The prediction of microstructural evolution during thermomechanical processing is a central challenge in materials science, as the final microstructure governs the mechanical and functional properties of metallic alloys. Numerical simulation has become an essential tool for exploring these transformations, enabling the prediction of grain growth, recrystallization, phase transformations, and the influence of second-phase particles under various processing conditions. Over the past decades, a wide spectrum of modelling strategies has emerged, ranging from phenomenological formulations to mean-field and full-field approaches1–3, each with specific advantages and limitations.
A critical distinction between these families of models lies in their balance between physical fidelity and computational cost. While phenomenological approaches provide fast but very simplified descriptions of microstructural changes, others such as full-field models, resolve microstructural features with high fidelity at the expense of prohibitive computational demands. Mean-field approaches offer an important intermediate solution by combining physical relevance with computational efficiency. This balance directly impacts the applicability of a given model, whether for industrial process simulations, high-throughput studies, or fundamental investigations. Within this context, comparing such approaches is essential to assess their predictive capabilities and computational efficiency. Such a comparison not only clarifies their respective domains of applicability but also highlights the current challenges in bridging the gap between realistic physical representation and reasonable computation times.
Among the prediction models of microstructural evolution reported in the literature, three representative approaches are currently implemented within the Transvalor software suite. The first model known as «JMAK» (Johnson-Mehl- Avrami-Kolmogorov), integrated in FORGE®, is based on a phenomenological approach4. The overall kinetics of recrystallization can typically be described by the well-known JMAK relationship. In this empirical model, the fraction of recrystallized material, denoted X, evolves according to the following relation:
Where b and n are the Avrami coefficients, whose values depend on the growth and nucleation rates. This model is able to globally describe nucleation, growth and impingement of the growing grains (see Figure 1). This curve shows the typical sigmoidal shape given by the Avrami’s equation.
Figure 1: Representative sigmoidal recrystallization curve obtained under isothermal conditions, consistent with the Avrami equation5.
The JMAK calculation is performed during FORGE®’s computation without any additional time cost. While it is practical, this model is limited by its assumption of large, constant nucleation and growth rates, which fails when the number of nuclei is small or when conditions deviate from the model’s assumptions6,7. These factors can lead to inaccuracies in certain experimental scenarios.
To maintain a physical description of the metallurgical phenomena, other solutions exist: full-field models such as the one employed in DIGIMU® which relies on the Level-Set method to predict microstructural evolution8. In DIGIMU®, the microstructure is explicitly described using a meshed microstructure with real grain shapes and a real neighbourhood. The heterogeneities of the material are accurately captured while preserving their topological features. The boundary conditions applied to the Representative Volume Element (RVE) reflect the thermomechanical loading experienced by a material point at the macroscopic scale. Using a Finite Element framework, the software simulates key physical mechanisms occurring during metal forming processes, including recrystallization, grain growth, and grain boundary pinning by second-phase particles9,10. While DIGIMU® offers more comprehensive results throughout the deformation process, the simulation on a single material point whose thermo-mechanical history has been captured by a sensor can take from a few hours to several hours, depending on the density of the microstructure.
A viable compromise lies in the use of a mean-field model, offering a balance between physical accuracy and computational efficiency. The mean-field solver proposed by Transvalor relies on an original class mean-field approach called NHM, which stands for “NeighbourHood Model”, developed by the MSR research team at CEMEF (Centre de Mise en Forme des Matériaux, Mines ParisTech). Unlike classical mean-field models, which assume a homogeneous equivalent medium, NHM introduces a more refined description by considering a statistical neighbourhood of each grain and allowing grain shapes to evolve from spherical to ellipsoidal. This better captures the physical changes in grain boundary surface area during deformation. A key advantage of NHM is that it relies on the same constitutive equations as a full-field model like in DIGIMU®, enabling direct comparison and validation between the two approaches. Moreover, the NHM can be used to efficiently identify model parameters through inverse analysis, making it both physically relevant and computationally practical11.
In this model, Maire et al.11 adopted a simplified statistical representation of the microstructure for the simulation of recovery and recrystallisation during hot-working processes. They represented the microstructure by defining classes of spherical grains, wherein each grain class is surrounded by a number of grains belonging to other classes which compose its neighbourhood. Each grain class is described by three state variables: the radius Ri of a class i, the frequency ηi of grains in class i, and the dislocation density ρi.
The evolution of the microstructure in this model is represented through volumetric exchanges based on the grain boundary migration equation:
where Sc(i,j) is the contact surface between class i and class j as illustrated in Figure 2, and is equal to the product of the contact probability p(i,j) and the “remaining surface” of grain class SRi, and dR(i,j) is the radius variation resulting from grain boundary migration, which accounts for both capillary pressure and the driving pressure related to stored energy. To learn more about the NHM you can refer to the thesis work of M. Roth12.
