Int J Adv Manuf Technol (2010) 49:949–959 DOI 10.1007/s00170-009-2435-7

ORIGINAL ARTICLE

Process parameters optimization of injection molding using a fast strip analysis as a surrogate model Peng Zhao & Huamin Zhou & Yang Li & Dequn Li

Received: 6 May 2009 / Accepted: 9 November 2009 / Published online: 28 November 2009 # Springer-Verlag London Limited 2009

Abstract Injection molding process parameters such as injection temperature, mold temperature, and injection time have direct influence on the quality and cost of products. However, the optimization of these parameters is a complex and difficult task. In this paper, a novel surrogate-based evolutionary algorithm for process parameters optimization is proposed. Considering that most injection molded parts have a sheet like geometry, a fast strip analysis model is adopted as a surrogate model to approximate the time-consuming computer simulation software for predicating the filling characteristics of injection molding, in which the original part is represented by a rectangular strip, and a finite difference method is adopted to solve one dimensional flow in the strip. Having established the surrogate model, a particle swarm optimization algorithm is employed to find out the optimum process parameters over a space of all feasible process parameters. Case studies show that the proposed optimization algorithm can optimize the process parameters effectively. Keywords Injection molding . Parameters optimization . Surrogate model . Evolutionary algorithm . Fast strip analysis . Particle swarm optimization

P. Zhao Institute of Advanced Manufacturing Engineering, Zhejiang University, Hangzhou 310027 Zhejiang, People’s Republic of China H. Zhou (*) : Y. Li : D. Li State Key Laboratory of Material Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan 430074 Hubei, People’s Republic of China e-mail: [email protected]

1 Introduction Injection molding is the most widely used process for producing plastic products. During this process, many parameters such as injection temperature, mold temperature, and injection time are very important, which have direct influence on the quality and cost of the products. However, the optimization of process parameters is a complex and difficult task [1–3]. An increase in injection temperature causes a decrease in melt viscosity, which results in reduced cavity pressure and shear stress. On the other hand, high injection temperature also increases cooling time which lowers productivity. Increasing mold temperature reduces heat losses, and the maximum temperature difference may be reduced. However, a high mold temperature increases cooling time. Too short injection time increases the cavity pressure and shear stress although it can reduce temperature difference. On the contrary, a long injection time will lead to a decrease in the flow front temperature, as well as an increase in melt viscosity and cavity pressure. It is clear that an optimization algorithm must trade off these conflicting process parameters to obtain optimum parameters and produce a high quality part at minimum cost. In the past, the optimization of the process parameters was considered to be a “black art,” which relied heavily on the experience and knowledge of experts and required a trial-and-error process [4]. This trial-and-error method is time-consuming, and there is no assurance that the optimum process parameters can be obtained. With advances in numerical modeling and computer simulation techniques, there have been tremendous efforts made to develop computer simulation software to facilitate injection molding design and process setups. Lam et al. [5] proposed a computer-aided system to

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optimize injection molding process parameters using a two-step exhaustive search strategy. However, this approach requires generating a finite element model and running a number of simulations in order to obtain the acceptable process parameters. As it could take an hour or more on a personal computer for a single run of a moderately complex part simulation with more than 104 mesh nodes, it may not be practical to perform the large number of simulations in shop-floor production environment [6]. In order to reduce the computational cost, some surrogate models were employed, such as response surface methodology, artificial neural network, support vector regression, and Gaussian process. These surrogate models were used to construct a mathematical approximation for replacing the time-consuming computer simulation software. Kurtaran and Erzurumlu [7] created a predictive model for warpage using response surface methodology. Shen et al. [8] combined artificial neural network and genetic algorithm to optimize process parameters for reduction of maximum volume shrinkage difference. Chen et al. [9] developed a self-organizing map plus a backpropagation neural network model for predicting product quality. Zhou et al. [10] used support vector regression to optimize process parameters. Gao and Wang [11] proposed an adaptive optimization method based on kriging surrogate model. Zhou and Turng [12] presented an integrated simulation-based optimization system using Gaussian process approach. It is a pity that the above mentioned surrogate models are needed to be trained first by plenty of computer simulation results. With the help of these models, the optimization task still cost several hours to be accomplished. In this paper, a novel surrogate-based evolutionary algorithm is proposed for process parameters optimization in injection molding. Considering that most injection molded parts have a sheet-like geometry, a fast strip analysis (FSA) model is adopted as a surrogate model to predicate the filling characteristics of injection molding with minimum computational cost, in which the original part is represented by a rectangular strip, and a finite difference method is adopted to solve one dimensional flow in the strip. Having established the surrogate model, a particle swarm optimization (PSO) algorithm is employed to evaluate the surrogate model to find out the optimum process parameters. The outline of this paper is as follows. Section 2 presents the optimization problem, including the selection of optimization objectives and the construction of objective function. In section 3, the surrogate-based evolutionary algorithm is described in detail. Section 4 proposes two case studies to validate the presented algorithm. Finally, the last section gives the conclusions.

