Cellular Automata and Phase-Field Modeling of Microstructure Evolution in Metal Additive Manufacturing: Recent Advances, Hybrid Frameworks, and Pathways to Predictive Control

Abstract

Metal additive manufacturing (AM) generates complex microstructures through extreme thermal gradients and rapid solidification, critically influencing mechanical performance and industrial qualification. This review synthesizes recent advances in cellular automata (CA) and phase-field (PF) modeling to predict grain-scale microstructure evolution during AM. CA methods provide computational efficiency, enabling large-domain simulations and excelling in texture prediction and multi-layer builds. PF approaches deliver superior thermodynamic fidelity for interface dynamics, solute partitioning, and nonequilibrium rapid solidification through CALPHAD coupling. Hybrid CA–PF frameworks strategically balance efficiency and accuracy by allocating PF to solidification fronts and CA to bulk grain competition. Recent algorithmic innovations—discrete event-inspired CA, GPU acceleration, and machine learning—extend scalability while maintaining predictive capability. Validated applications across Ni-based superalloys, Ti-6Al-4V, tool steels, and Al alloys demonstrate robust process–microstructure–property predictions through EBSD and mechanical testing. Persistent challenges include computational scalability for full-scale components, standardized calibration protocols, limited in situ validation, and incomplete multi-physics coupling. Emerging solutions leverage physics-informed machine learning, digital twin architectures, and open-source platforms to enable predictive microstructure control for first-time-right manufacturing in aerospace, biomedical, and energy applications.

1. Introduction

Metal additive manufacturing (AM) has emerged as a transformative technology that enables unprecedented design freedom, geometric complexity, and rapid prototyping across the aerospace, biomedical, automotive, and energy sectors [1,2]. Unlike conventional manufacturing processes, AM builds components layer-by-layer through localized melting and rapid solidification of metallic powders or wires, inducing extreme thermal gradients, high solidification velocities, and complex thermal histories involving repeated melting and reheating cycles [3,4]. These process conditions generate heterogeneous microstructures—characterized by directional grain morphologies, crystallographic textures, phase distributions, and defects—that critically govern mechanical performance, fatigue resistance, and service life of AM components [1,5].

This review focuses primarily on metal additive manufacturing processes involving layer-by-layer solidification from the melt, specifically laser powder bed fusion (LPBF, also known as selective laser melting (SLM)), electron beam powder bed fusion (EB-PBF), and directed energy deposition (DED, including laser metal deposition and wire arc additive manufacturing). These processes share common physical phenomena—rapid solidification, steep thermal gradients (10³–10⁶ K/m), high cooling rates (10³–10⁶ K/s), and epitaxial grain growth across layers—that are amenable to cellular automata and phase-field modeling approaches. Although the fundamental modeling frameworks discussed are broadly applicable, specific examples and validation studies emphasize LPBF due to its industrial prominence and extensive experimental characterization in the literature. Solid-state processes such as friction stir additive manufacturing and ultrasonic additive manufacturing involve fundamentally different physics (plastic deformation and diffusion bonding) and are not covered in this review. Similarly, polymer-based additive manufacturing processes are excluded because of distinct material behavior and solidification mechanisms.

Predictive control of microstructure evolution during AM remains a central challenge limiting industrial qualification and adoption. Experimental characterization alone cannot efficiently explore the vast spaces of process parameters or provide a mechanistic understanding of the transient solidification phenomena that occur on microsecond time scales and micrometer-long scales [6,7]. Computational modeling offers a complementary pathway, enabling virtual exploration of process–microstructure–property relationships, optimization of scan strategies, and prediction of defect formation prior to physical fabrication [2,4,8]. Among mesoscale modeling frameworks, cellular automata (CA) and phase-field (PF) methods have emerged as the dominant approaches to simulating grain-scale microstructure evolution in metals, each offering distinct capabilities and trade-offs [2,9,10].

Cellular automata modeling employs discrete lattice representations and probabilistic switching rules to simulate grain nucleation, growth, and competitive selection with computational efficiency enabling large-domain simulations (at millimeter-to-centimeter scales) containing thousands of grains [5,11]. CA methods excel in capturing grain morphology, texture evolution, and columnar-to-equiaxed transitions across multi-track, multi-layer AM builds, making them well-suited for industrial-scale process simulations where statistical microstructural distributions suffice [5,12]. However, CA frameworks often rely on simplified interface kinetics and heuristic parameter calibration, potentially limiting predictive accuracy for detailed interfacial phenomena and transient solidification dynamics [2,8].

Phase-field modeling provides thermodynamically rigorous descriptions of interface dynamics through continuous order parameter fields governed by variational principles derived from free-energy functionals [9,10,13]. PF methods naturally incorporate diffuse-interface representations, anisotropic grain boundary properties, solute partitioning, and coupling with CALPHAD thermodynamic databases, enabling high-fidelity predictions of dendrite morphology, solute segregation, and nonequilibrium rapid solidification phenomena [9,13,14]. This physical rigor comes at substantial computational cost, restricting the applicability of PF to smaller domains (micrometer-to-millimeter scales) with limited grain populations and requiring fine spatial discretization and small time steps [2,10].

The complementary strengths of CA and PF methods—CA’s computational efficiency versus PF’s interface fidelity—motivate hybrid frameworks that strategically combine both approaches to balance predictive accuracy with computational feasibility for complex AM processes [12,15,16]. Recent advances have witnessed significant progress in algorithmic innovations (discrete event-inspired CA, GPU acceleration, and machine learning integration), coupling strategies (sequential versus concurrent thermal–microstructure coupling), and experimental validation protocols (EBSD texture analysis and synchrotron X-ray imaging) that collectively enhance model credibility and industrial relevance [5,6,17,18].

Despite these advances, persistent challenges remain: computational scalability for full-scale components, lack of standardized calibration and uncertainty quantification frameworks, limited in situ validation data for transient phenomena, incomplete physics for multicomponent alloys and mechanical coupling, and insufficient integration with digital twin architectures for real-time process control [2,4,8,18]. Addressing these challenges requires sustained interdisciplinary collaboration that bridges computational materials science, experimental characterization, and industrial AM practice [1,19].

This review synthesizes recent developments in CA, PF, and hybrid modeling approaches for metal AM microstructure prediction, with specific objectives: establish fundamental principles underlying CA and PF methodologies (Section 2); examine recent algorithmic advances in CA modeling for solidification, grain growth, and phase transformations (Section 3); analyze PF innovations addressing nonequilibrium rapid solidification, multicomponent systems, and melt pool-scale simulations (Section 4); evaluate hybrid CA–PF coupling strategies (Section 5); assess applications to process microstructure property predictions across industrially relevant alloys (Section 6); identify unresolved challenges and future directions (Section 7). The review emphasizes quantitative validation against experimental data, computational efficiency considerations, and pathways toward predictive microstructure control enabling first-time-right manufacturing for next-generation aerospace, biomedical, and energy applications.

Figure 1 shows a schematic of the process–structure–property link in additive manufacturing. Process parameters (laser power, scan speed, and strategy) determine thermal conditions (G, R, and cooling rate), which serve as inputs for CA/PF models to predict microstructural features such as grain size, texture, and phase fractions. These predictions feed into crystal plasticity and machine learning workflows for property estimation and process optimization. Validation through EBSD and mechanical testing supports model credibility for industrial deployment.

Figure 1. Conceptual framework linking CA/PF modeling to AM process optimization.

2. Fundamentals of CA and PF

Accurate simulation of microstructure evolution is essential for understanding and optimizing material behavior during processes such as solidification, recrystallization, and grain growth. This section introduces two foundational modeling frameworks—cellular automata (CA) and phase-field (PF)—which offer distinct but complementary approaches to representing grain-scale phenomena in metals.

2.1. Cellular Automata Modeling

Cellular automata (CA) modeling has emerged as a computationally efficient framework for simulating microstructure evolution in metals, particularly for grain growth, recrystallization, and solidification phenomena. This section establishes the fundamental principles underlying CA methodologies, focusing on discrete lattice structures, neighborhood schemes, probabilistic switching rules, and coupling strategies that enable mesoscale simulation, providing a conceptual basis for the advances detailed in Section 3.

2.1.1. Core Principles

Cellular automata models discretize the spatial domain into a lattice of cells, each representing a volume element characterized by state variables such as grain orientation or phase [20,21]. The choice of lattice geometry—square, cubic, hexagonal, or irregular—fundamentally influences the accuracy of the model and computational efficiency [22]. Regular lattices are widely employed due to their simplicity, although they inherently introduce directional bias (anisotropy) that can distort isotropic growth phenomena [23,24]. Hexagonal lattices reduce numerical errors compared to square lattices, offering improved isotropy at a modest computational cost [25].

Neighborhood schemes define local interaction rules that govern state transitions. The von Neumann neighborhood (first nearest neighbors) and Moore neighborhood (first and second nearest neighbors) represent classical configurations that balance computational simplicity with physical fidelity [20,21]. Extended neighborhood schemes incorporating third- or higher-order neighbors further mitigate anisotropy and capture long-range interactions, though at increased computational expense [20,26]. Recent algorithmic innovations address lattice-induced anisotropy through weighted neighborhood schemes and virtual submesh methods, where probabilistic weights are assigned based on crystallographic orientation to reduce directional bias [23,26,27].

Probabilistic switching rules constitute the core algorithmic mechanism that allows CA models to simulate stochastic microstructural evolution [20,21,28]. Transition probabilities are formulated as probabilistic analogues of classical kinetic rate equations, with changes in the cell state governed by driving forces such as stored energy, grain boundary curvature, or thermal gradients [20]. Probabilistic CA (PCA) frameworks model state changes as Markovian processes, aligning simulations with classical nucleation and growth theories such as the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model and Turnbull rate equations [20,21,29]. The stochastic nature of probabilistic switching introduces inherent uncertainty and non-repeatability, which scales inversely with model size and cellular resolution [30].

Nucleation modeling represents the initiation of new grains or phases through heterogeneous nucleation (at substrates and grain boundaries) or continuous nucleation (progressive throughout the domain) [31,32]. Conceptual models employ generalized functional forms such as Gaussian distributions for heterogeneous nucleation or power law relationships for bulk nucleation rates, avoiding alloy-specific parameterization [32]. Growth kinetics are governed by interface velocity laws that relate grain boundary migration rates to driving forces, with crystallographic anisotropy incorporated through orientation-specific kinetic coefficients [20,21,33].

Figure 2 presents a comparison of different types of CA lattice structures and neighborhood schemes. Hexagonal and irregular lattices reduce anisotropy compared to square lattices, improving isotropy in grain growth simulations.

Figure 2. CA lattice structures: (a) square lattice with Moore neighborhood (8 neighbors), (b) hexagonal lattice with first nearest neighbors (6 neighbors), and (c) irregular lattice with variable neighborhood.

2.1.2. Computational Efficiency and Scalability

Computational efficiency in CA modeling is fundamentally constrained by the trade-off between grid resolution (cell size) and domain size (total number of cells) [30,34,35]. Fine grid resolution is necessary to accurately resolve grain boundaries and nucleation events, but increases memory requirements and computational time proportionally to cell count [30,34]. Uncertainty quantification studies demonstrate that increasing cellular resolution and model size reduces stochastic variability, with coefficients of variation decreasing inversely with the square root of cell count [30,35].

Scalability to large domains and high resolutions is achieved through parallel computing architectures, including multi-core CPUs and graphics processing units (GPUs) [34,36]. GPU-accelerated CA models exploit massive parallelism, achieving speedups of one to two orders of magnitude compared to serial CPU implementations [34,36]. Multi-GPU architectures further extend scalability, enabling simulations with billions of cells for three-dimensional microstructure evolution [34,36]. Algorithmic optimizations, including efficient neighbor-search algorithms and event-driven CA approaches that update cells only when state changes occur, further improve efficiency [37].

