Parallel Block Jacobi
(
PBJ
) 1 is an asynchronous spatial domain decomposition with application in solving the neutron transport equation due to its extendibility to massively parallel solution in ...unstructured spatial meshes (grids) without the use of the computationally complex and expensive sweeps required by the
Source Iteration
(
SI
) method in these applications. 2 However,
PBJ
iterative methods suffer a lack of iterative robustness in problems with optically thin cells, 1 which we have previously demonstrated to be a consequence of
PBJ
’s asynchronicity. To mitigate this effect, we have developed multiple
PBJ
/
SI
hybrid methods which employ a
PBJ
method (
Parallel Block Jacobi - Integral Transport Matrix Method
(
PBJ-ITMM
) or
Inexact Parallel Block Jacobi
(
IPBJ
)) along with
SI
. 3,4 In this work, we perform a parametric study to determine performance of numerous
PBJ
/
SI
hybrid methods as a function of multiple problem parameters. This parametric study reached 5 main conclusions: 1) our hybrid approach is more effective with
PBJ-ITMM
than with
IPBJ
, 2) for
PBJ-ITMM
, there is a hybrid method that mitigates the aforementioned iterative slowdown in optically thin cells without diminishing the method’s potential parallelism in unstructured grids, 3) this hybrid method is most effective in problems with large, continuous regions of very thin cells, 4) the best performing hybrid method consistently executes within a factor of ten slower than current state-of-the-art acceleration methods that are not efficiently extendable to the massively parallel regime, and 5) both
PBJ
-
ITMM
and
IPBJ
are observed to be viable approaches for our desired applications. In the pursuit of implementing
PBJ-ITMM
in unstructured grids, we conclude with a description of the
Green’s Function ITMM Construction
(
GFIC
) algorithm, which allows for the
ITMM
matrices to be constructed using the pre-existing
SI
sweep algorithm already present in unstructured grid
SN
transport codes.
The fuel in high-temperature reactors (HTRs) is made up of tiny fuel particles (called TRISO particles) dispersed randomly in a graphite matrix. For the commonly used fuels, namely, enriched uranium ...or low-enriched uranium/
233
U mixed with Th, this dispersion may be treated as a homogeneous mixture at almost all neutron energies except at the resonances of the major nuclide (
238
U or
232
Th) where the macroscopic cross section is so large that the neutron mean free path is comparable to the fuel kernel radius. In the equivalence theory approach used in many codes based on the WIMS-D library, this heterogeneity is accounted for by the escape cross section from a fuel kernel together with a carefully evaluated Dancoff correction factor.
In HTRs designed to burn plutonium, the fuel kernel contains oxides of various isotopes of Pu, and there is no fertile nuclide. Hence, the macroscopic cross section of the fuel is rather high in the thermal region, i.e., below 4 eV in the WIMS-D library (which also includes the two major resonances of
240
Pu and
239
Pu). For such fuels, the treatment of neutron transport in the thermal energy range must also include the random heterogeneous distribution of fuel in the graphite matrix. Since the resonance treatment method in the WIMS-D library, described above, is neither available nor applicable for thermal energies, a different method is clearly necessary.
HTR lattice codes such as WIMS and DRAGON use approximate collision probabilities between the various microscopic regions, namely, the fuel kernel, the coating layers, and the graphite matrix in the solution of the multigroup transport equation to account for this heterogeneity. In an earlier paper, we developed a simple though heuristic approach for solving this problem. In the present paper, we develop a more systematic theoretical method for solving the transport equation in a random dispersion by an exact evaluation of the collision probabilities in various regions of the lattice cell. The method has been incorporated in the collision probability code BOXER3. Comparison with benchmarks of Pu-based HTR fuels shows very good agreement thus validating the new approach. The method is likely to have application beyond the problem discussed in the paper.
