A hybrid level-set / embedded boundary method applied to solidification-melt problems

Abstract

In this paper, we introduce a novel way to represent the interface for two-phase flows with phase change. We combine a level-set method with a Cartesian embedded boundary method and take advantage of both. This is part of an effort to obtain a numerical strategy relying on Cartesian grids allowing the simulation of complex boundaries with possible change of topology while retaining a high-order representation of the gradients on the interface and the capability of properly applying boundary conditions on the interface. This leads to a two-fluid conservative second-order numerical method. The ability of the method to correctly solve Stefan problems, onset dendrite growth with and without anisotropy is demonstrated through a variety of test cases. Finally, we take advantage of the two-fluid representation to model a Rayleigh-Bénard instability with a melting boundary.

Adjoint-based optimization of two-dimensional Stefan problems

“…On the other hand, the Level Set method, where the interface is implicitly defined as a continuous function, has been proven to be well suited in continous adjoint-based optimization, specifically for Stefan problems [34]. Moreover, this method has been shown to accurately reproduce dendritic pattern formation [36,9]. An implicit signed distance strategy has therefore been used here to represent the interface.…”

“…where τ denotes a pseudo-time, S(φ 0 ) denotes a sign function, and φ a signed distance function (once steady-state is reached). There exist many numerical methods for solving the Eikonal equation and, here we have adopted that of Min et al [43], also recently used in the work of Limare et al [9]. This approach relies on a second order ENO spatial discretization with sub-cell resolution near the interface.…”

“…In order to accurately reproduce the crystal shapes produced by dendritic growth, we introduce a variable surface tension coefficient, similar to [47] and [9], in the Gibbs-Thomson relation (Eq. 2.3) to account for anisotropy effects,…”

“…In one dimension, this problem has been studied in depth using various numerical algorithms [6,7,8]. In higher dimensions, various methods have been used such as the level set method in Limare et al [9] and Osher et al [2] and front-tracking method in Juric et al [3]. One of the main challenges associated with modeling such problems in multiple dimensions is due to the unstable dentritic pattern formation [10,11,12].…”