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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

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