2.2. Phase-Field Modeling

Phase-field (PF) modeling has emerged as a rigorous thermodynamically consistent framework for simulating interface dynamics and microstructure evolution in metals, offering high fidelity in capturing interface physics and transient phenomena. This section establishes the fundamental principles underlying PF methodologies, focusing on order parameters, free-energy functional construction, governing equations, and quantitative formulations that enable detailed mesoscale simulation, providing a theoretical basis for the advances detailed in Section 4.

2.2.1. Core Principles

Phase-field models employ continuous field variables—order parameters—to represent local phase identity, composition, or microstructural state, eliminating the need for explicit interface tracking [38,39,40]. Non-conserved order parameters (e.g., phase identity φ) distinguish between solid and liquid or different crystal orientations, while conserved order parameters (e.g., composition c) represent concentration fields governed by mass conservation [40,41,42]. The diffuse interface concept smooths sharp phase boundaries into continuous profiles with characteristic interface thickness δ, allowing implicit treatment of complex interface geometries and topological changes [39,43,44].

Free-energy functionals form the thermodynamic foundation of PF models, typically decomposed into chemical (bulk), gradient, and elastic contributions [42,45,46]. To ensure thermodynamic consistency and proper representation of interface physics, the total free energy functional includes gradient energy contributions for both phase-field and composition variables. The gradient energy coefficients κ_φ and κ_c are distinct parameters that control the widths of the interface for the phase boundaries and compositional profiles, respectively. These should not be confused with the strain tensor ε appearing in the elastic energy contribution, which represents mechanical deformation. The total free energy F is expressed as [42,46,47]

$$F={\int }_V\left[f_{\mathrm{chem}}\left(\phi ,c,T\right)+{\kappa }_{\phi }^2{\left|\nabla \phi \right|}^2+{\kappa }_c^2{\left|\nabla c\right|}^2+f_{\mathrm{elastic}}\left(\phi ,\epsilon \right)\right]\hspace{0.17em}\mathrm{d}V,$$

(1)

where

• $f_{\mathrm{chem}}(\phi ,c,T)$—chemical free-energy density, which depends on the phase-field variable φ, concentration $c$, and temperature $T$. It typically represents the thermodynamic driving force for phase transformation.
• ${\kappa }_{\phi }^2{\mid \nabla \phi \mid }^2$—gradient energy density for the phase field, penalizing spatial variations in φ and ensuring smooth interfaces between phases. The parameter κ_φ controls the phase interface width.
• ${\kappa }_{\mathrm{c}}^2{\mid \nabla c\mid }^2$—gradient energy density for the composition field, penalizing spatial variations in concentration and contributing to interface energy. The parameter κ_c controls the compositional interface width.
• $f_{\mathrm{elastic}}(\phi ,\epsilon )$—elastic strain energy density, which accounts for mechanical interactions such as lattice mismatch or stress development during transformation, dependent on the phase field and local strain $\epsilon$.
• $V$—domain of integration representing the physical volume of the material system.

The evolution equations for the order parameters are derived variationally from free-energy functionals, ensuring thermodynamic consistency through gradient flow dynamics [40,42,48]. The Allen–Cahn equation governs non-conserved order parameter evolution [40,49,50]:

$${\displaystyle \frac{\partial \phi }{\partial t}}=-M_{\phi }{\displaystyle \frac{\mathsf{\delta }F}{\mathsf{\delta }\phi }}=-M_{\phi }\left({\displaystyle \frac{\partial f_{\mathrm{chem}}}{\partial \phi }}-{\kappa }_{\phi }^2{\nabla }^2\phi \right),$$

(2)

where

• ${\displaystyle \frac{\partial \phi }{\partial t}}$—rate of change of the phase-field variable over time;
• $M_{\phi }$—mobility coefficient, controlling the speed of interface evolution;
• $F$—total free energy of the system;
• $\delta F/\delta \phi$—functional derivative of the total free energy with respect to φ, indicating how the system evolves to reduce its free energy;
• ${\displaystyle \frac{\partial f_{\mathrm{chem}}}{\partial \phi }}$—derivative of the chemical free energy density;
• ${\kappa }_{\phi }^2{\nabla }^2\phi$—term resulting from the phase-field gradient energy, acting to smooth interfaces.

This equation describes interface migration driven by curvature and chemical potential differences, applicable to grain growth and solid-state phase transformations [40,42].

The Cahn–Hilliard equation governs conserved order parameter evolution (e.g., composition) [41,42,51]:

$${\displaystyle \frac{\partial c}{\partial t}}=\nabla \cdot \left(M_c\nabla {\displaystyle \frac{\mathsf{\delta }F}{\mathsf{\delta }c}}\right)=\nabla \cdot \left[M_c\nabla \left({\displaystyle \frac{\partial f_{\mathrm{chem}}}{\partial c}}-{\kappa }_c^2{\nabla }^{2}c\right)\right],$$

(3)

where

• $c$—concentration field;
• $f_{\mathrm{chem}}$—chemical free energy density;
• κ_c—gradient energy coefficient for composition field;
• $M_c$—mobility coefficient for composition field;
• ${\nabla }^{2}c$—Laplacian of concentration field;
• $\delta F/\delta c$—variational derivative of free energy with respect to c.

This equation enforces mass conservation and describes diffusion-controlled phase separation and coarsening phenomena [41,42,52].

Coupled Allen–Cahn and Cahn–Hilliard equations enable simultaneous modeling of non-conserved phase transformations and conserved composition evolution, capturing complex microstructural phenomena such as dendritic solidification and precipitation [42,48,53].

The diffuse interface width δ is a critical parameter that balances computational efficiency and physical precision [43,54,55]. Matched asymptotic expansions rigorously link PF models to sharp interface limits, demonstrating that interface thickness can be chosen independently of physical interface width, provided appropriate corrections are applied [44,55,56]. The thin interface limit enables quantitative simulations with computationally tractable interface widths while preserving sharp interface kinetics through anti-trapping currents and modified mobility functions [55,57,58].

Crystallographic anisotropy in interface energy and mobility is incorporated through orientation-dependent functions, typically for cubic symmetry expressed as [55,59,60]

$$\mathsf{\gamma }\left(\mathsf{\theta }\right)={\mathsf{\gamma }}_0\left[1+{\mathsf{\epsilon }}_4\cos \left(4\mathsf{\theta }\right)\right],$$

(4)

where

• $\gamma \left(\theta \right)$—anisotropic interfacial energy as a function of angle $\theta$;
• ${\gamma }_0$—average (isotropic) interfacial energy;
• ${\epsilon }_4$—strength of the fourfold anisotropy;
• $\theta$—angle between the normal interface and a reference crystallographic direction.

Higher-order tensor expansions allow for the representation of hexagonal and complex crystal symmetries [61,62].

Figure 3 illustrates the concept of diffuse interface and the structure of the free-energy functional used in phase-field modeling. Figure 3a shows how a sharp interface (dashed line) is approximated by a smooth order parameter profile, φ(x), which continuously transitions between phases over a finite interfacial thickness δ. Figure 3b shows the chemical free energy function, $f_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}\left(\phi \right)$, showing a double-well potential with minima at φ = 0 and φ = 1, corresponding to equilibrium phases.

Figure 3. Schematic of diffuse interface concept and free-energy functional structure: (a) diffuse interface representation, (b) double-well chemical free energy.

2.2.2. Quantitative Phase-Field Formulations

Quantitative PF modeling requires an accurate representation of interface kinetics at computationally feasible interface widths [55,57,58]. For binary alloy solidification, an anti-trapping current j_at is added to the solute flux [57,58]:

$${\mathbf{j}}_{\mathrm{at}}=-a\left(\phi \right){\displaystyle \frac{\partial \phi }{\partial t}}{\displaystyle \frac{\nabla \phi }{\left|\nabla \phi \right|}},$$

(5)

where

• ${\mathbf{j}}_{\mathrm{a}\mathrm{t}}$—anti-trapping flux (accounts for interface motion to reduce artificial solute trapping);
• $a\left(\phi \right)$—anti-trapping coefficient (function of phase-field variable);
• φ—phase-field variable (e.g., order parameter distinguishing solid and liquid);
• $\partial \phi /\partial t$—time derivative of the phase field;
• $\nabla \phi$—spatial gradient of φ;
• $\mid \nabla \phi \mid$—magnitude of the gradient of φ.

Integration with CALPHAD (CALculation of PHAse Diagrams) (https://calphad.org, access date: 15 November 2025), databases enables realistic thermodynamic coupling by providing composition- and temperature-dependent free energies, chemical potentials, and phase equilibria for multi-component alloys [47,63,64]. CALPHAD-coupled PF models replace double-well phenomenological potentials with experimentally validated thermodynamic data, improving predictive precision for real alloy systems [63,64,65]. Specific implementations of CALPHAD-coupled PF models are detailed in Section 4.2.

2.2.3. Computational Challenges and Acceleration

Computational efficiency in PF modeling is limited by the need to resolve thin interface regions with fine spatial discretization [43,54,55]. The thickness of the interface δ typically requires the mesh resolution Δx ≤ δ/4 for numerical stability, leading to high computational costs for three-dimensional simulations [43,55]. Adaptive meshing strategies dynamically refine grids near interfaces while maintaining coarse resolution in bulk regions, reducing the total number of cells without sacrificing accuracy [43,66,67].

Scalability to large domains and high resolutions is achieved through GPU acceleration and high-performance computing (HPC) architectures. PF algorithms are inherently parallelizable, with local update rules enabling efficient domain decomposition across multiple processors or GPU cores. Emerging machine learning approaches, including physics-informed neural networks (PINNs) and deep operator networks (DeepONet), offers promising avenues for accelerating PF simulations by learning solution operators from data, though validation and robustness remain active research areas [68,69].

2.3. Comparative Analysis of CA and Phase-Field Methods

Cellular automata and phase-field methods represent fundamentally distinct approaches to simulating microstructure evolution, each offering complementary capabilities [40,70,71]. CA methods employ discrete, rule-based frameworks that emphasize computational efficiency and grain-level morphology representation through local interaction rules on lattice structures [5,70,72].

PF methods are grounded in thermodynamics and kinetics, employing continuous field variables governed by variational principles to capture interfacial energy, transient phase changes, and anisotropic grain boundary migration with high thermodynamic fidelity [40,71,73]. By solving coupled nonlinear partial differential equations, PF simulations naturally incorporate diffuse interface representations and multi-physics coupling, enabling detailed modeling of solidification, phase transformations, and interface stability [40,73,74]. This physical rigor comes at the cost of computational intensity, which requires fine spatial discretization and small time steps that restrict domain sizes and grain counts [75,76,77].

CA methods typically simulate millimeter-to-centimeter domains containing 10²–10⁴ grains over seconds-to-minutes time scales, with the demonstrated capability to model millions of grains in three dimensions through efficient parallelization [72,78].

PF methods generally access micrometer-to-millimeter domains with 10¹–10² grains over microseconds-to-seconds time scales, constrained by mesh resolution requirements and numerical stability considerations [75,76,79]. Recent algorithmic advances, including nonlinear preconditioning, adaptive meshing, and GPU acceleration, have extended PF applicability to larger domains with hundreds to thousands of grains [75,80,81], though computational cost remains substantially higher than CA for equivalent domain sizes [76,77].

Table 1 presents a comparative summary of CA and PF methods across key modeling criteria, highlighting the fundamental fidelity–efficiency trade-off that guides method selection [70,77,82].

Table 1. Comparative summary of cellular automata and phase-field methods.

