State-of-the-Art Review of the Simulation of Dynamic Recrystallization

Abstract

The evolution of microstructures during the hot working of metallic materials determines their workability and properties. Recrystallization is an important softening mechanism in material forming that has been extensively researched in recent decades. This paper comprehensively reviews the basic methods and their applications in numerical simulations of dynamic recrystallization (DRX). The advantages and shortcomings of simulation methods are evaluated. Mean field models are used to implicitly describe the DRX process and are embedded into a finite element (FE) program for forming. These models provide recrystallization volume fraction and average grain size in the FE results without requiring extra computational resources. However, they do not accurately describe the microphysical mechanism, leading to a lower simulation accuracy. On the other hand, full field methods explicitly predict grain topology on a mesoscopic scale, fully considering the microscopic physical mechanism. This enhances the simulation accuracy but requires a significant amount of computational resources. Recently, the coupling of full field methods with polycrystal plasticity models and precipitation models has rapidly developed, considering more influencing factors of recrystallization on a microscale. Furthermore, integration with evolving machine learning methods has the potential to significantly improve the accuracy and efficiency of recrystallization simulation.

1. Introduction

Hot working is a crucial process for the structural components of metal and alloys with ideal strength and ductility. The external shape and internal microstructures are equally important for ensuring a qualified hot wrought component [1]. However, controlling the microstructures during hot forming is more challenging than controlling the shape. Recrystallization is a key in the evolution of microstructures in metallic materials, and it can be categorized into static recrystallization (SRX), dynamic recrystallization (DRX), and meta-dynamic recrystallization (mDRX) or post-recrystallization [2]. SRX occurs during the annealing of a material that has undergone plastic deformation. The nucleation and growth of new and strain-free grains take place during the annealing process. DRX occurs concurrently with plastic deformation. The nucleation and growth of new grains happen continuously during the deformation process (Figure 1). mDRX spans the plastic deformation and subsequent phase, where nucleation occurs during deformation, and crystal growth happens in the post-deformation phase. DRX receives more attention than SRX and mDRX because it significantly influences not only the service performance but also the workability of the deformed material. This process can lead to the refinement of the grain structure, thereby improving mechanical properties such as strength and toughness. At the same time, DRX affects processing technology itself because it is an important softening mechanism of metal and alloy. By the DRX of deformed materials, a reduced forming load and improved hot workability can be achieved. So, understanding and controlling recrystallization are essential for optimizing the hot working processes to achieve the desired microstructural characteristics and mechanical performance.

DRX is a process in which a new grain structure is generated by the formation and migration of high-angle grain boundaries (HAGBs) during plastic deformation at medium and high temperatures, driven by strain energy [3]. In single-pass cold forming, the amount of plastic deformation that metallic materials undergo is generally low. This limited deformation makes it challenging to achieve a uniform microstructure by only SRX during the subsequent annealing process [4]. Therefore, DRX in the hot forming of wrought metals and alloys has become an important means to refine grains and improve the mechanical properties of materials. The DRX mechanism is closely related to the deformation mechanism, which can be divided into four types: discontinuous dynamic recrystallization (DDRX), continuous dynamic recrystallization (CDRX), geometric dynamic recrystallization (GDRX) [2,5], and twin-induced recrystallization (TDRX) [6,7]. These recrystallization mechanisms play a crucial role in determining the final microstructure and properties of materials processed through plastic deformation. DDRX is favored by low stacking fault energy (SFE), intermediate to high temperatures, high strain rates, coarse initial grain size, and high deformation levels. CDRX occurs at high SFE, high temperatures, low to moderate strain rates, fine initial grain size, and low to moderate deformation. Both GDRX and TDRX depend on special conditions, for example, high strain or severe deformation, high temperatures, and high strain rates, and even TDRX only occurs in special materials with twin as the main deformation mechanism. The contribution of each mechanism to the overall DRX behavior of the material is affected by factors such as the type of alloy (possessing different SFE), initial grain size, deformation conditions, and strain degree [2,8].

To date, the DDRX mechanism has received more attention than other recrystallization mechanisms [8], particularly in the field of numerical simulation research. During the hot forming process of metals and alloys, DDRX is often observed, that is, the HAGBs protrude to form recrystallization nuclei, and then, the new nuclei grow by consuming the deformed matrix. At present, a large number of scholars have conducted in-depth analysis and research on the DDRX mechanism [2,8,9,10,11,12,13,14] and have reached the following consensus. The dislocation density of deformed metals and alloys reaching the critical value is a precondition for starting the nucleation of DDRX. DDRX nucleation is related to strain-induced grain boundary migration. During the plastic deformation process accompanying recrystallization, the sub-grains on both sides of the HAGBs have different dislocation densities, that is, there is an energy storage difference on both sides of the grain boundary. When the driving force for grain boundary migration provided by the energy storage difference is greater than the resistance generated by the interaction between the grain boundary and the lattice dislocation, the grain boundary migrates toward the grain with high dislocations, forming a grain boundary bulging. As deformation progresses, grain boundary sliding near the bulging leads to the appearance of additional inhomogeneous strains, which in turn induces the formation of small low-angle grain boundaries (LAGBs) at the root of the bulging. By capturing lattice dislocations, LAGBs gradually transform into HAGBs, thus completing the DDRX nucleation process. Subsequently, driven by the strain energy difference, the new nuclei with low dislocation density gradually grow toward the matrix with high dislocation density through grain boundary migration. The migration of HAGBs of recrystallized nuclei during DDRX captures and dissociates a large number of lattice dislocations generated during deformation. Compared to other DRX mechanisms with only limited grain boundary migration, DDRX softens the material to a greater extent [8].

The investigation of DRX through experimental methods has some inherent limitations. DRX happens quickly, and it is complicated to capture this process through deformation testing. The current experimental methods find it challenging to observe the DRX process that occurs alongside the plastic deformation of materials, particularly within the sample. Compared to the in situ observation of SRX reported recently [15,16,17], there are few similar reports on DRX. Therefore, numerical simulation emerges as an effective approach to studying DRX. The numerical models of microstructure evolution, especially those involving DRX, can be broadly categorized into two types based on their simulation output (Figure 2) [18,19,20].

(1)

Mean field models (statistical average results)

These models predict the statistical average properties of the microstructure, such as the average grain size, volume fraction of recrystallized grains, and dislocation density. They rely on simplified mathematical formulations to represent the evolution of these properties over time. They can be further subdivided into two types: empirical (phenomenological) models and internal state variable (ISV) models. Empirical models are often based on experimental observations and use empirical equations to describe the relationship between processing parameters (e.g., temperature, strain rate, and initial grain size) and microstructure evolution. Although they are relatively simple and computationally efficient, they may lack precision in capturing the detailed microstructure morphology. ISV models use internal state variables to represent the microstructural state and its evolution during deformation. They offer a balance between accuracy and computational efficiency by capturing the average behavior of microstructures without explicitly modeling their spatial distribution.

(2)

Full field models (Microstructure morphology)

These models simulate the detailed spatial and temporal evolution of microstructures, offering a more comprehensive representation of the microstructure, including its morphology and heterogeneity. They are more computationally intensive but provide richer and more accurate predictions. Full field models are subdivided into four types: the Monte Carlo (MC) model, vertex model, cellular automaton (CA) model, phase-field (PF) model, and level set (LS) model.

The MC model uses probabilistic rules to simulate the evolution of microstructures, particularly useful for modeling grain growth and recrystallization processes. It captures the complex interaction between grains during DRX. In the vertex model, the microstructure is depicted using polygons to represent grains. Each vertex of the polygon, known as the triple junction, corresponds to an intersection of three adjacent grains. During recrystallization simulation, the grain topology is updated by moving the vertices and edges. The main rule of the vertex model is to minimize the free energy. The CA model divides the material into discrete cells, each representing a small volume of material. The evolution of each cell’s state (e.g., grain orientation, dislocation density) is governed by local rules, allowing for the simulation of microstructural evolution over time. The PF model uses thermodynamic principles to describe the evolution of microstructures by solving partial differential equations. It is particularly powerful in capturing the dynamics of phase transformations and microstructural evolution during DRX. The LS model tracks the evolution of interfaces between different phases or grains within the material, useful for modeling complex morphologies and topological changes during DRX [20].

Mean field models offer simplicity and speed, making them suitable for large-scale simulations where detailed microstructural information is not critical. Full field models, on the other hand, provide detailed insights into the microstructural evolution but require significant computational resources. The choice between these models depends on the specific requirements of the study, such as the need for precision versus computational efficiency. In this paper, we review the numerical simulation methods of recrystallization, especially dynamic recrystallization, including their basic principles, equations, strengths, weaknesses, and recent advances and applications.

