Neutronic and Thermal-Mechanical Coupling Study of Thermal Expansion Reactivity Feedback in Small Fast Reactors

ZHANG Shenghao¹, LIU Shichang¹, LIANG Dongsheng¹, MA Yugao², WANG Jinyu², ZHANG Jingyu¹
(1. School of Nuclear Science and Engineering, North China Electric Power University, Beijing 100096, China;
2. Nuclear Power Institute of China, Chengdu 610213, China)

Abstract

Compared with traditional thermal neutron reactors, fast reactors feature higher fuel enrichment and a harder neutron spectrum, resulting in a weaker Doppler broadening effect dominated by U-238 during operation. Consequently, core structural deformation exerts a more significant influence on reactivity, accounting for a larger share of the overall reactivity feedback. Therefore, accurate assessment of thermal expansion reactivity feedback effects constitutes a critical component of fast reactor safety analysis. Focusing on fast neutron pulse reactors, this study employs Monte Carlo methods for criticality calculations of pre- and post-thermal expansion models and finite element analysis for thermal-mechanical coupling calculations, thereby conducting a neutronic-thermal-mechanical coupling analysis of the thermal expansion reactivity feedback effects for the Godiva-I and Godiva-IV pulse reactors. The calculated fuel reactivity temperature coefficients show good agreement with experimental values, with maximum relative errors within 5%, validating the applicability and accuracy of the coupling methodology.

Keywords: Small fast reactors; Thermal expansion reactivity feedback; Neutronic-thermal-mechanical coupling; Godiva-I; Godiva-IV

Received date: 16 April 2025; Revised date: 8 July 2025

Foundation items: National Natural Science Foundation of China (U2330117, 12175067); Natural Science Foundation of Hebei Province (A2022502008); Beijing Science and Technology Rising Star Program (20240484596); Fundamental Research Funds for the Central Universities (2024MS046)

Corresponding author: LIU Shichang, E-mail: liu-sc@ncepu.edu.cn

0 Introduction

Small nuclear reactors (micro nuclear reactors) have emerged as critical energy solutions for extreme environment missions such as deep-sea exploration and space exploration, owing to their advantages of high energy density, compact structure, and low maintenance requirements. They not only provide persistent and stable autonomous power for various equipment but also satisfy the flexible deployment needs of high-energy-consumption devices, making them a focal point of international nuclear energy research and development in recent years. Current mainstream micro-reactor technology routes primarily include helium-cooled high-temperature gas-cooled reactors, heat pipe reactors based on heat pipe heat transfer, sodium-cooled/lead-cooled fast reactors with liquid metal coolant, molten salt reactors with inherent safety characteristics, and technologically mature light water reactors[1].

The aforementioned mainstream micro-reactor types are predominantly fast neutron reactors. Compared with conventional thermal neutron reactors, fast reactors exhibit higher fuel enrichment and a harder neutron spectrum, resulting in a relatively weaker Doppler broadening effect dominated by U-238 during normal operation. Consequently, the influence of core structural deformation on reactivity accounts for a larger proportion of the overall reactivity feedback. Small fast reactors demonstrate more complex multi-physics coupling characteristics during operation than thermal neutron reactors: in addition to basic neutronic-thermal coupling, their compact core structure introduces two novel coupling mechanisms—neutronic-mechanical and thermal-mechanical coupling—forming a strongly coupled system of neutronics, thermal-hydraulics, and structural mechanics (referred to as neutronic-thermal-mechanical coupling).

Existing research has primarily focused on neutronic-thermal coupling for thermal reactors, where methods neglect geometric deformation feedback (such as density/dimensional changes), leading to reactivity assessment biases in fast reactors characterized by simplified geometric effects and insufficient dynamic coupling. Studies on neutronic-thermal-mechanical coupling for thermal expansion reactivity feedback effects in fast reactors remain limited. Edward Lum et al.[2] employed Monte Carlo and finite element methods to calculate the reactivity temperature coefficient of the Godiva-IV pulse reactor, using ANSYS to compute thermal expansion deformation before importing displacement data into MCNP for reactivity temperature coefficient calculations, thereby achieving only stepwise one-way coupling. Manuele Aufiero et al.[3] coupled the Serpent Monte Carlo code with OpenFOAM, utilizing dynamic mesh technology to simulate neutron transport and thermal-mechanical feedback for Godiva-I prompt criticality; however, their explicit coupling exhibited insufficient stability under large deformations, required manual handling of mesh distortion, and suffered from low computational efficiency. Chen Shuo et al.[4] performed neutronic-thermal-mechanical one-way coupling calculations of fission rate, temperature rise, and stress response for the Godiva-IV pulse reactor based on MCNP and ANSYS, but adopted experimentally measured reactivity temperature coefficients and implemented only one-way coupling.

