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Angularly Adaptive P1--Double P0 Diffusion Solutions of Non-Equilibrium Grey Radiative Transfer Problems in Planar Geometry

Description: The double spherical harmonics angular approximation in the lowest order, i.e. double P{sub 0} (DP{sub 0}), is developed for the solution of time-dependent non-equilibrium grey radiative transfer problems in planar geometry. The standard P{sub 1} angular approximation represents the angular dependence of the radiation specific intensity using a linear function in the angular domain -1 {le} {mu} {le} 1. In contrast, the DP{sub 0} angular approximation represents the angular dependence as isotropic in each half angular range -1 {le} {mu} < 0 and 0 < {mu} {le} 1. Neglecting the time derivative of the radiation flux, both the P{sub 1} and DP{sub 0} equations can be written as a single diffusion equation for the radiation energy density. Although the DP{sub 0} diffusion approximation is expected to be less accurate than the P{sub 1} diffusion approximation at and near thermodynamic equilibrium, the DP{sub 0} angular approximation can more accurately capture the complicated angular dependence near the non-equilibrium wave front. We develop an adaptive angular technique that locally uses either the DP{sub 0} or the P{sub 1} diffusion approximation depending on the degree to which the radiation and material fields are in thermodynamic equilibrium. Numerical results are presented for a test problem due to Su and Olson for which a semi-analytic transport solution exists. The numerical results demonstrate that the adaptive P{sub 1}-DP{sub 0} diffusion approximation can yield improvements in accuracy over the standard P{sub 1} diffusion approximation for non-equilibrium grey radiative transfer.
Date: June 6, 2005
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

Angularly Adaptive P1-Double P0 Flux-Limited Diffusion Solutions of Non-Equilibrium Grey Radiative Transfer Problems

Description: The double spherical harmonics angular approximation in the lowest order, i.e. double P{sub 0} (DP{sub 0}), is developed for the solution of time-dependent non-equilibrium grey radiative transfer problems in planar geometry. Although the DP{sub 0} diffusion approximation is expected to be less accurate than the P{sub 1} diffusion approximation at and near thermodynamic equilibrium, the DP{sub 0} angular approximation can more accurately capture the complicated angular dependence near a non-equilibrium radiation wave front. In addition, the DP{sub 0} approximation should be more accurate in non-equilibrium optically thin regions where the positive and negative angular domains are largely decoupled. We develop an adaptive angular technique that locally uses either the DP{sub 0} or P{sub 1} flux-limited diffusion approximation depending on the degree to which the radiation and material fields are in thermodynamic equilibrium. Numerical results are presented for two test problems due to Su and Olson and to Ganapol and Pomraning for which semi-analytic transport solutions exist. These numerical results demonstrate that the adaptive P{sub 1}-DP{sub 0} diffusion approximation can yield improvements in accuracy over the standard P{sub 1} diffusion approximation, both without and with flux-limiting, for non-equilibrium grey radiative transfer.
Date: December 13, 2005
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

Asymptotic Analysis of Time-Dependent Neutron Transport Coupled with Isotopic Depletion and Radioactive Decay

Description: We describe an asymptotic analysis of the coupled nonlinear system of equations describing time-dependent three-dimensional monoenergetic neutron transport and isotopic depletion and radioactive decay. The classic asymptotic diffusion scaling of Larsen and Keller [1], along with a consistent small scaling of the terms describing the radioactive decay of isotopes, is applied to this coupled nonlinear system of equations in a medium of specified initial isotopic composition. The analysis demonstrates that to leading order the neutron transport equation limits to the standard time-dependent neutron diffusion equation with macroscopic cross sections whose number densities are determined by the standard system of ordinary differential equations, the so-called Bateman equations, describing the temporal evolution of the nuclide number densities.
Date: September 27, 2006
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

Angularly Adaptive P1 - Double P0 Flux-Limited Diffusion Solutions of Non-Equilibrium Grey Radiative Transfer Problems

