Publications

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Stochastic media transport problems have long posed challenges for accurate modeling. Brute force Monte Carlo or deterministic sampling of realizations can be expensive in order to achieve the desired accuracy. The well-known Levermore-Pomraning (LP) closure is very simple and inexpensive, but is inaccurate in many circumstances. We propose a generalization to the LP closure that may help bridge the gap between the two approaches. Our model consists of local calculations to approximately determine the relationship between ensemble-averaged angular fluxes and the corresponding averages at material interfaces. The expense and accuracy of the method are related to how "local" the model is and how much local detail it contains. We show through numerical results that our approach is more accurate than LP for benchmark problems, provided that we capture enough local detail. Thus we identify two approaches to using ensemble calculations for stochastic media calculations: direct averaging of ensemble results for transport quantities of interest, or indirect use via a generalized LP equation to determine those same quantities; in some cases the latter method is more efficient. However, the method is subject to creating ill-posed problems if insufficient local detail is included in the model.

The efficiency of discrete-ordinates transport sweeps depends on the scheduling algorithm, domain decomposition, the problem to be solved, and the computational platform. Sweep scheduling algorithms may be categorized by their approach to several issues. In this paper we examine the strategy of domain overloading for mesh partitioning as one of the components of such algorithms. In particular, we extend the domain overloading strategy, previously defined and analyzed for structured meshes, to the general case of unstructured meshes. We also present computational results for both the structured and unstructured domain overloading cases. We find that an appropriate amount of domain overloading can greatly improve the efficiency of parallel sweeps for both structured and unstructured partitionings of the test problems examined on up to 105 processor cores.

We present an improved deterministic method for analyzing transport problems in random media. In the original method realizations were generated by means of a product quadrature rule; transport calculations were performed on each realization and the results combined to produce ensemble averages. In the present work we recognize that many of these realizations yield identical transport problems. We describe a method to generate only unique transport problems with the proper weighting to produce identical ensemble-averaged results at reduced computational cost. We also describe a method to ignore relatively unimportant realizations in order to obtain nearly identical results with further reduction in costs. Our results demonstrate that these changes allow for the analysis of problems of greater complexity than was practical for the original algorithm.

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This report provides a summary of notes for building and running the Sandia Computational Engine for Particle Transport for Radiation Effects (SCEPTRE) code. SCEPTRE is a general purpose C++ code for solving the Boltzmann transport equation in serial or parallel using unstructured spatial finite elements, multigroup energy treatment, and a variety of angular treatments including discrete ordinates. Either the first-order form of the Boltzmann equation or one of the second-order forms may be solved. SCEPTRE requires a small number of open-source Third Party Libraries (TPL) to be available, and example scripts for building these TPL’s are provided. The TPL’s needed by SCEPTRE are Trilinos, boost, and netcdf. SCEPTRE uses an autoconf build system, and a sample configure script is provided. Running the SCEPTRE code requires that the user provide a spatial finite-elements mesh in Exodus format and a cross section library in a format that will be described. SCEPTRE uses an xml-based input, and several examples will be provided.

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The master equation has been used to examine properties of transport in stochastic media. It has been shown previously that not only may the Levermore-Pomraning (LP) model be derived from the master equation for a description of ensemble-averaged transport quantities, but also that equations describing higher-order statistical moments may be obtained. We examine in greater detail the equations governing the second moments of the distribution of the angular fluxes, from which variances may be computed. We introduce a simple closure for these equations, as well as several models for estimating the variances of derived transport quantities. We revisit previous benchmarks for transport in stochastic media in order to examine the error of these new variance models. We find, not surprisingly, that the errors in these variance estimates are at least as large as the corresponding estimates of the average, and sometimes much larger. We also identify patterns in these variance estimates that may help guide the construction of more accurate models.

The linear Boltzmann transport equation is solved using a least-squares finite element approximation in the space, angular and energy phase-space variables. The method is applied to both neutral particle transport and also to charged particle transport in the presence of an electric field, where the angular and energy derivative terms are handled with the energy/angular finite elements approximation, in a manner analogous to the way the spatial streaming term is handled. For multi-dimensional problems, a novel approach is used for the angular finite elements: mapping the surface of a unit sphere to a two-dimensional planar region and using a meshing tool to generate a mesh. In this manner, much of the spatial finite-elements machinery can be easily adapted to handle the angular variable. The energy variable and the angular variable for one-dimensional problems make use of edge/beam elements, also building upon the spatial finite elements capabilities. The methods described here can make use of either continuous or discontinuous finite elements in space, angle and/or energy, with the use of continuous finite elements resulting in a smaller problem size and the use of discontinuous finite elements resulting in more accurate solutions for certain types of problems. The work described in this paper makes use of continuous finite elements, so that the resulting linear system is symmetric positive definite and can be solved with a highly efficient parallel preconditioned conjugate gradients algorithm. The phase-space finite elements capability has been built into the Sceptre code and applied to several test problems, including a simple one-dimensional problem with an analytic solution available, a two-dimensional problem with an isolated source term, showing how the method essentially eliminates ray effects encountered with discrete ordinates, and a simple one-dimensional charged-particle transport problem in the presence of an electric field.

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This report describes the theoretical background on modeling electron transport in the presence of electric and magnetic fields by incorporating the effects of the Lorentz force on electron motion into the Boltzmann transport equation. Electromagnetic fields alter the electron energy and trajectory continuously, and these effects can be characterized mathematically by differential operators in terms of electron energy and direction. Numerical solution techniques, based on the discrete-ordinates and finite-element methods, are developed and implemented in an existing radiation transport code, SCEPTRE.

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