Initial guess of rigid plastic finite element method in hot strip rolling

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1816–1825

journal homepage: www.elsevier.com/locate/jmatprotec

Initial guess of rigid plastic finite element method in hot strip rolling G.L. Zhang a , S.H. Zhang a,∗ , J.S. Liu a , H.Q. Zhang b , C.S. Li c , R.B. Mei c a b c

Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, PR China Shenyang University, Shenyang 110044, PR China State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110014, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history:

Although the rigid plastic finite element method (RPFEM) is extremely efficient and partic-

Received 20 November 2007

ularly suitable for analyzing the strip rolling, it is unavailable for online application due to

Received in revised form

the large computational time. During iterative solution of RPFEM, the convergence speed

18 March 2008

is greatly determined by the initial guess. In this paper, three different initial guesses are

Accepted 18 April 2008

constructed through Engineering Method, G Functional and Neural Network, respectively. Especially, the back propagation neural network model for predicting the relative velocity field (nodal velocities/roll speed) is trained from huge amounts of RPFEM results. Where-

Keywords:

after, the initial guess is calculated by multiplying the predicted relative velocity field by the

Rigid plastic finite element method

roll speed. The numerical examples of seven passes hot strip rolling are provided to show

Strip rolling

the solution efficiency and the accuracy of RPFEM code in the cases of different initial guess.

Initial guess

Compared with other two methods, the Neural Network has the remarkable advantages to

Neural network

reduce the CPU time and the iterations of RPFEM code. From the numerical results, it is found that the CPU time, stability and the accuracy of RPFEM code in the initial guess by the Neural Network can meet the requirements of online control completely in hot strip rolling. © 2008 Elsevier B.V. All rights reserved.

1.

Introduction

Steel manufacturers have to improve the qualities of products and productivities in order to maintain the competitiveness in the steel manufacturing market. The main way is to develop a kind of high efficient strip rolling technology with high precision, high speed and automation. Therefore, the online control models with high accuracy, reliability and flexibility are absolutely necessary for setting the control parameters (such as rolling force, temperature and rolling speed, etc.). Traditionally, the mathematical model and statistical model based on analytical or energy method of rolling theory have been employed widely for the online control in strip rolling

∗

Corresponding author. Tel.: +86 24 8397 8266; fax: +86 24 23906831. E-mail address: [email protected] (S.H. Zhang). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.04.038

(Sun, 2002). In recent years, neural networks with the better accuracy and flexibility have been developed and used for predicting the rolling force and the mechanical properties of the rolled material in strip rolling. For examples, the offline learning neural network (Lee and Lee, 2002) and the online learning neural network (Son et al., 2005) have been developed and used successfully for predicting the rolling force in hot strip rolling. The online learning is used to improve the accuracy of online prediction and the offline learning is good for the improvement of the flexibility in the prediction of the first coils. However, as Larkiola et al. (1998) reviewed the role of neural network in strip rolling, the credibility and successful application of neural network is based on the suf-

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ficient amount and the accuracy of measured learning data and a good understanding of the process, which also limit the application of neural network. At present, offline analysis of strip rolling is also necessary for the correction of online control models. The rigid plastic finite element method (RPFEM) was early developed by Kobayashi et al. (1989) and Osakada et al. (1982). It has been used widely in the analysis of metal-forming processes including strip rolling, which can provide the velocity field, stress field, strain rate field, and then the rolling force, forward slip and roll torque. The rigid plastic finite element method is extremely efficient and particularly suitable for the analysis of both the non-steady state rolling and the steady state rolling, and both the shape rolling and the strip rolling (Jiang and Tieu, 2001; Liu, 1994). Especially for the steady state rolling, the variational friction model has been proposed to analyze the cold strip rolling (Jiang and Tieu, 2004) and the hot strip rolling (Tieu et al., 2002). It was found that the accuracy and efficiency can be improved remarkably by this model. Through the finite element analysis, the rolling process can be optimized, and the control model also can be corrected. Moreover, the training data of neural network can be obtained from the finite element results which can cut down the amount of experiments. Although the rigid plastic finite element method has high accuracy and good flexibility, it is not allowed to be used in the online control in strip rolling. It is because the computational time of the FEM code is too large for the online application. In order to improve the solution efficiency and the robustness of RPFEM, most of researches were focused on the solution algorithm. Generally, the formulation of RPFEM is based on the minimum principle of the energy functional, which solved normally by Newton–Raphson (N–R). In recent years, some new methods such as linear programming (LP) and potential reduction interior point method (PM) were proposed by Xu et al. (2003) and Huang et al. (2003), respectively. Compared with the N–R method, LP and PM increased the storage of memory and variables, but PM reduced the CPU time of RPFEM code, while LP increased the CPU time of RPFEM code. It is because that the complex boundary condition can be dealt with PM method directly and without the convergence problem. Mori and Yoshimura (2000) and Mori et al. (2006) established the formulation of RPFEM using diagonal matrix and developed a parallel processing for the largescale simulation of metal-forming processes. However, for the iterative method (N–R or PM), the convergence is greatly determined by the initial guess. Traditionally, there were three main methods to construct the initial guess. It includes engineering method, elementary method, and establishment of similar functional which can produce the linear equations of initial velocity field (Liu, 1994). Jiang et al. (2000) compared the G function method with the elementary method for the convergence of RPFEM in strip rolling. It showed that less N–R iterations happened for G function method. But the computational time has not been analyzed. Recently, the radial basis function (RBF) neural network was used to construct the initial guess for RPFEM in strip rolling (Gudur and Dixit, 2008). It showed better ability of convergence compared with a traditional method which was not described in detail.

