The maximum principle is the most important property of solutions to diffusion equations. Violation of the maximum principle by the applied discretization scheme is the cause for severe numerical instabilities: the emergence of negative concentrations and, in the nonlinear case, the deterioration of the convergence of the Newton iteration. We compare finite volumes (FV) and finite elements (FE) in three dimensions with respect to the constraints they impose on the mesh to achieve a discrete maximum principle. Distinctive mesh examples and simulations are presented to clarify the mutual relationship of the resulting constraints: Delaunay meshes guarantee a maximum principle for FV, while the recently introduced dihedral angle criterion is the natural constraint for FE. By constructing a mesh which fulfills the dihedral angle criterion but is not Delaunay we illustrate the different scope of both criteria. Due to the lack of meshing strategies tuned for the dihedral angle criterion we argue for the use of FV schemes in three-dimensional diffusion modeling.

The article describes a novel class of arithmetic architectures for Galois fields GF(2/sup k/). The main applications of the architecture are public key systems which are based on the discrete logarithm problem for elliptic curves. The architectures use a representation of the field GF(2/sup k/) as GF((2/sup n/)/sup m/), where k=n/spl middot/m. The approach explores bit parallel arithmetic in the subfield GF(2/sup n/) and serial processing for the extension field arithmetic. This mixed parallel-serial (hybrid) approach can lead to fast implementations. As the core module, a hybrid multiplier is introduced and several optimizations are discussed. We provide two different approaches to squaring. We develop exact expressions for the complexity of parallel squarers in composite fields, which can have a surprisingly low complexity. The hybrid architectures are capable of exploring the time-space trade-off paradigm in a flexible manner. In particular, the number of clock cycles for one field multiplication, which is the atomic operation in most public key schemes, can be reduced by a factor of n compared to other known realizations. The acceleration is achieved at the cost of an increased computational complexity. We describe a proof-of-concept implementation of an ASIC for multiplication and squaring in GF((2/sup n/)/sup m/), m variable.

For wafer sizes in state of-the-art semiconductor manufacturing ranging up to 300 mm, the uniformity of processes across the wafer becomes a very important issue. We present a fully three-dimensional model for the feature scale simulation of continuum transport and reaction determined high-pressure chemical vapor deposition processes suitable for the investigation of such nonuniformities. The newly developed three-dimensional approach combines topography simulation, meshing, and finite element method tools, and allows simulations over arbitrary geometries such as structures resulting from nonuniform underlying physical vapor deposition films. This enables the examination of film profile variations across the wafer for multistep processes consisting of low- and high-pressure parts such as Ti/TiN/W plug-fills, Additionally, the model allows a very flexible formulation of the involved gas chemistry and surface reactions and can easily be extended to process chemistries including gas phase reactions of precursors as observed in deposition of silicon dioxide from tetraethylorthosilicate (TEOS). We show simulation examples for a tungsten deposition process, which is applied as last step in a Ti/TiN/W plug-fill. For filling of an L-shaped trench, we show the transition from transport to reaction limited process conditions.

This contribution introduces a new class of multipliers for finite fields GF((2/sup n/)/sup 4/). The architecture is based on a modified version of the Karatsuba-Ofman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the module reduction can be combined. This saves computation and area in VLSI implementations. The new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with KOA-based architectures with separate module reduction. The new multipliers lead to highly modular architectures and are, thus, well suited for VLSI implementations. Three types of field polynomials are introduced and conditions for their existence are established. For the small fields, where n=2,3,...,8, which are of primary technical interest, optimized field polynomials were determined by an exhaustive search. For each field order, exact space and time complexities are provided.