This work involves the development of a grid generation system and flow solver to improve the study of the microstructures that develop during the solidification of metal alloys. While the work is primarily of interest to material scientists, on a larger scale, it focuses more attention on the development of grid generation software needed at NIST and throughout the scientific and engineering community. The author has already received requests that the completed system be made available online as a testbed for adaptive/moving grid generation algorithms developed by other researchers.
Understanding the microstructures produced during solidification can help metallurgists improve the quality of metal products manufactured by casting and welding techniques and aid solid-state physicists in producing the pure semiconductor crystals needed for electronic devices. Metallurgists have looked at the stability of planar interfaces to determine under what conditions complex cellular and dendritic 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 development and testing of the grid generation code has been completed. The code is described in A Boundary Conforming Grid Generation System for Interface Tracking in the journal Computers and Mathematics with Applications, Vol. 29, No. 10, 1995. The author has written a flow solver based on the solutal model of directional solidification which assumes that the temperature field is linear and unaffected by changes in the interface shape. The code uses an iterative solver for large sparse linear systems developed by the Center for Numerical Analysis, University of Texas at Austin. Testing of the integrated grid generator/flow solver is ongoing.
Once the final testing is complete the author will look at the feasibility of making the code generally available as a testbed for other grid generation techniques. The author will also look at what modifications are necessary to make the grid generation code suitable for the study of phase field models of solidification where the solid-liquid interface is not tracked directly.