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Chapter 22.

Finite element modelling

M.R. Tonks1, R. Williamson1, R. Masson2 1Idaho National Laboratory, US,

2CEA, DEN, DEC, Centre de Cadarache, France

Abstract

The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to boundary value problems. While FEM is commonly used to solve solid mechanics equations, it can be applied to a large range of BVPs from many different fields. FEM has been used for reactor fuels modelling for many years. It is most often used for fuel performance modelling at the pellet and pin scale, however, it has also been used to investigate properties of the fuel material, such as thermal conductivity and fission gas release. Recently, the United Stated Department Nuclear Energy Advanced Modelling and Simulation Program has begun using FEM as the basis of the MOOSE-BISON-MARMOT Project that is developing a multi-dimensional, multi-physics fuel performance capability that is massively parallel and will use multi-scale material models to provide a truly predictive modelling capability.

Introduction

The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to Boundary Value Problems (BVP). In FEM, the domain is discretised into a number of small subdomains called elements, allowing for the generation of a system of discretised equations that can be combined to represent the continuous solution of the BVP. While FEM is commonly used to solve solid mechanics equations, it can be applied to a large range of BVPs from many different fields.

FEM has been used for reactor fuels modelling for many years. It is most often used for fuel performance modelling at the pellet and pin scale, and has been very successful. In addition, it has been used to investigate properties of the fuel material, such as thermal conductivity and fission gas release.

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More recently, FEM is the basis of the US Department of Energy’s MOOSE-BISON-MARMOT (MBM) suite of fuel performance codes under development at Idaho National Laboratory (INL). MOOSE (the multi-physics object-oriented simulation environment) is a finite element-based numerical framework for solving partial differential equations (PDEs) and forms the basis for BISON and MARMOT. BISON is the macroscale fuel performance code that can model a large range of fuel geometries and fuel materials in 1D, 2D and 3D. MARMOT is a mesoscale simulation tool that predicts the impact of microstructure evolution on fuel properties to develop more mechanistic material models of radiation damage for use in BISON.

In this report, we summarise the past use of FEM for fuel performance modelling. We then summarise work using FEM to investigate fuel properties. Finally, we describe the MBM suite of codes and the application to fuel performance modelling.

FEM for macroscale fuel performance modelling

The goal of a fuel performance code is to calculate the internal temperature and mechanical state of the fuel and cladding while under reactor conditions, in order to establish quantitative operation guidelines to mitigate fuel duty-related failures. Fuel operates in an extreme environment that induces complex and often tightly coupled multi-physics behaviour. Adding to this complexity, important aspects of fuel behaviour are inherently multi-dimensional, examples include pellet-clad mechanical interaction, fuel fracture and non-axisymmetric neutronics and cooling.

Early fuel performance codes, such as FRAPCON [1], TRANSURANUS [2] and ENIGMA [3], focused on LWR fuel rods and approximated this complex behaviour using an axisymmetric, axially-stacked, one-dimensional radial representation (often referred to as 1½D) and finite difference numerical methods. Similar 1½D codes evolved that used FEM, such as FEMAX [4] and CYRANO [5].

EPRI’s FALCON code was the first 2D FEM-based fuel performance code developed in the US [6,7]. It models the thermal, mechanical and chemical behaviour of a single fuel rod during irradiation and can be applied to steady or transient operation. CEA’s first-generation fuel performance code METEOR 1D code, as well as TOUTATIS 3D, has been implemented in the PLEIADES platform [8]. PLEIADES has been under development for ten years and includes the multi-dimensional finite element solver CAST3M. ALCYONE is the PWR fuel performance code in the PLEIADES platform, and 3D simulations with ALCYONE are used to investigate localised fuel behaviour such as pellet clad mechanical interaction [9].

While FEM is used to solve for the thermomechanical behaviour of the fuel, materials models are required to set the various material properties and their evolution throughout the lifetime of the fuel. Historically, many of these models were developed by fitting to experimental data. However, since the 1980s’ researchers have also used FEM simulations of the fuel microstructure to elucidate its properties and how they change with radiation damage.

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FEM for fuel property modelling

The earliest use of FEM as a tool for investigating UO2 fuel properties found by the authors was by Oguma [10]. In this paper, a 2D mechanics FEM code was used to investigate the fracture pattern in UO2 pellets after reactor start-up for various power levels. Since then, one paper has used FEM to investigate fission gas release using a 1D spherical approximation of a UO2 grain [11], where the diffusion equation was solved via FEM. The impact of porosity and Pu content on the fuel viscoplastic behaviour has also been studied with FEM analysis (see Section III.09). However, the most common use of FEM has been to investigate the impact of the bubble and grain structures within the fuel on the fuel thermal conductivity.

Bakker et al. have written two papers in which bubble structures were reconstructed from micrographs of irradiated fuel and meshed for FEM analysis. 2D heat conduction simulations were then conducted using FEM across the reconstructed meshes to determine the effective thermal conductivity [12,13] (an example of the FEM mesh is shown in Figure 1a). Microstructures from SEM and EDS data on high burn-up MOX fuel were reconstructed as 2D FEM meshes [14]. Again, heat conduction simulations were used to determine the effective thermal conductivity of the microstructures, including the impact of metallic precipitates (see Figure 1b).