This paper aims to provide a comprehensive review of the methods used to simulate recrystallization, and some applications especially in DDRX are emphasized. Section 2 reviews the mean filed methods. Section 3 reviews the full field methods. In Section 4, the multi-physical coupling models are reviewed. The numerical simulation models of DRX are summarized and outlooked in Section 5.

2. Mean Field Methods

2.1. Empirical Models

Mean field methods include the two models, the empirical model and the internal state variable (ISV) model. Empirical models of recrystallization simulation are typically based on experimental observations and provide practical ways to predict recrystallization behavior under various conditions. These models often involve mathematical expressions that relate recrystallization kinetics parameters (external state variables) like temperature, strain, and time.

The JMAK (Johnson–Mehl–Avrami–Kolmogorov) model, also called the Avrami equation, is one of the most widely used empirical models for implicitly describing the kinetics of phase transformations and recrystallization [21]. Before being used to predict the fraction of recrystallized material as a function of time, the JMAK model had been applied in other fields to model some processes with nucleation and growth [22]. Initially, the recrystallization process during isothermal annealing processes was modeled using the JMAK model. It is particularly useful in metals like steel [23,24,25,26,27,28,29,30,31], where recrystallization significantly affects mechanical properties. The kinetics of recrystallization comprise of two important outputs, the recrystallization volume fraction and the average recrystallized grain size.

The dynamic recrystallization volume faction is described as [26]

$$X(\epsilon )=1-\exp \left[-k{\left({\displaystyle \frac{\epsilon -{\epsilon }_{\mathrm{c}}}{{\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{c}}}}\right)}^n\right]$$

(1)

where $X$ is the recrystallized fraction at strain $\epsilon ,$ ${\epsilon }_{\mathrm{c}}$ is the critical strain triggering recrystallization, ${\epsilon }_{\mathrm{s}}$ is the strain corresponding to the steady-state stress, $k$ is a material constant, and $n$ is the Avrami exponent, which depends on the nucleation and growth mechanism. For simplicity, ${\epsilon }_{\mathrm{s}}$ is often replaced by ${\epsilon }_{0.5}$, which is a characteristic strain when the recrystallization volume fraction achieves 0.5 [23,25,27]. It needs to be noted that the strain terms of Equation (1) are altered to time in static recrystallization.

The critical strain model defining the critical strain (${\epsilon }_{\mathrm{c}}$) required for the onset of DRX is often empirically related to the Zener–Hollomon parameter (Z) using a power–law relationship [26,29,30]:

$${\epsilon }_{\mathrm{c}}=A{d}_0^{p_0}{Z}^m$$

(2)

where A, m, and $p_0$ are materials constants empirically calibrated by experiments, and $d_0$ is the initial grain size. The parameter Z, the Zener–Hollomon parameter, combines the effects of strain rate and temperature into a single parameter, allowing for the prediction of material behavior under various processing conditions [23].

$$Z=\dot{\epsilon }\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{Q_{\mathrm{d}\mathrm{e}\mathrm{f}}}{RT}}\right)$$

(3)

where $\dot{\epsilon }$ is strain rate, $Q_{\mathrm{d}\mathrm{e}\mathrm{f}}$ is the deformation activation energy, R is the gas constant, and T is the Kelvin temperature. Considering the effect of the loading speed, the strain rate can be added in Equation (2) [25,31].

The characteristic strain ${\epsilon }_{0.5}$ need to be determined by many processing conditions, as [24,27]

$${\epsilon }_{0.5}=\alpha d_0^{p_1}{\epsilon }^{q_1}{\dot{\epsilon }}^{m_1}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{Q_{\mathrm{d}\mathrm{r}\mathrm{x}}}{RT}}\right)+c_1$$

(4)

where $\alpha , p_1$, $q_1,m_1$, and $c_1$ are materials constants; $Q_{\mathrm{d}\mathrm{r}\mathrm{x}}$ is the active energy of DRX; and the other symbols have the same meaning as those mentioned above.

Equation (2) helps in determining the conditions under which new grains will form during deformation, which is crucial for controlling the microstructure and mechanical properties of the material.

The model of average size of DRX grains $d_{\mathrm{D}\mathrm{R}\mathrm{X}}$ is described as [24,32]:

$$d_{\mathrm{D}\mathrm{R}\mathrm{X}}=\beta d_0^{p_2}{\epsilon }^{q_2}{\dot{\epsilon }}^{m_2} \mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{Q_{\mathrm{d}\mathrm{r}\mathrm{x}}}{RT}}\right)+c_2$$

(5)

where $\beta ,p_2$, $q_2,m_2$, and $c_2$ are materials constants.

Thus, the average grain size $d$ of investigated region is expressed as

$$d={Xd}_{\mathrm{D}\mathrm{R}\mathrm{X}}+(1-X)d_0$$

(6)

The JMAK model has been modified and improved to properly simulate the DDRX under complex conditions in recent decades and applied in materials besides steels, for example, superalloys [32,33,34], aluminum alloys [35,36], zinc alloys [37], and titanium alloys [38]. It can be embedded into a finite element (FE) program for metal forming simulation, and the recrystallization volume fraction and grain size are calculated for every element integration point, without additional computational cost [23]. Figure 3 shows the simulation results of compression, including the strain, DRX fraction, and average grain size distribution inside workpieces. Like the state variables on the macroscale, the microstructure’s temporal and spatial evolution can be estimated along with FE calculation. Even so, it has an inherent shortcoming: the calculation accuracy of microstructure evolution is low due to lacking the physical basics. Three assumptions in the empirical model—a random distribution of nucleation sites, isotropic growth, and a constant growth velocity—lead to a low prediction confidence. The main cause is that the model depends on only the processing conditions (external state variables), including stress, strain, strain rate, temperature, and initial grain size, without considering microscopic mechanisms.

2.2. Internal State Variable (ISV) Models

Internal state variables (ISVs), describing the material’s behavior under certain conditions by tracking internal changes that are not directly observable, have been added to the constitutive models for plastic deformation, microstructure evolution, and damage prediction in recent decades. The dislocation density, an important ISV, is employed as the parameter of the DRX model, that is, the internal variable method is formed [39]. Additionally, the average grain size and DRX volume fraction are considered in the flow stress model of materials, which is called the unified constitutive model [40,41]. The plastic deformation history is not considered in the empirical models; only the external variables related to the deformation state (stress, strain, and strain rate) are involved. In ISV models, the stored deformation energy as the essential driving factor of DRX is considered through the dislocation density. From this perspective, ISV models are closer to the physical essence of DRX than empirical models.

Table 1. Comparisons between two models of the mean field method.

	Empirical Model	ISV Model
Basic strategy	Data-driven, fitting to experiment data (stress, strain, strain rate, initial grain size, temperature, etc.)	Physically-based, describing microscale mechanisms
Implementation complexity	Simpler, fewer variables	More complex, involves internal state variables (dislocation density)
Generalization Performance	Limited, only valid inside trained conditions	Good, can be extrapolated at a large range
Physical intension	Lacks physical meanings	Provides deep physical insights
Application universality	Suitable for specific conditions, not for other microstructure evolutions	Suitable for complex conditions, e.g., phase transformation
Computing cost	Inexpensive (hardly increase the FE costs)	More intensive (significantly increase computing costs)

compares empirical models and ISV models across five different criteria.

The evolution of dislocation density $\rho$ is generally described using the Kocks–Mecking (KM) equation that represents the competition of strain hardening (dislocation pile up) and softening (dynamic recovery) [42], as shown in Equation (7). There are other equations different from the KM equation [42,43].

$${\displaystyle \frac{\partial \rho }{\partial \epsilon }}=K_1\sqrt{\rho }-K_2\rho$$

(7)

where $\epsilon$ is the equivalent plastic strain, $K_1$ is a constant charactering the dislocation storage with the deformation progress, and $K_2$ is the dynamic recovery depending on the temperature and strain rate [44]. If the static recovery is considered (deformation at a strain rate lower than $0.001 {\mathrm{s}}^{-3}$), an additional term has to be added to Equation (7) [40].

Strain-based DRX onset criteria are applicable to a stable deformation. When the strain of hot-worked metals under non-steady conditions varies greatly in space and time, the critical strain is not applicable to determine the onset of DRX [39]. In ISV models, the critical dislocation density ${\rho }_{\mathrm{c}}$ is used for a condition triggering DRX nucleation, implying the storage deformation energy exceeds the capillary actions of the nucleus [45,46]. The critical dislocation density ${\rho }_{\mathrm{c}}$ is often expressed [40,43,45]:

$${\rho }_{\mathrm{c}}={\left({\displaystyle \frac{20\dot{\epsilon }G}{3bL{M}_{\mathrm{g}}{\tau }^2}}\right)}^{{\displaystyle \frac{1}{3}}}$$

(8)

where $\dot{\epsilon }$ is strain rate, $G$ is the grain boundary energy per unit area, b is the Burgers vector, $L$ is the dislocation mean-free-path length, $M_{\mathrm{g}}$ is the grain boundary mobility depending on the temperature, and $\tau$ is the dislocation line energy [2].

