Research and Verification of the One-Step Resonance and Transport Methods Based on the OpenMOC Code

Abstract

The one-step method in reactor physics has become one of the important research directions in recent two decades. Based on the open-source OpenMOC code, the following work was carried out. Firstly, the global–local resonance method with multi-group and continuous neutron libraries was researched and established. Next, based on the 2D and 3D MOC solver, the 2D/1D and the MOC/DD transport methods were realized in OpenMOC. Finally, verification of the transport and resonance methods was conducted using the C5G7 macro benchmark and the VERA micro benchmark. The numerical results demonstrated that the average eigenvalue deviation was 44 pcm and average maximum pin power distribution deviation was 0.37% in the VERA-2 benchmark, which showed the good accuracy of the resonance method. As for the transport method, the 3DMOC method exhibited better accuracy in strong anisotropic cases, but the computational time was 38 times that of the 2D/1D method.

1. Introduction

In recent years, with the continuous development of computer technology, the high-fidelity one-step method for the precise modeling and calculation of reactors has become one of hotspots in reactor physics research. Several high-fidelity codes have been developed around the world, including DeCART [1,2], MPACT [3], nTRACER [4], PROTEUS-MOC [5], OpenMOC [6,7], APOLLO-3 [8], STREAM [9], NECP-X [10,11], VITAS [12], and ALPHA [13]. The research area covers library, resonance, transport, burnup, kinetics, and multi-physics coupling, achieving one-step transport calculation for pressurized water reactors.

OpenMOC [14] is an open-source direct transport code with a method of characteristics (MOC) developed by the Massachusetts Institute of Technology (MIT). OpenMOC adopts a hybrid programming with Python3 and C++ languages to balance the good efficiency of C++ and good UI of Python. The constructive solid geometry (CSG) [15] method and long characteristic ray method [16] are applied to support the arbitrary geometry modeling and calculation ability. Spatial domain decomposition parallelism for memory and calculation has been used for large-scale cases. Moreover, coarse mesh finite difference (CMFD) [17] for accelerating iterative convergence and GPU/CPU hybrid computing [18] for MOC calculation have been supported in OpenMOC.

In this paper, a resonance module based on the global–local method [11] was added in OpenMOC. Moreover, a multi-group library and a continuous library were built for the global–local resonance calculation. In the transport module, based on the original solvers based on the 2DMOC and 3DMOC [19] methods, two three-dimensional solvers based on the 2D/1D method [20] and the MOC/DD (diamond difference) method [21] were expanded. Calculation accuracy and efficiency were compared for the whole-core direct transport method.

2. Framework of the OpenMOC Code

The framework of the OpenMOC code with the added module was shown in Figure 1. Firstly, as for the input/output module, the original structure and input logic were retained, utilizing CSG methods for high-fidelity geometric modeling and incorporating nuclear density input information for addressing real microscopic issues. Next, multi-group and continuous libraries were established for resonance calculation. Then, a global–local resonance calculation module was developed, employing the global neutron current method and local pseudo-nuclide subgroup method. Moreover, inflow transport corrections were applied to obtain macro cross-sections for each material region. Finally, direct transport calculations were performed with the 3DMOC, 2D/1D, and MOC/DD methods. The CMFD and parallel acceleration method were employed from the original OpenMOC code.

This study presents the first implementation in OpenMOC of a multi-group library, a global–local resonance calculation model, an inflow transport correction model, a 2D/1D transport module, and a MOC/DD transport model. Thereby, these added modules extend the OpenMOC code from a transport solver to a one-step three-dimensional transport calculation framework for microscopic problems. In this way, the paper presents the first side-by-side comparison of one-step transport schemes such as 3D-MOC, 2D/1D, and MOC/DD within a single code. This comparison eliminates numerous differences when comparing results with different numerical methods that are not implemented in the same software package, thus providing clearer insights into the differences between the numerical methods. At the same time, OpenMOC is able to perform one-step, pin-level transport calculations.

