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A hybrid of back propagation neural network and genetic algorithm
for optimization of injection molding process parameters
Fei Yin a, Huajie Mao a,⇑, Lin Hua b
a School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, Chinab School of Automobile Engineering, Wuhan University of Technology and Hubei Key Laboratory of Advanced Technology of Automotive Parts. Wuhan 430070, China
a r t i c l e i n f o
Article history:
Received 9 December 2010
Accepted 29 January 2011
Available online 24 February 2011
Keywords:
A. Polymers
C. Moulding
F. Defects
a b s t r a c t
This paper presents a hybrid optimization method for optimizing the process parameters during plastic
injection molding (PIM). This proposed method combines a back propagation (BP) neural network
method with an intelligence global optimization algorithm, i.e. genetic algorithm (GA). A multi-objective
optimization model is established to optimize the process parameters during PIM on the basis of the
finite element simulation software Moldflow, Orthogonal experiment method, BP neural network as well
as Genetic algorithm. Optimization goals and design variables (process parameters during PIM) are spec-
ified by the requirement of manufacture. A BP artificial neural network model is developed to obtain the
mathematical relationship between the optimization goals and process parameters. Genetic algorithm is
applied to optimize the process parameters that would result in optimal solution of the optimization
goals. A case study of a plastic article is presented. Warpage as well as clamp force during PIM are inves-
tigated as the optimization objectives. Mold temperature, melt temperature, packing pressure, packing
time and cooling time are considered to be the design variables. The case study demonstrates that the
proposed optimization method can adjust the process parameters accurately and effectively to satisfy
the demand of real manufacture.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Process parameters during Plastic Injection Modeling (PIM)
were mostly relied on the technicians’ personal experience in the
past. Although a better combination of process parameters can
be found with the help of the computer numerical simulation
technology nowadays, it is still hard to find the optimum combina-
tion of the processing parameters accurately and quickly. As a mul-
ti-objective and nonlinear optimization problem, process
optimization of PIM has attracted more and more attentions
worldwide. Many researches have been carried out to optimize
the process parameters during PIM.
In 2008, Gao et al. proposed an effective optimization method to
minimize the warpage in injection molding by using the Kriging
model. The warpage of a cellular phone cover was investigated,
and the warpage of the cellular phone cover was effectively de-
creased by the proposed optimization method [1]. Subsequently
they proposed an adaptive optimization method based on
Kriging-surrogate model to minimize the warpage of injection
molded parts in 2009 [2]. Deng et al. applied Taguchi’s parameter
design method, regression analysis, and the Davidon–Fletcher–
Powell method to propose an approach for determining the
optimal process parameter settings of plastic injection molding un-
der single quality characteristic considerations [3]. Zhang et al.
applied a mode-pursuing sampling (MPS) method for warpage
optimization by integrating injection molding simulation with
MPS, and by proposing a reinforced convergence criterion for the
optimization process, in an attempt to search for the optimal
process parameters of injection molding for minimizing warpage
defect [4]. Deng et al. presented an optimization method for min-
imizing the warpage of injection molded plastic parts based on
mode-pursuing sampling method and genetic algorithm (GA).
Warpage of a food tray plastic part was minimized by using the
proposed method [5]. Altan minimized the shrinkage of rectangu-
lar-shaped specimens by Taguchi, experimental design and the
analysis of variance (ANOVA) method. Neural network was also
used to predict the shrinkage of the part [6]. Hasan Kurtaran
et al. proposed an efficient minimization method of warpage on
thin shell plastic parts by integrating finite element (FE) analysis,
statistical design of experiment method, response surface method-
ology (RSM), and genetic algorithm [7]. Shen et al. minimized the
shrinkage of a plastic part by using the artificial neural network
and genetic algorithm [8]. Kurtaran et al. considered mold temper-
ature, melt temperature, packing pressure, packing time and cool-
ing time as the key process parameters during PIM and got the
optimum values of process parameters in injection molding of a
0261-3069/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.matdes.2011.01.058
⇑ Corresponding author. Tel.: +86 13807171614; fax: +86 027 87168391.
E-mail address: maohj@whut.edu.cn (H. Mao).