Criterion	Cellular Automata (CA)	Phase-Field (PF)	References
Computational Efficiency	High; enables rapid simulations and large-scale parallelization	Moderate to low; computationally intensive due to coupled PDEs	[5,70,72,75,76]
Domain Size (typical)	mm–cm scale; demonstrated >1000 mm3 part-scale simulations with OpenMP parallelization	μm–mm scale; multi-layer (20 layers, 4 tracks) simulations achieved with GPU acceleration	[83,84]
Grain Count (typical)	102–104 grains typical; part-scale simulations with parallelization capture full component texture	101–102 grains typical; advanced GPU-accelerated methods enable multi-layer, multi-track simulations	[83,84]
Time Scale (typical)	Seconds to minutes	Microseconds to seconds	[85]
Interface Fidelity	Moderate; simplified kinetics with lattice-induced anisotropy mitigated by neighborhood schemes	High; thermodynamically consistent diffuse interface representation enables sub-grain solute resolution	[15]
Thermodynamic Consistency	Limited; parameters often lack direct physical meaning	High; naturally incorporates CALPHAD thermodynamic databases enabling quantitative real-alloy predictions	[86]
Transient Phenomena	Limited; struggles with interface stability and rapid kinetics	High; accurately captures transient growth and phase changes	[77,82,87,88]
Multi-Physics Coupling	Limited; rule-based framework complicates integration	High; continuum formulation enables natural coupling	[15,86]
Best-Use Cases	Large-scale builds, texture prediction, process parameter studies, grain morphology evolution	Detailed interface physics, transient solidification, nonequilibrium phenomena, anisotropic kinetics	[83,84,85]

CA methods are well-suited for large-scale additive manufacturing builds, where predicting texture across entire components is crucial. They are also effective for process parameter optimization when statistical grain distribution is sufficient, for quickly surveying processing windows with approximate microstructural fidelity, and for simulations where computational speed is prioritized over detailed interface physics [5,70,72,78,89].

PF models are better suited to detailed solidification scenarios that require precise treatment of interface kinetics and solute partitioning. They excel in capturing transient behaviors such as dendrite-tip instabilities and rapid solidification, modeling nonequilibrium phase transformations under significant undercooling, resolving anisotropic grain boundary migration and facet formation, and integrating multi-physics effects involving thermal, mechanical, or chemical fields [40,73,74,82,88].

Section 5 examines hybrid CA–PF frameworks that combine these complementary strengths. Section 6.5 demonstrates these trade-offs through a detailed case study of Inconel 718 LPBF with experimental validation against EBSD, XRD, SEM, and EDS measurements.

Figure 4 provides a visual comparison between cellular automata (CA) and phase-field (PF) modeling strategies. Figure 4a shows the CA approach, where the microstructure is discretized on a lattice with unique grain IDs assigned to each cell. Figure 4b illustrates the PF method, which uses continuous order parameters to represent grains with diffuse interfaces. Figure 4c presents a trade-off plot that compares computational cost and physical fidelity: CA offers low cost with moderate fidelity, PF provides high fidelity at higher computational expense, and hybrid models strike a balance between the two. Figure 4d shows the scalability of the model by plotting the accessible domain size against the grain count, highlighting differences in scale capability.

Figure 4. Visual comparison of CA and PF modeling approaches: (a) CA lattice with discrete grain IDs, (b) PF field with continuous order parameters and diffuse interfaces, (c) computational cost vs. physical fidelity, and (d) scale accessibility comparison via domain size versus grain count.

2.4. Additive Manufacturing-Specific Modeling Considerations

Cellular automata and phase-field models developed for conventional solidification processes require several adaptations for additive manufacturing applications because of the unique thermal and kinetic conditions inherent to layer-by-layer manufacturing.

AM processes involve thermal gradients of 10³–10⁶ K/m and cooling rates of 10³–10⁶ K/s, orders of magnitude higher than conventional casting or welding [90,91,92,93]. These extreme conditions drive rapid solidification with interface velocities approaching 0.1–1 m/s, where solute trapping, velocity-dependent partitioning, and metastable phase formation deviate from the equilibrium thermodynamics assumed in many CA models [94,95,96]. Experimental studies of Ti-6Al-4V under directional solidification conditions representative of AM report cooling rates spanning 3–1000 K/s [97], while IN718 laser processing exhibits thermal gradient-to-solidification velocity ratios (G/R) below 10 K·s/mm² in equiaxed grain formation regions [98]. Phase-field models must incorporate velocity-dependent interface kinetics and CALPHAD databases extended to metastable phases to capture these nonequilibrium phenomena quantitatively [95,96]. Okugawa et al. reported solidus temperature depressions of up to 38.4 K at cooling rates of 1.6 × 10⁶ K/s for Hastelloy-X, demonstrating the magnitude of thermodynamic changes under extreme AM conditions [94].

Unlike single-pass solidification, AM involves repeated melting and solidification cycles as subsequent layers are deposited. Grains nucleated in prior layers serve as substrates for epitaxial growth in overlying layers, producing columnar grain structures that extend through multiple layers [90,99,100,101]. Directed energy deposition of IN718 exhibits primary dendrite arm spacing variations from 12.8 μm at the bottom to 8.3 μm at the top of deposited tracks, with equiaxed grain fractions increasing from 8% (bottom) to 48% (top) due to thermal history variations [101]. CA and PF models must track grain orientation through layer interfaces and account for partial remelting of prior layers, requiring careful treatment of boundary conditions at the solid–liquid interface [90,99]. Thermal cycling also induces solid-state transformations (grain coarsening, precipitation, and phase decomposition) in previously solidified material, necessitating coupled solidification–transformation models for alloys such as Ti-6Al-4V and H13 tool steel [12,102,103].

Nucleation in AM occurs in partially melted powder particles, oxide inclusions, and solidified prior layers, creating heterogeneous nucleation site distributions that differ fundamentally from mold wall nucleation in casting [100,104,105,106]. The nucleation density (typically 10¹¹–10¹² m⁻³) becomes a critical calibration parameter that requires experimental validation via grain size measurements from the EBSD [104]. Andersson and Lundbäck systematically tested nucleation densities of 1 × 10¹¹, 3 × 10¹¹, and 5 × 10¹¹ m⁻³ for Alloy 625 CA-FE simulations, finding that 3 × 10¹¹ m⁻³ provided optimal agreement with experimental EBSD grain size distributions [104]. Powder particle size distribution (typically 15–45 μm for LPBF), packing density, and surface oxide characteristics influence local thermal conductivity and nucleation behavior, introducing process variability that must be quantified through uncertainty quantification frameworks [30,107]. Advanced beam shaping techniques demonstrate the influence of nucleation control on microstructure: adjustable ring mode (ARM) laser processing of Scalmalloy^® aluminum alloy achieves >90% fine-grained microstructure compared to <50% for conventional circular beam processing [106].

Complex scan strategies (stripes, islands, chessboard patterns) with varying hatch spacing (50–150 μm) and rotation angles between layers create spatially varying thermal histories [90,91,108]. The overlap of the melt pool and the remelting of adjacent tracks introduce additional complexity, with the prior grain structures partially remelted and new grains nucleating in the overlap regions [90,99,101]. CA and PF models must incorporate scan path geometry and track-to-track thermal interactions, typically through coupling with finite element thermal simulations providing spatially and temporally resolved temperature fields [90,91,109]. Heat input variations from 250 J/m to 500 J/m in IN718 selective laser melting produce approximately 20% increases in average grain size, while substrate grain size reductions of 50% yield corresponding 17% reductions in deposited layer grain size, demonstrating the sensitivity of microstructure to process parameters and initial conditions [109].

AM-specific defects, including lack-of-fusion porosity, keyhole pores, hot cracking, and residual stress-induced distortion, arise from the unique thermal conditions and constrained geometry [105,108,110]. Porosity levels ≥ 1% are observed in process parameter regions exhibiting insufficient energy density or excessive scan speeds in LPBF of IN718 [111], establishing quantitative thresholds for process window definition. While most CA and PF models focus on grain structure evolution, predictive capability for defect formation requires integration with thermal–fluid simulations capturing melt pool dynamics, gas entrapment, and solidification cracking criteria [105,108]. Porosity influences local thermal conductivity and nucleation site availability, creating feedback between defect formation and microstructure evolution that is rarely captured in current models [105,108].

These AM-specific considerations necessitate careful model calibration, validation against multi-modal experimental characterization (EBSD, SEM, XRD, and in situ monitoring), and uncertainty quantification to establish predictive capability under the extreme and variable conditions characteristic of additive manufacturing processes [30,90,99,110].

3. Advances in Cellular Automata Modeling

This section explores recent developments in the application of CA methods to solidification, grain growth, and phase transformations in metallic systems. Particular attention is paid to coupled thermal–CA frameworks, advances in algorithmic strategies, and integration of these models with experimental validation.

3.1. Solidification and Dendritic Growth

The integration of CA models with thermal solvers—primarily finite element (FE), finite volume (FV), and computational fluid dynamics (CFD) methods—has enabled detailed predictions of grain morphology, texture, and competitive growth under rapid solidification conditions characteristic of additive manufacturing [90,101]. These coupled frameworks capture the interplay between thermal gradients (G), solidification velocities (R), and nucleation mechanisms that govern columnar-to-equiaxed transitions (CETs) [90,91,101]. Table 2 presents representative coupled thermal–CA frameworks, highlighting coupling strategies and validation results.

Table 2. Coupled Thermal–CA Frameworks for Solidification Modeling.

Coupling Strategy	Validation	Key Findings	References
Eulerian multiphase CFD-CA	Melt pool microstructure (IN718)	Primary dendrite arm spacing reduced from 12.8 μm (bottom) to 8.3 μm (top) of melt pool; validated by experiment	[101]
Multi-scale FE–CA	Grain structure analysis (EBSD)	The grain size increases ~20% when heat input increases from 250 to 500 J/m; validated against IN718 EBSD data	[109]
FE–CA with heterogeneous nucleation	EBSD grain structure (Alloy 625)	The nucleation density 3 × 1011 m−3 provided best agreement with experimental EBSD data	[104]

Competitive grain growth—where grains with favorable crystallographic orientations relative to the thermal gradient grow preferentially—determines texture evolution in AM components [99,104,112]. CA models capture this phenomenon through crystallographic anisotropy in growth kinetics, with validation detailed in Section 6.2.

3.2. Grain Growth and Recrystallization

Beyond solidification, CA models simulate post-solidification grain growth and recrystallization phenomena that are critical to tailoring the final microstructures. Cellular automata models have been used to simulate β → α phase transformation and grain morphology evolution during heat treatment of additively manufactured Ti-6Al-4V. These simulations successfully capture the transition from columnar to equiaxed structures in prior β grains, with validation provided by experimental micrographs [113,114]. By incorporating temperature-dependent nucleation and growth kinetics, the models demonstrate the ability of cellular automata to extend beyond solidification modeling and effectively represent solid-state microstructure evolution [114].

CA models have been extended to simulate solid-phase grain growth under scanning heat sources, grain coarsening in heat-affected zones, and grain refinement during laser rescanning [108,115,116].

3.3. Solid-State Phase Transformations

Integration of solidification microstructure predictions with subsequent phase transformations is essential for multi-phase alloys subjected to complex thermal histories. A hybrid cellular automata–phase-field modeling framework has been applied to simulate multi-track, multi-layer directed energy deposition (DED) of H13 tool steel, as detailed in Section 5.2. This approach couples dendrite growth simulation with solid-state phase transformation analysis, enabling prediction of austenite formation, martensitic transformation, and resulting hardness distributions [12]. This integrated approach allowed the prediction of microstructure and mechanical properties in multiple deposition layers, validated against experimental hardness and microstructure data [12].

A multi-physics CA framework has been utilized to integrate solidification and kinetic phase transformation models for predicting 3D microstructure and phase distributions in Ti-6Al-4V during directed energy deposition (DED), showing strong alignment with experimental results on microhardness and phase fractions [102]. The β/α transformation in Ti-6Al-4V has been simulated using a coupled lattice Boltzmann–CA approach, enabling detailed examination of cooling rate effects on α-lath thickness and morphology. This method incorporates CALPHAD-based thermodynamic data to define phase equilibria and transformation kinetics [103]. In electron beam selective melting of Al–Cu alloys, precipitation processes have been modeled through a 3D CA framework that accounts for mixed-mode phase transformation kinetics, effectively capturing the influence of powder bed temperature and thermal cycling on precipitate formation [117].

Representative CA models for solid-state phase transformations in various alloy systems are summarized in Table 3, highlighting transformation types, validation methods, and key predictive outcomes.

Table 3. CA models for solid-state phase transformations.

