21 research outputs found
Rayleigh quotient iteration with a multigrid in energy preconditioner for massively parallel neutron transport
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Review of Hybrid Methods for Deep-Penetration Neutron Transport
Methods for deep-penetration radiation transport remain important for radiation shielding, nonproliferation, nuclear threat reduction, and medical applications. As these applications become more ubiquitous, the need for accurate and reliable transport methods appropriate for these systems persists. For such systems, hybrid methods often obtain reliable answers in the shortest time by leveraging the speed and uniform uncertainty distribution of a deterministic solution to bias Monte Carlo transport and reduce the variance in the solution. This work reviews the state of the art among such hybrid methods. First, we summarize variance reduction (VR) for Monte Carlo radiation transport and existing efforts to automate these techniques. Relations among VR, importance, and the adjoint solution of the neutron transport equation are then discussed. Based on this exposition, the work transitions from theory to a critical review of existing VR implementations in modern nuclear engineering software. At present, the Consistent Adjoint-Driven Importance Sampling (CADIS) and Forward-Weighted Consistent Adjoint-Driven Importance Sampling (FW-CADIS) hybrid methods are the gold standard by which to reduce the variance in problems that have deeply penetrating radiation. The CADIS and FW-CADIS methods use an adjoint scalar flux to generate VR parameters for Monte Carlo radiation transport. Additionally, efforts to incorporate angular information into VR methods for Monte Carlo are summarized. Finally, we assess various implementations of these methods and the degree to which they improve VR for their target applications
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Exact Transport Representations of the Classical and Nonclassical Simplified PN Equations
We show that the recently introduced nonclassical simplified P equations can be represented exactly by a nonclassical transport equation. Moreover, we validate the theory by showing that a Monte Carlo transport code sampling from the appropriate nonexponential free-path distribution function reproduces the solutions of the classical and nonclassical simplified P equations. Numerical results are presented for four sets of problems in slab geometry. N
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Nonclassical particle transport in one-dimensional random periodic media
We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path length s), and models particle transport in homogenized random media in which the distance to collision of a particle is not exponentially distributed. To solve the nonclassical equation, one needs to know the s-dependent ensemble-averaged total cross section Σt(μ, s) or its corresponding path-length distribution function p(μ, s). We consider a one-dimensional (1-D) spatially periodic system consisting of alternating solid and void layers, randomly placed along the x-axis. We obtain an analytical expression for p(μ, s) and use this result to compute the corresponding Σt(μ, s). Then, we proceed to solve numerically the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions μ = ±1. To assess the accuracy of these solutions, we produce benchmark results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely used atomic mix model in most problems
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Exact Transport Representations of the Classical and Nonclassical Simplified PN Equations
We show that the recently introduced nonclassical simplified PN equations can be represented exactly by a nonclassical transport equation. Moreover, we validate the theory by showing that a Monte Carlo transport code sampling from the appropriate nonexponential free-path distribution function reproduces the solutions of the classical and nonclassical simplified PN equations. Numerical results are presented for four sets of problems in slab geometry
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Rayleigh quotient iteration with a multigrid in energy preconditioner for massively parallel neutron transport
Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy preconditioner. The multigroup Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. The new multigrid in energy preconditioner reduces iteration count for many problem types and takes advantage of the new energy decomposition such that it can scale efficiently. These two tools are useful on their own, but together they enable the RQI eigenvalue solver to work. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices that cannot converge without the multigrid in energy preconditioner. Using these methods together, RQI converged in fewer iterations and in less time than all PI calculations for a full pressurized water reactor core. It also scaled reasonably well out to 275,968 cores
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Weighted delta-tracking in scattering media
In this work, we expand the weighted delta-tracking routine to include a treatment for scattering. The weighted delta-tracking routine adds survival biasing to normal delta-tracking, improving problem figure of merit. In the original formulation of this method, only absorption events were considered. We have expanded the method to include scattering and investigated the method's effectiveness with two test cases: a pressurized water reactor pin cell and a fast reactor pin cell. We compare the figure of merit for calculating infinite flux and total cross-section while incrementally changing the amount of weighted delta-tracking used. We find that this new weighted delta-tracking (WDT) routine has strong potential to improve the efficiency of fast reactor calculations, and may be useful for light water reactor calculations
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Assessment of the Lagrange Discrete Ordinates Equations for Three-Dimensional Neutron Transport
The Lagrange Discrete Ordinates (LDO) equations, developed by Ahrens as an alternative to the traditional discrete ordinates formulation, have been implemented in Denovo, a three-dimensional radiation transport code developed by Oak Ridge National Laboratory. The LDO equations retain the formal structure of the classical discrete ordinates equations but treat particle scattering in a different way. Solutions of the LDO equations have an interpolatory structure such that the angular flux can be naturally evaluated at directions other than the discrete ordinates used in arriving at the solutions, and the ordinates themselves may be chosen in a strategic way for the problem under consideration. Of particular interest is that the LDO equations have been shown to mitigate ray effects at increased angular resolutions. In this paper we present scalar flux solutions of the LDO equations for a small number of test cases of interest and compare the results against flux solutions generated using standard quadrature types. The LDO equations’ flux solutions were found to be comparable to those resultant from the standard quadrature types in value; results from the LDO equations were also found to be commensurate with those of standard quadrature types when comparing the flux solutions in the context of the experimental benchmark test case examined