Mesh Generation and Energy Group Condensation Studies for the Jaguar Deterministic Transport Code

PHYSOR 2012 – Advances in Reactor Physics – Linking Research, Industry, and Education Knoxville, Tennessee, USA, April 15-20, 2012, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2012)

MESH GENERATION AND ENERGY GROUP CONDENSATION STUDIES FOR THE JAGUAR DETERMINISTIC TRANSPORT CODE

R.A. Kennedy, A.M. Watson, C.I. Iwueke, E.J. Edwards

Knolls Atomic Power Laboratory Bechtel Marine Propulsion Corporation

P.O. Box 1072, Schenectady, NY 12301-1072 [email protected]; [email protected]

ABSTRACT

The deterministic transport code Jaguar is introduced, and the modeling process for Jaguar is demonstrated using a two-dimensional assembly model of the Hoogenboom-Martin Performance Benchmark Problem. This single assembly model is being used to test and analyze optimal modeling methodologies and techniques for Jaguar. This paper focuses on spatial mesh generation and energy condensation techniques. In this summary, the models and processes are defined as well as thermal flux solution comparisons with the Monte Carlo code MC21. Key Words: Jaguar, SBA, Deterministic Transport, Benchmark, PWR.

1. INTRODUCTION The deterministic transport code Jaguar [1] has been under development at Knolls Atomic Power Laboratory (KAPL) since 2006. The developmental focus of Jaguar has been to provide reactor core design products for which the speed vs. accuracy is flexible. Meeting this goal will allow the design community to perform scoping studies quickly for core optimization while using Monte Carlo to provide the benchmark solutions on the same model. The Jaguar model building and analysis framework is being implemented in a way that is compatible with MC21, which is the Monte Carlo code under joint development by KAPL and Bettis Atomic Power Laboratory[2]. This paper demonstrates the first few steps of the modeling process being implemented for Jaguar by using a 2D representation of a portion of the PWR benchmark problem proposed by Hoogenboom, Martin, and Petrovic [3] for the purpose of measuring the performance of Monte Carlo codes on commercial reactor models. Specifically, the spatial mesh generation and the energy group condensation are presented. MC21 has recently been used to solve this benchmark problem. MC21 performance metrics on this benchmark have been documented and presented [4, 5]. The Jaguar user community has selected this performance benchmark as a representative PWR model for test purposes. This performance benchmark will provide the developers and users with a test medium to gauge Jaguar’s speed, accuracy and flexibility. The benchmark will also provide a platform for the presentation of methods development, methods implementation, and modeling techniques to the reactor physics community.

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1.1. Jaguar Jaguar is under development at KAPL for application to both reactor physics and shielding design. The code was designed to implement the Slice Balance Approach (SBA) [6] but has been extended to include other transport methods as well. Jaguar was constructed to support several transport solvers, including diffusion theory solvers. Currently only discrete ordinates has been implemented, but diffusion solvers designed to be used for acceleration are planned. Since SBA is a general-geometry transport framework, Jaguar can solve the transport equation on arbitrary polyhedral meshes in addition to traditional orthogonal meshes. This allows the modeling of curved geometric features, such as cylinders, as many-sided polygons with the ability to preserve the volume of the material as well as the approximate geometric shape. A higher degree of accuracy is achievable with this capability over traditional orthogonal meshes. Jaguar uses the multigroup method in energy, and thus requires some energy condensation consistent with all multigroup transport codes; these methods are discussed in Section 2.2 of this paper. Jaguar also has some restrictions on the type and dimension of the spatial mesh. These restrictions are relaxed compared to restrictions in other similar codes. For example, since the SBA exactly preserves the incoming angular fluxes, the restrictions on the number of mesh cells adjacent to a single mesh cell is lifted (e.g., a mesh cell may have ten incoming faces and one outgoing face without a loss of accuracy). However, due to current limitations in the implementation of SBA in Jaguar, all mesh cells must be convex. This will force a slight increase in the number of cells required to produce the results in this paper (~10%). 1.2. Hoogenboom-Martin Performance Benchmark Problem The Hoogenboom-Martin problem was first proposed in 2009 [7]. The problem considered in this paper is the revised version of the benchmark from 2011 [3]. The Hoogenboom-Martin benchmark problem consists of a full PWR core, radial reflectors, axial reflectors, and exterior structural components. Figure 1 provides both radial and axial slices of the full model geometry. The core consists of 241 fuel assemblies with a core height of 366 cm. Each assembly is composed of a 17×17 square-pitched array of fuel pins and guide tubes. The water in the core contains soluble boron for criticality control and the materials are modeled at a burn-up of 24,000 MWd/ton.