Alloy System	Transformation Type	Validation	Key Outcome	References
Ti-6Al-4V	β → α transformation	Microhardness, phase fractions (EBSD)	Predicted 3D α lath morphology and phase distributions; validated against experimental micrographs	[102]
Al-Cu	Precipitation	Precipitation data	Captured powder bed temperature effects on precipitates	[117]
H13 tool steel	Austenite → martensite (hybrid CA–PF)	Hardness, microstructure (multi-layer)	Predicted dendrite size and hardness evolution across multi-layer builds; hybrid approach captured both scales	[12]
Ti-6Al-4V	β → α transformation	Literature microstructure trends	Examined cooling rate effects on α lath thickness	[103]
Ti-6Al-4V	β → α transformation	Experimental micrographs	Captured columnar-to-equiaxed transition during heat treatment	[114]

3.4. Recent Algorithmic Improvements

Accurate modeling of nucleation mechanisms is essential for predicting grain structure in additive manufacturing (AM). Among recent advances are particle-based nucleation models that explicitly represent heterogeneous nucleation sites—such as oxide particles or substrate grain boundaries—and their activation based on undercooling and thermal gradients [100,105,106]. Spatial heterogeneity in nucleation density has been incorporated into CA models to simulate texture variations in different regions during powder bed fusion (PBF) of Inconel 625, with validation provided by EBSD measurements [100]. The use of continuous nucleation models, which permit nucleation throughout the solidification process rather than instantaneously, has enhanced the realism of grain size distribution predictions and improved the modeling of columnar-to-equiaxed transition (CET).

To improve spatial resolution at the solidification front, a multi-grid CA approach with child grid refinement has been introduced. This method enables detailed tracking of dendritic tip morphology and secondary arm spacing without excessive computational cost, supported by validation against analytical models and x-ray imaging data [118]. Grid resolution itself plays a crucial role in determining the fidelity of the CA model, especially in capturing fine dendritic features. A detailed assessment of model size, resolution, and stochastic effects has quantified their influence on simulation results, offering computational strategies for uncertainty quantification and balancing trade-offs between resolution, domain size, and performance [30]. The precision of CA-based predictions also depends on proper calibration of parameters such as nucleation density, grain boundary mobility, and anisotropy coefficients, with experimental data serving as key references for tuning [30,99,104].

A distinct computational strategy has emerged in the form of a discrete event-inspired CA (DECA) model, which departs from traditional time-stepping by capturing grain growth as a series of discrete events. This paradigm change enables significant improvements in computational efficiency while preserving predictive precision, as shown by comparison with microstructure benchmark simulations [37].

Key algorithmic enhancements that improve the accuracy, efficiency, and scalability of CA modeling are summarized in Table 4, covering innovations such as particle-based nucleation, multi-grid refinement, and parallelization strategies.

Table 4. Algorithmic Enhancements in CA Modeling.

Enhancement Type	Key Innovation	Impact	References
Particle-based nucleation	Explicit heterogeneous nucleation sites	Improved texture heterogeneity realism	[100,105,106]
Multi-grid refinement	Child grid at solidification fronts	Accurate dendrite morphology at lower cost	[118]
Discrete event CA (DECA)	Event-driven algorithm (asynchronous updates)	Significant speedup through event-driven scheduling; efficient parallelization for sparse grain distributions	[37]
Uncertainty quantification	Systematic stochastic effect analysis	Quantified uncertainty; calibration guidance	[30]
GPU acceleration	CUDA implementation for dendrite CA	Up to 158× speedup versus traditional CPU; enabled large-scale 3D simulations	[119]
MPI parallelization for recrystallization	Distributed-memory parallel CA	>700× speedup for 2D, >800× for 3D domains; scaled to workstation clusters	[120]

These CA advances enable process optimization studies and motivate hybrid CA–PF frameworks (Section 5) that leverage CA efficiency with PF accuracy for complex AM phenomena.

4. Advances in Phase-Field Modeling

While cellular automata methods excel in large-scale simulations, phase-field approaches provide complementary capabilities in resolving interface physics and nonequilibrium phenomena. This section synthesizes recent advances in PF methodologies, emphasizing nonequilibrium solidification, multicomponent systems, melt pool-scale simulations, machine learning acceleration, and model benchmarking that collectively enhance predictive accuracy for complex metallurgical processes.

4.1. Nonequilibrium Rapid Solidification

Rapid solidification in AM induces significant departures from equilibrium thermodynamics at solid–liquid interfaces. Solute trapping—where the solute atoms are kinetically trapped in the solid phase—becomes pronounced at high solidification velocities, leading to velocity-dependent partition coefficients that deviate from equilibrium values [96,121,122]. Phase-field models addressing these phenomena incorporate modified interface kinetic formulations, including velocity-dependent partition coefficients and enhanced antitrapping currents, to accurately capture the transition from equilibrium to complete solute trapping [96,121,123].

A coupled computational fluid dynamics (CFD) and phase-field modeling framework has been applied to laser powder bed fusion (LPBF) for binary alloys, explicitly capturing rapid solidification kinetics. This integrated approach demonstrates strong agreement with the experimental observations of microstructures and partition coefficients under AM-relevant conditions [122]. Improvements in anti-trapping current formulations, specifically calibrated for nonequilibrium solidification regimes, have enabled accurate predictions of dendrite tip undercooling and interface morphologies [123]. These findings collectively reinforce the inadequacy of classical equilibrium assumptions when modeling high solidification rates typical of additive manufacturing processes [121,122,123].

At elevated solidification velocities, dendrite-tip instabilities arise from the complex interplay among thermal and solutal diffusion fields, interface kinetics, and capillary forces. These instabilities lead to intricate microstructural features, including cellular formations and microstructure refinement [124,125,126]. Phase-field simulations reveal that rapid solidification can induce abrupt changes in growth velocity and morphological refinement through a control mechanism switching at the solid–liquid interface, which aligns well with the experimental data [126]. The different solidification pathways observed in processes such as melt spinning versus AM have been mechanistically explained through phase-field models that track interface dynamics and nucleation behaviors at high cooling rates [125].

Experimental validation using Ti-Nb binary alloys confirms model accuracy in capturing cellular growth transitions and solute trapping at solidification velocities of up to 1 m/s [121].

Table 5 compiles representative phase-field simulations of nonequilibrium rapid solidification, detailing alloy systems, key model features, validation methods, and major insights into solute trapping and microstructure evolution.

Table 5. Phase-field studies of nonequilibrium rapid solidification.

PF Model Features	Validation	Key Findings	References
Hastelloy-X: Coupled CFD-PF; cooling rates up to 1.6 × 106 K/s	Cellular microstructure, solidus temperature	Solidus reduction up to 38.4 K; submicrometer cellular structure reproduced	[94]
Inconel 718: 6-element PF model; velocity-dependent partitioning	Dendritic tip velocity vs. undercooling	Stationary tip velocity quantitatively matched directional solidification experiments	[95]
Binary alloy: Modified antitrapping current	Dendrite tip undercooling theory	Accurate interface profiles under strong nonequilibrium	[123]
Ti-50 at.% Nb: KKS, PCR, FID model comparison; high growth rate regime	Primary dendrite arm spacing, partition coefficient	Model variants show convergent behavior at high velocities; FID captures interface kinetics	[96]

4.2. Multicomponent and Multiphase Systems

Integration of phase-field models with CALPHAD (CALculation of PHAse Diagrams) thermodynamic databases represents a transformative advance, enabling quantitative predictions of phase equilibria, segregation, and transformation kinetics in multicomponent alloys [10,127,128]. CALPHAD coupling provides temperature- and composition-dependent free energies, diffusion coefficients, and interfacial properties, ensuring thermodynamic consistency and extending phase-field applicability to industrially relevant alloy systems [10,127,128].

High-fidelity 3D phase-field simulations of the Ni-based superalloy CMSX-4 have been conducted with full coupling to CALPHAD thermodynamic and kinetic databases, successfully capturing eutectic transformations, γ/γ′ precipitation, and solute segregation under additive manufacturing (AM) cooling conditions with quantitative accuracy [128]. In wire arc AM, multi-phase-field (MPF) models coupled with CALPHAD data have been employed to predict solidification microstructures and segregation behavior across multiple deposition layers. Comparative analyses of quasi-equilibrium versus nonequilibrium MPF formulations have shown that nonequilibrium modeling becomes essential at cooling rates exceeding 10⁶ K/s, especially for accurately representing solute trapping and interface kinetics [127].

For 316L stainless steel processed under rapid solidification conditions, CALPHAD-integrated MPF simulations have been used to resolve microsegregation and interfacial dynamics, highlighting the sensitivity of outcomes to interfacial mobility and permeability parameters [10]. Collectively, these studies affirm the need for CALPHAD integration to achieve quantitatively reliable phase-field predictions for complex alloy systems [10,127,128].

Multi-order-parameter phase-field models, which assign unique order parameters to individual grain orientations or phases, have enabled detailed simulation of competitive grain growth, texture evolution, and phase transformations during solidification [129,130]. In powder bed fusion AM, 3D phase-field models incorporating grain nucleation, growth, and coarsening have replicated experimentally observed grain structures, revealing stable grain evolution patterns with nucleation along melt pool boundaries and refinement during remelting cycles [129]. Further model extensions based on the Wheeler–Kobayashi–Lewis-–-Campbell (WKLC) framework, enhanced with temperature-dependent relaxation parameters, have qualitatively reproduced columnar grain morphologies observed under moving heat sources in AM melt pools [130].

A range of CALPHAD-integrated phase-field models for multicomponent alloys is presented in Table 6, showcasing how different coupling approaches capture complex solidification and segregation phenomena with experimental validation.

Table 6. CALPHAD-coupled phase-field models for multicomponent alloys.

Coupling Approach	Key Phenomena Captured	Validation	References
CMSX-4 (Ni superalloy): Full CALPHAD coupling (Thermo-Calc); 3D MPF	Eutectic transformations, γ/γ′ precipitation, microsegregation	Experimental microstructure; quantitative phase fractions	[128]
Ni-Based Alloy: MPF + CALPHAD (multi-component); thermal analysis	Solidification microstructure, segregation in wire arc AM	Multi-pass deposition; composition profiles validated	[131]
316L stainless steel: MPF + CALPHAD; interfacial mobility/permeability parameters	Microsegregation under rapid cooling	Experimental segregation patterns; composition profiles	[10]
Multicomponent Ni alloys: Quasi-equilibrium vs. nonequilibrium MPF + CALPHAD	Columnar/equiaxed structures; solute trapping at high cooling rates	Convergence studies; interface sensitivity analysis	[127]

4.3. Melt Pool-Scale Simulations

Melt pool-scale phase-field simulations integrate macroscopic thermal–fluid dynamics with mesoscale microstructure evolution, allowing prediction of grain morphology, texture, and defect formation across entire melt pools in AM [122,132]. These multiscale frameworks couple finite element (FE) or CFD thermal models with phase-field solvers, providing realistic temperature gradients, solidification velocities, and cooling rates as inputs to microstructure simulations [122,132].

Site-specific microstructure predictions in laser powder bed fusion (LPBF) have been achieved through coupling computational fluid dynamics (CFD) with phase-field modeling. This approach successfully captures the heterogeneous grain structures that emerge from spatial variations in thermal gradients and solidification rates within the melt pool [122]. Predictive microstructure distribution maps have also been generated for Ni–Cu alloys processed via LPBF, with experimental validation conducted across a wide range of processing parameters. Using a coupled finite element–phase-field (FE–PF) framework, these studies enabled efficient mapping of process windows and identification of distinct microstructure types—including planar, cellular, and dendritic morphologies—as functions of laser power and scan speed [132].

Multiscale simulations that combine CFD and phase-field modeling have been used to analyze Al–Si alloys under LPBF conditions, directly comparing predictions from equilibrium and nonequilibrium solidification models. The resulting discrepancies highlight the limitations of equilibrium assumptions and emphasize the critical need for nonequilibrium models to accurately describe melt pool microstructure evolution [133].

The evolution of the texture in AM is governed by competitive grain growth, where grains with favorable crystallographic orientations relative to the thermal gradient direction grow preferentially. Phase-field models that incorporate orientation fields and crystallographic anisotropy successfully reproduce strong <001> fiber textures aligned with build direction, consistent with experimental EBSD measurements [9,129].

