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Journal of Materials Processing Technology 211 (2011) 245–255

Contents lists available at ScienceDirect

Journal of Materials Processing Technology

journa l homepage: www.e lsev ier .com/ locate / jmatprotec

Void closure prediction in cold rolling using finite element analysis andneural network

J. Chena, K. Chandrashekharaa,∗, C. Mahimkarb, S.N. Lekakhb, V.L. Richardsb

a Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 65409, United Statesb Department of Material Science and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, United States

a r t i c l e i n f o

Article history:Received 19 June 2010Received in revised form 9 September 2010Accepted 19 September 2010

Keywords:Void closureCold flat rolling processNonlinear dynamic finite element modelNeural network

a b s t r a c t

Cold rolling is used to eliminate void defects in cast materials thus improving the material performanceduring service. A comprehensive procedure is developed using finite element analysis and neural net-work to predict the degree of void closure. A three-dimensional nonlinear dynamic finite element modelwas used to study the mechanism of void deformation. Experiments were conducted to investigate voidclosure during the cold flat rolling process. Experimental results are compared to the three-dimensionalfinite element predictions to validate the model. The void reduction predictions from finite elementanalysis are in good agreement with experimental findings. Plastic strain, principal stress distributionaround the void and void reduction ratio are presented for various case studies. As finite element sim-ulation is time-consuming, a back-propagation neural network model is also developed to predict voidclosure behavior. Based on the correlation analysis, the reduction in sheet thickness, the dimension ofthe void and the size of the rollers were selected as the inputs for the neural network. The neural networkmodel was trained based on results obtained from finite element analysis for various simulation cases.The trained neural network model provides an accurate and efficient procedure to predict void closurebehavior in cold rolling.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Ingots and centrifugally cast products have micro-size andmacro-size void type defects caused by shrinkage and gas evolu-tion during solidification. Void defects in the final metal productaffect the performance of the material and shortens its servicetime. Intensive plastic deformation can eliminate most of these voiddefects. The cold flat rolling process has been in use for centuriesto produce sheets, strips, and other flat products. The objective ofthe flat rolling process is to reduce the thickness of the work pieceto a predetermined level. Cold flat rolling can increase density ofcast materials and eliminate void defects through significant plas-tic deformation. The efficiency of void defect healing depends onthe local plastic deformation and material properties.

A number of studies have investigated void closure behaviorduring the rolling process. Wallero (1985) studied the closure of acentral void during the hot rolling process through experiments andfound that the forceful closure of a central pore should be based onheavy reductions secured by means of large rolls. Using the upperbound method, Stahlberg et al. (1980) and Keife and Stahlberg(1980) analyzed the influence of stress distribution on the closing of

∗ Corresponding author. Tel.: +1 573 341 4587; fax: +1 573 341 6899.E-mail address: [email protected] (K. Chandrashekhara).

a central longitudinal hole inside the steel sheet during the rollingprocess. However, highly simplified upper bound methods offeronly a rough means of investigation and cannot accurately predictsome process parameters. During recent decades, the finite elementmethod has proved to be a powerful computational tool to analyzethe material deformation process. Many numerical studies haveexamined void closure behavior. Dudra and Im (1990) used plane-strain finite element method to investigate void closure in open-dieforging. The deformation of the internal void was compared toexperimental results. Pietrzyk et al. (1995) predicted the behav-ior of voids in a steel slab during hot rolling by a rigid-plastic finiteelement model. The influence of the stress state on void behav-ior was investigated. Wang et al. (1996) studied strain and stressdistributions around the voids using an elasto-plastic plane-strainfinite element model. The simulation results agreed closely with hisexperimental results. Using a two-dimensional plane-strain finiteelement model, Hwang and Chen (2002) explored the mechanismof void deformation and stress–strain distributions around internalvoids in the sheet during cold rolling. He discussed the influencesof various rolling conditions, and the predictions of his simulationagreed closely with his experimental results. Chen (2006) used arigid-plastic finite element model to investigate the deformationof porous metal sheets containing internal void defects during theflat rolling process. However, the two-dimensional finite elementmodels used by these researchers account only for the performance

0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2010.09.016


246 J. Chen et al. / Journal of Materials Processing Technology 211 (2011) 245–255

of central longitudinal voids inside the sheet. Since real voids havecomplex shapes in three-dimensional frame, a three-dimensionalmodel is needed to investigate the void closure during rolling pro-cess.