The relationship between the stress and the dislocation density constructs the macro–micro connection. During a deformation simulation, the stress increment $∆\sigma$ of a representative grain $i$ is described as [43]

$$∆{\sigma }_i=M_{\mathrm{T}}\alpha \mu b\sqrt{{\rho }_i}$$

(9)

where $M_{\mathrm{T}}$ is the Taylor factor evolving with the deformation but often considered constant for simplicity, $\alpha$ is a constant, and $\mu$ is the shear modulus. The stress of the ith grain is the sum of the initial yield stress and $∆{\sigma }_i$.

The grain size and the DRX grain volume fraction in ISV model are the same as those in the empirical models.

The ISV method captures the complex phenomena associated with microstructural evolution during deformation, such as grain refinement, nucleation, and grain growth. Compared to purely empirical models (e.g., JMAK), the ISV models allow a more detailed and physically based description of DRX processes. ISV-based models can be integrated with macroscale simulation tools (such as finite element models) to predict both the mechanical response and microstructural evolution during complex forming processes, such as hot rolling or forging [47,48,49,50,51]. Derived from physical principles (e.g., dislocation dynamics, nucleation theories), ISV models provide a more fundamental understanding of microstructural changes, enabling better prediction and optimization of material processing conditions.

The unified ISV model generally consists of two parts: One is the constitutive model that describes the plastic deformation behavior of the material, which links the microstructural evolution with the macroscopic deformation behavior through the internal state variables that characterize the microstructural characteristics. The other is the model for calculating the evolution of the internal state variables. The unified ISV model can achieve simultaneous prediction of microstructural evolution and macroscopic flow stress. Puchi-Cabrera et al. [52] established an ISV model that takes into account the softening effect of DRX based on Anand’s modeling method for DRV and expressed the softening effect of DRX by normalized internal state variables related to the DRX volume fraction. Galindo-Nava et al. [53,54] proposed a new ISV model based on kinetics and thermal statistics to describe the hot deformation behavior of face-centered cubic (FCC) alloys. All parameters in the model have clear physical meanings, breaking through the limitation of the traditional empirical model that requires model parameters to be fitted. The DRX coefficient is used in their model instead of the DRX volume fraction to calculate the softening degree of DRX on the material. Although the above-mentioned physical ISV model has a high calculation accuracy, it does not include the prediction of the average grain size of DRX and cannot fully describe the DRX behavior of the material. Lin et al. [55,56] combined the viscoplastic constitutive model and the DRX physical model to establish a unified ISV model that can simultaneously calculate the DRX volume fraction, average grain size, and related mechanical responses. On this basis, Tang et al. [57] further added the influence of the Hall–Petch effect to the constitutive model and specifically considered the role of nucleation rate on DRX dynamics. The established ISV model has good accuracy. As shown in Figure 4, compared to the empirical model, the significant advantage of the unified ISV model with clear physical meaning is that the model has good extrapolation [58].

The ISV models typically require more extensive calibration using experimental data than the empirical models. The determination of material constants and the validation of the model can be time-consuming and complex, as the variables must represent real physical behaviors accurately. When applied to large-scale problems, finite element simulations coupling with the ISV approach can consume considerable computation resources. Additionally, the ISV model simulates the DRX implicitly, meaning that some spatial aspects of the microstructure, such as the topology of the grain boundary network, are not directly simulated. This can limit the accuracy in predicting DRX accompanying specific mechanisms that are material-dependent, such as solute drag effects, anisotropic, complex phase transformations, or second-phase particle interactions in alloys.

3. Full Field Methods

To explicitly simulate the process of recrystallization, full field methods were introduced to the research in the microstructure evolution of metals or alloys. Researchers have developed various numerical models to understand the recrystallization behavior of metals or alloys [59].

3.1. Monte Carlo Model

The MC model, also called the MC Potts model, utilizes probabilistic rules to simulate the thermodynamic and kinetic processes involved in recrystallization. The simulated region is discretized using a regular lattice, and every lattice site is labeled by an integer number representing the grain orientation. The MC simulation typically operates on a lattice site representing a portion of the material, such as an atom or a group of atoms. A set of lattice sites compose a grain and are assigned the same orientation, represented by a number between 1 and Q (total number of grains) [60,61]. In this way, the region under simulation is mapped into a series of grains. The simulation is driven by the minimization of system energy, where grain boundaries are considered regions of higher energy. The total energy of the system is typically calculated based on the sum of the boundary energies between neighboring grains. The MC model assumes that recrystallization reduces the overall energy by favoring the growth of grains with lower stored energy (e.g., new recrystallized grains) into regions of higher stored energy.

The MC simulation includes three steps, the nucleation, growth, and termination of recrystallized grains. First, new grains are introduced at randomly selected sites, representing the nucleation of recrystallized grains. These grains typically have lower energy (higher mobility) than the surrounding deformed material. Second, the new grains grow by consuming the surrounding high-energy, deformed material [60,61]. The MC algorithm simulates this by allowing the new grain orientations to propagate across the lattice, reducing the overall system energy. The model changes the orientation of the selected site to match one of its neighboring sites, simulating the movement of a grain boundary. The simulation continues through many MC steps until the system reaches a stable state, where no significant energy changes occur, representing the end of the recrystallization process.

In the Monte Carlo model, the microstructure is discretized by a two-dimensional lattice by assigning each lattice site an index $S$, which represents the crystallographic orientation of the material in the vicinity of the site. A crowd of sites with the same index is considered within one grain, while peripheral sites with different indices are separated by a grain boundary. Store deformation energy proportional to the local dislocation density is assigned to each site. The total energy of the system includes contributions from grain boundary energy and stored energy. Hence, for the case of isotropic grain boundary energy, the total energy of the system $E$ is [60]:

$$E={\displaystyle \frac{J}{2}}\sum _i^n\sum _j^m\left(1-{\delta }_{S_i{S}_j}\right)+\sum _i^n{H}_{S_i}$$

(10)

where the first term represents total grain boundary energy, the grain boundary energy $J$ is a function of misorientation and follows the Read–Shockley equation, $\delta$ is Kronecker delta function, $n$ is the total number of sites, and $m$ is the number adjacent sites of site $j$. Thus, nearest neighbor pairs for each site contribute $J$ to the system energy when they possess different orientations and are zero otherwise. The second term of Equation (1) is the stored energy associated with each site with orientation $S$.

To decide whether to accept a proposed change (e.g., the movement of a grain boundary), the Metropolis criterion is used. This criterion is based on the change in energy $\Delta E$ due to the proposed change. The proposed change is accepted t unconditionally, if $\Delta E\le 0$, or else, a random number $r$ between 0 and 1 is generated. The proposed change is accepted if $r\le P$, and $P$ is a certain transition probability [60].

$$P\left(\Delta E\right)=\left\{\begin{array}{ll}1& \Delta E\le 0\\ exp\left(-{\displaystyle \frac{\Delta E}{KT}}\right)& \Delta E>0\end{array}\right.$$

(11)

where $K$ is the Boltzmann’s constant, and $T$ is the simulation temperature.

Spatially, an MC simulation can topologically present the final distribution and size of grains after recrystallization. Temporally, the evolution of grain size, grain boundary mobility, and other microstructural features can also be analyzed over time.

The main steps of an MC simulation for DRX are as follows:

(1)

Discretization: Divide the analysis domain into lattice points.

(2)

Initial conditions: Assign initial parameters, such as grain orientation and dislocation density.

(3)

Monte Carlo steps:

a.

Select a site.

b.

Calculate energy.

c.

Update the state.

d.

Repeat the process.

e.

Initiate the nucleation of new grains.

f.

Allow for the growth and coarsening of grains.

(4)

Boundary conditions: Apply periodic boundary conditions to the simulation.

(5)

Postprocessing: Output and visualize the simulation results.

The MC model is relatively simple to implement and can capture qualitative features of recrystallization, such as grain growth and the effect of different initial conditions on the final microstructure [62]. It facilitated the development of full field methods in recrystallization simulation at the early stage [63,64,65,66,67,68,69]. However, it uses probabilistic rules to model recrystallization, relying on stochastic processes rather than direct physical laws. While effective for capturing grain growth and qualitative features, it does not inherently account for the actual driving forces (e.g., stored energy, curvature) governing the process. In MC simulations, time evolution is not directly tied to real physical time. Instead, it is based on “Monte Carlo steps”, which do not correspond to real units of time. This artificiality makes it challenging to relate the simulation results to real-world kinetics or timescales. It also struggles with accurately modeling complex phenomena and detailed microstructural features [65].