Therefore, to accurately assess the reactivity feedback effects of core thermal expansion in small fast reactors, this study constructs a neutronic-thermal-mechanical coupling framework using the Monte Carlo neutron transport software RMC and the finite element analysis software ANSYS Mechanical. This framework achieves automatic coupling iteration with cell-level temperature/density/displacement mapping and automatic geometry updating. The coupling framework is applied to perform neutronic-thermal-mechanical coupling calculations for the Godiva-I and Godiva-IV pulse reactors, comparing calculated fuel reactivity temperature coefficients with experimental values, analyzing the reactivity feedback effects of thermal expansion, and preliminarily validating the applicability and accuracy of the coupling methodology.

1.1 Neutronic-Thermal-Mechanical Coupling Framework

The neutronic-thermal-mechanical coupling analysis employs the Monte Carlo program RMC[5], independently developed by the Department of Engineering Physics at Tsinghua University, for neutron transport calculations, and the general-purpose finite element analysis software ANSYS Mechanical 17.2 for thermal-mechanical analysis. RMC serves as a professional Monte Carlo neutron transport simulation tool capable of accurately modeling neutron physics processes, while ANSYS Mechanical provides mature thermal-mechanical analysis capabilities[6]; their integration enables multi-physics coupling calculations. Detailed descriptions of the neutronic-thermal-mechanical coupling methodology are provided in reference[8]. The coupling calculation flowchart is presented in Figure 1 [FIGURE:1]. This study adopts a hybrid coupling strategy with RMC as the main program[9], utilizing the Picard iteration method. The specific coupling procedure is as follows: RMC first performs neutron transport calculations to obtain the fuel power distribution, which is then processed through a coupling interface and output as a fuel power density information file recognizable by ANSYS. Subsequently, ANSYS is invoked by RMC, reads the power density file, and conducts thermal-mechanical coupling calculations to obtain the core temperature, density, and displacement fields. These results are written to data files based on the spatial mapping relationship between the finite element mesh model and the CSG model. RMC then checks the signal file and calls an automatic modeling program, which reads the geometric displacement information output by ANSYS and automatically updates the RMC input deck. Finally, RMC reads the geometry-updated input deck and the temperature and density information from the ANSYS output data file, performs on-the-fly cross-section broadening and macroscopic cross-section calculations, and re-executes the neutron transport calculation. This process repeats until convergence criteria are met or the maximum iteration count is reached.

1.2 Spatial Mapping and Data Exchange Method

Figure 2 [FIGURE:2] (online color figure) illustrates the spatial mapping and data exchange methodology in neutronic-thermal-mechanical coupling. Both RMC and ANSYS Mechanical achieve cell-level solution accuracy. The figure uses a single fuel pellet as an example to demonstrate the approach. Since RMC constructs computational models using CSG while ANSYS employs finite element meshes, and the number of finite element elements typically exceeds that of cells, the spatial mapping relationship in neutronic-thermal-mechanical coupling is a "one-to-many" mapping between RMC cells and ANSYS finite element meshes, with solution accuracy largely dependent on the number of cells in the RMC model.

Regarding data exchange, the fuel power distribution tallied by RMC's counter function is processed through the coupling interface into fuel power density and directly assigned to the nodes of the corresponding ANSYS finite element mesh. After completing the thermal-mechanical coupling calculation, ANSYS computes the volume-averaged temperature and volume of the fuel pellet through post-processing. The temperature is used for Doppler broadening in RMC, while the volume is used to determine the average density based on mass conservation for macroscopic cross-section adjustment in RMC. The average temperature and density are calculated as follows:

$$T = \frac{1}{n}\sum_{i=1}^{n}T_i$$

$$\rho = \rho_{\text{Initial}}\frac{\sum_{i=1}^{n}V_{i,\text{Initial}}}{\sum_{i=1}^{n}V_i}$$

where $T$ and $\rho$ represent the average temperature and density of the cell; $T_i$ and $V_i$ denote the temperature and volume of the $i$-th mesh node/element mapped to the cell; $n$ is the total number of mapped mesh nodes/elements; and the subscript 'Initial' indicates initial values.

