Full-field approaches for austenite-ferrite phase transformation simulations
Mariusz Wermiński
*, Mateusz Sitko, Łukasz Madej
AGH University of Krakow, al. A. Mickiewicza 30, 30-059 Krakow, Poland
*corresponding author
DOI:
https://doi.org/10.7494/cmms.2025.4.1029
Abstract:
Understanding the local evolution of phase transformations in steels, particularly the γ (austenite) → α (ferrite) transformation, is crucial for controlling the microstructure and properties of steel components. Over recent decades, significant progress has been made in the numerical modeling of this complex phenomenon. This development has been driven by both scientific curiosity and industrial needs, especially in processes such as hot rolling, forging, thermal treatment, etc. The developed models have evolved from simple solutions based on local equilibrium to more complex approaches that consider local heterogeneities. Modern computational approaches, such as Phase-Field (PF), Level-Set (LS), Cellular Automata (CA), Monte Carlo (MC) or Vertex based simulations, allow for the precise reproduction of microstructural evolution considering local instabilities. They also enable the analysis of phase boundary motion in an explicit manner. These techniques also allow for direct integration with thermodynamic data and mechanical models, thereby better capturing the physical mechanisms of phase transformations, such as chemical composition, diffusion resistance, or the influence of deformation. An overview of the state of the art in this area is presented within the paper. The model’s concepts, assumptions, fundamental equations, advantages, limitations, and potential practical applications are summarized. Special attention is given to modeling the γ → α transformation by the Cellular Automata method. The importance of incorporating phenomena such as diffusion, nucleation, and growth is emphasized. The need for consistency between experimental results and simulations is also highlighted.
Cite as:
Wermiński, M., Sitko, M., & Madej, Ł. (2025). Full-field approaches for austenite-ferrite phase transformation simulations. Computer Methods in Materials Science, 25(4), – . https://doi.org/10.7494/cmms.2025.4.1029
Article (PDF):
Accepted Manuscript – final pdf version coming soon
Keywords:
Phase transformations, Full-field models, Austenite, Ferrite, Cellular automata
Publication dates:
Received: 22.09.2025, Accepted: 20.10.2025, Published: xx.11.2025
Publication type:
Review paper
References:
Barbe, F., & Quey, R. (2011). A numerical modelling of 3D polycrystal-to-polycrystal diffusive phase transformations involving crystal plasticity. International Journal of Plasticity, 27(6), 823–840. https://doi.org/10.1016/j.ijplas.2010.09.008
Bengochea, R., López, B., Gutierrez, I. (1998). Microstructural evolution during the austenite-to-ferrite transformation from deformed austenite. Metallurgical and Materials Transactions A, 29(2), 417–426. https://doi.org/10.1007/s11661-998-0122-1
Bernacki, M. (2024a). Kinetic equations and level-set approach for simulating solid-state microstructure evolutions at the mesoscopic scale: State of the art, limitations, and prospects. Progress in Materials Science, 142, 101224. https://doi.org/10.1016/j.pmatsci.2023.101224
Bernacki, M. (2024b). Level-set method for the modeling of microstructure evolution. In M. Bernacki, S. Forest (Coordinators), Digital Materials: Continuum Numerical Methods at the Mesoscopic Scale (pp. 203–260). Wiley. https://doi.org/10.1002/9781394332489.ch6
Bhadeshia, H. K. D. H. (2010). Phase transformations contributing to the properties of modern steels. Bulletin of the Polish Academy of Sciences: Technical Sciences, 58(2), 255–265. https://doi.org/10.2478/v10175-010-0024-4
Bhadeshia, H. K. D. H. (2021). Theory of Transformations in Steels. CRC Press.
