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The three-dimensional, discrete ordinates neutral particle transport code TORT: An overview



The centerpiece of the Discrete Ordinates Oak Ridge System (DOORS), the three-dimensional neutral particle transport code TORT is reviewed. Its most prominent features pertaining to large applications, such as adjustable problem parameters, memory management, and coarse mesh methods, are described. Advanced, state-of-the-art capabilities including acceleration and multiprocessing are summarized here. Future enhancement of existing graphics and visualization tools is briefly presented.
The Three-Dimensional, Discrete Ordinates Neutral Particle
Transport Code TORT: An Overview*
Y. Y. Azmy
Oak Ridge National Laboratory**
Oak Ridge, Tennessee 37831-6363
"The submitted manuscript has been authored by a
contractor of the U. S. Government under contract
DE-AC05-96OR22464. Accordingly, the U.S.
Government retains a nonexclusive, royalty- tree
license to publish or reproduce the publised form of
this contribution, or allow others to do so, for U.S.
Government purposes."
to be presented at the
OECD/NEA meeting on 3D Deterministic Radiation Transport Computer Programs,
Feature, Applications and Perspectives
December 2-3, 1996
* Research sponsored by the U.S. Department of Energy.
*Managed by Lockheed Martin Energy Research Corp. for the U. S. Department of Energy under
Contract DE-AC05-96OR22464.
Y. Y. Azmy
Oak Ridge National Laboratory
Box 2008, MS 6363
Oak Ridge, TN 37831
The centerpiece of the Discrete Ordinates Oak Ridge System (DOORS), the three-dimensional
neutral particle transport code TORT is reviewed. Its most prominent features pertaining to large
applications, such as adjustable problem parameters, memory management, and coarse mesh
methods, are described. Advanced, state-of-the-art capabilities including acceleration and multipro-
cessing are summarized here and detailed in other papers in these Proceedings. Future enhancement
of existing graphics and visualization tools is briefly presented.
The Discrete Ordinates Oak Ridge System (DOORS) is comprised of several computer codes
developed over the years at Oak Ridge National Laboratory to solve a wide variety of neutral parti-
cle transport problems arising in applications. The first release of DOORS 3.1 in 1995 was charac-
terized by a modernized installation procedure and an expanded list of peripheral codes included in
the distribution, as well as other new features.
Bringing the member codes together in DOORS is
the first step in establishing smoother connections between them which will eventually be manipu-
lated via a user friendly Graphical User Interface (GUI).
As the power and capacity of electronic computers expanded over the past three decades the
work horse of the transport calculations evolved in dimensionality from one-, to two-, to three-
dimensional. Time consuming cross section generation is often still performed using one-
dimensional models, and two-dimensional models still suffice for many applications. Hence
DOORS includes codes for these purposes; the focus of this paper, however, is the three-
dimensional code TORT.
TORT was conceived in the mid eighties to compute radiation dose profiles in large multistory
concrete buildings. While largely based on its two-dimensional predecessor, DORT, it quickly
evolved in an individual fashion suited to the difficulties associated with the problems it is applied
primarily size. Thus high order, coarse mesh methods, vectorization of the mesh sweep, discon-
tinuous mesh, a^nong others, are features developed and implemented exclusively for TORT.
Managed by Lockheed Martin Energy Research, Inc. under contract DE-AC05-96OR22464 with the U.S. Department of
This report was prepared as an account of work sponsored by an agency of the
United States Government Neither the United States Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied, or
assumes any legal liability or responsibility for the accuracy, completeness, or use-
fulness of any information, apparatus, product, or process disclosed, or represents
that its use would not infringe privately owned rights. Reference herein to any spe-
cific commercial product, process, or service by trade name, trademark, manufac-
turer, or otherwise does not necessarily constitute or imply its endorsement, recom-
mendation, or favoring by the United States Government or any agency thereof.
The views and opinions of authors expressed herein do not necessarily state or
reflect those of the United States Government or any agency thereof.