Materials and Design 32 (2011) 3457–3464
Contents lists available at ScienceDirect
Materials and Design
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s
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bus ceiling lamp base to achieve minimum warpage by using neu-
ral network model and genetic algorithm [9].
All the researches cited beforehand accomplished their pur-
poses of optimizing process parameters during PIM and improved
the quality of plastic parts greatly. However, a clear mathematical
model and a systematical optimization method that can be gener-
ally used in process optimization during PIM are still required. In
addition, previous researches solved the optimization problem just
as a single-objective programming. Warpage or shrinkage of the
parts were investigated and decreased greatly. However, in real
manufacture, quality of products is only one of the most important
factors to be considered. Energy consumption as well as the pro-
duction cycle and other factors during PIM should also be taken
into consideration. Based on finite element analysis software
Moldflow, Orthogonal experiment method, Back Propagation (BP)
neural network as well as genetic algorithm, a multi-objective
mathematical optimization model as well as a hybrid of BP/GA
optimization method of injection molding process parameters are
presented systematically in this paper. In addition, a plastic part
is utilized to demonstrate the efficiency and validity of the pro-
posed optimization method. Warpage of plastic as well as the
clamp force during PIM are investigated in a multi-objective func-
tion. A series of solutions are achieved by changing the weights of
the optimization objectives in the multi-objective optimization
function.
2. BP/GA hybrid method for optimization of injection molding
process parameters
2.1. The mathematical model
As a multi-objective and nonlinear optimization problem, pro-
cess optimization of PIM can be stated as follows:
Find X ð1Þ
Minimize F ð X Þ ¼Xn
i¼1kiobji ¼
Xn
i¼1ki f ið x1; x2; . . . ; xmÞ
Subject to X min 6 X 6 X max
Xn
i¼1
ki ¼ 1
where F ( X ) denotes the multi-objective optimization function of the
process optimization; obji stands for the ith optimization goal,
i = 1, 2, . . . , n; f i represents the functional relationship between obji
and the key process parameters; ki denotes the weight of the obji;
X = [ x1, x2, . . . , xm] stands for the matrix consists of injection model-
ing process parameters, mP 1; X max, X min stand for the upper and
lower bounds of the process parameters, respectively.
In addition, in order to eliminate the dimension of each objec-
tive, function (2) is used to preprocess each objective by normaliz-ing the inputs so that they fall in the interval [À1, 1]. The algorithm
can be expressed as follows:
pn ¼ 2ð p À minð pÞÞ=ðmaxð pÞ À minð pÞÞ À 1 ð2Þ
where p denotes matrix of input (column) vectors; pn represents
matrix of normalized input vectors.
During the injection process, a number of defects may occur to
the moldings, such as warpage, shrinkage, sink marks as well as
weld lines and so on. These defects greatly influence the quality
of the plastic and should be controlled seriously. In the proposed
optimization model, these defects can be selected to be the optimi-
zation objectives. Meanwhile, the energy consumption of the pro-
duction and some other requirements such as the production cycle,
and production cost can also be selected to be the optimizationobjectives of the optimization model. And the process parameters
during PIM such as the mold temperature, melt temperature, pack-
ing pressure, packing time and cooling time which contribute
greatly to the cause of these considerations can be specified as de-
sign variables.
2.2. Establishment of the objective functions based on BP neural
network
Establishment of objective functions is the key step of the opti-
mization model. However, it is very hard to express the relation-
ship between injection process parameters and optimization
objectives by explicit mathematical functions. Hence, a so-called
black-box function established by BP neural network is used in
the mathematical model.
Artificial neural network (ANN) is developed based on the
working principle of the nervous system of organism. As a kind
of information processing system, the network consists of a num-
ber of artificial neural cells. Each artificial neural cell is connected
by the connection weight just as the synapse of the nervous sys-
tem. A designed artificial neural network has the ability to obtain
the internal law of input information by learning and training pro-
cess. BP neural network is one of the most widely used and
acknowledged artificial neural networks nowadays [10–12]. Its
powerful ability of nonlinear interpolation is utilized in this paper
to obtain the relationship between process parameters and optimi-
zation goals. In this study, a multilayer BP neural network model is
designed by using the Matlab neural network toolbox. The struc-
ture of the BP neural network can be seen in Fig. 1.
