THE METHOD FOR OPTIMAL CONTROL OF FORGING

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					                                 International Symposium on Computer Science & Technology 2009



     THE METHOD FOR OPTIMAL CONTROL OF FORGING
THERMOMECHANICAL PARAMETERS BASED ON FEM AND OPTIMAL
                  CONTROL THEORY

                            Jianpeng Feng 1      Zijian Luo 2
   1.R&D Center for High-Tech of Forging and Casting,Northwestern Polytechnical University
                                  Xi’an 710072, P.R.China
       2.Ningbo Dahongying Polytechnics of Software, Ningbo 315175, P.R.China
   Abstract: The present study put forward a new method for optimal control of the thermomechanical parameters
during hot plastic-working processes of advanced materials, based on finite element method and modern optimal
control theory. The proposed method can be described as follows. First, the optimal trajectories of thermomechanical
parameters were defined from the grain size evolution model and the stable regions of thermomechanical parameters.
The stable regions were determined by combining the artificial neural networks (ANN) with the dissipative structure
theory. Second, the finite element models were transferred to suitable state-space models. Third, the optimal profile for
the process parameters can be fixed based on the state-space models and linear quadratic regulator (LQR) theory in
order that the thermomechanical parameters of selected locations within the forging are conformable to the optimal
trajectories and physical constraints. Applying the proposed method to an upsetting process of IN718 alloy, ram
velocity profile was determined to obtain high quality forgings. The proposed method laid the theoretical foundation for
open loop control of forging processes for difficult to deformation materials.
   Keywords: Optimal control, Forging thermomechanical parameters, ANN, Dissipative structure theory, IN718 alloy

1. Introduction                                                proposed the use of the state space model and optimal
   Advanced materials such as titanium alloys, wrought         control theory for optimal control of forging processes,
superalloys and composite materials are widely used for        being a pioneering work with regard to the transition of
aerospace components working at elevated temperature,          forging technology for difficult to deformation materials
because of their excellent properties. In order to maintain    to intelligent manufacturing.
consistency in properties and microstructures for the             IN718 alloy is a precipitation strengthened nicked-iron
forgings of the mention materials, their process window        based superalloy widely used in advanced aeroengine.
is usually very narrow. Therefore, these materials are         Because the matrix for IN718 alloy is alloyed austenite,
called difficult to deformation materials. With the rapid      the austenite grain size (called grain size for short later
advance in high technology, the lot for forgings of            on) is an important index for characterizing the
difficult to deformation materials becomes smaller and         microstructures and properties of IN718 forgings. In
smaller, and the forgings must be delivered in time.           present study, the isothermal upsetting process of IN718
Therefore, it is an inevitable tendency for the forging        alloy on press was chosen as an object of research and
technology of difficult to deformation materials               the ram velocity was selected as the control variable.
becoming the intelligent manufacturing.                        First, stable regions of IN718 were defined by combining
   During the past 3 decades, the application of finite        the DMM with artificial neural networks. Then
element method to the field of plastic working of metals       introducing the stable regions into grain size evolution
has been progressing so rapidly that determining the           model, the optimal trajectories of thermomechanical
distribution of thermomechanical parameters and their          parameters were determined by using the interactive
history within the billet presents no problem for hot          optimal control software developed by the authors. Last,
deformation processes. Up to date, the use of the results      the finite element model was transformed into
of finite element simulation to promote the forging            state-space model and the state equations were obtained.
technology of difficult to deformation materials being         According to the state equations and the optimal
intelligent manufacturing has become the frontier in the       trajectories, the ram velocity profile can be determined
plastic working field. The dynamical materials model           by using linear quadratic regulator (LQR).
(DMM) was developed by Gegel et al. on the basis of the
second law of thermodynamics [1], for determining the          2. Stable regions and grain size evolution equations
stable regions of thermomechanical parameters (called          for IN718 alloy
stable regions for short later on) in order to control the     2.1 Stable regions of thermomechanical parameters
forging quality. By using the dissipative structure theory        Because the forging process consisting of workpieces,
and artificial neural networks, DMM was improved by            dies and equipment can be regarded as a
the authors and the stable regions for several wrought         thermomechanical system far away from equilibrium, the
superalloys were defined [2]. Grandhi and Malas et al. [3]     dissipative structure theory can be used to define the
                                       International Symposium on Computer Science & Technology 2009

stable regions to ensure the forgings being consistency in           specimen for retention of specific glass lubricant.
microstructures and properties.                                         Isothermal constant strain rate compression tests were
   The criteria for the stable region have been developed            conducted      using      hot      working       simulator,
as follows:                                                          Thermecmastor-Z. The specimens after compression test
  m                                                                 were cut in half along longitudinal direction by wire
        0     0  m 1                               (1)
      
  ln                                                               spark erosion. The Quant(1)imet-500 image analyzer was
                                                                     employed to measure the grain size of specimens after
 S                                                                  etching.
       0         S 1                                      (2)         According to the experimental data, the stable regions
     
 ln                                                                were defined. The stable region for effective strain rate
where     m   ln          
                              ,is effective strain rate
                                                                     and effective strain under constant temperature was
               
                       ln   T ,                                 shown in figure 1.
                                     
