A method for the optimal control of forging process variables using the finite element method and control theory

doi:10.1016/S0924-0136(00)00654-3

Journal of Materials Processing Technology

Volume 108, Issue 1, 1 December 2000, Pages 40-44

https://doi.org/10.1016/S0924-0136(00)00654-3 Get rights and content

Abstract

In this paper, a new method is advanced for optimal control of the thermomechanical parameters during hot plastic-working processes of advanced materials, based on the finite element method and modern optimal control theory. The proposed method can be described as follows. First, the optimal trajectories of the thermomechanical parameters were defined from a grain size evolution model and the stable regions of the thermomechanical parameters. The stable regions were determined by combining 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 was 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 conform to the optimal trajectories and physical constraints. Applying the proposed method to an upsetting process of IN718 alloy, the ram velocity profile was determined to obtain high quality forgings. The proposed method lays the theoretical foundation for the open-loop control of forging processes for difficult-to-deform materials.

Introduction

Advanced materials such as titanium alloys, wrought superalloys and composite materials are used widely for aerospace components working at elevated temperature, because of their excellent properties. In order to maintain consistency in properties and microstructures for the forgings of the above-mentioned materials, their process window is usually very narrow. Therefore, these materials are called difficult-to-deform materials. With the rapid advance in high technology, the lot size for forgings of difficult-to-deform materials becomes smaller and smaller, and the forgings must be delivered in time. Therefore, there is an inevitable tendency for the forging technology of difficult-to-deform materials to become intelligent manufacturing.

During the past three decades, the application of the finite element method to the field of the plastic working of metals has been progressing so rapidly that determining the distribution of thermomechanical parameters and their history within the billet presents no problem for hot deformation processes. To date the use of the results of finite element simulation to promote the forging technology of difficult-to-deform materials into intelligent manufacturing has become the frontier in the plastic-working field. A dynamic materials model (DMM) was developed by Gegel et al. on the basis of the second law of thermodynamics for determining the stable regions of thermomechanical parameters (called stable regions for convenience hereafter) in order to control forging quality [1]. By using the dissipative structure theory and artificial neural networks (ANN), DMM was improved by the authors and the stable regions for several wrought superalloys were defined [2]. Grandhi et al. [3], Malas and co-workers [4], [5] proposed the use of the state–space model and optimal control theory for optimal control of forging processes, being a pioneering work with regard to the transition of forging technology for difficult-to-deform materials to intelligent manufacturing.

IN718 alloy is a precipitation strengthened nickel–iron based superalloy used widely in advanced aeroengines. Because the matrix for IN718 alloy is alloyed austenite, the austenite grain size (called the grain size for convenience hereafter) is an important index for characterizing the microstructures and properties of IN718 forgings. In the present study, the isothermal upsetting process of IN718 alloy in a press was chosen as the object of research and the ram velocity was selected as the control variable. First, stable regions of IN718 alloy were defined by combining the DMM with ANN. Then, introducing the stable regions onto grain size evolution model, the optimal trajectories of the thermomechanical parameters were determined using interactive optimal control software developed by the present authors. According to the state equations and the optimal trajectories, the ram velocity profile was determined using a linear quadratic regulator (LQR) [6].

Section snippets

Stable regions of thermomechanical parameters

Because the forging process consists of workpieces, dies and equipment, it can be regarded as a thermomechanical system far away from equilibrium and the dissipative structure theory can be used to define the stable regions to ensure that the forgings are consistent in microstructures and properties.

The criteria for the stable region have been developed as follows [7]: $∂m ∂ ln ε ̄ ̇ ≤0, 0≤m≤1$ $∂S ∂ ln ε ̄ ̇ ≤0, S≥1$ where $m=[∂ ln σ ̄ /∂ ln ε ̄ ̇]_{T,ε̄}$ is the strain-rate sensitivity of the flow stress $(σ ̄)$ ; $ε ̄$ the

The optimal trajectories for the thermomechanical parameters

In order to maintain the thermomechanical parameters in the stable regions and minimize the grain size within the specified region of the billet, regarding the stable regions defined by using the dissipative structure theory and ANN as physical constraints, the performance index can be formulated as follows: $J= ∫ 0 t_{f} (α_{1} D^{2} (t)+α_{2} F(ε ̄, ε ̄ ̇)) d t$ where α₁ and α₂ are the weight factors; $ε ̄ ̇_{min}$ and $ε ̄ ̇_{max}$ are the lower boundary and upper boundary of the stable region for $ε ̄ ̇$ , respectively, and F is the

State–space model for forging process

It is well known that the finite element method can provide detailed information with regard to the distribution of the thermomechanical parameters and their history within the billet. However, finite element models have a large number of degrees of freedom, which causes difficulty in integrating the finite element method with optimal control theory to maintain the thermomechanical parameters within the specified regions of the billet at the desired level. Therefore, the finite element models

Application

Suppose a cylindrical billet of IN718 alloy is subjected to isothermal upsetting on a press. The initial billet has a radius of 40 mm and a height of 200 mm. The finite element mesh, with 169 elements, is shown in Fig. 3, in which the shaded element is the element concerned. The friction factor at the die and billet interface is chosen as 0.30. The die, workpiece and environment temperatures are all 980°C. The initial ram velocity is selected as 120 mm/s in order that the nominal strain rate

Conclusions

1.
The stable regions can be determined by combining the ANN with dissipative structure theory.
2.
The optimal trajectories of thermomechanical parameters can be defined by using an appropriate optimal control algorithm based on a grain size evolution model and stable regions of thermomechanical parameters to ensure consistency in the properties and microstructures of forgings of difficult-to-deform materials.
3.
The profile for control variables can be obtained by combining the state–space models with

Acknowledgements

The authors wish to thank the National Natural Science Foundation of China for contract 59875071 under which the present work was made possible.

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