Although the finite element method is a powerful approachfor the investigation of material deformation behavior and theeffect of process parameters during the rolling process, it is time-consuming. Recently, neural networks have been a popular tool forsolving complex engineering problems due to their powerful non-linear and adaptive nature and their self-learning capacities (Hamand Kostanic, 2001; Haykin, 1999). Many researchers have studiedthe effect of process parameters on the rolling process in recentyears. Dixit and Chandra (2003) used neural networks to predictroll force and roll torque in a cold flat rolling process. Yang et al.(2003) predicted the roll force and torque during cold rolling pro-cess using finite element modeling and neural networks. Gudurand Dixit (2008) used a neural network to predict the velocity fieldand location of neural points on the sheet. His training data wereobtained from a rigid-plastic finite element model. Hsiang and Lin(2007) established a back-propagation neural network model andapplied simulated annealing algorithm to find the optimal process-ing parameters for caliber rolling. Kim et al. (2008) developed aneural network based approach to predict a mechanical property ofthe hot-rolled alloy strip. Among these researches, neural networkwas a favorite tool for analyzing the rolling process. Little research,however, has focused on neural network modeling to study processparameters in the investigation of void closure behavior during thecold rolling process.

The present work employs a three-dimensional nonlinear finiteelement model to analyze void closure behavior during the coldrolling process. The model was developed and implemented in thecommercial finite element code ABAQUS, and accounts for contactand material nonlinearities. A global-local finite element techniquewas used to obtain precise values of void closure behavior dur-ing the rolling process. Simulation results of central longitudinalcylinder voids were verified by comparison with the experimentalresults. The three-dimensional finite element model was extendedto study the spherical void closure during the cold rolling process.Compared to the computation burden imposed by the finite ele-ment model, the neural network approach is not time-consumingand can be used for online prediction and optimization. A back-propagation neural network model was developed to establish arelationship between the predicted void closure behavior and thecorresponding rolling conditions. The training data for neural net-work model were obtained from finite element simulation results,and a correlation analysis technique was used to filter the rollingparameters for significant inputs. The performance of the neuralnetwork was validated by considering cases not included in thetraining process. The neural network results were in good agree-ment with finite element predictions.

2. Finite element modeling

2.1. Cold flat rolling process with internal void

The rolling process eliminates the void defects through plasticdeformation. Because the width of the sheet changes little during

Fig. 1. Rolling process of sheet with an internal void.

the rolling process relative to changes in length and thickness, andbased on the assumption of homogeneous compression suggestingthat planes remain planes during the pass, the rolling process isusually modeled based on the plane-strain plastic flow of the sheetmaterial. Fig. 1 shows a typical scheme of flat rolling scheme with aninternal void. The initial internal void is assumed to be round, witha diameter of d0. The final height and length of the internal voidare h1 and l1, respectively. L represents the contact length betweenthe sheet and roller, X is the distance from the internal void to theexit of the roll gap, and t0 and t1 are the initial and final thicknessesof the sheet, respectively, the relative sheet thickness reduction ris defined as t1 − t0/t0. The friction coefficient between the rollersand the sheet is represented by f, and the diameter of the roller isD. The relative void height reduction rvoid is defined as h1 − d0/d0.

Two types of voids are considered in this study. The first type isa “cylinder void,” which has a longitudinal hole through the widthof the sheet. The behavior of this void during rolling can be investi-gated using a two-dimensional plane-strain finite element model.Because such a hole can easily be drilled inside the aluminum sheet,its closure behavior during the rolling process can be determinedexperimentally. The second type of is called a “spherical void,”which has a spherical shape inside the aluminum sheet. A three-dimensional finite element model is needed to study spherical voidclosure behavior. The two types of void are shown in Fig. 2.

2.2. Finite element simulation

A nonlinear finite element model in three-dimensional frameis developed to investigate void closure behavior during the coldrolling process. The three-dimensional formulation for dynamicanalysis can be expressed as:

[Me]{�̈e} + [Ke]{�e} = {Fe} (1)

where [Me] is the mass matrix, [Ke] is the stiffness matrix, and {Fe}is mechanical loading.

