Metals and Materials International

, Volume 25, Issue 5, pp 1246–1257 | Cite as

A Particle Swarm Optimization-Based Multi-level Processing Parameters Optimization Method for Controlling Microstructures of an Aged Superalloy During Isothermal Forging

  • Dong-Dong Chen
  • Y. C. LinEmail author


To obtain the designed target microstructures of an aged superalloy during isothermal forging, a multi-level processing parameters optimization method is developed based on particle swarm optimization (PSO) algorithm. In the developed method, the accurate material models are used to characterize the microstructural evolution. Based on the designed target microstructures, the global and local optimality criterions are constructed to alternately optimize global and local multi-level processing parameters by the PSO algorithm with a linear decreasing inertia weight strategy. The optimized initial volume fraction of δ phase (δVF), deformation temperature and strain rate are 12.95%, 1000 °C and 0.001 s−1, respectively. According to these optimized parameters, the recrystallization volume fraction, average grain size and δVF are 100%, 11.2 µm and 2.1%, respectively, which well agree with the designed targets. Additionally, the processing parameters optimized by the developed method and traditional processing maps are compared. It is found that the developed method is more effective to control microstructures for the studied superalloy.


Microstructure control Multi-level optimization Particle swarm optimization Superalloy 

List of Symbols


Recrystallization volume fraction


Initial grain size


Recrystallized grain size


Average grain size


Volume fraction of δ phase

\(\varepsilon_{ 0. 5}\)

Strain for 50% dynamic recrystallization volume fraction


True strain

\(\dot{\varepsilon }\)

Strain rate (s−1)


Critical strain for initiating dynamic recrystallization


Peak strain


Zener–Hollomon parameter


Universal gas constant


Deformation temperature (K)


Volume fraction of δ phase


Dissolution fraction of δ phase

kc, k, m, Q, Ah, mh, Qh, nd, Ld, Ad, kd and n

Material constants


Designed recrystallization volume fraction


Designed average grain size


Designed volume fraction of δ phase

α, β and γ

Weight coefficients


Local optimality criterion


Global optimality criterion


Number of local solutions

fmin and fmax

Minimum and maximum values of recrystallization volume fraction

gmin and gmax

Minimum and maximum values of average grain size

Vmin and Vmax

Minimum and maximum values of volume fraction of δ phase


Velocity of the ith particle


Position of the ith particle


Inertia weight


Best previous positions of the ith particle


Best previous positions of all particles

c1 and c2

Constants to determine the weights of pi and pg, respectively

r1 and r2

Random values uniformly distributed in the range of [0, 1]


Current iteration of algorithm


Maximum number of iterations

wmax and wmin

Upper and lower bounds of inertia weight, respectively



This work was supported by the National Natural Science Foundation Council of China (Grant No. 51775564), the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2016 JJ1017), and Program of Chang Jiang Scholars of Ministry of Education (Grant No. Q2015140), China.