The Monte Carlo (MC) model has recently seen advancements in modification and application. Jedrychowski et al. [70] simulated partial recrystallization during the short annealing of zirconium using the MC model. They introduced the strain-induced boundary migration (SIBM) recrystallization model to replace the classical nucleation approach. Their results indicated that the SIBM model simulations aligned more closely with experimental data than those using the classical nucleation model. Xu et al. [71] utilized the kinetic Monte Carlo (KMC) model to simulate abnormal grain growth in an iron–silicon (Fe-Si) alloy. In this study, a composite system with anisotropic grain boundary mobility and energy played a crucial role in achieving accurate calculations. Zheng et al. [72] proposed an MC-assisted PF model to simulate grain growth in Fe-Si alloy and low-carbon steel. Their simulations incorporated an experimental data-based kinetic model to correlate simulation time with real time. Chao et al. [73] introduced a modified MC model to describe the grain growth of two phases in a short-fiber-reinforced metal composite. The experimental results confirmed the validity of this model and analyzed the positive effect of short fibers on the growth and refinement of matrix alloy grains.

3.2. Vertex Model

Kawasaki and his colleagues introduced the vertex model to simulate grain growth in two- and three-dimensional spaces [74,75]. To date, the vertex model has been mostly applied in two-dimensional recrystallization simulation [76,77,78,79]. This model focuses on the dynamics of grain boundaries and the interactions between grains in a polycrystalline material. The microstructure is mainly represented by triple points, which are the intersections of three adjacent polygons (mostly hexagons) characterizing grains. These triple points are called vertices, and the edges of polygons linking two vertices are considered grain boundaries. Therefore, the microstructure is described as a network of edges connecting vertices. The evolution of the grain structure is depicted by the movements of the triple points. To discrete the grain boundary, virtual vertices are added between two triple junctions [76]. Unlike the MC model, the vertex model is purely deterministic. The behavior of grains is governed by strict and unique laws; so, each program execution with the same initial data leads to the same final result. The grain boundary energy and mobility are anisotropic in both models, meaning they depend on the crystal misorientation between adjacent grains.

The vertex model for recrystallization simulation primarily focuses on the movement and interaction of grain boundary vertices in a polycrystalline material. The model can be extended to incorporate partial differential equations (PDEs) to describe the dynamics of these vertices under the influence of various forces, such as curvature-driven motion and stored energy differences.

The PDE governing the motion of vertices is a Lagrange equation, shown in Equation (12) [76,80].

$${\displaystyle \frac{\partial R}{\partial v}}=-{\displaystyle \frac{\partial E}{\partial r}}$$

(12)

where $R$ is the dissipated free energy by the motion of the vertices with a velocity $v$ and $E$ is the potential energy (deformation energy) dominating from the grain boundary area for vertices with positions $r$. The dissipated free energy $R$ is expressed as $R=\gamma {\int }_{\mathsf{\Gamma }}\kappa \mathbf{v}·\mathbf{n}d\mathsf{\Gamma }$, where $\gamma$ is the grain boundary energy per unit length, $\kappa$ is the local curvature of the grain boundary, $\mathbf{v}$ is the velocity vector of the vertex, $\mathbf{n}$ is the unit normal vector to the grain boundary at the vertex, and $\mathsf{\Gamma }$ is the grain boundary segment [80]. The movement of grain boundary vertices is often driven by the curvature of the grain boundaries. The velocity $v$ of a vertex due to curvature can be expressed as: ${\displaystyle \frac{\partial r}{\partial t}}=v=\gamma \kappa n$, where $t$ is time.

Forces acting on vertices include those due to grain boundary tension T (curvature-driven forces) and stored energy differences across grain boundaries. Grain boundary tension T is expressed by the sum of the grain boundary energy per unit length ($\gamma$) and its second derivative with respect to the boundary inclination angle $\varphi$, that is, $T=\gamma +{\displaystyle \frac{{\partial }^2\gamma }{\partial {\varphi }^2}}$. The force due to stored energy differences is expressed as $F=\Delta E\mathbf{n}$, where ΔE represents the energy difference across the grain boundary. Thus, the velocity of the grain boundary is [80]:

$$v=m_{\mathrm{G}\mathrm{B}}T+m_{\mathrm{G}\mathrm{B}}F=m_{\mathrm{G}\mathrm{B}}\left[\kappa \left(\gamma +{\displaystyle \frac{{\partial }^2\gamma }{\partial {\varphi }^2}}\right)+\sum _i^{n-1}{\Delta E}_i{\mathit{n}}_{\mathit{i}}\right]$$

(13)

where $m_{\mathrm{G}\mathrm{B}}$ is the mobility of the grain boundary. The first item represents the force due to the curvature of the grain boundary. The curvature of the boundary κ creates a pressure that drives the boundary to reduce its curvature, leading to grain boundary migration. The grain boundary energy per unit length γ determines the magnitude of this driving force. A higher grain boundary energy results in a stronger driving force for boundary movement. The second item indicates that the stored energy difference across the grain boundary acts as another driving force. This energy difference arises from dislocations and defects within the grains. Grain boundaries tend to move towards regions with a lower stored energy, facilitating the reduction in the overall internal energy in the material.

The following steps are involved in completing a DRX simulation using the vertex model:

(1)

Initialization: Define the initial grain structure using vertices and their connections. Assign dislocation density and other microstructural properties within each grain.

(2)

Define energy functions: Establish the grain boundary energy function based on the boundary length and the misorientation angle between adjacent grains. Additionally, bulk energy functions related to dislocation density and other internal stresses should be established. Thus, total energy covering the two energy terms mentioned above is formed.

(3)

Vertex movements: Randomly move the vertices, calculate energy changes, and accept or reject these moves based on the Metropolis criterion.

(4)

Grain nucleation and growth: Introduce new grains at high-energy sites and move vertices to reduce the total energy while handling topological changes. Implement either periodic or fixed boundary conditions.

(5)

Time stepping: Increment the time, repeat the process, and check for convergence.

(6)

Output: Record and analyze the microstructural data, perform statistical analyses, and visualize the results.

Recently, the vertex models have been improved and developed. In the vertex model, the vertex velocities calculation is based on the total energy minimization and a grain boundary configuration changes at each step. Consequently, a global and complex system of equations needs to be solved at each step, which pushes up calculation costs. To simplify calculations, Piękoś et al. [81] proposed a stochastic vertex model, in which the movements of triple points were determined by the Monte Carlo method, instead of by the deterministic equation. The comparison between the simulated and experimental results of recrystallization indicated that it was an effective way. Sazo et al. [82] developed a fully-parallelizable matrix-free Graphical Processing Unit (GPU)-based algorithm to implement a 2D vertex model of recrystallization based on the stored energy formalism. To ensure a robust GPU implementation, they used a novel polling system to handle topological transitions. Mazzi [83] introduced a new approach for representing microstructure. In addition to the network of grain boundaries and vertices, the grain orientation is represented by a regular grid of points in the physical space. Each point is assigned three Euler angles to represent the grain orientation. This modified vertex model enhances the efficiency of reproducing the overall characteristics of the grain growth process.

The vertex model is relatively simple compared to other models, such as the phase field or cellular automaton models, which makes it computationally efficient. This model handles topological changes naturally. Vertex movement and the rearrangement of grain boundaries can be easily implemented, making it effective for studying grain growth and recrystallization, where grains frequently merge or split. However, the vertex model can sometimes oversimplify the physics of grain boundary movement and interactions. For example, it assumes that boundaries move according to local energy minimization principles, which might not capture all the complexities of real microstructural evolution. Some physical phenomena, such as anisotropic grain boundary energies and mobilities, are difficult to incorporate accurately into the vertex model. This can limit the model’s ability to simulate materials with complex grain boundary properties, especially for DRX. Lately, there has been a trend of combining the crystal plastic finite element method (CPFEM) with the vertex model for DRX simulations [84,85,86]. Sai Deepak Kumar [87] developed a coupled phase-field and vertex dynamics model for simulating grain growth. This model switches between the phase-field and vertex dynamics depending on the growth-controlling mechanism. The microstructures generated by the phase-field model can be used as input for the vertex dynamics, replacing Voronoi dissolution. This reduces the computational costs and allows for more realistic and complex microstructure simulations. This trend involves coupling the vertex model with other mesoscale simulation methods, such as the crystal plasticity model, to address its limitations.

3.3. Cellular Automaton (CA) Model

Since the end of the 1990s, the CA method has become a powerful tool for simulating DRX. The CA is a discrete model that couples multiple time and space elements with transition rules to represent a continuous process. The CA model replaces complex partial differential equations with discrete interactions of state variables between neighboring elements, making it an effective model. When used to simulate the DDRX, the nucleation model and grain growth model are functions of the temperature and strain rate, coupled with which the CA is capable of describing the microstructure evolution in different forming conditions. At the same time, the reliability of the CA model depends on the evolution rules. With the development of computer technology and comprehension of recrystallization behavior, the recrystallization CA model has been advanced in both width and accuracy.