Portions of this document may be illegible
in electronic image products. Images are
produced from the best available original
and other unique features of the code made it an invaluable computational tool for large problems
in a wide variety of applications. A high level of confidence in TORT's reliability and computa-
tional efficiency has been established over the years through the experiences of many researchers
and engineers working in different areas.
While TORT is a general purpose transport code it has gained particular notoriety for accu-
rately solving large problems with complex configurations. At present the largest TORT application
employs 3.6 million computational cells, Sj
angular quadrature, P
anisotropic scattering expansion,
and 11 energy groups, a monumental challenge to the computational resources typically available at
research institutions. The code's production attributes, such as robustness, efficiency, accuracy, and
reliability, resulted in its proliferation into many areas of application new to deterministic transport
methodologies. These include medical applications,
charged particle transport,
and time depen-
dent calculations,
in addition to the traditional reactor physics and shielding applications; see for
example Refs. 8,9.
In the remainder of this paper we present an overview of TORT emphasizing its most salient
features that make it particularly suitable for large production level applications. We start with a
brief description of DOORS and its member code, then we discuss adjustable problem parameters
including discontinuous mesh, and the memory management capability. The three spatial approxi-
mations available in TORT are reviewed next, then acceleration schemes for the iterative process
are described, followed by a brief section on multiprocessing with TORT. The last section is dedi-
cated to graphical routines currently available and visualization tools planned for the future.
Discrete Ordinates Oak Ridge System (DOORS)
DOORS is a collection of codes built around the discrete-ordinates codes ANISN, DORT, and
TORT gives the user a wide range of capabilities to pre- and post-process data in transition from
one computation to the next. These codes represent a total investment of many man-years of effort
contributed by many researchers over the past three decades. While most codes included in the
present version of DOORS are in their original condition, they have been packaged and will be
maintained in unisome in the future. DOORS also introduced modern maintenance and installation
procedures, e.g. makefiles, that facilitate updating activities and extend the package's compatibility
into the future. ORNL supports DOORS on UNIX-based platforms, Cray supercomputers, and a
variety of workstations; PC versions of some of the member codes have been reported and are
available from RSICC.
A complete list of the codes comprising the DOORS package at present is shown in Table 1.
Among these perhaps the best known are the one-, and two-dimensional sister codes ANISN, and
DORT, respectively. While TORT has a one-, and two-dimensional solution capability, these codes
are easier to use, and have additional features, e.g. some curvilinear geometry options, which have
been more thoroughly tested over the years. Furthermore, because their lower dimensionality is
hardwired into them, ANISN and DORT utilize computational resources more efficiently and are
more likely to provide faster execution. For this reason they are often used in scoping calculations
that require repetitive solution of slightly differing problem configurations. In this regard the
adjoint capability of all DOORS transport codes sometimes aides the search for optimal
The TORSED and TORSET codes implement splicing and bootstrapping techniques that per-
mit the solution of problems that are too large to solve with TORT in their entirety. Essentially
these methods amount to a split of the problem domain into two, or more, subdomains that are
loosely coupled in only one direction from the source, primary, to the observable, secondary,
configurations, with weak feedback. For details of these techniques and their implementation in
important example which has proven extremely valuable in solving very large problems with sub-
stantial geometric detail is the discontinuous mesh option in TORT. This feature allows the user to
employ more computational cells where fine geometric structure exists, and fewer cells elsewhere.
This is achieved by allowing the user, within some constraints, to adjust the number and boundaries
of individual cells within a row, not necessarily coinciding with those in adjacent rows. The fact
that the solution algorithm in TORT reduces the most general problem to a sequence of sweeps
along a row of computational cells in the x -dimension in a given direction allows each row to be
processed individually then the boundary angular flux incoming to adjacent rows are inter- or extra-
polated to that row's cell structure.
Other adjustments to problem parameters not related to phase space size include, for example,
setting iteration number by group, etc. These typically pertain more to reducing execution time by
avoiding tight convergence of groups that are inconsequential to the purpose of the computation.
Recent research aimed at providing TORT with the capability to account for heterogeneous
material, combinatorial geometry objects by manipulating the spatial weights in the discrete-van able
equations is detailed in Ref. 11.