  1. 1.
    Y.C. Lin, X.M. Chen, A critical review of experimental results and constitutive descriptions for metals and alloys in hot working. Mater. Des. 32, 1733–1759 (2011)CrossRefGoogle Scholar
  2. 2.
    M.S. Chen, W.Q. Yuan, H.B. Li, Z.H. Zou, New insights on the relationship between flow stress softening and dynamic recrystallization behavior of magnesium alloy AZ31B. Mater. Charact. 147, 173–183 (2019)CrossRefGoogle Scholar
  3. 3.
    H. Kim, D. Kim, K. Ahn, D. Yoo, H.S. Son, G.S. Kim, K. Chung, Inverse characterization method for mechanical properties of strain/strain-rate/temperature/temperature-history dependent steel sheets and its application for hot press forming. Met. Mater. Int. 21, 874–890 (2015)CrossRefGoogle Scholar
  4. 4.
    Q.W. Wang, Y.C. Lin, X.G. Liu, Y.Q. Jiang, X.Y. Zhang, D.D. Chen, C. Chen, K.C. Zhou, Precipitation behavior of a β-quenched Ti–5Al–5Mo–5V–1Cr–1Fe alloy during high-temperature compression. Mater. Charact. 151, 358–367 (2019)CrossRefGoogle Scholar
  5. 5.
    D. Samantaray, S. Mandal, A.K. Bhaduri, A critical comparison of various data processing methods in simple uni-axial compression testing. Mater. Des. 32, 2797–2802 (2011)CrossRefGoogle Scholar
  6. 6.
    D. Samantaray, S. Mandal, M. Jayalakshmi, C.N. Athreya, A.K. Bhaduri, V.S. Sarma, New insights into the relationship between dynamic softening phenomena and efficiency of hot working domains of a nitrogen enhanced 316L(N) stainless steel. Mater. Sci. Eng. A 598, 368–375 (2014)CrossRefGoogle Scholar
  7. 7.
    M.S. Ghazani, B. Eghbali, G. Ebrahimi, Kinetics and critical conditions for initiation of dynamic recrystallization during hot compression deformation of AISI 321 austenitic stainless steel. Met. Mater. Int. 23, 964–973 (2017)CrossRefGoogle Scholar
  8. 8.
    L.X. Li, L.Y. Zheng, B. Ye, Z.Q. Tong, Metadynamic and static recrystallization softening behavior of a bainite steel. Met. Mater. Int. 24, 60–66 (2018)CrossRefGoogle Scholar
  9. 9.
    M. Rezayat, M.H. Parsa, H. Mirzadeh, J.M. Cabrera, Dynamic deformation response of Al–Mg and Al–Mg/B4C composite at elevated temperatures. Mater. Sci. Eng. A 712, 645–654 (2018)CrossRefGoogle Scholar
  10. 10.
    H. Mirzadeh, M. Roostaei, M.H. Parsa, R. Mahmudi, Rate controlling mechanisms during hot deformation of Mg–3Gd–1Zn magnesium alloy: dislocation glide and climb, dynamic recrystallization, and mechanical twinning. Mater. Des. 68, 228–231 (2015)CrossRefGoogle Scholar
  11. 11.
    P. Zhou, Q.X. Ma, Dynamic recrystallization behavior and constitutive modeling of as-cast 30Cr2Ni4MoV steel based on flow curves. Met. Mater. Int. 23, 359–368 (2017)CrossRefGoogle Scholar
  12. 12.
    Y.Q. Ning, B.C. Xie, C. Zhou, H.Q. Liang, M.W. Fu, Strain-rate sensitivity of powder metallurgy superalloys associated with steady-state drx during hot compression process. Met. Mater. Int. 23, 350–358 (2017)CrossRefGoogle Scholar
  13. 13.
    M. Zhang, G.Q. Liu, H. Wang, B.F. Hu, Modeling of thermal deformation behavior near γ′ solvus in a Ni-based powder metallurgy superalloy. Comput. Mater. Sci. 156, 241–245 (2019)CrossRefGoogle Scholar
  14. 14.
    A. He, G. Xie, X.Y. Yang, X.T. Wang, H.L. Zhang, A physically-based constitutive model for a nitrogen alloyed ultralow carbon stainless steel. Comput. Mater. Sci. 98, 64–69 (2015)CrossRefGoogle Scholar
  15. 15.
    X.M. Chen, Y.C. Lin, D.X. Wen, J.L. Zhang, M. He, Dynamic recrystallization behavior of a typical nickel-based superalloy during hot deformation. Mater. Des. 57, 568–577 (2014)CrossRefGoogle Scholar
  16. 16.
    F. Chen, Z.S. Cui, H.G. Ou, H. Long, Mesoscale modeling and simulation of microstructure evolution during dynamic recrystallization of a Ni-based superalloy. Appl. Phys. A 122, 890 (2016)CrossRefGoogle Scholar