3. Global–Local Resonance Method

For the whole-core resonance calculation, the main research approach is to accurately simulate various resonance phenomena while ensuring acceptable computational efficiency. The global–local resonance method separates resonance phenomena into global effects and local effects, balancing computational efficiency and accuracy. It has achieved good computational results in thermal spectrum reactor. The basic theory of the global–local method will be divided into libraries, the global method, and the local method.

3.1. Library

This paper uses the classical WIMS-69 group structure [22]. Based on the WIMS library, a new multi-group library is expanded to improve the resonance effect calculation capability and a continuous energy library is built to match the global–local resonance method.

3.1.1. Multi-Group Library

For thermal spectrum reactors, the mainstream one-step calculation programs generally use multi-group library energy group structures with fewer than a hundred groups. Among them, the WIMS database is an internationally renowned open-source library, and its 69-group energy group format is widely used in pressurized water reactor calculations. Based on the WIMS library, improvements have been made in several aspects such as the resonance energy range, resonance nuclides, and resonance integral tables to create a multi-group database suitable for the global–local resonance method.

In terms of the resonance energy range, the original resonance range of WIMS is 4 eV to 9.118 keV, which cannot consider the resonance peaks of Pu nuclides below 4 eV as shown in Figure 2. This theoretically causes some deviation in the calculation of MOX fuel. Additionally, the ENDF/B-VII.0 evaluated nuclear library provides distinguishable resonance parameters for U-238 above 9.118 keV, and the original resonance energy range division did not consider this part of the resonance self-shielding phenomenon. To address the above issues, the multi-group library used in this paper extends the resonance energy range to 0.625 eV~24.78 keV.

Regarding nuclides, ²⁴²Pu has a strong resonance effect in the thermal energy region. The WIMS library approximates the resonance self-shielding effect of ²⁴²Pu for high-enriched and low-enriched MOX assembly separately. The multi-group library used in this paper expands the resonance energy range, which can explicitly and accurately simulate the resonance peaks of ²⁴²Pu. Furthermore, based on the 28 resonance nuclides in the WIMS database, the NJOY program [23] is used to create multi-group databases for Zr, Xe, Sm, Am, Eu, and other nuclides, increasing the number of resonance nuclides to 66.

Moreover, the resonance integral tables are expanded in two dimensions: temperature and background cross-section points. The temperature points are increased from 1100 K to 1800 K, and the number of temperature points is increased from 4 to 6. The background cross-section points were limited to 10 by the NJOY program in the original WIMS library. By resonance integral tables, the number of background cross-section points is increased to 34. The above improvements further enhance the fidelity of resonance integral table interpolation [24].

3.1.2. Ultra-Fine Group Library

Considering that the resonance pseudo-nuclide method requires the solution of neutron moderation equations online, an ultra-fine group continuous library is needed. The ultra-fine group cross-section library should include the ultra-fine energy group cross-sections of various nuclides at various temperature points. The amount of data is too large to be stored directly. Therefore, in the continuous energy library, only cross-sections are stored. Using the BROADR module of the NJOY program, the energy points, total cross-sections, elastic scattering cross-sections, and fission cross-sections in the output files are processed into binary files. A continuous energy library has been created for 66 resonance nuclides at 32 temperature points. The library size can be decreased to 1.1 GB, which is acceptable for storage.

3.2. Global Method

In global calculations, for a global system with multiple fuel and moderator regions, after introducing isotropic approximation, the neutron balance equation can be written as follows:

$${\mathsf{\Sigma }}_f\left(u\right){\varphi }_f\left(u\right)V_f={\displaystyle \sum _{f\in F}{P}_{f^{\prime }\to f}\left(u\right)S_{f^{\prime }}\left(u\right)V_{f^{\prime }}}+{\displaystyle \sum _{m\in M}{P}_{m\to f}\left(u\right)S_m\left(u\right)V_m}$$

(1)

where f is the index of the fuel region; m is the index of the moderator region; $\mathsf{\Sigma }\left(u\right)$ is the macroscopic cross-section; $\varphi \left(u\right)$ is the neutron flux density; V is the fuel and moderator density; $S\left(u\right)$ is the neutron source; $P_{f^{\prime }\to f}\left(u\right)$ is the probability of neutrons generated from an isotropic neutron source in fuel region f′ undergoing their first collision in fuel region f; and $P_{m\to f}\left(u\right)$ is the probability of neutrons produced in moderator region m undergoing their first collision in fuel region f.

