A Newton method with adaptive finite elements for solving phase-change problems with natural convection
We present a new numerical system using finite elements with mesh adaptivity for the simulation of solidliquid phase change systems. In the liquid phase, the natural convection flow is simulated by solving the incompressible NavierStokes equations with Boussinesq approximation. A variable viscosity model allows the velocity to progressively vanish in the solid phase, through an intermediate mushy region. The phase change is modeled by introducing an implicit enthalpy source term in the heat equation. The final system of equations describing the liquidsolid system by a single domain approach is solved using a Newton iterative algorithm. The space discretization is based on a P2P1 TaylorHood finite elements and mesh adaptivity by metric control is used to accurately track the solidliquid interface or the density inversion interface for water flows. The numerical method is validated against classical benchmarks that progressively add strong non-linearities in the system of equations: natural convection of air, natural convection of water, melting of a phase-change material and water freezing. Very good agreement with experimental data is obtained for each test case, proving the capability of the method to deal with both melting and solidification problems with convection. The presented numerical method is easy to implement using FreeFem++ software using a syntax close to the mathematical formulation.
newton method finite element mesh adaptivity navierstokes boussinesq melting water freezing solidification freefem++ pcm