1 ∂ϕ ∂ϕ ∂ϕ ∂ϕ G (ϕ) = ϕds (13) + + dxdydz − 2 ∂x ∂y ∂z ∂n V A potential function ϕ(x, y, z), which varies linearly inside each hexahedron element is defined as; 8



ϕ x, y, z = Ni x, y, z .ϕi

(14)

i=1

where ϕi is electrical potential for each node and Ni is the shape function for each node and for trilinear hexahedron element in global coordinate system. Substitution

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Figure . Hexahedron with  nodes and the mapped cube.

of Eq. (14) into Eq. (13) and then minimization of G(ϕ) give the element stiffness matrix (Sun et al., 2006);

 K [e] [i] j =



   Ve

∂Nie ∂x



∂N ej ∂x





+

∂Nie ∂y



∂N ej ∂y





+

∂Nie ∂z



∂N ej ∂z



dV,

(15) where K is the element stiffness matrix, e is the element number. Solution of Eq. (15) requires very long CPU time. Therefore, in this study, numerical solution is made in natural coordinate system as an isoparametric element to decrease the analysis time. This method has been used for solving two- and three-dimensional finite-element problems with great success (Rao, 2011). With using a natural (or intrinsic) coordinate system ξ , η, ζ that is defined by element geometry, the isoparametric element equations are formulated. In other words, ξ , η, ζ are attached to the solid element to describe the length of the element along the axial coordinate. For each element of a specific structure, there is a relationship between the natural coordinate system ξ , η, ζ and the global coordinate system x, y, z which must be used in the element equation formulations (Gu and Gennert, 1991). The reason of choosing these particular limits is to simplify the Gaussian quadrate formula for the hexahedron elements. Type of FEM is chosen as hexahedron element which has six faces and 8 nodes. Figure 6 shows the mapped cube of a hexahedron element. In Figure 6, ξ goes from face 3267 to face 4158, ζ goes from face 4378 to face 1265 and η goes from face 1234 to face 5678. The intersection of the two medians is the center of a face. Node coordinates in natural coordinate system change between −1 and 1 as usual. According to these data, the shape functions of this element can be explained in Eq. (16). Ni =

1 (1 + ξ ξi ) (1 + η ηi ) (1 + ζ ζi ) , 8

(16)

where ξi , ηi , and ζi denote the coordinates of ith node in natural coordinates. The element stiffness matrix in Eq. (15) can be written as follows:  1 1 1 Be .Be T . det [J] dV (17) K [e] = −1

−1

−1

where “e” is the element number and det[J] is the determinant of the Jacobian matrix, and Jacobian describes the relation between the length of an element in the

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global coordinate system to the length of an element in the natural coordinate system. In general, [ J ] is a function of ξ , η, ζ and depends on the numerical values of the nodal coordinates (Logan, 2007). According to the polynomial completeness theory, Jacobian matrix is shown as follows: ⎡ ∂X ∂Y ∂Z ⎤ e

∂ξ ⎢ ∂X ⎢ J = ⎣ ∂ηe ∂Xe ∂ζ

e

∂ξ ∂Ye ∂η ∂Ye ∂ζ

e

∂ξ ∂Ze ∂η ∂Ze ∂ζ

⎥ ⎥ ⎦

(18)

where e is the element number, Xe , Ye , Ze are the transformation functions for mapping the hexahedron element from local coordinate system to natural coordinate system. Be is the matrix that includes derivative of shape functions and for one degree of freedom, it can be written as; ⎤ ⎡ ∂N e ∂N8e 1 · · · · · · · · · ∂x ∂x ⎥ ⎢ ⎢ ∂N1e .. e ∂N8e ⎥ B = ⎢ ∂y · · · (19) . · · · ∂y ⎥ ⎦ ⎣ ∂N e ∂N1e · · · · · · · · · ∂z8 ∂z ∂N e

∂N e

∂N e

where ∂x1 , ∂y1 , ∂z1 are the derivatives of shape functions, and they can be obtained using the inverse of the Jacobian matrix for each element as follows: ⎡ ∂N e ⎤ ⎡ e⎤ ∂Ni ∂x

i

⎢ ∂ξ e ⎥ ⎢ e⎥ ⎢ ∂Ni ⎥ −1 ⎢ ∂Ni ⎥ ⎢ ∂y ⎥ = J ⎢ ∂η ⎥ ⎦ ⎣ e⎦ ⎣ ∂Nie ∂z

(20)