In this paper, firstly, the rigid plastic finite element model with the consideration of shear deformation work rate is employed to cut down the unknown variables of the system. Secondly, three different initial guesses are constructed by the Engineering Method (EM), the G Function Method (GF) and the Back Propagation Neural Network (BP), respectively. The Engineering Method proposed by Liu (1994) is based on the rolling boundary conditions and three assumptions. In G Function Method, the initial guess is calculated by linear equations deduced by the minimum of the G functional which is similar to the energy functional. Especially, the multi-layer perception (MLP) neural network with Back Propagation algorithm has been used to predict the initial guess. In order to obtain the huge amount of reliable learning datasets, a united model which can describe all the strip rolling conditions has been well established by analyzing the relations between the input rolling parameters and the outputs of neural network model. Finally, the performances such as CPU time, N–R iterations and accuracy of RPFEM have been investigated and discussed in the three different initial guesses. Then the advantages of each method for constructing the initial guess to the improvement of solution efficiency have been clarified clearly.

2.

Finite element formulation

The slightly compressible method in 2D rigid plastic FEM is employed in this paper. According to the variational principle, the real velocity field must minimize the following functional: 1  =  + + + = gm + 1 p

f

k



 k Vk dl ±

+ Lk





t

T1 v dl

¯ ε¯˙ dA + A

f |Vf | dl Lf

(1)

Lv

where the first term on the right side represents the work rate of plastic deformation (p ), ¯ the equivalent stress, ε¯˙ the equivalent strain rate and gm is the strain rate sensitivity index. The second term on the right side f represents the work rate of friction, Vf is the relative slip velocity at the interface between the workpiece and the roller,  f is the frictional shear stress. The third term on the right side k is the work rate of shear deformation on the section with the discontinuous velocity, here,  k is the shear yield stress, Vk is the relative slip velocity on the interface with discontinuous velocity. k can be replaced by adding extreme thin film (Jiang and Tieu, 2001). Here, in order to reduce the number of unknown variables and speed up the solution of RPFEM, the material before bitten is regarded as the rigid zone and it is neglected in the finite element model. Therefore, this simplification must consider the term k . The last term on the right side is the work rate of applied tension, T1 is the tension applied on the interface of the entry and the exit of the deformation zone, v represents the velocities, ‘−’ indicates the forward tension, and ‘+’ indicates the backward tension. The relative slip velocity Vf can be given by Vf = vx sec ˇ − VR

(2)

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Table 1 – Velocity boundary conditions (see Fig. 1) Position

vx vy vy vy

OA OC BC AB

where vx is the velocity component in the rolling direction, ˇ the angular position of the node on the surface between the roller and the workpiece (see Fig. 1), and VR is the tangential velocity of the roll. The frictional shear stress  f can be represented by the constant frictional shear stress model as

2 

 tan

−1

|Vf |

 (3)

Cv

where m is the friction factor,  s the yield stress, and Cv is a small constant. Then, the work rate of friction can be expressed as

  f = − Lf

Vf

0

2

ms √ 3



 tan−1

|Vf | Cv

 dVf dl

(4)

The equivalent stress for the slightly compressible material ¯ can be expressed as

