An adaptive mesh refinement scheme for solidification problems
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INTRODUCTION
MODELING of solidification has important applications in casting, crystal growth, and other processes where products are made directly from the melt. In this article, we focus on efficient numerical methods for resolving large scale motions of the liquid-solid interface. We do this by selectively refining the computational grid near the interface, while allowing it to be relatively coarse further away. A number of researchers, dating back to Stefan,[~] have studied solidification problems with a unique freezing temperature. These problems fall into a broader class called free-boundary problems, where the position of the interface is found as part of the solution. A number of analytical solutions have been reported, limited to simple geometries and constant material properties (cf. References 2 and 3 for a review). For more realistic problems, numerical techniques have been developed. The principal difficulty addressed in these methods is the need to apply boundary conditions on a surface whose position is a priori unknown. Two general types of numerical methods have evolved: those which deform the grid so that the boundary and the grid coincide, and those that solve the problem on a fixed grid, reconstructing the position of the interface from the solution. The deforming grid methods typically iterate for solution by first estimating the location of the interface, applying one boundary condition to solve for the temperature, and finally using the residual error in the remaining boundary condition to update the interface position. These techniques are most useful when the movement of the interface from its initial position is small; otherwise, severe mesh distortion may result. [4] Fixed grid techniques tend to be more robust when large motions of the interface occur. However, these methods re-
NAGENDRA PALLE, formerly Graduate Student, Department of Mechanical and Industrial Engineering, University of Illinois, is Engineering Specialist, Scientific Research Laboratory, Ford Motor Company, Dearborn, MI 48121. JONATHAN A. DANTZIG, Associate Professor of Mechanical Engineering, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801. This article is based on a presentation made at the "Analysis and Modeling of Solidification" symposium as part of the 1994 Fall meeting of TMS in Rosemont, Illinois, October 2-6, 1994, under the auspices of the TMS Solidification Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
quire that solidification be allowed to occur over a range of temperatures, even for pure materials, leading to a loss of resolution of the interface' position. In particular, there is an undesirable interaction between resolution of the interface, temperature gradients, and mesh size.t51 Increasing the mesh density sufficiently to resolve the interface can lead to grids with so many nodes that even small problems become intractable. This problem is especially acute in the solidification of dilute alloys, where resolution of concentration boundary laye
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