  17. 17.
    F. Chen, H. Wang, H. Zhu, H. Zhu, F. Ren, Z.S. Cui, High-temperature deformation mechanisms and physical-based constitutive modeling of ultra-supercritical rotor steel. J. Manuf. Process. 38, 223–234 (2019)CrossRefGoogle Scholar
  18. 18.
    G.Z. Quan, G.C. Luo, J.T. Liang, D.S. Wu, A. Mao, Q. Liu, Modelling for the dynamic recrystallization evolution of Ti–6Al–4V alloy in two-phase temperature range and a wide strain rate range. Comput. Mater. Sci. 97, 136–147 (2015)CrossRefGoogle Scholar
  19. 19.
    Y.C. Lin, X.M. Chen, M.S. Chen, Y. Zhou, D.X. Wen, D.G. He, A new method to predict the metadynamic recrystallization behavior in a typical nickel-based superalloy. Appl. Phys. A 122, 1–14 (2016)Google Scholar
  20. 20.
    Y.C. Lin, Y.X. Liu, M.S. Chen, M.H. Huang, X. Ma, Z.L. Long, Study of static recrystallization behavior in hot deformed Ni-based superalloy using cellular automaton model. Mater. Des. 99, 107–114 (2016)CrossRefGoogle Scholar
  21. 21.
    Y.V. Prasad, H.L. Gegel, S.M. Doraivelu, J.C. Malas, J.T. Morgan, K.A. Lark, D.R. Barker, Modeling of dynamic material behavior in hot deformation: forging of Ti-6242. Metall. Mater. Trans. A 15, 1883–1892 (1984)CrossRefGoogle Scholar
  22. 22.
    B.J. Jang, H.S. Park, M.S. Kim, High temperature deformation behavior of Al–Zn–Mg-based new alloy using a dynamic material model. Met. Mater. Int. 24, 1249–1255 (2018)CrossRefGoogle Scholar
  23. 23.
    Y.C. Lin, G. Liu, Effects of strain on the workability of a high strength low alloy steel in hot compression. Mater. Sci. Eng. A 523, 139–144 (2009)CrossRefGoogle Scholar
  24. 24.
    A. Jenab, A.K. Taheri, Experimental investigation of the hot deformation behavior of AA7075: development and comparison of flow localization parameter and dynamic material model processing maps. Int. J. Mech. Sci. 78, 97–105 (2014)CrossRefGoogle Scholar
  25. 25.
    D.X. Wen, Y.C. Lin, H.B. Li, X.M. Chen, J. Deng, L.T. Li, Hot deformation behavior and processing map of a typical Ni-based superalloy. Mater. Sci. Eng. A 591, 183–192 (2014)CrossRefGoogle Scholar
  26. 26.
    D.X. Wen, Y.C. Lin, J. Chen, J. Deng, X.M. Chen, J.L. Zhang, M. He, Effects of initial aging time on processing map and microstructures of a nickel-based superalloy. Mater. Sci. Eng. A 620, 319–332 (2015)CrossRefGoogle Scholar
  27. 27.
    A. Momeni, K. Dehghani, G.R. Ebrahimi, S. Kazemi, Developing the processing maps using the hyperbolic sine constitutive equation. Metall. Mater. Trans. A 44, 5567–5576 (2013)CrossRefGoogle Scholar
  28. 28.
    S.M. Abbasi, A. Momeni, Hot working behavior of Fe–29Ni–17Co analyzed by mechanical testing and processing map. Mater. Sci. Eng. A 552, 330–335 (2012)CrossRefGoogle Scholar
  29. 29.
    X.Y. Shu, S.Q. Lu, K.L. Wang, G.F. Li, Optimization of hot working parameters of as-forged Nitinol 60 shape memory alloy using processing maps. Met. Mater. Int. 21, 726–733 (2015)CrossRefGoogle Scholar
  30. 30.
    C.R. Anoop, A. Prakash, S.K. Giri, S.V.S. Narayana Murty, I. Samajdar, Optimization of hot workability and microstructure control in a 12Cr–10Ni precipitation hardenable stainless steel: an approach using processing maps. Mater. Charact. 141, 97–107 (2018)CrossRefGoogle Scholar
  31. 31.
    F.C. Ren, F. Chen, J. Chen, X.Y. Tang, Hot deformation behavior and processing maps of AISI 420 martensitic stainless steel. J. Manuf. Process. 31, 640–649 (2018)CrossRefGoogle Scholar
  32. 32.
    A. Rudra, S. Das, R. Dasgupta, Constitutive modeling for hot deformation behavior of Al-5083 + SiC composite. J. Mater. Eng. Perform. 28, 87–99 (2019)CrossRefGoogle Scholar
  33. 33.
