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Some functional properties of engineering materials, i.e. physical, mechanical and thermal ones, depend directly on the microstructure, which is a result of processes occurring in the material during the forming and thermomechanical processing. The proper microstructure can be obtained in many cases by the phase transformation. This phenomenon is one of the most important processes during hot forming and heat treatment. The purpose of this paper is to develop a new comprehensive hybrid model for modeling diffusion phase transformations. A problem has been divided into several tasks and is carried out on several stages. The purpose of this stage is a development of the structure of a hybrid model, development of an algorithm used in the diffusion module and one-dimensional heat flow and diffusion modeling. Generally, the processes of phase transformations are studied well enough but there are not many tools for their complex simulations. The problems of phase transformation simulation are related to the proper consideration of diffusion, movement of phase boundaries and kinetics of transformation. The proposed new model at the final stage of development will take into account the varying grain growth rate, different shape of growing grains and will allow for proper modeling of heat flow and carbon diffusion during the transformation in many processes, where heating, annealing and cooling can be considered (e.g. homogenizing and normalizing).

One of the most suitable methods for modeling of microstructure evolution during the phase transformation is cellular automata (CA), while lattice Boltzmann method (LBM) suits for modeling of diffusion and heat flow. Then, the proposed new hybrid model is based on CA and LBM methods and uses high performing parallel computations.

The first simulation results obtained for one-dimensional modeling confirm the correctness of interaction between LBM and CA in common numerical solution and the possibility of using these methods for modeling of phase transformations. The advantages of the LBM method can be used for the simulation of heat flow and diffusion during the transformation taking into account the results obtained from the simulations. LBM creates completely new possibilities for modeling of phase transformations in combination with CA.

The studies are focused on diffusion phase transformations in solid state in condition of low cooling rate (e.g. transformation of austenite into ferrite and pearlite) and during the heating and annealing (e.g. transformation of the ferrite-pearlite structure into austenite, the alignment of carbon concentration in austenite and growth of austenite grains) in carbon steels within a wide range of carbon content. The paper presents the comprehensive modeling system, which can operate with the technological processes with phase transformation during heating, annealing or cooling.

A brief review of the modeling of phase transformations and a description of the structure of a new CA and LBM hybrid model and its modules are presented in the paper. In the first stage of model implementation, the one-dimensional LBM model of diffusion and heat flow was developed. The examples of simulation results for several variants of modeling with different boundary conditions are shown.

The holistic numerical model based on cellular automata (CA) and lattice Boltzmann method (LBM) are being developed as part of an integrated modelling approach applied to study the interaction of different physical mechanisms in laser-assisted additive layer manufacturing (ALM) of orthopaedic implants. Several physical events occurring in sequence or simultaneously are considered in the holistic model. They include a powder bed deposition, laser energy absorption and heating of the powder bed by the moving laser beam, leading to powder melting or sintering, fluid flow in the melted pool and flow through partly or not melted material, and solidification. The purpose of this study is to develop a structure of the holistic numerical model based on CA and LBM applicable for studying the interaction of the different physical mechanisms in ALM of orthopaedic implants. The model supposed to be compatible with the earlier developed CA-based model for the generation of the powder bed.

The mentioned physical events are accompanied by heat transfer in solid and liquid phases including interface heat transfer at the boundaries. The sintering/melting model is being developed using LBM as an independent numerical method for hydrodynamic simulations originated from lattice gas cellular automata. It is going to be coupled with the CA-based model of powder bed generation.

The entire laser-assisted ALM process has been analysed and divided on several stages considering the relevant physical phenomena. The entire holistic model consisting of four interrelated submodels has currently been developed to a different extent. The submodels include the CA-based model of powder bed generation, the LBM-CA-based model of heat exchange and transfer, the thermal solid-liquid interface model and the mechanical solid-liquid interface model for continuous liquid flow.

The results obtained can be used to explain the interaction of the different physical mechanisms in ALM, which is an intensively developing field of advanced manufacturing of metal, non-metal and composite structural parts, for instance, in bio-engineering. The proposed holistic model is considered to be a part of the integrated modelling approach being developed as a numerical tool for investigation of the co-operative relationships between multiphysical phenomena occurring in sequence or simultaneously during heating of the powder bed by the moving high energy heat source, leading to selective powder sintering or melting, fluid flow in the melted pool and through partly (or not) melted material, as well as solidification. The model is compatible with the earlier developed CA-based model for the generation of the powder bed, allowing for decrease in the numerical noise.

