2.1 DRX Theoretical Modeling
The CA method can be used to simulate the growth of newly recrystallized equiaxial grains, calculate the DRX nucleation rate, and describe the dislocation density variation and growth kinetics of dynamically recrystallized grains (all these processes are closely associated with actual hot deformation parameters). The nucleation and grain growth are two important aspects of DRX that affect material microstructure and are closely related to the dislocation density.
The following two assumptions developed by Ding et al. [13] are utilized for constructing the CA model:

(1)
The initial dislocation density of all primary grains is uniform and equal to 10^{9}/m^{2}. When the dislocation density reaches a critical value, DRX occurs.

(2)
The DRX nucleation occurs only at grain boundaries. Hence, only interface cells can become possible nucleation sites.
2.1.1 Dislocation Evolution
The dislocation evolution during DRX mainly includes work hardening, dynamic recovery, and recrystallization processes. A phenomenological approach (the KM model) is utilized to predict variations of the dislocation density with strain [14, 15]
$$\frac{\text d\rho }{\text d\varepsilon } = k_{1} \rho^{{\frac{1}{2}}}  k_{2} \rho ,$$
(1)
where \(\rho\) represents the cell dislocation evolution, \(k_{1} = \frac{{2\theta_{0} }}{\alpha ub}\) is the work hardening parameter, and \(k_{2} = \frac{{2\theta_{0} }}{{\sigma_{s} }}\) is the softening parameter. \(\alpha\) is the dislocation interaction term, which is normally set to 0.5 for metals, \(\mu\) is the shear modulus, and b is the Burgers vector mode. \(\theta_{0}\) is the hardening rate, which can be obtained from different values of strain rate \(\dot{\varepsilon }\) and temperature T ranging from 0.05 s^{−1} to 5 s^{−1} (for \(\dot{\varepsilon }\)) and from 850 °C to 1150 °C (for T). \(\sigma_{\text{s}}\) is the saturated stress calculated by using the following equation
$$\dot{\varepsilon } = A\sigma_{s}^{n} \exp (  \frac{{Q_{act} }}{RT}) .$$
(2)
Using previous experimental and calculation results, the values of A and n are set to \(1. 4 8 4\; \times \; 1 0^{ 1 5}\) and 5.619, respectively.
2.1.2 Nucleation Rate
In our proposed model, the nuclei formation during DRX is observed at primary and dynamically recrystallized grain boundaries, which is similar to the model developed by Ding et al. [13]. The DRX nucleation rate per unit grain boundary area can be expressed as a function of both temperature and strain rate by using the following equation
$$\dot{n}(\dot{\varepsilon },T) = C\dot{\varepsilon }^{m} \exp (  \frac{{Q_{act} }}{RT}) ,$$
(3)
where \(\dot{\varepsilon }\) is the strain rate, \(\dot{n}\) is the nucleation rate, C is the material constant, \(Q_{act}\) is the nucleation activation energy, and the exponent m is set to 1. R is the molar gas constant, and T is the temperature.
The critical dislocation density for the grain boundary nucleation can be calculated by taking into account the energy change, as proposed by Roberts and Ahlblom [15]:
$$\rho_{c} = (\frac{{20\gamma \dot{\varepsilon }}}{{3blM\tau^{2} }})^{{\frac{1}{3}}} ,$$
(4)
where \(\tau = \frac{{\mu b^{2} }}{2}\) is the dislocation line energy, and \(l = \frac{{K_{1} \mu b}}{\sigma }\) is the dislocation mean free path. \(K_{1}\) is the constant equal to about 10 for metals, \(\gamma\) is the grain boundary energy, and M is the grain boundary mobility.
2.1.3 Grain Growth Modeling
The dislocation density of the newly recrystallized grains is much smaller than that of the initial grains due to DRX. The driving force for the DRX grain growth originates from the dislocation density difference between the dynamically recrystallized and initial grains. The relationship between the growth velocity V and the driving force f can be described by the following formula [14]:
where \(\lambda\) is the hinder parameter for an alloy element. The driving force per unit area is \(f = \tau (\rho_{m}  \rho_{ij} )  \frac{2\gamma }{r}\). \(\tau = \frac{{\mu b^{2} }}{2}\)is the dislocation line energy, \(\rho_{m}\) is the dislocation density of parent grains, \(\rho_{ij}\) is the cell dislocation evolution parameter with coordinates i,j, and r is the grain radius. The grain boundary mobility M can be calculated from the equation provided by Chen and Cui [1]:
$$M = \frac{{\delta D_{ob} b}}{KT}\exp (  \frac{{Q_{b} }}{RT}) ,$$
(6)
where \(\delta\) is the characteristic grain boundary thickness, \(D_{ob}\) is the boundary selfdiffusion coefficient at absolute zero, \(Q_{b}\) is the boundary diffusion activation energy, and K is Boltzmann’s constant.