Figure 1. FEM-based simulations investigating the impact of radiation damage on fuel thermal conductivity, where (a) is a FEM mesh with reconstructed bubble structure [12] and (b) shows the reconstructed microstructure [14] where metallic precipitates are in red, porosity in yellow

and UO2 in blue

MOOSE-BISON-MARMOT

The United States Department Nuclear Energy Fuels Modeling and Simulation (NEAMS) Program recognised a need for a modern, advanced 3D fuel performance capability that can model LWR accident conditions and new fuel and reactor concepts. The legacy US fuel performance codes have been under development for many years, and thus use older, serial code architectures. In addition, they are typically restricted to 1½D or 2D representations of LWR geometry. Finally, they employ materials models that are

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empirical fits to experimental data and thus cannot be extrapolated outside the test conditions. Thus, NEAMS has funded a team of researchers from the Idaho and Los Alamos National Laboratories to develop the MOOSE-BISON-MARMOT, or MBM, suite of fuel performance codes.

MOOSE (Multi-physics Object-Oriented Simulation Environment) is a numerical framework that facilitates the rapid development of multi-physics simulation tools that solve systems of partial differential equations using FEM [15]. MOOSE uses modern object-oriented architecture and is massively parallel without requiring the user to write any parallel code. Though initially developed for fuel performance modelling, MOOSE now has users at laboratories, universities and industry across the world to develop advanced FEM-based simulation tools.

BISON is the NEAMS flagship fuel performance code and is based on the MOOSE framework [16]. BISON is a multi-dimensional code that can be run in 1D, 2D or 3D to represent desired fuel behaviour. It has been used to model LWR fuel, as well as TRISO [17] and metallic fuel. BISON models the thermomechanical behaviour of the fuel and cladding, as well as species diffusion. The pellet-cladding interaction is modelled with advanced contact algorithms that function with parallel computing and implicit time integration. BISON has been coupled to neutronics and coolant flow codes for full core simulations and to mesoscale codes to represent the impact of radiation damage. An example of a 3D BISON simulation is shown in Figure 2a.

BISON was initially developed using traditional materials models, however advanced mechanistic models are under development using multi-scale modelling and simulation to supplement difficult to perform experiments. Simulations at the atomistic and mesoscales are used to predict the radiation-induced microstructure evolution and the impact of the microstructure on the fuel properties. These results are then used in the development of mechanistic materials models correlated to evolving variables that describe the current state of the microstructure [18]. When complete, these new materials models will provide a predictive fuel performance modelling capability.

The MOOSE-based MARMOT code has been developed to model the co-evolution of microstructure and properties due to applied load, temperature and radiation damage. The microstructure evolution is modelled with the phase-field method fully coupled with computational mechanics and heat conduction equations. This set of multi-physics equations is solved with FEM using MOOSE [19]. The models are quantitative and use material properties from experiments or from atomistic simulation (molecular dynamics and density functional theory). MARMOT has been used to investigate microstructure evolution, including bubble growth and migration [19,20] and grain boundary migration and grain growth [21,22]. It has also been used to investigate the impact of GB bubbles on the fuel thermal conductivity [18,23]. An example of a MARMOT simulation is shown in Figure 2b.

Though the MBM codes are still under development, they are already being used to understand and predict fuel performance for a wide range of different types of reactors. As this work progresses, it will become a powerful tool for modelling fuel performance from its initial assembly, lifetime in the reactor and final disposition.

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Figure 2. Examples of simulations using the MBM suite of codes, where (a) shows a three-dimensional BISON simulation of a five pellet rodlet with a manufacturing defect in the centre pellet

[16] and (b) shows a three-dimensional MARMOT calculation that predicts the effective thermal conductivity during bubble growth in UO2 [19]

Conclusion and future challenges

FEM is a powerful numerical tool that has been used for many years in a large range of applications. For fuel modelling, FEM has been effectively used to model fuel performance and to investigate the impact of radiation damage on the fuel material behaviour. Now, the US NEAMS project is using FEM as the basis of the MBM project that is developing a multi-dimensional, multi-physics fuel performance capability that is massively parallel and will use multi-scale materials models to provide a truly predictive modelling capability. Though FEM has been successfully applied to fuels modelling, there are challenges. For macroscale fuel performance modelling, one challenge is in directly modelling the fragmentation of the fuel pellet, as discrete crack formation and propagation is difficult with FEM. Generalist FE codes, like CAST3M FE, can model this behaviour to some extent, as demonstrated in [24] in which a viscoplastic law for creep is coupled with a multi-surface plastic softening law for cracking to represent crack development in PWR fuel pellets. A challenge comes in being able to perform massively parallel simulations with such models. Another challenge is modelling the contact between fuel and cladding, especially in 3D. An additional challenge arises as efforts are made to concurrently couple microstructure FEM models with fuel performance codes, due to difficulties in bridging the disparate length and timescales.