¯ =



3  1  ·  + 2 2 ij ij g m

 (5)

where ij is the deviatoric stress tensor,  m the hydrostatic stress, and g is a small positive constant which indicates a small degree of compressibility. Generally, g = 0.0001–0.01. The equivalent strain rate for the slightly compressible material ε¯˙ can be expressed as ε¯˙ =



2 1 ε˙ · ε˙ + ε˙ 2 3 ij ij g v

 (6)

where ε˙ ij is the strain rate tensor and ε˙ v = ε˙ x + ε˙ y is the volumetric strain rate. Therefore, the stress tensor can be written as follows: ij =

¯ ε˙

2 3

ε˙ ij + ıij

1 g

−

2 9

  ε˙ v

(7)

where ıij is the Kronecker delta. Therefore the specific energy consumption  can be expressed as the function of the nodal velocity field. The modified Newton–Raphson iteration to solve the minimum of functional  can be written as ∇ 2 (v(k) )v + ∇(v(k) ) = 0. vk+1 = vk + vk

= vx1 =0 = 0, vx = vxn = −vx tan ˇ

where  is the relaxation factor, it can be searched by the golden method. There are two convergent criteria. The first one is according to the change of energy functional :

Fig. 1 – FE mesh used for fast simulation of strip rolling process.

ms f = √ 3

Velocity components

(8)

 ≤ εe 

(9)

The second is according to the updating of velocity field: ||v|| ≤ εv ||v||

(10)

where εe and εv are small positive constant, which are set to 1 × 10−5 The rolling force can be calculated by

 P=b

l

y dx

(11)

0

where b is the strip width, l the projected length of the contact arc on the rolling direction, and  y is the tensor stress along thickness direction of the rolled material.

3.

Simulation condition

3.1.

FE mesh

As the deformation in the width direction is very small compared with that along the rolling and the thickness directions, thus it can be simplified to 2D plane strain. Furthermore, the deformation is symmetric about the central plane (Fig. 1, OC), half of workpiece, as shown in Fig. 1, is studied. h0 and h1 are one-half of thickness of the workpiece before rolling and after rolling, respectively. R is the roller radius and ˛ is the bite angle. In order to reduce the number of variables in RPFEM, the regions of the workpiece before rolling and after rolling are treated as the rigid zone. Only the region (ABCO in Fig. 1) of workpiece under the contact surface is meshed. Therefore, the distribution for velocity component vy at the nodes on cross section OA (Fig. 1) is discontinuous. They are zero when the nodes belong to the rigid region, while they are unknown variables when the nodes are in the deformation region. Therefore, the work rate of shear deformation (k ) due to discontinuous velocity distribution is considered on cross section OA. Isoparametric quadrilateral elements with four Guass points are employed for meshing.

3.2.

Boundary conditions

The velocity boundary conditions are shown in Table 1. The velocity components vx on the boundary OA are considered to

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be the same and set to one variable. It is the same condition on the boundary BC. On the boundary AB, the velocity component vy can be expressed by vx . Thus it can be eliminated in the velocity variables. OC is the symmetric plane where the velocity component vy equals to zero.

where ˛ is the bite angle (see Fig. 1), f is the friction coefficient. Then one-half of the thickness at the central point can be calculated as

3.3.

The velocity components at the nodes can be calculated according to the above assumptions

hr = 2R(1 − cos ) + h1

Material model

The material flow model of carbon steel which has been used widely in the online control is as follows (Sun, 2002):

vxi =

hr vxr , hxi

vyij =

(20)

vxi (− tan ϕ)2yij

⎧    ε˙ gm  ε gn  ε  ⎪ ⎨ 2.8 exp 5.0 − 0.01 1.3 − , t0 ≥ td0 0.05  10  0.2  0.2  T5.0 [C] +0.01  = gm ε˙ ε gn ε ⎪ ⎩ 2.8g exp − 1.3 − , t0 < td0 Td

[C] + 0.05

10

0.2



 gm =

[C] + 0.49 [C] + 0.42

2

+

0.027 (0.081[C] − 0.154)T + (−0.019[C] − 0.207) + , [C] + 0.32

td0 = 950

[C] + 0.41 − 273 [C] + 0.32

+ 273 1000

t0

Td =

(13)

(−0.019[C] + 0.126)T + (0.075[C] − 0.05),

gn = 0.41 − 0.07[C]

T=

[C] + 0.06 [C] + 0.09

(12)

0.2

where g = 30.0([C] + 0.090) T − 0.95

where vxi is the velocity component along the x direction at the nodes of column i, vyij is the velocity component along the y direction at the nodes of column i and row j. Therefore, the velocity field can be calculated easily by the formula (21). t0 ≥ td0

(14)

t0 < td0

(15)

4.2.

