Abstract
For the prediction of the internal physical process of SIMONS (Small Innovative heliumxenon cooled MObile Nuclear power Systems), this research created a coupled threedimensional highfidelity calculation platform of the neutronics/ thermoelasticity analysis called FEMAS (FEM based Multiphysics Analysis Software for Nuclear Reactor). This platform allows for the multiphysics coupling calculations of neutron diffusion/ transport, thermal diffusion, and thermal elasticity. It is based on the opensource Monte Carlo code OpenMC and the opensource finite element codes Dealii and Fenics. In this paper, a simplified SIMONS reactor core is analyzed using the coupling platform. The results demonstrate that the coupling platform is capable of accurately predicting the effective multiplication factor change curve, power and temperature distribution, and thermal expansion phenomenon of SIMONS. With 240 kW of thermal power, the local temperature difference of the whole reactor is 390.1 K, and thermal stressrelated deformation occurs at a rate of 2.4%. The reactivity feedback caused by the monolith’s heating and thermal expansion is 30.5 pcm. Leveraging the highprecision computing hardware, this platform can assess the core performance to ensure that the core design satisfies the design criteria of ultralong life and inherent safety.
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Keywords
 Multiphysics Coupling
 Numerical Simulation
 Mobile Nuclear Power Systems
 Thermal Expansion
 Finite Element Method
1 Introduction
In some circumstances, a micro transportable nuclear reactor power system can function as an independent microgrid that can offer a longterm, highpower emergency power supply. It will lessen the loss of life and property due to power outages, helping in communication, transportation, the military, medicine, natural disaster rescue and many other industries, offering a wide range of application prospects in the future.
At present, the United States has carried out projects such as the Prometheus [1], eVinci [2], and Holos reactors [3]. Reactors have been designed using a wide variety of cuttingedge technologies aiming at output powers ranging from kilowatts to megawatts. In China, Tsinghua University has designed a 200kW gascooled reactor with power called IGCR200 [4].
Designed to be packaged in an ISO container and transported by truck or train, the design of the core of the micro transportable nuclear reactor power system is particularly important. Compared with regular reactors, mobile reactors have strict restrictions for size, weight, powerdensity and lifetime. Besides, it has been demonstrated that the thermal expansion effect constitutes a significant portion of the reactivity feedback for micro nuclear reactors [5]. Conventional methods of research and analysis based on a single physical field and static geometry can hardly meet these needs.
In order to precisely forecast the internal physical processes of SIMONS, this research developed FEMAS (FEMbased Multiphysics Analysis Software for Nuclear Reactor), a coupled threedimensional highfidelity computation platform for neutronics/thermoelasticity analysis. For neutronics, it employs a highfidelity 3D multigroup neutron diffusion/ transport model with unstructured grids to estimate the neutron flux field and heat release distribution of the nuclear reactor core. For thermoelasticity, it can establish a 3D thermal diffusion model and stress analysis model to accurately obtain the temperature field and deformation. This multiphysics platform enables us to carry out highprecision analysis of neutron physics, heatconduction, deformation, providing technical assistance for the core design of the mobile nuclear reactor power system.
2 Methodology
2.1 Method
2.1.1 Neutronics
The neutronics model of FEMAS is based on the finite element method (FEM) [6] to support a highfidelity geometrical modeling of SIMONS. The FEM is a basic strategy based on “discrete approximation”, a numerical method for solving Partial Differential Equations (PDEs). As an illustration, consider the steadystate neutron diffusion equation of a twogroup [7]:
where
where \(\phi\) is the neutron flux, \(D\) is the diffusion factor, \(\Sigma_{s}\) is the scattering cross section, \(\nu \Sigma_{f}\) is the neutron production cross section, and \(\chi\) is the fission spectrum. The rest notations are conventional in the reactor physics field, thus will not be elaborated here.
By introducing the finite element shape function and the Galerkin method, the steadystate neutron diffusion equation of twogroup in finite element form can be obtained as follows:
where
and the matrices elements (a, b) are given by
where \({\mathcal{N}}_{i}\) and \({\mathcal{N}}_{j}\) are the shape functions in the finite element. Similarly, this principle can be applied to the multigroup neutron diffusion equation.
2.1.2 Thermal Conduction
The differential equation of thermal conduction in Cartesian coordinates is:
This equation can be cast into the variational form:
Equation (1.4) can be easily solved with the opensource finite element platform Fenics, as will be illustrated in Sect. 2.2.
2.1.3 ThermalElasticity
The thermalelasticity model in FEMAS is based on a perfectly elastic isotropic body assumption. The relationship between the deformation component (strain: \(\varepsilon\)) and the stress component (stress: \(\sigma\)) is [8]:
where \(\lambda\) and \(\mu\) are Lame constants. The transformation relationship between them and elastic modulus \(E\), Poisson’s ratio \(v\) is:
Therefore, the expression for stress can be written as:
where \(I\) denotes the identity tensor. The expression for strain can be written as:
The stress governing equation of the whole system is:
where \(\sigma = \lambda {\text{tr}} (\varepsilon )I + 2\mu \varepsilon\), \(\varepsilon = \frac{1}{2}\left( {\nabla u + (\nabla u)^{T} } \right)\), and \(f\) is the external force per unit volume in the entire system. Similar as Sect. 2.1.2, the variational form of the stress equation is:
Temperature variations induce deformations in elastically unrestrained solids. Therefore, mechanical and thermal processes form the global strain field. In the context of the theory of linear small deformation, the total strain can be decomposed into the sum of mechanical and thermal components as:
For isotropic materials,
where \(\alpha\) is the linear thermal expansion coefficient of the material.
Therefore, the expression for the total strain is:
Thus, the linearized thermoelastic constitutive equation can be given by:
Similar to the elasticity equation, the weak form of the thermal expansion equation can be obtained as:
2.2 Coupling Framework
In order to achieve highprecision simulation of full reactor, this research developed the multiphysics calculation platform FEMAS, integrated opensource codes (OpenMC, Dealii and Fenics) based on the external coupling framework.
The iteration process goes as follows:

(1)
Use OpenMC to perform neutron transport simulation and obtain a crosssection library.

(2)
Based on the preset temperature distribution and geometric parameters in cold state, use Dealii to perform multigroup neutron diffusion calculation, thus getting the spatial distribution of power.

(3)
Given the boundary conditions of the heliumxenon cooling channel and the spatial distribution of power, the temperature distribution was calculated by solving the thermal diffusion equation using Fenics.

(4)
Combined with the temperature field and the mechanical boundary conditions, the structural displacement is obtained by solving the thermoelasticity equation using Fenics.

(5)
Update the geometry, density and cross sections of the model.

(6)
Dealii then performs the neutron diffusion calculation again. The steps above are repeated until certain physical quantities meet the convergence criteria or the execution reaches the maximum number of iteration steps (Fig. 1).
2.3 Model
2.3.1 Model
To meet the demand for terrestrial transportable nuclear reactor power supply, this research takes the overall conceptual design of SIMONS as the test case. SIMONS is designed to be an intelligent micro landbased transportable nuclear reactor featuring small size, high power density and inherent safety. Its thermal power is 20 MWe and it can operate continuously for 3300 EFPDs without refueling.
As a preliminary test of the coupling platform, a simplified scaling model of the SIMONS core is employed to reduce the computational cost. The scaled core exhibits a prismatic core design, using graphite as the monolith material. There are several holes in the monolith with a hexagonal arrangement to accommodate fuel rods and coolant. The overall height of the core is 30 cm, and the overall radius is 23 cm. The axialradial schematic diagram of the core is shown in Fig. 2.
The operating parameters, geometric information and material selection of the simplified SIMONS modeling are shown in Table 1.
2.3.2 Neutronics
The coupling calculations employ a highfidelity modelling approach by explicitly describing all the rods and holes using threedimensional unstructured meshes. Homogenized cross sections are generated using twodimensional OpenMC calculations for every material zone. This research uses a fourgroup structure which is divided as:

1)
Group1 (497.87 keV–20 meV)

2)
Group2 (9.11882 keV–497.87 keV)

3)
Group3 (0.625 eV–9.11882 meV)