sensitivity of flow stress (  );  is effective strain rate;
      is    effective     strain; T is temperature; S=
 (ln  )  (1 T ) T  ,   ,is temperature sensitivity of
flow stress.
   While using the dissipative structure theory to define
the stable regions of thermomechanical parameters, the
constitutive relationship used as the dynamic equation
for the dissipative structure plays the key role in ensuring
the stable regions conformable to practice. At present
time, the widely used methods to establish the
constitutive relationships are linear regression methods
based on Arrhenius-type equations. According to the
authors’ experiences, the constitutive relationships
established by above methods were too complex to be                                                    
                                                                        Fig.1. The stable region for  and  (T=980℃)
taken as the dynamics equations for dissipative structures.
The back propagation network (BP) can be looked on as                2.2 State-space model for grain size evolution
high non-linear map from input to output, thus BP is                    The relationship between grain size and
suitable to representation of the high non-linear                    thermomechanical parameters can be determined by
constitutive relationships for difficult to deformation              using non-linear regression programs as follows:
materials. However, the constitutive relationships
formulated by using BP are implicit, therefore, the                                                                     
                                                                     D  exp 0.002(ln Z )1.1185  1.208( ) 0.5954  2.395   (3)
criteria (Eq. (1) and (2)) should be solved numerical
differentiation to define the stable regions.                                      
                                                                     where Z   exp(Q RT ) , is Zener-Hollom parameter;
Table 1 Monthly quasi orders of tobacco manufacture                  R  8.314J / mol  K , is gas constant; Q is activation
factory                                                              energy, kJ, determined by following equation:
                               Start   Due            Equipment
    Orders    Quantity                        Brand
                               Time    Time             Type
     W1         1150             1      10     B1        E1                                
                                                                     Q  1.5973exp(5.5531 )  406.57                       (4)
     W2         170              1       2     B2        E1
     W3        23000             1      30     B3        E2             The state-space models for grain size evolution can be
     W4         750              1       2     B4        E3
     W5         2500             1       5     B5        E2
                                                                     derived in accordance with Eq.(3) and (4) as follows
     W6         6380             1      11     B6        E3             
                                                                       D   f (T ,  ,  ) 
                                                                                     
     W7         1300             2      13     B7        E1                              
     W8         4200             2      13     B8        E3                                                        (5)
                                                                      T   
                                                                                            
                                                                          Cp 
     W9         6160             5      18     B9        E2
     W10        160             10      11     B10       E1
     W11        1270            11      21     B11       E1                               
                                                                     where D and T are the time rates of grain size D and
                                                                     temperature T;  is a coefficient that determines how
     W12        4000            11      18     B12       E3
     W13        1000            13      21     B13       E1
     W14        400             13      14     B14       E3          much of the mechanical work is converted into heat and
     W15        6380            14      30     B15       E3          contributes to the increase in temperature of the billet;
     W16
     W17
                4000
                5400
                                18
                                18
                                        30
                                        27
                                               B16
                                               B17
                                                         E2
                                                         E3
                                                                       is the density of workpiece and Cp is the specific
     W18        1150            21      30     B18       E1                                               
                                                                     heat. It is necessary to note that   u is the system
     W19        1020            21      30     B19       E1
                                                                     input in the present case.
     W20        2240            27      30     B20       E3
   Cylindrical specimens of 8mm diameter and 12 mm                   3. The optimal trajectories for the thermomechanical
height were machined from the rod. Shallow grooves of                parameters
0.2mm depth were machined on both ends of the                           In order to maintain the thermomechanical parameters
                                                International Symposium on Computer Science & Technology 2009

