Analysis of Stefan Problem with Level Set Method

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76235/1/AIAA-2002-2874-847.pd

Monolithic simulation of convection-coupled phase-change - verification and reproducibility

Phase interfaces in melting and solidification processes are strongly affected by the presence of convection in the liquid. One way of modeling their transient evolution is to couple an incompressible flow model to an energy balance in enthalpy formulation. Two strong nonlinearities arise, which account for the viscosity variation between phases and the latent heat of fusion at the phase interface. The resulting coupled system of PDE's can be solved by a single-domain semi-phase-field, variable viscosity, finite element method with monolithic system coupling and global Newton linearization. A robust computational model for realistic phase-change regimes furthermore requires a flexible implementation based on sophisticated mesh adaptivity. In this article, we present first steps towards implementing such a computational model into a simulation tool which we call Phaseflow. Phaseflow utilizes the finite element software FEniCS, which includes a dual-weighted residual method for goal-oriented adaptive mesh refinement. Phaseflow is an open-source, dimension-independent implementation that, upon an appropriate parameter choice, reduces to classical benchmark situations including the lid-driven cavity and the Stefan problem. We present and discuss numerical results for these, an octadecane PCM convection-coupled melting benchmark, and a preliminary 3D convection-coupled melting example, demonstrating the flexible implementation. Though being preliminary, the latter is, to our knowledge, the first published 3D result for this method. In our work, we especially emphasize reproducibility and provide an easy-to-use portable software container using Docker.Comment: 20 pages, 8 figure

Comparison of the Adomian decomposition method and the variational iteration method in solving the moving boundary problem

AbstractIn this paper, a comparison between two methods: the Adomian decomposition method and the variational iteration method, used for solving the moving boundary problem, is presented. Both of the methods consist in constructing the appropriate iterative or recurrence formulas, on the basis of the equation considered and additional conditions, enabling one to determine the successive elements of a series or sequence approximating the function sought. The precision and speed of convergence of the procedures compared are verified with an example

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2

Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs{Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by #wt + � %(') 't = r: (b(')rw) ; c a � %(')w = " � � �(r') '

An efficient numerical method for estimating the average free boundary velocity in an inhomogeneous Hele-Shaw problem

We develop a numerical method to estimate the average speed of the free boundary in a Hele-Shaw problem with periodic coefficients in both space and time. We test the accuracy of the method and present a few examples. We show numerical evidence of flat parts (facets) on the free boundary in the homogenization limit.Comment: 18 page