Description: The double spherical harmonics angular approximation in the lowest order, i.e. double P{sub 0} (DP{sub 0}), is developed for the solution of time-dependent non-equilibrium grey radiative transfer problems in planar geometry. Although the DP{sub 0} diffusion approximation is expected to be less accurate than the P{sub 1} diffusion approximation at and near thermodynamic equilibrium, the DP{sub 0} angular approximation can more accurately capture the complicated angular dependence near a non-equilibrium radiation wave front. In addition, the DP{sub 0} approximation should be more accurate in non-equilibrium optically thin regions where the positive and negative angular domains are largely decoupled. We develop an adaptive angular technique that locally uses either the DP{sub 0} or P{sub 1} flux-limited diffusion approximation depending on the degree to which the radiation and material fields are in thermodynamic equilibrium. Numerical results are presented for two test problems due to Su and Olson and to Ganapol and Pomraning for which semi-analytic transport solutions exist. These numerical results demonstrate that the adaptive P{sub 1}-DP{sub 0} diffusion approximation can yield improvements in accuracy over the standard P{sub 1} diffusion approximation, both without and with flux-limiting, for non-equilibrium grey radiative transfer.
Date: August 8, 2006
Creator: Brantley, P. S.
Partner: UNT Libraries Government Documents Department

Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks

Description: An artificial neural network (ANN) method is developed for treating the spatial variable of the one-group slab-geometry discrete ordinates (S{sub N}) equations in a homogeneous medium with linearly anisotropic scattering. This ANN method takes advantage of the function approximation capability of multilayer ANNs. The discrete ordinates angular flux is approximated by a multilayer ANN with a single input representing the spatial variable x and N outputs representing the angular flux in each of the discrete ordinates angular directions. A global objective function is formulated which measures how accurately the output of the ANN approximates the solution of the discrete ordinates equations and boundary conditions at specified spatial points. Minimization of this objective function determines the appropriate values for the parameters of the ANN. Numerical results are presented demonstrating the accuracy of the method for both fixed source and incident angular flux problems.
Date: March 23, 2001
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

A Variational P1 -DP0 Diffusion Theory for Planar Geometry

Description: A variational analysis is used to derive a mixed P{sub 1}DP{sub 0} (spherical harmonics-double spherical harmonics) angular approximation to the time-independent monoenergetic neutron transport equation with linearly anisotropic scattering in one-dimensional planar geometry. This mixed approximation contains a space-dependent weight factor {alpha}(x) that controls the local angular approximation used: {alpha}(x) = 1 yields the standard P{sub 1} (diffusion) approximation, {alpha}(x) = 0 gives the standard DP{sub 0} approximation, and 0 < {alpha}(x) < 1 produces a mixed approximation. The diffusion equation obtained differs from the standard P{sub 1} diffusion equation only in the definition of the diffusion coefficient. The variational analysis shows that both the scalar flux and the current are continuous at material interfaces regardless of the value of {alpha}(x). Standard Marshak boundary conditions are also obtained via the variational analysis. In this paper, they examine the use of this mixed angular approximation to more accurately treat material interfaces and vacuum boundaries. Numerical results from a mixed-oxide fuel test problem are presented to demonstrate that significant improvements in accuracy can be obtained using this method. For this test problem, the mixed P{sub 1}-DP{sub 0} angular approximation with {alpha} = 0.25 is found to be more robust than the standard DP{sub 0} approximation ({alpha} = 0) for treating the material interfaces and vacuum boundaries.
Date: April 6, 2002
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

