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.
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.
We examine the modeling of charged-particle transport when both collision processes with background media and electromagnetic effects are important using the Boltzmann-Vlasov equation. We derive and transform the Boltzmann-Vlasov equation into a form very similar to the standard linear Boltzmann equation with additional operators. We apply the discontinuous finite element methods for discretization in the spatial, energy, and angular variables. An implementation of these methods demonstrates correct transport behavior for fixed electric and magnetic fields. We also demonstrate coupling to Maxwell’s equations with a simple electromagnetic solver to generate self-consistent fields.
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.
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.
The well-known "sweep" algorithm for inverting the streaming-plus-collision term in first-order deterministic radiation transport calculations has some desirable numerical properties. However, it suffers from parallel scaling issues caused by a lack of concurrency. The maximum degree of concurrency, and thus the maximum parallelism, grows more slowly than the problem size for sweeps-based solvers. We investigate a new class of parallel algorithms that involves recasting the streaming-plus-collision problem in prefix form and solving via cyclic reduction. This method, although computationally more expensive at low levels of parallelism than the sweep algorithm, offers better theoretical scalability properties. Previous work has demonstrated this approach for one-dimensional calculations; we show how to extend it to multidimensional calculations. Notably, for multiple dimensions it appears that this approach is limited to long-characteristics discretizations; other discretizations cannot be cast in prefix form. We implement two variants of the algorithm within the radlib/SCEPTRE transport code library at Sandia National Laboratories and show results on two different massively parallel systems. Both the "forward" and "symmetric" solvers behave similarly, scaling well to larger degrees of parallelism then sweeps-based solvers. We do observe some issues at the highest levels of parallelism (relative to the system size) and discuss possible causes. We conclude that this approach shows good potential for future parallel systems, but the parallel scalability will depend heavily on the architecture of the communication networks of these systems.