The present results are original and new for the study of the complex relationships between multiphysical phenomena occurring during ALM process based on selective laser sintering or melting, including fluid flow and heat transfer, identified as crucial for obtaining the desirable properties.

A fundamental principle of materials engineering is that the microstructure of a material controls the properties. The phase transformation is an important phenomenon that determines the final microstructure. Recently, many analytical and numerical methods were used for modeling of phase transformation, but some limitations can be seen in relation to the choice of the shape of growing grains, introduction of varying grain growth rate and modeling of diffusion phenomena. There are also only few comprehensive studies that combine the final microstructure with the actual conditions of its formation. Therefore, the objective of the work is a development of a new hybrid model based on lattice Boltzmann method (LBM) and cellular automata (CA) for modeling of the diffusional phase transformations. The model has a modular structure and simulates three basic phenomena: carbon diffusion, heat flow and phase transformation. The purpose of this study is to develop a model of heat flow with consideration of enthalpy of transformation as one of the most important parts of the proposed new hybrid model. This is one of the stages in the development of the complex model, and the obtained results will be used in a combined solution of heat flow and carbon diffusion during the modeling of diffusion phase transformations.

Different values of overheating/overcooling affect different values in the enthalpy of transformation and thus the rate of transformation. CA and LBM are used in the hybrid model in part related to heat flow. LBM is used for modeling of heat flow, while CA is used for modeling of the microstructure evolution during the phase transformation.

The use of LBM and CA in one numerical solution creates completely new possibilities for modeling of phase transformations. CA and LBM in comparison with commonly used approaches significantly simplify interface and interaction between different parts of the model, which operates in a common domain. The CA can be used practically for all possible processes that consist of nucleation and grains growth. The advantages of the LBM method can be well used for the simulation of heat flow during the transformation, which is confirmed by numerical results.

The developed heat flow model will be combined with the carbon diffusion model at the next stage of work, and the new complex hybrid model at the final stage will provide new solutions in numerical simulation of phase transformations and will allow comprehensive modeling of the diffusional phase transformations in many processes. Heating, annealing and cooling can be considered.

The paper presents the developed model of heat flow (temperature module), which is one of the main parts of the new hybrid model devoted to modeling of phase transformation. The model takes into account the enthalpy of transformation, and the connection with the model of microstructure evolution was obtained.

The well-known discrete methods of computational fluid dynamics (CFD), lattice Boltzmann method (LBM), cellular automata (CA), volume-of-fluid (VoF) and others rely on several parameters describing the boundary or the surface. Some of them are vector normal to the surface, coordinates of the point on the surface and the curvature. They are necessary for the reconstruction of the real surface (boundary) based on the values of the volume fractions of several cells. However, the simple methods commonly used for calculations of the vector normal to the surface are of unsatisfactory accuracy. In light of this, the purpose of this paper is to demonstrate a more accurate method for determining the vector normal to the surface.

Based on the thesis that information about the volume fractions of the 3 × 3 cell block should be enough for normal vector determination, a neural network (NN) was proposed for use in the paper. The normal vector and the volume fractions of the cells themselves can be defined on the basis of such variables as the location of the center and the radius of the circumference. Therefore, the NN is proposed to solve the inverse problem – to determine the normal vector based on known values of volume fractions. Volume fractions are inputs of NNs, while the normal vector is their output. Over a thousand variants of the surface location, orientations of the normal vector and curvatures were prepared for volume fraction calculations; their results were used for training, validating and testing the NNs.

The simplest NN with one neuron in the hidden layer shows better results than other commonly used methods, and an NN with four neurons produces results with errors below 1° relative to the orientation of the normal vector; for several cases, it proven to be more accurate by an order of magnitude.

The method can be used in the CFD, LBM, CA, VoF and other discrete computational methods. The more precise normal vector allows for a more accurate determination of the points on the surface and curvature in further calculations via the surface or interface tracking method. The paper contains the data for the practical application of developed NNs. The method is limited to regular square or cuboid lattices.

The paper presents an original implementation of NNs for normal vector calculation connected with CFD, LBM and other application for fluid flow with free surface or phase transformation.