The modified XFEM for solving problems of a phase change with natural convection

This paper presents an extended finite element method applied to solve phase change problems taking into account natural convection in the liquid phase. It is assumed that the transition from one state to another, e.g., during the solidification of pure metals, is discontinuous and that the physical properties of the phases vary across the interface. According to the classical Stefan condition, the location, topology and rate of the interface changes are determined by the jump in the heat flux. The incompressible Navier-Stokes equations with the Boussinesq approximation of the natural convection flow are solved for the liquid phase. The no-slip condition for velocity and the melting/freezing condition for temperature are imposed on the interface using penalty method. The fractional four-step method is employed for analysing conjugate heat transfer and unsteady viscous flow. The phase interface is tracked by the level set method defined on the same finite element mesh. A new combination of extended basis functions is proposed to approximate the discontinuity in the derivative of the temperature, velocity and the pressure fields. The single-mesh approach is demonstrated using three two-dimensional benchmark problems. The results are compared with the numerical and experimental data obtained by other authors.