Spatially Continuous Mixed P2-P1 Solutions for Planar Geometry

Description: Even-order Legendre polynomial (P{sub N}) expansion approximations of the neutron transport equation have historically seen only limited practical application. Research in the last decade has resolved one of the historical theoretical objections to the use of even-order PN approximations in planar geometry, namely the ambiguity in the prescription of boundary conditions as a result of an odd number of unknowns. This research also demonstrated the P{sub 2} approximation to be more accurate than the P{sub 1} approximation in planar geometry away from boundary layers and material interfaces. Neither the P{sub 1} nor the P{sub 2} approximation is convincingly more accurate near material interfaces. This progress motivated the reexamination of the multidimensional simplified P{sub 2} (SP{sub 2}) approximation, the development of P{sub 2} approximations for planar geometry stochastic transport problems, and the examination of the P{sub 2} and SP{sub 2} approximations as a synthetic acceleration technique for the discrete ordinates equations. The major remaining objection to even-order PN approximations is that the scalar flux distributions obtained using these approximations can exhibit large spatial discontinuities at material interfaces and source discontinuities. In contrast, the odd-order PN approximations typically utilized give spatially continuous scalar flux distributions at these locations. In this paper, we propose a mixed P{sub 2}-P{sub 1} angular approximation designed to take advantage of the improved accuracy of the P{sub 2} approximation in the interior of material regions and near external boundaries while retaining the continuous solutions obtained by the P{sub 1} approximation near material interfaces and source discontinuities. We present numerical results from a series of eigenvalue calculations to demonstrate the accuracy of the mixed P{sub 2}-P{sub 1} angular approximation.
Date: June 4, 2004
Creator: Brantley, P S
Partner: UNT Libraries Government Documents Department

Monte Carlo Particle Transport Capability for Inertial Confinement Fusion Applications

Description: A time-dependent massively-parallel Monte Carlo particle transport calculational module (ParticleMC) for inertial confinement fusion (ICF) applications is described. The ParticleMC package is designed with the long-term goal of transporting neutrons, charged particles, and gamma rays created during the simulation of ICF targets and surrounding materials, although currently the package treats neutrons and gamma rays. Neutrons created during thermonuclear burn provide a source of neutrons to the ParticleMC package. Other user-defined sources of particles are also available. The module is used within the context of a hydrodynamics client code, and the particle tracking is performed on the same computational mesh as used in the broader simulation. The module uses domain-decomposition and the MPI message passing interface to achieve parallel scaling for large numbers of computational cells. The Doppler effects of bulk hydrodynamic motion and the thermal effects due to the high temperatures encountered in ICF plasmas are directly included in the simulation. Numerical results for a three-dimensional benchmark test problem are presented in 3D XYZ geometry as a verification of the basic transport capability. In the full paper, additional numerical results including a prototype ICF simulation will be presented.
Date: November 6, 2006
Creator: Brantley, P S & Stuart, L M
Partner: UNT Libraries Government Documents Department

Impact of Spherical Inclusion Mean Chord Length and Radius Distribution on Three-Dimensional Binary Stochastic Medium Particle Transport

Description: We describe a parallel benchmark procedure and numerical results for a three-dimensional binary stochastic medium particle transport benchmark problem. The binary stochastic medium is composed of optically thick spherical inclusions distributed in an optically thin background matrix material. We investigate three sphere mean chord lengths, three distributions for the sphere radii (constant, uniform, and exponential), and six sphere volume fractions ranging from 0.05 to 0.3. For each sampled independent material realization, we solve the associated transport problem using the Mercury Monte Carlo particle transport code. We compare the ensemble-averaged benchmark fiducial tallies of reflection from and transmission through the spatial domain as well as absorption in the spherical inclusion and background matrix materials. For the parameter values investigated, we find a significant dependence of the ensemble-averaged fiducial tallies on both sphere mean chord length and sphere volume fraction, with the most dramatic variation occurring for the transmission through the spatial domain. We find a weaker dependence of most benchmark tally quantities on the distribution describing the sphere radii, provided the sphere mean chord length used is the same in the different distributions. The exponential distribution produces larger differences from the constant distribution than the uniform distribution produces. The transmission through the spatial domain does exhibit a significant variation when an exponential radius distribution is used.
Date: March 2, 2011
Creator: Brantley, P S & Martos, J N
Partner: UNT Libraries Government Documents Department