Biglari, M., & Mittemeijer, E. J. (2013). Simulation of the massive austenite–ferrite transformation under uniaxial loading. Computational Materials Science, 77, 214–223. https://doi.org/10.1016/j.commatsci.2013.04.041
Biglari, M., & Mittemeijer, E. J. (2014). Mobility of the austenite–ferrite interface under various states of loading. Philosophical Magazine, 94(8), 801–813. https://doi.org/10.1080/14786435.2013.868941
Bos, C., Sommer, F., & Mittemeijer, E. J. (2006). Multi-lattice kinetic Monte Carlo simulation of interphase kinetics for an iron fcc to bcc transformation. Modelling and Simulation in Materials Science and Engineering, 14(2), 273–282. https://doi.org/10.1088/0965-0393/14/2/010
Bos, C., Mecozzi, M. G., & Sietsma, J. (2010). A microstructure model for recrystallisation and phase transformation during the dual-phase steel annealing cycle. Computational Materials Science, 48(3), 692–699. https://doi.org/10.1016/j.commatsci.2010.03.010
Bzowski, K., Rauch, Ł., Pietrzyk, M., Kwiecień, M., & Muszka, K. (2021). Numerical modeling of phase transformations in dual-phase steels using level set and SSRVE approaches. Materials, 14(18), 5363. https://doi.org/10.3390/ma14185363
Cha, P.-R., Kim, J.-Y., Kim, W.-T., & Kim, S. G. (2010). Phase field study on the austenite/ferrite transition in low carbon steel. Materials and Manufacturing Processes, 25(1–3), 106–110. https://doi.org/10.1080/10426910903153166
Chafia, Z., Yvonnet, J., Bleyer, J., Vincent, S., & El Ouafa, S. (2024). Massively parallel phase field fracture simulations on supercomputers: Towards multi-billion degree-of-freedom computations. Advanced Modeling and Simulation in Engineering Sciences, 11(1), 25. https://doi.org/10.1186/s40323-024-00280-4
Chandrappa, N. (2024). A Global Level-Set Based Full-Field Numerical Framework to Simulate Diffusive Solid-State Phase Transformation in Polycrystalline Metallic Materials [PhD thesis, Université Paris Sciences et Lettres]. ParisTech. https://pastel.hal.science/tel-04702550
Chandrappa, N., & Bernacki, M. (2023). A level-set formulation to simulate diffusive solid/solid phase transformation in polycrystalline metallic materials – Application to austenite decomposition in steels. Computational Materials Science, 216, 111840. https://doi.org/10.1016/j.commatsci.2022.111840
Chandrappa, N., & Bernacki, M. (2024). High-fidelity level-set modeling of diffusive solid-state phase transformations for polycrystalline materials. Computational Materials Science, 243, 113142. https://doi.org/10.1016/j.commatsci.2024.113142
Cheloee Darabi, A., Rastgordani, S., Khoshbin, M., Guski, V., & Schmauder, S. (2023). Hybrid data-driven deep learning framework for material mechanical properties prediction with the focus on dual-phase steel microstructures. Materials, 16(1), 447. https://doi.org/10.3390/ma16010447
Chen, H., & Zwaag, S., van der (2014). A general mixed-mode model for the austenite-to-ferrite transformation kinetics in Fe–C–M alloys. Acta Materialia, 72, 1–12. https://doi.org/10.1016/j.actamat.2014.03.034
Chen, H., Liu, Y., Yan, Z., Li, Y., & Zhang, L. (2010). Consideration of the growth mode in isochronal austenite-ferrite transformation of ultra-low-carbon Fe–C alloy. Applied Physics A, 98(1), 211–217. https://doi.org/10.1007/s00339-009-5375-z
Chen, L. (2002). Phase-Field Models for Microstructure Evolution. Annual Review of Materials Research, 32(1), 113–140. https://doi.org/10.1146/annurev.matsci.32.112001.132041
Chen, L., & Zhao, Y. (2022). From classical thermodynamics to phase-field method. Progress in Materials Science, 124, 100868. https://doi.org/10.1016/j.pmatsci.2021.100868
Deffrennes, G., Terayama, K., Abe, T., Ogamino, E., & Tamura, R. (2023). A framework to predict binary liquidus by combining machine learning and CALPHAD assessments. Materials & Design, 232, 112111. https://doi.org/10.1016/j.matdes.2023.112111
Enomoto, M., Kamiya, M., & Nagai, T. (1996). Computer simulation of two-dimensional grain growth with anisotropic grain boundary energy and mobility by vertex model. Materials Science Forum, 204–206, 71–82. https://doi.org/10.4028/www.scientific.net/MSF.204-206.71
Ervik, Å., Lervåg, K. Y., & Munkejord, S. T. (2014). A robust method for calculating interface curvature and normal vectors using an extracted local level set. Journal of Computational Physics, 257(A), 259–277. https://doi.org/10.1016/j.jcp.2013.09.053
Fuchizaki K., & Kawasaki K. (1995). Time evolution of three-dimensional cellular systems: Computer modeling based on vertex-type models. Physica A: Statistical Mechanics and Its Applications, 221(1–3), 202–215. https://doi.org/10.1016/0378-4371(95)00224-U
Furuhara, T., Kikumoto, K., Saito, H., Sekine, T., Ogawa, T., Morito, S., & Maki, T. (2008). Phase transformation from fine-grained austenite. ISIJ International, 48(8), 1038–1045. https://doi.org/10.2355/isijinternational.48.1038
Gamsjäger, E., Militzer, M., Fazeli, F., Svoboda, J., & Fischer, F. D. (2006). Interface mobility in case of the austenite-to-ferrite phase transformation. Computational Materials Science, 37(1–2), 94–100. https://doi.org/10.1016/j.commatsci.2005.12.011
Gaude-Fugarolas, D., & Bhadeshia, H. K. D. H. (2003). A model for austenitisation of hypoeutectoid steels. Journal of Materials Science, 38(6), 1195–1201. https://doi.org/10.1023/A:1022805719924
Gawąd , J., Paszyński, M., Matuszyk, P., & Madej, L. (2008). Cellular automata coupled with HP-adaptive Finite Element Method applied to simulation of austenite-ferrite phase transformation with a moving interface, In M. Pietrzyk et al. (Eds.), Metal Forming 2008. Proceedings of the 12th International Conference on Metal Forming, Akademia Górniczo-Hutnicza, Kraków, Poland (vol. 2, pp. 579–586). Verlag Stahleisen.
