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Parallel Simulations of the Linear Boltzmann Equation for Models in Microelectronics Manufacturing

Matthias K. Gobbert and Michael Reid, Department of Mathematics and Statistics, UMBC, and Timothy S. Cale, Process Evolution, Ltd. and School of Materials, Arizona State University

Production steps in the manufacturing of microelectronic devices involve gas flow at a wide range of pressures. We develop a kinetic transport and reaction model based on a system of time-dependent linear Boltzmann equations. These kinetic equations have the property that velocity appears as an independent variable, in addition to position and time. A deterministic numerical solution for realistic three-dimensional application problems requires the discretization of the three-dimensional velocity space, the three-dimensional position space, and time.

We design a spectral Galerkin method to discretize the velocity space by specially chosen basis functions. The basis functions in the expansion lead to a system of hyperbolic conservation laws with constant diagonal coefficient matrices for each of the linear Boltzmann equations. These systems of conservation laws are solved using the discontinuous Galerkin finite element method. Stability and convergence of the method are verified analytically and demonstrated numerically. As an application example, we simulate chemical vapor deposition at the feature scale in two and three spatial dimensions and analyze the effect of pressure. Finally, we present parallel performance results which indicate that the implementation of the method possesses excellent scalability on a Beowulf cluster with a high-performance Myrinet interconnect.


  1. Michael J. Reid and Matthias K. Gobbert, Parallel Performance Studies for a Hyperbolic Test Problem, Technical Report number HPCF-2008-3, UMBC High Performance Computing Facility, University of Maryland, Baltimore County, 2008. (HPCF machines used: hpc and kali.) PDF