DeDevvelopment and identification of the cellular aut elopment and identification of the cellular automata phase tr omata phase transfansformation model ormation model

AAbstrbstractact.. The development and identification of the complex microscale austenite to ferrite transformation model during continuous cooling based on the Cellular Automata method and DigiCore library is the main goal of the work. The model is designed to predict phase transformation from a fully austenitic range and involves nucleation of ferrite grains with their further growth. The major driving force for the CA grain growth is based on the carbon concentration differences across the microstructure. The model parameters are identified with the inverse analysis method with the goal function defined on the basis of the dilatometric investigation. The basic assumption of the developed model, experimental procedure, as well as subsequent identification stages, are presented within the work.


Intr Introduction oduction
The development of reliable material models for metal forming simulations has been in scientists' interest for a number of years [1][2][3].The main challenge in modeling metal forming operations that the industry is currently facing is to realistically describe phenomena typically occurring at lower length scales under deformation or subsequent heat treatment conditions and incorporate them into the continuum-based approaches.One of the difficulties in this regard is the fact that many micro-scale phenomena are stochastic in nature.This is particularly important when modern metallic materials, including new steel grades, are under investigation.Most of these materials have multiphase microstructures with various microscale features directly responsible for the macroscopic in-use properties.In this case, the complex microstructure has a direct impact on the behavior of the final component under exploitation conditions, which, most of the time, are very demanding.One of the available options to address that issue is the development of full-field, discrete modeling methods, e.g., Monte Carlo (MC), Cellular Automata (CA) etc., that have the capability of explicit representation of investigated microstructure features during the numerical simulation.
Therefore, the development and identification of the complex microscale austenite to ferrite transformation model during continuous cooling based on the mentioned Cellular Automata method and DigiCore library is the main goal of the work.The classical cellular automata method is based on a regular, discrete computational domain composed of a set of CA cells aligned along two or three directions of the 2D, or 3D space, respectively [4].The type of space discretization depends on the CA cell's selected shape, e.g., squares, triangles, hexagons etc., as presented for the 2D case in Fig. 1. where: N(i) -neighbourhood of the CA cell, Yi -state of the CA cell, t -time step, Λ = Λ (p, q, Yi, Yj) -logical function within the transition rules also depends on the internal q q and external p p variables.The neighborhood that is considered during the evaluation of the transition rules can be of a different nature, as seen in Fig. 2. With the presented assumptions, the cellular automata model is expressed by a quadrupole: Development and identification of the cellular automata phase transformation model 2640/2 where: ω -cellular automata space, Y Y -set of cell variables, N N -set of neighbours, f -transition rules.
Additionally, the stochastic material behavior at the microstructure scale can be incorporated into the CA models through the definition of non-deterministic transition rules and the random type neighborhood.Finally, to develop a robust CA model, the appropriate boundary conditions have to be created.There are three different types of boundary conditions in the classical CA method: absorbing, where the state of cells located at the edges of the computational domain are adequately fixed with a specific state to absorb moving quantities; reflective, where the state of cells located at the edges are adequately fixed to reflect moving quantities; and most commonly used in material science application the periodic boundary conditions.In the latter case, the CA neighborhood takes into account cells located at the opposite edges of the computational domain and assume their interaction.
As presented in the approach, the system's complex evolution emerges only from local interactions as the long-range communications of the CA cells are neglected.It should also be mentioned that all the available CA models for the material science applications are based on the same common major elements, e.g., CA space as a computational domain, type of neighborhood, transition rule definitions, and also additional components like absorbing or periodic boundary conditions [5][6][7].Therefore, to facilitate the development of the new model, a DigiCore library that encapsulates all the CA approach's common elements is being intensively developed as presented in [8].The DigiCore library is based on the unified data-structures, allowing all the developed microstructure evolution models to be easily coupled for simulation of complex thermo-mechanical operations.