In the CA model, the 2D/3D simulation domain is discretized into several cells, and a set of state variables is assigned to each cell. The distribution of the gains in the simulated domain can be described by specifying the value of each cell state variable. The cell state variables change with time and space according to the established evolution rules to simulate the spatiotemporal evolution of the microstructure. Evolution rules typically dictate how to determine the state variables of cell grid points at the next time step. The change in the state of a cell is determined by the current state of the cell itself and the states of its neighboring cells.

The CA model of DRX was first established by Goetz et al. [88]. The model includes four modules—microstructure initialization, dislocation density evolution, DRX nucleation, and grain growth—which laid the foundation for the process framework of the CA-DRX model. The model can reproduce the basic characteristics of DRX, such as the necklace-like microstructure and the transition of flow stress from single peak to multi-peak with the increase in the initial grain size. However, the key DRX parameter values in the model, such as the number of nuclei and growth rate, are mostly specified artificially and are not related to the thermal deformation process of the actual material. To this end, Ding and Guo [89] developed a CA model that combines the physical metallurgical principles of DRX. The nucleation rate $\dot{n}$ is considered the function of temperature T and strain rate $\dot{\epsilon }$, as shown in Equation (14) [89].

$$\dot{n}=C{\dot{\epsilon }}^m\mathit{exp}\left(-{\displaystyle \frac{Q_{\mathrm{act}}}{RT}}\right)$$

(14)

where C is a material constant, $Q_{\mathrm{act}}$ is the activation energy of recrystallization, and m is the exponent of strain rate (for most materials, m is close to 1 [89]). Thus, the nucleation rate is almost linearly proportional to the strain rate. The nucleation rate $\dot{n}$ is used to estimate the volume fraction of DRX $\eta$.

$$\eta =\dot{n}{\displaystyle \frac{\epsilon }{\dot{\epsilon }}}{\displaystyle \frac{4}{3}}\pi r^3$$

(15)

where r is the average radius of all recrystallized grains considered spherical particles.

After DRX nucleation, the nucleus continues to grow under the action of the driving force. The grain growth rate v can be expressed by the product of the grain boundary mobility $m_{\mathrm{G}\mathrm{B}}$ and the driving force per unit grain boundary area P, that is, $v=m_{\mathrm{G}\mathrm{B}}P$. The grain boundary mobility $m_{\mathrm{G}\mathrm{B}}$ is expressed as:

$${\mathrm{m}}_{\mathrm{G}\mathrm{B}}={\displaystyle \frac{\delta D_{0\mathrm{b}}b}{kT}}\mathit{exp}\left(-{\displaystyle \frac{Q_{\mathrm{b}}}{RT}}\right)$$

(16)

where $\delta$ is the characteristic grain boundary thickness, $D_{0\mathrm{b}}$ is the grain boundary self-diffusion coefficient, $k$ is the Boltzmann constant, and $Q_{\mathrm{b}}$ is the grain boundary self-diffusion activation energy. The driving force per unit grain boundary area is calculated as

$$P=\tau \left({\mathsf{\rho }}_{\mathrm{m}}-{\rho }_{\mathrm{R}}\right)-\gamma \kappa$$

(17)

where $\tau$ is the dislocation line energy, ${\rho }_{\mathrm{R}}$ is the dislocation density of the recrystallized grain, ${\rho }_{\mathrm{m}}$ is the dislocation density of the adjacent grain, $\kappa$ is the curvature of the grain boundary segment, and $\gamma$ is the grain boundary energy.

The numerical implementation of a CA-DRX model is carried out through the following steps:

(1)

Discretization: Divide the simulation domain into cells and set initial conditions, including dislocation density and grain orientation for each cell, etc.

(2)

Transition rules: Define the transition rules and determine the energy states and state variables for the cells.

(3)

Nucleation and growth: Introduce new grains at high-energy sites based on nucleation criteria. Allow the recrystallized grains to grow by converting adjacent cells.

(4)

Neighborhood and updates: Establish a neighborhood for each cell and create rules for state changes. Randomly select cells, calculate their energy, and update their states based on a probabilistic approach.

(5)

Time increment and convergence: Increment the time, repeat the process, and check for convergence. Apply periodic or fixed boundary conditions as needed.

(6)

Results output: Record and analyze the microstructure data, perform statistical analyses, and visualize the results.

Parameters such as dislocation density, nucleation rate, and grain boundary migration rate in the model are calculated according to physical or phenomenological formulas, which can simulate the DRX behavior of actual materials under different thermal deformation conditions. On this basis, in order to more accurately predict the DRX behavior or make the model more suitable for a specific material, researchers have made various improvements to the Ding–Guo model. The initial microstructure has a great influence on the DRX dynamics. To accurately mimic the initial microstructure in the CA model, Chen et al. [90] established the CA rules for generating initial grains based on the principles of physical metallurgy. Wang et al. [91] used multi-step nucleation in the CA program for initial grain generation to ensure the consistency of the initial grain size distribution with the experiment. The topological shape of the grain changes with deformation. Xiao et al. [92] realized the topological deformation of the grain by calculating the cell coordinate vector. Chen et al. [90] simultaneously established an invariant cell coordinate system and a variable material coordinate system and simultaneously ensured the equiaxed growth and topological deformation of the grain by mapping the cells between the cell coordinate system and the material coordinate system. To make the CA model truly reproduce the DRX process, it is necessary to adopt reasonable model descriptions and evolution rules. Xu et al. [93] proposed a description of the uneven distribution of dislocation density within the grains. Jin and Cui [94] established nucleation rules that take into account the influence of dislocation density. Chen et al. [95] introduced a multilevel DRX nucleation module into the traditional CA model to make the nucleation process closer to the actual physical mechanism. Lee and Im [96] introduced coefficients into the grain boundary mobility equation to characterize the solute drag effect in pure copper. The simulation results using this model had a better agreement with experimental data than the conventional CA model without consideration of the solute drag effect.

Since Goetz et al. [88] first introduced the CA to the simulation of DDRX of pure copper, the CA has been applied in the analysis of the DDRX or dSRX (discontinuous static recrystallization) of various materials, such as Al alloys [97], Mg alloys [91,98], steels [99,100,101,102], Ti alloys [103,104,105], pure Ti [106], and pure Cu [89,107], etc. Additionally, continuous recrystallization CA models were developed and applied in severe plastic deformation conditions [108]. Figure 5 shows the CA-simulated grains of magnesium alloy undergoing different processing conditions [91]. The accuracy of the calculations was confirmed by comparing the microstructure topologies in cellular automaton (CA) simulation with experimental observations. This was made possible by generating the actual initial grain morphologies in the CA models, instead of using a random distribution of initial grain sizes that only depicted the average grain size. After decades of research, CA models have achieved vast success in the simulation of SRX or DRX, contributing to both academic research and industrial application.

CA methods are based on physical principles and directly incorporate key factors such as grain boundary energy, storage energy, and nucleation, making them more accurate than the MC methods in simulating real-world recrystallization processes [105,106]. Compared to the phase-field method, the CA method can efficiently simulate large-scale microstructures, especially in computationally intensive cases. The CA method is highly flexible and can be adjusted to simulate different processes such as static and dynamic recrystallization, grain growth, and phase transformations. Being superior to the MC method, the CA method can be coupled with realistic physical time scales. This allows for more accurate modeling of recrystallization dynamics and facilitates the comparison of the simulation results with experimental data. Figure 6 shows 3D CA-simulated grain distribution in a drawn wire of micro-alloyed steel [108]. The micro-alloyed steel was subjected to severe plastic deformation, including continuous Accumulative Angular Drawing (AAD) in combination with wire drawing (WD) and wire flattening (WF) processes. The forming was simulated using Abaqus, and the FEM results were used as input for the CA model. The coupled simulation showed changes in the number, size, and shape of grains and subgrains, dislocation cells, and pole figures. The experimental observations confirmed the effectiveness of the simulation results.

The CA models commonly use discretized cell meshes that are typically regular shapes, such as squares or cubes. This regularity can lead to grains growing preferentially along the mesh directions, resulting in less realistic anisotropy. Additionally, the accuracy of the CA model is influenced by the size of the cell mesh. To break through these limitations, researchers have proposed several innovative approaches to CA models. For instance, Sitko et al. [109] developed a multi-mesh CA-Finite Element Method (CA–FEM) model to simulate dynamic recrystallization (DRX) during the rolling process. In this model, the dislocation density distribution within each CA cell was calculated using a finer sub-mesh for FEM calculation, which was used in the simulation of the grain growth in the relatively coarse CA cell model. Afterward, this model has evolved into a random cellular automaton (RCA) and has been applied in various simulations of DRX [110,111,112,113,114]. Furthermore, Chen et al. [90,115] introduced an approach that describes deformed grain boundaries using a material coordination system in addition to the standard cellular coordinate system, which is referred to as the updated topology deformation model. Guo et al. [116] integrated the independent component analysis (ICA) with CA model to solve overlap problems of grain topological structures during DRX simulations.

