Modeling and simulation of recrystallization in metals with spheroidal grain structures using hybrid cellular automata

ABSTRACT

Hybrid Cellular Automata (HCA) are robust computational tools that extend the classical Cellular Automata (CA) framework by integrating both discrete and continuous modeling principles. This hybrid nature enables more accurate, flexible, and efficient simulations of complex microstructural phenomena. In this work, a three-dimensional HCA approach is employed to simulate the recrystallization of metallic materials with anisotropic spheroidal grains, specifically, prolate and oblate morphologies. The proposed model introduces geometric and mathematical formulations that enable control over grain shape and spatial orientation during growth, thus extending previous models limited to spherical symmetry. The implementation is entirely computational, developed in Python, and structured for modularity and efficiency. Several simulations were conducted under varying conditions, including single and multiple nuclei, different grid sizes, and controlled anisotropic orientations. These simulations demonstrate the model’s capacity to generate realistic and heterogeneous microstructures, with adjustable morphological and directional features. Quantitative analyses of grain volumes and growth patterns further validate the model’s precision and consistency. Additionally, the work explores spatial rotation techniques to simulate heterogeneous anisotropic textures within a single matrix, enabling the modeling of complex structures. The flexibility, ease of implementation, and qualitative accuracy of this HCA model establish it as a powerful tool for predictive analysis in computational materials science.

Keywords:
Recrystallization; Spheroidal grains; Hybrid cellular automata; Microstructural modeling; Anisotropic microstructures

1. INTRODUCTION

The study of microstructural phenomena such as recrystallization in metallic materials remains of central importance in both academic and industrial contexts, given its substantial influence on manufacturing processes and the resulting mechanical, thermal, and physical properties of metals. Metallic microstructures typically are composed of polycrystalline grains with varying crystallographic orientations, separated by boundaries that directly affect the overall material behavior.

While equiaxed recrystallization has traditionally been a focus of simulation and analysis due to its relevance in isotropic systems [¹,²,³], other forms of grains could be relevant. For example, many metallurgical processes, such as rolling, extrusion, forging, induce or are affected by anisotropic behavior, leading to grains that deviate markedly from spherical symmetry. In some cases, recrystallized grains could assume spheroidal geometries, among others [⁴,⁵,⁶,⁷].

The anisotropy introduced by processing conditions or intrinsic material properties plays a pivotal role in determining the final grain morphology and texture, which in turn affects material performance. Therefore, the accurate modeling and simulation of non-equiaxed, spheroidal grains, including the directional dependence of growth kinetics and boundary mobility, is essential for advancing predictive capabilities in computational materials science.

In this context, the present work extends previous approaches by explicitly considering the formation and evolution of anisotropic spheroidal grains. Through three-dimensional simulations, we investigate how shape anisotropy and directional growth conditions influence microstructural evolution, providing insights relevant to both fundamental understanding and practical applications in Materials Engineering.

The spheroid, or ellipsoidal spherical shape, is a surface of revolution of an ellipse, in which two of its three semi-axes (a, b, and c) are equal (a = b). If c > a, it is called a prolate spheroid, whereas if c < a, it is called an oblate spheroid. If c = a, the spheroid degenerates into a sphere. These spheroids can be intuitively visualized along their main axis as either an elongated sphere, in the case of the prolate spheroid, or a flattened sphere, in the case of the oblate spheroid. Figure 1 shows illustrative representations of both.

Figure 1
Illustrative representations of the two spheroidal grain morphologies considered in this study: (left) oblate spheroid, characterized by compression along the polar axis (c < a); and (right) prolate spheroid, characterized by elongation along the polar axis (c > a).

Hybrid Cellular Automata (HCA) stand out as a promising alternative due to their ability to combine high-quality simulation results with relatively simple computational implementation. This potential has already been demonstrated in the context of spherical grain modeling [⁸]. Building upon these foundations, the present study aims to systematically present all the necessary steps and methodological adaptations required to apply the HCA approach to the recrystallization modeling of metallic microstructures composed of spheroidal grains. The objective is not only to simulate the morphological evolution during recrystallization, but also to ensure reproducibility and flexibility for further investigations involving more complex grain geometries.

2. METODOLOGY

This study was carried out through computational modeling, using exclusively numerical simulations. The entire implementation was developed in Python (version 3.12.1), leveraging libraries such as NumPy and Matplotlib. The coding followed a procedural structure, focusing on modular clarity and execution efficiency. The simulation algorithm was executed on a personal computer running Microsoft Windows^® 10 (64-bit), equipped with an Intel^® Core™ i7-2600 CPU @ 3.40 GHz and 12.0 GB of RAM. Although execution times were reasonable on standard hardware, no systematic benchmarking was conducted, since efficiency analysis was not the focus of this study.

