Applied Mathematics, 2013, 4, 1531-1536
Published Online November 2013 (http://www.scirp.org/journal/am)
Open Access AM
Sensivity Analysis of the Cellular Automata Model for
Austenite-Ferrite Phase Transformation in Steels
Rafal Golab*, Daniel Bachniak, Krzysztof Bzowski, Lukasz Madej
AGH University of Science and Technology, Kraków, Poland
Email: *firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com
Received September 18, 2013; revised October 18, 2013; accepted October 25, 2013
Copyright © 2013 Rafal Golab et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The main goal of the present research is to realize a sensitivity analysis of the developed complex micro scale austenite
(γ) to ferrite (α) phase transformation model. The proposed solution is implemented in the developed Cellular Automata
Framework that facilitates implementation of various microstructure evolution models. Investigated model predicts
phase transformation progress starting from the fully austenitic or two-phase regions. Theoretical background of the
implemented austenite-ferrite phase transformation model is presented in the paper. The defined transition rules for ini-
tiation and subsequent growth as well as internal variables for each particular CA cell are also discussed. Examples of
results obtained from the developed model, as well as model capabilities are shown. Finally sensitivity analysis using
Morris OAT Design is also presented and discussed.
Keywords: Sensitivity Analysis; Cellular Automata; Austenite; Ferrite; Phase Transformation
Small, medium and large industrial companies aim to de-
velop an innovative and efficient manufacturing tech-
nology to make products that are competitive on the global
market. However, development of these new technolo-
gies based on conventional approach composed of indus-
trial test and trials is very expensive and time consuming.
As a result, industry cannot quickly adapt to the rapidly
changing demands of the market. Thus, recently, industry
more often relies on technology development supported
by a series of numerical calculations and computer simu-
lations. These approaches provide possibility to accurately
describe material behavior not only at the macro scale
level but also can predict changes in the microstructure at
the mezo or micro scale levels. As a result, significant
drop in the amount of laboratory or industrial tests is
possible. This creates an opportunity to save valuable
time and also decrease R&D costs. However, access to
appropriate hardware and software is crucial in this ap-
proach; otherwise it is impossible to take full advantage
of the computer aided technology design.
Conventional numerical approaches are usually based
on the finite element method (FEM) combined with stan-
dard closed-form micro scale models that can provide
general information about the state of the material e.g.
temperature, average phase volume fraction, grain sizes,
stress-strain state etc. However, exact information on ma-
terial microstructure and morphology is also crucial when
development of novel manufacturing technology is un-
dertaken by a company. Unfortunately, these fast con-
ventional models fail to provide such detailed data. The
solution to this limitation is replacing the conventional
models with modern discrete models based on e.g. Monte
Carlo (MC), Cellular Automata (CA) etc. The CA model
besides standard information on material behavior can
also provide information about the final morphology of
the material after, but also during, the process as shown
In the last decade, series of microstructure evolution
models based on the cellular automata method were cre-
ated [1-4]. Unfortunately, these models were designed as
in-house codes and have the capability to solve only se-
lected problem e.g. the simulation of dynamic or static
recrystallization. If a new CA model e.g. austenite to
ferrite phase transformation is required, the algorithmic
and programming work has to be initiated from scratch
and that significantly rises model development time.
To solve this issue, authors decided to develop a uni-
versal framework for the cellular automata method. All
the basic cellular automata algorithmic solutions are avail-
R. GOLAB ET AL.
Figure 1. Schematical illustration of advantages provided
by the cellular automata material models.
able to each user. This allows focusing the main burden
of the work on development of the appropriate transition
rules that replicate physical phenomena and not on im-
plementation aspects. As a result, the use of the frame-
work to develop microstructure evolution models will be
possible for material scientists without extensive knowl-
edge of programming and algorithmic solutions. The CAF
(Cellular Automata Framework) is based on the applica-
tion-defined programming interface (Application Program-
ming Interface, API), which allows for quick creation of
a ready-made applications and reuse of developed codes.
The framework was implemented based on the object-
oriented technology, so that it was possible to make it in
the form of the linked dynamic library for other develop-
ers and available interfaces which are shown in Figure 2.
Details on this solution can be found in earlier authors
Based on the Cellular Automata Framework function-
ality, the phase transformation model to predict changes
from austenite to ferrite during continuous cooling was
developed. Details of the model can be found in ,
while major assumptions and examples of obtained re-
sults are described in the following chapter.