The neutronic-thermal-mechanical coupling feedback mechanism is implemented through dynamic updating of temperature fields, density fields, and geometric parameters in RMC. For thermal feedback, on-the-fly cross-section broadening technology is employed to treat material Doppler effects, utilizing the Target Motion Sampling (TMS) method[10] for the resolved resonance energy region and probability table interpolation for the unresolved resonance region. For mechanical feedback, geometric deformation is realized by updating the computational model. In this study, for the Godiva-I model, geometric expansion of the metal sphere is described by updating the general spherical surface equation; for the Godiva-IV model, an elliptical equation describes radial geometric expansion of fuel rings, while plane equations describe height variations. Density reactivity effects are implemented through macroscopic cross-section correction, with the calculation formula as follows:

$$\Sigma = N\sigma = \frac{\rho N_A}{M}\sigma$$

where $\Sigma$ is the macroscopic cross-section, $N$ is the nuclide number density, $\sigma$ is the microscopic cross-section, $\rho$ is the material density, $N_A$ is Avogadro's number, and $M$ is the molar mass.

1.3 Convergence Criteria

Convergence of the neutronic-thermal-mechanical coupling calculations in this study is based on keff convergence, with temperature and displacement variations also serving as criteria for thermal-mechanical coupling convergence. The relative change in keff between adjacent iteration steps is calculated as follows:

$$\epsilon = \frac{|k_{\text{eff}}^{n+1} - k_{\text{eff}}^n|}{k_{\text{eff}}^n}$$

where the superscripts $n-1$, $n$, and $n+1$ denote iteration step numbers; the subscript $i$ indicates mesh index; $N$ represents the total number of meshes; and $\epsilon$ is the tolerance.

1.4 Reactivity Temperature Coefficient Calculation

The reactivity temperature coefficient is defined as:

$$\alpha_T = \frac{d\rho}{dT}$$

In this coupling methodology, RMC in the first iteration step performs criticality calculations using initial geometry and temperature, while at coupling iteration convergence, RMC employs the temperature and geometry obtained from ANSYS thermal-mechanical coupling calculations for criticality computations. The reactivity temperature coefficient at coupling convergence is then obtained by dividing the difference between the keff values from the initial and converged iterations by the average temperature rise.

2.1 Godiva-I Pulse Reactor Coupling Calculation and Validation

The Godiva-I pulse reactor was developed by the Los Alamos National Laboratory (LANL) and features a bare spherical geometry design without reflector. The fuel consists of metallic uranium with an enrichment of 93.8%, a fuel radius of 8.7407 cm, and a total loading of 52.42 kg of ²³⁵U, exhibiting typical fast neutron spectrum characteristics[11][12]. Due to the extremely short operation time of the Godiva-I pulse reactor—a transient process on the microsecond scale—the average fuel temperature rise is only 17 K. Within this small temperature rise range, property variations such as the thermal expansion coefficient (1.39×10⁻⁵ K⁻¹) and thermal conductivity (27.5 W·m⁻¹·K⁻¹) of metallic uranium fuel are less than 1%, and geometric deformation dominates the reactivity feedback. Therefore, the constant property assumption adopted in ANSYS thermal-mechanical coupling calculations is reasonable. The material properties used in ANSYS calculations are listed in Table 1 [TABLE:1][13][14].

The RMC and ANSYS models for the Godiva-I pulse reactor are illustrated in Figure 3 [FIGURE:3]. RMC can receive temperature, density, and geometric dimension information output by ANSYS, while ANSYS can receive power distribution information output by RMC. The RMC criticality calculation employs vacuum boundary conditions with 500,000 particles per generation, 150 inactive generations, and 350 active generations, yielding an average standard deviation of approximately 0.000057. Both ANSYS thermal analysis and mechanical calculations utilize a transient solver, with a total mesh count of approximately 170,000, a total simulation time of 5.0×10⁻⁴ s, a time step of 5.0×10⁻⁵ s, and 10 time steps. This study adopts the power at an initial period of 29.5 μs as the total thermal power, with an initial thermal analysis temperature of 300 K and adiabatic boundary conditions applied to the metal sphere surface. The mechanical calculation uses the temperature field generated by thermal analysis as the loading condition, with free expansion boundary conditions applied to the metal sphere surface and zero initial displacement.