The modeling approach applied herein is based on a deterministic, three-dimensional cellular framework, organized over a non-periodic cuboidal grid composed of discrete, cubic elements. In this structure, boundary conditions are closed, meaning the outermost cells do not interact with an external environment.

Only first-neighbor interactions through face adjacency (Face Neighborhood, FN) are considered for grain expansion. This choice was made due to its computational simplicity and superior performance. Although FN introduces geometric distortions, given its inherent octahedral bias when applied to cubic grids, this limitation is mitigated in this work due to the hybrid nature of the model and the geometric parameterization adopted for the grains, now treated as spheroidal.

The simulation engine, termed [⁸], is organized into three decoupled modules: the grain growth module, the post-processing/parameterization module, and the 3D visualization module. These modules operate independently and do not interfere with the simulation logic once the process is initiated.

In the grain growth module, the virtual metallic material is represented by a 3D matrix with uniform initial conditions, i.e., all cells initialized with the same scalar value. The nucleation follows a site saturation mechanism, where a predefined number of grains (seeds) is inserted prior to the onset of the simulation. Each seed occupies one cell and is labeled with a unique state to differentiate it from both the original matrix and other seeds.

As the simulation progresses, only the active cells, those constituting the grain boundaries, are iteratively processed. In each time step, the algorithm examines the face-adjacent neighbors of active cells. If a neighboring cell belongs to the original structure (i.e., has the base state), it is reclassified with the state of the grain from which the search originated, thereby extending the grain’s domain.

This search-propagation routine repeats until the entire matrix is transformed, that is, until there are no more base-structure cells adjacent to any active grain boundary. Unlike the previous version of the model, in this study the growth of grains is conceptualized and analyzed considering spheroidal morphology, enabling more realistic simulations of anisotropic growth tendencies.

During execution, the grain evolution is saved at user-defined intervals as structured data files, allowing later analysis. No interactive control or real-time decision-making is applied during simulation runs. The parameterization module processes the saved files and extracts quantitative descriptors related to both temporal progression and geometric evolution of the grains. This includes grain size distributions, volume calculations, and temporal growth curves. Selected graphical outputs may also be generated as part of this post-processing.

The simulations were carried out in a non-periodic cubic domain. It is well recognized that grains intersecting external surfaces may suffer biases [⁹, ¹⁰], either by artificial truncation or by the absence of external neighbors that would otherwise limit their growth. Different strategies have been proposed in the literature to minimize such boundary effects. Periodic boundary conditions are among the most frequently adopted solutions, as they reduce, but do not fully eliminate, surface artifacts. In this work, we opted for a simpler and more conservative approach: all grains intersecting the cube surfaces were excluded from the statistical analyses of size and morphology. This ensures that the reported distributions reflect only fully developed interior grains, thereby minimizing boundary-induced artifacts [⁸].

Finally, the visualization module is responsible for generating three-dimensional renderings of the simulated microstructure. It can display the full matrix or orthogonal cross-sections (XY, YZ, and XZ planes) for qualitative inspection. All visualizations use the default Cartesian coordinate convention, as shown in [⁸]. All reported dimensions in the figures are expressed in normalized units, corresponding to the number of voxels in the cubic matrices (e.g., 200³ cells), without direct scaling to physical micrometers. This choice highlights the morphological evolution rather than absolute dimensional calibration.

In summary, the implemented HCA model operates through a fully deterministic, stepwise iterative process, in which each voxel evolves according to local neighborhood rules and geometric constraints defined by the axial ratio (K_a). This procedure effectively reproduces the physical evolution of recrystallization fronts under controlled anisotropic conditions, forming the basis for the spheroidal adaptation described in the following section.

3. RESULTS AND DISCUSSION

3.1. Mathematical foundations

The main analytical relationships and equations that describe spheroids are well established. By convention, the two distinct semi-axis lengths are denoted as a and c, with the spheroid symmetrically oriented along the z-axis. Accordingly, its parametric representation is given by:

(1) $$x=a\sin \theta \cos \varphi$$

(2) $$y=a\sin \theta \sin \varphi$$

(3) $$z=c\cos \varphi$$

Where:

a is the semi-axis corresponding to the radius of the equatorial section, i.e., the x and y axes.

c is the semi-axis corresponding to the elongation along the z-axis.

θ is the polar angle, where θ ∈ [0, π].

ϕ is the azimuthal angle, where ϕ ∈ [0, 2π].