∂Ni ∂ζ

By assembling all element stiffness matrices, global stiffness matrix can be obtained. The correlation between the potential and global stiffness matrix is shown as follows; ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ k1 1 k2 1 0 ϕ1 C ... 0 ⎢ k1 2 k2 2 . . . ⎢ ⎢ ⎥ ⎥ ... . . . ⎥ ⎢ ϕ2 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ 0 ⎢ ⎥ ⎢ ⎥ (21) ... ... ... ... ⎥ ⎢ ⎥ . ⎢ . . . ⎥ = ⎢. . .⎥ ⎣ . . . . . . . . . kt−1 t−1 kt−1 t ⎦ ⎣ . . . ⎦ ⎣. . .⎦ 0 ϕt 0 . . . . . . kt t−1 kn n where t is the layer number, k is the symmetric stiffness matrix, ϕ1 and ϕn are the electric potentials on anode and cathode surfaces, respectively. ϕ2 − ϕt−1 are the electric potentials at layers that are divided in the gap domain. C is the column vector that is shown as follows: ⎡ ⎤ c1 ⎢ c2 ⎥ ⎢ ⎥ ⎥ (22) C=⎢ ⎢ c3 ⎥ ⎣. . .⎦ cm

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where m is the node number at anode surface and cm is the surface area integration of Eq. (10) and can be written as: cm =

∂ϕ .S ∂n

(23)

where “S” describes the surface area of a hexahedron element. For a free-form surface, it can be calculated as follows:   |Su (u, v ) x Sv (u, v )| du dv (24) S= su

sv

where su and sv are the boundaries for an element along u and v directions. According to the conditions that are discussed above, the Eq. (21) can be solved as follows: ϕ2 = −k−1 12 . (k11 .ϕ1 − C) ϕi+1 = −k−1 i i+1 . (ki i−1 .ϕi−1 + ki i .ϕi )

(25) i>1

(26)

ki j are the matrix component of the global stiffness matrix for each layer and i and j are the layer number. For one element and one layer, k11 , k21 , k12 , k22 are shown in Eq. (27). From Eqs. (25) and (26), the potential distribution inside the gap domain can be obtained.

As can be seen from Eq. (27), stiffness matrix components are written as a 3-D array due to advantages like easy to understand, searching and sorting the elements, and easy to type the algorithm. First dimension of “K” describes the element number, second dimension is the column, and third dimension is the row of stiffness matrix. Case study: A free-form cathode surface design

In this section, the developed mathematical model is implemented for a simple freeform surface to create a corresponding cathode surface. Anode surface, shown in Figure 7, was modeled using the control points of the free-form surface. To obtain FEM results, the points on the surface were obtained with using a 3-D surface modeling software. Mesh quality has a significant role to get good accuracy and stability for the numerical computation. In this study, the aspect ratio and skewness of the

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Figure . Sample free-form surface.

elements are chosen to check the mesh quality. Skewness can be defined as follows:   γmax − γe γe − γmin (28) , Max 180 − γe 180 − γe where γmax is the largest angle in the cell, γmin is the smallest angle in cell, and γe is the angle for an equiangular cell (90° for hexahedron element). Table 1 gives the cell quality and the corresponding range of skewness values. The range of skewness value of this work is 0.0108 < γ < 0.012. The other main factor that affects the mesh quality is the aspect ratio. An element aspect ratio is the ratio of its maximum to its minimum width. To obtain good numerical results, aspect ratio must be near to 1. The aspect ratio range of this study is taken between 1.00013 ≤ AR ≤ 1.032. Due to these rules, anode surface and the mesh points are shown in Figure 8. The anode surface mesh points that have good mesh quality are imported from a 3-D modeling software to Mathematica . With the help of the mesh points, surface knot vectors along u and v directions, B-Spline basis functions, NURB surface model S(u, v ) are obtained, and normal vectors of all points are calculated through S(u, v ). After all these calculations, layer points are determined for a given gap distance that is shown in Table 2 and Eqs. (3)–(5).

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Table . Skewness values for mesh quality. Value of skewness Mesh quality

Excellent

Good

Acceptable

Poor

Sliver

Degenerate

–.

.–.

.–.

.–.

.–.

.–.

Table . ECM process parameters.

Electrolyte type NaNO

Electrical conductivity (k) (mS/cm)

Electrochemical machinability mm3 (kv ) ( 100A.min )

Scalar potential on anode (U) (V)

Feed rate (Vf ) (mm/min)

Equilibrium gap (mm)

.







.

ECM, electrochemical machining.

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Figure . Free-form surface with mesh points.