The grain boundary energy \(\gamma\) can be expressed as a function of misorientation by using the Read–Shockley equation [12]
$$\gamma = \left\{ {\begin{array}{*{20}l} {\gamma_{m} \frac{\theta }{{\theta_{m} }}(1  In\frac{\theta }{{\theta_{m} }})\begin{array}{*{20}l} {} \\ \end{array}, \theta < 15^{ \circ } }, \\ \quad \quad {\gamma_{m}, \begin{array}{*{20}l} {} \quad {} \quad {} & {} & {} \\ \end{array} \theta \ge 15^{ \circ } } ,\\ \end{array} } \right.$$
(7)
where \(\theta\) is the grain boundary misorientation angle, and \(\theta_{m} = 15^{ \circ }\) is the misorientation limit for lowangle boundaries. \(\gamma_{m} = \frac{{\mu b\theta_{m} }}{4\pi (1  v)}\) is the boundary energy, and \(v\) is Poisson’s ratio.
2.2 CA Rules for Initial Microstructure Generation
In accordance with the results of previous studies reported by Liu et al. [16], Hua et al. [17], Guan et al. [18], and Chen et al. [1, 14], which take into account the problem of grain boundary overlapping, the following CA state transition rules for austenitic grain growth before DRX were established based on the related thermodynamic movement mechanism, activation energy values, and curvaturedriven mechanism.
Rule 1: Considering the temperature dependence of the austenitic grain growth, the cell orientation changes when the energy of grain boundary cells exceeds the activation barrier value. The probability of this event can be described by the following expression:
$$P_{i} = C\frac{{T  T_{Aci} }}{{T_{m}  T_{Aci} }}\exp (  \frac{{Q_{b} }}{RT}) ,$$
(8)
where T is the current austenitizing temperature, \(T_{Aci}\) is the initial austenitizing temperature, \(T_{m}\) is the material melting point, and \(Q_{b}\) is the boundary diffusion activation energy. The value of constant C can be determined by making T equal to \(T_{m}\) and \(P_{i}\) equal to 1.
Rule 2: As shown clearly in Figure 2, if five or more neighbors of cell \(C_{j}\) during the current CAS have the same state according to Moore’s classification, the state of cell \(C_{j}\) assumes that of its neighboring cells during the next CAS.
$$\xi_{Cj}^{t + \Delta t} = f(\xi_{Cj  4}^{t} ,\xi_{Cj  3}^{t} ,\xi_{Cj  2}^{t} ,\xi_{Cj  1}^{t} ,\xi_{Cj}^{t} ,\xi_{Cj + 1}^{t} ,\xi_{Cj + 2}^{t} ,\xi_{Cj + 3}^{t} ,\xi_{Cj + 4}^{t} )$$
(9)
Rule 3: In Figure 3, if the states of any three \(C_{j  3}\), \(C_{j  1}\), \(C_{j + 1}\), or \(C_{j + 3}\) neighbors of cell \(C_{j}\) have the same state during the current CAS, the state of cell \(C_{j}\) assumes that of its neighboring cells during the next CAS.
$$\xi_{Cj}^{t + \Delta t} = f(\xi_{Cj  3}^{t} ,\xi_{Cj  1}^{t} ,\xi_{Cj + 1}^{t} ,\xi_{Cj + 3}^{t} )$$
(10)
Rule 4: In Figure 4, if any three \(C_{j  4}\), \(C_{j  2}\), \(C_{j + 2}\), or \(C_{j + 4}\) neighbors of cell \(C_{j}\) have the same state during the current CAS, the state of cell \(C_{j}\) assumes that of its neighboring cells during the next CAS.
$$\xi_{Cj}^{t + \Delta t} = f(\xi_{Cj  4}^{t} ,\xi_{Cj  2}^{t} ,\xi_{Cj + 2}^{t} ,\xi_{Cj + 4}^{t} )$$
(11)
Rule 5: If neither of the previous rules apply, the state of cell \(C_{j}\) assumes the state of an optional neighboring cell \(C_{k}\). The difference between the obtained grain boundary energies calculated from the deformation transformation of the Hamilton function can be described by the following formula:
$$\Delta E_{j \to k} = E_{k}  E_{j} = J\sum\limits_{n}^{R} {(1  \delta_{DkDn} )}  J\sum\limits_{n}^{R} {(1  \delta_{DjDn} )}$$
(12)
where J is the grain boundary measurement parameter, which is normally set to 1, and δ is the Kronecker symbol. R is the total number of neighbors for cell \(C_{j}\), n is the nth neighbor of cell \(C_{j}\), \(D_{j}\) is the orientation number for cell \(C_{j}\), and \(D_{n}\) is the nth neighbor orientation number for cell \(C_{j}\).