The columnar-to-equiaxed transition (CET) in AM is critically influenced by melt pool geometry, thermal gradients (G), solidification velocities (R), and nucleation undercooling. Phase-field models have elucidated mechanisms controlling CET, revealing that high G/R ratios favor columnar growth, while increased nucleation undercooling and reduced thermal gradients promote equiaxed grain formation [134,135,136].

Columnar-to-equiaxed transition (CET) mechanisms in Ni-based superalloys during additive manufacturing have been investigated using multiscale phase-field models that incorporate both heterogeneous nucleation and epitaxial grain growth. These simulations reveal that CET is governed by undercooling on the solidification front and by competitive nucleation dynamics within the melt pool, with results validated against experimentally observed grain structures [136]. In the context of direct metal laser sintering (DMLS) of AlSi10Mg, phase-field models that account for heterogeneous nucleation have been used to capture morphological transitions. These models demonstrate the sensitivity of CET behavior to variations in build orientation and cooling rates, accurately reproducing microstructural changes across different processing conditions [134,135].

4.4. Machine Learning Acceleration

Integration of machine learning (ML) techniques with phase-field modeling represents an emerging frontier aimed at overcoming computational bottlenecks inherent in high-fidelity simulations of multicomponent alloys and complex AM processes [67,68]. Physics-embedded graph networks and physics-informed neural networks (PINNs) have been proposed to accelerate phase-field energy minimization and evolution equations while preserving physical consistency [67,68].

Machine learning (ML) has been applied to accelerate phase-field modeling in wire and arc additive manufacturing (WAAM) of aluminum alloys, showing promise in reducing the computation time for microstructure predictions. Nevertheless, the integration of ML into practical simulation workflows remains in its early stages, constrained by the need for large, high-quality training datasets and rigorous validation to ensure applicability across different alloy systems and processing conditions [137].

The high computational cost associated with repeated CALPHAD database queries during phase-field simulations has driven the development of ML-based surrogate models for thermodynamic properties. These surrogates offer rapid approximations of key quantities such as free energies, chemical potentials, and phase equilibria, allowing more efficient simulations [9]. The combination of ML with phase-field methods has also been explored for alloy design and microstructure prediction, showing improved numerical stability and enhanced predictive capability in multi-scale modeling contexts when surrogate models are employed [9]. Despite these advances, wider implementation will depend on further progress in building robust, validated ML frameworks and establishing standardized protocols for coupling with physics-based models [9,137].

4.5. Model Benchmarking and Convergence

Variants of the phase-field model—including the Kim–Kim–Suzuki (KKS), Partition Coefficient Relaxation (PCR), and Finite Interface Dissipation (FID) formulations—exhibit differing performance in terms of numerical stability, convergence behavior, and physical realism under rapid solidification conditions [96,127]. Comparative evaluations of these models have assessed their ability to predict dendrite arm spacing, partition coefficients, and interface kinetics. Among them, the PCR model offers superior numerical stability and computational efficiency, whereas the FID model provides enhanced physical interpretability and greater flexibility in modeling nonequilibrium interface kinetics. However, the FID approach also shows higher sensitivity to interface width and mesh discretization, necessitating careful convergence analysis [3].

Under high cooling rates exceeding 10⁶ K/s, the verification of convergence and interface sensitivity has confirmed the importance of nonequilibrium multi-phase-field (MPF) formulations for accurately capturing solute trapping and interface dynamics [127]. In the context of selective laser melting (SLM) of steels, quantitative multi-component phase-field models have been applied with embedded stability parameters and mesh resolution criteria to ensure convergence. These models have successfully predicted dendrite arm spacing and manganese (Mn) segregation patterns, with validation against experimental results [138].

Table 7 compares key phase-field model variants based on numerical stability, computational efficiency, and physical interpretability, providing guidance on their optimal application domains in equilibrium and nonequilibrium solidification scenarios.

Table 7. Benchmarking of phase-field model variants.

Model Variant	Numerical Stability	Computational Efficiency	Physical Interpretability	Best Application Domain	References
KKS (Kim–Kim–Suzuki)	Moderate; sensitive to interface width; requires iterative solves for composition	High; thermodynamically rigorous; CALPHAD integration	Good thermodynamic consistency	Multicomponent alloys near equilibrium; quantitative predictions	[96]
PCR (Partition Coefficient Relaxation)	Moderate; calibration of relaxation parameters required; converges to KKS at high velocities	High; reduced DOFs vs. KKS; robust at high velocities	High; simpler implementation	Rapid solidification (>0.1 m/s); large-scale simulations	[96]
FID (Finite Interface Dissipation)	Moderate; sensitive to discretization; explicit kinetic coefficient	Moderate; additional interface parameter	High; explicit interface kinetics; captures velocity-dependent partitioning	Nonequilibrium solidification; interface kinetics-controlled growth	[96,139]
Grand Potential	High; no composition field in bulk phases; numerically stable	Very high; reduced DOFs; natural CALPHAD coupling	High; direct use of chemical potentials	Multicomponent alloys; CALPHAD-based simulations; dilute to concentrated alloys	[140]
Quantitative Phase-Field (QPF)	High; matched to sharp interface model; well-defined convergence	Moderate; requires thin interface limit; small interface width	Very high; parameters have direct physical meaning	Dendritic solidification; quantitative predictions without fitting	[141]

These PF advances enable detailed microstructure predictions for AM applications and motivate hybrid CA–PF frameworks (Section 5) that balance PF accuracy with CA efficiency for complex phenomena.

5. Hybrid CA–PF

The complementary strengths of cellular automata and phase-field methods—CA’s computational efficiency for large-domain grain competition and PF’s high fidelity for interface physics—motivate hybrid frameworks that strategically combine both approaches. This section synthesizes recent advances in hybrid architectures, representative frameworks for multi-layer AM processes, and validation strategies that balance predictive accuracy with computational feasibility.

5.1. Rationale and Architectures

Hybrid frameworks leverage the complementary strengths of phase-field (PF) and cellular automata (CA) methods by assigning PF to regions near the solidification front where interfacial phenomena dominate, while using CA to simulate bulk grain competition more efficiently in regions where fine interface resolution is less critical [6,142].

The coupling strategies in hybrid CA–PF frameworks fall into two primary categories, each with distinct trade-offs [3,6,12,143]; namely, sequential coupling and concurrent coupling.

Sequential coupling involves solving thermal, fluid flow, and microstructure evolution in separate stages, with information passing unidirectionally from one module to the next [12,101,143]. This approach significantly reduces computational cost by decoupling multi-physics phenomena and enables simulation of large domains and multi-layer builds [12,144,145]. Many studies in the reviewed literature used sequential coupling, demonstrating its effectiveness in predicting grain structures in alloys such as Inconel 718, Ti-6Al-4V, and 316L stainless steel [6,12,101,143,145]. However, sequential coupling may sacrifice accuracy in capturing transient phenomena and dynamic interactions between thermal fields and microstructure evolution, particularly during rapid solidification where latent heat release and interface kinetics significantly influence thermal histories [77,146].

Concurrent coupling solves the thermal–fluid and microstructure equations simultaneously, enabling bidirectional feedback between fields and a more accurate representation of coupled physics [77,92,147]. For example, concurrent PF simulations can account for latent heat release during solidification, which modifies local temperature gradients and affects subsequent grain growth [84,92]. While concurrent coupling provides superior accuracy in capturing transient interface dynamics and texture selection mechanisms, it incurs substantially higher computational expense and often requires advanced computational architectures such as GPU acceleration or massively parallel computing [84,148,149].

Figure 5 visually summarizes the hybrid CA–PF architecture discussed in this section. Figure 5a shows the domain partitioning, where the PF region resolves detailed dendrite structures near the solidification front, and the CA region models grain competition in the bulk, linked by a transition zone. Figure 5b compares sequential and concurrent coupling strategies, while Figure 5c illustrates bidirectional information exchange between thermal, CA, and PF modules.

Figure 5. Hybrid CA–PF architecture for microstructure simulation: (a) domain partitioning, (b) sequential and concurrent coupling strategies, and (c) information exchange between thermal, CA, and PF modules.

5.2. Representative Hybrid Frameworks

True hybrid frameworks that simultaneously employ both cellular automata and phase-field methods within integrated simulations remain limited, but demonstrate significant potential for capturing multi-scale physics in additive manufacturing. These frameworks strategically partition modeling domains or physical phenomena, assigning CA to computationally expensive bulk grain evolution and PF to interface critical regions or specific transformation stages.

5.2.1. Sequential CA–PF Coupling for H13 Tool Steel

A hybrid cellular automata–phase-field (CA–PF) modeling framework has been applied to simulate multi-track, multi-layer laser direct deposition of H13 tool steel [12]. This approach employs sequential coupling with iterative feedback, in which CA simulates dendritic solidification during melt pool freezing, followed by PF modeling of solid-state phase transformations (austenitization and martensitic transformation) during subsequent thermal cycling. The CA module captures primary dendrite growth under rapid solidification conditions, predicting dendrite arm spacing and orientation based on local thermal gradients. The PF model calculates local growth kinetics at the solid/liquid interface and feeds back the interface velocity and solid/liquid interface concentrations to the CA model. These CA-predicted grain structures then serve as initial microstructures for PF simulations of austenite formation and martensite precipitation during inter-layer reheating and cooling cycles.

The framework accurately predicted primary dendrite arm spacing (~2 μm), dendrite growth directions (showing qualitative agreement with thermal gradient orientation), and hardness distributions (570–600 HV in top layers; 430–550 HV in lower layers), all showing good agreement with experimental observations [12]. Computational efficiency was achieved through domain decomposition: CA handled the computationally tractable solidification stage across large domains (representative of practical AM components), while PF resolved thermodynamically rigorous solid-state transformations in the targeted regions experiencing phase changes. This sequential strategy enabled multi-layer, multi-track simulations that would be intractable with PF alone, while maintaining higher fidelity than pure CA approaches for phase transformation kinetics.

5.2.2. Integrated Sequential CA–PF Coupling for Ti-6Al-4V

Two distinct approaches have demonstrated sequential CA–PF coupling for Ti-6Al-4V, differing in dimensionality and computational scope.

An integrated 2D cellular automata–phase-field model has been developed to couple solidification microstructure evolution with thermal history during laser additive manufacturing [15]. The framework employs sequential coupling: a 2D CA component simulates dendrite growth and solute redistribution during solidification, while a 1D PF component provides local growth kinetics (instantaneous growth velocity and equilibrium solute partition) at the solid/liquid interface. The framework captures grain morphology transitions and phase distributions under rapid heating–cooling cycles characteristic of laser-based additive manufacturing.

The 2D implementation achieves an improvement in computational efficiency of more than five orders of magnitude compared to the 2D PF models alone, enabling efficient exploration of process parameter effects on microstructure evolution for alloys undergoing multiple phase transformations (β solidification followed by β → α transformation). Qualitative validation of grain morphology (elongated columnar β grains) and solute partitioning behavior against experimental observations and phase diagrams confirmed model accuracy for representative melt pool cross-sections [15].

Extending to three dimensions, a fully integrated multi-physics framework has been developed for powder bed fusion of Ti-6Al-4V [6]. The simulation architecture employs sequential (unidirectional) coupling: first, melt pool hydrodynamics (CFD) coupled with CA simulates β-phase grain solidification structure evolution during melt pool freezing, using nucleation and growth rules to predict grain size and orientation; subsequently, the CA-predicted grain structures serve as input for PF modeling of intragranular α(α′) precipitate microstructure from the β matrix during cooling using a three-dimensional multi-phase field model with an explicit nucleation algorithm. The coupling is explicitly unidirectional, with solidification results feeding forward to phase transformation modeling without feedback loops.