G function method

(16)

(17)

G function method is one of the methods to establish a functional similar to the real energy functional  and is easy to be differentiated. A functional G is written as

 

td0 + 273

(18)

1000

(21)

hxi

where T is the temperature (◦ C), [C] the carbon content in the steel (wt.%); ε the true strain and ε˙ is the strain rate. This material model can be used under the conditions: 0.01% < [C] < 1.16%, 973 K < T < 1473 K, ε < 0.7, ε˙ ranges from 0.1 to 100 s−1 .

2 (¯ ε¯˙ ) dA +

G= A



 2

2

(f Vf ) dl + Lf

(k Vk ) dl

(22)

Lk

4.

Construction of initial guess

where the first term on the right side is the work rate of plastic deformation. The second term on the right side is the work rate of friction. Especially, the work rate of shear deformation on the cross section with the discontinuous velocity is expressed as the third term on the right side. To minimize the functional G with the velocity variables, sequentially, the linear equations of velocity variables can be obtained

4.1.

Engineering method

[KInt ]{v}i = {CInt }

According to the characteristics of strip rolling and rolling theory, the engineering method is a simple and efficient way to establish the initial guess based on the boundary conditions. It can be described as follows (Liu, 1994). Assumptions: (1) the velocity components vx at the nodes on the same cross section are set to the same value; (2) the distributions of velocity components vy along y direction are considered as linear; (3) the material volume passes through the arbitrary cross section per second is the same. According to the engineering method, the central angle for the central cross section where the velocity of the roller and the workpiece is same can be calculated as =

˛ 2



1−

˛ 2f

 (19)

(23)

where [KInt ] is the stiffness matrix, and it is unsymmetrical, {v}i the vector of velocity variables, and {CInt } represents the vector of external force. The solving of Eq. (23) can obtain the initial guess which meets well with the boundary conditions. However, the stiffness matrix is unsymmetrical. Thus the CPU time for solving Eq. (23) is very long. The algorithm to solve the large sparse matrix equations is used to calculate Eq. (23) in this paper.

4.3.

Back propagation neural network

4.3.1.

Analysis of BP neural network

Neural network has the good nonlinear mapping capability, which can establish the relationship between the independent variables and the dependent variables. The training of neural

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the reduction rate equals to (h0 − h1 )/h0 × 100%. Among these parameters, R/h1 , r and h1 determine the profile of the workpiece, gm and gn describe the material properties, m and VR express the boundary conditions. In order to reduce the number of input variables, the correlations between the input variables and the velocity field should be analyzed on the bases of statistical theory. The ranges of the input parameters for this analysis are as follows: R/h1 : 10–100; Fig. 2 – A diagram of MLP neural network.

gm : 0.01–0.41;

m : 0.1–0.9; gn : 0.01–0.41;

r : 5–55%; h1 : 5–35 mm;

VR : 1000–2000 mm/s

network need a huge amount of dataset and much time, but the simulation of trained neural network is fast and can be applied for the online control in strip rolling. RBF neural network has been trained from less than 24 datasets and used to set up the initial guess of RPFEM in strip rolling, it shows good ability of nonlinear mapping (Gudur and Dixit, 2008). However, RBF neural network is a local approaching method which has to determine the centers of the active function in the hidden layer. However, multi-layer perception back propagation algorithm is a global approaching method and has the good nonlinear mapping capacity. Compared with RBF neural network, the BP neural network with one layer of hidden neurons is simpler and has fewer parameters. A MLP neural network with one layer of neurons is shown in Fig. 2. It indicates that the neural network consists of three layers: input layer, hidden layer and output layer. The training of MLP neural network is to adjust the weights so that the network error between the network output and required output is minimized. The back propagation algorithm that performs a gradient descent minimization of the network error is an algorithm to update the weights. In order to improve the convergence and the stability of BP algorithm, the momentum method is used to update the learning rate. More details can be seen in Haykin’s work (2004). Here, linear function is used as the output activation function g(x), the hidden activation function f(x) is tanh function. The procedure for predicting the initial guess by the BP neural network can be seen in Fig. 3. The first step is the offline training of neural network. The second step is the online setup of the initial guess for FE simulation using trained neural network. The accuracy of the trained BP neural network is mainly dependent on the amount of training data. Also, the neural network has poor ability of extrapolation. Thus, huge amount of reliable training data should be obtained through FE simulation, and they should contain the rolling conditions as much as possible to increase its generality. The training and predicting efficiency of neural network is depended on the number of input parameters, output variables, and hidden neurons which is adjustable in the BP neural network. Therefore, in order to establish a high accurate and efficient neural network model, the input data should be analyzed firstly.