Ghosh, S., Newman, C. K., & Francois, M. M. (2022). Tusas: A fully implicit parallel approach for coupled phase-field equations. Journal of Computational Physics, 448, 110734. https://doi.org/10.1016/j.jcp.2021.110734
Ghumman, U. F., Fang, L., Wagner, G. J., & Chen, W. (2023). Calibration of Cellular Automaton Model for Microstructure Prediction in Additive Manufacturing Using Dissimilarity Score. Journal of Manufacturing Science and Engineering, 145(6), 061002. https://doi.org/10.1115/1.4056690
Gomez, H., Bures, M., & Moure, A. (2019). A review on computational modelling of phase-transition problems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377(2143), 20180203. https://doi.org/10.1098/rsta.2018.0203
Gouné, M., Danoix, F., Ågren, J., Bréchet, Y., Hutchinson, C. R., Militzer, M., Purdy, G., Zwaag, S., van der, & Zurob, H. (2015). Overview of the current issues in austenite to ferrite transformation and the role of migrating interfaces therein for low alloyed steels. Materials Science and Engineering: R: Reports, 92, 1–38. https://doi.org/10.1016/j.mser.2015.03.001
Hackenberg, R. E. (2012). 1 – The historical development of phase transformations understanding in ferrous alloys. In E. Pereloma, D. V. Edmonds (Eds.), Phase Transformations in Steels (vol. 1: Fundamentals and Diffusion-Controlled Transformations, pp. 3–55). Woodhead Publishing. https://doi.org/10.1533/9780857096104.1.3
Hallberg, H. (2011). Approaches to modeling of recrystallization. Metals, 1(1), 16–48. https://doi.org/10.3390/met1010016
Hallberg, H., Håkansson, P., & Ristinmaa, M. (2010). Thermo-mechanically coupled model of diffusionless phase transformation in austenitic steel. International Journal of Solids and Structures, 47(11–12), 1580–1591. https://doi.org/10.1016/j.ijsolstr.2010.02.019
Hansen, P. B. (1993). Parallel cellular automata: A model program for computational science. Concurrency: Practice and Experience, 5(5), 425–448. https://doi.org/10.1002/cpe.4330050504
Hillert, M. (1999). Solute drag, solute trapping and diffusional dissipation of Gibbs energy. Acta Materialia, 47(18), 4481–4505. https://doi.org/10.1016/S1359-6454(99)00336-5
Hodgson, P. D., Beladi, H., & Kelly, G. L. (2005). The development of ultrafine grained steels through thermomechanical processing. Canadian Metallurgical Quarterly, 44(2), 179–186. https://doi.org/10.1179/cmq.2005.44.2.179
Hu, Y., Wang, K., & Spatschek, R. (2024). A data-driven strategy for phase field nucleation modeling. Npj Materials Degradation, 8(1), 109. https://doi.org/10.1038/s41529-024-00529-8
Huang, C.-J., Browne, D. J., & McFadden, S. (2006). A phase-field simulation of austenite to ferrite transformation kinetics in low carbon steels. Acta Materialia, 54(1), 11–21. https://doi.org/10.1016/j.actamat.2005.08.033
Iwamoto, T., Cherkaoui, M., & Busso, E. P. (2008). A finite element-based level-set method of an interface motion driven by a diffusion field: Application to a phase transformation problem. Computational Materials Science, 44(2), 792–801. https://doi.org/10.1016/j.commatsci.2008.05.030
Jacobsson, E., Hallberg, H., Hektor, J., Iyengar, S., & Ristinmaa, M. (2024). A level set approach to modelling diffusional phase transformations under finite strains with application to the formation of Cu 6 Sn 5. Computational Materials Science, 244, 113284. https://doi.org/10.1016/j.commatsci.2024.113284
Jacot, A., & Rappaz, M. (1997). A two-dimensional diffusion model for the prediction of phase transformations: Application to austenitization and homogenization of hypoeutectoid Fe-C steels. Acta Materialia, 45(2), 575–585. https://doi.org/10.1016/S1359-6454(96)00199-1
Kawasaki, K., Nagai, T., & Nakashima, K. (1989). Vertex models for two-dimensional grain growth. Philosophical Magazine B, 60(3), 399–421. https://doi.org/10.1080/13642818908205916
Khedkar, K., Mamaghani, A. C., Ghysels, P., Patankar, N. A., & Bhalla, A. P. S. (2025). Preventing mass loss in the standard level set method: New insights from variational analyses. Journal of Computational Physics, 520, 113495. https://doi.org/10.1016/j.jcp.2024.113495
Kobayashi, R. (1993). Modeling and numerical simulations of dendritic crystal growth. Physica D: Nonlinear Phenomena, 63(3–4), 410–423. https://doi.org/10.1016/0167-2789(93)90120-P