The explicit method was used to solve the nonlinear dynamicsimulation in ABAQUS. Although the implicit method is most effi-cient for solving smooth nonlinear problems, the explicit methodis the clear choice for the cold rolling problem as it includes largedeformations, contact and material nonlinearity. The equations ofmotion are integrated using the explicit central-difference integra-

(a) Cylinder void (b) Spherical void

Fig. 2. Voids types inside the aluminum sheet.


J. Chen et al. / Journal of Materials Processing Technology 211 (2011) 245–255 247

Fig. 3. Global finite element model for cold rolling process.

tion rule:

{�̈e}(i) = [Me]−1({Fe}(i) − {Ie}(i)) (2)

{�̇e}(i+1/2) = {�̇e}(i−1/2) + �t(i+1) + �t(i)

2{�̈e}(i) (3)

{�e}(i+1) = {�e}(i) + �t(i+1){�̇e}(i+1/2) (4)

where {Ie}(i) represents the internal force vector, which is equalsto [Ke]{�e}(i). The subscript i refers to the increment number in anexplicit dynamics step.

A lumped mass matrix was used because its inverse is simpleto compute. An elasto-plastic material model was used to describethe aluminum sheet deformation during the cold rolling process. Apower law was used to express the relationship between the stressand the amount of plastic strain:

� = K(εp)n (5)

where � is the stress, εp is the plastic strain, K is the strength coeffi-cient, and n is the strain hardening exponent. The contact constraintwas applied when the clearance between two surfaces becomeszero. The contact pair algorithm was used to investigate the impactof the rollers on the sheet. Hard contact was introduced to describethe contact behavior normal to the surfaces. For the purposes ofthe simulation, the rollers were considered as rigid bodies. A con-stant friction coefficient was assumed between the roller and steelsheet. The friction of the contact areas was defined by the Coulombfriction law as:

�crit = �p (6)

where �crit is the critical shear stress, � is the friction coefficient,and p is the contact pressure between the two surfaces.

The small void inside the sheet had to be modeled in substan-tial detail to obtain accurate deformation values. If the void wasmodeled together with the whole sheet, the resulting element sizewould be too small and the model would be too large, demandingtremendous computational cost. This problem was resolved by aglobal-local finite element analysis strategy. The global model withcoarse mesh was used to obtain the displacement solution aroundthe void region. The local model of the region surrounding the voidwas created with a reasonably fine mesh.

The dimensions of the aluminum sheet and rollers consideredin the present analysis were based on the parameters of the exper-

Table 1Material properties of aluminum.

Material AA1050 (≥99.50%)

Density (kg/m3) 2700Young’s Modulus (GPa) 75Poisson’s ratio 0.33Power law for plasticity behavior K = 187.7 MPa, n = 0.079

imental tests. Table 1 lists the material properties of aluminum.Because the deformation of the rollers can be disregarded, a quar-ter model and three-dimensional rigid elements were used forrollers to minimize computational time. The aluminum sheet wasmodeled using an 8-node solid brick element. Reduced integrationimproved the computational efficiency and avoided the shear lock-ing of elements. Fig. 3 shows the schematic of the mesh for the finiteelement model. A fine mesh was used near the void. To accuratelyevaluate the void behavior, a local finite element model (shownin Fig. 4) was created for a small volume of a sheet containingthe spherical void. The 8-node brick element with second-orderaccuracy was used to investigate void behavior during cold rolling.The mesh density in the local model was much higher than that inthe global model. The boundary conditions of the local model weredriven by the global analysis solution.

Fig. 4. Local finite element model for spherical void.



Fig. 5. Experimental test results of void closure during cold rolling process.

Table 2Parameters for cold flat rolling process.

Roller diameter 406.4 mm

Friction between the rollers and the sheet 0.6Rolling speed 5.0894 rad/s (0.54 rotations/s)Sheet dimension 120 mm × 100 mm × 18.75 mm

2.3. Comparison with experimental results for cylindrical void

The three-dimensional finite element model described here wasverified with the results of experiments performed on the cylindervoid closure during cold rolling. The experiments were conductedin the cold rolling facility at the Missouri University of Science andTechnology. Parameters for the cold rolling process are listed inTable 2. Sheet specimens were AA 1050 aluminum. Three throughholes were drilled inside the aluminum sheet, as shown in Fig. 5(a).Two kinds of thickness reductions were considered to investigatethe deformed shape of the voids after rolling as shown in Fig. 5(b)and (c). A micrometer unit was used to measure the size of thedeformed voids.

