# A strategy to control microstructures of a Ni-based superalloy during hot forging based on particle swarm optimization algorithm

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## Abstract

In this study, a strategy based on the particle swarm optimization (PSO) algorithm is developed to control the microstructures of a Ni-based superalloy during hot forging. This strategy is composed of three parts, namely, material models, optimality criterions, and a PSO algorithm. The material models are utilized to predict microstructure information, such as recrystallization volume fraction and average grain size. The optimality criterion can be determined by the designed target microstructures and random errors. The developed strategy is resolved using the PSO algorithm, which is an intelligent optimal algorithm. This algorithm does not need a derivable objective function, which renders it suitable for dealing with the complex hot forging process of alloy components. The optimal processing parameters (deformation temperature and strain rate) are obtained by the developed strategy and validated by the hot forging experiments. Uniform and fine target microstructures can be obtained using the optimized processing parameters, which indicates that the developed strategy is effective for controlling the microstructural evolution during the hot forging of the studied superalloy.

## Keywords

Processing parameters Microstructure Particle swarm optimization (PSO) algorithm Superalloy## List of symbols

- \(v_{i}\)
Velocity of the

*i*th particle- \(x_{i}\)
Position of the

*i*th particle- \(w\)
Inertia weight

- \(p_{i}\)
Best previous positions of the

*i*th particle- \(p_{\text{g}}\)
Best previous positions of all particles

- \(c_{1}\), \(c_{2}\)
Constants to determine the weights of \(p_{i}\) and \(p_{\text{g}}\), respectively

- \(r_{1}\), \(r_{2}\)
Random values uniformly distributed in the range of [0, 1]

- \(t\)
Current iteration of algorithm

- \(t_{{\mathrm{max}}}\)
Maximum number of iterations

- \(w_{{\mathrm{max}} }\), \(w_{{\mathrm{min}} }\)
Upper and lower bounds of inertia weight, respectively

- \(\sigma_{\text{sat}}\)
Saturation stresses

- \(\sigma_{\text{ss}}\)
Steady stresses

- \(\sigma_{\text{rec}}\)
Flow stress when dynamic recovery is the dominant softening mechanism

- \(\varepsilon\)
True strain

- \(\sigma\)
True stress

- \(\dot{\varepsilon }\)
Strain rate (s

^{−1})- \(\varepsilon_{\text{c}}\)
Critical strain for initiating dynamic recrystallization

- \(\varepsilon_{\text{p}}\)
Peak strain

- \(\sigma_{\text{p}}\)
Peak stress

- \(\sigma_{0}\)
Yield stress

- \(\varOmega\)
Coefficient of dynamic recovery

- \(K_{\text{d}}\)
Material constant

- \(Z\)
Zener-Hollomon parameter

- \(R\)
Universal gas constant

- \(T\)
Deformation temperature (K)

- \(\rho\)
Material density (kg/m

^{3})- \(C_{\text{v}}\)
Specific heat \(\left( {{\text{J}}\cdot\left( {{\text{kg}}\cdot {^\circ{\text{C}}}} \right)^{ - 1} } \right)\)

- \(\eta (\dot{\varepsilon })\)
Adiabatic correction factor

- \(f\)
Recrystallization volume fraction

- \(g_{ 0}\)
Initial grain size

- \(g_{\text{drx}}\)
Recrystallized grain size

- \(g\)
Average grain size

- \(\varepsilon_{ 0. 5}\)
Strain for 50% dynamic recrystallization volume fraction

- \(J_{\text{f}}\)
Optimality criterion for recrystallization volume fraction

- \(J_{\text{g}}\)
Optimality criterion for average grain size

- \(J_{\text{r}}\)
Optimality criterion for random error

- \(f_{\text{des}}\)
Designed recrystallization volume fraction

- \(g_{\text{des}}\)
Designed average grain size

- \(r\)
Random value in the range of [0, 1]

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 51775564), and the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2016JJ1017).

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