The microstructure evolution during the casting of metal products significantly impacts the quality of solidified material. Therefore, predicting microstructure evolution is crucial for designing and producing high-quality castings for scientific, medical, and industrial use. This postdoctoral project aims to enhance the physical modelling of the casting, especially the direct-chill casting of aluminium alloys and the continuous casting of steel. The project’s primary focus is the upgrade of the already implemented physical models for predicting microstructure evolution during the casting. Another essential aim of the project is upgrading the adaptive meshless solution procedure for accurate and efficient modelling of microstructure evolution.

The project represents a logical continuation of the completed applied research projects: L2-6775 Simulation of industrial solidification processes under the influence of electromagnetic fields, L2-9246 Multiphysics and multi-scale numerical modelling for competitive continuous casting, MARTINA Materials and technologies for new applications, MARTIN Modelling thermo-mechanical processing of aluminium alloys for top-most products. The project leader participated in the mentioned completed projects as a PhD student. He co-developed the microscopic and mesoscopic physical models for predicting grain structure’s evolution during aluminium alloys and steel casting.

The project’s first goal is to upgrade the adaptive algorithm for the solution of the phase-field (PF) models. The main philosophy behind the adaptive algorithm is to dynamically ensure the highest space-time resolution at the evolving solid-liquid interface and the lowest resolution in the bulk of phases. That way, we reduce the problem’s computational complexity while sustaining accuracy. The meshless 2-D quadtree-based space-time adaptive solution procedure is currently used for PF modelling. The algorithm will be upgraded to 3-D in the framework of the project using the octree-based adaptivity.

The project’s second goal is to implement a polycrystalline phase-field model for the solidification of metallic alloys. The 2-D cellular automaton model currently simulates polycrystalline solidification. The 2-D PF model will be implemented first, followed by the implementation of the 3-D model.

The project’s final goal is to incorporate the 3-D polycrystalline PF model into the multi-physics multi-scale simulation system for a detailed prediction of solidification phenomena during the casting of metal products. The system considers mass, momentum, heat, and species conservation equations on the macroscopic scale. The 2-D cellular automaton and PF models apply to simulate microstructure evolution at mesoscopic and microscopic scales. The macroscopic model provides temperature and chemical history to mesoscopic and microscopic models. In the framework of the postdoctoral project, the 3-D PF model will replace the current 2-D mesoscopic and microscopic models. The developed octree-based space-time adaptive algorithm will allow the computationally demanding 3-D PF simulations on the mesoscopic scale to be performed in acceptable computation times.

The main originality of the postdoctoral project is in coupling 3-D octree-based and 2-D quadtree-based space-time adaptive algorithms with the local meshless methods for an accurate and computationally efficient PF modelling of polycrystalline dendritic solidification for the first time. This unique numerical approach will represent a state-of-the-art simulation tool for predicting microstructure evolution during the casting of metal products. The project results will be published in high-ranked scientific journals and presented at international conferences in the field of numerical modelling and solidification.

The project is divided into four work packages (WP).

WP 1: Development of a 3D spatio-temporally adaptive solution procedure – overall completion of project activities: 100%

WP 1.1: Implementation of an octree

Implementation: An octree data structure was implemented, based on the recursive subdivision and aggregation of cubes or cuboids, along with the corresponding recursive algorithms for searching neighboring subdomains.

WP 1.2: Implementation of algorithms for node generation, PDE discretization, and communication between neighboring subdomains

Implementation: The algorithms for node generation and PDE discretization were initially implemented for a single 3D subdomain. Subsequently, an algorithm for communication between 3D subdomains was implemented. During the implementation, it was found that discretization using meshless methods is highly sensitive to the treatment of boundary conditions. Within the project, we determined that the accuracy and stability of the employed numerical scheme are significantly improved by the use of so-called additional nodes (ghost nodes).

Figure 1: Comparison between different implementations of boundary conditions in meshless methods (source).

WP 1.3: Verification of the spatio-temporally adaptive model

Implementation: The model was successfully verified and indirectly also validated. The verification is based on a comparison with reference results for the growth of a dendritic crystal in a single-component melt.

Figure 2: Example of a simulation result using the 3D spatio-temporally adaptive solution procedure (source).

WP 2: Implementation of a phase-field model for polycrystalline solidification – overall completion of project activities: 100%

WP 2.1: Implementation in 2D

Implementation: A model of polycrystalline solidification in 2D was implemented. It was tested on the case of polycrystalline solidification of a supersaturated binary alloy.

Figure 3: Concentration field during the polycrystalline solidification of a dilute binary alloy (source).

WP 2.2: Implementation in 3D

Implementation: The model from Section 2.1 for 2D was also applied in 3D. The only difference is the definition of the preferential growth direction, which in 3D is determined by three Euler angles, compared to a single angle in 2D.

WP 3: Integration of the polycrystalline phase-field model into a simulation system for direct-chill casting of aluminium alloys – overall completion of project activities: 100%

WP 3.1: Comparison between cellular automata and phase-field models

Implementation: A benchmark case was defined, addressing the growth of a dendritic grain from an undercooled melt. The test case was computed using the cellular automata method in 2D and the phase-field method in both 2D and 3D.