Figure 2: 3D representation of the microstructure for the Maire model as illustrated in the work of Roth12.
This efficient model has been coupled to FORGE® since its earlier version NxT 3.0, thanks to a new tool called DynamiX and its user-ready interface. This tool allows users to visualize microstructural evolution on selected cross-sections of a workpiece, at a specific time during the process, by coupling a mean-field solver to FORGE®’s internal sensors. To generate these visualizations, the use of a metallurgical post-processor is required: users may either connect an external mean-field solver or employ the NHM solver, which is accessible through a DIGIMU® license. This design provides a flexible and user-oriented tool providing a micro-macro link between process and microstructural modeling.
Once a FORGE® simulation is finalized, the user can exploit DynamiX to post-process the microstructure of the part on an already defined cutting plane (see Figure 3(a)). The first step is then to define through the DynamiX GUI a sensor grid on the cutting plane at the targeted location on the workpiece (see Figure 3(b)). The initial microstructure can then be defined through a grain size distribution, typically represented in the form of a histogram. The material on the other hand is characterized by a set of parameters that govern its response, including the strain hardening constant (K₁), the dynamic recovery constant (K₂), the dislocation line energy (τ), among others (see Figure 3(c)).
Figure 3: (a) User interface of Dynamix in FORGE®, (b) definition of the cutting plane (via normal vector data), and of the sensor grid density, (c) the initial microstructure is defined by a grain size distribution, and the material is characterized by a set of parameters.
The evolution of the microstructure is thus computed for each point along its thermo-mechanical history. The resulting final microstructural properties are then visualized as contour plots on the selected cutting plane as illustrated in Figure 4, which shows a representative result obtained under conditions within the validity range of the JMAK model. With minimal user inputs, these interpolated results can be visualized within few minutes over the entire cutting plane of the component, providing a representation of the microstructure of the workpiece on the cutting plane. This methodology represents a substantial advancement over conventional JMAK-based models, particularly when applied to complex, multistep processing sequences, owing to the exploitation of more predictive mean-field models in combination with enhanced accurate spatial resolution.
Figure 4: Microstructural results (left: recrystallized fraction, right: average grain diameter) interpolated from the sensors on the cutting plane using DynamiX (top) and the JMAK model integrated in FORGE® (bottom).
The coupling of DynamiX and NHM (or any mean field model) can therefore be regarded as a powerful tool that achieves an effective balance between accuracy and computational cost. Nevertheless, when the objective is to resolve fine-scale heterogeneities and obtain the most precise representation of the evolving microstructure, full-field approaches like DIGIMU® remain indispensable, complementing the intermediate efficiency offered by the mean field.
One of the main distinctions between the two approaches lies in the treatment of idealized surfaces. In NHM, these surfaces are smaller than in DIGIMU®, with the geometric factor set to π/2. Although both solutions rely on comparable models in terms of parameters and values, several differences exist, notably concerning the mobility M0, the nucleation parameter Kg, and the treatment of Smith–Zener pinning. In DIGIMU®, a grain is represented with local curvatures and an explicit neighbourhood, whereas in NHM each class is described by an averaged curvature and is embedded within a global statistical neighbourhood, as illustrated in Figure 5. To compensate for this simplification, the mobility in NHM is corrected by a factor of 1.5. Similarly, the nucleation rate is adjusted by a factor of 2.5, ensuring that NHM generates fewer nuclei than DIGIMU®.
Figure 5: (a) Representation of a grain with local curvatures and an explicit neighbourhood in a DIGIMU® simulation, compared with (b) the averaging of curvatures and the statistical neighbourhood in NHM.
As a comparative illustration, a first series of simulations was performed with NHM and DIGIMU® to investigate grain growth and dynamic recrystallization (DRX) in Inconel 718, considering both cases with and without the introduction of secondary phase particles (SPP). The first series of simulations, addressing grain growth without SPP at different temperatures, was conducted to validate the corrections applied to NHM parameters, particularly the mobility adjustment. The results illustrated in Figure 6 demonstrate satisfactory agreement, with a mobility correction factor of 1.5 appearing to be appropriate, although slight refinement may be achieved by slightly increasing this factor.
Figure 6: Comparison of grain size results during grain growth, obtained with NHM (red curves) and DIGIMU® (green curves) at different temperatures.