The CA model accuracy is affected by the cell mesh size. Large cell meshes are adverse to capturing the microstructural details and reduce calculation accuracy, while small cell meshes increase computational significantly. In addition, the CA method usually requires careful calibration of model parameters, which is time-consuming and requires a large amount of experimental data [117,118].

3.4. Phase-Field (PF) Model

Based on thermodynamics and diffuse-interface description, the PF model has proved to be powerful in capturing the microstructure evolution without having to track interfaces explicitly, as is the case with conventional sharp-interface models [119]. PF model can simulate phenomena like recrystallization [120,121,122,123,124,125,126,127,128,129,130,131], grain growth [132,133,134,135], phase transformations [136,137,138,139], and solidification [140,141], which have been greatly developed in the past two decades [142].

In the PF model, the microstructure is represented by a set of phase-field variables (order parameters) $\eta$ that are continuous functions of space $r$ and time $t$. For single-phase polycrystalline microstructure, the order parameter of a point within the $i$th grain equals 1, ${\eta }_i\left(r,t\right)=1$ [121]. The order parameter of the point within other grains equals 0, ${\eta }_i\left(r,t\right)=0$. The order parameter of the points on the boundary of the $i$th grain $i$ is between 0 and 1, $0<{\eta }_i\left(r,t\right)<1$ [121]. The time-dependent kinetic equations for microstructural evolution are defined over the whole domain. The method also allows many physical phenomena including applied stress, temperature, electrical, and magnetic fields to be treated simultaneously.

In the PF model, order parameters are essential to describe the microstructural evolution of materials, for they represent the state of the material at each point in space and time. To analyze the evolution of the order parameters, time-dependent partial differential equations typically derived from thermodynamic principles have to be built. For conserved order parameters (like a concentration in non-linear diffusion), the most common form of these equations is the Cahn–Hilliard equation (Equation (18)) [119]. The Allen–Cahn equation (Equation (19)) is for non-conserved order parameters (like a phase or grain orientation) [119].

$${\displaystyle \frac{\partial {\eta }_i(r,t)}{\partial t}}=\mathsf{\nabla }·\left(M_{ij}\mathsf{\nabla }{\displaystyle \frac{\partial F}{\partial {\eta }_j}}\right)\hspace{1em}\hspace{1em}i,j=1, 2,\cdots n$$

(18)

$${\displaystyle \frac{\partial {\eta }_i\left(r,t\right)}{\partial t}}=-L_{ij}{\displaystyle \frac{\partial F}{\partial {\eta }_j}}\hspace{1em}\hspace{1em}i,j=1, 2, \cdots ,n$$

(19)

where $t$ is time; $M_{ij}$ is the diffusivities of different conserved variables; $L_{ij}$ is the mobilities of different non-conserved variables; and $F$ is the free energy functional of the system. The application of PF method to different phenomena is mainly reflected in the description of free energy functional. In DRX simulation, the Allen–Cahn equation is often employed.

For simulating DRX process, the thermodynamic free energy of the system $F$ is composed of contributions from grain boundary energy $F_{\mathrm{gb}}$ and the stored deformation energy $f_{\mathrm{def} }$ [121,122].

$$F_{\mathrm{tot}}=F_{\mathrm{gb}}+F_{\mathrm{def}}={\int }_V{f}_{\mathrm{gb}}dV +{\int }_V{f}_{\mathrm{def}}dV$$

(20)

where $f_{\mathrm{gb}}$ and $f_{\mathrm{def}}$ are the grain boundary (gradient) energy density and deformation (bulk free) energy density, respectively. V is the simulation domain. The researcher proposed different formulations describing two classes of energy density for the DRX simulation [121,122,123].

The PS model of DRX is achieved as follows:

(1)

Initialization: Define the simulation domain and set initial conditions. Establish order parameters to distinct the grain and grain boundary.

(2)

Defining free energy functional: Including bulk and gradient energy density functions.

(3)

Building governing equations: Derive and formulate PS and coupled evolution equations (the time-dependent Ginzburg–Landau equations). These equations minimize the total free energy of the system.

(4)

Numerical implementation: Discretize the governing PDEs using appropriate numerical methods, such as finite difference, finite element, or spectral methods. Choose a time integration scheme.

(5)

Nucleation and growth: Periodically introduce new grains by altering the order parameters at nucleation sites.

(6)

Boundary conditions: Apply periodic or fixed boundary conditions.

(7)

Time stepping: Increment time, solve the equations, and check for convergence.

(8)

Output: Record, analyze, and visualize the microstructure data.

Based on thermodynamic and kinetic principles, the PF model is capable of capturing the complex interplay between various driving forces in DRX, such as stored energy, grain boundary mobility, and nucleation. It can model the evolution of microstructure without explicitly tracking interfaces, which allows for the natural formation and movement of grain boundaries and the representation of complex microstructural features like curved boundaries and triple junctions. The PF model can simulate the competition between nucleation and growth of new grains during deformation, allowing for a detailed study of how microstructure evolves under dynamic conditions. The model allows for spatial and temporal variation in nucleation rates, which can be critical in capturing the inhomogeneous nature of DRX.

Figure 7 shows the PF simulated dynamic recrystallization process with the deformation process (strain) of magnesium alloy AZ80 [121]. The “necklace” grain microstructure occurring at the early stage of DDRX was replicated using the PS model. A comparison with the experiment’s measurements demonstrated that the predicted grain structure and mechanical response were consistent with the experimental data.

Elder et al. [124] proposed the phase-field crystal (PFC) model, which is a computational method used to simulate the atomic-scale structure and dynamics of crystalline materials over long time scales. It combines elements of PF modeling and density functional theory to describe the spatial distribution of atomic density, allowing for the study of phenomena like grain boundary formation, defect dynamics, and crystal growth. The PFC model is particularly useful for capturing the microstructural evolution of materials while accounting for both elastic and plastic deformations. Zhao et al. [125] applied a phase-field crystal model to simulate the interaction between dislocation and grain boundary of nano polycrystalline composites under tension. The simulation results indicated that the temperature influenced the mode and rate of dislocation entering the grain boundary. It transpired that the method regulates the mechanical properties of nano-polycrystalline materials.

Additionally, the PF model developed in recent years can be coupled with other physical processes, such as local inhomogeneous deformation, phase transformation, precipitation, heat transfer, and diffusion. This is particularly important in DRX, where deformation influences microstructural evolution and vice-versa. They will be reviewed in the next section.

In the PF method, solving time-dependent partial differential equations over fine grids uses significant computational resources, especially for high-resolution simulations or large time scales, potentially limiting its practical application in some cases. The accuracy of PF simulations heavily depends on the correct calibration of model parameters, such as grain boundary mobility, nucleation rates, and free energy densities. These parameters often require detailed experimental data or additional modeling, which can be time-consuming and complex to obtain. The grain boundary width in a PF model is only a numerical parameter and is determined not physically. It may introduce artificial effects if not carefully managed.

3.5. Level-Set (LS) Model

The LS method is a numerical tool used to capture the moving interfaces with the time in a computational domain, which possesses an advantage in solving complex boundaries that may change topology over time, such as merging or splitting. Similar to the PF method, the LS method also avoids interface tracking which is considered as difficult in microstructure evolution simulation. Bernacki and Logé et al. [143,144,145] first proposed a framework that combined the LS and finite element (FE), that is, LS-FE, to model recrystallization in 2008 and 2009, focusing on the primary recrystallization. In 2019, Hallberg [146] began to simulate the DRX process using a modified LS approach. Recently, the LE-FE simulations of DRX have progressed in a large step [147,148,149,150,151,152,153,154].

The core idea of the LS method is to represent the interface (or boundary) between different phases or grains as the zero LS of a higher-dimensional function, typically denoted by $\varphi \left(\mathbf{x},t\right)$, where $\mathbf{x}$ represents the spatial coordinates and t represents time. The LS function $\varphi \left(\mathbf{x},t\right)$ is usually a signed distance function:

$\varphi \left(\mathbf{x},t\right)$ > 0 for points inside the region of interest;

$\varphi \left(\mathbf{x},t\right)$ < 0 for points outside the region;

$\varphi \left(\mathbf{x},t\right)$ = 0 for points on the interface (the actual boundary between regions).