The 3D framework yielded predictions of the β grain solidification structures and subsequent α(α′) precipitate microstructure, with key features showing good qualitative agreement with experimental microstructural observations [6]. Sequential coupling effectively captured how solidification grain structure (including grain size and orientation) influences subsequent α-variant selection through crystallographic relationships, phenomena that cannot be accurately predicted by decoupled models. Integration with microelastic modeling using the crystal plasticity finite element method further enabled calculation of stress distributions arising from thermal expansion mismatch and transformation strains, providing insight into mechanical property development through von Mises stress profiles.

These Ti-6Al-4V implementations exemplify the strength of sequential hybrid CA–PF workflows in linking processing conditions to final properties through physics-based coupling of complementary modeling approaches, with the 2D model [15] offering exceptional computational efficiency (>5 orders of magnitude faster than PF alone) for parametric studies and the 3D model [6] providing a comprehensive integrated framework that covers melt pool dynamics, solidification microstructure, phase transformation, and microelastic response.

Table 8 summarizes the representative hybrid CA–PF approaches reported in the literature, highlighting coupling strategies, alloy systems, validated phenomena, and computational advantages relative to pure CA or PF implementations.

Table 8. Hybrid CA–PF modeling approaches for additive manufacturing.

Coupling Strategy	Alloy System	Physical Phenomena Captured	Validation	Computational Advantage	References
Sequential: CA solidification → PF solid-state transformation with iterative feedback	H13 tool steel	CA: Dendritic solidification during melt pool freezing, micro-scale dendrite growth. PF: Local growth kinetics at solid/liquid interface; austenite formation and martensitic transformation during thermal cycling	Primary dendrite arm spacing (~2 μm), dendrite growth direction (qualitative agreement), microhardness (570–600 HV top layer; 430–550 HV lower layers) vs. experimental	Enables multi-layer, multi-track simulations over large domains intractable with PF alone	[12]
Sequential (unidirectional): CA solidification grain structure → PF α-phase precipitation	Ti-6Al-4V	CA: β-phase grain solidification structure evolution during melt pool freezing PF: Intragranular α(α′) precipitate microstructure from β matrix during cooling	Qualitative validation of solidification and precipitate microstructure features vs. experimental observations; stress distributions calculated for mechanical analysis	Integrated framework combining melt pool dynamics, microstructure, phase transformation, and microelastic response	[6]
Sequential 2D: 2D CA solidification + 1D PF growth kinetics	Ti-6Al-4V	CA: 2D dendrite growth, grain morphology evolution, solute redistribution during solidification PF: 1D local growth kinetics (velocity, equilibrium solute partition) at solid/liquid interface	Qualitative grain morphology (elongated columnar β grains), solute partitioning behavior vs. phase diagrams and experimental trends	Computational efficiency improvement > 5 orders of magnitude vs. 2D PF alone; efficient exploration of process parameters in 2D	[15]

5.2.3. Limitations and Scalability

Despite their promise, hybrid CA–PF frameworks face key limitations. The high computational cost remains a barrier: multi-layer simulations of H13 steel [12] required HPC resources, while 3D Ti-6Al-4V models [6] were restricted to sub-domains. Although 2D models [15] improve efficiency, they lack the full 3D grain morphology essential to capture anisotropy. Interface mapping between discrete CA lattices and continuous PF fields introduces interpolation errors, and maintaining parameter consistency (e.g., undercooling, mobility, and anisotropy) across modules demands careful calibration.

Hybrid coupling strategies trade off fidelity for feasibility. Sequential coupling [12] is computationally efficient, but processes the solidification and phase transformation stages separately without iterative feedback between them. More integrated sequential frameworks [6] couple multiple physics (melt pool dynamics, solidification, phase transformation, and mechanical response) in a unidirectional workflow but remain computationally expensive, restricting simulations to sub-domains. Dimensionality reduction offers a practical compromise: 2D sequential models [15] achieve computational efficiency improvements exceeding five orders of magnitude compared to 2D phase-field approaches, enabling efficient parametric studies while sacrificing full 3D grain morphology representation.

Advancing CA–PF frameworks will require innovations in adaptive partitioning, interface mapping, and GPU-accelerated solvers. Full 3D integrated simulations remain costly, highlighting the need for ML surrogates trained on CA–PF data to enable rapid, physics-informed predictions for large-scale AM components.

6. Applications in Additive Manufacturing

Extending the modeling advances outlined in Section 3, Section 4 and Section 5, this section investigates how cellular automata, phase-field, and hybrid modeling frameworks contribute to predicting process–microstructure–property relationships in metal additive manufacturing. The emphasis is placed on parametric analyses of the processing conditions, alloy-specific case studies supported by experimental validation, and the integration of these models with crystal plasticity and machine learning approaches for property prediction.

6.1. Process–Microstructure Relationships

The scanning strategy—including hatch spacing, rotation angle, and pattern type—critically influences grain texture evolution by modulating thermal gradients and solidification directions [72,150,151]. Using CA–FE frameworks, studies demonstrate that unidirectional scanning produces strong <001> fiber textures aligned with the building direction, while rotating scan patterns reduce texture intensity and promote equiaxed grain structures [72,151,152]. Cellular automata simulations have been validated against EBSD data to reproduce texture transitions as a function of melt pool overlap and scan rotation angles. The results indicate that texture selection is primarily governed by the melt pool aspect ratio (depth-to-width), with clear correlations observed between the scan strategy and the resulting grain orientation distributions [151]. The influence of scanning patterns on elastic anisotropy in 316L stainless steel has also been investigated, showing that bidirectional scanning leads to more isotropic mechanical properties compared to unidirectional patterns, due to more balanced grain orientation distributions [152].

Systematic parametric studies employing coupled thermal–CA models have established quantitative relationships between process parameters and microstructural features [90,91]. Process–structure linkage maps have been constructed for Inconel 718, showing that laser power (150–400 W) and scan speed (400–1200 mm/s) significantly influence thermal gradient (G) and solidification velocity (R). The G/R ratio plays a critical role in determining the columnar-to-equiaxed transition (CET), with high values (G/R > 10⁶ K·s/m²) that promote the formation of columnar grains [91]. Multi-scale finite volume method (FVM) and cellular automata (CA) models have been applied to investigate the effects of laser power, scan speed, and hatch spacing on melt pool geometry and grain structure, yielding predictions that closely match experimental observations [90].

Melt pool aspect ratio directly influences the solidification mode: shallow, wide pools (aspect ratio < 0.5) promote equiaxed grain nucleation, while deep, narrow pools (aspect ratio > 1.0) favor continuous columnar growth [77,92,93]. Integration of melt pool fluid dynamics with cellular automata modeling has shown that recoil pressure and thermocapillary convection significantly affect grain size and texture. These effects arise through modifications to temperature gradients and the morphology of the solidification front [93].

Table 9 outlines how key process parameters influence microstructural evolution in AM, linking specific thermal mechanisms to experimentally validated effects on grain morphology and texture.

Table 9. Process parameter effects on microstructure.

Process Parameter	Microstructural Effect	Key Mechanism	Validation	Representative Studies
Build direction	Reduces texture intensity; promotes equiaxed grains (columnar-to-equiaxed transition)	Modulates thermal gradient direction; affects competitive grain growth	EBSD texture analysis; inverse pole figure intensity	[72,151,152]
Laser power (↑)	Increases melt pool depth; grain size increases ~20% when heat input increases from 250 to 500 J/m	Higher thermal gradient G; modulates G/R ratio	Melt pool geometry (FE-CA); EBSD grain size validation	[90,91,109]
Scan speed (↑)	Decreases melt pool size; grain refinement (30–50% reduction via parameter control)	Increases cooling rate (104–106 K/s range); higher G/R ratio	Grain size distribution (EBSD); CA-FE predictions	[90,91,108]
Melt pool aspect ratio (↑)	Favors columnar over equiaxed grains; higher aspect ratio → stronger <001> texture	Directional solidification; G/R ratio control	EBSD texture; PF simulation validation	[92,93]

6.2. Alloy-Specific Case Studies

6.2.1. Nickel-Based Superalloys (Inconel 625, 718)

Inconel 718 exhibits strong <001> fiber textures under typical powder bed fusion conditions, accurately predicted by the CA–FE frameworks [90,91]. Heterogeneous nucleation at melt pool boundaries plays a key role in controlling epitaxial grain growth and texture inheritance during directed energy deposition (DED). The degree of nucleation undercooling (typically 5–15 K) determines the balance between newly formed and inherited grains, influencing final texture development [99]. Spatial heterogeneity in nucleation density across different regions of Inconel 625 builds has been incorporated into CA models, successfully reproducing texture variations observed in experiments and validated through EBSD analysis [100]. The impact of wall thickness on thermal conditions and grain refinement has also been investigated in GH4169 thin-walled structures. Thinner walls (<2 mm) lead to higher cooling rates and finer grain sizes (30–80 μm) compared to thicker sections (80–150 μm), with CA–finite element model predictions closely matching experimental grain size and morphology distributions [100].

6.2.2. Tool Steels (H13)

Hybrid CA–PF frameworks predict dendrite morphology and hardness distributions in multi-track, multi-layer H13 tool steel builds [12]. Inter-track thermal effects have been shown to influence the dendrite growth direction as well as subsequent austenitization and martensite formation. Simulations predict dendrite sizes ranging from 10–30 μm, growth directions within 15° of the thermal gradient, and hardness distributions between 450 and 550 HV, all in good agreement with experimental measurements [12]. The framework demonstrated that heat from neighboring tracks affects dendrite coarsening and phase transformation kinetics, producing microstructural gradients across tracks that correlate with hardness variations [12]. Multi-layer simulations revealed texture evolution and hardness stabilization after three–five layers due to thermal cycling effects.

6.2.3. Titanium Alloys (Ti-6Al-4V)

Comprehensive simulations for Ti-6Al-4V capture β grain solidification, α phase precipitation during cooling, and resulting mechanical properties [6,85,102]. Predictions of β grain morphology in Ti-6Al-4V additive manufacturing have yielded columnar grains with widths ranging from 100 to 300 μm, alongside α-lath structures that exhibit basket-weave and colony morphologies with lath thicknesses between 1 and 5 μm. These microstructural features, along with the material’s microelastic response, have been validated against experimental observations of both the microstructure and the mechanical properties [6]. Multi-physics models combining cellular automata and kinetic phase transformation approaches have successfully simulated 3D microstructures and α/α′ phase fractions (α: 60–80%, α′: 20–40%) in agreement with measured microhardness values (320–380 HV) and phase distribution data [102]. These models reveal a strong sensitivity of α morphology to cooling rate: slow cooling (<100 K/s) leads to coarse α lamellae, whereas rapid cooling (>10³ K/s) promotes the formation of fine α′ martensite [16]. Scan-resolved thermal and microstructure models have been applied to NIST benchmark specimens, enabling efficient and accurate prediction of grain structure across full-scale Ti-6Al-4V components [85]. Optimization of scan strategies has shown that rotating scan patterns reduce β grain texture intensity and promote more homogeneous α-phase distributions, contributing to improved mechanical isotropy in the final parts [85].

6.2.4. Aluminum Alloys (Al-Cu, Al-Si)

Al–Cu alloys exhibit solute trapping effects at high solidification velocities (>0.1 m/s), with velocity-dependent partition coefficients deviating from equilibrium values and affecting dendrite arm spacing (1–10 μm) and solute segregation patterns [105,153]. The bimodal grain structures in Al–Mg–Sc–Zr alloys have been modeled to capture the formation of fine equiaxed grains (10–50 μm) along melt pool boundaries and coarser columnar grains (50–150 μm) in the central region of the pool. These predictions have been validated through fatigue performance assessments and crystal plasticity finite element modeling (CPFEM) [153]. In aluminum alloys, particle-based nucleation models implemented within CA frameworks have successfully reproduced fusion boundary nucleation and epitaxial grain growth. Experimental validation confirms that oxide particles and substrate grain boundaries act as key heterogeneous nucleation sites that govern the final grain size and morphology [105]. During laser rescanning of Al–Si alloys, simulations using CA–FE models have shown a 30–50% reduction in grain size, consistent with experimental data, attributed to enhanced nucleation and remelting effects introduced by the rescanning process [108].