Fig. 1 illustrates the structure of the BP neural network designed
in this study. It can be generally applied to the optimization of
injection molding process parameters. The network consists of
one input layer with m neurons standing for the process parame-
ters x1, x2, . . . , xm, respectively; z hidden layers with several neu-
rons each and one output layer having n neurons representing
optimization goals obj1, obj2, . . . , objn, respectively.
The input of each neuron comes from the output of the neurons
contained in the preceding layer by the transition function shownas follows:
net i ¼XN j¼0
xij x j ð3Þ
where net i is the total input of the ith neuron in the computing
layer; N denotes the number of the neuron in the forward layer;
xij stands for the connection weight of the jth neuron in the for-
ward layer and ith neuron in the computing layer; x j represents
the output of the jth neuron in the forward layer. Output of the
ith neuron in the computing layer (out i) is generated by processing
Fig. 1. Structure of the designed BP neural network.
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the input (net i) through a transfer function f s. The function can be
described as follows:
out i ¼ f sðnet iÞ ¼1 À eÀnet i
1 þ eÀnet ið4Þ
Mathematical relationship between process parameters and
objectives can be gained by training the experimental data got
from FE simulations on the commercial software Modelflow plat-form. During training process, the connection weights are calcu-
lated to minimize the error between the predictive data and
experimental data. The objective function contained in the trained
BP neural network can be expressed approximatively as follows:
obji ¼ f ið X Þ ¼ f lX
w z þ1 f s Á Á Á f s
Xw2 f
sX
w1 X
ð5Þ
where obji stands for the ith optimization objective; X = [ x1,
x2, . . . , xm] denotes the matrix consists of the values of process
parameters; f l is the liner transfer function between hidden layer
z and output layer; f s is the transfer function between input layer
and hidden layer 1, as well as hidden layer i and hidden layer
i + 1, i = 1, 2, . . . , z À 1; w1, w2, . . . , w z +1 represent the connection
weights between input layer and hidden layer 1, hidden layer 1
and hidden layer 2. . .
, hidden layer z and output layer, respectively.
2.3. Solution of the mathematic model
An intelligence global optimization algorithm, i.e. genetic algo-
rithm, is employed to solve the mathematical model established in
this paper. Genetic algorithm simulates biological evaluation pro-
cess: Darwin’s ‘‘survival of the fittest’’ and has been widely used
in engineer application [13,14]. At the beginning of the solution,
a set of potential solutions are randomly selected as the initial
chromosomes. The entire set of these chromosomes constitutes a
population. Then on the basis of the ‘‘survival of the fittest’’ theory,
new generations are generated through copy, crossover or muta-
tion method. The new chromosomes are then evaluated via a cer-
tain fitness criteria and the best ones are kept while the others are
discarded. After several generations, the fitness of the chromo-
somes will be increased. And the chromosome having the best
fitness is taken as the best solution of the problem. Fig. 2 illustrates
the solution procedure of the GA.
The entire technical line of the hybrid BP/GA process optimiza-
tion method for PIM can be seen in Fig. 3.
3. Case study
In this paper, a plastic part is utilized to demonstrate the
efficiency and validity of the proposed optimization method. As
one of the most common and prominent defects of plastic, warpage
affects both the usage and the appearance of the part and is consid-
ered to be one of the most critical considerations for the produc-
tion of a quality plastic part. Hence, warpage minimization is
specified to be one of the optimization objectives in this paper.
Besides, capacity of the equipment and energy consumption are
considered in this paper.
Ozcelik et al. stated that packing pressure is the most influential
parameter on the warpage of PC/ABS material [15]. In addition,
Huang et al. also pointed out that the packing pressure has the
greatest influence on the warpage, and with the increase of the
packing pressure; the warpage of plastic can be decreased [16].Namely, warpage of plastic can be greatly decreased by increasing
packing pressure during PIM. However, considering the capacity of
the equipment and cost of production, packing pressure can not be
increased without limit. Hence, the maximum clamp force greatly
determined by packing pressure is specified to be the other optimi-
zation objective. Mold temperature, melt temperature, packing
pressure, packing time as well as the cooling time are considered
to be design variables.