A Hybrid Monte Carlo-Deterministic Method for Global Binary Stochastic Medium Transport Problems

Description: Global deep-penetration transport problems are difficult to solve using traditional Monte Carlo techniques. In these problems, the scalar flux distribution is desired at all points in the spatial domain (global nature), and the scalar flux typically drops by several orders of magnitude across the problem (deep-penetration nature). As a result, few particle histories may reach certain regions of the domain, producing a relatively large variance in tallies in those regions. Implicit capture (also known as survival biasing or absorption suppression) can be used to increase the efficiency of the Monte Carlo transport algorithm to some degree. A hybrid Monte Carlo-deterministic technique has previously been developed by Cooper and Larsen to reduce variance in global problems by distributing particles more evenly throughout the spatial domain. This hybrid method uses an approximate deterministic estimate of the forward scalar flux distribution to automatically generate weight windows for the Monte Carlo transport simulation, avoiding the necessity for the code user to specify the weight window parameters. In a binary stochastic medium, the material properties at a given spatial location are known only statistically. The most common approach to solving particle transport problems involving binary stochastic media is to use the atomic mix (AM) approximation in which the transport problem is solved using ensemble-averaged material properties. The most ubiquitous deterministic model developed specifically for solving binary stochastic media transport problems is the Levermore-Pomraning (L-P) model. Zimmerman and Adams proposed a Monte Carlo algorithm (Algorithm A) that solves the Levermore-Pomraning equations and another Monte Carlo algorithm (Algorithm B) that is more accurate as a result of improved local material realization modeling. Recent benchmark studies have shown that Algorithm B is often significantly more accurate than Algorithm A (and therefore the L-P model) for deep penetration problems such as examined in this paper. In this research, we ...
Date: March 4, 2010
Creator: Keady, K P & Brantley, P
Partner: UNT Libraries Government Documents Department

Errors associated with standard nodal diffusion methods as applied to mixed oxide fuel problems

Description: The evaluation of the disposition of plutonium using light water reactors is receiving increased attention. However, mixed-oxide (MOX) fuel assemblies possess much higher absorption and fission cross- sections when compared to standard UO2 assemblies. Those properties yield very high thermal flux gradients at the interfaces between MOX and UO2 assemblies. It has already been reported that standard flux reconstruction methods (that recover the homogeneous intranodal flux shape using the converged nodal solution) yield large errors in the presence of MOX assemblies. In an accompanying paper, we compare diffusion and simplified PN calculations of a mixed-oxide benchmark problem to a reference transport calculation. In this paper, we examine the errors associated with standard nodal diffusion methods when applied to the same benchmark problem. Our results show that a large portion of the error is associated with the quadratic leakage approximation (QLA) that is commonly used in the standard nodal codes.
Date: July 24, 1998
Creator: Brantley, P. S., LLNL
Partner: UNT Libraries Government Documents Department

Spatially Continuous Mixed Simplified P2-P1 Solutions for Multidimensional Geometries