By introducing the reciprocity relation, neglecting the global resonance interference effects and global temperature distribution effects, and employing a narrow resonance approximation for the source term in the moderator region, the neutron balance equation can be simplified as follows:

$${\mathsf{\Sigma }}_f\left(u\right){\varphi }_f\left(u\right)=\left[1-P_{f\to M}\left(u\right)\right]S_f\left(u\right)+P_{f\to M}\left(u\right){\mathsf{\Sigma }}_f\left(u\right)$$

(2)

Assuming there is only absorption and no scattering in the fuel region, and the source term is zero, the equation can be simplified as Equation (3).

$$P_{f\to M}\left(u\right)={\varphi }_f\left(u\right)$$

(3)

Introducing the isolated rod approximation to place the fuel rod in an infinitely large moderator, the escape probability $P_{\mathrm{e}}\left(u\right)$ equals the collision probability from the fuel to the moderator $P_{f\to M}\left(u\right)$. $P_{\mathrm{e},f}\left(u\right)$ is the escape probability from the fuel region. Let the flux in the isolated rod fuel region be ${\varphi }_{f,0}\left(u\right)$ and the flux in the original system fuel region be ${\varphi }_{f,1}\left(u\right)$, and the gray Dancoff correction factor [25] can be obtained:

$$C_f\left(u\right)={\displaystyle \frac{P_{\mathrm{e},f}\left(u\right)-P_{f\to M}\left(u\right)}{P_{\mathrm{e},f}\left(u\right)}}={\displaystyle \frac{{\varphi }_{f,0}\left(u\right)-{\varphi }_{f,1}\left(u\right)}{{\varphi }_{f,0}\left(u\right)}}$$

(4)

To avoid the complexity of the grey Dancoff calculation introduced by the energy terms and further reduce the computational amount, the concept of the black Dancoff factor is introduced. Through the theoretical derivation of the collision probability $P_{f\to M}\left(u\right)$, the calculation formula can be obtained as follows:

$$P_{f\to M}\left(u\right)=P_{\mathrm{e}}\left(u\right)\left\{1-{\displaystyle \frac{C_{\mathrm{b}}\overline{l}{\mathsf{\Sigma }}_f\left(u\right)P_{\mathrm{e}}\left(u\right)}{1-C_{\mathrm{b}}\left[1-\overline{l}{\mathsf{\Sigma }}_f\left(u\right)P_{\mathrm{e}}\left(u\right)\right]}}\right\}$$

(5)

where $\overline{l}$ is the average chord length of fuel region, $P_{\mathrm{e}}\left(u\right)$ is the escape probability, and $C_{\mathrm{b}}$ is black Dancoff factor. It can be observed that the consistence of $P_{f\to M}\left(u\right)$ will be ensured if the black Dancoff factor is conserved for the fuel and moderator system. Therefore, as for the entire problem to be solved, black Dancoff factors for each pin cell need to be solved for establishing an equivalent one-dimensional model. By introducing the black body approximation, where the cross-section of fuel region is infinitely large, the black Dancoff factor can be obtained as:

$$\underset{{\mathsf{\Sigma }}_f\to \infty }{\lim }{C}_{\mathrm{b}}=\underset{{\mathsf{\Sigma }}_f\to \infty }{\lim }{\displaystyle \frac{P_{\mathrm{e}}-P_{f\to M}}{P_{\mathrm{e}}}}={\displaystyle \frac{{\varphi }_{f,0}-{\varphi }_{f,1}}{{\varphi }_{f,0}}}$$