Memory Management
In spite of the flexibility afforded the user in concentrating the computational effort at regions
of high sensitivity to the level of detail it is customary in scientific research and engineering design
to constantly push the capability limits in order to make progress. Thus the insatiable demand for
finer detail kept pace with the phenomenal growth in memory size and external storage options
available. Virtual memory is a generic solution to this problem available on many platforms that is
designed to conduct the I/O activity at the operating system level thereby relieving the programmer
from the burden of foreseeing and accommodating a wide variety of hardware configurations and
run time conditions. Nevertheless, the price of this flexibility is paid by the user in the form of
long execution times because the generic I/O scheme does not take into account natural breakpoints
in the solution algorithm of transport problems.
In recognition of this fact, and also to accommodate a variety of platforms with a disparate
range of memory sizes TORT attempts to meet a memory objective set by the user at run time.
First TORT tries to fit the entire problem in the memory objective; if this is not possible it attempts
to it one group at a time within the memory objective and I/O flux and cross section data to scratch
files. If this too fails, the code breaks up the geometric configuration into blocks of planes, each of
which can fit within the memory objective, and I/O to scratch files is used to maintain and update
the data during the calculation. If a single plane does not fit in the memory objective TORT tries to
obtain additional memory beyond that specified by the memory objective, assuming it is less than
all that is available on the machine, and if this fails it informs the user of its attempts then ter-
minates execution unsuccessfully. Clearly this sequence of attempts is designed to minimize the
adverse effect of I/O on performance while enabling the solution of ever larger problems.
Spatial Discretization Methods
TORT is^ based on the discrete ordinates approximation of the independent angular variable,
and the multigroup discretization of the energy variable. Three approximation methods for the spa-
tial dependence are available in the code to accommodate a broad spectrum of applications.
The oldest is the 9-weighted method, which was originally implemented in DORT, represents
a whole set of methods parametrized by the single parameter 0 set by the user at run time in the
Table 1. Member Codes in the DOORS Package
Solve one-dimensional transport problems
anisn with group band option for thermal upscatter
Solve two-dimensional transport problems
Solve three-dimensional transport problems
Couple primary to secondary tort calculations
Couple rz-dort to xyz-tort calculations
Prepare torsed input from dort output file
Maintenance of cross section library
Prepare cross section library
Estimate uncollided flux and first collided source
Project last collision source to point detector
Convert internal boundary flux to internal boundary source
Convert flux moment file across formats
Generate contour plot of flux or response
Extract planar slice from tort output to plot with isoplot
Execute sequence of calculations
Driver module called by jdos
Maintenance of code system*
Graphics library*
TORT see Ref. 10.
* Public domain software from Sandia National Laboratory
Adjustable Problem Parameters
Perhaps the factor that most contributes to the difficulty of transport calculations is the large
size of the discrete variable system of equations that must be solved numerically. This is a direct
consequence of the high dimensionality of phase space, a fact that is easily illustrated by consider-
ing a a steady state transport problem in <i-dimensional geometry. In such case the phase space is
of dimension Id: d variables representing physical space, d-\ representing the particles direction
of motion (discrete ordinates), and one energy variable. It is critical, therefore, to conserve discrete
variables as much as possible without jeopardizing the accuracy and reliability of the solution. This
is accomplished in TORT by permitting the user to adjust the level of detail in a variety of problem
parameters according to the anticipated local rate of change in the solution.
For exanfple, sharp flux and cross section anisotropies are more notable at the high energy end
of the spectrum. Hence TORT permits the user to select a high order angular quadrature set and
high order P, expansion of the cross sections in the high energy groups, and lower orders in the
low energy groups thereby reducing the problem size without sacrificing accuracy. Another
range [0,1). This is a weighted diamond difference method spanning the range from the diamond
difference scheme (optically thin cells) to the Step method (optically thick cells) where the weights
are computed to ensure positivity of the outgoing angular flux given positive incoming flux and
volumetric source. Due to the relative simplicity of its equations and its low order approximation,
the 9-weighted method is the least computationally intensive option among TORT's spatial approx-
imation methods. Also, it is the least accurate on a given mesh.