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2 Optimization problem definition The optimization of process parameters for injection molding is a constraint optimization problem as following: Find X ¼ ½x1 x2 x3 

ð1Þ

Minimize FðX Þ ¼ ½f1 ðX Þ; f2 ðX Þ;    ; fn ðX Þ ðlÞ

ðuÞ

Subject to : xi  xi  xi ði ¼ 1; 2; 3Þ where X is a set of process parameters including injection temperature T0, mold temperature Tw, and injection time tinj. ðlÞ ðuÞ The variables of xi and xi are the lower and upper limits of xi, respectively. F(X) denotes the objective function, and fn(X) represents different optimization objectives. 2.1 Optimization objectives There are many optimization objectives that might be included in the objective function in order to provide a comprehensive quality evaluation [1]. Besides the quality objectives, the cost objectives are also taken into account, which aim at increasing productivity, as well as lowering capital investment in the molding machine and the mold. In this paper, three typical optimization objectives are selected and described as follows. Cavity pressure at the end of filling, Pcavity, is both quality and cost objective. A lower cavity pressure corresponds to a lower shear stress, which reduces the chance of molding defects such as flash, warpage, and mechanical sticking. Meanwhile, minimizing the cavity pressure can lower the capital investment of the molding machine and the mold. Temperature difference at the end of filling, Td, is a quality objective for the uniformity of melt temperature. A nonuniform temperature distribution will cause shrinkage or warpage during the cooling stage. The temperature difference can be measured by the difference between the maximum melt temperature Tmmax and the minimum melt temperature Tmmin at the end of filling. It can be expressed as: Td ¼ Tmmax  Tmmin

ð2Þ

Cooling time, tc, is cost objective, which refers to the time needed for the melt to cool from a high melt temperature to the ejection temperature. The cooling time should be minimized, so as to reduce cycle time and increase productivity. 2.2 Objective function According to the above discussion, the optimization of process parameters is a multiobjective optimization problem, and it aims at lower cavity pressure, more uniform

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Fig. 1 Schematic diagram of a surrogate-based evolutionary algorithm

temperature distribution, and shorter cooling time. In this paper, multiobjective optimization problem is transformed into a single objective problem by means of weighting method, and each objective value is normalized to a range between 0 and 1 by its upper limit before applying a weight to it. Then the weighted objective function F(X) can be expressed as: ðuÞ

ðuÞ

FðX Þ ¼ w1  Pcavity =Pcavity þ w2  Td =Td þ w3  tc =tcðuÞ

ð3Þ where w1, w2, and w3 are three weights. All the weights range from 0 to 1, and their sum is 1.

3 Surrogate-based evolutionary algorithm A surrogate-based evolutionary algorithm for process parameters optimization of injection molding is illustrated as Fig. 1. The keys for the algorithm are surrogate model and evolutionary algorithm. Surrogate model is established to approximate the time-consuming computer simulation software for predicating the filling characteristics of injection molding including cavity pressure, temperature difference, and cooling time, with minimum computational cost. Evolutionary algorithm is used to search for optimum process parameters over a space of all feasible process parameters according to a fitness determined by their performance.

