# Mixed finite elements for convection-coupled phase-change in enthalpy form: Open software verified and applied to 2D benchmarks

## TL;DR

This paper presents a verified open-source mixed finite element software for accurately simulating convection-coupled phase-change processes in 2D, with a novel continuation approach and validation against experimental data.

## Contribution

It introduces a separation of mesh refinement from nonlinear regularization and a continuation procedure for robust solutions, along with a verified open-source implementation.

## Key findings

- Code accurately solves 2D phase-change problems with verified convergence.
- Benchmark simulations match experimental data and literature results.
- Numerical errors are effectively reduced through parameter sensitivities.

## Abstract

Melting and solidification processes are often affected by natural convection of the liquid, posing a multi-physics problem involving fluid flow, convective and diffusive heat transfer, and phase-change reactions. Enthalpy methods formulate this convection-coupled phase-change problem on a single computational domain. The governing equations can be solved accurately with a monolithic approach using mixed finite elements and Newton's method. Previously, the monolithic approach has relied on adaptive mesh refinement to regularize local nonlinearities at phase interfaces. This contribution instead separates mesh refinement from nonlinear problem regularization and provides a continuation procedure which robustly obtains accurate solutions on the tested 2D uniform meshes. A flexible and extensible open source implementation is provided. The code is formally verified to accurately solve the governing equations in time and in 2D space, and convergence rates are shown. Two benchmark simulations are presented in detail with comparison to experimental data sets and corresponding results from the literature, one for the melting of octadecane and another for the freezing of water. Sensitivities to key numerical parameters are presented. For the case of freezing water, effective reduction of numerical errors from these key parameters is successfully demonstrated. Two more simulations are briefly presented, one for melting at a higher Rayleigh number and one for melting gallium.