The Linear Nodal (LN) method was installed in TORT to enable using optically thick cells
while retaining high accuracy of the computed angular flux. This is achieved by computing the first
spatial moment of the angular and scalar flux, in addition to the average quantities computed in the
0-weighted method.
Many modifications to the original method have been implemented over
the years in order to improve method accuracy and efficiency, and solution positivity to the extent
that, at present, LN is the recommended method for most large applications with optically thick
regions in TORT.
The Linear Characteristic (LC) method
also computes the first spatial moment of the flux on
each cell's surfaces and volume using the exact characteristic paths from incoming to outgoing sur-
then projecting the resulting expression onto the basis functions (constant and linear). As far
as accuracy is concerned LC is competitive with LN with each method gaining an edge ov£r the
other for some, but not all, problems. LC executes faster than LN on scalar machines, but due to
the high level of vectorization of the LN it is about four times faster on Cray computers.
On a given mesh LN and LC run longer, require more memory, and consume larger disk
space but provide more accurate solutions than the 6-weighted method. However, for a fixed accu-
racy the latter method typically requires eight times as many computational cells than either of the
linear schemes. Ultimately for the same solution accuracy requirement, the linear methods end up
utilizing less computational resources, i.e. CPU time, memory, and disk space, than the 9-weighted
Iteration Acceleration Methods
The recommended method for accelerating the iterative convergence of the inner iterations in
TORT is the Partial Current Rebalance (PCR) method.
This method is based on reinforcing the
balance of neutrons over each computational cell using the cell-surface partial currents resulting
from the latest mesh sweep. The discrete variable equation resulting from PCR has the same cell-
coupling stencil as a discretized cell-centered diffusion equation but does not necessarily possess
some of its important features like diagonal dominance, etc. The PCR matrix equation is solved via
a Successive Over-Relaxation (SOR) scheme with the relaxation factor computed numerically from
the SOR iterates.
For most TORT applications PCR has proven robust and efficient. However, recent advances
in the analysis of the spectral properties of iterative procedures for solving the transport equation
have provided the basis for powerful acceleration operators that, at least theoretically, exceed the
performance of PCR. The most notable example of such new methods is Diffusion Synthetic
Acceleration (DSA) whose spectral radius is bounded from above by 0.25 for model problem
configurations, i.e. homogeneous material composition and uniform mesh. The most serious limita-
tion of the class of unconditionally stable DSA operators as far as large three dimensional applica-
tions is concerned is that they are edge-centered. Since there are many more surfaces than compu-
tational cells the DSA matrix equation can be prohibitively large; this difficulty is compounded
further in high order methods, i.e. LN and LC, if the first spatial moments of the flux are to be
accelerated also.
A more general framework for acceleration schemes, the Adjacent-cell Preconditioning
method, has been implemented recently in TORT and is reviewed in Ref. 15. AP is cell-centered
and has the same coupling scheme as a discretized cell-centered diffusion equation but its elements
are not based on the diffusive properties of a computational cell. Rather, the preconditioner ele-
ments are set to provide a vanishing spectral radius of the flat eigenmode of the homogeneous
model problem, with reciprocal averaging across material heterogeneity interfaces. Therefore, the
preconditioning stage of the iterative process is comprised of a system of discrete-variable equations
that is amenable to solution via the same SOR routines in TORT used to solve the PCR equations.
Testing of the AP in TORT demonstrated its effectiveness in reducing the number of iterations
required to achieve convergence for all members of the Burre Suite of Test Problems (BSTeP) cov-
ering a wide range in parameter space. In fact AP converged in fewer iterations than PCR or
TWODANT's DSA for the vast majority of cases comprising BSTeP. However, these tests illus-
trated the deterioration in spectral properties in cases with sharp material discontinuity contradicting
behavior predicted by the homogeneous model problem analysis. Furthermore, this undesirable
behavior seems to be commonly shared with other acceleration schemes including PCR and DSA.
Analysis of a model problem with material discontinuity, the Periodic Horizontal Interface (PHI),
has been analyzed to verify and investigate this phenomenon.