The polar angle (θ) represents the inclination with respect to the z-axis (analogous to latitude). At the north pole (z = c), θ = 0; at the equator (z = 0), θ = π⁄2; and at the south pole (z = –c), θ = π. The azimuthal angle (ϕ) is analogous to longitude: along the positive x-axis, ϕ = 0; along the positive y-axis, ϕ = π⁄2; along the negative x-axis, ϕ = π; and along the negative y-axis, ϕ = 3π⁄2. Figure 2 illustrates the spatial disposition of the azimuthal and polar angles in relation to the x, y, and z coordinates.

Figure 2
Representation of the azimuthal (ϕ) and polar (θ) angles over the x, y, and z coordinates.

Thus, sin θ controls the position within the horizontal (xy) plane, cos θ controls the position along the vertical (z) axis, and cos ϕ and sin ϕ define the direction within the xy plane to determine the x and y coordinates.

The geometric anisotropy ratio (or axial ratio) is given by:

(4) $$K_a\text{=}c/a$$

The Cartesian equation of the spheroid is given by:

(5) $$\frac{x^2+y^2}{a^2}\text{+}\frac{z^2}{c^2}\text{= 1}$$

Additionally, the volume and surface area of a spheroid can be determined by:

(6) $$V_{Esferoide}\text{=}\frac{4}{\text{3}}\pi a^{2}c$$

(7) $$S_{Ep}\text{= 2}\pi a^{\text{2}}\left[1+\frac{c}{a{e}_p}arcsin(e_p)\right]$$

(8) $$e_p\text{=}\sqrt{\text{1}-\frac{a^2}{c^2}}$$

(9) $$S_{Eo}\text{= 2}\pi a^2\left(1+\frac{1-e_o{}^2}{2e_o}{\tanh }^{-1}(e_o)\right)=\text{2}\pi a^2\left\{1+\left(\frac{1-e_o{}^2}{2e_o}\right)\left[\frac{1}{\text{2}}\text{In}\left(\frac{1+e_o}{\text{1}-e_o}\right)\right]\right\}$$

(10) $$e_o\text{=}\sqrt{\text{1}-\frac{c^2}{a^2}}$$

Where:

V_Esferoide is the volume of the spheroid,

S_Ep is the surface area of a prolate spheroid,

S_Eo is the surface area of an oblate spheroid,

e_p is the eccentricity of the prolate spheroid,

e_o is the eccentricity of the oblate spheroid

3.2. The HCA adapted to the spheroid

In essence, the HCA technique remains the same as that previously presented for spherical grain growth [⁸], with adaptations made to accommodate the new geometric shape in the form of spheroids. When dealing with prolate spheroids (K_a > 1), the Cartesian Equation (5) must be rearranged to isolate c; and in the case of oblate spheroids (K_a < 1), it must be rearranged to isolate a. The resulting forms of the equations are as follows:

(11) $$c\text{=}\sqrt{k_a{}^2(x^2+y^2)+z^2}$$

(12) $$a\text{=}\sqrt{x^2+y^2+k_a{}^2{z}^2}$$

Thus, in its Python code the Cartesian equation of the sphere [⁸] is replaced by the Cartesian equation of the spheroid, where Equation (11) is used in simulations involving prolate spheroids, and Equation (12) is used for oblate spheroids.

The simulation step value adopted is 0.5, as previously used in the spherical modeling [⁸], in order to ensure full hybridization with computational efficiency. However, since the spheroid lacks the perfect symmetry of the sphere, the step value is applied to control the growth along the most prominent axis. Thus, for prolate spheroids (K_a > 1), the step progression is applied to control the evolution of the c axis; whereas for oblate spheroids (K_a < 1), the step progression controls the evolution of the a axis.

This approach is necessary to ensure a more restrictive condition along the less prominent axis. If the step control were applied to the minor axis, a fixed step of 0.5 could induce large “jumps” along the major axis at each simulation loop, compromising both the hybridization and the smooth evolution of the spheroid.

To illustrate the application of the HCA technique to the spheroidal geometry, simple simulations were performed using a single seed placed at the center of the environment. Two grid sizes were tested: 21³ and 201³ cells. Simulations were carried out for a range of axial ratios (K_a) including 8, 4, 2, 1/2, 1/4, and 1/8, thereby covering a broad spectrum of elongation and flattening scenarios.

