CC BY
8
MSC 30-04
DOI: 10.14529/mmp190101
COMPARISON OF QUASI-STATIONARY AND NON-STATIONARY SOLUTIONS OF ELECTROCHEMICAL MACHINING PROBLEMS APPLYING TO PRECISION CUTTING WITH PLATE ELECTRODE-TOOL
V.P. Zhitnikov1, N.M. Sherykhalina1, S.S. Porechny1
:Ufa State Aviation Technical University, Ufa, Russian Federation E-mails: [email protected], [email protected], [email protected]
The quasi-stationary problem for modelling the process of electrochemical cutting with a plate electrode-tool is formulated. The formulation of the problem is based on the use of a stepwise function of current efficiency from the current density. Thus three areas with various conditions are formed on the machined surface. The usual stationarity condition is used in the area of high current densities. In the area of low current densities the dissolution is absent and the initial form of the boundaries remains. In the intermediate zone, the current density at each point is equal to the critical value. The presence of boundary conditions on each section of the machined surface allows to formulate a boundary problem for the analytical function of the complex variable and to find the shape of the boundary at any moment, regardless of the background. The solutions of quasi-stationary and non-stationary problems are compared, and the range of existence of quasi-stationary solutions is found.
Keywords: electrochemical shaping; stepwise function; quasi-stationary model; error estimation.
Introduction
Problems of modelling of electrochemical machining (ECM) has been solved by many authors. However, more attention was paid to the study of stationary solutions with different current efficiency dependencies on the current density (a part of the current participating in the reaction of metal dissolution) [1-14]. The solution of non-stationary problems allows to determine many additional parameters that are not described by a stationary solution, including the time of establishing of a stationary process with a given accuracy. However, the need for computational resources for solving such problems is much greater [15-21].
In the simulation of many processes a non-stationary problem can be simplified by using the quasi-stationary approximation. In this case, a solution for every moment of time can be found independently of the others. There is no need to decrease the time step to maintain the accuracy and stability. But it is necessary to study the range of applicability of the quasi-stationary approximation within the limit of errors.
The choice of current efficiency n (j) on current density j is offered in [22] and justified on the basis of analysis of experimental data in [14]
(1)
for applying in modelling the ECM process of the Faraday's law
Vecm = kr/-, k = Ks/p1 (2)
K
where Vecm is the rate of electrochemical dissolution, e is an electrochemical equivalent, p is a density of dissolved metal, k is an electrolyte conductivity, j1 is the critical value of current density, restricted the dissolution region.
If we use models (1), (2), the quasi-stationary approximation is formed by specifying the usual stationary condition on the parts with a high current density and condition j =j1 on other parts [23]. The dissolution is absent in areas where j<j1 and the original shape of the workpiece is retained. The approximation is to use the stationarity condition, since in reality the stationary solution is established asymptotically. Condition j =j1 in non-stationary processes fulfills dynamically when the current efficiency changes abruptly from zero to no and conversely, therefore does not lead to inaccuracy.
In this work we apply to a quasi-stationary model of electrochemical vertical cutting by a plate electrode-tool. The stationary problem of this type is solved in [14]. The electric field, as in [14], is considered to be potential and solenoidal. Therefore, the methods of the theory of functions of complex variables [24, 25] are used to solve the problems.
1. The Quasi-Stationary Problem
Let us consider the shaping problem of machining surface on machining by the plane semi-infinite vertical electrode-tool (ET) A'CB' that moves vertically downwards with constant velocity Vet. The cut of inter-electrode space (IES) is shown in Fig. 1 a. Let U is potential difference between the equipotential anode and cathode. Then a band with width U (Fig. 1 b) is the image of IES on complex potential plane W = $ + is
electric field potential, ^ is the stream function).
The time variation is binding in the given problem statement with edge shift of ET C in the direction of the machining surface. It means that L = Vett is the deepening of ET downwards under the initial surface.
b
Fig. 1. The cut of inter-electrode space: a - the physical plane; b - the complex potential plane. HDE is the active dissolution area; EMG, FH are the areas of constant (critical) value of strength; AF, GB are the insoluble (rectilinear) boundaries
The dependence of current efficiency on strength is simulated by stepwise function (1). In this case the tree domains with tree types of boundary conditions are formed on machining surface [22].
Dissolution is absent on horizontal parts AF and GB, where | E | < E1. Vertical line segment GB corresponds to it on strength hodograph plane E = dW/dZ (Fig. 2 a), as far as the strength vector is directed along the normal to the equipotential surfaces from the cathode to the anode. By symmetry, the right half of plane domain EE is considered.