A multitasking capability at the macrotasking level is available in TORT for execution on
multiprocessor Cray computers running the UNICOS operating system. It is based on a coarse-
grained angular domain decomposition which typically produces good parallel efficiency due to the
relatively large computation load to parallelization overhead ratio. Since angular domain decompo-
sition in Cartesian geometry is intrinsic there is a one-to-one correspondence (within arithmetic pre-
cision) between the sequential and parallel intermediate and final results, so that the number of
iterations required to achieve convergence is independent of the number of concurrent processes.
The multitasking option, selected by the user at run time, has been tested on Cray Y/MP, C90, and
J90 systems and have exhibited significant wall clock speedup factors even for modestly large prob-
The problem with quantifying parallel performance on time-shared computers such as the
Cray is its sensitivity to machine loading during execution. Nevertheless it was possible to quantify
the parallelization overhead for two test problems and conclude the necessity to reduce it sharply.
Recently, a performance model was constructed and validated ibr the parallelization overhead
as a function of the number of participating tasks and other problem parameters. Parametric studies
with this model identified the major contributors to parallelization penalty, and efforts were made to
reduce their effect on parallel speedup.
As a result the present multitasking algorithm incurs only
of the parallelization penalty previously measured, and improvement in wall clock speedup
of up to 50% has been observed in some cases.
Graphics and Visualization
Earlier development of graphics capabilities for DORT utilized a commercial library to pro-
vide elementary graphics constructs. More recently ORNL has opted for public-domain-based
development (?f our neutronic codes graphics capabilities in order to facilitate distribution of the
package in a self-contained form for the users' convenience. For this purpose we adopted Sandia
National Laboratory's RSCORS graphics library which is distributed with future releases of
see Table 1.
The user
generate plots
activities from formatted
unformatted files
by the
code using
ISPL3D codes.
dimensional plots
are gen-
selected planes
in the
TORT geometry,
and can be
viewed on-screen
for a
or in
hardcopy. Several options
plotted quantity: color
grayscale shading, symbol
line contours.
geometric configuration
of the
selected plane
can be
on the
Another ORNL graphics capability that
originally developed
for use
torial geometry models
of the
MASH shielding code
is the Oak
Ridge Geometry Analysis
Modeling Interface (ORIGAMI). ORIGAMI
XWindows based
and has
been developed
on IBM
primary function verification
computed results. ORIGAMI
is not
in the
present release
in the
future. Finally, users with specific interfacing needs
develop their
own pre- and
The seemingly endless increase
computational power,
storage size,
caused applications
size resulting
extremely large data sets that
becoming harder
to manage.
for an
interactive Graphical User Interface
(GUI) is
and we
begin constructing
one for
DOORS soon.
W. A.
Azmy, "Three-dimensional
Calculations with
the Oak
Ridge TORT Code," Proc.
Int. Conf. on
Computations, Reactor Physics,
and Environmental Analyses Portland, Oregon, April
30 - May 4, 1995, Vol. 1, p. 480,
ican Nuclear Society, LaGrange Park,
Y. Y.
Azmy, "Recent Advances
Neutral Particle Transport Methods
in Proc. Fifth
Int. Conf. on
Nuclear Techniques, Crete, Greece, June
W. A.
and D. B.
TORT Three-Dimensional Discrete Ordinates
Neutron/Photon Transport Code," ORNL/TM-13221,
to be
D. W.
et. al.,
three-dimensional deterministic radiation transport
theory dose distribution analysis
boron neutron capture therapy," Medical Physics,
18, p.
R. A.
Lillie, "BNCT Filter Optimization Using
Two- and
Three-Dimensional Ordinates," these
6. Clifton
Drumm, "Multidimensional electron-photon transport with standard discrete ordi-
nates codes," Proc.
Mtg. on
Radiation Protection
April 21-25,
Sedat Goluoglu
and Lee
Deterministic Method
Transient, Three-Dimensional
Neutron Transport,"
8. A. J. J. Bos, J. E. Hoogenboom, T. M. John, P. F. A. de Leege, "Application of TORT to a
Complicated N-Gamma Shielding Problem," these proceedings.