The fuel in high-temperature reactors (HTRs) consists of a large number of tiny (a few hundred microns in size) TRISO-coated particles dispersed randomly in a graphite matrix. At the resonances of ...the major nuclide (
238
U or
232
Th) of the fuel, the neutron mean free path in the fuel is often comparable to the kernel dimensions, and hence the dispersion of particles in the graphite matrix must be treated as a heterogeneous medium for obtaining self-shielded cross sections in the resonance (epithermal) groups. HTRs containing only plutonium as fuel (as may be the case in reactors designed for burning plutonium) have high concentrations of the isotopes
239
Pu and
240
Pu in the kernels. This fact together with their rather large resonance cross section in the thermal groups results in a very short neutron mean free path in the fuel that is comparable to the kernel dimensions. In such cases the fuel zone must be treated as a heterogeneous medium in the thermal energy region as well. However, the resonance treatment method in libraries such as the WIMS library does not cover the two large resonances of plutonium lying in the thermal region. Instead, a large number of groups are used to cover the details of cross-section variation in this region.
The heterogeneity of the fuel region together with the heterogeneous distribution of fuel region, graphite moderator, and coolant is referred to as the double heterogeneity of HTRs. The paper describes work we carried out for addressing these problems. A new method for generating a random distribution of TRISO particles in the fuel zone of a pebble or fuel compact and Monte Carlo calculation of the Dancoff factor in the heterogeneous random medium, required for calculating the self-shielded resonance group cross sections, is presented. Dancoff factors obtained by our method are compared with values available in published literature on the subject. The paper also discusses a new methodology developed to solve the double-heterogeneity problem at the stage of the multigroup transport theory solution, which is particularly important in the thermal region occurring in high-content Pu-fueled HTRs. These features have been incorporated in the WIMS library-based lattice code BOXER3. An option for handling the spherical geometry of the lattice cell of a pebble bed reactor has been added in the code. Results of analysis of a number of HTR lattice cell benchmark problems are presented.
A comprehensive investigation of the Ronen method is performed in homogeneous and heterogeneous slab problems from the Sood benchmark, considering isotropic and linearly-anisotropic problems. Three ...finite differences implementations are exercised and compared. The results are compared to reference solutions using one and two energy groups. The validation is performed for the criticality eigenvalue and the fundamental neutron flux distribution. The results demonstrate the significantly improved accuracy achievable by the Ronen method using a broad set of problems. For standard convergence tolerances, the maximal deviation in criticality eigenvalue is less than ten pcm, and the maximal deviation in the spatial distribution of the flux is less than 2%, always located near sharp interfaces or vacuum boundaries.
A new method for calculating time-dependent integral neutron transport single collision fluxes has been developed for one-dimensional infinite media problems using dimensionless variables and the ...reduced collision formulation. By transforming the integral scalar flux equation into dimensionless variables, one may immediately integrate over the time domain and reduce the problem complexity to integrations strictly over the spatial domain, significantly reducing the computational work required to obtain a solution. The results have been compared to published benchmark solutions for validation. Further, reformulating the scalar flux as a Green’s function has allowed for the determination of an analytical expression for the relative error of any collision level scalar flux to the summed scalar flux. This provides the absolute number of collisions required for convergence at any given mean free time, using previous work in one-dimensional Cartesian transport and derived work herein for the one-dimensional spherical transport.
•A Dimensionless Variable Time-Dependent Integral Transport method was developed.•Benchmark type results were generated for one-dimensional slab geometry.•Benchmark type results were generated for ...one-dimensional spherical geometry.•A method for estimating the number of collisions for convergence was developed.
A new method for calculating time-dependent integral neutron transport single collision fluxes has been developed for one-dimensional infinite media problems using dimensionless variables and the reduced collision formulation. By transforming the integral scalar flux equation into dimensionless variables, one may immediately integrate over the time domain and reduce the problem complexity to integrations strictly over the spatial domain, significantly reducing the computational work required to obtain a solution. The results have been compared to published benchmark solutions for validation. Further, reformulating the scalar flux as a Green’s function has allowed for the determination of an analytical expression for the relative error of any collision level scalar flux to the summed scalar flux. This provides the absolute number of collisions required for convergence at any given mean free time, using previous work in one-dimensional Cartesian transport and derived work herein for the one-dimensional spherical transport.
The mutual information, a quantity measuring the dependency of two random variables, is studied in the context of the fission source of neutron transport calculations. Both Monte Carlo and integral ...transport methods are developed to compute the mutual information in 1-D and 2-D Cartesian geometry. The mutual information of the fission source may be connected to the correlation of fission sources of different iterations within a Monte Carlo calculation and the resulting underestimation of tally uncertainties. Results suggest that a reasonable global correction of tally uncertainties may be obtained. The calculations demonstrate the Monte Carlo estimate of the mutual information has a positive bias that is proportional to the square of the number of discrete regions divided by the number of active histories, agreeing with theoretical behavior. This asymptotic form may be used to get a reasonable estimate of the mutual information of the fission source in Monte Carlo calculations.