4.3.2.

Firstly, the influence of the roll speed VR on the velocity field is analyzed. In the case that other variables are fixed, the node velocity can be written as a function of the roll speed VR , as shown in Fig. 4. The velocities of three nodes are investigated, where Node 1 is at the inlet zone, Node 2 is at the exit zone and Node 3 is at the center under the contact zone. The linear function y = Bx is used to regress the curves of the node velocity with the roll speed. As a result, both the velocity components vx and vy show good linear relation with the roll speed. The errors of constant B in the linear function are ±0.00306 for vx and ±0.00270 for vy . Obviously, the relative velocities vx /VR or vy /VR for the nodes equals to B which is a constant when the other rolling parameters are fixed except the roll speed. Thus, if the relative velocity field vx /VR and vy /VR are defined as the output variables of the neural network, the roll speed can be removed from the input variables. The correlation analysis of the relations between input parameters and relative velocity field are carried out based on statistical theory. Each input parameter is taken at two levels—low and high. Totally, 27 = 128 dataset have been obtained from FEM code. The correlations between the input parameters and the relative velocity field at Nodes 1–3 are shown in Table 2. The parameter R/h1 presents significant cor-

Input data analysis

The velocity field can be expressed as a function of rolling parameters including: R/h1 , m, r, gm, gn, h1 and VR . r is

Fig. 3 – Procedure for predicting the initial guess by neural network.

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Fig. 4 – Regression curves of nodal velocities with roll speed: (a) vx ; (b) vy .

relation at the 0.01 level or at the 0.05 level for relative nodal velocities except that it indicates insignificant correlation for vx /VR at Node 1. The reduction rate r makes significant correlation at the 0.01 level for both vx /VR and vy /VR at all the three nodes. The fiction factor m presents insignificant correlation only for the vx /VR at Node 1 and vy /VR at Node 2, but it presents significant correlation at the 0.01 level or the 0.05 level for others. The correlation of strain rate sensitivity index gm is significant only at the 0.05 level for vy /VR at Node 1. The parameters gn, h1 and VR are found to be insignificant for the relative velocities at the three nodes. Therefore, gn, h1 and VR can be removed from the neural network model.

4.3.3.

Neural network modeling

Neural network model to predict the relative velocity field is established, where the input parameters are R/h1 , r, m and gm. Furthermore, some nodal velocities can be obtained easily from the boundary conditions, and then they are also eliminated from the output variables. The initial guess can be gotten by multiplying the relative nodal velocities to roll speed VR . Fig. 3 illustrates the procedure to offline train the neural network model and to online predict the initial guess for RPFEM. The procedure of offline training of the neural network model can be seen from the first part in Fig. 3. The software for training the BP neural network is developed with FORTRAN language and integrated with the RPFEM code of

strip rolling. Huge amounts of reliable training datasets can be obtained from the finite element simulation of strip rolling which the initial guess is constructed by EM or GF. Here, 10,000 datasets are calculated by randomly choosing and covered all the ranges of input parameters R/h1 , r, m and gm. The input parameters gn, h1 and VR are set to constants in the production of training datasets from RPFEM code. Subsequently, BP neural network with one hidden layer contains 10 hidden neurons is trained from the 10,000 datasets. Due to the input and output variables lie in different ranges, thus the normalization has been carried out for the variables so that their values lie in the range of −1 to 1. After training, the weights and the range of input and output variables are recorded for the online prediction of the initial guess. The procedure of online simulation of RPFEM code with the initial guess by the trained neural network (NN) model can be seen from the second part in Fig. 3. The parameters of the trained neural network model (weights and range of input and output variables) are read from the recorded file according to the mesh of FEM. Whereafter, with the input parameters R/h1 , r, m and gm, the relative velocity field is predicted by the NN model. After that, the initial guess is calculated by multiplying the relative velocity field by the roll speed VR . Subequently, the Newton–Raphson iteration begins from the initial guess. Finally, real velocity field, strain rate field and stress field can be calculated, and then rolling force, rolling torque and forward slip can be quickly obtained.