Krielaart, G. P., Sietsma, J., & Zwaag, S., van der (1997). Ferrite formation in Fe-C alloys during austenite decomposition under non-equilibrium interface conditions. Materials Science and Engineering: A, 237(2), 216–223. https://doi.org/10.1016/S0921-5093(97)00365-1
Kumar, M., Sasikumar, R., & Kesavan Nair, P. (1998). Competition between nucleation and early growth of ferrite from austenite—Studies using cellular automaton simulations. Acta Materialia, 46(17), 6291–6303. https://doi.org/10.1016/S1359-6454(98)00243-2
Kundu, S., Dutta, M., Ganguly, S., & Chandra, S. (2004). Prediction of phase transformation and microstructure in steel using cellular automaton technique. Scripta Materialia, 50(6), 891–895. https://doi.org/10.1016/j.scriptamat.2003.12.007
Łach, Ł., & Svyetlichnyy, D. (2020). Development of hybrid model for modeling of diffusion phase transformation. Engineering Computations, 37(8), 2761–2783. https://doi.org/10.1108/EC-02-2019-0059
Łach, Ł., & Svyetlichnyy, D. (2023). 3D model of heat flow during diffusional phase transformations. Materials, 16(13), 4865. https://doi.org/10.3390/ma16134865
Lan, Y. J., Li, D. Z., Huang, C. J., & Li, Y. Y. (2004a). A cellular automaton model for austenite to ferrite transformation in carbon steel under non-equilibrium interface conditions. Modelling and Simulation in Materials Science and Engineering, 12(4), 719–729. https://doi.org/10.1088/0965-0393/12/4/012
Lan, Y. J., Li, D. Z., & Li, Y. Y. (2004b). Modeling austenite–ferrite transformation in low carbon steel using the cellular automaton method. Journal of Materials Research, 19(10), 2877–2886. https://doi.org/10.1557/JMR.2004.0397
Lan, Y. J., Li, D. Z., & Li, Y. Y. (2005a). Mesoscale simulation of ferrite transformation from deformed austenite during continuous cooling in a C–Mn steel using a cellular automaton method. Computational Materials Science, 32(2), 147–155. https://doi.org/10.1016/j.commatsci.2004.07.005
Lan, Y. J., Xiao, N. M., Li, D. Z., & Li, Y. Y. (2005b). Mesoscale simulation of deformed austenite decomposition into ferrite by coupling a cellular automaton method with a crystal plasticity finite element model. Acta Materialia, 53(4), 991–1003. https://doi.org/10.1016/j.actamat.2004.10.045
Lépinoux, J., Weygand, D., & Verdier, M. (2010). Modeling grain growth and related phenomena with vertex dynamics. Comptes Rendus. Physique, 11(3–4), 265–273. https://doi.org/10.1016/j.crhy.2010.07.015
Lervåg, K. Y., & Ervik, Å. (2013). Curvature calculations for the level-set method. In A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, M. V. Tretyakov (Eds.), Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011 (pp. 209–217). Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_23
Leeuwen, Y., van, Sietsma, J., & Zwaag, S., van der (2003). The influence of carbon diffusion on the character of the γ–α phase transformation in steel. ISIJ International, 43(5), 767–773.
Li, D. Z., Xiao, N. M., Lan, Y. J., Zheng, C. W., & Li, Y. Y. (2007). Growth modes of individual ferrite grains in the austenite to ferrite transformation of low carbon steels. Acta Materialia, 55(18), 6234–6249. https://doi.org/10.1016/j.actamat.2007.07.041
Liang, C., Song, G., Wang, W., Sohn, I., & Zeng, J. (2024). In situ observation of the austenite to ferrite transformation in low-carbon steels from different initial phases at defined cooling rates. Journal of Materials Research and Technology, 28, 2116–2126. https://doi.org/10.1016/j.jmrt.2023.12.157
Liang, X., Bos, C., Hermans, M., & Richardson, I. (2023). An Improved Cellular Automata Solidification Model Considering Kinetic Undercooling. Metallurgical and Materials Transactions B, 54(3), 1088–1098. https://doi.org/10.1007/s11663-023-02742-3
Liu, C., Garner, A., Zhao, H., Prangnell, P. B., Gault, B., Raabe, D., & Shanthraj, P. (2021). CALPHAD-informed phase-field modeling of grain boundary microchemistry and precipitation in Al-Zn-Mg-Cu alloys. Acta Materialia, 214, 116966. https://doi.org/10.1016/j.actamat.2021.116966
Liu, Y., Sommer, F., & Mittemeijer, E. (2006). The austenite–ferrite transformation of ultralow-carbon Fe–C alloy; transition from diffusion- to interface-controlled growth. Acta Materialia, 54(12), 3383–3393. https://doi.org/10.1016/j.actamat.2006.03.029