•Methods for calculating the mutual information of the fission source with Monte Carlo and integral transport are developed.•A connection of the mutual information to the correlation of tallies in a Monte Carlo eigenvalue calculation is shown.•An asymptotic form of the bias of the mutual information is shown to give a reasonable estimate.
The BOXER3 code was developed in the Bhabha Atomic Research Centre during the 1980s as a three-dimensional code for the analysis of a pressurized heavy water reactor supercell containing fuel, ...moderator, and a reactivity device inserted perpendicular to the fuel channel, with options for carrying out calculations in a general two-dimensional geometry (infinite and homogeneous in one direction) and a one-dimensional plane geometry. Taking into account the computing resources available then, the code was run in few groups after obtaining condensed group cross sections for various materials from a one-dimensional multigroup calculation.
In this paper, we describe various developments carried out recently for enabling its use as an assembly-level lattice-burnup code. In addition to the collision probability method originally available, the method of characteristics for solving the multigroup transport equation has been added. This development permits the treatment of anisotropic scattering wherever necessary and available in cross-section libraries. Other developments include coupling of the code to the WIMS 69/172-group library, a method for the evaluation of the pin-dependent Dancoff factor, and the introduction of burnup. The transport equation in the collision probability method is cast in a form more suitable for iterations as well as for the method of renormalization of collision probabilities used in the work. The analysis of several benchmark problems has been carried out and the results obtained using the new code are presented.
•Development of whole core code TRANPIN.•Use of interface current method based on 2D collision probability method.•Using double P2 expansion of angular flux in hexagonal geometry.•Analysis of OECD ...VVER-1000 MOX Core Computational Benchmark.•Using ultra fine 172 group WIMS Library for core calculation.
The advances in computer processing power have made it possible to perform a detailed pin-by-pin calculation of the whole reactor core. A transport theory code TRANPIN has been developed to perform a whole reactor core pin-by-pin calculation in 2D hexagonal geometry. This code employs the interface current method based on 2D collision probability for reactor core calculation. The present code divides each lattice cell location in the fuel assembly (FA) into finer regions. The subdivided regions inside the lattice cell are connected using the 2D collision probabilities. The coupling of lattice cells within the assembly and assembly to assembly coupling is achieved using interface currents. The interface currents are obtained by expanding the angular flux leaving or entering the lattice cell surface into double orthonormal P2 polynomials. The interface current method is traditionally used to perform lattice calculations and its application to core calculations in hexagonal geometry is not reported in literature to the best knowledge of authors. The present code can perform the calculation in any multigroup energy structure. The code TRANPIN has been validated against two benchmark problems i) a simplified high temperature test reactor (HTTR) benchmark problem and ii) OECD VVER-1000 MOX Core Computational Benchmark. It may be mentioned that the OECD benchmark is analyzed using ultra fine WIMS library in 172 energy groups. This approach may be considered as unprecedented since the full reactor core calculations are normally performed with cross sections prescribed in few energy groups by some prior transport simulations. The present paper describes the interface current methodology incorporated in TRANPIN. The comparison of results of benchmark analyses with published results is also presented.
•One hexagonal assembly is divided into 6 triangular prisms in order to get accurate flux distributions.•Transmission probability method is applied to solve the integral neutron transport ...equation.•The neutron flux and source are expanded spatially by a set of second order orthogonal polynomials.•The neutron flux at the interface is approximated with simplified P1 approximation.
A new project has been started recently at KIT to develop a code able to treat hexagonal-z geometries with low density regions. The mathematical method chosen for that purpose is the Transmission Probability Method (TPM) for solving the integral neutron transport equation. In this model, one hexagonal prism is divided into six or more triangular prisms in order to get accurate flux distributions. Within each triangular prism, the neutron source is assumed to be isotropic, the scalar flux and source being approximated in space with a set of second order orthogonal polynomials. The neutron flux at the interfaces is constant in space and approximated with the simplified P1 approximation in angle. A new code, TPM-HEXZ, based on the described model is developed and some benchmarks are used to verify the code, the results are in good agreement with reference ones.
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