9. E. Botta, J. R. Galando, P. Neuhold, G. Saiu, "Three Dimensional Reactor Pressure Vessel
Fast Neutron Fluence Calculations for the AP600 using TORT," these proceedings.
D. B. Simpson, "Splicing and Bootstrapping Methods for Coupling Primary DORT/TORT
Models to Secondary TORT Models," these proceedings.
T. J. Burns, "A Hybrid Technique for 3-Dimensional Discrete Ordinates Analysis of Combina-
torial Geometry Models," these proceedings.
W. F. Walters, "Augmented weighted-diamond form of the linear-nodal scheme for Cartesian
coordinate systems," Nuclear Science and Engineering, 92, pp. 192-196, 1986.
R. L. Childs and W. A. Rhoades, "Theoretical basis of the linear nodal and linear characteris-
tic methods in the TORT computer code," ORNL/TM-12246, 1993.
R. L. Childs, "Numerical methods for the flux solution," Workshop for the DORT and TORT
Radiation Transport Codes, No. Falmouth, MA, April 21, 1996.
Y. Y. Azmy, "Analysis and Performance of Adjacent-Cell Preconditioners for Accelerating
Multidimensional Transport Calculations," these proceedings.
Y. Y. Azmy, "Recent Improvements in the Performance of the Multitasked TORT on Time-
shared Cray Computers," these proceedings.
C. O. Slater, "The XTORID and ISPL3D codes for plotting TORT activities," ORNL Internal
March 25, 1996.
J. E. Hoogeriboom, T. M. John, A. Hersman, P. F. A. de Leege, "Pre- and Post Processing of
TORT Data and Preliminary Experience with TORT Version 3," these proceedings.
... Reactor dynamics codes like DYN3D and PARCS were extended by implementing a higher order solution of the neutron transport equations than the diffusion approximation (Beckert and Grundmann, 2008a,b;Grundmann, 2009;Downar et al., 2006), e.g. the transport approximation SP 3 (Brantley and Larsen, 2000). On the other hand, discrete ordinates codes such as TORT-TD (Azmy, 1996;Seubert, 2011) are being improved for coupling with thermal-hydraulics codes (CFD, sub-channel and system codes). ...
... In this document we report our verification of this hypothesis in a simplified two-dimensional setting using the renowned neutral particle transport code DORT, [2] and companion codes in the Discrete Ordinates Oak Ridge System (DOORS) package. [3] The conclusion of this preliminary investigation is that realistic models of the workbench and tools can indeed result in a 40% reduction in the dose rate, thus justifying further, more comprehensive modeling of individual workers and their surroundings. ...
... If inadequacies are observed, the user must then determine their impact on the quantities of interest, and if necessary must repeat the solution process with modified settings in pursuit of an improved solution or code performance. In addition, peripheral codes accompanying TORT in the DOORS package supplement this capability by affording the user means to break up the problem into pieces that can be solved in stages feeding into one another (Azmy, 1996). ...
Design of the Target Service Cell (TSC) of the Spallation Neutron Source (SNS) requires computing detailed profiles of the gamma-ray dose rate outside its structure in order to abide by exposure limits and plan access control for the facility's personnel. Three-dimensional spatial distributions of the dose rate inside the TSC are also necessary to optimize the locations of electronic instruments and verify their design criteria at selected sites. For these reasons, in addition to the deep penetration feature typical of shielding calculations, a deterministic transport method solution, namely discrete ordinates, is preferred over Monte Carlo. Even then, the computation is complicated by the large size of the structure, large volume of air (internal void), optical thickness of the enclosing walls, and multiplicity of radiation sources. Furthermore, severe ray effects observed in preliminary calculations throughout the TSC internal cavity that persist in the transport through the concrete walls require special treatment. The computational model for conducting this complex calculation using Oak Ridge National Laboratory's TORT code, with support from peripheral codes in the Discrete Ordinates Oak Ridge System (DOORS), is presented. Successful elimination of primary ray effects via the newly developed three-dimensional uncollided flux and first collision source code GRTUNCL3D is also illustrated.