Loginova, I., Odqvist, J., Amberg, G., & Ågren, J. (2003). The phase-field approach and solute drag modeling of the transition to massive γ → α transformation in binary Fe-C alloys. Acta Materialia, 51(5), 1327–1339. https://doi.org/10.1016/S1359-6454(02)00527-X
Loginova, I., Ågren, J., & Amberg, G. (2004). On the formation of Widmanstätten ferrite in binary Fe–C – phase-field approach. Acta Materialia, 52(13), 4055–4063. https://doi.org/10.1016/j.actamat.2004.05.033
Madej, L., Sitko, M., & Pietrzyk, M. (2016). Perceptive comparison of mean and full field dynamic recrystallization models. Archives of Civil and Mechanical Engineering, 16(4), 569–589. https://doi.org/10.1016/j.acme.2016.03.010
Málek, M., Mičian, M., & Moravec, J. (2020). Determination of phase transformation temperatures by dilatometric test of S960MC steel. Archives of Foundry Engineering, 21(2), 57–64. https://doi.org/10.24425/afe.2021.136098
Marek, M. (2013). Grid anisotropy reduction for simulation of growth processes with cellular automaton. Physica D: Nonlinear Phenomena, 253, 73–84. https://doi.org/10.1016/j.physd.2013.03.005
Mason, J. K., Lind, J., Li, S. F., Reed, B. W., & Kumar, M. (2015). Kinetics and anisotropy of the Monte Carlo model of grain growth. Acta Materialia, 82, 155–166. https://doi.org/10.1016/j.actamat.2014.08.063
Mecozzi, M. G., Sietsma, J., & Zwaag, S., van der (2005). Phase field modelling of the interfacial condition at the moving interphase during the γ→α transformation in C–Mn steels. Computational Materials Science, 34(3), 290–297. https://doi.org/10.1016/j.commatsci.2005.03.002
Mecozzi, M. G., Militzer, M., Sietsma, J., & Zwaag, S., van der (2008). The role of nucleation behavior in phase-field simulations of the austenite to ferrite transformation. Metallurgical and Materials Transactions A, 39(6), 1237–1247. https://doi.org/10.1007/s11661-008-9517-2
Mecozzi, M. G., Bos, C., & Sietsma, J. (2011). 3D cellular automata modelling of solid–state transformations relevant in low–alloy steel production. Solid State Phenomena, 172–174, 1140–1145. https://doi.org/10.4028/www.scientific.net/SSP.172-174.1140
Militzer, M., Mecozzi, M. G., Sietsma, J., & Zwaag, S., van der (2006). Three-dimensional phase field modelling of the austenite-to-ferrite transformation. Acta Materialia, 54(15), 3961–3972. https://doi.org/10.1016/j.actamat.2006.04.029
Moelans, N., Blanpain, B., & Wollants, P. (2008). An introduction to phase-field modeling of microstructure evolution. Calphad, 32(2), 268–294. https://doi.org/10.1016/j.calphad.2007.11.003
Mourad, H. M., & Garikipati, K. (2006). Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics. Computer Methods in Applied Mechanics and Engineering, 196(1–3), 595–607. https://doi.org/10.1016/j.cma.2006.06.005
Murgas, B., Flipon, B., Bozzolo, N., & Bernacki, M. (2022). Level-set modeling of grain growth in 316L stainless steel under different assumptions regarding grain boundary properties. Materials, 15(7), 2434. https://doi.org/10.3390/ma15072434
Mushongera, L. T., Kubendran Amos, P. G., Nestler, B., & Ankit, K. (2018). Phase-field simulations of pearlitic divergence in Fe-C-Mn steels. Acta Materialia, 150, 78–87. https://doi.org/10.1016/j.actamat.2018.02.059
Mushongera, L. T., Kubendran Amos, P. G., Schoof, E., Kumar, P., & Nestler, B. (2020). The non-steady-state growth of divergent pearlite in Fe–C–Mn steels: A phase-field investigation. Journal of Materials Science, 55(12), 5280–5295. https://doi.org/10.1007/s10853-019-04307-9
Ning, H., Qian, S., & Zhou, T. (2023). Applications of level set method in computational fluid dynamics: A review. International Journal of Hydromechatronics, 6(1), 1. https://doi.org/10.1504/IJHM.2023.129126
Olshanskii, M., Reusken, A., & Schwering, P. (2024). A narrow band finite element method for the level set equation (no. arXiv:2407.02950). arXiv. https://doi.org/10.48550/arXiv.2407.02950
Opara , J., & Kuziak, R. (2020). Study of phase transformations in complex phase steel using a mesoscale cellular automaton model Part I: Modeling Fundamentals. Journal of Metallic Materials, 72(3), 17–31.