2. Description of Austenite—Ferrite
The developed CA model is designed to simulate austen-
ite to ferrite transformation during cooling in the 2D
space. Each CA cell is described by several states and
internal variables in order to properly describe physical
state of the material. The cell can be in three different
states (Figure 3):
Figure 2. Interfaces of the CA framework.
Open Access AM
R. GOLAB ET AL. 1533
Figure 3. Illustration of the nucleus of the ferrite phase and
the surrounding cells in the ferrite -auste nite (α/γ) state.
austenite (γ) and,
The last state describes CA cells located at the phase
interface between austenite and ferrite grains. Addition-
ally a series of internal variables were defined for each
CA cell to describe other necessary microstructure fea-
tures e.g. how many ferrite phase is in a particular cell
Fi,j, what is the carbon concentration in each cell Ci,j, the
growth length li,j of the ferrite-austenite cell into the aus-
tenite cell or the growth velocity vi,j of the interface cell.
These internal variables are then used in the transition
rules to replicate mechanisms of phase transformation.
Similar solutions with different level of complexity are
also available in the literature [8,9].
The developed model takes into account two major
mechanism occurring during transformation, namely nu-
cleation and growth of the ferrite grains into the austen-
itic matrix. Due to the fact that nucleation process is con-
sidered as stochastic in nature, various approaches to de-
scribe this process can be used e.g. . In the present
model, to replicate the stochastic character of nucleation,
a number of nuclei n is calculated in a probabilistic
manner at the beginning of each time step:
where: B0, Bi—mean amount of γ cells at the beginning
and i-th time step, respectively, N—total number of α
nuclei, a1, a2, a3—model parameters, Ae3—start tempera-
ture of the austenite—ferrite transformation, x, y— width
and height of the CA space, P—probability, τ—time step.
Also locations of grain nuclei in the CA space are also
generated randomly along initial austenite grain bounda-
ries. When a cell is selected as a nuclei, the state of this
cell changes from austenite (γ) to ferrite (α). At the same
time all the neighboring cells of the ferrite (α) cell change
their state to ferrite-austenite (α/γ) (Figure 3). The car-
bon content, which was in the austenite cell is then di-
vided between all neighboring cells (the Moore neighbor-
hood is used), which are in the state α/γ. Nucleation
process has a continuous character and it occurs during
the entire CA simulation until the end of transformation.
When a nucleus occurs in the CA space, the growth of
ferrite phase is calculated in the following steps.
Ferrite growth is controlled by the carbon diffusion,
thus the carbon distribution across the microstructure is
evaluated by the solution of the diffusion equation on the
basis of the finite difference (FD) method:
The transition rules describing growth of ferrite grains
during phase transformation are designed to replicate
experimental observations of mechanisms responsible for
this process [8,10]. Because, newly formed ferrite nuclei
grow into the austenite phase, the velocity of the α/γ in-
terface is assumed to be a product of the mobility M and
the driving force for interface migration F:
where: M0—mobility coefficient, T—absolute tempera-
ture, Qact—activation energy.
The driving force for the phase transformation F in the
present work is based only on chemical driving force
Fchem. The driving force related with accumulated energy
is neglected. The chemical driving force is a result of the
differences in the carbon concentration in equilibrium
conditions and carbon concentration in each cell:
chem ,eqii j
where: β—model coefficient, Ceq—equilibrium carbon
concentration calculated using ThermocalcTM software,
Ci,j—carbon concentration in the (i, j) CA cell.
Equilibrium carbon concentration is calculated based
on the ThermocalcTM data as schematically shown in
A set of transition rules is proposed in the model to
replicate the phenomena occurring at the austenite-ferrite
boundary. When the ferrite phase is present in the mate-
rial, the CA ferrite cells grow into the austenite phase. In
the current time step t the growth length of the austen-
Open Access AM
R. GOLAB ET AL.
Figure 4. FeC equilibrium diagram.
ite-ferrite cell with indexes (i, j) towards an austenite
neighbouring cell with indexes (k, l) is described as:
where: t0—time when the CA cell (i, j) changed into the
ferrite state, vi,j—the growth velocity of the CA cell (i, j).
The growth velocity v is obtained from (5) and then
the ferrite volume fraction in the CA cell (k, l) is calcu-
lated as a result of the ferrite growth:
where: Fk,l—total ferrite volume fraction in the CA cell
(k, l), as a contribution from all the neighbouring austen-
ite-ferrite CA cells, LCA—dimension of the CA cell in the
CA space, t—time step.
Based on these calculations the transition rules are de-
fined as follows:
where: ,—state of the cell (k, l) in the time step t,
—state of the Moore neighboring (M) cell (i, j) in
the time step t, Fcr—critical value of the volume fraction
of ferrite in the CA cell, temp—temporary state.