Description: The simplified P{sub N} (SP{sub N}) angular approximation to the neutron transport equation has received renewed interest in recent years [1-3]. The even-order simplified P{sub 2} angular approximation has been shown to yield improved accuracy over the P{sub 1} (diffusion) angular approximation for problems in which the P{sub 1} approximation is reasonably accurate [1]. In particular, integral quantities such as reaction rates and eigenvalues computed by the SP{sub 2} approximation are typically more accurate. The SP{sub 2} scalar flux spatial distributions are also often more accurate than the P{sub 1} distributions away from material interfaces and source discontinuities. However, the SP{sub 2} scalar flux distributions are spatially discontinuous at material interfaces and source discontinuities. These spatial discontinuities are unphysical, aesthetically displeasing, and can lead to large inaccuracies near interfaces. Brantley [4] has recently investigated combining the P{sub 1} and P{sub 2} angular approximations in planar geometry to eliminate the discontinuities in the P{sub 2} scalar flux distributions near material interfaces and source discontinuities. In essence, the P{sub 1} angular approximation is used at material interfaces and source discontinuities to obtain spatially continuous scalar flux distributions; the P{sub 2} angular approximation is utilized elsewhere to take advantage of its improved accuracy. The combined angular approximation smoothly transitions between the P{sub 1} and P{sub 2} approximations. In this paper, we extend this planar geometry mixed P{sub 2}-P{sub 1} angular approximation approach to the multidimensional SP{sub 2} angular approximation, i.e. we develop a spatially continuous mixed SP{sub 2}-P{sub 1} angular approximation. We present numerical results from a two-dimensional test problem to demonstrate the accuracy of this spatially-continuous mixed SP{sub 2}-P{sub 1} angular approximation.
Date: January 14, 2005
Creator: Brantley, P S & Tomasevic, D I
Partner: UNT Libraries Government Documents Department

Artificial Neural Network Solutions of Slab-Geometry Neutron Diffusion Problems

Description: Artificial neural network (ANN) methods have been researched extensively within the nuclear community for applications in systems control, diagnostics, and signal processing. We consider here the use of multilayer perceptron ANNs as an alternative to finite-difference and finite-element methods for obtaining solutions to neutron diffusion problems. This work is based on a method proposed by van Milligen et. al. to obtain solutions of the differential equations arising in plasma physics applications. This ANN method has the potential advantage of yielding an accurate, differentiable approximation to the solution of diffusion problems at all points in the spatial domain.
Date: June 12, 2000
Creator: Brantley, P. S.
Partner: UNT Libraries Government Documents Department

MCAPM-C Generator and Collision Routine Documentation

Description: This report documents the C version of the MCAPM (Monte Carlo All Particle Method) generator and collision physics library of subroutines. The Monte Carlo data generator (gen2000) reads cross sections and distributions that describe in-flight reactions from a binary library and creates an internal data library. The collision routines (bang2000) use this internal library to perform the physics of interaction of particles with the background material. Particles modeled with MCAPM-C are neutrons, charged particles (p, d, t, {sup 3}He, and {alpha}), and {gamma}'s. MCAPM-C is written in (nearly) standard C, and versions exist for Sun Solaris, Compaq Unix, IBM AIX, SGI Irix, and Linux. The library and its data files are installed on LC's Compass, TC, Forest, Blue, and Sky machines. This report describes the contents and format of the library, physics assumptions made, and the interface to the library's subroutines.
Date: November 22, 2000
Creator: Brantley, P S; Hagmann, C A & Rathkopf, J A
Partner: UNT Libraries Government Documents Department

Computational Challenges in Nuclear Weapons Simulation

Description: After a decade of experience, the Stockpile Stewardship Program continues to ensure the safety, security and reliability of the nation's nuclear weapons. The Advanced Simulation and Computing (ASCI) program was established to provide leading edge, high-end simulation capabilities needed to meet the program's assessment and certification requirements. The great challenge of this program lies in developing the tools and resources necessary for the complex, highly coupled, multi-physics calculations required to simulate nuclear weapons. This paper describes the hardware and software environment we have applied to fulfill our nuclear weapons responsibilities. It also presents the characteristics of our algorithms and codes, especially as they relate to supercomputing resource capabilities and requirements. It then addresses impediments to the development and application of nuclear weapon simulation software and hardware and concludes with a summary of observations and recommendations on an approach for working with industry and government agencies to address these impediments.
Date: August 29, 2003
Creator: McMillain, C F; Adams, T F; McCoy, M G; Christensen, R B; Pudliner, B S; Zika, M R et al.
Partner: UNT Libraries Government Documents Department