Five iteration calculations were performed on the aforementioned model, with the variation of coupling parameters versus iteration step or time shown in Figure 4 [FIGURE:4]. Figure 4a presents the convergence criterion keff as a function of iteration step, demonstrating that keff begins to converge from the fourth iteration. The thermal-mechanical coupling calculation yields a reactivity feedback of -0.0731 $, while the average reactor temperature rise is 17 K, therefore the fuel reactivity temperature coefficient is -0.0043 $/K, corresponding to a relative deviation of -2.4% from the experimental value[12] of -0.0042 $/K. Potential reasons for the calculated fuel reactivity temperature coefficient being lower than the experimental value include: (a) the RMC model employs vacuum boundary conditions on the sphere surface, failing to simulate environmental neutron reflection under experimental conditions, thereby overestimating neutron leakage; and (b) the "one-to-many" mapping between RMC cells and ANSYS meshes leads to averaging of feedback parameters, indirectly affecting reactivity.

Figure 4b illustrates the volume-averaged temperature rise and relative volume expansion versus pulse reactor operation time from the ANSYS thermal-mechanical coupling calculation at iteration convergence. The maximum calculated temperature rise is 17 K, differing by 1 K from the literature value[3] (6.25% relative deviation). The maximum calculated displacement is 2×10⁻³ cm, differing by less than 1 μm from the literature value[3] (3.5% relative deviation). The absolute deviations in temperature rise and displacement from literature values are minimal and do not significantly impact engineering evaluation of reactivity coefficients; therefore, these deviations can be neglected.

2.2 Godiva-IV Pulse Reactor Coupling Calculation and Validation

The Godiva-IV fast neutron pulse reactor employs a cylindrical fuel configuration, with the core model shown in Figure 5 [FIGURE:5]. Godiva-IV features three equally spaced clamps securing a cylinder composed of highly enriched uranium and steel. The cylinder can be further decomposed into six fuel rings, with a main shaft and base screwed together to secure the fuel rings. Each fuel ring consists of uranium enriched to 93% ²³⁵U alloyed with 1.5 wt% molybdenum. Within the rings are three cylindrical blocks of similar composition, designated from top to bottom as the upper subassembly plate, middle subassembly plate, and safety block. The interior of these components comprises a safety block base and main shaft made of stainless steel[2]. The main shaft and safety block base maintain the entire cylindrical assembly. Similar to the Godiva-I pulse reactor, Godiva-IV operates on the microsecond scale with an average fuel temperature rise of 49 K. Within this small temperature rise range, property variations such as the thermal expansion coefficient (1.48×10⁻⁵ K⁻¹) and thermal conductivity (29.4 W·m⁻¹·K⁻¹) of the uranium-molybdenum alloy fuel are less than 2%, and geometric deformation dominates reactivity feedback. Therefore, the constant property assumption in ANSYS thermal-mechanical coupling calculations is reasonable. The material properties for the U-1.5%Mo alloy used in ANSYS calculations are listed in Table 2 [TABLE:2].

Based on the aforementioned model description, the fuel rings, safety block, subassembly plates, and control rod materials are all fuel components, and when assembled, they form a nearly gapless ring of highly enriched uranium. Therefore, for neutron transport calculations, it is unnecessary to retain all modeling details of the Godiva-IV model. The safety block, subassembly plates, and control rod are treated as parts of the fuel rings, while the C-clamps serving as fixtures are neglected to simplify the neutron transport model. The RMC coupling model includes six fuel rings (incorporating the corresponding portions of the safety block, subassembly plates, and control rod), the main shaft, and the safety block base. Conversely, the ANSYS model requires detailed modeling with retention of gap and free space details, as gaps in the model close or contract during thermal expansion, thereby affecting component thermal expansion and thermal stress. The complete model comprises six fuel rings, safety block, subassembly plates, control rod, main shaft, main shaft bolts, buffer pads, support pads, safety block base, safety block shims, C-clamps, and gaps between components. Figure 6 [FIGURE:6] illustrates the RMC and ANSYS coupling models for Godiva-IV, with the ANSYS model containing approximately 480,000 elements.

The six fuel rings in the RMC model can individually receive temperature, density, height change, and inner/outer radius change information output by ANSYS. The RMC criticality calculation employs vacuum boundary conditions with 100,000 particles per generation, 150 inactive generations, and 350 active generations, yielding an average standard deviation of approximately 0.000128. The ANSYS thermal analysis calculation uses measured temperature values for the 4th fuel ring and safety block[2] as input loading, with air convection boundary conditions applied to all external surfaces. The mechanical calculation uses the temperature field generated by thermal analysis as the loading condition, with boundary conditions set as fixed center of the C-clamp and free expansion for all other components, and zero initial displacement.