Full-text available
An overview of Oak Ridge National Laboratory's (ORNL) 3D neural particle transport code, TORT, is presented. Special features of the code that make it invaluable for large applications are summarized for the prospective user. Advanced capabilities currently under development and installation in the production release of TORT are discussed in some detail. These include: multitasking on Cray platforms running the UNICOS operating system; adjacent-cell preconditioning acceleration scheme; and graphics codes for displaying computed quantities such as the flux. Further developments for TORT and its companion codes to enhance its present capabilities, as well as expand its range of applications will be discussed. Speculation on the next generation of neutral particle transport codes at ORNL, especially regarding unstructured grids and high order spatial approximations, will also be mentioned.
The equations of the high-order linear-nodal numerical scheme are cast in an augmented weighted-difference form for three-dimensional Cartesian nodes. The coupling exhibited by these equations indicates that this new algorithm is simpler and, hence, faster than previous nodal schemes of this degree of accuracy. A well-logging problem and a fast reactor problem are examined. The new scheme developed is compared with the classical linear-linear nodal scheme and the diamond-difference scheme. For the well-logging problem, it is found that the new scheme is both faster and simpler than the classical linear-linear nodal scheme while sacrificing little in accuracy. Even though the new scheme is more accurate than the diamond-difference scheme for the reactor problem, the results indicate that state-of-the-art acceleration methods are needed for nodal schemes.
A deterministic method for solving the time-dependent, three-dimensional Boltzmann transport equation with explicit representation of delayed neutrons has been developed and evaluated. The methodology used in this study for the time variable is the improved quasi-static (IQS) method. The position, energy, and angle variables of the neutron flux are computed using the three-dimensional (3-D) discrete ordinates code TORT. The resulting time-dependent, 3-D code is called TDTORT. The flux shape calculated by TORT is used to compute the point kinetics parameters (e.g., reactivity, generation time, etc.). The amplitude function is calculated by solving the point kinetics equations using LSODE (Livermore Solver of Ordinary differential Equations). Several transient 1-D, 2-D, and 3-D benchmark problems are used to verify TDTORT. The results show that methodology and code developed in this work have sufficient accuracy and speed to serve as a benchmarking tool for other less accurate models and codes. More importantly, a new computational tool based on transport theory now exists for analyzing the dynamic behavior of complex neutronic systems.
TORT calculates the flux or fluence of neutrons and/or photons throughout three-dimensional systems due to particles incident upon the system`s external boundaries, due to fixed internal sources, or due to sources generated by interaction with the system materials. The transport process is represented by the Boltzman transport equation. The method of discrete ordinates is used to treat the directional variable, and a multigroup formulation treats the energy dependence. Anisotropic scattering is treated using a Legendre expansion. Various methods are used to treat spatial dependence, including nodal and characteristic procedures that have been especially adapted to resist numerical distortion. A method of body overlay assists in material zone specification, or the specification can be generated by an external code supplied by the user. Several special features are designed to concentrate machine resources where they are most needed. The directional quadrature and Legendre expansion can vary with energy group. A discontinuous mesh capability has been shown to reduce the size of large problems by a factor of roughly three in some cases. The emphasis in this code is a robust, adaptable application of time-tested methods, together with a few well-tested extensions.
Coarse-grained angular domain decomposition of the mesh sweep algorithm has been implemented in ORNL`s three dimensional transport code TORT for Cray`s macrotasking environment on platforms running the UNICOS operating system. A performance model constructed earlier is reviewed and its main result, namely the identification of the sources of parallelization overhead, is used to motivate the present work. The sources of overhead treated here are: redundant operations in the angular loop across participating tasks; repetitive task creation; lock utilization to prevent overwriting the flux moment arrays accumulated by the participating tasks. Substantial reduction in the parallelization overhead is demonstrated via sample runs with fixed tunning, i.e. zero CPU hold time. Up to 50% improvement in the wall clock speedup over the previous implementation with autotunning is observed in some test problems.