Osher, S., & Fedkiw, R. P. (2001). Level set methods: An overview and some recent results. Journal of Computational Physics, 169(2), 463–502. https://doi.org/10.1006/jcph.2000.6636
Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79(1), 12–49. https://doi.org/10.1016/0021-9991(88)90002-2
Osher, S., & Tsai, R. (2003). Level set methods and their applications in image science. Communications in Mathematical Sciences, 1(4), 1–20. https://doi.org/10.4310/CMS.2003.v1.n4.a1
Pariser, G., Schaffnit, P., Steinbach, I., & Bleck, W. (2001). Simulation of the γ-α-transformation using the phase-field method. Steel Research, 72(9), 354–360. https://doi.org/10.1002/srin.200100130
Pohjonen, A. (2023). Full field model describing phase front propagation, transformation strains, chemical partitioning, and diffusion in solid–solid phase transformations. Advanced Theory and Simulations, 6(3), 2200771. https://doi.org/10.1002/adts.202200771
Preis, T., Virnau, P., Paul, W., & Schneider, J. J. (2009). GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model. Journal of Computational Physics, 228(12), 4468–4477. https://doi.org/10.1016/j.jcp.2009.03.018
Qin, R. S., & Bhadeshia, H. K. (2010). Phase field method. Materials Science and Technology, 26(7), 803–811. https://doi.org/10.1179/174328409X453190
Raabe, D. (2000). Scaling Monte Carlo kinetics of the Potts model using rate theory. Acta Materialia, 48(7), 1617–1628. https://doi.org/10.1016/S1359-6454(99)00451-6
Rozo Vasquez, J., Arian, B., Kersting, L., Homberg, W., Trächtler, A., & Walther, F. (2023). Detection of Phase Transformation during Plastic Deformation of Metastable Austenitic Steel AISI 304L by Means of X-ray Diffraction Pattern Analysis. Metals, 13(6), 1007. https://doi.org/10.3390/met13061007
Sarkar, P. (2000). A brief history of cellular automata. ACM Computing Surveys, 32(1), 80–107. https://doi.org/10.1145/349194.349202
Scholtes, B., Ilin, D., Settefrati, A., Bozzolo, N., Agnoli, A., & Bernacki, M. (2016). Full field modeling of the Zener pinning phenomenon in a level set framework – discussion of classical limiting mean grain size equation. In M. Hardy, E. Huron, U. Glatzel, B. Griffin, B. Lewis, C. Rae, V. Seetharaman, S. Tin (Eds.), Superalloys 2016. Proceedings of the 13th International Symposium on Superalloys (pp. 497–503). Wiley. https://doi.org/10.1002/9781119075646.ch53
Schönfisch, B., & de Roos, A. (1999). Synchronous and asynchronous updating in cellular automata. Biosystems, 51(3), 123–143. https://doi.org/10.1016/S0303-2647(99)00025-8
Schoof, E., Schneider, D., Streichhan, N., Mittnacht, T., Selzer, M., & Nestler, B. (2018). Multiphase-field modeling of martensitic phase transformation in a dual-phase microstructure. International Journal of Solids and Structures, 134, 181–194. https://doi.org/10.1016/j.ijsolstr.2017.10.032
Shi, L., Xu, S., Lu, H., Chen, C., Shuai, S., Hu, T., Kao, A., Wang, J., & Ren, Z. (2022). A High-efficiency Virtual Submesh Cellular Automata Method for Solidification Simulation with Low Mesh Anisotropy. ISIJ International, 62(8), 1674–1683. https://doi.org/10.2355/isijinternational.ISIJINT-2022-066
Shirzad, K., & Viney, C. (2023). A critical review on applications of the Avrami equation beyond materials science. Journal of The Royal Society Interface, 20(203), 20230242. https://doi.org/10.1098/rsif.2023.0242
Sietsma, J. (2012). 14 – Nucleation and growth during the austenite-to-ferrite phase transformation in steels after plastic deformation. In E. Pereloma, D. V. Edmonds (Eds.), Phase Transformations in Steels (vol. 1: Fundamentals and Diffusion-Controlled Transformations, pp. 505–526). Woodhead Publishing. https://doi.org/10.1533/9780857096104.4.505
Sietsma, J., & Zwaag, S., van der (2004). A concise model for mixed-mode phase transformations in the solid state. Acta Materialia, 52(14), 4143–4152. https://doi.org/10.1016/j.actamat.2004.05.027
Sitko, M., & Madej, Ł. (2016). Evaluation of efficient computational work division in parallel Monte Carlo grain growth algorithm. Computer Methods in Materials Science, 16(3), 113–120. https://doi.org/10.7494/cmms.2016.3.0579
Sitko, M., & Madej, L. (2021). The role of the cellular automata cell size and time step length in the microstructure evolution model – The static recrystallization case study. Journal of Computational Science, 54, 101437. https://doi.org/10.1016/j.jocs.2021.101437
Spencer, P. J. (2008). A brief history of CALPHAD. Calphad, 32(1), 1–8. https://doi.org/10.1016/j.calphad.2007.10.001