Finite element solutions are also obtained using a two-dimensional plane-strain model. Table 3 compares the experimen-tal and simulation results. The results of the three-dimensionalmodel match the experimental results better than those of thetwo-dimensional plane-strain model, indicating that the three-dimensional model can better predict void closure behavior.

Fig. 6 shows the void reduction results yielded by the exper-iments and the two models. The curve of the three-dimensionalmodel matches that of the experimental results. Further investi-gation of the three-dimensional model (shown in Fig. 7) indicatesthat the void reduction through the width of the sheet was not uni-form; the two-dimensional plane-strain model showed no changeof void reduction at all. The results also indicate that the voidfirst closed at the center in the direction of the width. Becausethe materials were subjected to different stress conditions at thecenter than on the side of the sheet (shown in Fig. 8). The mate-rial on the side of the sheet had no constraint on one side in thedirection of width. The material flow determined by the differentstress conditions created various void reductions along the widthdirection.

Table 3Comparison of simulation and experimental results.

Case no Void diameter (mm) Thickness reduction (%) Void reduction (experiment) Void reduction (simulation) Error (%)

1 2.537 20.32 −0.6117 2D Model −0.7696 25.83D Model −0.6028 1.46

2 2.537 40.69 −1 2D Model −1.00 03D Model −1.00 0

3 3.166 20.32 −0.5556 2D Model −0.7331 32.03D Model −0.5617 2.08

4 3.166 40.69 −0.9315 2D Model −1.00 7.403D Model −0.9004 3.33

5 4.231 20.32 −0.5332 2D Model −0.6409 20.203D Model −0.5287 0.84

6 4.231 40.69 −0.9076 2D Model −1.00 10.20



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Rel

ativ

e vo

id re

duct

ion

Thickness reduction

2.537 mm(experiment)2.537 mm(2D model)2.537 mm(3D model)3.166 mm(experiment)3.166 mm(2D model)3.166 mm(3D model)4.231 mm(experiment)4.231 mm(2D model)4.231 mm(3D model)

Fig. 6. Void reduction versus sheet thickness reduction.

2.4. Simulation for spherical void closure usingthree-dimensional finite element model

As experimental results for spherical void closure are not avail-able, a three-dimensional finite element model was used to studythe spherical void closure behavior. The diameter of the spheri-cal void was 2.5 mm and the void was located at the center of thealuminum sheet. The thickness of the sheet was reduced by 20%;other rolling process parameters and material properties were thesame as listed before. To investigate the deformed void in a three-dimensional frame, three planes were defined, as shown in Fig. 9.Figs. 10 and 11 show the void shape, plastic strain, and maximumprincipal stress distributions around the void in the three planesafter rolling. The spherical void inside the aluminum sheet was

0 2 4 6 8 10 12 14 16 18-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

Width direction(mm)

Rel

ativ

e vo

id re

duct

ion

d=2.537 mm, r=20.32%d=3.166 mm, r=20.32%d=4.231 mm, r=20.32%

Fig. 7. Void reduction along the width.

deformed to an elliptical shape. The void was reduced primarilyin terms of thickness.

Figs. 12 and 13 show plastic strain and principal stress distribu-tions along the paths around the void in three planes. The plasticstrain around the void was much higher along the length thanalong the height or width. This specific strain distribution forcedthe void to deform as an elliptical shape during the rolling process.The maximum plastic strain occurred around the void in the X–Zplane, implying that the void would be eliminated in this plane.Moreover, the principal stress distribution shows that the materi-als around the void were subjected to compression along its lengthand width, and tension along its height. Thus, the void would beclosed in the X–Z plane.

Fig. 14 shows the relative void closure reduction versus the ratioof X to L in all three directions. During the cold rolling process,

xσxσ

zσ

zσyσ

zσ

xσxσ

Yσ

X

Y

Z

Fig. 8. Components of stress near the void.

Z-Y plane

X-Y plane

X-Z plane

Rolling direction X

Y

Z

Fig. 9. Definitions of planes for spherical void.



Fig. 10. Plastic strain distribution around spherical void after cold rolling process.

both the width and height of the void were reduced, and its lengthincreased as the void flowed out of the roller gap. Further, rate ofvoid reduction at the entrance of the roller gap was larger than thatat the exit of the roller gap.