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Figure 1: 2D (a) radial and (b) axial cross section view of the Hoogenboom-Martin Performance Benchmark problem.

1.3. Overview and Motivation Reactor physics problems are traditionally solved using an orthogonal mesh [8] when deterministic transport is applied. This paper proposes a series of arbitrary polyhedral meshes to be used in the modeling of the Hoogenboom-Martin PWR Performance Benchmark problem for Jaguar. This paper also defines the cross section preparation process and modeling procedure. The results presented provide an initial comparison between the Jaguar solutions on pin and assembly models and Monte Carlo results from MC21. This mesh study attempts to answer a series of questions regarding arbitrary polyhedral meshes and cross section data: What are the advantages in mesh count, flexibility, accuracy, and solution speed? What is the optimum representation of the circular geometry given the constraints in the mesh? How should the group structure be represented for an eigenvalue calculation? Which spatial methods should be applied using the different meshes? The quadrature set and scattering approximations are left for subsequent analysis.

2. DETERMINISTIC AND MONTE CARLO PIN CELL MODELING

2.1. Pin Cell Model Geometries The benchmark core is composed of zircaloy-clad fuel pins and zircaloy guide tubes. The MC21, Jaguar, and PARTISN [9] models of the benchmark are constructed in a hierarchical manner using two unit cell models. The pin unit cell model is defined as a single fuel pin in a square of borated water (at the hot water temperature of 535K) with a length and width of 1.26

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cm (referred to as the model pitch). The guide tube unit cell model is defined as a single tube in a square of borated water (also at the hot water temperature) with a length and width of 1.26 cm. Table I provides the dimensions of the components as well as the pitch of each unit cell model. The materials are defined in Reference [3].

Table I. Unit Cell Model Dimensions

Dimension Fuel Pin Guide Tube Inner Radius (cm) 0.41 0.56 Outer Radius (cm) 0.475 0.62 Model Pitch (cm) 1.26 1.26

The two unit cell models were constructed for three different solvers in 2D: Jaguar, MC21, and the deterministic transport code PARTISN. The MC21 model of the pin cell was used as the reference multiplication factor, k. The PARTISN pin cell model was constructed for Jaguar validation. The PARTISN flux results were used to provide the collapsing fluxes because the few group energy condensation job stream is partially automated with PARTISN. Reflective boundary conditions were used on all sides of the pin cell models. Descriptions of the solver specifications and modeling techniques are defined for MC21, PARTISN, and Jaguar in Sections 2.1.1, 2.1.2, and 2.1.3 respectively. The eigenvalues from PARTISN and Jaguar are compared to the MC21 multiplication factor, k. For the unit cell analysis this quantity is used as an integral indicator of accuracy for the spatial discretization/method and energy group condensation; it also provides a gross estimate of the reasonableness of the chosen group structure. A more detailed comparison of flux distributions will be performed for the assembly level models in subsequent sections of this paper. 2.1.1. MC21 pin cell model The MC21 pin cell model was used as the reference model for testing spatial and energy mesh in the deterministic codes. Figure 2 provides a diagram of the MC21 pin cell model. The pin cell model was constructed exactly according to the dimensions specified in Table I with reflective boundary conditions on all sides of the model. Point-wise/continuous energy cross section data was used to compute the k of the pin cell model with an upper energy cutoff of 20 MeV. The eigenvalue calculation was performed using 20,000 histories per cycle for 1000 cycles and 200 cycles were discarded to allow for source convergence. The pin cell model k was 1.0067 ± 2.9×10−4.



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Figure 2: 2D x-y cross section view of the problem geometry for the MC21 pin cell model.

2.1.2. PARTISN pin cell model PARTISN is used to provide a comparison to Jaguar results for validation purposes. The PARTISN pin cell model is depicted in Figure 3. A 0.19 mm × 0.19 mm mesh was applied to the pin cell geometry. The mesh count total was 4356 cells. This mesh preserved the volume of the pin to within one hundredth of a percent. The PARTISN eigenvalue calculations used an S16 quadrature set. A P3 approximation was used to represent the cross section data. The PARTISN pin cell model was primarily used to provide the collapsing flux for the energy group condensation method. The energy group condensation method is described in Section 2.2 of this paper. Table II defines the different number of groups tested using the PARTISN pin cell model. The table also provides the number of thermal groups (as well as upscatter groups) and the iteration count.