3.1.1 Mathematical model FSA model is based on a geometric approximation of the original part by a rectangular edge-gated strip. For a given geometry and gate locations, the flow length L and average thickness 2b can be calculated, and the approximation is done so that the length, thickness, and volume of the strip equal the flow length, average thickness, and volume of the part, respectively. The approximated rectangular edge-gated strip is illustrated in Fig. 2, and the basic assumptions for the FSA model are: (a) the polymer is incompressible and purely viscous; (b) inertia is neglected as compared to the viscous force; (c) the velocity components other than that in the main flow direction (x) are negligible; (d) longitudinal thermal conduction (in the x and y directions) and transverse convection (in the y and z directions) are neglected; and (e) the melt density, thermal conductivity, and specific heat are assumed constant. With the above approximations, the governing equations can be written as
 @ @u @P h ¼0 ð4Þ  @z @z @x
rCp

@T @T þu @t @x Z

Q¼W 3.1 Surrogate model Considering that most injection molded parts have a sheetlike geometry, with the thickness much smaller than the other dimensions of the parts [13], a FSA model is adopted as the surrogate model in this paper.

Fig. 2 Schematic diagram of simplified geometry for the cavity

b

udz b




2 @2T @u ¼K 2 þh @z @z

ð5Þ

ð6Þ

where u is the velocity component in the x direction; P, T, t, and Q are pressure, temperature, time, and volume flow rate, respectively; η, ρ, C p , and K represent shear viscosity, density, specific heat, and thermal conductivity, respectively.

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For simplification, it is assumed that the velocity and temperature are symmetric about the centerline (z=0), and the boundary conditions can be described as: T ¼ Tw ; u ¼ 0

at z ¼ b

ð7Þ

@T @u ¼ ¼0 @z @z

at z ¼ 0

ð8Þ

T ¼ T0

at x ¼ 0

ð9Þ

According to governing equations and boundary conditions, the fluidity S, pressure gradient Λ, shear rate g_, and velocity u can be expressed as: Z b 2 z S¼ dz ð10Þ 0 h Λ¼

@P Q ¼ @x 2WS



g ¼ Λz=h Z

b

u¼



g dz

ð11Þ

method to compute these equations numerically. The finite difference grid in the half-gap thickness is sketched in Fig. 3. The discrete nodes in the grid are identified as the intersections of grid lines. If the gap from z=0 to z=b is divided into jmax –1 intervals withj=1 at the centerline and j=jmax at the wall, and Ti;j;k ¼ T xi ; zj ; tk denotes the temperature of node (xi, zj) at time step tk, the discrete form of Eqs. 10–13 are listed as follows: Si;kþ1 ¼

jmax X zi;j 2 $z h j¼1 i; j; k

Λi;kþ1 ¼



Q 2WSi; kþ1

ð17Þ

Λi;kþ1 zi;j hi;j;k

ð18Þ

g i; j; kþ1 ¼ ð12Þ ð13Þ

z

The Cross-WLF viscosity model is used for the nonNewtonian polymer, as: h0 ðT ; P Þ ð14Þ h¼   1n 1 þ h0 g =t * where η0 is zero shear rate viscosity, n and τ* are viscosity model constants.   A1 ðT  D2  D3 PÞ h0 ¼ D1 exp ð15Þ A2 þ T  D2

ui; j;kþ1 ¼

ð16Þ

jmax 1  X



g i; jþ1;kþ1 þ g i; j;kþ1

2$z

ð19Þ

j

In the x direction, the upwind differencing method is used, and in the z direction, the central finite difference is adopted. Thus, Eq. 5 can be rewritten as: Ti; j;kþ1  Ti; j;k 1 ð20Þ ¼ rCp $t
 2 Ti; jþ1;kþ1  2Ti; j; kþ1 þ Ti; j1; kþ1  K þ h g i; j; kþ1 i; j; kþ1 $z2

 ui; j; kþ1 þ ui1:j; kþ1 Ti; j; kþ1  Ti1; j; kþ1  2 $x

where D1, D2, D3, A1, and A2 are viscosity model constants. 3.1.2 Numerical implementation Equations 10–15 form a set of relationships between velocity, temperature, pressure, and viscosity, which cannot be solved directly. This paper presents a finite difference