A second series focused on DRX without SPP at 980°C, 1050°C, and 1120°C, with strain rates of 0.001 s⁻¹, 0.01 s⁻¹, 0.1 s⁻¹, and 1 s⁻¹, as illustrated in Figure 7. At 980°C, results from NHM and DIGIMU® are generally consistent, except at the lowest strain rate (0.001 s⁻¹). A similar trend is observed at 1050°C, whereas at 1120°C divergences appeared at 0.001 s⁻¹ and 0.01 s⁻¹. Overall, for moderate to high strain rates, the mean-field predictions show good agreement with full-field simulations. However, at low Zener-Hollomon values (corresponding to slow deformation and high temperatures), the results tend to diverge due to the limited number of microstructural classes. To address this limitation, an automatic «dynamic class adaptation» algorithm is currently under development. This algorithm will allow dynamic refinement or coarsening of the classes throughout the simulation and thus improve the model’s accuracy across the entire thermo-mechanical domain.
Figure 7: Dynamic recrystallization (DRX) at 980°C, 1050°C, and 1120°C under strain rates of 0.001–1 s⁻¹, without SPP, comparing grain size results from NHM (red curves) and DIGIMU® (green curves).
A third simulation series was realized to investigate partial DRX with post-dynamic recrystallization (PDRX) at 980°C in the presence of SPP in the microstructure, comparing both NHM and DIGIMU® for identical precipitate populations with a diameter of 0.6 µm and a fraction of 0.5%. During the first 12 seconds of DRX at 0.1 s⁻¹, and in the first seconds of PDRX, no significant effect of SPP is visible. However, when 100% of recrystallized fraction is reached and all the energy of the grains is consumed, the grain growth is clearly slowed down by the SPP to reach a final equivalent mean diameter value below 20 microns. Further simulations were then performed by varying the SPP volume fraction while maintaining the same process parameters. Visible and consistent effects were obtained for a fraction of 0.5%, as illustrated in Figure 8. Although the results were qualitatively satisfactory, adjusting the Smith-Zener pinning parameters for a given SPP fraction can lead to an even better accuracy.
Figure 8: Comparison of results obtained with DIGIMU® (green curves) and NHM (red curves) during DRX–PDRX at 980°C for an SPP fraction of 0.5%, in terms of average equivalent diameter (a) without SPP and (b) with SPP, and recrystallized fraction (c) without SPP and (d) with SPP.
Finally, Figure 9 presents an animation comparing grain size evolution during dynamic recrystallization of Inconel 718 at 1050°C as predicted by NHM and DIGIMU®. The comparison highlights differences in recrystallization kinetics and grain size distribution evolution. In the early stages, DIGIMU® indicates approximately 12% recrystallization after 5 s (0.5 strain), whereas NHM predicts 25%, reflecting faster early-stage recrystallization in the mean-field approach. By 10 s (1 strain), the recrystallized fractions predicted by both approaches converge, although the NHM distribution remains slightly too narrowly centered. At 20 s (2 strain), the final grain size distributions are broadly similar, with NHM still showing a tendency to over-concentrate around a single value compared to DIGIMU® and experimental expectations. This slightly scattered evolution in NHM results from a limited number of grain classes during growth, particularly in the absence of SPP. Overall, despite early-stage differences, the recrystallization behaviour predicted by NHM remains reasonably consistent with the full-field simulation and could be further improved by adjusting model parameters.
Figure 9: Grain size evolution during dynamic recrystallization (DRX) in Inconel 718 at 1050°C as predicted by NHM (blue) and DIGIMU® (purple). Inset maps show grain cartographies generated with DIGIMU®, highlighting detailed microstructural predictions of the full-field simulation.
In this work, different models for grain size prediction in a nickel-based Inconel 718 superalloy were assessed. The phenomenological JMAK model, though practical, is limited by its simplified microstructure description, and its assumption of constant nucleation and growth rates. The mean-field NHM, coupled to DynamiX, provides low computation times and is well suited for large-scale studies, but its reliance on macroscopic descriptors and empirical laws imposes certain limitations. In contrast, the full-field model in DIGIMU® relies on physics-based evolution laws, offering more accurate microstructural descriptions.
Recent joint advances in DynamiX and NHM (GUI, enhanced stability, automatic parameter adaptation …) have extended the modelling capabilities, particularly for Inconel 718 in the forging range (940–1080°C) with Smith–Zener pinning. Comparisons with DIGIMU® demonstrate the feasibility of transferring material data between solvers, highlighting their complementarity. Mean-field and full-field models thus represent two complementary strategies: one optimized for efficiency, the other for physical accuracy. Together, they address industrial demands for faster and more reliable microstructural predictions.
Explore both DynamiX (NHM) and DIGIMU® to identify the best solution for your specific microstructural simulations!
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