    M. Hu, L.M. Dong, Z.Q. Zhang, X.F. Lei, R. Yang, Y.H. Sha, A novel computational method of processing map for Ti–6Al–4V alloy and corresponding microstructure study. Materials 11, 1599 (2018)CrossRefGoogle Scholar
  34. 34.
    W.G. Frazier, J.C. Malas, E.A. Medina, S. Medeiros, S. Venugopal, W.M. Mullins, A. Chaudhary, R.D. Irwin, Application of control theory principles to optimization of grain size during hot extrusion. Mater. Sci. Techol. Lond. 14, 25–31 (1998)CrossRefGoogle Scholar
  35. 35.
    J.C. Malas, W.G. Frazier, S. Venugopal, E.A. Medina, S. Medeiros, R. Srinivasan, R.D. Irwin, W.M. Mullins, A. Chaudhary, Optimization of microstructure development during hot working using control theory. Metall. Mater. Trans. A 28, 1921–1930 (1997)CrossRefGoogle Scholar
  36. 36.
    S. Venugopal, E.A. Medina, J.C. Malas, S. Medeiros, W.G. Frazier, W.M. Mullins, R. Srinivasan, Optimization of microstructure during deformation processing using control theory principles. Scr. Mater. 36, 347–353 (1997)CrossRefGoogle Scholar
  37. 37.
    D. Recker, M. Franzke, G. Hirt, Fast models for online-optimization during open die forging. CIRP Ann. Manuf. Technol. 60, 295–298 (2011)CrossRefGoogle Scholar
  38. 38.
    Y.C. Lin, D.D. Chen, M.S. Chen, X.M. Chen, J. Li, A precise BP neural network-based online model predictive control strategy for die forging hydraulic press machine. Neural Comput. Appl. 29, 585–596 (2018)CrossRefGoogle Scholar
  39. 39.
    D.G. He, Y.C. Lin, J. Chen, D.D. Chen, J. Huang, Y. Tang, M.S. Chen, Microstructural evolution and support vector regression model for an aged Ni-based superalloy during two-stage hot forming with stepped strain rates. Mater. Des. 154, 51–62 (2018)CrossRefGoogle Scholar
  40. 40.
    Y.C. Lin, F.Q. Nong, X.M. Chen, D.D. Chen, M.S. Chen, Microstructural evolution and constitutive models to predict hot deformation behaviors of a nickel-based superalloy. Vacuum 137, 104–114 (2017)CrossRefGoogle Scholar
  41. 41.
    N. DuyTrinh, Y. Shaohui, N.N. Tan, P.X. Son, L.A. Duc, A new method for online monitoring when grinding Ti–6Al–4V alloy. Mater. Manufact. Process. 34, 39–53 (2019)CrossRefGoogle Scholar
  42. 42.
    G.Z. Quan, Z. Zou, H. Wen, S. Pu, W. Lv, A characterization of hot flow behaviors involving different softening mechanisms by ANN for as-forged Ti–10V–2Fe–3Al alloy. High Temp. Mater. Proc. 34, 651–665 (2015)Google Scholar
  43. 43.
    A. Forcellese, F. Gabrielli, M. Simoncini, Prediction of flow curves and forming limit curves of Mg alloy thin sheets using ANN-based models. Comput. Mater. Sci. 50, 3184–3197 (2011)CrossRefGoogle Scholar
  44. 44.
    A. Jenab, I.S. Sarraf, D.E. Green, T. Rahmaan, M.J. Worswick, The use of genetic algorithm and neural network to predict rate-dependent tensile flow behaviour of AA5182-O sheets. Mater. Des. 94, 262–273 (2016)CrossRefGoogle Scholar
  45. 45.
    D.D. Chen, Y.C. Lin, Y. Zhou, M.S. Chen, D.X. Wen, Dislocation substructures evolution and an adaptive-network-based fuzzy inference system model for constitutive behavior of a Ni-based superalloy during hot deformation. J. Alloys Compd. 708, 938–946 (2017)CrossRefGoogle Scholar
  46. 46.
    L.Y. Wang, L. Li, Z.H. Zhang, Accurate descriptions of hot flow behaviors across β transus of Ti–6Al–4V alloy by intelligence algorithm GA-SVR. J. Mater. Eng. Perform. 25, 3912–3923 (2016)CrossRefGoogle Scholar
  47. 47.
    Y.C. Lin, F. Wu, Q.W. Wang, D.D. Chen, S.K. Singh, Microstructural evolution of a Ni–Fe–Cr-base superalloy during non-isothermal two-stage hot deformation. Vacuum 151, 283–293 (2018)CrossRefGoogle Scholar
  48. 48.
    A. Seret, C. Moussa, M. Bernacki, N. Bozzolo, On the coupling between recrystallization and precipitation following hot deformation in a γ-γ′ nickel-based superalloy. Metall. Mater. Trans. A 49, 4199–4213 (2018)CrossRefGoogle Scholar