(6)

In practical calculations, considering the above black body approximation, corresponding cross-section values are set to solve the single-group fixed source equation on a global scale, obtaining the average flux ${\varphi }_{f,1}$ for each fuel rod in the fuel region. For each fuel rod, considering the isolated rod approximation in an infinitely large moderator, the Carlvik method is used to calculate the first collision probability $P_{f\to M}$, denoted as ${\varphi }_{f,0}$. In this way, the black Dancoff factor for each fuel rod can be calculated in global calculation.

3.3. Local Method

Based on gray Dancoff factors obtained from global calculations, an equivalent one-dimensional model is established as shown in Figure 3. This transformation enables the conversion of global calculations into local calculations for individual fuel rods.

Firstly, searches for the moderator radius of each fuel rod in the one-dimensional model are conducted based on Dancoff factors. Maintaining the fuel radius and fuel escape probability $P_{\mathrm{e}}$ constant according to Equation (6), a binary search is employed to compute the collision probability for various moderator radii, aiming to find the moderator radius that aligns with the pin-cell Dancoff factor.

Next, all resonance nuclides are transformed into a pseudo-nuclide based on nuclear density ratios. Pseudo-nuclide cross-section tables and subgroup probability tables are established online for each equivalent one-dimensional model. As for the pseudo-nuclide cross-section tables, the total cross-section, absorption cross-section, and scattering cross-section under various state points are calculated with the zero-dimensional ultrafine-energy-group method. The input parameters of the interpolation table include background cross-sections, actual temperatures, average temperatures, and burnup depths, as shown in Equation (7). As for the subgroup probability table, subgroup parameters such as the number of subgroups and subgroup cross-sections are determined using the Padé approximation and fitting method for pseudo-nuclides.

$${\sigma }_{x,g}=f_{x,g}\left({\sigma }_0,T,T_{\mathrm{ave},g},Bu\right)$$

(7)

Subsequently, traditional subgroup methods are employed to solve each equivalent one-dimensional model. This involves establishing subgroup fixed source equations based on subgroup parameters, solving for subgroup flux using collision probability methods, and calculating effective self-shielding cross-sections.

Finally, the super homogenization factor (SPH) method is utilized to address multi-group equivalent effects. Iterative processes are conducted to iteratively correct the effective self-shielding cross-sections until convergence is achieved for final SPH factors.

4. Direct Transport Method

For direct one-step transport calculations, the MOC method has become mainstream because of its excellent geometric description capability and good computational accuracy. Several techniques have been developed based on the MOC method, including the 3DMOC method, the 2D/1D method, and the MOC/DD axial expansion method.

The 3DMOC method directly extends the MOC method into three-dimensional space for accurate solutions. It is considered the ultimate method for direct transport calculations. However, it suffers from low computational efficiency and high memory consumption. The 2D/1D method introduces approximations to transform three-dimensional characteristic rays into multi-layer two-dimensional rays, significantly improving computational efficiency. However, it encounters instability issues and accuracy problems on strong anisotropic cases. The MOC/DD method also transforms three-dimensional characteristic rays into multi-layer two-dimensional rays while introducing differential relationships along the axial direction to eliminate leakage term isotropic approximations.

Based on the existing two-dimensional MOC calculation module in the OpenMOC program, new modules for the 2D/1D method and the MOC/DD method have been developed. The existing coarse mesh finite difference (CMFD) acceleration method and spatial domain decomposition parallelization techniques have been retained to enhance the computational capabilities of direct transport calculations.

5. Numerical Results

5.1. Macro C5G7 Benchmark

C5G7 [26] is a mainstream benchmark for validating heterogeneous transport calculation codes, including unrodded, half-rodded (rodA), and fully-rodded (rodB) configurations. For the unrodded case of the C5G7 benchmark, sensitivity analyses of computational parameters were conducted separately for the 2D/1D method and the 3DMOC method. Calculations were performed using the same set of computational parameters for the 3DMOC method, the 2D/1D method, and the MOC/DD method.