Fig. 2. Solution of the GA. Fig. 3. Technical line of hybrid BP/GA process optimization method for PIM.
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The commercial injection molding Computer-Aided Engineering
(CAE) software Moldflow is employed to simulate the injection
molding process. Warpage defects as well as the maximum clamp
force can be retrieved from the simulation results.
3.1. Problem description
Geometry of the plastic part utilized in this study is shown in
Fig. 4a. Its width, length and maximum part thickness are
200 mm, 200 mm and 2 mm, respectively. The material of the part
is PP. And the material mode of PP with the trade name of BP
Amoco 1046 and manufactured by BP Chemicals which is from
the library of the moldflow software database was employed as
the material of the part during simulations. The detailed material
properties can be seen in Table 1. Fig. 4b shows the CAE analysis
model of the plastic established under Moldflow environment.
The part is meshed in fusion mesh method and the meshed part
includes 7696 elements. Mesh condition as well as the filling sys-
tem and cooling system can also be seen in Fig. 4b.
Five key process parameters are selected as the design variables
in the mathematical model. These are mold temperature (T mold),
melt temperature (T melt ), packing pressure (P p), packing time (t p)
as well as the cooling time (t c ). The upper and lower bounds of
the process parameters are set based on the recommended values
provided by Moldflow software, and the ranges of the process
parameters can be seen in Table 2.The mathematical model of the multi-objective optimization
problem can be formulated as follows:
Find X ¼ ½T melt ; T mold;P p; t p; t c � ð6Þ
Minimize k1 f nðW Þ þ k2 f
nðF C Þ
Subject to : 30 6 T mold 6 60 220 6 T melt 6 260 50
6 P p 6 120 5 6 t p 6 15 5 6 t c 6 20
Fig. 4. (a) Geometry and (b) FE model of the plastic cover.
Table 1
Material properties of PP.
Material properties Performance
Melt density (g/m3) 0.7751
Solid density (g/m3) 0.92889
Eject temperature (°C) 93
Maximum shear stress (MPa) 0.26Maximum shear rate (sÀ1) 24000
Thermal conductivity (W/m°C) 0.15
Elastic module (MPa) 1340
Poisson ratio 0.392
Table 2
Ranges of the process parameters.
Process parameters Ranges
Mold temperature (°C) 30–60
Melt temperature (°C) 220–260
Packing pressure (% injection pressure) 50–120
Packing time (s) 5–15
Cooling time (s) 5–20
Fig. 5. The structure of the BP neural network used in this case.
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where W denotes the warpage of the part; F C represents the maxi-
mum clamp force during the injection molding process; f n repre-
sents the normalized function as shown in function (2); k1, k2
represent the weight of warpage and clamp force, respectively.
3.2. Implementation of the proposed BP/GA optimization procedure
3.2.1. Objective functions founded by BP neural network
A 5-9-9-2 BP network is used to gain the mathematical relation-
ship between optimization objectives and process parameters. The
structure of the designed network can be seen in Fig. 5.
The function of the designed neural network is to predict the
warpage of the part as well as the clamp force during PIM under
a specified combination of the process parameters. In order to
make the designed network acquiring the ability of prediction, it
should be trained by a number of samples first. To save the com-
puting resource and improve the coverage of the samples, orthog-
onal experiment method is employed to conduct the FE
simulations under Moldflow environment. Sixteen samples de-
signed by the orthogonal experiment method as well as other 44
samples randomly generated by computer, 60 samples in sum,
are used to train the designed network. The distribution of the
samples can be seen in Fig. 6.
During training process, the learn rate of the network is set as
0.03 and the mean square error of the training data is set as
0.0001. The training process takes about half an hour on HP per-
sonal workstation. Fig. 7 shows the training process of the network,
it can be seen from Fig. 7 that with the updating of the connection
weights, the mean square error between the network prediction
data and training data declines gradually and converges to
0.0001 interminably within 900,000 epochs.
Six groupsof processparametersnot used in the training process
are used to test theaccuracy and reliabilityof thepredictivesystem.