3. Neural network modeling

3.1. Back-propagation neural network method

A three-layer neural network model with back-propagationtraining algorithm was developed to predict the void reductionduring the cold rolling process. The neural network has three lay-ers: an input layer, a hidden layer, and an output layer. Each layerconsists of several neurons in parallel (Hu et al., 2009a,b). Inputand output target patterns were used to train the network until itcould approximate the relationship between the input data and theoutput target.

In the first stage of the training process, the network synap-tic weights and neuron biases were initialized randomly. Fromthe set of training inputs, the neural network model calculatedthe output patterns. This step is called feed forward. In the sec-ond stage, the desired network output patterns were comparedto the actual output of the network and all the local errors cal-culated. In the third stage, the weights of the network wereupdated based on these errors using the gradient descent equa-tions. This step is called back propagation. Finally, the trainingprocess repeated the former two steps until the least error wasreached.

3.2. Correlation analysis

The relationship between the void closure and the four factors(thickness reduction, void dimension, friction coefficient, and roller



Fig. 11. Maximum principal stress distribution around spherical void after cold rolling process.

size) was investigated to determine the dependence of the void clo-sure behavior on these four main impact factors. Because two-leveldesigns are sufficient for evaluating most production processes,each factor for void closure took two value levels (low and high). Forsimulation design with four factors, a two-level full factorial designproduces 16 simulation cases. However, considering the large num-ber of three-dimensional elements and the small size of elementsin the local model, the three-dimensional finite element analysis ofvoid closure during the cold flat rolling process is computationallycostly. Fractional factorial designs use a fraction of the runs requiredby full factorial designs. For four factors, a fractional factorial analy-sis with 8 cases can be carried out to study the relationship betweenthe factors and the output. Table 4 lists the results of 8 simulationcases.

Correlation analysis is used to determine whether a relationshipexists between variables. When there is no correlation betweenthe two variables, there is no tendency for the value of one vari-able to increase or decrease with the value of the second variable.The correlation coefficient matrix, which is used to calculate thenormalized measurement of the strength of the linear relationshipbetween variables, is defined as follows:

rX,Y = E(XY) − E(X)E(Y)√E(X2) − E2(X)

√E(Y2) − E2(Y)

(7)

where X and Y are the two variables and E is the expected valueoperator. The range of the correlation coefficient is from −1 to 1.Values close to 1 suggest that there is a positive linear relationship



Table 4Fractional factorial analysis for void closure simulation.

No. Thickness reduction r (%) Void diameter d0 (mm) Friction coefficient f Roller diameter D (mm) Void reduction rvoid (height)

Range 10–30 1.0–4.0 0.3–0.9 203.2–812.8 0 to −1.01 10 1.0 0.3 812.8 −0.23432 10 1.0 0.9 203.2 −0.22893 10 4.0 0.3 203.2 −0.22134 10 4.0 0.9 812.8 −0.20465 30 1.0 0.3 203.2 −0.81856 30 1.0 0.9 812.8 −0.92957 30 4.0 0.3 812.8 −0.65748 30 4.0 0.9 203.2 −0.7530

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Angle (degree)

Pla

stic

stra

in

Path in X-Y planePath in X-Z planePath in Z-Y plane

Void

Angle

Fig. 12. Plastic strain distribution along the paths around the void.

0 50 100 150 200 250 300 350-150

-100

-50

0

50

100

Angle (degree)

Path in X-Y planePath in X-Z planePath in Z-Y plane

Void

Angle

Max

imum

prin

cipa

l stre

ss (M

Pa)

Fig. 13. Maximum principal stress distribution along the paths around void.

00.10.20.30.40.50.60.70.80.91-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

X/L

Rel

ativ

e vo

id re

duct

ion

Length reductionHeight reductionWidth reduction

Fig. 14. Spherical void reduction during cold rolling process.

r

D

0d voidr

1hb

2hb

hNb

hf

hf

hf

of Toutf

Tinf

ob

r′

0d ′

D′

voidr′

11hW

11oW

12oW

1oNW

3hNW

Input layer Output layer Hidden layer

Fig. 15. Neural network architecture for void closure during the cold rolling process.

between two variables; values close to −1 suggest that there is anegative linear relationship between two variables; values close to0 suggest that there is no linear relationship between two variables.