Part HDE is characterized by the stationarity condition. The strength is greater than Ei because of proximity to ET. As in the stationary problem on the plane of strength hodograph circle arc [14] with radius E0/2 and center at point iE0/2 (where E0 = Vet/ (kn0) is the strength at point D) corresponds to section (Fig. 2 a).
A circle arc with radius E1 < E0 and center at the origin of coordinates corresponds to transitional section |E| = E1 EMG (M is boundary inflection point). The existence of the area of a constant strength equal to the critical value is the main condition of quasi-stationary model construction.
Note, border part EG can have a cut on plane E as far as this boundary is not stationary in contrast to problem [14]. But border part DE cannot have a cut on plane E because there is no dissolution and the stationarity condition is violated if |E | < E1. So, the absence of cut on DE is the condition of the quasi-stationary solution existence.
Thus, the problem is reduced to finding a conformal mapping of the region of W-plane to the domain of plane E. Since the forms on both domains are known, then, in case of known correspondence to three points on the boundaries of these domains (in given problems B, C, D), according to Riemann's theorem on conformal mapping [24, 25], the solution of this problem exists and it is unique. The length of cut EM determines the depth of embedment L of edge ET C in the workpiece.
The problem is solved in a parametric form. Let us introduce parametric planes t\ = Eq/E and which domains are shown in Fig. 2 b, c. Conformal mapping t\ (£) is more convenient to find with the help of numerical integration of the derivative
= (C2 + 72)1/2(C2 + ^2)3/2^ (0 ' (3)
as far as the derivative has real values on boundary parts DE, DC, BG and imaginary values on part BC. In this case, first factor (3) takes into account the power singularities of the derivative. Second factor (3) ^ (Z) must have purely real values on horizontal and vertical radiuses Z• Thus, according to the principle of symmetry, the function can be analytically continued to the whole circle and can be represented as the Taylor series converging inside circle [24]
» (Z) = E c2mZ2m.
m=0
Upper half-plane t2 = -0, 5 (Z2 + Z-2) is used for obtaining of mapping W(Z) (Fig. 3).
Fig. 3. The domain shape on parametric plane t2 Function W (t2) is defined with help of Schwartz-Christoffel transformation [24]
W (t2) = -i- In V =-v, =• (4)
* y/tt-i (72 + 7"2) + y/i (/?2 + P~2) - è (72 + 7"2)
Function Z(Z) is obtained by integration
C
Z
1 1 /" iir can dti
¿7 ¿7 %Y
The calculation of machining surface shape is realized by numerical integration (5) by the method of parabolas. Function t\ (Z) is computed by numerical integration (3) with condition ii (0) = — i.
The problem is solved by collocation method, n+1 summands preserved in sum (3). Coefficients c2m, m = 0,... ,n, and also parameters fl, 7 are determined from condition |ti (e%a)| = a = E0/Ei given in discrete set of knots am = mn/ (2n), m = 1,... ,n. In addition, condition of closure Re ti (i) = 0 and the geometric conditions Im ti (7) = 0, Im Z (i) = L must be satisfied. Resulting system of n + 3 nonlinear equations is solved by the Newton method with step control by minimizing of the residuals squares sum.
Let us denote typical dimension as S = U/E0 = kn0U/Vet, parameter a = E0/Ei, a > 1
Z Vet , (a\
z = ~s-> r = (6)
(t is the time, t is dimensionless time). As far as L = Vett, then t = L/S.
The numerical results for a=2 and t=3; 4; 5; ... are given in Fig. 4 a in comparison with stationary solution (curve 1) obtained in [14]. The shapes are shown at coordinates
0 1 2 3 x 2,8 3,0 3,2 x
a b
Fig. 4. Computations results for a=2: a - the machining surface shapes; b - the part of the surface close to insoluble area
system connected with moved ET. It allows visualization of the limit shape establishment near ET border.
This solution is compared with the limit solution appearing nearby insoluble part of surface (GB) at fixed coordinates system in Fig. 4 b. Known problem [26] of flow from under a linear shield (curve 2) is this limit solution
Filtration method [27] of numerical data series obtained for Tj=3; 4; 5; 6; 7; 8; 9 is applied for estimation of the limit parameters values for t ^ ro. Front gap Sf and abscissa of point G (in which the ordinate of quasi-stationary surface is equal to zero) are used as characteristic parameters. A priori model of the process is used and tested
Sj = s^ + de-piTj + C2e-p2Tj + ... + cje-pjTj + A (Tj). (8)
Here Sj are the calculated gap values; s^ is the required magnitude of the steady gap; ck and pk are unknown coefficients; A (Tj) is an error that consists of not included terms of the expansion in the sum, the rounding error and other components due to the features of the algorithm and software. Filtration [27] consists in the repeated application of the formulas of Aitken [28] and Richardson [29]
« si-*-si-\ SW = s + (9)
sj_i - sj j 3 epi - 1 w
After applying of transformation (9) to all values sj, new sequence sj1^ is obtained, the model of which is similar to (8), but does not contain term c1 e-piTj. Then transformation (9) can be applied again to eliminate term c2e-p2Tj, and so on.