Figure 3: 2D x-y cross section view of the mesh for the PARTISN pin cell model.

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Table II. PARTISN Pin Cell Model Results

Groups* k Δk** Iterations 171 (47/47) 1.0066 0.0002 25103

21 (8/7) 1.0069 -0.0002 3423 9 (4/0) 1.0060 0.0007 1978 9 (4/3) 1.0050 0.0018 2129 8 (3/0) 1.0083 -0.0016 1965 7 (3/0) 1.0085 -0.0018 1639 5 (1/0) 1.0058 0.0009 1129

*(thermal/upscatter) **Δk defined as kMC21−kPARTISN

The performance of the group structure is defined as a balance of timing vs. accuracy. The few-group data set with five groups (one thermal) showed good agreement with the MC21 pin cell model solution. These PARTISN eigenvalues indicate that it is possible to obtain accurate eigenvalue solutions for few-group models.

Figure 4: 2D x-y cross section view of the meshes for the Jaguar pin cell model. 2.1.3. Jaguar pin cell models As Jaguar is unable to use geometries modeled with curved surfaces, the pin and tube cylinders were converted to polygons by preserving the volumes of the materials. Sixteen, eight, six, and four sided polygons were used to represent the pin geometry. This was accomplished by computing an effective radius for each polygon in a way which preserves the volume of the pin. Figure 4 illustrates each Jaguar pin cell model. The 16-sided pin cell model contains 164 mesh



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cells. The user selected the maximum mesh cell dimension to be less than the smallest mean free path of a particular material for the 171-group cross section data. The eight-, six-, and four-sided pin cell models were not meshed to meet the mean free path requirements. These models were meshed to test the impact of a coarse mesh on the solution accuracy. The perimeter of the polygonal representation of the pin is closest to the circumference of the pin for the higher mesh count models. For each mesh illustrated in Figure 4, four different group structures and two spatial methods were tested. The first spatial method tested was SBA-Diamond Difference (SBA-DD) [10]. The second spatial method tested was a Linear Discontinuous (LD) finite element method. The Jaguar eigenvalue calculations used an S16 quadrature set. A P3 approximation was used to represent the cross section data. A global convergence criterion of 1×10−5 was applied in the solution of the problem. The Jaguar eigenvalue solution for each pin cell model is provided in Tables III–VI. The Jaguar eigenvalues are compared to the k from MC21. Iteration count and timing information (for a single processor) are also provided for each group structure, mesh, and spatial method. As the Jaguar runs were performed in an uncontrolled computing environment, the repeatability is not guaranteed.

Table III. Jaguar Sixteen-Sided Pin Cell Model Results

Groups* k Δk** Iterations Time (min)

171 (47/47) 1.00737 -0.0006 88038 102.0 21 (8/7) 1.00764 -0.0009 1289 5.3 9 (4/0) 1.01010 -0.0034 1140 3.0 5 (1/0) 1.00667 0.0001 1041 2.0

171 (47/47), LD 1.00663 0.0001 59487 430.0 9 (4/0), LD 1.00686 -0.0001 2406 11.8 5 (1/0), LD 1.00625 0.0005 1646 8.5

*(thermal/upscatter) **Δk defined as kMC21−kJaguar

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Table IV. Jaguar Eight-Sided Pin Cell Model Results

Groups* k Δk** Iterations Time (min) 171 (47/47) 1.00934 -0.0026 94746 24.4

21 (8/7) 1.00878 -0.0020 1226 <1 9 (4/0) 1.01092 -0.0042 972 <1 5 (1/0) 1.00720 -0.0005 668 <1

171 (47/47), LD 1.00626 0.0005 111235 94.2 9 (4/0), LD 1.00679 -0.00005 1615 1.3 5 (1/0), LD 1.00619 0.0005 1462 1.2


Table V. Jaguar Six-Sided Pin Cell Model Results


21 (8/7) 1.00929 -0.0025 1118 <1 9 (4/0) 1.01101 -0.0043 720 <1 5 (1/0) 1.00715 -0.0004 514 <1

171 (47/47), LD 1.00582 0.0009 98580 48.4 9 (4/0), LD 1.00878 -0.0020 1330 <1 5 (1/0), LD 1.00712 -0.0005 1064 <1


Table VI. Jaguar Four-Sided Pin Cell Model Results


21 (8/7) 1.00989 -0.0032 752 <1 9 (4/0) 1.01173 -0.0050 602 <1 5 (1/0) 1.00761 -0.0009 450 <1