Table 5 shows the correlation coefficient values of the fourfactors related to void reductions. It indicates that the thicknessreduction of sheet along with void diameter and roller diametersignificantly influence void closure. The friction coefficient betweenthe roller and the sheet surface apparently has little influence onvoid closure. Therefore, based on the results of correlation analysis,three factors (thickness reduction, void diameter, and roller diam-eter) were selected as neural network inputs, and the void heightreduction was chosen as the output of the neural network.

3.3. Neural network model for void closure during the cold rollingprocess

A feed-forward back-propagation neural network model wasdeveloped to predict void closure behavior during cold rolling. Thenetwork includes one hidden layer and one output layer (Fig. 15).Each layer consists of a number of processing units, with an activa-tion function, known as neurons. For faster training and a robustneural network, inputs were scaled to a desirable range by adesigned transfer function f T

inbefore entering the input layer. Then

normalized inputs were passed through weighted connections tothe hidden layer whose outputs were the inputs of the third layer(output layer). The output of output layer is the network responsevector. Finally, the output r′

voidwas scaled back to the value of void

reductions by a transfer function f Tout . The number of neurons in

the hidden layer was determined by training relatively large neuralnetwork architectures.

The input transfer function is defined as:

f Tin(Ii) = (nmax − nmin)

(Ii − Imini

)

(Imaxi

− Imini

)+ nmin (8)



where Ii is the input variables (i = 1, 2, and 3, corresponding to r, d0,and D, respectively) and [nmin nmax] is the desired scale range andis taken as [−1 1].

Normalized input variables I′i

are given by

I′i = f Tin(Ii) (9)

The activation function in the hidden layer is the Log-sigmoid func-tion

f h(x) = 11 + e−x

(10)

The activation function in the output layer is the Pureline function

f o(x) = x (11)

The normalized void reductions r′void

is given by:

r′void =

N∑j=1

Wo1jf

h

(3∑

i=1

Whji I

′i + bh

j

)+ bo (12)

where N is number of neurons in the hidden layer, Wo1j

is weights

in the output layer, Whji

is weights in the hidden layer, bo is bias in

the output layer, and bhj

is biases in the hidden layer.The output transfer function is expressed as:

f Tout(r

′void) = (rmax

void − rminvoid)

r′void

− nmin

nmax − nmin+ rmin

void (13)

The final void reductions rvoid is given by

rvoid = f Tout(r

′void) (14)

Training patterns are a set of known input–output pairs. The back-propagation learning process consists of two passes through thenetwork: a forward calculation pass and an error back-propagationpass. In the forward pass, input patterns are transferred from theinput layer to the output layer to generate the actual networkresults. The errors between the actual outputs of networkand the designed outputs are then obtained. During the errorback-propagation pass, weights in each node are iteratively

Table 5Correlation coefficients rX,Y for void closure simulation.

Parameter r d0 f D rvoid

r 1.00 0.00 0.00 0.00 −0.9575d0 0.00 1.00 0.00 0.00 0.2073f 0.00 0.00 1.00 0.00 −0.0392D 0.00 0.00 0.00 1.00 −0.1940rvoid −0.9575 0.2073 −0.0392 −0.1940 1.00

Start

Establish training database

3D finite element model • Global-local technology • Explicit method • Elasto-plastic material model

Train the BP neural network model

Mean square error < 0.0001

Use trained neural network for prediction, and verify prediction by testing data

Error < 8%

End

• Change the neuron number • Change the initial parameters • Adjust the training pattern

Testing data

Y

N

Y

N

Fig. 16. Flow chart of neural network training and verification procedure.



-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

3D finite element simulation results

Neu

ral n

etw

ork

resu

lts

Fig. 17. Comparison of relative void reduction between 3D finite element modeland neural network.

adjusted based on the errors using the gradient descent equations.The procedure is repeated until convergence is achieved. TheLevenberg–Marquardt back-propagation algorithm was used inthe training process. The input–output training sets were obtainedfrom finite element simulation results. The neural network modelwas implemented and trained using MATLAB neural network toolbox.