WP 3.2: Replacement of old models with newly developed ones

Implementation: An environment was prepared for one-way coupling of the macroscopic model with the phase-field model. The macroscopic model solves the conservation equations for momentum, mass, and energy. The output data of the macroscopic model are used as input data for the microscopic model. In addition, the input data of the microscopic model include alloy material properties and numerical parameters. The output data of the microscopic model are microsegregation profiles over representative volumes at different locations of the casting.

WP 4: Dissemination of results – overall completion of project activities: 100%

Implementation:

The article Improved finite difference method for phase-field modelling of dendritic solidification was published in the journal Journal of Computational Physics on the topic of simulating dendritic growth using a space-time adaptive solution procedure.

The article A coupled domain–boundary type meshless method for phase-field modelling of dendritic solidification with fluid flow was published in the journal International Journal of Numerical Methods for Heat & Fluid Flow on the topic of coupling two meshless methods.

The article A study on different implementations of Neumann boundary conditions in the meshless RBF-FD method for the phase-field modelling of dendrite growth was published in the journal Engineering Analysis with Boundary Elements on the topic of analysing different implementations of boundary conditions.

The article A meshless multiscale and multiphysics slice model for continuous casting of steel was published in the journal Metals on the topic of applying the developed numerical methods in industry.

The article A hybrid radial basis function–finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 3: Application to thermo-mechanical modelling of continuous casting of steel billets was published in the journal Engineering Analysis with Boundary Elements on the topic of applying the developed numerical methods in industry.

At the conference MCWASP 2023: 16th International Conference on Modelling of Casting, Welding and Advanced Solidification Processes (18/06/2023 – 23/06/2023, Banff, Canada), the contribution Application of a meshless space-time adaptive approach to phase-field modelling of polycrystalline solidification was presented and published in IOP Conference Series: Materials Science and Engineering on the topic of modelling polycrystalline solidification.

Three articles were published in Journal of Physics: Conference Series as part of the Eurotherm 2024 conference, held in Bled from 10–13 June 2024.

The project leader attended the ICASP 7  – 7th International Conference on Advances in Solidification Processes (June 10–13, 2025, Madrid, Spain), where he presented the project results in a lecture entitled Assessment of the meshless RBF-FD method for 3D phase-field modelling of dendrite growth.

Figure 4: Phase field without fluid flow (top left), phase field with fluid flow (top right), temperature field (bottom left), and magnitude of velocity (bottom right) (source).

[1] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. Improved finite difference method for phase-field modelling of dendritic solidification. Journal of Computational Physics. 2026. [COBISS.SI-ID 267350019]

[2] VUGA, Gašper, MAVRIČ, Boštjan, DOBRAVEC, Tadej, ŠARLER, Božidar. A hybrid radial basis function-finite difference method for modelling two-dimensional thermo-elasto-plasticity, part 3: application to thermo-mechanical modelling of continuous casting of steel billets. Engineering analysis with boundary elements. 2026. [COBISS.SI-ID 263639811]

[3] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. A study on different implementations of Neumann boundary conditions in the meshless RBF-FD method for the phase-field modelling of dendrite growth. Engineering analysis with boundary elements. 2025. [COBISS.SI-ID 227519491]

[4] ŠARLER, Božidar, MAVRIČ, Boštjan, DOBRAVEC, Tadej, VERTNIK, Robert. A meshless multiscale and multiphysics slice model for continuous casting of steel. Metals. 2025. [COBISS.SI-ID 248938755]

[5] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ZAHOOR, Rizwan, ŠARLER, Božidar. A coupled domain–boundary type meshless method for phase-field modelling of dendritic solidification with the fluid flow. International journal of numerical methods for heat & fluid flow. Jun. 2023. [COBISS.SI-ID 154935811]

[6] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. Application of a meshless space-time adaptive approach to phase-field modelling of polycrystalline solidification. IOP conference series, Materials science and engineering. [COBISS.SI-ID 153282819]

[7] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. On different implementations of boundary conditions in the meshless RBF-FD method for phase-field modelling of dendritic solidification. Journal of Physics: Conference Series. Jun. 2024. [COBISS.SI-ID 197919747]

[8] MAVRIČ, Boštjan, DOBRAVEC, Tadej, ŠARLER, Božidar. Lessons from accelerating an RBF-FD phase-field model of dendritic growth on GPUs. Journal of Physics: Conference Series. Jun. 2024. [COBISS.SI-ID 208869891]

[9] VUGA, Gašper, DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. A new hybrid local radial basis function collocation method for 2.5D thermo-mechanical modelling of continuous casting of steel. Journal of Physics: Conference Series. Jun. 2024. [COBISS.SI-ID 199016963]

[10] DOBRAVEC, Tadej, MAVRIČ, Boštjan, ŠARLER, Božidar. Assessment of the meshless RBF-FD method for the 3-D phase-field modelling of dendrite growth. V: ICASP-7 : 7th International Conference on Advances in Solidification Processes : Madrid, Spain, June 10-13, 2025 : abstract booklet. [COBISS.SI-ID 242746627]