References

[1] Berna, G.A., C.E. Beyer, K.L. Davis, D.D. Lanning (1997), FRAPCON-3: A Computer Code for the Calculation of Steady-state, Thermal- Mechanical Behavior of Oxide Fuel Rods for High Burnup, Technical Report NUREG/CR-6534 Vol. 2, PNNL-11513.

[2] Lassmann, K., A. Schubert, J. van de Laar (2003), TRANSURANUS Handbook, Technical Report Document Number Version 1 Modification 1, (V1M1J2003).

[3] Kilgour, W.J., J.A. Turnbull, R.J. White, A.J. Bull, P.A. Jackson, I.D. Palmer (1992), “Capabilities and validation of the ENIGMA fuel performance code”, Proceedings of the ENS Meeting on LWR Fuel Performance, Avignon, France.

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[4] Ito, K., M. Ichikawa, T. Okubo, Y. Iwano (1983), “FEMAXI-III, a computer code for fuel rod performance analysis”, Nuclear Engineering Design, 76, 3-11.

[5] Baron, D., P. Thevenin, R. Largenton, R. Masson, S. Pujet, R. Arnaud (2008), “Cyrano 3 the EDF fuel performance code especially designed for engineering applications”, Water Reactor Fuel Performance Meeting Proceeding, Seoul.

[6] Rashid, Y., R. Dunham, R. Montgomery (2004), Fuel Analysis and Licensing Code: FALCON MOD01, Technical Report EPRI 1011308, Electric Power Research Institute.

[7] Rashid, J., S. Yagnik, R. Montgomery (2011), “Light water reactor fuel performance modeling and multi-dimensional simulation”, JOM-International Journal of Minerals, Metallurgy and Materials S, 63, 81–88.

[8] Michel, B., C. Nonon, J. Sercombe, F. Michel, V. Marelle (2013), “Simulation of pellet-cladding interaction with the PLEIADES fuel performance software environment”, Nuclear Technology, 182(2), 124-137.

[9] Thouvenin, G., B. Michel, J. Sercombe (2007), “Multidimensional modeling of a ramp test with the PWR fuel performance code ALCYONE”, Proceedings of the 2007 International LWR Fuel Performance Meeting, San Francisco, California, US.

[10] Oguma, M. (1983), “Cracking and relocation behavior of nuclear fuel pellets during rise to power”, Nuclear Engineering and Design, 76, 35-45.

[11] Ito, K., R. Iwasaki, Y. Iwano (1985), “Finite element model for analysis of fission gas release from UO2 fuel”, Journal of Nuclear Science and Technology, 22(2) 129-138.

[12] Bakker, K. et al. (1995), EHP Cordfunke, “Determination of a porosity correction factor for the thermal conductivity of irradiated UO2 fuel by means of the finite element method”, Journal of Nuclear Materials, 226, 128-143.

[13] Bakker, K. (1997), “Using the finite element method to compute the influence of complex porosity and inclusion structures on the thermal and electrical conductivity”, International Journal of Heat and Mass Transfer, 40(15) 3503-3511.

[14] Teague, M., M.R. Tonks, S. Novascone, S Hayes (2013), “Microstructure modeling of thermal conductivity of high burn-up mixed oxide fuel”, Journal of Nuclear Materials.

[15] Gaston, D., C. Newman, G. Hansen, D. Lebrun-Grandie (2009), “MOOSE: A parallel computational framework for coupled systems of nonlinear equations”, Nuclear Engineering and Design, 239, 1768-1778.

[16] Williamson, R., J. Hales, S. Novascone, M. Tonks, D. Gaston, C. Permann, D. Andrs, R. Martineau, J. Nucl. Mater. 2012, 423, 149–163.

[17] Hales, J.D. et al. (2013), “Multidimensional multiphysics simulation of TRISO particle fuel”, Journal of Nuclear Materials, 443, 531-543.

[18] Tonks, M.R. et al. (2013), “Multiscale development of a fission gas thermal conductivity model: Coupling atomic, meso and continuum level simulations”, Journal of Nuclear Materials, 440, 193-200.

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[19] Tonks, M.R. et al. (2012), “An object-oriented finite element framework for multiphysics phase field simulations”, Computational Materials Science, 51, 20-29.

[20] Zhang, L. et al. (2012), “Phase-field modeling of temperature gradient driven pore migration coupling with thermal conduction”, Computational Materials Science, 56, 161-165.

[21] Tonks, M.R. et al. (2013), “Guidance to design grain boundary mobility experiments with molecular dynamics and phase-field modeling”, Acta Materialia, 2013, 61, 1373-1382.

[22] Tonks, M.R., Y. Zhang, X. Bai, P.C. Millett (2013), “Demonstrating the temperature gradient impact on grain growth in UO2 using the phase field method”, Materials Research Letters, 2013, in press.

[23] Millett, P.C. et al. (2013), “Three-dimensional calculations of the effective Kapitza resistance of UO2 grain boundaries containing intergranular bubbles”, Journal of Nuclear Materials, 439, 117-122.

[24] Michel, B. et al. (2008), “3D fuel cracking modelling in pellet cladding mechanical interaction”, Engineering Fracture Mechanics, 75(11), 3581-3598.

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