The filtration process is controlled by the results visualization. Relative errors are estimated by the way of comparison with a single standard which is elected as one of
z
values obtained by filtration. To verify the correctness of the standard choice, a sequence of differences sjk) — sj+)1 is constructed, which does not contain . For this sequence the standard is zero, and its filtration is performed according to the specified rules. In the case of the correct choice of the standard the lines on the graph, corresponding to the comparison with the standard are superimposed on the lines obtained by pairwise subtraction.
The decimal logarithms (with negative sign) of relative error estimates 5 are put on the ordinate axis in Fig. 5. It shows the number of valid digits of a result and allows estimating the error decrease rate. The values of dimensionless time t are put on the abscissa axis. The results of filtration of data differences that correspond to the sequential values t are shown in Fig. 5 a as the bold lines. It facilitates the standard choice with which all computed and filtered values are compared. Line 0 corresponds to the differences of computed values; lines 1 and 2 correspond to the filtered values. The results of standard choice for Sf=0,4982668 and comparison of computed and filtered values with it are presented in Fig. 5 a as thin lines put onto bold ones. The value of order 3-10-7 is obtained as the result of error estimation of the standard. An additional verification of the error estimate obtained is carried out by comparison with the value of stationary gap Sf=0,49826686, obtained in [14] with greater accuracy. The comparison shows, that the determined difference is less than 10-7, and it confirms the estimate.
The filtration of expression
Pi
A
1 +
sf (t) — S
is fulfilled for definition of coefficient p1 and value p1=0,7882±10 4 is obtained (Fig. 5 b)
a b
Fig. 5. The filtration results of numerical data sequence (lateral gap): a - the estimate of the limiting value ; b - coefficient p1 estimate
The similar estimates for abscissa of point G is obtained =3,273244± 10-5. As far as the accurate limiting value of parameter a (n + 2)/n is known from solution of the problem of flow from under a linear shield (7), we can find the accurate error value 5-10-6. It isn't contradicting the estimate obtained without accurate value application. As for Sf (t) coefficient q1 of exponential xG dependence as (8) c1 e-qiT is determined q1 =0,7923±10-4.
The particular case is a=1 (E0 = E1), when circle arc EMG takes up arc DE on plane E (Fig. 2 a). The analytical solution of this problem is obtained in [23]. The asymptotical
expression particularly follows from this solution
S/ - * In 2 + ~ - [2 + 7r - f2 + e-/2
n ne n L \2J
2. The Comparison with Non-Stationary Solution
The non-stationary problem is solved by the method described in [17, 18]. It is defined the conformal mapping of band x = —iW/ (2U) (parametric variable) onto physical plane z = Z/S and also partial derivative dz fdr (x, t) of dimensionless coordinates with respect to dimensionless time (6) for determination the boundary shift on each step of algorithm. The dimensionless velocity of ET is equal to 1 for the given way of dimensionlessness (6).
The partial derivative is determined from solution of the equation as Polubarinova -Galin [30], which follows from Faraday law (2)
f dz dz\
г ~v
dw
dz
dw . ,
Ьо (10)
where a = Rex, w = W/U, dw/da = 2i.
The application of the stepwise function (1) leads to dissolution switch on or off under passage of strength modulo value through the critical value E\. It is realized with multiplication of the right part of (10) by 1 or 0.
The function conformally mapping a band of parametric variable plane x on IES area of the physical plane at the fixed relative to the workpiece body coordinate system is defined as sum
z (x,T)= g (t)sinh nx + za (x,T) , (11)
where function zo (x) = g sinh nx for g>0 conformally maps the band of plane x on the upper half-plane with a cut; zA(x,t) is an analytical function which determines the distinctions of the machining surface formfromthe linear one (in this case Im z (to, t) = 0). The ordinate of point ET is defined from the equation
Z T) = 9 ^ + ZA (l' T) = ~ T
(s0 is the dimensionless initial gap value). Then the derivative
dg d (i Tr = ~1~YrZA U'r
We introduce the grid for parameters a and t:
am = mAa, m = 0, 1, ..., n; Tj = jAT, j = 0, 1, ..., M.