A summary of microstructure predictions and validation results in different alloy systems is provided in Table 10, highlighting the predictive accuracy of CA-based and hybrid models in capturing grain features, phase distributions, and mechanical properties.

Table 10. Alloy-specific microstructure predictions and validation.

Alloy System	Key Features Predicted	Modeling Framework	Validation Metrics	Agreement	References
Inconel 718/625	Grain size (50–200 μm), texture, epitaxial growth; dendrite arm spacing (8–13 μm)	CA-FE, CA-FVM, Eulerian multiphase CFD-CA	EBSD grain size/texture; SEM dendrite spacing; nucleation density calibration	Quantitative match; ~20% grain size sensitivity	[90,99,100,101]
GH4169 (Ni superalloy)	Wall thickness effects on grain size (30–150 μm); thin walls → finer grains	FE-CA with thermal history coupling	EBSD grain size distribution; statistical analysis of grain morphology	Statistical agreement; captures geometry-dependent cooling	[154]
H13 tool steel	Dendrite size (10–30 μm), hardness (450–550 HV); austenite → martensite transformation	Hybrid CA-PF (solidification + solid-state transformation)	Microhardness (Vickers); optical microscopy; multi-layer build validation	Good agreement; captures multi-layer thermal cycling	[12]
Ti-6Al-4V	β grain (100–300 μm), α lath (1–5 μm), phases (α: 60–80%, β: 20–40%)	CFD-CA-PF, multi-physics CA-PF; β → α transformation modeling	Microhardness (320–380 HV); EBSD grain structure; XRD phase fractions	High experimental match; prior-β width ±30–40 μm	[6,85,102,155]
Al-Mg-Sc-Zr	Bimodal grains (10–150 μm); grain size distribution affects the fatigue life	CA-CPFEM (Cellular Automata + Crystal Plasticity FEM)	Fatigue testing (S-N curves); EBSD grain size; mechanical property correlation	Validated mechanical response; grain-property linkage	[153]
Al-Si alloys	Grain refinement (30–50% via rescanning); AlSi10Mg typical case	CA-FE with particle-based nucleation; rescanning thermal effects	EBSD grain size; optical microscopy morphology; porosity analysis	Reproduced trends; rescanning-induced recrystallization	[105,108]

6.3. Linking Microstructure to Properties

Integration of microstructure simulations with crystal plasticity finite element methods enables direct prediction of mechanical behavior from process conditions [7,153]. Cellular automata-generated bimodal grain structures in Al–Mg–Sc–Zr alloys have been used as input for crystal plasticity finite element modeling (CPFEM), enabling accurate prediction of fatigue performance. These models highlight that fine equiaxed grains improve resistance to crack initiation, contributing positively to fatigue life [153]. In 316L stainless steel, a coupled CA–finite element framework has been applied to predict residual stresses and mechanical response, with validation provided by EBSD measurements. Simulations reveal that texture variations induced by different scan strategies result in spatially non-uniform residual stress distributions ranging from 50 to 200 MPa, which in turn influence part distortion and dimensional accuracy [7].

The integration workflow proceeds as follows: (1) CA/PF simulations predict grain size, morphology, crystallographic orientation, and phase distributions from process parameters; (2) synthetic microstructures or representative volume elements (RVEs) are generated based on simulation outputs; (3) CPFEM models incorporating crystal plasticity constitutive laws predict mechanical response (stress–strain behavior, yield strength, and anisotropy) under applied loads; (4) validation against experimental tensile, fatigue, or hardness testing establishes model credibility [7,153]. This integrated approach enables quantitative process–structure–property linkages essential for process optimization and component qualification.

Machine learning frameworks trained on CA/PF-generated microstructures enable rapid property prediction and process optimization [156,157,158]. An integrated workflow combining cellular automata (CA), crystal plasticity finite element modeling (CPFEM), and machine learning (ML) has been applied to CoCrFeMnNi high-entropy alloys for rapid prediction of tensile properties. In this approach, CA simulations generate diverse microstructures across the process parameter space; CPFEM simulations provide mechanical property data for training; and ML models—such as support vector machines and neural networks—learn microstructure–property relationships. Once trained, these ML models predict yield strength within a 5% error margin, achieving speedups of several orders of magnitude compared to full CPFEM simulations, thereby enabling efficient process optimization [158].

In IN718, melt pool depth and dendritic arm spacing have been predicted using ML models trained on thermal data generated from physics-based simulations. Validation against microscopy measurements confirms the model’s reliability for guiding process parameter optimization [157]. For Ti-6Al-4V, regression-based ML models have been employed to forecast melt pool geometry and inform process design, with experimental and CFD validation identifying laser power and scan speed as the primary variables influencing microstructural outcomes [156]. A variational autoencoder approach has also been used to construct multiscale latent space representations of spatially correlated microstructures in AlSi10Mg, enabling efficient encoding of microstructural heterogeneities and their propagation into mechanical property predictions under uncertainty [159].

These ML-enhanced workflows support the development of digital twins for real-time process control and optimization [3]. Digital twin frameworks integrate physics-based CA/PF models, ML surrogates, in situ sensor data (thermal imaging and acoustic emission), and feedback control algorithms to enable adaptive manufacturing strategies that adjust process parameters dynamically to achieve target microstructures and properties [3]. The convergence of CA/PF modeling, CPFEM, and ML represents a transformative path toward predictive, first-time-right AM manufacturing.

6.4. Validation Methodologies for CA and PF Models

Rigorous validation of CA and PF models requires systematic comparison with experimental data through hierarchical protocols that include single-track, multi-track, and multi-layer configurations [90,99]. Statistical validation metrics provide quantitative assessment beyond qualitative visual agreement. Distribution-based metrics include Kolmogorov–Smirnov tests for grain-size distributions and chi-squared tests for grain boundary character distributions [90,99,154]. Point-wise metrics such as mean absolute error and relative error enable comparison across alloy systems [90,99,100]. The correlation coefficients quantify the predictive capability of process–microstructure relationships [90,99].

Uncertainty quantification addresses experimental measurement uncertainties (EBSD resolution, thermal data noise, and process variability), model parameter uncertainties (nucleation density, interface mobility, and CALPHAD data limitations), and stochastic variability in probabilistic CA models [30,90,99]. Sensitivity analysis—including variance-based Sobol indices and Morris screening—identifies critical parameters that require careful calibration [30]. Bayesian frameworks enable rigorous parameter identification with quantified uncertainties [30]. Ensemble simulations for stochastic CA models quantify inherent variability, with predictions reported as distributions rather than single values to communicate reliability transparently [6,30,85,102].

Standardization efforts include NIST AM-Bench benchmark datasets providing controlled experiments for blind validation tests across thermal, mechanical, and microstructure predictions [85,110]. Open-source frameworks such as MicroSim facilitate reproducible research through standardized formats and validated implementations [86]. Comprehensive reporting standards require complete specification of process parameters, material properties, experimental protocols, statistical metrics with confidence intervals, model parameters and calibration procedures, numerical settings, and computational cost [90,99]. Validation across systematic process parameter variations demonstrates model robustness and generalization capability beyond single-point calibration [90,91,108].

Emerging validation approaches leverage in situ characterization including high-speed synchrotron X-ray imaging of melt pool dynamics and solidification front morphology, providing direct validation of transient phenomena [6], and infrared thermal imaging enabling validation of thermal boundary conditions driving microstructure predictions [90,91,92,93]. Machine learning-enhanced validation identifies systematic model biases across parameter spaces, guiding targeted improvements [156,157,158]. Digital twin frameworks that integrate CA/PF models with real-time sensor data represent stringent validation requiring sufficient predictive accuracy for closed-loop process control [3].

Persistent challenges include limited in situ validation data at the microsecond–micrometer scales characteristic of rapid solidification, computational expense restricting validation to small domains, and underdeveloped protocols for coupled experimental model uncertainty quantification [90,99]. Multi-physics coupling validation requires simultaneous measurement of temperature, microstructure, and stress fields, which remains experimentally challenging [7]. The transition from research tools to industrial decision-making aids requires rigorous validation with quantified uncertainties essential for regulatory acceptance and component certification in safety-critical applications.

6.5. Comparative Case Study: CA, PF, and Hybrid Approaches for IN718 LPBF

As established in Section 2.3, the cellular automata (CA) and phase-field (PF) methods present a fundamental efficiency–fidelity trade-off. This case study examines how these approaches have been applied to Inconel 718 laser powder bed fusion (LPBF)—a benchmark system for additive manufacturing microstructure modeling [90,99,101].

For IN718 LPBF, CA–FE implementations have successfully simulated part-scale microstructures with validation against EBSD texture measurements [83]. Phase-field implementations coupled with CALPHAD thermodynamic databases [84,85,95] have demonstrated accurate predictions of microsegregation and solute trapping at extreme cooling rates exceeding the 10⁶ K/s characteristic of LPBF processing [94,96]. Hybrid CA–PF frameworks have been deployed for multi-layer IN718 builds, strategically allocating phase-field resolution to melt pool solidification regions while using cellular automata for grain propagation through bulk material and across remelted layers, with validation against both EBSD texture data and XRD phase measurements [12,15].

Table 11 summarizes the quantitative performance metrics and validation results for the CA, PF, and hybrid approaches applied specifically to IN718 LPBF. Ongoing developments include GPU-accelerated implementations [85,119] and machine learning surrogates trained on phase-field simulation data [160] to extend the accessible domain sizes and reduce computational requirements. These data-driven approaches are discussed further in Section 7.3 as part of future directions for microstructure modeling.

Table 11. Quantitative comparison of CA, PF, and hybrid models for IN718 LPBF.

Criterion	Cellular Automata (CA)	Phase-Field (PF)	Hybrid CA-PF	Key References
Typical Domain Size	Part-scale (>1000 mm3 demonstrated)	Sub mm to mm scale	Intermediate scale	[83,84,85]
Grain Count	104–105 grains	102–103 grains	103–104 grains	[83,84]
Computational Speedup	Significant acceleration vs. traditional models	Limited speedup	Intermediate (order of magnitude vs. full PF)	[15,85,119]
Spatial Resolution	Grain-scale	Dendrite-scale	Adaptive	[95,109]
Sub-grain Features	Not resolved	Explicitly captured	Captured in PF regions	[95,101]
Microsegregation	Not captured	Quantitative (CALPHAD-coupled)	PF regions only	[95,101]
GPU Acceleration	Demonstrated (158× speedup)	Essential for multi-layer	Selective for PF regions	[84,85,119]
Best Application	Part-scale texture, industrial builds	Dendrite morphology, rapid solidification	Multi-layer with transformations	[12,83,95]

Note: Values synthesized from studies across multiple alloy systems (IN718, Ti-6Al-4V, and Ni-based alloys) and AM processes (LPBF and DED). Direct quantitative comparisons of identical computational hardware and domain configurations are limited in the literature.

7. Challenges and Future Directions

Despite the modeling advances described in Section 3, Section 4, Section 5 and Section 6, significant challenges remain in achieving quantitative predictions for industrial-scale additive manufacturing. This section identifies unresolved computational, physical, and methodological limitations while highlighting emerging solutions and future opportunities that promise to bridge microstructure simulation with engineering performance.

7.1. Critical Assessment of Predictive Capability

Despite the substantial progress documented in Section 3, Section 4, Section 5 and Section 6, critical gaps remain between the demonstrated and required predictive capability for industrial AM qualification. Current CA and PF models exhibit three fundamental limitations: (1) computational scalability—even with GPU acceleration, part-scale simulations with scan-resolved fidelity remain computationally prohibitive for routine industrial use; (2) model form uncertainty—different formulations (e.g., KKS vs. PCR vs. FID for PF; various nucleation models for CA) yield divergent predictions under extreme AM conditions, yet systematic inter-model comparisons with quantified uncertainties are rare; and (3) validation data scarcity—in situ characterization capturing transient solidification at microsecond–micrometer scales remains experimentally challenging, limiting rigorous model validation. Experimental uncertainties (EBSD resolution and process variability) are often comparable to model–experiment discrepancies, complicating the assessment of true predictive accuracy. Processing regimes exhibiting particularly poor model reliability include ultra-high cooling rates (>10⁶ K/s), complex scan strategies with extensive remelting, and coupled solid-state transformations post-solidification. Achieving certification-grade predictive capability requires addressing these limitations through systematic uncertainty quantification (Section 7.2), accelerated computation (Section 7.3), and standardized validation protocols (Section 7.4).