171 (47/47), LD 1.00251 0.0042 99366 33.7 9 (4/0), LD 1.00706 -0.0003 1237 <1 5 (1/0), LD 1.00697 -0.0002 919 <1




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Results indicated good agreement with MC21 for all of the pin cell model meshes. The application of an arbitrary polyhedral mesh in Jaguar significantly reduced the number of mesh cells required to represent the pin. As a result, the solution time for the pin cell models using SBA-DD was significantly shorter than the PARTISN model (with DD) without a loss of accuracy. The four-sided pin representation indicated a loss of accuracy in the k results compared to the other three representations. The 171-group LD cases demonstrate the effect of the mesh shape and size most appropriately (without introducing a loss of accuracy from other sources, such as energy group condensation or negative flux values in mesh cells that are too large for the cross section data). The discrepancy between the Jaguar k and the MC21 k grows as the pin cell is represented with fewer sides. The error ranges from 1×10−4 for the 16-sided representation to 42×10−4 for the four-sided representation. The 5-group cases demonstrated good agreement (within 10−4 of the MC21 k) on all four meshes. Other group structures were not as accurate. This difference highlights the sensitivity of the solution to the structure of the few group data. The LD spatial method demonstrated improved accuracy on coarser meshes. However, LD is approximately four times slower than SBA-DD because there are four times as many unknowns. In some cases, LD was beneficial using the 171-group data. For example, using 171 groups and the eight-sided pin cell model with LD improved the difference between MC21 and Jaguar k values by from 26.0×10−4 to 5.0×10−4. This particular example demonstrates that, since the coarse mesh (eight-sided pin model) had fewer than 25% of the mesh cells compared to the 16-sided pin cell model, it is beneficial to use the coarser mesh with the higher order differencing scheme.

2.2. Cross Section Data Condensation

Flux-weighted cross sections were computed for different energy group structures. Multi-group structures containing 5, 7, 8, 9, 21, and 171 groups were considered. The group structures were selected by visually grouping the fine (171) group bins based on the pin cell model total spectrum as computed with MC21. This spectrum is provided in Figure 5. The 5-group structure (used in the assembly level analysis of Section 3.2) is indicated by the divisions on Figure 5. Cross section data were processed using the NJOY system [11]. A generic PWR energy spectrum was used to collapse the point-wise cross section data to 171 fine groups. The flux solution from the spatially-detailed meshed PARTISN model was applied as the collapsing flux to condense the 171 fine groups to the few-group structure. For the pin cell models, a collapsing flux was used for each of the fuel, clad and water regions. The single pin unit cell model spectrum is used as the collapsing flux for the assembly as an appropriate approximation because the spectrum will be similar throughout the assembly model.

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Figure 5: The spatially integrated 171-group flux spectrum from the single pin unit cell

model as computed using MC21. The 5-group structure is indicated by the plot divisions.

3.0 ASSEMBLY MODELS Two dimensional single assembly models were constructed for MC21 and Jaguar. Section 3.1 describes the MC21 assembly model and Section 3.2 describes the Jaguar assembly model. The thermal flux solutions are compared on a pin-wise basis between the two codes. The k values are also compared between MC21 and Jaguar for different polyhedral mesh representations of the pins. Two dimensional models were considered appropriate for this analysis because the problem heterogeneities primarily exist in the x-y plane. 3.1. MC21 Assembly Model The single assembly model was constructed in a hierarchical manner. The lattice of pins and guide tube models was generated using copies of a single pin/guide tube and translating the small model definitions into the array indicated in Figure 6. The pin and guide tube models were constructed according to the dimensions specified in Table I. Reflective boundary conditions were applied to all sides of the assembly model. The eigenvalue calculation for the assembly was performed with 20,000 histories per cycle. The first 200 of 1200 total cycles were discarded. The k for the assembly model was 1.0113 ± 2.5×10−4. Pin-wise tallies were constructed for the model. The flux solution was tallied in the 5 energy group bins defined in Figure 5. The error on each pin tally was less than ~1%. The total computation time was 20 minutes on 64 processors.