Sun, F., Mino, Y., Ogawa, T., Chen, T.-T., Natsume, Y., & Adachi, Y. (2023). Evaluation of austenite–ferrite phase transformation in carbon steel using bayesian optimized cellular automaton simulation. Materials, 16(21), 6922. https://doi.org/10.3390/ma16216922
Suwanpinij, P., Rudnizki, J., Prahl, U., & Bleck W. (2009). Investigation of the effect of deformation on γ-α phase transformation kinetics in hot-rolled dual phase steel by phase field approach. Steel Research International, 80(9), 616–622. https://doi.org/10.2374/SRI09SP047
Svoboda, J., Fischer, F. D., Fratzl, P., Gamsjäger, E., & Simha, N. K. (2001). Kinetics of interfaces during diffusional transformations. Acta Materialia, 49(7), 1249–1259. https://doi.org/10.1016/S1359-6454(01)00012-X
Svyetlichnyy, D. S., & Mikhalyov, A. I. (2014). Three-dimensional frontal cellular automata model of microstructure evolution – phase transformation module. ISIJ International, 54(6), 1386–1395. https://doi.org/10.2355/isijinternational.54.1386
Syha, M., & Weygand, D. (2010). A generalized vertex dynamics model for grain growth in three dimensions. Modelling and Simulation in Materials Science and Engineering, 18(1), 015010. https://doi.org/10.1088/0965-0393/18/1/015010
Szeliga, D., Bzowski, K., Rauch, Ł., Kuziak, R., & Pietrzyk, M. (2020). Mean field and full field modelling of microstructure evolution and phase transformations during hot forming and cooling of low carbon steels. Computer Methods in Materials Science, 20(3). https://doi.org/10.7494/cmms.2020.3.0727
Thiessen, R. G., Sietsma, J., Palmer, T. A., Elmer, J. W., & Richardson, I. M. (2007). Phase-field modelling and synchrotron validation of phase transformations in martensitic dual-phase steel. Acta Materialia, 55(2), 601–614. https://doi.org/10.1016/j.actamat.2006.08.053
Tong, M., Li, D., Li, Y., & Ni, J. (2002). Modeling the austenite-ferrite isothermal transformation in an Fe-C binary system and experimental verification. Metallurgical and Materials Transactions A, 33(10), 3111–3115. https://doi.org/10.1007/s11661-002-0295-y
Tong, M., Li, D., & Li, Y. (2004a). Modeling the austenite–ferrite diffusive transformation during continuous cooling on a mesoscale using Monte Carlo method. Acta Materialia, 52(5), 1155–1162. https://doi.org/10.1016/j.actamat.2003.11.006
Tong, M., Li, D., Li, Y., Ni, J., & Zhang, Y. (2004b). Monte Carlo-method simulation of the deformation-induced ferrite transformation in the Fe-C system. Metallurgical and Materials Transactions A, 35(5), 1565–1577. https://doi.org/10.1007/s11661-004-0263-9
Tong, M., Li, D., & Li, Y. (2005). A q-state Potts model-based Monte Carlo method used to model the isothermal austenite-ferrite transformation under non-equilibrium interface condition. Acta Materialia, 53(5), 1485–1497. https://doi.org/10.1016/j.actamat.2004.12.002
Torres, C., Emelianenko, M., Golovaty, D., Kinderlehrer, D., & Ta’asan, S. (2015). Numerical analysis of the vertex models for simulating grain boundary networks (no. arXiv:1410.5809). arXiv. https://doi.org/10.48550/arXiv.1410.5809
Tourret, D., Liu, H., & LLorca, J. (2022). Phase-field modeling of microstructure evolution: Recent applications, perspectives and challenges. Progress in Materials Science, 123, 100810. https://doi.org/10.1016/j.pmatsci.2021.100810
Ubhi, H. S., Parsons, J., Othen, N., Campbell, S., Poole, R., & Gholinia, A. (2014). In-situ EBSD phase transformation and recrystallisation. Journal of Physics: Conference Series, 522, 012011. https://doi.org/10.1088/1742-6596/522/1/012011
Van der Ven, A., & Delaey, L. (1996). Models for precipitate growth during the γ → α + γ transformation in Fe–C and Fe–C–M alloys. Progress in Materials Science, 40(3), 181–264. https://doi.org/10.1016/0079-6425(96)00002-3
Varma, M. R., Sasikumar, R., Pillai, S. G. K., & Nair, P. K. (2001). Cellular automaton simulation of microstructure evolution during austenite decomposition under continuous cooling conditions. Bulletin of Materials Science, 24(3), 305–312. https://doi.org/10.1007/BF02704927
Wang, B. X., Liu, X. H., & Wang, G. D. (2009). Effect of deformation of austenite and cooling rates on transformation microstructures in a Mn–Cr gear steel. Materials & Design, 30(6), 2198–2204. https://doi.org/10.1016/j.matdes.2008.08.021
Wang, K., Wu, H., Lv, S., Dong, L., Zhang, C., Wang, S., Wu, G., Gao, J., Zhu, J., & Mao, X. (2025). Multiphase field modeling of austenite to pearlite–ferrite transformation in hypoeutectoid steel. International Journal of Minerals, Metallurgy and Materials, 32(6), 1427–1440. https://doi.org/10.1007/s12613-024-2993-6
