Comparison of Regression and Bounding for Richardson Extrapolation-Based Discretization Error Estimators
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The ORIGEN module in SCALE is a common method of performing quick depletion and irradiation calculations of various materials in light-water reactors. However, these analyses depend on the neutron spectra in the ARP libraries being accurate for the situation presented. When using insertable burnable absorbers in a LWR with Li-6 in order to produce tritium, the neutron spectrum is highly different than those in the default libraries in ORIGEN, which leads to significant mispredictions in tritium production. Therefore, custom libraries are required, and when comparing to lattice physics codes (CASMO5 and WIMS10) with ORIGEN, using custom libraries is much more accurate than using the default libraries.
Many areas of science exhibit physical processes that are described by high dimensional partial differential equations (PDEs), e.g., the 4D, 5D and 6D models describing magnetized fusion plasmas, models describing quantum chemistry, or derivatives pricing. Such problems are affected by the so-called “curse of dimensionality” where the number of degrees of freedom (or unknowns) required to be solved for scales as N D where N is the number of grid points in any given dimension D. A simple, albeit naive, 6D example is demonstrated in the left panel of Figure 1. With N = 1000 grid points in each dimension, the memory required just to store the solution vector, not to mention forming the matrix required to advance such a system in time, would exceed an exabyte - and also the available memory on the largest of supercomputers available today. The right panel of Figure 1 demonstrates potential savings for a range of problem dimensionalities and grid resolution. While there are methods to simulate such high-dimensional systems, they are mostly based on Monte-Carlo methods, which rely on a statistical sampling such that the resulting solutions include noise. Since the noise in such methods can only be reduced at a rate proportional to $\sqrt{N_p}$ where N p is the number of Monte-Carlo samples, there is a need for continuum, or grid/mesh-based methods for high-dimensional problems, which both do not suffer from noise and bypass the curse of dimensionality. We present a simulation framework that provides such a method using adaptive sparse grids.
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This report summarizes a recent project aimed at developing and validating the necessary tools to enable more accurate modeling of denser and more complex particle flows in next-generation particle receivers used in concentrating solar power towers. A newly developed CFD/DEM simulation capability was created by coupling existing the modeling and simulation tools Sierra and LAMMPS. This new capability permitted the inclusion of additional physics for particle drag and particle collisions to model
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The streaming operator of the transport equation is derived for spherical coordinates by starting from Newton’s second law for a free particle expressed in spherical coordinates. We shall show that the partial derivatives with respect to the velocity variables of the particle, which are absent in the Cartesian coordinate formulation of the transport equation, arise in the spherical coordinate formulation of the transport equation in response to the centrifugal force which prevents a free particle from ‘falling into the origin’ of the coordinate system.
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