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Figure 6: 2D x-y cross section view of the problem geometry for the MC21 assembly model. 3.2. Jaguar Assembly Models The eight-sided and four-sided pin cell models were used to construct a set of single assembly models for Jaguar. Figure 7 illustrates these meshes with a detailed inset in the upper right-hand corner. The four-sided pin cell assembly (Figure 7 – left mesh) has 2400 radial mesh cells and the eight-sided pin cell assembly (Figure 7 – right mesh) has 7500 radial mesh cells. Both models are one mesh cell thick in the axial direction. In the four-sided pin assembly model the guide tube is homogenized with the water to reduce the number of total meshes in the model. The guide tubes are modeled as eight-sided polygons in the eight-sided pin assembly model. In both the eight-sided and four-sided pin assembly models SBA-DD was the spatial method selected. The assembly model cross section data uses the 5-group structure indicated in Figure 5.

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Figure 7: Jaguar assembly meshes (with detailed inset). The four-sided pin cell mesh is on

the left and the eight-sided pin cell mesh is on the right.

The Jaguar eigenvalues for the assembly models are provided in Table VII. The difference in k, with respect to MC21, and timing information are included in this table. The assembly models are approximately 30×10−4 different in k from the MC21 result. The timing information provided for Jaguar is on a single processor. The eight-sided pin assembly model was run with the guide tube homogenized in one case, and modeled as an eight-sided polygon in the other case. The difference in the Jaguar eigenvalue between these two cases represents the error introduced by this homogenization.

Table VII. Jaguar Assembly Eigenvalue Results

Model: k Δk Iterations Time(min) Four-Sided Pin Model (5 groups, SBA-DD) 1.0084 0.0029 435 8.67 Eight-Sided Pin Model* (5 groups, SBA-DD) 1.0082 0.0032 466 34.38 Eight-Sided Pin Model (5 groups, SBA-DD) 1.0079 0.0034 466 34.78 Eight-Sided Pin Model* (171 groups, LD) 1.0119 -0.0005 25,893 23,000 *Guide tube is explicitly modeled

The pin cell model analysis demonstrated that using 171 groups and LD on the eight-sided pin cell model was the most accurate and efficient fine group analysis (the 171-group case with SBA-DD on the 16-sided pin cell was slower because there was more than four times the number of mesh cells). For this reason, the eight-sided pin was selected using 171 groups and LD to quantify the bias introduced in k by the few-group condensation. This 171-group, LD solution on the eight-sided pin assembly model showed excellent agreement with the MC21 calculation. The results indicate that the energy group condensation to 5 groups introduced a bias in k of approximately 30×10−4. To converge this model to 1×10−5 the calculation required approximately 12 hours on 32 processors.



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A comparison between the Jaguar eight-sided pin assembly model and the MC21 model of the thermal flux (group 5) is presented in Figure 8. The flux comparison is presented by pin number, where Pin #1 is in the bottom left corner of the model, Pin #2 is the first pin in row 2, and Pin #264 is in the top right corner of the model. The error bars on the MC21 data points represent the 95% confidence interval on the flux tally. Two cosine-like shapes are visible in the results. Each row is visible in the small cosine distribution along the axis (effectively x-direction in Figure 7), of which there are 17. The cosine in the y-direction (Figure 7) is the large cosine shape which spans the entire axis of Figure 8. The percent difference between the pin-wise MC21 flux solution and the pin-wise Jaguar solution is represented in Table VIII. The maximum difference is less than 5% for two edge pins in the four-sided pin assembly and less than 4% for all pins in the eight-sided pin assembly. Approximately 90% of the pins fall within 3% with respect to the MC21 tally results, and half of the pins fall within 2% for both Jaguar models.

Figure 8: Comparison of Jaguar and MC21 thermal flux solution by pin number.

Table VIII. Percentage of Pin Thermal Flux Results that Agree within 2%, 3%, 4%, and 5% of MC21 Solution*

Model: 2% 3% 4% 5%

8-Sided Pin Assembly (5-group) 67.80% 91.67% 100% 100% 4-Sided Pin Assembly (5-group) 63.64% 85.98% 99.2% 100%

* The 95% uncertainty on each MC21 pin flux tally is less than ~1%.

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One possible source of discrepency in the flux distribution may be the quadrature set, the impact of which could be defined in a quadrature set analysis. In addition, further analysis would be necessary to understand the correlation between the model boundary layer and the mesh. The impact of the multigroup approximation and low order spatial method on the flux distribution inaccuracy could be quantified prior to more sophisticated model construction.