Wei, L., Lin, X., Wang, M., & Huang, W. (2016). Low artificial anisotropy cellular automaton model and its applications to the cell-to-dendrite transition in directional solidification. Materials Discovery, 3, 17–28. https://doi.org/10.1016/j.md.2016.06.001
Weigel, M. (2017). Monte Carlo methods for massively parallel computers (no. arXiv:1709.04394). arXiv. https://doi.org/10.48550/arXiv.1709.04394
Wermiński, M., Perzyński K., Sitko M., & Madej Ł. (2023). Evaluation of data transfer influence in coupled Monte Carlo finite element model on microstructure evolution predictions. In L. Madej, M. Sitko, K. Perzynski (Eds.), Material Forming: ESAFORM 2023. The 26th International ESAFORM Conference on Material Forming – ESAFORM 2023 – held in Kraków, Poland, April 19–21, 2023 (pp. 1551–1558). Materials Research Forum. https://doi.org/10.21741/9781644902479-167
Wermiński, M., Sitko M., & Madej Ł. (2025). Reducing the influence of the computational domain discretisation on grain growth in the cellular automaton austenite-to-ferrite transformation model. In P. Carlone, L. Filice, D. Umbrello (Eds.), Material Forming: ESAFORM 2025. The 28th International ESAFORM Conference on Material Forming – ESAFORM 2025 – Paestum, Italy, 7–9 May, 2025 (pp. 1953–1961). Materials Research Forum. https://doi.org/10.21741/9781644903599-210
Williamson, A., & Delplanque, J.-P. (2015). Strain energy limitations in Monte Carlo Potts modeling of grain growth. Computational Materials Science, 108(A), 48–55. https://doi.org/10.1016/j.commatsci.2015.05.032
Wolfram, S. (1984). Universality and complexity in cellular automata. Physica D: Nonlinear Phenomena, 10(1–2), 1–35. https://doi.org/10.1016/0167-2789(84)90245-8
Wright, S. A., Plimpton, S. J., Swiler, T. P., Fye, R. M., Young, M. F., & Holm, E. M. (1997). Potts-model grain growth simulations: Parallel algorithms and applications (no. SAND97–1925). Sandia National Laboratories. https://doi.org/10.2172/522745
Wu, H. (2022). A review on the Cahn–Hilliard equation: Classical results and recent advances in dynamic boundary conditions. Electronic Research Archive, 30(8), 2788–2832. https://doi.org/10.3934/era.2022143
Xiao, N., Tong, M., Lan, Y., Li, D., & Li, Y. (2006). Coupled simulation of the influence of austenite deformation on the subsequent isothermal austenite–ferrite transformation. Acta Materialia, 54(5), 1265–1278. https://doi.org/10.1016/j.actamat.2005.10.055
Xu, Z., & Meakin, P. (2008). Phase-field modeling of solute precipitation and dissolution. The Journal of Chemical Physics, 129(1), 014705. https://doi.org/10.1063/1.2948949
Yamanaka, A. (2023). Phase-field modeling and simulation of solid-state phase transformations in steels. ISIJ International, 63(3), 395–406. https://doi.org/10.2355/isijinternational.ISIJINT-2022-343
Zhang, Jiantao, Hu, X., & Chou, K. (2020). In-situ environmental transmission electron microscopy investigation of the phase transformation austenite → ferrite in duplex stainless steel. Materials Letters, 264, 127259. https://doi.org/10.1016/j.matlet.2019.127259
Zhang, Jin, Chadwick, A. F., Chopp, D. L., & Voorhees, P. W. (2023). Phase field modeling with large driving forces. Npj Computational Materials, 9(1), 166. https://doi.org/10.1038/s41524-023-01118-0
Zhang, L., Zhang, C. B., Wang, Y. M., Wang, S. Q., & Ye, H. Q. (2003). A cellular automaton investigation of the transformation from austenite to ferrite during continuous cooling. Acta Materialia, 51(18), 5519–5527. https://doi.org/10.1016/S1359-6454(03)00416-6
Zhang, X., Chen, J.-S., & Osher, S. (2008). A multiple level set method for modeling grain boundary evolution of polycrystalline materials. Interaction and Multiscale Mechanics, 1(2), 191–209. https://doi.org/10.12989/IMM.2008.1.2.191
Zhang, Z., Ge, P., Li, J. Y., Ren, D. X., & Wu, T. (2022). Monte Carlo simulations of solidification and solid-state phase transformation during directed energy deposition additive manufacturing. Progress in Additive Manufacturing, 7(4), 671–682. https://doi.org/10.1007/s40964-021-00253-8
Zheng, C., Raabe, D., & Li, D. (2012). Prediction of post-dynamic austenite-to-ferrite transformation and reverse transformation in a low-carbon steel by cellular automaton modeling. Acta Materialia, 60(12), 4768–4779. https://doi.org/10.1016/j.actamat.2012.06.007
Zhi, Y., Jiang, Y., Ke, D., Hu, X., & Liu, X. (2024). Review on cellular automata for microstructure simulation of metallic materials. Materials, 17(6), 1370. https://doi.org/10.3390/ma17061370
Zyska, A., Boroń, K., & Kordas, P. (2018). Modeling of dendritic structure evolution during solidification of Al-Cu alloy. Archives of Foundry Engineering, 18(4), 87–92. https://doi.org/10.24425/afe.2018.125174