The CA cell changes the state from austenite into aus-
tenite-ferrite when ferrite volume fraction in this cell
exceeds the critical value Fcr. Otherwise the cell remains
in the austenite state see Equation (9). If the CA cell
change state from austenite to austenite-ferrite state every
von Neuman neighboring cells change state to temp state
according to (10). After that transition rule described as
(11) is initiated—every old austenite/ferrite cells change
state to ferrite, and the cells on temp state change state to
austenite-ferrite. When the cell changes its state to aus-
tenite-ferrite, all the neighboring cells in the austenite-
ferrite state change their states into the ferrite. When a
change in the cell state occurs, the corresponding carbon
concentration changes according to the FeC diagram. To
obtain stable solution of the FDM and CA calculation
and save computational time at the same time an effec-
tive time step refinement/derefinement procedure was
implemented into the CAF.
Examples of obtained results on the basis of the model
implemented in the CA framework are presented in Fig-
ure 5. Figure 5 shows the carbon distribution changes,
corresponding phase fraction and microstructure mor-
phology evolution during cooling with the cooling rate
1˚C/s. The size of the investigated CA space is assumed
to be 200 × 200 cells what corresponds to the physical
size 200 × 200 µm.
As seen in Figure 5, the growth of newly formed fer-
rite nuclei is directly related with carbon concentration
(a) (b) (c)
Figure 5. (a) Carbon distribution (b) phase fraction (c) mi-
crostructure morphology obtained during cooling with the
cooling rate 1˚C/s.
Open Access AM
R. GOLAB ET AL. 1535
controlled by the diffusion process.
As presented, developed model is based on several
model coefficients that can directly influence quality of
obtained results. Thus to evaluate level of influence of
these model parameters on obtained results a sensitivity
analysis based on Moris One At a Time (OAT) design
3. Morris Oat Design
The Morris design belongs to the class of screening
methods . Screening designs, as a part of the sensi-
tivity analysis methods , deal with the question which
factors of the physical model or computer simulation are
really important. The factor means either parameter, which
describes properties of the model or input variable, which
is directly observable in the corresponding real system.
Screening methods estimate qualitative statistic of the
factors in order of their importance, i.e. they state that
one factor is more important than another, but they do
not provide the quantitative information of the factors
Screening designs widely use the OAT approach. Meth-
ods based on the OAT technique investigate the impact
of the variation of each factor in turn. The OAT design
developed by Morris  is called the global sensitivity
analysis, because the algorithm explores the entire space
over which the factors vary. In this algorithm the main
effect of the factor is estimated by computing the as-
sumed number of local measures at different points in the
input space and next the average value is taken. These
points are selected in such a way that each factor covers
the whole interval in which it was defined.
Sensitivity analysis was performed for 6 major pa-
rameters of the CA model: 3 parameters of nucleation
model: a1, a2, a3, mobility coefficient M0, activation en-
ergy Qact and β parameter. These parameters can be
found in Equations (2)-(6). Additionally, to validate model
functionality a 7th parameter describing sensitivity to
cooling rates was added to the investigation. Overall 800
calculations were realized during the investigation. Model
output was defined as temperature of initiation and end
of the transformation process.
As expected the model is sensitive to cooling rates,
what proves physical nature of obtained results. It seems
from Figure 6, that the most important model parameter
is mobility coefficient that is directly related with the
phase boundary movement. Thus, this coefficient has to
be particularly well identified during the model identifi-
cation procedure. Sensitivity of other model parameters
remains on similar level.
Based on the presented research it can be stated that:
Figure 6. Sensitivity effect of different parameters for tem-
perature ferrite start and ferrite stop.
Implementation of the austenite-ferrite phase trans-
formation model based on the developed CA frame-
work is possible;
The nucleation and growth parts of the model are
directly related with changes in carbon concentration.
Incorporation of the FDM into the CAF is a robust
solution. The use of FDM for solving carbon diffu-
sion provided a possibility for detailed reproduction
of reality in the transformation of austenite-ferrite;
Sensitivity analysis revealed that the model is mostly
sensitive to the mobility coefficient and this parame-
ter has to be carefully identified using inverse analy-
sis. Sensitivity to other coefficients remains on simi-
Future work will focus on identification of the CA
model parameters based on inverse technique as well as
further extension the CAF capabilities. Other microstruc-
ture evolution models e.g. recrystallization models will
be also implemented and combined in the CAF using
Financial assistance of the NCN, project no. 2011/01/
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