  49. 49.
    S.P. Zhu, Q. Liu, J. Zhou, Z.Y. Yu, Fatigue reliability assessment of turbine discs under multi-source uncertainties. Fatigue Fract. Eng. M 41, 1291–1305 (2018)CrossRefGoogle Scholar
  50. 50.
    S.P. Zhu, Y. Liu, Q. Liu, Z.Y. Yu, Strain energy gradient-based LCF life prediction of turbine discs using critical distance concept. Int. J. Fatigue 113, 33–42 (2018)CrossRefGoogle Scholar
  51. 51.
    M.S. Chen, K.K. Li, Y.C. Lin, W.Q. Yuan, An improved kinetics model to describe dynamic recrystallization behavior under inconstant deformation conditions. J. Mater. Res. 31, 2994–3003 (2016)CrossRefGoogle Scholar
  52. 52.
    Y.C. Lin, X.Y. Wu, X.M. Chen, J. Chen, D.X. Wen, J.L. Zhang, L.T. Li, EBSD study of a hot deformed nickel-based superalloy. J. Alloys Compd. 640, 101–113 (2015)CrossRefGoogle Scholar
  53. 53.
    M. Zhang, G.Q. Liu, B.F. Hu, Effect of microstructure instability on hot plasticity during thermomechanical processing in PM Nickel-based superalloy. Acta Metall. Sin. 53, 1469–1477 (2017)Google Scholar
  54. 54.
    A. Loyda, L.A. Reyes, G.M. Hernández-Muñoz, F.A. García-Castillo, P. Zambrano-Robledo, Influence of the incremental deformation during rotary forging on the microstructure behaviour of a nickel-based superalloy. Int. J. Adv. Manuf. Technol. 97, 2383–2396 (2018)CrossRefGoogle Scholar
  55. 55.
    D.X. Wen, Y.C. Lin, J. Chen, X.M. Chen, J.L. Zhang, Y.J. Liang, L.T. Li, Work-hardening behaviors of typical solution-treated and aged Ni-based superalloys during hot deformation. J. Alloys Compd. 618, 372–379 (2015)CrossRefGoogle Scholar
  56. 56.
    D.G. He, Y.C. Lin, M.S. Chen, J. Chen, D.X. Wen, X.M. Chen, Effect of pre-treatment on hot deformation behavior and processing map of an aged nickel-based superalloy. J. Alloys Compd. 649, 1075–1084 (2015)CrossRefGoogle Scholar
  57. 57.
    Y.C. Lin, D.G. He, M.S. Chen, X.M. Chen, C.Y. Zhao, X. Ma, Z.L. Long, EBSD analysis of evolution of dynamic recrystallization grains and δ phase in a nickel-based superalloy during hot compressive deformation. Mater. Des. 97, 13–24 (2016)CrossRefGoogle Scholar
  58. 58.
    M. Irani, M. Joun, Determination of JMAK dynamic recrystallization parameters through FEM optimization techniques. Comput. Mater. Sci. 2018(142), 178–184 (2018)CrossRefGoogle Scholar
  59. 59.
    M.S. Chen, W.Q. Yuan, H.B. Li, Z.H. Zou, Modeling and simulation of dynamic recrystallization behaviors of magnesium alloy AZ31B using cellular automaton method. Comput. Mater. Sci. 136, 163–172 (2017)CrossRefGoogle Scholar
  60. 60.
    D.X. Wen, Y.C. Lin, Y. Zhou, A new dynamic recrystallization kinetics model for a Nb containing Ni–Fe–Cr-base superalloy considering influences of initial δ phase. Vacuum 141, 316–327 (2017)CrossRefGoogle Scholar
  61. 61.
    D.X. Wen, Microstructural formation mechanism and processing planning method of GH4169 superalloy during die forging, Ph.D. dissertation Central South University 2017Google Scholar
  62. 62.
    D.X. Wen, Y.C. Lin, X.H. Li, S.K. Singh, Hot deformation characteristics and dislocation substructure evolution of a nickel-base alloy considering effects of δ phase. J. Alloys Compd. 764, 1008–1020 (2018)CrossRefGoogle Scholar
  63. 63.
    A. Nickabadi, M.M. Ebadzadeh, R. Safabakhsh, A novel particle swarm optimization algorithm with adaptive inertia weight. Appl. Soft Comput. 11, 3658–3670 (2011)CrossRefGoogle Scholar

Copyright information

© The Korean Institute of Metals and Materials 2019

Authors and Affiliations

  1. 1.School of Mechanical and Electrical EngineeringCentral South UniversityChangshaChina
  2. 2.State Key Laboratory of High Performance Complex ManufacturingChangshaChina

Personalised recommendations