3.4. Neural network results and discussion

The flow chart of neural network training and the verificationprocedure are shown in Fig. 16. The neural network was trainedby 64 pattern sets from finite element simulation results with var-ious thickness reductions, void sizes and roller diameters. Sevenneurons were employed in the hidden layer of the network, andthe learning rate was set to 0.01. The network was trained until themean squared error between the target output and network outputdecreased to 1 × 10−4. Once the network converged, it was capableof predicting the void reduction based on thickness reduction, voidsize, and roller diameter values. Fig. 17 compares the void reduc-tions between the training data from finite element simulation andthe trained neural network predictions. The trained neural networkresults are in good agreement with the training data, implying thatthe neural network is an effective approach to describe the rela-tionship between the inputs and output of training data.

To evaluate the performance of neural network prediction, ninetesting cases (shown in Table 6) which are not previously used fortraining were studied. Figs. 18–20 compare test data and the neuralnetwork predictions. The difference between the neural networkprediction and test data is below 8%. The trained back-propagationneural network, therefore, can be used to predict any void reductionwithin the trained ranges of the sheet thickness reduction, void size,and roller diameter.

4. Conclusion

A comprehensive three-dimensional finite element model wasdeveloped to predict the void closure behavior inside an aluminumsheet during flat cold rolling. Global-local analysis strategy wasused to obtain deformation, strain and stress distributions aroundthe void. The deformation around the void region was determinedusing a coarse mesh from the global model. The local model ofthe void region with fine mesh was used to determine strains,stresses, and void closure simulation results. Two types of voidswere considered: a longitudinal void and a spherical void. Theclosure behavior of the longitudinal void was investigated using

0 1 2 3 4 5

-1

-0.8

-0.6

-0.4

-0.2

0

Void size (mm)

Voi

d re

duct

ion

Simulation resultNNk prediction

Fig. 18. Comparison of neural network (NNk) predictions and simulation results fordifferent void sizes (r = 30%, D = 406.4 mm).

10 15 20 30 40 45

-1

-0.8

-0.6

-0.4

-0.2

0

Thickness reduction (%)

Voi

d re

duct

ion


Fig. 19. Comparison of neural network (NNk) predictions and simulation results fordifferent sheet thickness reductions (d0 = 2.0 mm, D = 406.4 mm).

100 203.2 304.8 400 500 600 700 812.8

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Roller diameter (mm)

Voi

d re

duct

ion


Fig. 20. Comparison of neural network (NNk) predictions and simulation results fordifferent roller diameters (r = 20%, d0 = 2.0 mm).



Table 6Neural network testing cases and error in prediction.

Testing case no. NNk inputs NNk prediction Simulation Error (%)

r (%) d0 (mm) D (mm) rvoid (%) rvoid (%)

1 30 1.0 406.4 −0.9108 −0.9256 −1.602 30 2.5 406.4 −0.8834 −0.9193 −3.913 30 4.0 406.4 −0.8739 −0.8987 −2.764 15 2.0 406.4 −0.3739 −0.3664 2.055 30 2.0 406.4 −0.9019 −0.8552 5.186 40 2.0 406.4 −1.0641 −1.0000 6.417 20 2.0 203.2 −0.4945 −0.4866 1.628 20 2.0 304.8 −0.5144 −0.4939 4.159 20 2.0 812.8 −0.5326 −0.5126 3.90

both a two-dimensional plane-strain model and three-dimensionalmodel. The three-dimensional finite element results matched wellwith experimental findings. Spherical void closure behavior dur-ing the cold flat rolling process was investigated using the verifiedthree-dimensional finite element model. Principal stress distribu-tion, plastic strain distribution, and the void reduction during therolling process were reported. The void closure was maximumalong the height direction.

As finite element simulation is a time-consuming process, afeed-forward back-propagation neural network model was devel-oped to establish the relationship between void closure behaviorand corresponding rolling conditions. Correlation analysis indi-cated that the sheet thickness reduction, void size, and rollerdiameter are significant parameters that influence void closure, andtherefore these three parameters were chosen as the inputs of theneural network. The training data for the neural network modelwere obtained finite element simulation. The difference betweenthe neural network predictions and the test simulation data wasbelow 8%. Therefore, the neural network model developed here canbe used as an efficient approach to predict void closure behaviorduring the cold flat rolling process.

Acknowledgements

We would like to thank the U.S. Army Benet Labs for fundingthis research. The conclusions and opinions expressed are those ofthe authors and not of Benet Laboratories.

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