DigiCor DigiCore libr e library ary
The DigiCore library is being developed within the object-oriented C++ programming language under the clear modularity concept.Therefore, the library's main part contains a set of classes and methods, which are the essential building elements to be inherited and extended by other libraries.In this case, each of the additional dynamic link libraries implements a particular microstructure evolution algorithms e.g.static, dynamic or metadynamic recrystalizaiton within the predefined cellular automata space, as presented in Fig.In the sequence diagram (Fig. 4), additional modules are located between the wrappers and the core of the system because they cannot act as separate independent programs but only based on the MSMcoreLib.This application structure ensures the system's required modularity, i.e., is created on a single base library that can be extended by further system components independent of each other but working within the single workflow allowing simulation of microstructure evolution subjected to various thermomechanical processing conditions.The driving force, in this case, is due to the differences in the carbon concentration in equilibrium conditions (Fig. 5) and actual carbon concentration in each CA cell: where: β -model coefficient, Ceq -equilibrium carbon concentration, Ci,j -carbon concentration in the (i,j) CA cell.As the major driving force for grain growth is based on the differences in the carbon concentration across the microstructure, its evaluation becomes an important factor.This issue is addressed by the solution of the diffusion equation on the basis of the finite difference method: When the growth velocity v is obtained from ( 5) then the ferrite volume fraction in the CA cell (k,l) is calculated as: where: fk,l -total ferrite volume fraction in the CA cell (k,l), as a contribution from all the neighboring austenite-ferrite CA cells, LCA -dimension of the CA cell in the CA space, t -time step, t0 -time when the CA cell (i,j) changed into the ferrite state, vi,j -the growth velocity of the CA cell (i,j).
Finally, based on these calculations, a transition rule controlling the evolution of the CA cells states was defined as follows: the CA cell changes the state from austenite-ferrite (interface) into ferrite when ferrite volume fraction in this cell exceeds the critical value fcr set to 1.0 in the current study.If this condition is not fulfilled, the CA cell remains in the austenite-ferrite state.At the same time, when the CA cell changes the state from austenite-ferrite to ferrite, all the neighboring cells in the austenite state change their states into the austenite-ferrite interface.
However, prior to application of the developed model for the numerical predictions of the phase transformation progress in the investigated steel subjected to a particular cooling cycle, its parameters have to be properly identified to reflect the characteristic behavior of a particular steel grade.
3 Identification of CA model par 3 Identification of CA model paramet ameters ers

Experimental data Experimental data
The investigated material is micro-alloyed steel with a chemical composition presented in Table 1.
T Table 1.Chemical composition of the in able 1.Chemical composition of the inv vestig estigat ated st ed steel, wt%.eel, wt%.
To obtain information on phase transformation progress in a range of slow cooling rates between 0.1 and 1 ºC/s a series of dilatometric tests with the DIL 805 apparatus was performed.Samples dimensions were set as Φ4×7 mm.
Identified characteristic start/stop temperatures of austenite to ferrite phase transformation are gathered in Table 2, range between 90% and 80% for the low and high cooling rates, respectively.
T Table 2. Start and st able 2. Start and stop of select op of selected phase tr ed phase transf ansformations.ormations.The two important equilibrium temperatures Ae1 and Ae3 were also calculated by the ThermoCalc software and are equal to 731ºC and 871.0ºC, respectively, as seen in Fig. 7.

In Inv verse anal erse analy ysis sis
The presented set of experimental data, as well as identified equilibrium parameters, were used as boundary conditions for the CA phase transformation model during the model identification procedure based on the inverse analysis [9].
The classical inverse analysis algorithm [10] based on: the CA model from chapter 2.3 used as a direct problem model; experimental data presented in chapter 3.1; and non-gradient optimization method was used during the investigation.
The measured phase transformation start and stop temperatures, as well as volume fractions of ferrite, were used to define the goal function: The final set of CA model parameters identified within the current work is summarized in Table 3.
T Table 3. were also accurately predicted.Some overestimation of the start temperatures is visible in this case, which may be attributed to the interpretation of experimental observations.In the case of the slowest cooling rate, the model also shows a small deviation with respect to experimental observations.In this case, the probabilistic aspects of the CA model can play a role.This issue will be the subject of future work.

Development and identification of the cellular automata phase transformation model 2640/ 4 Fig. 4 .
Fig. 4. Sequence diagr Fig. 4. Sequence diagram pr am presenting the r esenting the role of the wr ole of the wrapper in the application.apper in the application.

Fig. 7 .
Fig. 7. Part of the calculated equilibrium diagram with characteristic temperatures.

ESAFORM 2021 .
Fig. 8. Agr Fig. 8. Agreement betw eement between calculat een calculated and measur ed and measure start and st e start and stop t op temper emperatur atures.es.

Identified phase tr able 3 .
Identified phase transf ansformation model par ormation model paramet ameters.ers. 4 Conclusions 4 Conclusions This work is a step towards the development of the full-field cellular automata phase transformation model in steels focused on transformation from austenite to ferrite phase under continuous colling.During the research, a set of dilatometric experiments was conducted to reveal the behavior of investigated material during cooling in a small cooling range regime.These experiments supported by the ThermoCalc calculations provided the required set of CA model boundary conditions and data for the definition of the goal function.Finally, the application of the inverse analysis technique provided a set of identified phase transformation model coefficients that can be used during further numerical simulations.The model qualitatively predicted the grain size and final volume fractions what is one of the major benefits of the developed full-field model.The start and stop temperatures of austenite to ferrite transformation