The choice of these axial ratios (K_a) was intended to cover the widest possible morphological spectrum, from highly elongated (K_a = 8) to highly flattened (K_a = 1/8) spheroids. These values were not meant to reproduce exact experimental measurements, but rather to test the flexibility and robustness of the HCA formulation under extreme anisotropy conditions. At the same time, they are qualitatively related to metallurgical practice: prolate spheroids (K_a > 1) are consistent with the elongated grains commonly observed after rolling or extrusion [¹¹, ¹²], whereas oblate spheroids (K_a < 1) are analogous to flattened grains generated by compressive processes such as forging or stamping [¹³, ¹⁴].

Figures 3 to 6 present selected visualizations of these simulations, and Table 1 summarizes some of the parameters obtained.

Figure 3
Images at 3% transformation from simulations of prolate spheroids initiated from a single seed placed at the center of environment (normalized units, 213 cells). The seed and transformed cells are shown in blue.

Figure 4
Images at 3% transformation from simulations of oblate spheroids initiated from a single seed placed at the center of environment (normalized units, 213 cells). The seed and transformed cells are shown in blue.

Figure 5
Images at 3% transformation from simulations of prolate spheroids initiated from a single seed placed at the center of environment (normalized units, 2013 cells). The seed and transformed cells are shown in blue.

Figure 6
Images at 3% transformation from simulations of oblate spheroids initiated from a single seed placed at the center of environment (normalized units, 2013 cells). The seed and transformed cells are shown in blue.

Table 1
Complete simulations performed in environments of 21³ cells and 201³ cells, each initiated from a single cell located at the center, for various values of K_a.

ENVIRONMENT (NO. OF CELLS)(1)	Ka	TOTAL NUMBER OF LOOPS	TOTAL NUMBER OF CELLS SEARCHED
			Nº	(%)(2)
213 (9,261)	8	115	349,038	3,769
	4	83	242,130	2,615
	2	60	165,738	1,790
	1/2	49	153,126	1,653
	1/4	85	347,958	3,757
	1/8	163	753,846	8,140
2013 (8,120,601)	8	1,149	329,288,490	4,055
	4	825	227,551,722	2,802
	2	600	155,564,394	1,916
	1/2	490	141,869,454	1,747
	1/4	849	323,145,750	3,979
	1/8	1,625	705,629,622	8,689

• Note: ⁽¹⁾In all simulations, 100% of the cells in the environment were transformed.
• ⁽²⁾(Total Number of Cells Searched / Total Number of Cells in the Environment) × 100.

3.3. The resulting microstructures

To illustrate the effect of spheroidal geometry and the axial ratio (K_a) on the appearance of the generated microstructures, simulations were performed in an environment consisting of 200³ cells. These simulations were conducted with 100 randomly distributed initial seeds, considering Kₐ values of 8, 4, and 1/4 to cover a representative range of geometric configurations.

Figures 7 and 8 show selected visualizations from these simulations. Additionally, in the same environment but with 1000 randomly distributed initial seeds, a simulation was conducted with K_a = 4. The resulting images are presented in Figure 9. After removing the grains that touch the surface, according to the procedure detailed earlier [⁸], the final processed images are shown in Figure 10.

Figure 7
3D visualization of microstructures generated by spheroidal grains with 100 randomly distributed initial seeds in a 2003 cell environment (normalized units).

Figure 8
Central plane views of the microstructures generated by spheroidal grains from the simulations in Figure 7 (100 randomly distributed initial seeds in a 2003 cell environment - normalized units).

Figure 9
3D views and central plane slices of microstructures generated by prolate spheroidal grains (K_a = 4), with 1000 randomly distributed initial seeds in a 2003 cell environment (normalized units).

Figure 10
3D views and central plane slices of the microstructures generated in the simulation from Figure 9 (prolate spheroidal grains, K_a = 4, with 1000 randomly distributed initial seeds in a 2003 cell environment – normalized units), after removing the grains that touch the surface (482 grains retained and 518 grains eliminated).

Considering r as 0.1 mm, and analyzing the volume distribution of the grains obtained from the microstructure in Figure 10, the histograms shown in Figure 11 were generated, along with the results presented in Table 2.

Figure 11
Histograms of the grain volume distribution and normalized grain volume distribution for the simulation from Figure 10 (482 grains retained).

Table 2
Mean and selected values obtained for grain volume (GV) and normalized grain volume (NV).

PARAMETER	SIMULATION	MEAN VALUE	STANDARD DEVIATION	MEDIAN	MINIMUM VALUE	MAXIMUM VALUE
GV (mm 3 )	Figure 10	8.00	3.75	7.46	1.05	28.81
NV		1	0.47	0.93	0.13	3.60

• Note: Simulations with 482 final grains that do not touch the boundaries of an original matrix of 1000 grains.