The evolution of the LS function $\varphi \left(\mathbf{x},t\right)$ over time t is governed by a Hamilton–Jacobi-type PDE [143].

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial \varphi }{\partial t}}+F\left|\nabla \varphi \right|=0\\ {\varphi }_i\left(t=0,\mathbf{x}\right)={\varphi }_i^0\left(\mathbf{x}\right)\end{array}\right.$$

(21)

where ${\displaystyle \frac{\partial \varphi }{\partial t}}$ is the time derivative of the LS function, $F$ is the speed of the interface in the direction normal to itself (this can be a function of space, time, curvature, or other physical quantities), and $\left|\nabla \varphi \right|$ is the magnitude of the gradient of $\varphi$, ensuring that the evolution happens in the normal direction to the interface. The normal velocity F can depend on various factors, such as interface curvature (mean curvature flow), external forces, or any other physics governing the movement of the interface.

Implementing an LS model for calculating DRX involves representing the interfaces between different grains as level sets of a higher-dimensional function and solving the corresponding partial differential equations (PDEs) that govern the evolution of these level sets. The following are the main steps for implementing such a model:

(1)

Initialization: Define the simulation domain and set initial conditions.

(2)

Define level set functions: Establish level set functions to represent grain boundaries. Use multiple level set functions to represent multiple grains in the microstructure.

(3)

Define energy functional: Define the grain boundary energy proportional to the curvature of the grain boundary and bulk energy related to dislocation density and stored energy within the grains.

(4)

Governing equations: Derive and formulate the level set evolution equations, that is, the Hamilton–Jacobi-type PDE (Equation (21)) that governs the evolution of the level set function. Calculate the normal velocity as a function of local curvature, stored energy, and other relevant parameters.

(5)

Numerical implementation: Discretize the equations and choose appropriate numerical methods (such as finite difference or finite element methods).

(6)

Nucleation and growth: Introduce new grains based on nucleation criteria. Periodically introduce new level set functions to represent new grains at nucleation sites.

(7)

Boundary conditions: Apply periodic or fixed boundary conditions.

(8)

Time stepping: Increment time, solve the equations, and check for convergence.

(9)

Output: Record, analyze, and visualize the microstructure data.

The LS method can naturally handle the topology changes of the interface, which is one of its advantages. This method can automatically deal with these changes as the simulation progresses, for the interface is implicitly represented by the LS function. The computational domain is discretized into a grid, and the LS function $\varphi$ is calculated at each grid point. So, the LS can naturally be combined with the finite element method [143,144,145]. The time evolution of Hamilton–Jacobi PDE is solved using numerical schemes such as finite differences. In the LS model, the grid has to be refined to capture moving interfaces. In microstructure evolution simulation, there are considerable grain boundaries as interfaces, and they dynamically appear, move, and vanish. So, dealing with the local mesh adaptation process is necessary and tremendous calculation resources have to be spent [148]. This also limits the development of LS methods to some extent. Compared to other methods simulating DRX at the mesoscale, LS is a newly developed method, but it has shown powerful capabilities. Bernacki provided a detailed review of the LS method in microstructure evolution in a recently published paper [154].

4. Multi-Physics Coupling Models

4.1. Coupled with Crystal Plasticity

During the forming of crystalline materials, the heterogenous deformation resulting from the distinct crystallographic orientation of every grain has to be considered to obtain accurate simulations of microstructure evolution. More and more crystal plasticity (CP) models have been combined with CA, PF, and LS models and applied in recrystallization simulation in the last decade. A full field method simulates recrystallization in mesoscale and the grain topologies are explicitly described. Thus, the CPFEM is naturally combined with the simulations of recrystallization, where the deformation energy (dislocation density) of grains is calculated more accurately than when using the conventional model [155].

Figure 8 displays the LS-simulated grain morphology and the distributions of crystalline orientation, dislocation density, and stress of 304 L stainless steel [151]. In this simulation, LS functions were used to describe the velocity of the grain boundaries under thermodynamic driving pressures. The number of nucleation sites was defined as a function of the dislocation density and misorientation. When coupled with the crystal plasticity finite element method (CPFEM), the LS simulation provided a detailed description of the dynamic recrystallization (DRX) phenomenon.

Raabe and Becker [155] first performed the coupling of the CP and CA by translating the state variables from the CPFEM to the CA models. The positions of the FE integration points were mapped to the CA’s quadratic meshes by the Wigner–Seitz mapping algorithm. Meanwhile, the grain topologies and stored energy data calculated by CP-FEM were translated into the cellular automaton model. The length scale, time step, and local switching probability of the CA model were determined depending on the obtained cell size, maximum driving force, and maximum grain boundary mobility in the region. Wu et al. [156] combined the CA model and the CPFEM model to simulate the two-dimensional hot compression of $\alpha +\beta$ titanium alloy. The calculated strain, dislocation density, phase state variable, and crystal orientation during the deformation process by CPFEM were input into the CA model in each simulation step to investigate the effect of non-uniform deformation on the DDRX. Popova et al. [157] simulated the DRX of magnesium alloy using the CA−CPFEM framework using a new nucleation criterion developed based on the local mismatch in dislocation density. To accurately capture the morphological characteristics of DRX, Li et al. [158] established a 3D model through the full coupling of CA and CPFEM. During the modeling, the CA algorithm accounting for the DRX evolution was built into the CPFEM framework that accounts for multiscale inhomogeneous deformation. The full coupling of the CA and CPFEM was achieved because the CA-simulated softening effects from DRX were returned to the CPFEM calculation. Sun et al. [159] proposed the sub-mesh ISV-based CA−CPFEM model. This model can accurately describe DRX nucleation and the volume fraction of the DDRX grain in the sub-mesh scale. Legwand and Madej et al. [160,161] proposed and applied a random cellular automata finite element (RCA-FE) method to fully couple the FE and CA models, where the CA cells directly correspond with FE integration points. Recently, the CA−CPFEM models have been applied in the DRX simulations of the heat-assisted incremental sheet forming of Ti−6Al−4V alloy [162], the high-temperature compression test of 304LN stainless steel [163], and high-speed rolling of micro-alloyed steel [164]. Park et al. described in detail their multiscale framework of the CA−CPFEM model (Figure 9) [163]. The multiscale framework was applied to simulate the DRX of the hot-compressed AISI 304 LN stainless steel at various temperatures and strain rates. The predicted behaviors were compared with the experimental results.

Similarly, the PF method has been combined with CP theory to consider the inhomogeneous deformations in grain scale and investigate their impacts on recrystallization [165]. An important step of the CPFEM−PF method is mapping the calculation results from the FE mesh to the PF regular grid used for finite difference calculations [166]. In general, except for displacement, the calculated results using CPFEM are not continuous between elements; thus, the data are infinitely sharp at the grain boundary. The PF method has to describe the smooth-distributed driving force (dislocation density) of recrystallization inside the interface region (grain boundary). So, before mapping data from CA into PF, the data must be smoothed [165,166]. Figure 10 displays the mapping of the calculated dislocation density distribution from the CPFEM model to the PF model [166]. Takaki et al. [167] developed a numerical model for SRX by coupling the CPFEM and multi-phase-field (MPF) method. Their SRX simulation included three steps: the CPFEM calculation of room-temperature plastic deformation of polycrystalline metal; the deformation prediction in subgrain structure from the CPFEM calculated results; and the MPF simulation of subgrain growth in the SRX process. Chen et al. [168] coupled the fast Fourier transformation (FFT)-based PF model and the CP−FFT model to model the SRX. The high efficiency of the FFT solver achieved the recrystallization simulation of polycrystalline materials in 3D. The calculated plastic strain field by the CP−FFT approach was transformed into the FFT-based PF model, and the driving forces for recrystallization were determined. The FFT algorithm used in both MPF and CP facilitates their integration and enhances computational efficiency. A computational framework integrating CPFEM and PF was applied in the SRX simulations [169,170,171,172,173]. Muramatsu et al. [174] simulated DRX using the dislocation CP model and multi-phase-field model; the former estimated inhomogeneous deformation in polycrystalline materials, and the latter calculated the nucleation and growth of recrystallized grains. Recently, the CPFEM−PF method was applied to DRX simulations [175,176,177].

Zhou et al. [178] integrated a viscoplastic self-consistent (VPSC) model with an empirical DRX model for coupling CP and DRX. Unlike the CPFEM−PF method, this integrated model is the mean field method without grain topology. By means of a dislocation density evaluation, the DRX nucleation and growth criteria are added to the VPSC model.

