Non-equilibrium phase field model using thermodynamics data estimated by machine learning for additive manufacturing solidification

A multi-phase field method using finite interface dissipation model proposed by Steinbach et al. is applied to simulate solidification microstructure evolution of stainless steel composition in the non-equilibrium condition of high cooling rate and temperature gradient of additive manufacturing. The calculation is performed for quinary system in order to simulate solidification of engineering composition. Thermodynamic calculation using CALPHAD database in this multi-phase field method calculation is replaced by machine learning prediction procedure to reduce calculation time. The microstructure evaluated by using machine learning parameter is good agreement with one directly coupled with CALPHAD database. This calculation is approximately five times faster than the direct CALPHAD calculation method. Finally, it is confirmed that this multi-phase field method can be applicable to simulate non-equilibrium phase transformation of additive manufacturing condition with high numerical stabilization. Introduction Many authors apply Cellular Automata method or Monte Carlo model to simulate rapid solidification texture and grain evolution of additive manufacturing (AM) [1, 2]. However, it seems that these methodologies could not provide proper dendrite morphology according to the constitutional undercooling because of lack of interface driving force estimation derived from chemical potential. S. Ghosh et al. discussed microstructure evolution of solidification of Ni alloy in the condition of AM, high cooling rate and temperature gradient, on the basis of the classical solidification theory using three-dimensional finite element thermal analysis and phase-field method [3, 4]. It was confirmed that the relation between the primary dendrite arm spacing and cooling rate and temperature gradient follow the solidification theory equation. Furthermore, they discussed segregation behavior by comparing with the theoretical solute petitioning equation for rapid solidification interface. However, these studies were performed for binary, Ni-Nb, system. It is considered that development of multi-phase field method (MPFM) for multi-element, that is free of quasi-equilibrium in the interface for adapting rapid solidification, should be developed for applying these discussion to engineering alloy. Multi-phase field method that proposed by Steinbach about twenty years ago has been applied to many numerical simulation fields [5]. MPFM coupled with CALPHAD database using TQ-Interface of Thermo-Calc [6] provides us to simulate microstructure evolution for many engineering metal alloy processes [7]. MPFM coupling with solving solute diffusion is based on the diffuse-interface model, in which equal diffusion potential or quasi-equilibrium is assumed in the interface region. It is also generally called as KKS model [8]. However, in practical calculation using CALPHAD database coupling, it consumes long time to obtain quasi-equilibrium concentration at all grid points in the interface region. Steinbach and Zhang proposed finite interface dissipation model, which was based on non-equal diffusion potential after that [9, 10]. It is called non-equilibrium multiphase field model (NEMPFM) in this study. This method has another advantage to avoid iterating calculation obtaining quasi-equilibrium composition in the interface region. It leads to decreasing calculation time by comparing with that of MPFM. We developed CALPHAD database coupled NEMPFM program for practical engineering steel system [11]. 1875 Solid Freeform Fabrication 2018: Proceedings of the 29th Annual International Solid Freeform Fabrication Symposium – An Additive Manufacturing Conference Reviewed Paper However, NEMPFM using TQ-Interface spends huge time for the thermodynamics calculation for NEMPFM yet. Furthermore, it can perform only on Thermo-Calc usage permitted platform. In this study, we attempt to accelerate NEMPFM thermodynamics calculation by replacing CALPHAD database coupling method by linear prediction procedure of neural network as a machine-learning. The thermodynamics calculation consists of obtaining Gibbs free energy and chemical potential that are varied as functions of temperature and composition in the interface region. The mini-batch gradient descent method [12] is employed for neural network training, of which data, these values, are obtained from CALPHAD database coupled NEMPFM calculation result. The identified neural network parameters, bias and weight, are adapted to the linear prediction procedure in NEMPFM program. This method is applied to solidification simulation for quinary, Fe-Cr-Ni-Mo-C, stainless steel in conditions of high cooling rate and thermal gradient of AM. Non-equilibrium Multi-Phase Field Model In NEMPFM that is called as finite interface dissipation model [9], the reference volume (RV) is defined as small area in the phase field interface region. In practical calculation of discretized numerical method e.g. finite difference method, its grid is selected as RV, as illustrated in Fig.1 It is assumed that diffusion is not considered between RVs. Concentration partition rate is assumed to be function of total free energy variation by concentration: