A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Abstract

We present and distribute a new numerical system using classical finite elements with mesh adaptivity for computing two-dimensional liquid-solid phase-change systems involving natural convection. The programs are written as a toolbox for FreeFem++ (www3.freefem.org), a free finite-element software available for all existing operating systems. The code implements a single domain approach. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. This model describes naturally the evolution of the liquid flow which is dominated by convection effects. To make it valid also in the solid phase, a Carman-Kozeny-type penalty term is added to the momentum equations. The penalty term brings progressively (through an artificial mushy region) the velocity to zero into the solid. The energy equation is also modified to be valid in both phases using an enthalpy (temperature-transform) model introducing a regularized latent-heat term. Model equations are discretized using Galerkin triangular finite elements. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretizations are possible. The coupled system of equations is integrated in time using a second-order Gear scheme. Non-linearities are treated implicitly and the resulting discrete equations are solved using a Newton algorithm. An efficient mesh adaptivity algorithm using metrics control is used to adapt the mesh every time step. This allows us to accurately capture multiple solid-liquid interfaces present in the domain, the boundary-layer structure at the walls and the unsteady convection cells in the liquid. We present several validations of the toolbox, by simulating benchmark cases of increasing difficulty: natural convection of air, natural convection of water, melting of a phase-change material, a melting-solidification cycle, and, finally, a water freezing case. Other similar cases could be easily simulated with this toolbox, since the code structure is extremely versatile and the syntax very close to the mathematical formulation of the model.

Parallel finite-element codes for the simulation of two-dimensional and three-dimensional solid–liquid phase-change systems with natural convection

“…(17) a penalty term on the pressure. An extensive discussion of the role of this parameter in reinforcing the incompressibility constraint and in the stabilization of the method is provided in Rakotondrandisa et al (2020). We recall the main lines of this discussion.…”

“…Resolving all the scales in the liquid region using the Navier-Stokes-Boussinesq equations and accurately capturing the solid-liquid interfaces are the main challenges for a numerical system addressing these problems. In our recent contribution (Rakotondrandisa et al, 2020) we presented an in-depth review of physical and numerical models dealing with solid-liquid phase-change problems with convection. The approach retained in Rakotondrandisa et al (2020) was based on the widely used enthalpy-porosity single-domain model, also called the fixed-domain model (Brent et al, 1988).…”

“…In our recent contribution (Rakotondrandisa et al, 2020) we presented an in-depth review of physical and numerical models dealing with solid-liquid phase-change problems with convection. The approach retained in Rakotondrandisa et al (2020) was based on the widely used enthalpy-porosity single-domain model, also called the fixed-domain model (Brent et al, 1988). In the present contribution, we extend this model to parallel computations using domain-decomposition (DD) methods, as illustrated in Fig.…”

“…The enthalpy-porosity single-domain model was implemented in Rakotondrandisa et al (2020) using an adaptive finite-element method. The corresponding Toolbox for FreeFem++(a free software under LGPL license) 1 was distributed with that paper.…”

“…Consequently, the main purpose of the present contribution is to extend the singledomain model of Rakotondrandisa et al (2020) to parallel computing using domaindecomposition methods. For two-dimensional (2D) configurations, the new Toolbox can be used to reduce the computational time on personal computers with multi-core processors.…”