So, values (am,Tj) are determined on every time step for known z^{am,Tj) and (<7m,Tj) with the help of solving of linear equations system, which are obtained by substitution aj in (10). And zA is varied by the Euler method
(o"m, Tj+1) = ZA (am, Tj) + (am, Tj) AT
Then -j^ (crm,Tj) are calculated once more and so on.
Thus, it is necessary to find two analytical functions of a complex variable on every time step
d z \
fl Tj) = zA (z, Tj), /2 (z, Tj) = — (z, Tj).
Functions f1 (z, Tj) and f2 (z, Tj) are found as follows. We find the solution at mesh points am(m=0,... ,n) on boundary x = a• Values Im fk (am, Tj) = /m (k=1, 2) are given. We assume Im fk (an,Tj) = 0, as far as Im fk (a,Tj) decreases fast (as an exponent) for a ^ ±œ. Values Im fk (a, Tj) at points between mesh points are found with help of cubic splines Sk (a,Tj) having two continuous derivatives.
As far as fk (x, Tj) are analytical functions with pure imaginary values on straight line Imx=1/2, we continue these functions up to a band of unit width. Then Im fk (a + i, Tj) = Im fk (a + i0, Tj) by virtue of symmetry principle [24]. The Schwarz formula is applied for reconstruction of functions fk (x,Tj) (k=1, 2), taking into account that Im fk (a,Tj) due to symmetry are the even functions with respect to a [17, 18]
r f N . , f Sk (a,Tj ) da f Sk (a,Tj ) da
/fe(x^) = sinh7TX / -r--—r-+ smh tï\ / -r-^—r-. (12)
J cosh na — cosh nx J cosh na + cosh nx
0 0
Derivative values Im are equal to the coefficients of spline Si(a1Tj)
constructed according to mesh values = Im za (crm,Tj), and derivative Re ^^ (cfm^j) is defined by computation of the coefficients of the spline constructed according to mesh values xm = Re f1 (am,Tj) obtained from (12).
The shapes of non-stationary surface in fixed coordinates system is shown in Fig. 6 a. The step of dimensionless time At is equal to 1. The dimensionless rate of ET is equal to 1. The images of these curves on the strength hodograph plane (At = 0, 5) are presented in Fig. 6 b. One can see the formation of quasi-stationary mode (see Fig. 2 a).
a b
Fig. 6. Computations results: a - machining surface shapes (At = 1); b - machining surface images on the strength hodograph plane (At = 0, 5)
3. On Error Estimation
It is detected that shape error in non-stationary problem solving is the highest near the dissolution absent area. The surface shapes neighborhood point G in comparison with the solution of problem of fluid flow from under a shield (curve 1) are shown in Fig. 7. One can see that absolute error exceed 0,1 for nodes number n=144 and it decreases to 0,02 approximately (the relative error is smaller by order) for n=729.
-0,125
-0.125
2,8
3.2 x
3,0 b
Fig. 7. The surface part near the insoluble area: a - n = 144; b - n = 729
The solution of differential equation (10) with the discontinuous right part leads to questions appearance about method accuracy order. The numerical filtration of results obtained for different At and n, described in [27], is applied for error estimation.
The error investigation with the filtration application of the numerical data, obtained for L=5, n=144, 216, 324, 486, 729 and At = (1, 0, 5, 0, 25, 0,125, 0,0625, 0,03125) • 10-3 is realized. The ordinate of non-stationary surface at point G (at which the ordinate of quasi-stationary surface is equal to zero) is used as a magnitude characterizing the error. The filtration results of numerical data obtained for different At for n=729 are shown in Fig. 7 a. The decimal logarithms of relative error with negative sign are put on the ordinate axis. The decimal logarithms of relation of the dimensionless time step At to At0 = 0, 01 are put on the abscissa axis. The filtration results of subtraction of the data corresponding to the sequential values At are shown by bold lines. The choice results of the standard that equal to 1,6-10-3 and comparison results of the computed and filtered data with the standard are presented in Fig. 8 a by thin lines.