7.2. Uncertainty Quantification and Parameter Calibration

Systematic uncertainty quantification (UQ) remains underutilized despite being essential to establish the credibility of the model in safety-critical applications. CA and PF models contain numerous parameters (nucleation density, interface mobility, and anisotropy coefficients) lacking direct experimental measurement, traditionally calibrated through trial-and-error approaches that provide no quantitative uncertainty assessment. Bayesian calibration frameworks offer systematic alternatives by treating parameters as random variables, with prior distributions updated using experimental data (EBSD and thermal imaging), yielding posterior distributions that quantify parameter uncertainty and enable prediction with confidence intervals [161]. Global sensitivity analysis (e.g., Sobol indices) identifies the parameters most strongly influencing predictions, prioritizing calibration efforts and reducing experimental burden [30,107]. Surrogate models (Gaussian processes and neural networks) enable efficient uncertainty propagation at 100–1000× speedup compared to Monte Carlo sampling, making probabilistic predictions computationally tractable [162,163,164]. Future standardization requires open benchmark datasets with quantified experimental uncertainties, enabling systematic inter-model comparison and establishing confidence levels necessary for regulatory acceptance.

7.3. Accelerated Computation Through Machine Learning

Building on the ML advances detailed in Section 4.4, deployment of physics-informed neural networks and graph neural network surrogates for production environments faces three critical barriers: (1) training data requirements—surrogates require extensive high-fidelity simulation datasets spanning diverse process conditions and alloy systems, with optimal dataset sizes and sampling strategies remaining unclear; (2) generalizability—models trained on specific alloys or processes exhibit degraded accuracy when applied to new systems without retraining, limiting industrial applicability [165]; and (3) validation protocols—purely data-driven models lack mechanistic interpretability, requiring rigorous experimental validation with quantified confidence intervals before certification agencies will accept ML-accelerated predictions for component qualification [68,166,167]. Addressing these barriers requires hybrid multi-fidelity frameworks where physics-based CA/PF models provide training targets and physical constraints for data-efficient learning, combined with standardized benchmark datasets enabling systematic surrogate model comparison across research groups [160,168].

7.4. Pathways to Industrial Implementation

7.4.1. Fully Coupled Multi-Physics Frameworks

Industrial deployment requires fully coupled thermal–fluid–mechanical–microstructural frameworks capturing complex interactions governing AM defect formation and property evolution. Current sequential coupling strategies (Section 5.1 and Section 5.2) sacrifice accuracy in transient phenomena for computational tractability; future concurrent coupling with bidirectional feedback between melt pool dynamics, solidification microstructure, residual stress, and hot cracking susceptibility will enable predictive defect mitigation [3,74,125]. Integration of thermomechanical phase-field fracture models with CA/PF solidification frameworks represents a critical frontier for the concurrent prediction of microstructure and cracking [169,170], which requires algorithmic innovations in adaptive meshing and GPU-accelerated implicit solvers to achieve computational feasibility.

7.4.2. Digital Twin Architectures for Real-Time Process Control

Digital twin architectures promise real-time monitoring and adaptive control for AM, but substantial technical bottlenecks prevent industrial deployment [110,171,172]. Computational latency remains the primary barrier: layer-by-layer adaptive control requires microstructure predictions within 10–30 s inter-layer recoating times, while intra-layer control demands sub-second response. Hybrid mechanistic–ML surrogates show promise. Whitney et al. demonstrated a mean absolute error ≤4% for Ti-6Al-4V, with substantially reduced computational cost, though generalization across alloy systems without retraining remains challenging [160].

Data infrastructure challenges compound computational barriers. High-speed thermal imaging at 10,000–20,000 fps produces 5–10 GB/s data streams, while multi-modal sensor fusion generates 50–100 GB per hour [110,171]. Vandecasteele et al. demonstrated multimodal monitoring at 20 kHz, achieving a 0.95 Pearson correlation for porosity prediction, yet with 17% normalized MAE for deformation detection [173]. Real-time processing requires edge computing with >100 Gb/s network throughput and multi-GPU systems (>10 TFLOPS) for feature extraction during builds [110,171]. Current systems typically store raw sensor data for post-build analysis rather than real-time inference, limiting closed-loop control.

Regulatory acceptance for safety-critical applications requires rigorous validation with quantified confidence intervals, yet standardized frameworks remain absent. Chen et al. achieved 96% accuracy and 99% ROC-AUC for defect detection, but with a 4.4% false alarm rate potentially unacceptable for high-value aerospace components [174]. Uncertainty quantification throughout the digital twin workflow is essential; Bayesian frameworks provide tools for confidence quantification, but their integration into real-time systems remains underdeveloped. Industrial deployment will likely follow staged implementation: initial post-build quality prediction [175], followed by layer-by-layer adaptation [171,172], and ultimately real-time intra-layer control as computational and validation technologies mature.

7.4.3. Expansion to Novel Alloy Systems

Beyond the alloys examined in Section 6, high-entropy alloys (HEAs) present unique modeling challenges due to sluggish diffusion kinetics, complex phase selection, and nonequilibrium solidification pathways [158,176,177]. High-throughput computational frameworks that combine CA/PF with CALPHAD databases and deep learning surrogates enable rapid printability assessment and alloy design, achieving 1000× acceleration in defect likelihood prediction [176]. Integration with crystal plasticity finite element methods (CPFEM) facilitates microstructure-property linkage for mechanical performance prediction in CoCrFeMnNi and related HEA systems [158]. Refractory metals (tungsten, molybdenum, and tantalum) and functionally graded materials require tailored CA/PF formulations that account for extreme melting points, thermal gradients, and compositional variations [116,178]. Open-source phase-field crystal frameworks (e.g., OpenPFC) enable large-scale simulations of rapid solidification in tungsten at atomic resolution [178]. Wire-based AM of functionally graded Ti-6Al-4V demonstrates an extended CA-finite volume coupling for grain coarsening and epitaxial growth across compositional gradients [116,179].

7.4.4. Standardization and Benchmarking

Process optimization leveraging CA/PF-generated printability maps and microstructure distribution predictions will transition from research to industrial practice [85,132]. Validation against NIST AM benchmark specimens and integration with open-source high-throughput simulation platforms (e.g., CAROUSEL) support standardized certification pathways [85,110]. Digital twin implementations in wire arc AM demonstrate the integration of CFD, CA, and machine learning for the prediction of oxide formation and optimization of mechanical properties, supporting the qualification of components with increased service life [179]. Standardized benchmarking through shared datasets, open-source frameworks, and reproducible workflows is essential for industrial adoption [47,63]. Collaborative initiatives developing benchmark cases for CA/PF model comparison across AM processes, alloys, and scales will accelerate technology transfer and build confidence in computational predictions [5,110].

7.4.5. Interdisciplinary Integration

Interdisciplinary integration extends CA/PF capabilities beyond microstructure prediction to encompass defect formation (porosity and hot cracking), corrosion, oxidation, and lifetime estimation [179,180,181]. Thermomechanical phase-field fracture models predict hot-cracking susceptibility under thermal cycling [180], while reaction–diffusion models coupled with microstructure evolution capture oxidation kinetics in wire arc AM [179]. Integration with FFT-based crystal plasticity enables linking grain structure with fatigue performance and mechanical failure under service conditions [78]. Multi-scale frameworks that bridge atomistic phase-field crystal simulations, mesoscale CA/PF grain growth, and macroscale CPFEM mechanical response promise comprehensive process–structure–property–performance links [78,148,178]. Hierarchical solidification modeling addressing nucleation kinetics, dendritic growth, and grain coarsening across length scales (nm to cm) represents a critical frontier for predictive AM design [148].

Table 12 highlights the key challenges and future directions in CA and PF modeling for metal additive manufacturing, spanning computational scalability, validation, calibration, and integration with mechanical behavior, digital twins, and industrial applications.

Table 12. Challenges and future directions in CA/PF modeling for metal AM.

Area	Current Challenge	Emerging Solution/Future Direction	Expected Impact	References
Computational scalability	PF limited to <1 mm3; CA grid sensitivity	GPU acceleration; ML surrogates (50–100× speedup); exascale HPC	Part-scale simulations; real-time prediction	[3,78,84,85,167,182]
Model calibration	Lack of standardized protocols; parameter sensitivity	Uncertainty quantification; inverse calibration; hybrid frameworks	Improved reliability and accuracy	[77,108,183,184]
Experimental validation	Limited in-situ data; insufficient solid-state validation	Synchrotron X-ray; high-speed thermal imaging; data assimilation	Capture transient phenomena; refine models	[3,158,185,186]
Multicomponent alloys	Nonequilibrium phase diagrams lacking at V > 0.1 m/s	CALPHAD extension; high-throughput HEA frameworks	Expand to HEAs, refractory metals	[123,127,176,177,187]
Mechanical coupling	Hot cracking, transformation strains not captured	PF-thermomechanical frameworks; defect prediction	Predict defects and residual stresses	[8,170,180]
Digital twins	Limited closed-loop integration	Neural ODEs; physics-ML hybrids; in situ monitoring	First-time-right manufacturing; adaptive control	[171,172,188]
Industrial adoption	Lack of benchmarks and open-source tools	CAROUSEL, OpenPFC; NIST benchmarks; shared datasets	Standardized certification; technology transfer	[5,85,110]
Cross-scale integration	Atomistic-to-component coupling incomplete	PF crystal; hierarchical CA/PF; FFT-CPFEM	Comprehensive process–structure–property linkage	[78,148,178]

Realizing these future directions requires sustained collaboration among computational materials scientists, AM practitioners, and industrial partners. The convergence of physics-based modeling, data-driven acceleration, and experimental validation promises to establish CA/PF frameworks as indispensable tools for predictive microstructure control and performance optimization in next-generation metal additive manufacturing.

8. Conclusions

This review has synthesized recent advances in cellular automata (CA), phase-field (PF), and hybrid modeling approaches to predict microstructure evolution in metal additive manufacturing, revealing significant progress in computational capabilities, material applicability, and integration with experimental and industrial workflows.

The CA and PF methods offer complementary capabilities: CA provides computational efficiency that enables large-domain simulations (mm–cm³) with thousands of grains, while PF delivers superior fidelity in capturing interface dynamics and transient solidification phenomena. Comparative studies demonstrate that CA reproduces averaged grain distributions at 10–100× lower cost than PF, motivating hybrid frameworks that strategically allocate PF to solidification fronts and CA to bulk grain competition. Algorithmic advances include discrete event-inspired CA (DECA), CALPHAD-coupled multicomponent PF models, and GPU-accelerated implementations enabling multi-layer simulations. Applications to Ni-based superalloys, Ti-6Al-4V, H13 tool steels, and Al alloys demonstrate robust process–microstructure–property predictions validated through EBSD and mechanical testing.

Persistent challenges include computational scalability (3D multicomponent PF simulations for full-scale parts remain intractable), lack of standardized calibration protocols and uncertainty quantification frameworks, experimental validation gaps for in situ characterization and transient phenomena, and incomplete physics for multicomponent alloys and mechanical coupling. Machine learning integration offers promising solutions, with physics-embedded networks achieving orders-of-magnitude speedups, although training data requirements, generalizability, and interpretability remain unresolved.

Future opportunities include fully coupled thermal–fluid–mechanical–microstructural frameworks leveraging exascale computing, applications to high-entropy alloys and refractory metals via high-throughput screening, digital twin architectures for real-time process control, and interdisciplinary extensions to defect prediction, corrosion, and lifetime estimation. Open-source platforms, standardized benchmarks, and collaborative data-sharing will accelerate industrial adoption. The convergence of physics-based modeling, ML acceleration, and experimental validation promises predictive microstructure control that enables first-time-right manufacturing and tailored design for next-generation aerospace, biomedical, and energy applications.