13,008 research outputs found
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
We present a mathematical framework for constructing and analyzing parallel
algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting
algorithms have the capacity to simulate a wide range of spatio-temporal scales
in spatially distributed, non-equilibrium physiochemical processes with complex
chemistry and transport micro-mechanisms. The algorithms can be tailored to
specific hierarchical parallel architectures such as multi-core processors or
clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms
are controlled-error approximations of kinetic Monte Carlo algorithms,
departing from the predominant paradigm of creating parallel KMC algorithms
with exactly the same master equation as the serial one.
Our methodology relies on a spatial decomposition of the Markov operator
underlying the KMC algorithm into a hierarchy of operators corresponding to the
processors' structure in the parallel architecture. Based on this operator
decomposition, we formulate Fractional Step Approximation schemes by employing
the Trotter Theorem and its random variants; these schemes, (a) determine the
communication schedule} between processors, and (b) are run independently on
each processor through a serial KMC simulation, called a kernel, on each
fractional step time-window.
Furthermore, the proposed mathematical framework allows us to rigorously
justify the numerical and statistical consistency of the proposed algorithms,
showing the convergence of our approximating schemes to the original serial
KMC. The approach also provides a systematic evaluation of different processor
communicating schedules.Comment: 34 pages, 9 figure
Avalanches, loading and finite size effects in 2D amorphous plasticity: results from a finite element model
Crystalline plasticity is strongly interlinked with dislocation mechanics and
nowadays is relatively well understood. Concepts and physical models of plastic
deformation in amorphous materials on the other hand - where the concept of
linear lattice defects is not applicable - still are lagging behind. We
introduce an eigenstrain-based finite element lattice model for simulations of
shear band formation and strain avalanches. Our model allows us to study the
influence of surfaces and finite size effects on the statistics of avalanches.
We find that even with relatively complex loading conditions and open boundary
conditions, critical exponents describing avalanche statistics are unchanged,
which validates the use of simpler scalar lattice-based models to study these
phenomena.Comment: Journal of Statistical Mechanics: Theory and Experiment, 2015, P0201
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