Mellbin et al. [84,85,86] established a framework for finite strain plasticity using the crystal plasticity model and developed a graph-based vertex algorithm to trace the topology of the grain boundary network and the grain inter-connectivity. Thus, misorientation-dependent grain boundary energy and mobility and the appropriate identification of nucleation sites for recrystallization were determined. The simulation results for copper rolling showed that the model was capable of capturing many of the prominent features of DRX, such as the effects of strain rate, nucleation rate, and initial grain size on the flow stress behavior, as well as describing the texture evolution. McElfresh et al. [179] simulated SRX during the high-temperature annealing of α-Fe polycrystals. Two important inputs of deformed microstructures, the dislocation density and misorientations of the deformed, resulted from the crystal plasticity finite element calculation of cold forming. Grain boundary evolution is based on a two-dimensional vertex model of the polycrystal microstructure.

4.2. Coupled with Precipitation

Depending on the alloy composition, grain morphology, and processing conditions, the second-phase particles inside alloys possess different forms and roles. During the hot deformation and subsequent annealing of alloys, some second-phase particles are prone to dispersing in the matrix and no longer precipitate or dissolve. In contrast, some second-phase particles are apt to precipitate from the matrix, influencing the microstructure and properties of alloys. Some precipitated second-phase particles interact with other microstructure evolution mechanisms, such as DRX. Precipitated second-phase particles can achieve strengthening, toughening, or provide other desired properties for the alloy. Some methods of DRX, such as PF, are also powerful tools for simulating precipitation [180].

To date, most of the recrystallization models that consider the influence of second-phase particles have not explicitly simulated the dynamically precipitating process during the DRX process [113,180,181,182,183]. In these simulations, the influences of second-phase particles on DRX were pre-set in the model as the constants, which were closely related to the volume fraction, size, and number density of the second phase. Under certain conditions, second-phase dynamic precipitates (DPs) with DRX progress and interact with DRX, and microstructure evolution laws become more complicated than those ignoring DPs. As the precipitation progresses, the volume fraction, size, and number density of the precipitated phase evolve, and its influence on recrystallization also changes accordingly. At the same time, the precipitation will also alter the solute concentration in the matrix, thereby affecting the solute drag effect, and the occurrence of recrystallization will also cause changes in the precipitation process. In addition, it is also necessary to consider the role of plastic deformation of the material in promoting precipitation, etc. In this case, modeling becomes more complicated, and there are currently few reports on modeling studies on simultaneous precipitation and recrystallization behavior.

For simulating SRX and static precipitation (SP) during annealing, Zurob et al. [184] established a model to describe the temporal evolution of characteristic state variables during the annealing of Nb steel. The model was composed of three modules: static recovery (SRV), SRX, and SP, and the modules were coupled to each other to represent the interaction of the three microstructural evolution mechanisms. The SRX dynamics in the model were described by the JMAK formula in the saturated nucleation form. The driving force of the recrystallized grain growth changed with the decrease in dislocation density caused by the recovery of and the change in pinning force calculated by the volume fraction and radius of the precipitated phase. The recrystallized grain boundary mobility was affected by the decrease in the Nb solute concentration $C_{\mathrm{N}\mathrm{b}}$ caused by precipitation. SP was considered to occur only on dislocations, and the evolution of its number density and radius was calculated by the classical nucleation and growth theory formula. The reduction in the dislocation density caused by recovery reduced the nucleation rate of the precipitate phase and reduced the solute equivalent diffusion coefficient, thereby slowing down the growth and coarsening of the precipitate phase.

Schäfer et al. [185,186] developed a mesoscale model to predict the evolution of SRX and SP during post-rolling annealing. The model was composed of multiple coupled sub-models, with the core being the CA module that simulates the growth of SRX grains. Firstly, the deformation texture and dislocation density of the material after cold rolling were predicted, and the initial microstructure information was input into the CA model of SRX. The SRX nucleated on grain boundary and particle-induced nucleation was also considered. Grain growth was simulated by the CA model, in which the effects of recovery, precipitation, and preferred growth direction on grain growth were added. The precipitation process was simulated by a model based on classical nucleation and growth theory. This coupling model can not only show microstructure evolution under the complex interactions of SRX and SP but also predict the recrystallization texture.

Compared to SRX and SP occurring simultaneously during annealing, the simulation of DRX and DP occurring simultaneously during hot deformation is not very developed. L’ecuyer and L’espérance [187] built a model for the microstructural evolution during the hot deformation of HSLA steel and predicted the evolution of statistical average values, such as flow stress, DRX volume fraction, and precipitate volume fraction. The DRX kinetics in the model were calculated using the JMAK formula, taking into account the hindering effect of precipitate pinning on DRX nucleation and growth. For the DP process, the model assumed that once precipitate nucleation occurred, the precipitate particles of a specific size were produced, and the further growth and coarsening of the precipitate were ignored. This assumption was based on the experimental phenomenon that the precipitate in HSLA steel always remains at a small size. The DP nucleation rate is calculated according to the classical nucleation formula.

Zhu et al. [188] constructed a PF model for the AZ80 magnesium alloy that takes into account the effect of precipitates on DRX. The model generated precipitates on the grain boundaries of the initial grains based on experimental observations. The size and volume fraction of the precipitates are consistent with the experimental values. The simulated microstructural topology and macroscopic flow stress showed good consistency with the experimental results. However, their model assumed that the precipitates were pre-existing, ignoring the physical process of precipitation and its changes with strain, strain rate, and temperature. In addition, the relevant parameters of the precipitates need to rely on the experimental measurement results. Shuai et al. [189] conducted an in-depth analysis of the effect of dynamic precipitates on DRX nucleation and growth in hot-extruded Mg-Y-Zn alloys based on PF simulation. The differences in the number density and size of precipitates at different temperatures were considered according to the experimental results. The research results provide an important basis for understanding the mechanism of the effect of DP on DRX in magnesium alloys. However, the model also assumed that the DP precipitates remained constant during deformation and did not involve a detailed description of the DP physical process. Recently, He et al. [190] developed a multilevel cellular automaton (CA) model to simulate the DRX and DP of magnesium alloy. Mechanisms including work hardening, DRV, DRX, DP, and solute diffusion were integrated and interconnected by their mutual effects. To improve modeling accuracy, a novel local pinning model was proposed to reflect the uneven retardation of a precipitate to grain boundary migration. The computational framework of multilevel CA is shown in Figure 11, and the simulated and experimental results are compared in Figure 12. Cheng et al. [191] established a unified VPSC−DRX−DPN model that comprehensively considered the hot deformation mechanisms of aluminum alloy. In their model, CDRX was especially focused on, and DP was considered to hinder the low angle boundary (LAB) formation and rotation during CDRX. This effect was modeled through the definition of a relaxation parameter.

5. Summary and Outlook

The basic methods of dynamic recrystallization simulations and their state-of-the-art applications are reviewed in this paper. The characteristics and development tendencies of these methods are as follows:

(1)

Combined with FE simulation, the mean field method rapidly predicts the average size and volume fraction distribution of the recrystallized grain size. Though the physical mechanisms are rarely considered in the model, the mean field method plays an irreplaceable role in recrystallization simulation, especially when it is necessary to predict the overall recrystallization trend of the workpiece at a larger scale. Compared to empirical models, ISV models have broader development and application prospects, especially coupled with the full field model.

(2)

The MC model uses random sampling techniques to simulate the evolution of microstructures based on energy minimization principles. Simplified rules for grain boundary movements and virtual time scales make MC difficult to model actual processing. Vertex model effectively captures the evolution of grain structures by representing grain boundaries as dynamic vertices. However, it can be limited by assumptions about isotropy and simplicity in boundary dynamics. CA models handle complex phenomena like nucleation, growth, and impingement of grains straightforwardly. So, it is most widely applied in recrystallization simulation among all full-field methods. PF models are proper to simulate complex microstructural evolutions with high accuracy, but have high computational expenses. PF models are a powerful tool for simulating not only recrystallization but also other phase transformations in metals and alloys. Thus, PF can be used in multi-physical coupling models. Developed for interface tracking problems, the level set method became relevant for recrystallization simulations due to its ability to handle evolving boundaries and complex geometries. LS represents interfaces implicitly and could naturally accommodate changes in topology, such as the merging and splitting of grains. Compared to other full-field methods, LS has been newly applied in recrystallization simulation, but its high computational costs hinder its development to some extent.

(3)

Coupled multi-physics models were recently developed, coupling recrystallization simulations with other physical phenomena (local inhomogeneous deformation, phase transformation, precipitation, etc.) Coupled models provide a more comprehensive understanding of the recrystallization process under various conditions, which will become the tendency of the recrystallization simulations.

(4)

The high computational demands of the full field method are supposed to be relieved by new algorithms, for example, the utilizing of graphic processing unit (GPU) [82,84,85,86] and machine learning (ML). Surrogate modeling is an important ML and has been applied in microstructure analysis, for accurate and fast predictions of recrystallization and grain growth [192,193]. New research involving the integration of recrystallization with ML has been reported [194,195,196,197,198].