Bonita V. Saunders, ACMD
Metallurgists know that the quality of a crystal alloy is affected by the complex cellular and dendritic microstructures that can develop at the solid/liquid interface during the solidification process. To gain a better understanding of these microstructures, metallurgists have looked at the stability of planar interfaces to determine under what conditions the more complex microstructures appear. Several researchers have examined slightly nonplanar microstructures in great detail, but few have examined the deeper bulb-shaped cells seen in experiments because of the need for efficient and flexible curvilinear coordinate systems that can adjust to dramatic changes in the interface shape.
To study these cells the author has developed a curvilinear coordinate system defined by a tensor-product grid generation mapping composed of cubic B-splines. Although Robert Brown and colleagues at MIT have had some success in modeling the deeper cells, their techniques involve breaking up the domain into simpler sections or using two step procedures involving different types of transformations. Saunders uses a single mapping for the entire domain. The coefficients for the mapping are initially chosen so that the grid fits a rectangular outer boundary and the interface. The smoothness and orthogonality of the grid are enhanced by modifying some of the coefficients to minimize a functional involving the variation in Jacobian values at nearby grid points and the dot product of tangent vectors to grid lines. A hyperbolic sine control function is used to concentrate the grid points near the interface.
The initial phase of the project, the development of the grid generation code, has been completed. The grid generation technique is discussed in "A Boundary Conforming Grid Generation System for Interface Tracking'' which is to appear in the journal Computers and Mathematics with Applications.
Currently, the author is writing the code to solve the equations that govern the transport of solute and determine the interface shape. The author is using the solutal model of directional solidification which assumes that the temperature field is linear and unaffected by changes in the interface shape. This code will be integrated with the grid generation code to create an adaptive system. Once this code is thoroughly tested the author will look at the more general problem obtained by taking away the linear temperature assumption and adding convective terms to the solutal and temperature field equations.