If \(\Delta E_{j \to k} < 0\), the probability of the cell orientation change is 1. If \(\Delta E_{j \to k} \ge 0\), the probability of the cell orientation change is 0.
$$\begin{gathered} \xi _{{Cj}}^{{t + \Delta t}} = {\mkern 1mu} f(\xi _{{Ck}}^{t} ), \hfill \\ \quad \quad \quad {\kern 1pt} (k = j  4,j  3,j  2,j  1,j + 1,j + 2,j + 3,j + 4), \hfill \\ \end{gathered}$$
(13)
Rule 6: For the purpose of making nonoverlapping grains occupy the entire space, ICA is performed by optimizing grain topological structures until they become mutually independent. The correlation coefficient is used to evaluate the degree of grain crosscorrelation. For example, when the correlation coefficient \(r_{ij}\) for grains i and j is equal to 1, the two grains overlap completely. When \(r_{ij}\) equals 0, the two grains are independent. When \(0 < \left {r_{ij} } \right < 1\), the two grains partially overlap. When \(0 < \left {r_{ij} } \right < 0.3\), the two grains are weakly interdependent [19, 20].
If a particular crosscorrelation criterion is not adapted, an error will occur during CA simulations. In order to minimize the influence of the crosscorrelation factor on the simulation results and ensure computational efficiency and precision, we selected \(0.1 \le \left {r_{ij} } \right \le 1\), \(0.2 \le \left {r_{ij} } \right \le 1\), and \(0.3 \le \left {r_{ij} } \right \le 1\) as grain crosscorrelation criteria for testing the described CA model. If the correlation coefficient \(r_{ij}\) meets the specified conditions, we assume that the grain overlap occurred and utilize the ICA method to mitigate it. Thus, the ICAbased grain topology deformation method and the related grain boundary mapping between two different coordinate systems are described in Section 2.3. The simulation error parameter \(E_{r} = 1  \frac{{A_{transf} }}{{A_{0} }}\) is used to characterize the mean correlation coefficient \(\left {r_{mean} } \right\) (here A_{0} and A_{transf} are the grain areas before and after mapping, respectively). The grain area can be evaluated from the cell number for a particular grain [19].
As shown in Figure 5, the magnitudes of the error parameter \(E_{r}\) calculated at different crosscorrelation values increase with an increase in strain. The bigger is the lower limit of the crosscorrelation criterion, the smaller is the error change with a strain increase. When \(0.3 \le \left {r_{ij} } \right \le 1\), the obtained error values are close to zero. Therefore, we chose \(0.3 \le \left {r_{ij} } \right \le 1\) as a crosscorrelation criterion, which would be further utilized in this study.
Every 10 CAS (at \(0.3 \le \left {r_{ij} } \right \le 1\)), the ICA procedure is applied to ensure mutual independence between grains. The optimized grain topological structure can be described by
$$CC^{*} = W_{m}^\text {T} CC = \sum {_{m} } V^\text {T} ,$$
(14)
where CC is the m×n matrix of cell state variables, and the values of m and n are set to 400 each. \(W_{m}\) is the singular value decomposition matrix of mth order. \(\sum {_{m} } = I_{m \times m} \sum {}\), where \(I_{m \times m}\) is the square unit diagonal matrix of mth order. Σ is the m×n rectangular nonnegative real diagonal matrix, and V is the \(CC^\text {T} CC\) characteristic vector matrix of the nth order.
2.3 ICAbased Grain Topology Deformation Method
In 2012, Chen and Cui from Shanghai Jiao Tong University developed a new grain topology deformation technique with a double coordinate system, which was based on the previously used model [1, 14]. The substance coordinate system was adopted to describe grain deformations and changes in cell sizes with an increase in strain. The cellular coordinate system was used to describe grain nucleation and growth of recrystallized grains (the related cell sizes were not affected by the strain increase). Such a model ensures that the new generation of recrystallized grains exhibit equiaxial growth and thus can describe the compression deformation process more accurately.