Five iteration calculations were performed on the aforementioned model, with keff variation versus iteration step or time shown in Figure 7 [FIGURE:7]. The figure indicates that neutronic-thermal-mechanical coupling converges at the third iteration, with the converged keff taken as the average of iterations 3 through 5. Compared with the initial iteration, reactivity decreases by 0.1517 $, while the average fuel ring temperature rise is 49 K, therefore the fuel reactivity temperature coefficient is -0.003096 $/K, corresponding to a relative deviation of -3.2% from the experimental value[15] of -0.003 $/K. Potential reasons for the calculated fuel reactivity temperature coefficient being lower than the experimental value include: (a) the RMC model employs vacuum boundary conditions on the sphere surface, failing to simulate environmental neutron reflection under experimental conditions, thereby overestimating neutron leakage; and (b) although the ANSYS model retains fuel ring gaps, the RMC model simplifies the geometry and does not simulate gap closure at elevated temperatures, leading to overestimated radial expansion and excessively strong negative reactivity feedback.

Figure 8 [FIGURE:8] displays the temperature and displacement field contours from the ANSYS thermal-mechanical coupling calculation. The maximum calculated fuel temperature is 345 K, differing by 3 K from the literature value (-0.87% relative deviation). The maximum calculated fuel displacement is 1.26×10⁻² cm, differing by 11 μm from the literature value (-8.7% relative deviation). The absolute deviations in temperature rise and displacement from literature values are minimal and do not significantly impact engineering evaluation of reactivity coefficients; therefore, these deviations can be neglected.

3 Feedback Effect Comparison

To investigate the impact of thermal expansion on reactivity, a comparative study was conducted by separately varying geometry and temperature. The fuel reactivity feedback coefficients were calculated and compared for two cases: considering temperature feedback alone and considering both temperature and geometric expansion feedback simultaneously, thereby determining the contribution of thermal expansion effects to the overall core reactivity feedback. Table 3 [TABLE:3] lists the fuel reactivity temperature coefficients for the Godiva-I pulse reactor in the temperature range of 300 K to 317 K, calculated by considering temperature change alone and by considering both temperature and geometric changes. The results show that in the 300 K to 317 K temperature range, the reactivity feedback considering only the Doppler effect is -0.0031 $, while when considering both Doppler broadening and thermal expansion effects, the negative reactivity feedback is -0.0731 $/K. This indicates that approximately 95% of the negative feedback effect resulting from fuel temperature rise originates from thermal expansion, with only about 5% from Doppler broadening.

Table 4 [TABLE:4] lists the fuel reactivity temperature coefficients for the Godiva-IV pulse reactor in the temperature range of 292 K to 341 K, calculated under the same two considerations. The results show that in the 292 K to 341 K temperature range, the reactivity feedback considering only the Doppler effect is -0.0215 $, while when considering both Doppler broadening and thermal expansion effects, the reactivity feedback is -0.1517 $/K. This demonstrates that approximately 85% of the negative feedback effect from fuel temperature rise arises from thermal expansion, with only about 15% from Doppler broadening.

4 Conclusion

This study establishes a neutronic-thermal-mechanical coupling methodology centered on the RMC Monte Carlo neutron transport code and ANSYS finite element software. Through a closed-loop iterative framework of "power distribution → temperature/displacement fields → geometry/cross-section update → neutron transport," the method achieves dynamic coupling of neutronics, thermal-hydraulics, and mechanics. Neutronic-thermal-mechanical coupling calculations and fuel reactivity feedback coefficient validation were performed for the Godiva-I and Godiva-IV pulse reactors, with comparative analyses of thermal expansion and Doppler broadening effects on reactivity feedback by separately varying geometry and temperature.

For the Godiva-I pulse reactor, the calculated fuel expansion reactivity temperature coefficient shows a -2.4% relative deviation from the experimental value; the maximum calculated fuel temperature rise differs by 1 K from the literature value (6.25% relative deviation); and the maximum calculated displacement differs by less than 1 μm from the literature value (3.5% relative deviation). For the Godiva-IV pulse reactor, the calculated fuel expansion reactivity temperature coefficient exhibits a -3.2% relative deviation from the experimental value; the maximum calculated fuel temperature rise differs by 3 K from the literature value (-0.87% relative deviation); and the maximum displacement differs by 11 μm from the literature value (-8.7% relative deviation).