5.1.1. Sensitivity Analysis of Computational Parameters

The 2D/1D method includes parameters such as ray width, number of azimuthal and polar angle, number of reflector and fuel cell flat source regions, and axial layer height. Sensitivity analysis results are shown in

Table 1. Sensitivity analyses of the 2D/1D method for the C5G7 unrodded case.

Cases	Input Parameters					keff	PinPower Max Diff	PinPower RMS Diff
	Ray Width/cm	Azi/pol Angles	Reflector FSRs	Fuel FSRs	Axial Height/cm
1	0.05	32/6	10 × 10	4 × 8	3.57	1.14297	2.96%	1.01%
2	0.03	32/6	10 × 10	4 × 8	3.57	1.14296	2.94%	1.01%
3	0.01	32/6	10 × 10	4 × 8	3.57	1.14296	2.94%	1.01%
4	0.03	64/6	10 × 10	4 × 8	3.57	1.14318	2.96%	1.03%
5	0.03	128/6	10 × 10	4 × 8	3.57	1.14338	2.96%	1.04%
6	0.03	64/6	8 × 8	4 × 8	3.57	1.14324	3.63%	1.17%
7	0.03	64/6	5 × 5	4 × 8	3.57	1.14331	4.30%	1.36%
8	0.03	64/6	8 × 8	4 × 16	3.57	1.14328	3.56%	1.15%
9	0.03	64/6	8 × 8	6 × 16	3.57	1.14329	3.56%	1.15%
10	0.03	64/6	8 × 8	4 × 16	7.14	1.14366	3.28%	1.08%
11	0.03	64/6	8 × 8	4 × 16	1.785	1.14323	3.63%	1.17%

. Ray width was analyzed with cases 1~3. The number of angles was analyzed with cases 2, 4, and 5. The reflector and fuel flat source region divisions were analyzed with cases 4, 6, and 7 and cases 6, 8, and 9 separately. The axial layer height was analyzed with cases 8, 10, and 11. According to these sensitivity analyses, it can be concluded that the influence of input parameters was relatively small and the maximum deviation from the reference solution was only 63 pcm on case 10. Input parameters, especially the reflector flat source region division, had a more significant impact on pin power distribution results. The maximum pin power difference was improved from 4.30% to 2.96% with the refinement of the reflector flat source region in cases 4, 6, and 7. The default parameters for subsequent calculations were set to be 0.03 cm ray width, 32 and 6 azimuthal and polar angles, 100 and 32 flat source regions for moderator and fuel pin cells, and 3.57 cm axial height.

As for the 3DMOC method, based on the default parameters from the 2D/1D method, sensitivity analyses were conducted, including the polar ray spacing, axial layer height, linear source method, and number of angles. The results are shown in

Table 2. Sensitivity analyses of the 3DMOC method for the C5G7 unrodded case.

Cases	Input Parameters				keff	PinPower Max Diff	PinPower RMS Diff
	Axial Spacing/cm	Axial Source Region Height /cm	Linear Source	Azi/pol Angles
1	1.5	0.714	Yes	32/6	1.14287	7.23%	2.23%
2	0.75	0.714	Yes	32/6	1.14277	2.57%	0.91%
3	0.25	0.714	Yes	32/6	1.14276	2.25%	0.82%
4	0.25	1.428	Yes	32/6	1.14279	2.18%	0.73%
5	0.25	3.57	Yes	32/6	1.14309	4.02%	0.98%
6	0.25	0.714	No	32/6	1.14287	3.64%	1.24%
7	0.75	0.714	Yes	64/6	1.14290	2.60%	0.95%

. The polar ray spacing needs to be 0.75 cm in cases 1–3. The axial height needs to be less than 1.428 cm in cases 3–5. The number of angles has little influence in cases 2 and 7. Therefore, considering the convergence of computational parameters on the accuracy of calculations, the default calculation parameters were set to be 0.75 cm polar ray spacing and 0.714 cm axial height, with the linear source method.