3.4. Rotating the spheroids

Any three-dimensional object can be spherically rotated within a discrete environment using a specific matrix transformation process [¹⁵,¹⁶,¹⁷]. However, in the case of spheroids, it is additionally possible to derive analytical expressions that enable their arbitrary rotation in space, which, in a certain way, simplifies the process.

Through a matrix transformation, the following expressions can be obtained for a rotated spheroid:

(13) $$x\prime =x\sin \theta \cos \varphi +y\sin \theta \sin \varphi +z\cos \theta$$

(14) $$y\prime =x\cos \theta \cos \varphi +y\cos \theta \sin \varphi –z\sin \theta$$

(15) $$z\prime =x\sin \varphi –y\cos \varphi$$

Where x′, y′, and z′ are related to the new coordinate axes of the rotated system, which forms an angle θ with the z-axis and an angle ϕ in the XY plane. By directly substituting these expressions into the respective Cartesian equations, one obtains:

(16) $$c=\sqrt{A^2+k_a{}^{{}^2}\left(B^2+C^2\right)}$$

(17) $$a=\sqrt{A^2+B^2+k_a{}^2{C}^2}$$

Where:

(18) $$A=x\cos \varphi \sin \theta +y\sin \varphi \sin \theta +z\cos \theta$$

(19) $$B=x\cos \varphi \cos \theta +y\sin \varphi \cos \theta –z\sin \theta$$

(20) $$C=x\sin \varphi –y\cos \varphi$$

To illustrate some examples of rotation, Figures 12 and 13 present simulations conducted in a 201³ cell environment, using prolate spheroids (K_a = 4) and oblate spheroids (K_a = 1/4). In Figure 12, a single seed is placed at the center of the environment, while in Figure 13, 100 seeds are randomly distributed.

Figure 12
3D views and central plane slices of spheroidal grains generated in simulations within a 2013 cell environment (normalized units), using a single central seed and 2% transformation.

Figure 13
Environment with 100 randomly distributed seeds, all generating prolate spheroidal grains (K_a = 4), with an azimuthal angle (ϕ) of 7π⁄12 radians and a polar angle (θ) of π⁄4 radians (2003 cells, normalized units).

3.5. Anisotropic heterogeneity

As presented, textural anisotropy is a property of metallic materials in which crystallographic features, such as grain orientation, are not uniformly distributed. This results from the formation of a crystallographic texture, which refers to the preferential alignment of specific crystallographic planes in certain directions. Such textures typically arise from plastic deformation, thermal treatments, or a combination of both [¹⁸, ¹⁹].

The anisotropic morphologies generated in the present HCA simulations are qualitatively consistent with experimental microstructures reported in the literature. Electron backscatter diffraction (EBSD) analyses of rolled and extruded alloys reveal elongated grains aligned with the deformation direction [²⁰], closely resembling the prolate spheroidal morphologies obtained in our simulations (K_a > 1). Likewise, EBSD studies of compressed and hot-deformed Al and Mg alloys show grain flattening under compressive strain [²¹–²²], consistent with the oblate configurations reproduced by the model (K_a < 1). Furthermore, multi-scale investigations of texture evolution in severely deformed silver highlight heterogeneous grain shapes and orientations [²³], features that align with the rotational and orientation modules in our approach.

Such directional dependence affects physical and mechanical properties, such as tensile strength, ductility, electrical conductivity, and thermal expansion, depending on the direction considered within the material. In rolled and drawn metals [¹⁰, ²⁴], sheet metal forming [¹¹], or even in more advanced deformation routes [²⁵], textural anisotropy is common and has a significant influence on the material’s behavior in both structural and functional applications [²⁶].

The purpose of using spheroidal grains in this context is to simulate anisotropy originating from recrystallization. However, an additional and particularly interesting case arises when the anisotropy exhibited by a metallic material is not homogeneous throughout its volume. This anisotropic heterogeneity may manifest in regions with distinct crystallographic orientations, for example, an area with one dominant grain growth direction and another region with a different orientation or boundary condition [⁹, ²⁷].

Such variation, often associated with plastic deformation or annealing processes, can occur across multiple spatial scales—from microscopic to macroscopic levels. As a result, several textural zones may form within the material, each with its own dominant crystallographic orientation, directly affecting the material’s mechanical and functional properties. A classic example of macroscopic heterogeneity is Damascus steel, whose microstructure is notably distinctive [²⁸].

Other recent studies reinforce the role of recrystallization in modulating anisotropy across different metallic systems [²⁶]. Examples include hot-rolled aluminum alloys [²⁰, ²²], Ti/Mg laminated composites [¹⁴], and AZ31 magnesium alloys [²¹]. Although a direct quantitative validation (e.g., recrystallization kinetics or orientation distribution functions) was not performed, these qualitative comparisons are consistent with the morphological features observed experimentally. Future research will explore quantitative validation strategies, potentially integrating EBSD or other microstructural datasets to assess model fidelity.

