Accurate and parallel simulation of the anisotropic dendrite crystal growth by Lagrangian data assimilation with directional operator splitting

Dendritic crystal growth is a prevalent natural phenomenon that generates a crystalline structure resembling a tree during a phase transition. In practical computations, inaccuracies in model parameters and initial conditions can introduce observation errors, even leading to inaccurate results. To enhance the precision and efficiency of the numerical simulation, the data assimilation method with parallel simulation are considered in this study. Firstly, based on the phase-field dendritic crystal growth model, a Lagrangian data assimilation method, which adds Lagrange multiplier terms into the phase-field partial differential equations (PDEs), is presented to integrate the observed data of physical information to modify the numerical solution, thereby improving simulation accuracy. Secondly, to achieve efficient data assimilation, a parallel directional operator splitting method is presented to solve the modified data assimilation PDEs. Thirdly, in the section of numerical experiments, we investigate the validity of the method and assess the impact of various factors such as the Lagrange multiplier parameter, spatio-temporal sampling rate and parameter perturbation ratio on the effectiveness of data assimilation. The evaluation is conducted for two distinct problem categories: initial observation errors and model parameter errors. Experimental results demonstrate that our method can effectively assimilate experimental observations in simulations, thereby enhancing more accurate dendritic crystal growth processes.