A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation |
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Authors: | Matthias K Gobbert Samuel G Webster Timothy S Cale |
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Affiliation: | 1.Department of Mathematics and Statistics,University of Maryland,Baltimore,USA;2.Focus Center – New York, Rensselaer: Interconnections for Hyperintegration, Isermann Department of Chemical and Biological Engineering,Rensselaer Polytechnic Institute,Troy,USA;3.Department of Mathematics and Computer Science,Hillsdale College,Hillsdale,USA |
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Abstract: | Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase
onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow
to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction
model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic
numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each
two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral
Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical
choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation
basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions.
The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical
vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability
and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel
performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory
cluster with a high-performance Myrinet interconnect. |
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Keywords: | Boltzmann transport equation spectral Galerkin method discontinuous Galerkin method cluster computing chemical vapor deposition |
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