Materials processing systems are often characterized by the presence of a number of distinct materials and phases with significantly different thermophysical and transport properties. They may also contain irregular boundaries, moving interfaces and free surfaces. An understanding of the complex transport phenomena in these systems is of vital importance for the design and fabrication of equipment and the optimization and control of the manufacturing process.
We are particularly interested in the transport phenomena which occur during the Czochralski (CZ) crystal growth processes. It has been known that these phenomena have a profound influence on the formation of defects and inhomogeneities in the crystals. This recognition has led to vigorous research in the areas of heat and mass transfer, fluid flow, and phase change dynamics in Czochralski growth processes [1-7]. In recent years, by combining realistic mathematical models, innovative numerical algorithms and recent advances in computer hardware, much progress has been made.
In most of these cases, however, further progress often depends on realistic, large scale numerical simulations. For example, very fine grids are needed to resolve the various temporal and spatial scales involved in an unsteady, often turbulent, flow field. Moreover, in many cases, three-dimensional simulations are necessary to yield a reliable description of the flow behavior. Furthermore, a large number of calculations are required in parametric studies to explore the solution dependence on process parameters. These simulations, which require extensive computer resources such as computational speed and processor memory, pose many challenges both in terms of computing hardware and software. Recent advances in computer architecture and algorithm development for massively parallel systems make parallel computation an attractive approach to perform these calculations.
In the last several years, there has been significant progress in developing parallel algorithms for many industrial and engineering problems involving complex fluid flow phenomena. However, few attempts have been made in using parallel computations in crystal growth simulations. Peric et al. [8] applied a parallel multigrid finite volume method for an axisymmetric model of Czochralski crystal growth. In that computation, only a regular cylindrical geometry has been considered and the effect of crucible bottom curvature or the moving phase interfaces were not considered. Xiao et al. [9, 10] have applied a three-dimensional parallel finite element method for the simulation of Czochralski oxide growth systems. They employed the Galerkin finite element method coupled with the backward Euler technique for the discretization of the field equations, and they employed the Newton-Raphson as well as GMRES (generalized minimal residual) iterative schemes to solve the resulting discretized equations.
In this paper, we present a parallel algorithm that is an extension of a serial algorithm, MASTRAPP, developed recently for the simulation of transport and phase-change processes in multizone systems [11]. The serial algorithm is based on multizone adaptive grid generation (MAGG) and curvilinear finite volume (CFV) discretization to study the transport phenomena associated with crystal growth processes. The algorithm allows the computational domain to consist of various materials in different phases with significantly different thermophysical and transport properties. It uses a set of unified governing equations for two-dimensional, transient processes involving diffusion and convection of heat and mass, and radiative heat transfer. An efficient scheme has also been devised for fast and accurate interface movement with respect to the phase change rate and flow oscillations, as well as for clustering of grids in interface regions as the solutions progress. The parallel algorithm is based on the overlapping domain decomposition approach. Communication between processors is required for the update of the boundary data. It is designed for MIMD (Multiple Instructions Multiple Data) architectures. Numerical results are presented to demonstrate the feasibility and potential of parallel computation for crystal growth process simulation.