Table 2 – Correlations of input variables for relative velocity field Input variables

R/h1 r m gn gm h1 VR ∗ ∗∗

Node 1

Node 2

Node 3

vx /VR

vy /VR

vx /VR

vy /VR

vx /VR

vy /VR

−0.049 −0.979** 0.054 0.006 0.043 −0.002 0.018

0.690** −0.552** −0.120* 0.077 0.112* 0.013 0.011

0.115* 0.259** 0.309** −0.099 −0.001 −0.029 0.027

0.545** −0.718** −0.084 0.062 0.024 0.010 0.011

0.138* 0.642** 0.479** −0.027 0.071 −0.019 0.032

0.579** −0.673** −0.188** 0.056 0.036 0.007 0.007

Correlation is significant at the 0.05 level (two-detailed). Correlation is significant at the 0.01 level (two-detailed).

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Fig. 5 – Contour maps of initial guess and real velocity field—(1) initial guess by EM: (a) vx , (b) vy ; (2) initial guess by GF: (c) vx , (d) vy ; (3) initial guess by NN: (e) vx , (f) vy ; (4) real velocity field: (g) vx , (h) vy .

5.

Results and discussion

The RPFEM code of strip rolling has been developed using FORTRAN 90. In order to investigate the reliability and the solution efficiency of RPFEM when applying the three methods to construct the initial guess, the seven passes hot strip rolling of a Chinese steel factory has been analyzed by RPFEM. The RPFEM code was run on the PC with Intel Pentium 4 CPU (3.0 GHz) and with 1 GB RAM. The comparisons have been carried out between the initial guess and initial energy by EM, GF and NN and the real value by RPFEM. And the comparisons about the N–R iterations and CPU time for RPFEM solution by the adapting the three methods have been also performed.

Fig. 6 – Difference between initial energy and final energy.

5.1.

Comparison of initial guess and real velocity field

In general, for the iterative solution of RPFEM, the closer the initial guess to the real velocity field, the faster the convergence will be obtained. From the initial guess, the RPFEM begins Newton–Raphson iteration, and gets the convergence in a limited number of iterations. Although there are three different initial guesses, they must converge to the same real velocity field. There are very little differences among the convergent velocity fields of RPFEM iterated from the three initial guesses. Therefore, the convergent velocity field of RPFEM iterated from any of the above three initial guesses can represent the real velocity field. Fig. 5 illustrates the initial guesses obtained by the three methods and real velocity field obtained by RPFEM in the first pass of hot strip rolling. The real velocity field in Fig. 5 is the convergent velocity field of RPFEM iterated from the initial guess by EM. By comparing the results shown in Fig. 5(a) and (b) with that shown in Fig. 5(g) and (h), it can be seen that the initial guess obtained by EM is far away from real velocity field calculated by RPFEM. It is because of simplification and many assumptions. From the vx distribution obtained by GF (Fig. 5(c)), it indicates that the backward slip zone (the velocity at the surface is faster than that at the center) and the forward slip zone (the velocity at the surface is slower than that at the center) can be distinguished slightly. Thus the vx distribution of the initial guess obtained by GF is slightly close to the real velocity field. However, compared the initial guess obtained by GF and the real velocity field in Fig. 5, it is clear that the difference between them is obvious, especially in the inlet zone. It can be clearly seen from Fig. 5(e) and

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the initial energy obtained by NN is larger than that obtained by GF. This is due to the initial guess predicted by NN is near to the real velocity field. A functional G (formula (21)) is established by GF, which approaches to the energy functional  (formula (1)). Thus the minimum of functional G and the minimum of energy functional  are almost equivalent. Therefore, the initial guess by GF is easy to reach the energy convergent criteria, but it is a little hard to reach the velocity convergent criteria. With the initial guess predicted by NN, the iteration of RPFEM can reach the convergence criteria quickly both for energy functional and velocity convergent.

