Numerical solution of the Stefan problem

The paper describes an algorithm for solving the Stefan problem by the finite element method for modeling heat and mass transfer processes in a fluid with a phase transition. Modeling of heat and mass transfer is based on the solution of the Navier-Stokes equations for an incompressible fluid in the Boussinesq approximation and the heat conduction equation for a solid phase. The solution was made according to the explicit-implicit scheme of the matrixless finite element method using a moving finite element mesh. The mobility of the grid nodes is due to the variable geometry of the solution region due to the motion of the melt-crystal interface. The new positions of the nodes of the moving mesh were calculated by the model of elastic media, providing the approximate equality of the volumes of the mesh cells. The grid nodes belonging to the moving boundary between the melt and the growing crystal moved in accordance with Stefan’s conditions. The auxiliary systems of algebraic equations for the nodal values of the desired functions were solved by the matrixless conjugate gradient method with preconditioning by using the diagonal approximation of the stiffness matrix. An example of the application of the described finite element implementation of the Stefan problem for modeling of process for semiconductor single crystal growth by the Bridgman method taking into account the rotations of crystal, crucible and heater-vibrator is given.