in the stable regions and minimize grain size within the                         It is well known that finite element methods can
specified region of the billet, regarding the stable regions                  provide detailed information with regard to the
defined by using the dissipative structure theory and                         distribution of thermomechanical parameters and their
artificial neural networks as physical constraints, the                       history within the billet. However, finite element models
performance index can be formulated as follows:                               have a large number of degrees of freedom, which causes
                                                                              difficulty in integrating finite element method with
         tf
                                                                            optimal control theory to maintain the thermomechanical
J            (1D 2 (t )   2 F ( ,  ))dt                        (6)
         0                                                                    parameters within the specified regions of billet at
                                                                              desired level. Therefore, the finite element models
where  1 and  2 are weight factors; D is grain size;                        should be transformed into suitable state-space models in
                                                                            order to control the thermomechanical parameters of the
 min and  max are the upper boundary and lower                              specified elements by using the results of finite element
                                
boundary of stable region for  , respectively;. F is the                     simulation.
constraint function to constrain the effective strain rate                       The finite element models can be generally expressed
                                                                            as follow [7]:
within  min and  max . The constraint function for
present study is as following:                                                KV  F                                                   (8)

                                    
F  1 ( ) exp(1 ( ) 2   2 ( )   3 ( ))   2 ( )        (7)      where K is the stiffness matrix, being function of V ;
                                                                               F is the load vector; and V is the nodal velocity vector.
where 1 , 2 , 1 ,  2 , 3 are coefficients in terms of                       If V and V  V represent the nodal velocity vectors
effective strain  . In a word, the discussed matter has                      for time t and t  t , respectively, Eq.(8) can be
become an optimal control problem, comprised of Eq. (5)                       linearized by Taylor expansion of Eq.(8) around V as
and Eq.(6). In order to solve the problem, the authors                        following:
developed an interactive program by integrating several
widely used optimal control algorithms into a package. It                     K s V  K t V  F                                       (9)
is very convenient for users to select the suitable
algorithm to solve their problems and alternate the                           where K s and K t are called secant stiffness and
algorithm during computation. The optimal trajectories                        tangent stiffness matrix, respectively
of thermomechanical parameters for IN718 alloy were                              In order to eliminate the unnecessary degrees of
shown in figure 2.                                                            freedom for facilitating the establishment of state-space
                                                                              model, the velocity vectors V were divided into five
                                                                              kinds, namely, Vt and Vn are the tangential and normal
                                                                              components of nodal velocities at the die-billet interface,
                                                                              respectively; Vc are the nodal velocities at the
                                                                              symmetric boundaries; Vi are the node velocities of the
                                                                              elements concerned and Vb are the nodal velocities of
                                                                              the rest elements. Suppose the isothermal upsetting
                                                                              process was conducted on press and the ram velocity,
                                                                              V d , was selected as control variable. From the linearized
                                                                              finite element model, classification of nodal velocities
                                                                              and boundary conditions, the state-space model can be
                                                                              derived from Eq. (9) as follows:

                                                                              dVs
                                                                                   A v Vs  B vVd  Wv                               (10)
                                                                               dt

                                                                              where    Vs  [ Vt Vi ]T , the state variables vector; A v is
                                                                              the plant matrix; B v is the input matrix; Wv is the
                                                                              constant perturbation vector.
                                                                              4.2 Optimal control method
Fig.2. Optimal trajectories of strain and strain rate for
                                                                                 For the case in which an isothermal upsetting process
minimizing the grain size for IN718 alloy
                                                                              was performed on press, the ram velocity was selected as
                 (a)  vs t        
                               (b)  vs t                                     the control variable and the strain rate within the
4. Optimal control of forging thermomechanical                                specified regions was kept at the desired values, the
parameters                                                                    performance index can be formulated as follows:
4.1 State-space model for forging process
                                                  International Symposium on Computer Science & Technology 2009

         t i  t
J      ti
                                                  
                    (( (t )   d (t ))T Q( (t )   d (t ))

                                               Vd (t )RVd (t ))dt
                                                  T
                                                                      (11)

       
where  (t ) is the strain rate of the elements concerned
                                             
over the time interval t  t  t  t ;  (t ) is the
                                          i           i          d

desired strain rate over the time interval t i  t  t i  t ;
Q, R are the state and control weight matrix, respectively,
which have to be semi-definite and positive-definite,
respectively.
   In light of optimal control theory, the objective for the
optimal control of thermomechanical parameters was to                                          Fig.4.   Strain rate profile
define the optimal trajectories for control variables
through minimizing the performance shown as Eq.(11)
and taking Eq.(10) as the constrain.