The numerical data filtration results obtained for different n and previously filtered with respect to At are demonstrated in Fig. 8 b. The bold lines show the filtration results of data subtractions corresponding to the sequential values n for the standard choice. The curve 0 corresponds to the subtraction of the computed values, curves 1 and 2 correspond to the values filtered one and two times. The thin lines show the choice results of the standard equal to A0 = -0, 5 • 10-3 and comparison results of the computed and filtered data with the standard. Note, that the line 1 is used for error estimation. The line 2 is used for this estimate approval only. One can see that the certain accuracy estimate shows hardly more than 3 digits. It makes to consider this result A0 doubtful. Thus, it has not succeeded to find a difference between non-stationary and quasi-stationary solutions exceeding the computational error.
a b
Fig. 8. Filtration results: a - for At; b - for n
Conclusion
The problem of quasi-stationary ECM is formulated and solved in this paper. The application of the stepwise function of current efficiency on current density has leaded to the appearance on the machining surface the segments with the three types of boundary conditions (stationary, transient and insoluble). It allows explanation of real process nature by comparison of the machining surface shape on the transient segments and the shape of free surface in hydrodynamic problem about flow from under a linear shield (particularly, by infinite curvature of the surface at the separation point).
The numerical investigations results allow obtaining the characteristics of non-stationary processes of formation. It is found that variation of characteristic sizes (for example: front gap) of non-stationary solution corresponds to the exponential low. There are found concrete dependences of these parameters on time. They are important for technologies development of precision electrochemical machining.
The obtained results allow to compare non-stationary problems solutions with stationary and quasi-stationary solutions. It simplifies error estimating and increases numerical results reliability. And also the diapasons of simplified methods applicability of investigation of non-stationary processes (quasi-stationary and limiting-quasi-stationary) are found.
The condition of quasi-stationary process formation is monotonously-increasing of the strength at the machining surface points. For example, it is correct for machining by a vertical plane and angular ET and it is not correct for machining by point ET and horizontal plane ET (for a long time). In these two cases and other similar ones the final mode forms on the machining surface part bordering on insoluble zone. The dissolution is absent and the form remains invariable for this region.
The numerical investigations show that in the case of quasi-stationary process the limiting area of machining surface forms rather quickly. This area is restricted by the condition of the strength modul equality to the critical value. The stationary area forms more slowly therefore a quasi-stationary solution is possible for t > t0, where t0 = t0(a).
The comparison of the obtained non-stationary and quasi-stationary shapes (in the existence range of consistent solutions of quasi-stationary problems) shows the absence of differences exceeding error of non-stationary problem solution (about 10-3).
Acknowledgements. This work was financially supported by the Russian Foundation for
Basic Research (Project no. 17-07-00356).
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Received April 15, 2018
УДК 621.9.047 DOI: 10.14529/mmp190101
СРАВНЕНИЕ КВАЗИСТАЦИОНАРНОГО И НЕСТАЦИОНАРНОГО РЕШЕНИЙ ЗАДАЧ ЭЛЕКТРОХИМИЧЕСКОГО
ФОРМООБРАЗОВАНИЯ ПРИМЕНИТЕЛЬНО К ПРЕЦИЗИОННОЙ РЕЗКЕ ПЛАСТИНЧАТЫМ ЭЛЕКТРОДОМ-ИНСТРУМЕНТОМ
В.П. Житников1, Н.М. Шерыхалина1, С.С. Поречный1
1 Уфимский государственный авиационный технический университет, г. Уфа,
Российская Федерация
Сформулирована квазистационарная задача для моделирования процесса электрохимической резки пластинчатым электродом-инструментом. Постановка задачи основана на использовании ступенчатой функции выхода по току от плотности тока. При этом на обрабатываемой поверхности образуются три зоны с различными условиями. В области высоких плотностей тока используется обычное условие стационарности. В области малых плотностей тока растворение отсутствует и сохраняется начальная форма границ. В промежуточной зоне плотность тока в каждой точке равна критическому значению. Наличие краевых условий на каждом участке обрабатываемой поверхности позволяет сформулировать краевую задачу для аналитической функции комплексного переменного и найти форму границы в любой момент времени, независимо от предыстории. Проведено сравнение решений квазистационарной и нестационарной задач, и найден диапазон существования квазистационарного решения.
Ключевые слова: электрохимическое формообразование; ступенчатая функция; квазистационарная модель; оценка погрешности.
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Владимир Павлович Житников, доктор физико-математических наук, профессор, кафедра «Вычислительная математика и кибернетика», Уфимский государственный авиационный технический университет (г. Уфа, Российская Федерация), [email protected].
Наталия Михайловна Шерыхалина, доктор технических наук, доцент, кафедра «Вычислительная математика и кибернетика», Уфимский государственный авиационный технический университет (г. Уфа, Российская Федерация), [email protected].
Сергей Сергеевич Поречный, кандидат физико-математических наук, доцент, кафедра «Высокопроизводительные вычислительные технологии и системы:», Уфимский государственный авиационный технический университет (г. Уфа, Российская Федерация), [email protected].
Поступила в редакцию 15 апреля 2018 г.
