Fixed-Grid Front-Tracking Algorithm for Solidification Problems, Part I: Method and Validation

Abstract

This paper presents a method for solving moving boundary problems associated with phase-change. This method is an explicit interface-tracking scheme that involves the reconstruction and advection of the moving interface on a fixed grid. Three distinct steps are undertaken to handle the movement of the interface: advection and reconstruction (tracking); calculation of normal velocities; and the solution of the governing equations for different phases.Details of each step and its implementation is provided. The transient heat diffusion equation in two space dimensions is the governing equation for energy transport. In order to validate the approach, results obtained from the front tracking scheme are compared with exact analytical solutions for melting problems in Cartesian and cylindrical coordinates. It is shown that the numerical and analytical results are in excellent agreement. Finally, problems involving the multi-dimensional solidification of a pure aluminum ingot subject to bi-directional heat extraction are presented. The results indicate that the algorithm was able to accurately track the front under these aggressive conditions which caused large curvature. In Part II, the front-tracking approach described herein is applied to directional solidification problems influenced by melt convection.

Direct numerical simulation of solidification with effects of density difference

“…The average error is less than 0.044% for St = 0.1, and less than 0.06% for St = 5. As compared with the results reported in [17], the present method produces the solutions that are more accurate, as shown in Fig. 3.…”

“…Results for the front location compared with the exact solution and with the computational results reported in [17], for St = 0.1 and for St = 5 are shown in Fig. 3a and Fig.…”

“…Other popular methods for tracking interface and moving boundaries in the Eulerian frame are the volume of fluid method [10][11][12], the level set method [13] and the phase field method [14]. Hybrid methods [15][16][17] between the Lagrangian and Eulerian approaches have also been proposed for phase change problems. In the variable-domain formulation, body-fitted grid methods [18] have been widely used.…”

Order verification of a Bridgman furnace front tracking model in steady state

“…FTMs have been used to track the solidification interface in simple phase change problems involving pure materials [11], and also in more complex problems involving binary alloys [12], for example, where columnar dendritic growth occurs [13], or to predict columnar to equiaxed transition [14]. In this paper, we will verify the Bridgman furnace FTM (or BFFTM) of Mooney et al [15], for steady state Bridgman solidification of high purity titanium.…”

Numerical and Experimental Investigation of Solidification Shrinkage

“…While considerable research has been targeted at developing front tracking methods for free/moving surface problems, such as level set, phase field, and other methods [15][16][17][18][19][20], limited effort has been devoted to the study of solidification shrinkage.…”