5.3. Fig. 7 – Change of energy with iteration step during solution of RPFEM in the first pass hot strip rolling.

(f) that the initial guess predicted by NN is very close to the real velocity field calculated by RPFEM both for vx distribution and vy distribution. The forward and backward slip (Fig. 5(e)) and the severe gradient vy distribution at inlet zone (Fig. 5(f)) due to shear deformation are accurately predicted by NN. As a result, according to the velocity convergent criteria (formula (10)), the RPFEM code with the initial guess by NN will get to the convergence swiftly through few steps of iteration.

5.2.

Comparison of initial energy and real energy

The solution of RPFEM is the minimum of energy functional. The convergence of RPFEM should meet both the velocity convergent criteria and the energy convergent criteria. From the initial guess, the initial energy can be calculated by the formula (1). Fig. 6 illustrates the comparison between initial energy obtained by EM, GF and NN and final energy calculated by RPFEM. It can be seen that the difference between the initial energy obtained by EM and the final energy calculated by RPFEM is very large, whereas the initial energy obtained by GF and NN is almost equal to the final values calculated by RPFEM. Fig. 7 illustrates the change of energy with iteration step during solution of RPFEM for the first pass hot strip rolling. It can be seen from Fig. 7 that the energy decreases greatly when the iteration step is less than 4, but it changes slightly in the following iteration steps. Moreover, from Fig. 7, it also can be seen that the energy of RPFEM for the initial guess predicted by NN is less than for that by GF after one step iterating while

Comparison of solution efficiency and accuracy

The N–R iterations and the CPU time of RPFEM code with the different initial guess obtained by the three methods have been obtained and listed in Table 3. In order to cut down the occasional error to obtain the CPU time of the RPFEM code, the RPFEM code has been run repeatedly for 100 times. The CPU time listed in Table 3 is the average of the one hundred times. Two types of mesh are applied to deal with the different rolling conditions in the seven passes hot strip rolling. From Table 3, it can be seen that the RPFEM code with initial guess predicted by NN reaches convergence quickly with less than 11 N–R iterations and less than 102 ms CPU time. Compared with that in the initial guesses obtained by EM and GF, the N–R iterations and the CPU time are greatly reduced in the initial guess obtained by NN. The N–R iterations by averaging the seven passes can be reduced by more than six steps (compared with that by EM) and by near five steps (compared with that by GF), respectively. And the CPU time by averaging the seven passes can be saved by more than 138 ms (compared with that by GF) and by more than 50 ms (compared with that by EM), respectively. Compared with that in initial guess obtained by EM, the RPFEM in the initial guess obtained by GF costs less N–R iterations, but it costs more CPU time. The phenomena mentioned above can be explained as follows. The initial guess and the initial energy predicted by NN almost equal to the final value calculated by RPFEM as discussed in Sections 5.1 and 5.2. Thus the iteration of the RPFEM code is very stable and fast to get the convergence. Moreover, there is little difference between the initial energy obtained by GF and the final value calculated by RPFEM. Thus compared with EM, the GF has the merit to improve the convergence abil-

Table 3 – N–R iterations and computational time of RPFEM code in three setup methods of initial guess Roll pass

Mesh

GF Iterations

1 2 3 4 5 6 7

16 × 8 16 × 8 24 × 6 24 × 6 16 × 8 24 × 6 16 × 8

14 15 12 16 14 11 15

EM Time (ms) 191 198 228 260 190 230 241

Iterations 15 16 14 14 15 12 22

NN Time (ms) 128 134 124 128 127 108 171

Iterations 9 9 7 11 8 9 10

Time (ms) 78 77 68 102 72 84 85

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Table 4 – Predicted rolling force by RPFEM and measured results Roll pass

1 2 3 4 5 6 7

Predicted rolling force (kN) EM

GF

19,405 20,465 15,437 16,205 12,850 12,456 6,922

19,678 20,532 16,285 16,870 12,876 12,436 6,780

Measured rolling force (kN)