Fig. 3 Schematic diagram of finite difference grid in the computational domain

The above discrete form equations can be solved by numerical techniques. Figure 4 shows the implementation procedure of FSA model. Based on the distribution of pressure and temperature at the end of filling, the cavity pressure and temperature difference can be calculated easily, and the cooling time can be calculated from the

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Fig. 4 Flowchart of implementation procedure for FSA model

empirical equation given by Ballman and Shusman[14], shown as Eq. 21.
 B2 p T 0  Tw  tc ¼ ln 4 Teject  Tw 2pa

ð21Þ

where B is the maximum thickness of the original part, α denotes the thermal diffusivity, and Teject represents the ejection temperature. 3.2 Evolutionary algorithm Evolutionary algorithms are very powerful techniques for finding solutions to many real-world search and optimization problems. In this paper, PSO algorithm is employed to solve the optimization problem discussed in Section 2 because of its potential advantage as evolutionary algorithm. PSO algorithm was first proposed by Kennedy and Eberhart [15], inspired by social behavior of bird flocking, fish schooling, and swarm theory. Although being relatively new, PSO algorithm has already been successfully applied in quite a number of research and application areas [16]. In PSO algorithm, multiple feasible solutions, called particles, fly around in a multidimensional search space. A particle status in the search space is characterized by two factors: velocity and position. The velocity and position

of the i particle in the D-dimensional search space  at generation τ can be   represented as Vit ¼ vti;1 ;    ; vti;D and Sit ¼ sti;1 ;    ; sti;D , respectively. Pit ¼ pti;1 ;    ; pti;D is the best position of the i particle searched so far at generation τ. The  global best particle is denoted as Pgt ¼ g1t ;    ; gDt , which represents the best position found so far at generation τ in the whole swarm. The i particle updates its position according to Eqs. 22 and 23.  Vitþ1 ¼ w  Vit þ c1  r1  Pit  Sit þ c2  r2    Pgt  Sit

ð22Þ

Sitþ1 ¼ Sit þ Vitþ1

ð23Þ

where w is inertia weight, c1 and c2 are constants called acceleration coefficients, r1 and r2 represent two independent random numbers with the range [0,1]. The implementation procedure of PSO algorithm can be illustrated with the flowchart in Fig. 5. First, an initial population is generated with random positions and velocities. Next, the performance for each particle is evaluated according to a fitness function, and all positions of particles are adjusted by Eqs. 22 and 23. This process is repeated until stopping criterion is satisfied. The stopping criterion is either that a given number of generations have been reached

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evolutionary algorithm can be used to find the upper limit of the single objective while assuming the other objectives are absent. The value of the single objective can be used as fitness function, and the fitness value at final generation is the required upper limit. Step 4 Optimize process parameters using surrogate-based evolutionary algorithm. In this optimization problem, the objective function expressed in Eq. 2 is a minimum problem, and its value varies from 0 to 1. Hence, the fitness function fit(X) can be defined as:

fitðX Þ ¼ 1  FðX Þ

ð24Þ

Based on the above optimization procedure, a program for the optimization of process parameters is developed using Visual C++.

4 Case studies Two case studies have been carried out as part of this paper to validate the presented optimization algorithm. 4.1 Case study 1 Fig. 5 Flowchart of implementation procedure for PSO algorithm

already, or that the population has become uniform. Finally, the global best particle is obtained. PSO algorithm requires little knowledge about the problem being solved, and it is easy to implement and robust due to the lack of cross and mutation. 3.3 Implementation procedure According to the current optimization problem and the proposed surrogate-based evolutionary algorithm, the procedure for process parameters optimization in injection molding is summarized as the following steps. Step 1 Set the lower and upper limits of process parameters including injection temperature, mold temperature, and injection time Step 2 Establish the FSA model based on the geometric approximation of original part. The flow length of the original part can be calculated by the “pseudoflow” algorithm [3], and the average thickness of the original part can be estimated from the average distance of all matched nodes. Step 3 Calculate the upper limit of each optimization objective in objective function. Surrogate-based

The first case study has been conducted to compare the cavity pressure predicted by surrogate model with experimental data. As shown in Fig. 6, the experimental cavity is a box part with dimensions 160×100×30 mm and average thickness of 2.6 mm. The position indicated by the “black circle” is a pressure measurement point. A GYY-7 II strain induction pressure sensor is selected as pressure sensor, and a K-803B pretreatment instrument is used to pretreat the signal. Then a K-810 signal collector is chosen to transform the signal to digital data and save the data in a computer.