Comparative analysis of Doppler broadening versus thermal expansion effects reveals that thermal expansion reactivity feedback dominates in both Godiva-I and Godiva-IV pulse reactors. The combined reactivity feedback from Doppler broadening and thermal expansion in Godiva-I is approximately -0.0731 $, with thermal expansion accounting for 95% of the feedback effect; for Godiva-IV, the reactivity feedback from Doppler broadening and thermal expansion is approximately -0.1517 $/K, with thermal expansion contributing 85%. These results clearly demonstrate that geometry deformation induced by thermal expansion plays a decisive role in reactivity regulation in fast neutron pulse reactor systems, while the Doppler effect—traditionally considered significant—contributes relatively less in such compact metal-fueled reactors. This finding carries important implications for understanding the safety characteristics of fast reactor systems.

References

[1] Du Shuhong, Li Yonghua, Sun Tao, et al. Research on the Development Trend of Micro Nuclear Reactor Technology[J]. Nuclear Power Engineering, 2022, 43(04): 1-4. (in Chinese)
[2] Edward Lum, Chad Pope. GODIVA-IV reactivity temperature coefficient calculation using finite element and Monte Carlo techniques[J]. Nuclear Engineering and Design, 331 (2018): 116-124.
[3] M Aufiero, C Fiorina, A Laureau, et al. Serpent-OpenFOAM coupling in transient mode: simulation of a Godiva prompt critical burst[C]// Joint International Conference on Mathematics and Computation (M&C), Supercomputing in Nuclear Applications (SNA) and the Monte Carlo (MC) Method. Nashville, Tennessee: 2015.
[4] Chen Shuo. Analysis of safety limits and constraints of fast neutron pulse reactors[D]. China Academy of Engineering Physics, 2019. (in Chinese)
[5] WANG K, LI Z, SHE D. RMC – A Monte Carlo code for reactor core analysis[J]. Annals of Nuclear Energy, 2015(82): 121-129.
[6] BHASHYAM G R. ANSYS Mechanical: A powerful nonlinear simulation tool[M]. Canonsburg: ANSYS Inc, 2002: 1.
[7] Ma Yugao, Liu Minyun, Yu Hongxing, et al. Neutronic/Thermal-Mechanical Coupling in Heat Pipe Cooled Reactor[J]. Nuclear Power Engineering, 2020, 41(4): 191-196. (in Chinese)
[8] Yugao Ma, Minyun Liu, Biheng Xie, et al. Neutronic and thermal-mechanical coupling analyses in a solid-state reactor using Monte Carlo and finite element methods[J]. Annals of Nuclear Energy, 2021, 151: 107923.
[9] Juanjuan Guo, Shichang Liu, Xiaotong Shang, et al. Coupled neutronics/thermal-hydraulics analysis of a full PWR core using RMC and CTF[J]. Annals of Nuclear Energy, 2017, 109: 327-336.
[10] LIU S, YUAN Y, YU J K, et al. Development of on-the-fly temperature-dependent cross-sections treatment in RMC code[J]. Annals of Nuclear Energy, 2016(94): 144-149.
[11] PETERSON R E, NEWBY G A. Lady Godiva: an unreflected uranium-235 critical assembly: LA-1614[R]. Los Alamos: Los Alamos National Laboratory, 1953.
[12] WIMETT T F, ENGLE L B, GRAVES G A, et al. Time behavior of Godiva through prompt critical[R]. Los Alamos: Los Alamos Scientific Laboratory, 1956: 1-39.
[13] AUFIERO M, FIORINA C, LAUREAU A, et al. Serpent--OpenFOAM coupling in transient mode: of a Godiva prompt critical burst[C]//Proceedings of Joint International Conference on Mathematics and Computation, Supercomputing in Nuclear Applications and the Monte Carlo Method. Nashville: ANS, 2015.
[14] WANG Y Q, SCHUNERT S, ORTENSI J, et al. Demonstration of MAMMOTH strongly-coupled multiphysics simulation with the Godiva benchmark problem[C]//International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering. Jeju, Korea: Idaho National Lab., 2017.
[15] Joetta M Goda, John A Bounds, William L Myers, 2013. Module 8: Godiva-IV Critical Assembly Demonstration, LA-UR-13-28223. LANL, Los Alamos.