A Fluid–Thermal Coupling Solution Method for the Shoulder Expansion Process of Sapphire Grown by the Kyropoulos Method

To address the low computational efficiency associated with the large deformation and transient evolution of the solid–liquid interface during the shoulder growth stage of sapphire crystals grown by the Kyropoulos method, a lightweight explicit interface-tracking method is developed. This study proposes a two-stage time stepping method for interface evolution. At each time level, the flow and temperature fields are solved based on the current solid–liquid interface configuration, and the effective interfacial growth velocity is evaluated using characteristic thermal gradients and the thermodynamic equilibrium relation. The solid–liquid interface, crystal profile and melt surface position are then updated via sequential growth and pulling sub-steps. The updated geometric configuration is delivered to the next time level to achieve continuous tracking of the moving interface during shoulder growth. Implemented on a fixed background mesh, the analytical geometry-update scheme effectively reduces the computational cost caused by frequent dynamic mesh remeshing. Based on this method, parametric simulations and experimental validation were carried out for different seed diameters. Compared with conventional strategies, the proposed method reduces the total computation time by approximately 30%. Industrial experiments confirm that the predicted results agree well with the observed macroscopic morphology, providing an efficient numerical approach for optimizing the shoulder growth process parameters.