However, the majority of grain topological studies are focused on describing the 2D or 3D size distributions and do not offer a comprehensive solution for the problem of overlapping grains in a topological structure. In order to reduce the grain overlap and better arrange them to fill as much space as possible, an optimized topology deformation technique based on Chens’ model and ICA is proposed in this study.
It is well known that, under certain conditions, new grains are formed at grain boundaries and grow in the equiaxial mode when the strain in the deformation area gradually approaches a critical value. Figure 6 describes the proposed topology deformation optimization model in more detail. In the substance coordinate system (shown in panel M1), the initial grains and grain boundaries are denoted by square cells. In panel M2, new grains and grain boundaries are formed with a strain increase. The dimensions of the deformed grains can be estimated according to \(v_{x} = ue^{\varepsilon }\) and \(v_{y} = ue^{  \varepsilon }\), where u is the side length of a square cell, \(\varepsilon\) is the strain, \(v_{x}\) is the compressed long side, and \(v_{y}\) is the compressed short side. During this process, thickening and overlapping of the grain boundaries inevitably occur. In panel M3, abnormal grain sizes and grain overlapping are mitigated by ICA; when the strain reaches the critical value, new grains are formed at the grain boundaries. In panel C1, the grain boundaries in the substance coordinate system are mapped to the cellular coordinate system to ensure that the recrystallized grains grow in the equiaxial mode. The abnormal grain shapes and grain boundaries observed during recrystallized grain growth in panel C2 are subsequently corrected by the ICA procedure depicted in panel C3. Panel M4 shows how the grain boundaries in the cellular coordinate system are mapped back to the substance coordinate system. In panel M5, new grains and grain boundaries are formed with a strain increase; as a result, the entire grain size changes followed by the next round of DRX.
2.4 CA Model Flowchart for the DRX Process
Figure 7 describes the CA model flowchart for the DRX process, which assumes normal grain growth in the initial topological structure. The dislocation density for the initial grain structure is set to 10^{9}/m^{2}. According to Kugler’s suggestion made in 2004 [21], the shortest time of a cell growth is adopted as the CAS for the material and thermal deformation properties, which allows estimating the strain increment \(\Delta \varepsilon_{CA} = \dot{\varepsilon } \cdot CAS\) and loop number \(n = \frac{{\varepsilon_{total} }}{{\Delta \varepsilon_{CA} }}\) parameters for the DRX simulations (\(\varepsilon_{total}\) is the total strain). When the dislocation density \(\rho_{ij}\) of a cell reaches the critical dislocation density \(\rho_{c}\), the cell size in the substance coordinate system changes (provided that the sum of the strain increment \(\sum {\Delta \varepsilon_{CA} }\) is proportional to the change in strain increment \(\Delta \varepsilon_{transf}\)). Afterwards, the grain boundaries in the substance coordinate system are mapped to the cellular coordinate system.
The cellular space is then scanned to update grain boundary variables and cell sizes. When cells are formed at the grain boundaries, the boundary variables are equal to 1. When the cell dislocation density at the grain boundaries exceeds \(\rho_{c}\), the cell recrystallization nucleation occurs. The initial dislocation density of the nucleated cells is set to 10^{−10}/m^{2}, and the initial grain orientations are selected randomly. The recrystallization number increases by 1 with every new nucleation event, while the DRX grains grow according to the CA state transition rules. As a result, compressed parent and equiaxially grown DRX grains are produced. When the grain correlation coefficients \(\left {r_{ij} } \right\) of the cell state variables are in the range between 0.3 and 1, the ICA is activated to make the cells statistically independent. The \(\left {r_{ij} } \right\) values can be determined by
$$\left {r_{ij} } \right = \left {\frac{{n\sum {c_{i} c_{j}  \sum {c_{i} \sum {c_{j} } } } }}{{\sqrt {n\sum {c_{i}^{2}  (\sum {c_{i} )^{2} } \sqrt {n\sum {c_{j}^{2}  (\sum {c_{j} )^{2} } } } } } }}} \right ,$$
(15)
where c_{i} and c_{j} are the ith and the jth cell state variables, respectively, and n is the grain internal cell number.
When the present cells do not meet the DRX requirements in the cellular space, the grain boundaries in the cellular coordinate system are mapped back to the substance coordinate system. If the change sum of the strain increment \(\sum {\Delta \varepsilon_{transf} }\) is equal to the total strain value \(\varepsilon_{total}\), the cell state variables are sent to the output indicating that the current simulation loop is complete.