5.1.2. Results of the C5G7 Benchmark with Three Direct Transport Methods

Based on the sensitivity analyses of the calculation parameters mentioned above, the verification calculation of the C5G7 benchmark was conducted with the 3DMOC method, the 2D/1D method, and the MOC/DD method. All cases were calculated with 162 CPUs, and the results are shown in

Table 3. Results of the C5G7 benchmark with three direct transport methods in OpenMOC.

	3DMOC	2D/1D	MOC/DD
Unrodded
Eigenvalue difference	−27 pcm	−32 pcm	−17 pcm
Pin power RMS difference	0.91%	0.83%	1.04%
Pin power MAX difference	2.57%	2.32%	2.86%
Calculation time	11,323 s	314 s	1281 s
Outer iteration times	33	85	47
RodA
Eigenvalue difference	−2 pcm	−28 pcm	−19 pcm
Pin power RMS difference	0.80%	0.78%	1.02%
Pin power MAX difference	2.14%	2.14%	2.79%
Calculation time	11,305 s	295 s	1289 s
Outer iteration times	33	85	47
RodB
Eigenvalue difference	40 pcm	−35 pcm	−25 pcm
Pin power RMS difference	0.64%	0.77%	0.98%
Pin power MAX difference	1.71%	2.27%	2.90%
Calculation time	11,364 s	284 s	1323 s
Outer iteration times	34	85	48

.

In terms of calculation accuracy, as for the unrodded and rodA cases, the eigenvalue and pin power differences were similar. The eigenvalue differences were less than 50 pcm, and the maximum pin power difference was less than 3%. All three methods demonstrated good calculation accuracy. As for the RodB case with the strong angular anisotropic effect, the 3DMOC method showed better accuracy performance on the pin power results, reducing the maximum difference from 2.27% to 1.71%.

Regarding the calculation efficiency, the average calculation time for the 3DMOC method was 11331 s, which is 38 times longer than the average time 298 s for the 2D/1D method and 8.7 times longer than the 1298 s for the MOC/DD method. The outer iteration times of the 2D/1D method are 2.50 times those of the 3DMOC method and 1.77 times those of the MOC/DD method. The phenomenon showed the poor convergence performance of the 2D/1D method, which is caused by the 2D and 1D iteration procedures.

5.2. Micro VERA-1 Benchmark

Geometric and material parameters for VERA benchmark problems are all derived from the Watts Bar pressurized water reactor nuclear power plant. VERA-1 is a series of pin-cell benchmarks with five cases. The VERA-1 benchmark was conducted for validating the global–local resonance module added in the OpenMOC code. The calculation conditions of OpenMOC included 0.01 cm ray spacing, 64 azimuthal angles, and six polar angles. The eigenvalue results are shown in

Table 4. Eigenvalue results of the VERA-1 benchmark with OpenMOC.

Cases	Description	keff (KENO-VI)	keff (OpenMOC)	keff Difference/pcm
1A	Fuel, 565 K	1.18704 ± 5 cpm	1.18734	30
1B	Fuel, 600 K	1.18215 ± 7 cpm	1.18247	32
1C	Fuel, 900 K	1.17172 ± 7 cpm	1.17187	15
1D	Fuel, 1200 K	1.16260 ± 7 cpm	1.16281	21
1E	IFBA, 600 K	0.77170 ± 7 cpm	0.77073	−97

.

5.3. Micro VERA-2 Benchmark

VERA-2 is a series of lattice benchmarks with cases 2A to 2Q in the VERA benchmark package. Descriptions and the OpenMOC results are shown in

Table 5. Eigenvalue and pin power results of the VERA-2 benchmark with OpenMOC.