For obtaining the cathode geometry with the boundary conditions in Eqs. (9)– (11), the mathematical model is obtained. The FEM parameters and ECM parameters are shown in Tables 2 and 3. According to the boundary conditions of this model, the potential of the cathode surface must be zero. Due to this, the potential distribution must be obtained for all layers. In this study, fifty layers are used. After obtaining the layer points that are discussed above, the transformational calculations (from global coordinate system to natural coordinate system) and assembling of global stiffness matrix are made by the developed program. The program can be used for all surface types. For this example, the cathode surface that satisfies the boundary conditions is the 45th layer, and the potential distribution is shown in Table 4. For manufacturing the cathode, calculated surface points can be exported as a .dxf file format by Mathematica and solid model of the cathode can be obtained as shown in Figure 9. To compare the results of this study with the literature data, the model which is developed by Li and Diu (2016) is improved by adding anode and cathode layers in ANSYS Workbench as shown in Figure 10. During ECM process, the electrolyte flows between anode and cathode. Therefore, electrolyte geometry does not perfectly fit to the geometry of anode and cathode. To investigate the effect of geometric idealization, second finite-element model is developed by extending the electrolyte as shown in Figure 11. The analysis type is selected as Electric and more than 400 thousands hexahedral solid elements are used in both models. The distribution of voltage potential along

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Table . FEM parameters for the freeform surface. Number of layer

Element type

Number of nodes

Number of elements



Hexahedron

,

,

FEM, finite-element method.

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Table . Potential distribution inside the gap domain. Point number Layer number st layer nd layer rd layer th layer th layer th layer th layer th layer th layer th layer th layer





















 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

 − . − . − .  − . − . − . − . − . − .

 − . − . − . − . − . − . − . − . − . − .

the electrolyte thickness in normal direction is compared in Figure 12. It is shown that the linearity in potential distribution is changed when the finite-element model is used with extended electrolyte. So, an extended model can give more realistic cathode design.

Figure . Solid model of cathode surface.

Figure . Finite element model of anode, cathode, and electrolyte developed in ANSYS (Model ).

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Figure . Finite element model of anode, cathode, and extended-electrolyte developed in ANSYS (Model ).

Figure . Potential distribution along the electrolyte thickness.

Conclusion A mathematical model was developed to design a cathode surface for ECM of freeform surfaces. The main aim of this work is to obtain an accurate cathode surface that satisfies the Laplace equation to account for some boundary conditions that arise when using FEM in ECM. Thus, a computer program was developed for easy and practical usage in solving the encountered equations. A case study was conducted for a free-form surface, and the cathode surface coordinates were obtained. Two different ANSYS Workbench models were used considering with the anode and cathode surfaces. This work was also verified by comparing the results with the literature (considering the linearity situation) and ANSYS Workbench. It was also shown that a cathode surface can easily be obtained using zero or near zero potential points that were obtained from the results of the developed program. The developed theoretical model proposes robust FEM solutions, best-fit cathode surface, and it will lead to machine correct form of free-form surfaces. Experimental and workshop applications are being conducted and the verification tests are to be future works.

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Funding This work was supported by the Gaziantep University Scientific Research Project (BAP) Department under the grant number MF.14.19.

Nomenclature C e J K k, l ki i kc kv LC m N n Pi j Re S S(u, v ) Su (u, v ) Sv (u, v ) T t U u, v V Vf wi j Xe , Ye , Ze xa xc ya yc za zc α

Column vector for anode surface control points Element number Jacobian matrix Stiffness matrix Orders of B-Spline basis functions Global stiffness matrix component Electrical conductivity of the electrolyte Electrochemical machinability of the anode material Characteristic length of the geometry Node number Shape function Normal vector of control points Three-dimensional control net vertices Reynolds number Surface area of a hexahedron element NURBS surface function Derivative of S(u, v ) along u direction Derivative of S(u, v ) along v direction Nonrotational basis function Layer number Applied voltage The biparametric directions Upstream velocity of the fluid Feed rate of cathode The weight of the Pi j Transformation functions for mapping the hexahedron element Coordinate of control points on anode surface along x coordinates Coordinate of control points on trial cathode surface along x coordinates Coordinate of control points on anode surface along y coordinates Coordinate of control points on trial cathode surface along y coordinates Coordinate of control points on anode surface along z coordinates Coordinate of control points on trial cathode surface along z coordinates Angle between the normal vector’s project on x − y plane and y direction

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γ θ μ ξ , η, ζ ρ ϕ

Skewness ratio of a hexahedron element Angle between the z direction and normal vector Dynamic viscosity of the fluid Global coordinate system directions Density of the fluid Electric potential

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