Given this, it becomes evident that simulating such distinct anisotropic regions computationally is feasible using spheroidal grains, provided that different grain orientation patterns can be assigned across the matrix within a single simulation.

To illustrate this capability, a simulation was conducted in a three-dimensional domain of 200³ cells, containing 1000 randomly distributed initial seeds, all with prolate spheroidal shapes (K_a = 4). Grains located in the lower half of the domain (relative to the z-axis) share the same orientation, while grains in the upper half have a distinct orientation. In this example, the upper half was assigned ϕ = 0 and θ = 0, and the lower half ϕ = 7π⁄12 radians and θ = π⁄4 radians.

Thus, the differing orientations between the two halves clearly demonstrate the heterogeneity resulting from this hypothetical simulation, where two distinct anisotropic regions are formed. Figures 14 and 15 illustrate this simulation, showing a clearly heterogeneous microstructure. Figure 16 presents the final stage of the microstructure from Figure 15, but with the surface-touching grains removed.

Figure 14
Simulation in a 2003 cell environment (normalized units) with 1000 randomly distributed initial seeds, all forming spheroidal grains (K_a = 4). The upper half of the matrix has the base orientation (ϕ = 0, θ = 0), while the lower half is assigned a directed orientation (ϕ = 7π⁄12 rad, θ = π⁄4 rad).

Figure 15
Central plane views from the simulation shown in Figure 14.

Figure 16
3D views and central plane slices of the microstructures generated in the simulation from Figure 15, after removing the grains that touch the surface (414 grains retained and 586 grains eliminated). Note: Prolate spheroidal grains (K_a = 4) with two distinct orientations within the matrix: the upper half has the base orientation (ϕ = 0, θ = 0), while the lower half has a directed orientation (ϕ = 7π⁄12 rad, θ = π⁄4 rad), with 1000 randomly distributed initial seeds in a 2003 cell environment.

Finally, a natural next step would be to extend the HCA approach developed here for spheroidal grains to other geometries, such as general ellipsoids, with all three semi-axes distinct (a ≠ b, b ≠ c, a ≠ c), or even to more complex and exotic shapes, including cylinders and cones.

4. CONCLUSIONS

This work presented an adaptation of the Hybrid Cellular Automata (HCA) framework to simulate recrystallization in metals with spheroidal and anisotropic grains. The proposed formulation introduced geometric parameterization and orientation control, enabling the generation of prolate (K_a > 1) and oblate (K_a < 1) morphologies that are qualitatively consistent with those reported in rolled, extruded, and forged metals. The model demonstrated computational simplicity and robustness, allowing the simulation of large normalized voxel-based domains while maintaining morphological fidelity. Although the present study focused on qualitative analysis, the results reinforce the HCA method as a flexible and powerful approach for representing anisotropic grain structures. Future developments will aim at extending the model toward quantitative validation using experimental microstructural data and the incorporation of physically calibrated anisotropy parameters.