Fig. 6 Box part molded with injection molding

Int J Adv Manuf Technol (2010) 49:949–959 Table 1 Predicted cavity pressure and experimental data

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Process parameters T0 (°C) 210 220 230 240 220 220 220 220 220 220

Cavity pressure (MPa)

Relative Error (%)

Tw (°C)

tinj (s)

Experimental

Predicted

40 40 40 40 20 30 50 40 40 40

1.7 1.7 1.7 1.7 1.7 1.7 1.7 0.9 1.2 2.8

10.1 9.47 8.82 8.28 10.04 9.8 9.16 8.32 8.78 11.03

10.6 9.69 8.81 7.95 10.12 9.91 9.46 9.02 9.25 10.65

The computer acts as a data processor. The selected polymer is GPPS. The polymer parameters of ρ, Cp, and K are 954.12 kg/m3, 1,700 J/(kg°C), and 0.14 W/(m°C), respectively, and the viscosity model constants of n, τ*, D1, D2, D3, A1, and A2 are 0.1, 61,400 Pa, 2.32×109 Pa s, 373.15 K, 0 K/Pa, 21.363, and 51.6 K, respectively. The comparison of the cavity pressure between experimental data and prediction results of surrogate model under different process parameters are listed in Table 1, which shows that the predicted results agree well with the experimental data, with the relative error being less than ±8.41%. 4.2 Case study 2 In this case study, the process parameters of a cellular phone cover have been optimized by the proposed optimization algorithm. The mesh of the cellular phone cover is shown in Fig. 7, with one gate marked by a “triangle.” Its cavity volume, flow length, average thickness, and maximum thickness are 10.25 cm3, 93.40 mm,

−4.95% −2.32% 0.11% 3.99% −0.80% −1.12% −3.28% −8.41% −5.35% 3.45%

1.50 mm, and 2.54 mm, respectively. The selected polymer is Cycoloy C1200 HF. The polymer parameters of ρ, Cp, K, and Teject are 1,017 kg/m3, 2,133 J/(kg°C), 0.24 W/(m°C), and 115°C, respectively, and the constants corresponding to the n, τ*, D1, D2, D3, A1, and A2 of the viscosity model are 0.3999, 39,700 Pa, 1.43×109 Pa s, 417.15 K, 0 K/Pa, 20.127, 51.6 K, respectively. 4.2.1 Verification of surrogate model In order to verify the accuracy of surrogate model, the cavity pressure, temperature difference, and cooling time have been predicted by surrogate model and commercial computer simulation software, Moldflow. When one parameter like injection temperature, mold temperature, or injection time varies within the range as listed in Table 2, other process parameters use the recommended values. The comparison of surrogate model results and Moldflow results with different injection temperatures, mold temperatures, and injection time are shown in Figs. 8, 9, and 10, respectively. As shown in the Figs. 8, 9, and 10, the surrogate model has a good performance, and it can map the relationship between the optimization objectives (cavity pressure, temperature difference, and cooling time) and the process parameters (injection temperature, mold temperature, and injection time). Another comparison between surrogate model and Moldflow software is the execution time. Both of them are running on Intel P4 2.4GHz processor with 1 GB RAM Table 2 Process parameters for the cellular phone cover Process parameters Lower limit Upper limit Recommended value

Fig. 7 Geometry model of a cellular phone cover

T0 (°C) Tw (°C) tinj (s)

255 60 0.1

295 80 2

275 70 1

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Fig. 8 Result comparison between surrogate model and Moldflow with different injection temperatures: a cavity pressure, b temperature difference, c cooling time

Int J Adv Manuf Technol (2010) 49:949–959

Fig. 9 Result comparison between surrogate model and Moldflow with different mold temperatures: a cavity pressure, b temperature difference, c cooling time