NN 19,387 20,295 15,986 16,506 12,726 12,748 6,996

ity of RPFEM code. However, the GF must cost much additional time on solving the unsymmetrical matrix equations (formula (23)), thus the CPU time is more than that in the initial guess by EM as listed in Table 3. For the application of RPFEM on online control, both the computational time and the stability of iteration should be considered, especially, the computational time must less than 500 ms or less. In the present example, the computational time of one pass finite element simulation in the initial guess by NN is less than 102 ms and the iterations is less than 11, which is very fast and stable for the online application. However, with the increase of the number of elements in FEM, the number of outputs variables goes up. Then the neural network becomes more complicated, and it has to spend more additional time on predicting the initial guess. Therefore, in order to speed up the solution of RPFEM code, the setup of initial guess by NN is limited by the number of elements. In the case of initial guess by EM, the computational time changes from 108 ms to 171 ms and the iterations ranges from 14 to 22, it is fast but its iteration is not very stable for the online control. In the case of initial guess by GF, the computational time is a little long for the online control which ranges from 190 ms to 260 ms. Therefore, both for the computational time and the stability of iteration, the initial guess by NN is the optimal way for the online control of RPFEM. After the neural network has been trained according to a certain number of elements, it will cover all the range of strip rolling conditions and it is easy to construct the initial guess, thus it is acceptable and practical for online control. Due to the initial guess and initial energy by EM is far away from the real velocity field and final energy, the iteration of RPFEM is unstable to get the convergence, thus it is unreliable for online control. The predicted rolling force by RPFEM has been compared with the measured rolling force. For the seven passes hot strip rolling, the predicted rolling force and measured rolling force are given in Table 4. The data of measured rolling force were obtained from the rolling force measurement systems of the roll mills in the production line of hot strip rolling. It can be found in Table 4 that there are little differences among the predicted rolling forces calculated by RPFEM in the different initial guesses. The maximum error among them is 5.49%. The errors between predicted value and measured value are less than 4.83% for initial guess by NN and 6.78% for initial guess by GF and 3.30% for initial guess by EM, which are quite accurate for the online application. Therefore, the high accuracy of RPFEM is the most advantage for the online application when compared with traditional mathematical model. Furthermore,

19,454 19,824 15,251 16,032 12,895 12,310 6,701

Error (%) EM

GF

NN

0.25 3.23 1.22 1.08 0.35 1.18 3.30

1.15 3.57 6.78 5.23 0.14 1.02 1.18

0.34 2.38 4.82 3.78 1.31 3.01 4.40

the RPFEM has the great flexibility for dealing with the complicated rolling conditions and great suitability for various kinds of strip products.

6.

Conclusions

In this paper, three different initial guesses have been constructed successfully for RPFEM through Engineering Method, G Functional Method and Neural Network, respectively. Particularly, the back propagation neural network trained from 10,000 datasets of RPFEM results has been successfully used for predicting the initial guess in hot strip rolling. The relative velocity field (velocity field/roll speed) has been well established as the output variables of the neural network model. The correlations of input parameters and output variables have been analyzed using statistical method. The input parameters: R/h1 , r, m, gm with significant influence on relative velocity field are included in neural network model. And the insignificant input parameters: gn, h1 , VR are excluded from neural network model, thus the size of the model has been reduced remarkably. The comparisons of the N–R iterations and the CPU time of RPFEM code have been performed among the different initial guesses obtained by the three methods. Compared with Engineering Method and G Functional Method, the Neural Network has remarkable advantages to reduce both the N–R iterations and the CPU time of RPFEM code. It is because that the initial guess and initial energy obtained by the Neural Network is almost equivalent to the final values calculated by RPFEM code. Compared with Engineering Method, G Functional Method has the advantage to reduce the N–R iterations, but it has the disadvantage to increase CPU time. It is because that the initial energy obtained by G Functional Method is very close to final value calculated by RPFEM, but it has to spend additional CPU time on the solution of the unsymmetrical matrix equation. It is easy for the Engineering Method to construct the initial guess, while it is unstable due to the large number of N–R iterations. In the case of G Functional Method, the iteration of RPFEM is stable with less number of N–R iterations, but the computational time is very long. For the method of Neural Network, after the neural network model has been trained completely, the initial guess for RPFEM can be constructed easily and quickly, it is the most practical and stable method for online control due to very less number of N–R iterations and very short computational time among the three methods.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1816–1825

Moreover, the rigid plastic finite element model of strip rolling has been well established and simplified by the consideration of shear deformation work rate. The RPFEM code for predicting the rolling force of hot strip rolling has been developed using FORTRAN language. The predicted results have been compared with the measured values, and it shows high accuracy. From the numerical results such as predicted rolling force, N–R iterations and CPU time, it is found that the RPFEM code has high stability, fast solution and high accuracy in the initial guess predicted by Neural Network, which can meet the requirement of online control in strip rolling.

Acknowledgement The authors gratefully acknowledge the financial support from the key project of National Natural Science Foundation of China (Grant No. 50534020).

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