Cases	Description	keff (KENO-VI)	keff (OpenMOC)	keff Diff /pcm	Pin Power Diff
					Max/%	RMS/%
2A	565 K	1.18218 ± 3 cpm	1.18268	50	0.20	0.07
2B	600 K	1.18336 ± 3 cpm	1.18392	56	0.18	0.05
2C	900 K	1.17375 ± 3 cpm	1.17430	55	0.16	0.05
2D	1200 K	1.16559 ± 3 cpm	1.16604	45	0.15	0.05
2E	600 K, 12Pyrex	1.06963 ± 2 cpm	1.07051	88	0.17	0.04
2F	600 K, 24Pyrex	0.97602 ± 3 cpm	0.97692	90	0.23	0.07
2G	600 K, 24AIC	0.84770 ± 3 cpm	0.84653	−117	0.36	0.13
2H	600 K, 24B4C	0.78822 ± 3 cpm	0.79153	331	0.54	0.14
2I	Instrument, 600 K	1.17992 ± 2 cpm	1.18044	52	0.19	0.05
2J	Instrument, 900 K, 24Pyrex	0.97519 ± 3 cpm	0.97617	98	0.20	0.07
2K	Zoned, 1200 K, 24Pyrex	1.02006 ± 3 cpm	1.02112	106	0.24	0.07
2L	80IFBA	1.01892 ± 2 cpm	1.01860	−32	0.28	0.07
2M	128IFBA	0.93880 ± 3 cpm	0.93855	−25	0.23	0.07
2N	104IFBA, 20WABA	0.86962 ± 3 cpm	0.86926	−36	0.29	0.08
2O	12 Gd	1.04773 ± 2 cpm	1.04788	15	1.03	0.18
2P	24 Gd	0.92741 ± 2 cpm	0.92759	18	1.20	0.29
2Q	Zircaloy Grid	1.17194 ± 2 cpm	1.17143	−51	0.57	0.14
Average				44	0.37	0.10

. Ray spacing was set to be 0.03 cm in all cases except for 0.01 cm in IFBA cases. The average eigenvalue difference of the VERA-2 benchmark was 44 pcm. The average maximum and RMS differences were 0.37% and 0.10% separately. Except for the 2H case with 24 B₄C absorber rods, all eigenvalue results were less than 120 pcm. Except for 2O and 2P cases with Gd absorber rods, the maximum pin power difference was less than 0.6%. All these results showed the good accuracy of the newly-developed global–local resonance module in OpenMOC.

To further study the larger eigenvalue and pin power differences on B₄C and Gd absorber cases, the results of 2O and 2P cases with the Gd absorber were analyzed. The geometry modeling of these two cases is shown in Figure 4, where the red rod was a burnable absorber. The detailed pin power differences are shown in Figure 5 and Figure 6. It can be concluded that the global–local resonance method undervalued the pin power results of burnable absorber rod, which needs further research in numerical algorithm.

6. Conclusions

Based on the OpenMOC open-source MOC transport calculation code, the global–local resonance method and direct transport methods were researched. A resonance calculation module, as well as multi-group and continuous-energy libraries, was developed in the framework of OpenMOC to provide multi-group cross-sections. A direct transport module with the 3DMOC method, 2D/1D method, and MOC/DD method was developed to provide multiple choices for calculation. The macro C5G7 benchmark and micro VERA benchmark were applied for the verification and calculation performance comparisons. As for the macro C5G7 benchmark, three kinds of direct transport methods were compared for accuracy and efficiency with a code system. The numerical results showed the better accuracy of the 3DMOC method on stronger anisotropic cases. However, the computation time for the 3DMOC was 38 times longer than that of the 2D/1D method. As for the VERA benchmark, average eigenvalue and maximum pin power differences were 44 pcm and 0.37%, which showed the good accuracy of resonance method. The resonance method can provide accurate multi-group cross-sections for the OpenMOC direct transport solver.