5. BIBLIOGRAPHY

• ^[1] GUO, Y.-N., LI, Y.-T., TIAN, W.-Y., et al, “Combined cellular automaton model for dynamic recrystallization evolution of 42CrMo cast steel”, Chinese Journal of Mechanical Engineering, v. 31, n. 85, pp. 1–11, 2018. doi: https://doi.org/10.1186/s10033-018-0284-8.
  » https://doi.org/10.1186/s10033-018-0284-8
• ^[2] CONTIERIA, R.J., ZANOTELLOB, M., CARAM, R., “Simulation of CP-Ti recrystallization and grain growth by a cellular automata algorithm: simulated versus experimental results”, Materials Research, v. 20, n. 3, pp. 688–701, 2017. doi: https://doi.org/10.1590/1980-5373-mr-2016-0459.
  » https://doi.org/10.1590/1980-5373-mr-2016-0459
• ^[3] ZHU, H., CHEN, F., ZHANG, H., et al, “Mesoscale modeling of dynamic recrystallization: multilevel cellular automaton simulation framework”, Metallurgical and Materials Transactions. A, Physical Metallurgy and Materials Science, v. 51, n. 3, pp. 1286–1303, 2021. doi: http://doi.org/10.1007/s11661-020-05912-3.
  » https://doi.org/10.1007/s11661-020-05912-3
• ^[4] GODIKSEN, R.B., RIOS, P.R., VANDERMEER, R.A., et al, “Three-dimensional geometric simulations of random anisotropic growth during transformation phenomena”, Scripta Materialia, v. 58, n. 4, pp. 279–282, Feb. 2008. doi: https://doi.org/10.1016/j.scriptamat.2007.10.003.
  » https://doi.org/10.1016/j.scriptamat.2007.10.003
• ^[5] ROMANOVA, V.A., BALOKHONOV, R.R., BORODINA, A., et al, “Crystal plasticity finite-element simulations for quasistatic deformation of polycrystals in terms of explicit dynamics”, PNRPU Mechanics Bulletin, n. 5, pp. 57–73, 2023. doi: https://doi.org/10.15593/perm.mech/2023.5.06.
  » https://doi.org/10.15593/perm.mech/2023.5.06
• ^[6] VANDERMEER, R.A., JENSEN, D.J., “Microstructural path modeling of primary recrystallization”, Journal of Physics: Conference Series, v. 2635, n. 1, pp. 012027-012028, 2023. doi: https://doi.org/10.1088/1742-6596/2635/1/012027.
  » https://doi.org/10.1088/1742-6596/2635/1/012027
• ^[7] VILLA, E., RIOS, P.R., “On volume and surface densities of dynamical germ-grain models with ellipsoidal growth: a rigorous approach with applications to Materials Science”, Stochastic Analysis and Applications, v. 38, n. 6, pp. 1134–1155, 2020. doi: https://doi.org/10.1080/07362994.2020.1773276.
  » https://doi.org/10.1080/07362994.2020.1773276
• ^[8] BRAGA, H.C., SILVA, S.N., “Study of recrystallization in metals through equiaxial growth of nuclei by computational simulation with three-dimensional hybrid cellular Automata”, Materials Research, v. 28, pp. e20250204, 2025. doi: https://doi.org/10.1590/1980-5373-MR-2025-0204.
  » https://doi.org/10.1590/1980-5373-MR-2025-0204
• ^[9] ŁACH, Ł., “Modeling of microstructure evolution during deformation processes by cellular automata — boundary conditions and space reorganization aspects”, Materials (Basel), v. 14, n. 6, pp. 1377, 2021. doi: https://doi.org/10.3390/ma14061377. PubMed PMID: 33809120.
  » https://doi.org/10.3390/ma14061377
• ^[10] PERZYŃSKI, K., “The effect of model size and boundary conditions on the representativeness of digital material representation simulations of ferritic-pearlitic sample compression”, Computer Methods in Materials Science, v. 22, n. 2, pp. 59–66, 2022. doi: https://doi.org/10.7494/cmms.2022.2.0780.
  » https://doi.org/10.7494/cmms.2022.2.0780
• ^[11] SIDOR, J., MIROUX, A., PETROV, R., et al, “Microstructural and crystallographic aspects of conventional and asymmetric rolling processes”, Acta Materialia, v. 56, n. 11, pp. 2495–2507, 2008. doi: https://doi.org/10.1016/j.actamat.2008.01.042.
  » https://doi.org/10.1016/j.actamat.2008.01.042
• ^[12] BARONY, N.B., JORGE, I.C.S., SILVA, Â.S., et al, “Comportamento mecânico e textura de chapas finas das ligas de alumínio 5052 e 5050C”, Revista Matéria, v. 29, n. 2, pp. e20230321, 2024. doi: https://doi.org/10.1590/1517-7076-RMAT-2023-0321.
  » https://doi.org/10.1590/1517-7076-RMAT-2023-0321
• ^[13] LUIZ, V.D., AMARAL, E.C., SOUZA, V.P., et al, “Influência da velocidade de estampagem e anisotropia no comportamento tribológico e fratura de uma chapa de aço AISI 430Nb”, Revista Matéria, v. 27, n. 1, pp. e202148196, 2022. doi: https://doi.org/10.1590/1517-7076-RMAT-2021-48196.