4)
Group4 (0 eV–0.625 eV)
The cross sections are stored as data tables with considering different fuel and monolith temperatures. During the neutronics calculations, the cross sections for each material zone are updated using linear interpolation method. The cross sections of fuel are only related to the temperature, while those of monolith are dependent on both temperature and density.
Due to the simplification of the core design, setting vacuum boundary conditions will make the k_{eff} too small. Therefore, the reflective boundary condition is imposed on the outer boundaries in the calculation. Figure 3 shows the mesh used in the neutronics calculation.
2.3.3 ThermoElasticity
Figure 4 depicts the mesh used in thermoelasticity calculations. In the heat diffusion calculation, it is assumed that the thermal conductivities of the fuel and the monolith are constant. Currently, the coolant channels are treated as Dirichlet boundaries, and subsequent work will consider the heat transfer portion of the flow in the channel. Based on the temperature distribution obtained by solving the thermal diffusion equation, the thermal expansion calculation can be performed.
3 Results
3.1 Coupling Results
In the coupling calculation, the maximum number of iterations is set to 10. In fact, between the second and third steps, the k_{eff} error dropped below the predetermined convergence threshold (1e−5). However, the iteration step is set to 10 in order to study the characteristics of the coupled calculation results in greater detail. The results of various fields of neutronics/thermoselasticity coupling calculations are shown in the Fig. 5.
The 3D fourgroup neutron flux distribution is shown in Fig. 6. It can be observed that the fast neutron flux are generated in the fuel rods, while thermal neutrons presents at the periphery of the active core due to the strong thermalization effect of the Be reflectors.
For the fuel region, based on the aforementioned neutron flux distribution and the combined total power, the heat release rate then can be calculated. Through thermal diffusion calculation, the temperature distribution of the simplified SIMONS can be obtained, shown in Fig. 7. Maximum temperature difference across the whole reactor is 390.1 K.
According to the temperature field, the effective temperature of the fuel rod can be calculated, shown in Fig. 8. The results indicate that the fuel rods on the periphery have the highest temperatures due to the thermalization of neutrons in the reflector. Within the fuel rod, the middle of it has the highest temperature. This is because given the reflective boundary condition, the neutron fluxes of the third and fourth groups are predominantly spread outside the core, resulting in a higher power here. Moreover, The hot spot locations are also attributed to less numbers of surrounding coolant channels than other fuel locations.
Figure 9 depicts the thermal expansion calculation results. It reveals that the maximum displacement is about 0.09 cm, and the core expands radially from the center to the periphery. Graphite’s linear thermal expansion coefficient is quite small, hence the thermal expansion impact is not as pronounced compared to other solid reactors. However, it still creates a succession of reactive feedback, geometric expansion, and other effects worthy of our consideration.
3.2 Thermal Results Analysis
Figure 10 illustrates the convergence diagram and monolith geometrical parameters for multiphysics calculations. At the first step, the k_{eff} is calculated by directly applying the OpenMCgenerated cross section in the diffusion calculation, yielding a value of 1.65501. Meanwhile, at the last step, the value of k_{eff} is 1.65471. Comparing their k_{eff} values reveals that the adoption of thermoelasticity calculation results in a 30.5 pcm decrease in reactivity because of the enhanced neutron leakage effect. Regarding the monolith’s geometric specifications, its radius expanded from 13 cm to 13.08 cm, and its height increased from 10 cm to 10.06 cm. Consequently, its density is lowered to 97.6% of its original value.
4 Conclusion
In response to the research and development requirements of innovative heliumxenon cooled mobile nuclear power systems, this research developed a threedimensional highfidelity multiphysics coupling platform FEMAS. This platform is built on the Picard iteration and incorporates the open source OpenMC, Dealii, and Fenics codes, which enables the multiphysics modeling of neutronics/ thermoelasticity.
Simultaneously, based on the scaled model of SIMONS, this research preliminarily concludes a series of simulations and analyses, including from the OpenMC crosssection generation, reactor modeling, and multiphysics simulations. The results demonstrate that the coupling platform can predict the power distribution, temperature distribution and thermal expansion of SIMONS. Given 240kW of thermal power, the local temperature difference of the whole reactor is 390.1 K, and the deformation rate caused by thermal stress is 2.4%, and the reactivity feedback due to heat conduction and thermal expansion is 30.5 pcm.
In the era of highprecision computing, this platform can evaluate the core performance of the core to guarantee that the design of the core meets the standards for ultralong life and inherent safety.
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Acknowledgement
This research is supported by National Key R&D Program of China under grant number 2020YFB1901900, and National Natural Science Foundation of China (NSFC) [12175138].
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Li, X., Liu, X., Chai, X., Zhang, T. (2023). Preliminary Multiphysics Coupled Simulation of Small HeliumXenon Cooled Mobile Nuclear Reactor. In: Liu, C. (eds) Proceedings of the 23rd Pacific Basin Nuclear Conference, Volume 1. PBNC 2022. Springer Proceedings in Physics, vol 283. Springer, Singapore. https://doi.org/10.1007/9789819910236_59
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DOI: https://doi.org/10.1007/9789819910236_59
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