5. Application
   Suppose a cylindrical billet of IN718 alloy was
isothermal upsetting on press. The starting billet has a
radius of 40 mm and height of 200 mm. The finite
element mesh with 169 elements is shown in Fig.3. The
friction factor between the die and billet interface is
chosen as 0.30. The die, workpiece and environment
temperatures are all 980℃. The initial ram velocity was                                        Fig.5. Ram velocity profile
selected as 120 mm/s in order that the nominal strain rate                         The distributions of grain size and effective strain
within the specified element is 1.5s-1. The black element                       were shown in Fig.6. and Fig.7. for optimal control case
in fig. 3 was the element concerned.                                            and constant ram velocity case, respectively. In view of
                                                                                the data presented in these two figures, the grain size of
                                                                                the billet for optimal control case was improved while
                                                                                the distribution of effective strain makes not much
                                                                                difference for the two cases.




                Fig.3. Finite element mesh
    Fig.2 indicates that the optimal trajectory for strain
rate varied slightly. For accelerating the computation
procedure, the strain rate of the element concerned was
kept at 1.5 s-1during the process.
   The effective strain rate profile and the ram velocity
profile that determined by using the proposed method
were compared with those for constant ram velocity in
Fig. 4 and Fig.5, respectively. In these figures, the solid
line stands for the optimal control case and the doted line
represents the constant ram velocity case. For the case of                        Fig.6.   The contour of grain size within billet(50%)
constant ram velocity 120 mm/s, the strain rate varies                                     (a) optimal control of ram velocity ;
dramatically, and is up to 3.9 s-1 when reaching the end                                         (b) constant ram velocity
of stroke (Fig.4). It can be seen from Fig.5 that the strain
rate of the element concerned is kept at the values of
1.5s-1 for the optimal control case.
                                 International Symposium on Computer Science & Technology 2009

                                                                     Element Method”, Int.J.Numer.Methods Eng.,.
                                                                     36(No.12), 1993: 1967-1986
                                                               [4]   J.C.Malas, W.G.Frazier, S.Venugopal, et al.,
                                                                     “ Optimization of Microstructure Development
                                                                     during Hot Working Using Control Theory ” ,
                                                                     Metallurgical and Materials Trans. A, 28A(9),
                                                                     1997:1921-1930
                                                               [5]   W.G.Frazier, J.C.Malas, E.A.Medina et al.,
                                                                     “Application of Control Theory Principle to
                                                                     Optimization of Grain Size during Hot Extrusion,”
                                                                     Materials Science and Technology, 14(1), 1998:
                                                                     25-31
                                                               [6]   Feng Jianpeng, Zhang Maicang, Luo Zijian et al.,
                                                                     “Using Artificial Neural Network to Characterize
                                                                     Constitutive Relationship for Wrought Superalloys”,
                                                                     Forging & StampingTech.,23(1), 1998:31-35 (in
                                                                     Chinese)
                                                               [7]   S.Kobayashi, S.I.Oh and T.Altan, Metal Forming
                                                                     and the Finite Element Method, Oxford University
                                                                     Press, New York, 1989
                                                               [8]   B.D.O.Anderson and J.B.Moore, Optimal Control—
                                                                     Linear Quadratic Methods, Prentice-Hall, N.J., 1990
Fig.7. The contour of effective strain within billet (50%)
          (a) Optimal control of ram velocity;
               (b) Constant ram velocity

6. Conclusion
   (1) Stable regions can be determined by combining the
artificial neural network with dissipative structure theory.
   (2) The optimal trajectories of thermomechanical
para-meters can be defined by using appropriate optimal
control algorithm based on grain size evolution model
and stable regions of thermomechanical parameters to
guarantee the consistency in properties and
microstructures for forgings of difficult to deformation
materials.
   (3) The profile for control variables can be obtained by
combining the state-space models with the desired
optimal trajectories of thermomechanical parameters.
   (4) The proposed method to control the
thermomechanical parameters for obtaining the desired
grain size laid the theoretical foundation for open loop of
forging processes for difficult to deformation materials.
Acknowledgments
  The authors wish to thank the National Natural
Science Foundation of China for contract 59875071
under which the present work was possible.
References
[1] H.L.Gegel, Synthesis of Atmoistics and Continuum
    to Describe Microstructure, Computer Simulation in
    Material Science, Metals Park, OH: ASM,1987
[2] Feng Jianpeng, Constitutive Relationships and their
    Application to wrought Superalloys, Master thesis,
    Northwestern Polytechnical University, 1997.3 (in
    Chinese)
[3] R.V.Grandhi, A.Kumar, A.Chaudhary, J.C.Malas,
    “State-Space Representation and Optimal Control of
    Non-Linear Material deformation Using the Finite