4. CONCLUSIONS Overall, excellent agreement is evident between the Jaguar thermal flux solution and the MC21 thermal flux solution. The speed of the assembly calculation on one processor indicates significant potential for Jaguar to provide fast and accurate scoping studies for reactor design. The arbitrary polyhedral mesh used to model the pin cell geometry demonstrates significant potential in reducing mesh cell count without a loss of accuracy. The arbitrary polyhedral meshes were able to preserve the volumes of the pins exactly – with total mesh cell counts at least an order of magnitude smaller than the orthogonal pin cell mesh. One purpose of the analysis described in this paper was to determine a few-group structure which would provide significant speed-up in the eigenvalue calculations while quantifying the bias introduced from the energy group condensation. The few-group structures examined in the analysis indicated certain biases at the unit cell level that were consistent (although more pronounced) in the assembly level calculations. The 5-group structure created a negative eigenvalue bias in Jaguar compared to MC21 (for the eight-, six-, and four-sided pin assembly models). Discrepencies in k between MC21 and Jaguar most likely arise from the energy condensation to few-groups. Modifications to the modeling process would focus on improving the collapsing flux for the guide tube regions and optimizing the quadrature set applied to the model as well as the scattering approxmation. In the current process, the collapsing flux for the guide tube is taken from the water surrounding the pin in the unit cell model. A more accurate collapsing flux should be obtained from the guide tube region of a double unit cell model which contains one pin and one guide tube. This modification might also improve the thermal flux comparison between MC21 and Jaguar in the pins near guide tubes. The largest differences in the flux distributions arise in the corners of the assembly model; a quadrature set study may improve the comparison in this region of the model. Subsequent analyses will continue to focus on the optimization of the run time as full core models are constructed.

REFERENCES 1. A.M. Watson, R.E. Grove. and M.T. Shearer, "Effective Software Design for a Deterministic

Transport System," Transactions of the American Nuclear Society, 97, 482–484 (2007). 2. T.M. Sutton, et al., “The MC21 Monte Carlo Transport Code,” Proc. Joint Int. Topical

Meetings on Math. And Comp. and Supercomputing in Nucl. Anal., American Nuclear Society, April 15-19, Monterey, CA, USA, on CD-ROM (2007).



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3. J. Eduard Hoogenboom, W.R. Martin, and B. Petrovic, “Monte Carlo Performance Benchmark for Detailed Power Density Calculation in a Full Size Reactor Core, Revision 1.2,” http://www.oecd-nea.org/dbprog/MonteCarloPerformanceBenchmark.htm (2011).

4. D.J. Kelly, et al., “MC21 Monte Carlo Analysis of the Hoogenboom-Martin Full-Core PWR Benchmark Problem,” Proceedings of PHYSOR-2010, American Nuclear Society, May 9-14, Pittsburgh, PA, USA, on CD-ROM (2010).

5. D.J. Kelly and T.M. Sutton, “MC21 Monte Carlo Analysis of the Hoogenboom-Martin Full-Core PWR Benchmark Problem,” Proceedings of PHYSOR-2012, April 15-20, Knoxville, TN, USA, (2012).

6. R.E. Grove, “A Characteristic-Based Multiple Balance Approach for Solving the SN Equations on Arbitrary Polygonal Meshes,” Ph.D. Dissertation, The University of Michigan, February (1996).

7. J. Eduard Hoogenboom, W.R. Martin, “A Proposal for a Benchmark to Monitor the Performance of Detailed Monte Carlo Calculations of Power Densities in a Full Size Reactor Core,” Proc. 2009 International Conference on Math, Comp., Meth., and Reactor Physics,” American Nuclear Society, May 3-7, 2009 Saratoga Springs, NY, USA on CD-ROM (2009).

8. E.E. Lewis and W.F. Miller, “Computational Methods of Neutron Transport,” American Nuclear Society, La Grange Park, IL, USA (1993).

9. R.E. Alcouffe, R.S. Baker, J.A. Dahl, S.A. Turner, R. Ward, “PARTISN: A Time-Dependent, Parallel Nuetral Particle Transport Code System,” LA-UR-05-3925 (2005).

10. R.E. Grove, A.M. Watson, M.T. Shearer, “A Diamond-Difference-Like SBA Scheme (SBA-DDL) for Polyhedral Meshes,” Transactions of the American Nuclear Society, 97, 479-481 (2007).

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Mesh Generation and Energy Group Condensation Studies for the Jaguar Deterministic Transport Code