  » https://doi.org/10.1590/1517-7076-RMAT-2021-48196
• ^[14] MO, T.Q., XIAO, H.Q., LIN, B., et al, “Improving ductility and anisotropy by dynamic recrystallization in Ti/Mg laminated metal composite”, Acta Metallurgica Sinica, v. 35, n. 12, pp. 1946–1958, 2022. doi: https://doi.org/10.1007/s40195-022-01442-2.
  » https://doi.org/10.1007/s40195-022-01442-2
• ^[15] AMMERAAL, L., ZHANG, K., Computer Graphics for Java Programmers, 3rd ed. Springer, 2017. doi: https://doi.org/10.1007/978-3-319-63357-2.
• ^[16] DE BERG, M., CHEONG, O., KREVELD, M., et al, Computational geometry: algorithms and applications, 3rd ed., Berlin, Springer, 2008. doi: https://doi.org/10.1007/978-3-540-77974-2.
  » https://doi.org/10.1007/978-3-540-77974-2
• ^[17] HUGHES, J.F., DAM, A.V., MCGUIRE, M., et al, Computer Graphics: principles and practice, 3rd ed., London, Addison-Wesley, 2014.
• ^[18] MISHIN, Y., MEHL, M.J., PAPACONSTANTOPOULOS, D., “Phase stability in the Fe–Ni system: Investigation by first-principles calculations and atomistic simulations”, Acta Materialia, v. 53, n. 15, pp. 4029–4041, 2005. doi: https://doi.org/10.1016/j.actamat.2005.05.001.
  » https://doi.org/10.1016/j.actamat.2005.05.001
• ^[19] BRAHME, A., ALVI, M.H., SAYLOR, D., et al, “3D reconstruction of microstructure in a commercial purity aluminum”, Scripta Materialia, v. 55, n. 1, pp. 75–80, 2006. doi: https://doi.org/10.1016/j.scriptamat.2006.02.017.
  » https://doi.org/10.1016/j.scriptamat.2006.02.017
• ^[20] LIU, Y., JIANG, J., ZHANG, Y., et al, “Recrystallization of hot-rolled 2A14 alloy during semi-solid temperature annealing process”, Materials (Basel), v. 16, n. 7, pp. 2796, 2023. doi: https://doi.org/10.3390/ma16072796. PubMed PMID: 37049090.
  » https://doi.org/10.3390/ma16072796
• ^[21] NIU, W., WANG, D., WANG, G., et al, “Recrystallization and anisotropy of AZ31 magnesium alloy by asynchronous rolling”, Metals, v. 13, n. 9, pp. 1631, 2023. doi: https://doi.org/10.3390/met13091631.
  » https://doi.org/10.3390/met13091631
• ^[22] LIU, L., CHEN, C., HUANG, K., et al, “Hot deformation behavior and microstructure evolution of Al-7.92Zn-1.64Mg-2.00Cu alloy”, Metals, v. 14, n. 2, pp. 176, 2024. doi: https://doi.org/10.3390/met14020176.
  » https://doi.org/10.3390/met14020176
• ^[23] SUWAS, S., SKROTZKI, W., SCHEERBAUM, N., et al, “Multi-scale investigation of microstructure and texture evolution during equal channel angular pressing of silver”, Journal of Materials Science, v. 59, n. 14, pp. 5698–5716, 2024. doi: https://doi.org/10.1007/s10853-023-09266-w.
  » https://doi.org/10.1007/s10853-023-09266-w
• ^[24] CARDOSO, G.S., DIAS, V.W., ROCHA, A.S., “Os efeitos do processo de corte cisalhante em barras oriundas do processo de trefilação combinada”, Revista Matéria, v. 22, n. 3, pp. e11853, 2017. doi: https://doi.org/10.1590/S1517-707620170003.0187.
  » https://doi.org/10.1590/S1517-707620170003.0187
• ^[25] MORAIS, F.L.D., CORRÊA, E.C.S., LOPES, W., “Estudo da conformabilidade do aço inoxidável AISI 409 sob rotas complexas de deformação”, Tecnologica em Metalurgia, Materiais e Mineração, v. 18, pp. e2336, 2021. doi: https://doi.org/10.4322/2176-1523.20212336.
  » https://doi.org/10.4322/2176-1523.20212336
• ^[26] GOTTSTEIN, G., Physical foundations of materials science, Berlin, Heidelberg, Springer Berlin Heidelberg, 2004. doi: https://doi.org/10.1007/978-3-662-09291-0.
• ^[27] BOURDIN, F., CASTAÑO, S.L., HÉMERY, S., et al, “Understanding the relationship between heterogeneous microstructure and tensile properties through characterization of Plastic 246 strains localization in Ti-6Al-4V processed by WDED”, In: Proceedings of the 15th World Titanium Conference 2023, pp. 1–5, Edinburgh, Jun. 2023.
• ^[28] VERHOEVEN, J.D., PETERSON, D.T., “What is a damascus steel?”, Materials Characterization, v. 29, n. 4, pp. 335–341, 1992. doi: https://doi.org/10.1016/1044-5803(92)90105-Q.
  » https://doi.org/10.1016/1044-5803(92)90105-Q

Publication Dates

• Publication in this collection
  02 Feb 2026
• Date of issue
  2026

History

• Received
  06 Sept 2025
• Accepted
  08 Dec 2025