It can be seen from Fig. 8 that the predictive values are in good
agreement with the experimental values. The predictive error is
less than 5% on average. Hence, the trained network can be used
as the surrogate of the objective functionin the optimization model.
3.2.2. Solution of GA
In the GA optimization process, the operation parameters
needed to be specified in GA are adapted. The population size,
the crossover rate, the mutation rate and the generation size are
set as 100, 0.5, 0.1 and 60, respectively. Four groups of weights dur-
ing optimization are given to the objective function:
Case 1: k1 ¼ 1 k2 ¼ 0.
Case 2: k1 ¼ 0:8 k2 ¼ 0:2.
Fig. 6. (a) Warpage and (b) Clamp force distribution of the samples.
Fig. 7. Training process of the 5-9-9-2 BP neural network.
Fig. 8. Testing results of the BP neural network.
Table 3
Process parameters optimized by the proposed BP/GA method under different
weights of optimization objectives.
Process parameters Case 1 Case 2 Case 3 Case 4
Mold temperature (°C) 50.911 36.419 54.971 34.149
Melt temperature (°C) 226.656 232.13 241.53 252.58
Packing pressure (% injection
pressure)
116.116 100.36 71.281 53.558
Packing time (s) 12.361 10.402 11.835 10.457
Cooling time (s) 8.065 5 16.471 16.059
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Case 3: k1 ¼ 0:5 k2 ¼ 0:5.
Case 4: k1 ¼ 0:2 k2 ¼ 0:8.
It takes about half an hour of each solution on HP workstation
platform. Process parameters optimized by the proposed optimiza-tion method can be seen in Table 3.
4. Results and discussion
Process parameters are set as recorded in Table 3 on the
Moldflow platform, warpage results and clamp force results of
the optimized process parameters can be seen in Fig. 9. In addition,to validate and compare the optimization results, we also obtain
Fig. 9. Warpage and clamp force analysis results of case 1–4.
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the warpage result as well as the clamp force result of the part by
employing the Moldflow recommended values for the process
parameters in the FE simulation. The recommended process
parameters are set as follows:
T mold ¼ 60; T melt ¼ 240; P p ¼ 100; t p ¼ 8; t c ¼ 15
The warpage and clamp force analysis results are 3.307 mm,
63.63 ton, respectively. And they can be seen in Fig. 10.In this study, to illustrate the efficiency and flexibility of the
proposed BP/GA optimization method, different weights of optimi-
zation objectives are specified to the multi-objective optimization
function. Fig. 9a–d shows the optimized results of case 1, 2, 3 and
4, respectively. And the comparison between the optimized results
and the recommended analysis results can be seen in Table 4
clearly.
Different optimized results are obtained by setting different
weights to the optimization objectives. In case 1, warpage is con-
sidered to be the only optimization objective. It can be seen from
Fig. 9a-1 and Table 4 that the optimized warpage value is
1.092 mm, which has been reduced by 66.9% comparing with the
warpage result, 3.307 mm, obtained by using Moldflow recom-
mended process parameters. Meanwhile, because the clamp forceis not taken into consideration during optimization, packing pres-
sure is just limited by the upper bound of packing pressure speci-
fied in the mathematical model. Hence, it can be increased without
limit to decrease the warpage of the part. Results show that the
optimized packing pressure of case 1 recorded in Table 3 is
116.116 MPa, which is very close to the upper bound of packing
pressure 120 MPa. The clamp force shown in Fig. 9a-2 is 79.01
ton, which has been increased by 24.1% comparing with the recom-
mended analysis result 63.63 ton.
In case 2, warpage is specified to be the main optimization
objective. Meanwhile, the clamp force during PIM is taken into
consideration. It can be seen in Fig. 9b-1 that the optimized war-
page value is 1.286 mm, which has been reduced by 61.1%, fewer
than 66.9% of case 1, comparing with the warpage result obtainedby using Moldflow recommended process parameters. At the same
time, the clamp force shown in Fig. 9b-2 is only increased slightly
comparing to the recommended analysis result.