The formal development of the Adjacent-cell Preconditioner (AP) and its implementation in the TORT code are briefly reviewed. Based on earlier experience with diffusion type acceleration, and excellent results in slab geometry the reciprocal averaging formula is used to mix the preconditioner elements across material and mesh discontinuities. Numerical testing of the method employing the Burre Suite of Test Problems (BSTeP), a collection of 144 cases covering a wide range in parameter space, using AP, Partial Current Rebalance (PCR), and TWODANT`s Diffusion Synthetic Acceleration (DSA) is presented. While AP outperforms the other two methods for the majority of the cases included in BSTeP it consumes many more iterations than can be explained by spectral analysis of the homogeneous model problem in cases with sharp material discontinuity. In order to verify this undesirable behavior and explore potential remedies a model problem, the Periodic Horizontal Interface (PHI), is developed that permits discontinuity of nuclear properties and cell height across the interface. Fourier mode decomposition is applied to AP with the reciprocal averaging mixing formula for the PHI configuration and shown to possess a spectral radius that approaches unity as the material discontinuity gets larger. The question of whether an unconditionally stable AP exists for PHI is tackled and preliminary indications are negative. Novel preconditioners that have nontraditional cell-coupling schemes that remain stable in these regimes may have to be sought.
TORT has been in service more than 10 years now. Although the original version was developed for the single purpose of calculating the penetration of radiation into large concrete buildings, more general versions have achieved widespread use and acceptance in many other applications. This paper discusses current features and capabilities of TORT and related peripheral codes.
Novel numerical procedures for solving the Boltzmann equation have been added to the Three Dimensional Oak Ridge Discrete Ordinates Transport Code (TORT). These procedures produce much more accuracy in theflux solutions for a given mesh size, or allow a smaller mesh to be used in order to reduce costs. The first method is a special adaptation of the linear nodal method proposed by Walters and O'Dell. The basic method has been extensively adapted in order to avoid numerical distortions that may occur in shielding problems. The second method is a characteristic procedure with linear expansion of sources and boundary flows. These methods are in widespread use in the TORT code.
A method is described for generating electron cross sections that are compatible with standard discrete ordinates codes without modification. There are many advantages to using an established discrete ordinates solver, e.g., immediately available adjoint capability. Coupled electron-photon transport capability is needed for many applications, including the modeling of the response of electronics components to space and synthetic radiation environments. The cross sections have been successfully used in the DORT, TWODANT, and TORT discrete ordinates codes. The cross sections are shown to provide accurate and efficient solutions to certain multidimensional electron-photon transport problems. The key to the method is a simultaneous solution of the continuous-slowing-down and elastic-scattering portions of the scattering source by the Goudsmit-Saunderson theory. The resulting multigroup-Legendre cross sections are much smaller than the true scattering cross sections that they represent. Under certain conditions, the cross sections are guaranteed positive and converge with a low-order Legendre expansion.
Conference Paper
This research develops an improved methodology (and corresponding code) for solving the time-dependent, 3-d Boltzmann Transport Equation with explicit representation of delayed neutrons. These improvements are incorporated in a modified version of the code TDKENO, entitled TDKENO-M. Specifically, these improvements are: (1) incorporate the improved quasistatic methodology into an existing quasistatic framework, specifically, include the flux shape derivative in the fixed source term instead of being neglected, also, compute the point kinetics parameters deterministically by their inner product definitions; (2) incorporate a hierarchy of three different integration time intervals for the numerical solution of the coupled set of ordinary differential equations, the shape function is assumed to vary linearly over the largest time interval, the second large time interval is used for determining the point kinetics parameters, finally, the smallest time step is used for solving the point kinetics equations; (3) apply TDKENO-M to benchmark problems to determine the accuracy of the method, particularly, TDKENO-M is applied to 1-D and 3-D benchmark problems to evaluate its capabilities; (4) combine input requirements into a single input file so that TDKENO-M is less cumbersome to execute; (5) develop the ability to restart a calculation at an intermediate problem time; and (6) develop a user-friendly manual for using TDKENO-M which describes in detail the input requirements as well as the output files, subroutines, modules, and the calculational flow.