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Fig. 11 The evolution process of parameters optimization

and Windows XP OS. Surrogate model takes less than one thousandth of the time of Moldflow software, only 0.02%. 4.2.2 Process parameters optimization The limits of each process parameter are listed in Table 2. The upper limit of each optimization objective can be generated by surrogate-based evolutionary algorithm, and the upper limit of Pinj, Td, and tc are 87.34 MPa, 24.38°C, and 14.62 s, respectively. In addition, different weights are used with a lager weight given to the temperature difference. Hence, the optimization problem for the cellular phone cover can be defined as follows: Find

 X ¼ T0 Tw tinj

ð25Þ

Minimize FðX Þ ¼ 0:3  Pcavity =87:34 þ 0:4  Td =24:38v þ 0:3  tc =14:62 Subject to: 255 C  T0  295 C 60 C  Tw  80 C 0:1s  tinj  2:0s

Fig. 10 Result comparison between surrogate model and Moldflow with different injection time: a cavity pressure, b temperature difference, c cooling time

The above optimization problem has been solved by realcoded PSO algorithm. The number of particles is 20, and the maximum number of generations is 50. The constants used in Eq. 22 are c1 =1 and c2 =1, and the inertia constant w is 0.8. The evolution process of parameters optimization using the PSO algorithm is shown in Fig. 11. From the figure, it can be seen that the fitness value improve over the generations. The optimum process parameters are as follows: injection temperature is 295°C, mold temperature is 60°C, and injection time is 0.189 s. The optimum injection temperature is high, which is at the upper limit of the injection temperature range, while the optimum mold temperature is at the lower limit of the mold temperature range. The optimum injection time is relatively short, which is close to the lower limit of the corresponding injection time range. Figure 12 shows the effect of each

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the difference in units and quantities, the objective values are divided by their upper limits, which are given in Eq. 25. From Fig. 12, it can be noted that the optimum process parameters agree well with the defined optimization problem. A lower cavity pressure requires a high melt temperature, and a low mold temperature results in a lower cooling time, while the temperature difference is reduced by decreasing injection time since short injection time reduces heat loss during the filling. With the help of the presented optimization algorithm, the overall optimization task can be accomplished in minutes (about 7.5 min running on Intel P4 2.4GHz processor with 1 GB RAM and Windows XP OS). These results show that the proposed optimization algorithm can effectively reduce the objective function value and obtain optimum process parameters with lower computational cost.

5 Conclusions The complexity of the optimization for process parameters is not just due to the interrelationships of these parameters. It is also due to the conflicting optimization objectives. In this paper, a novel surrogate-based evolutionary algorithm is proposed, which combines FSA model and PSO algorithm. The following conclusions can be made:

Fig. 12 The optimization objective values generated by Moldflow with different a injection temperature, b mold temperature, c injection time

process parameter on the optimization objective values under the process where all other parameters are kept at their optimum values. In this figure, the objective values are generated by Moldflow software, and in order to normalize

(1) Besides the quality objectives, the cost objectives are also need to be involved in objective function in order to provide a comprehensive quality evaluation. (2) The presented FSA model does not need to be trained by plenty of computer simulation results, and it can be applied for any polymer and any complex mold geometry. The predicted results of the FSA model agree well with both the experimental data and the computer simulation software simulation results. Moreover, the required time of the FSA model is several magnitudes less than that of computer simulation software. (3) PSO algorithm requires little knowledge about the problem being solved, and it is easy to implement and robust. It can be used as a search tool to find optimum process parameters according to the value of objective function that must be minimized. (4) The proposed surrogate-based evolutionary algorithm can optimize the process parameters in short time, and it does not rely on any experience or knowledge of molding process. With the help of this optimization algorithm, engineers can quickly determine suitable process parameters to produce perfect products at minimum cost.

Int J Adv Manuf Technol (2010) 49:949–959 Acknowledgements The authors would like to acknowledge financial support from the National Natural Science Foundation of China (grant no. 50875095) and the National 863 Program of China (grant no. 2009AA03Z104).

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