In case 3, warpage and clamp force are specified the same
weight during optimization. Analysis results shown in Fig. 9c-1, 2
illustrate that warpage of the part as well as the clamp force during
PIM are both reduced and comparing with the analysis results ob-
tained by using Moldflow recommended process parameters, theyare decreased by 54.2% and 33.6%, respectively.
In case 4, the clamp force during PIM is specified to be the main
optimization objective. Results show that the clamp force during
PIM is decreased greatly, that is 55.6% comparing to the recom-
mended analysis result. At the same time, the warpage of the part
is also decreased.
In addition, it can be seen from Table 4 that all the cases have
decreased the warpage of the part. And with the decrease of the
weight of warpage in the multi-objective optimization function
from case 1 to case 4, the decrement of warpage is declined grad-
ually. Meanwhile, clamp force is increased in case 1 and case 2, be-
cause higher packing pressure is needed to decrease the warpage
of the part, which shows a good agreement with the conclusions
of the cited literatures [15,16]. However, with the increase of the
weight of clamp force, clamp force plays a more and more impor-
tant role in the multi-objective function and is limited more and
more seriously. That is why increment of clamp force in case 2,
6.1%, is fewer than that in case 1, 24.1%. In case 3 and case 4, the
weight of clamp force is large enough to impact or dominate the
optimization. Hence, the clamp force in case 3 and 4 are decreased
instead of increased. Warpage is decreased just by adjusting other
process parameters, such as packing time and cooling time, result-
ing in a longer production cycle.
Comparing with the researches of the cited literatures [1–6], the
proposed BP/GA optimization method in this investigation takes
both the warpage of the plastic part and energy consumption dur-
ing PIM into consideration. In addition, the proposed optimization
method has the advantage of flexibility. The optimization objec-
tives can be optimized in different degrees by specifying differentcombination of the weights. And each solution has its advantages
and disadvantages. Case 1 extremely decreased the warpage of
the part, while its clamp force is the largest, which means process
of case 1 will consume more energy than other cases; cases 2, 3
and 4 take both warpage and clamp force into consideration, while
the ratio of the weights is different. Hence, the warpage and the en-
ergy consumption during PIM are optimized in different degree.
And a fittest solution can be selected by the demands and objective
factors of real manufacture.
5. Conclusions
In this study, a hybrid of BP/GA optimization method of injec-tion molding process parameters is presented systematically on
Fig. 10. (a) Warpage and (b) clamp force results by using Moldflow recommended process parameters.
Table 4
Comparison between the optimized analysis results and the recommended analysis
results.
Analysis results Rate of change (%)
Warpage
(mm)
Clamp force
(Ton)
Warpage Clamp
force
Recommended 3.307 63.63 – –Case 1 1.092 79.01 À66.9 +24.1
Case 2 1.286 67.51 À61.1 +6.1
Case 3 1.513 42.28 À54.2 À33.6
Case 4 1.765 28.26 À46.6 À55.6
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the basis of finite element analysis software Moldflow, Orthogonal
experiment method, BP neural network and genetic algorithm. The
mathematical model and technique line for optimization of process
parameters during PIM are established clearly in this paper.
As an application, a plastic part is employed in this paper.
Warpage as well as the clamp force during PIM are chosen to be
the optimization objectives. Mold temperature, melt temperature,
packing pressure, packing time as well as the cooling time are con-
sidered to be design variables. A series of combination of weights
of optimization objectives are specified to the multi-objective opti-
mization model. After implementing the proposed BP/GA method,
warpage as well as the clamp force of the part is optimized in dif-
ferent degrees comparing with the warpage result obtained by
using Moldflow recommended process parameters. The fittest
combination of process parameters can be selected by the require-
ments of the real manufacture.
For its efficiency and flexibility, the proposed BP/GA optimiza-
tion method can be used generally to optimize defects of plastic
as well as other considerations such as production cycle, cost
and so on during PIM. In addition, the crystallization tempera-
ture of the injected materials plays an important role in shrink-
age and warpage of the plastic part during PIM, hence, it can be
taken into consideration during the optimization for PIM in the
future.
Acknowledgment
The work was supported by a grant from National Science Fund
for Distinguished Young Scholars (No. 50725217). The supports are
gratefully acknowledged.
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