A hybrid moving Kriging-based element-free Galerkin-finite element method for the solution of non-linear advection–conduction heat transfer problems with solid–liquid phase change

A novel hybrid approach combining the finite element method (FEM) and the moving Kriging (MK)-based element-free Galerkin (EFG) method is introduced for the numerical solution of non-linear advection–conduction heat transfer problems with solid–liquid phase change. In this framework, the FEM is used in regions characterised by milder thermal gradients, whereas the EFG method is strategically applied to enhance the approximation near the phase-change zones. Given the intrinsic Kronecker delta property of the MK approximations, the coupling between the FEM and EFG domains is seamlessly achieved through a shared-nodes boundary. The highly non-linear problem arising from the latent heat release/absorption during phase change are handled in a deferred manner, cast as thermal load vectors within a Picard iterative scheme. The reliability and suitability of the proposed methodology are validated through several benchmark and applied problems, demonstrating also its efficacy for the thermal modelling of industrial metallurgical processes such as continuous and direct chill casting. The results reveal that this technique effectively overcomes the challenges associated with severe phase-change non-linearities, providing accurate and stable solutions within a remarkably straightforward implementation.