# Introduction

Bison (Williamson et al., 2012) is a finite element-based nuclear fuel performance code applicable to a variety of fuel forms including light water reactor fuel rods, TRISO particle fuel (Hales et al., 2013), and metallic rod (Medvedev, 2012) and plate fuel. It solves the fully-coupled equations of thermomechanics and species diffusion, for either 1D spherical, 2D axisymmetric or 3D geometries. Fuel models are included to describe temperature and burnup dependent thermal properties, fission product swelling, densification, thermal and irradiation creep, fracture, and fission gas production and release. Plasticity, irradiation growth, and thermal and irradiation creep models are implemented for clad materials. Models are also available to simulate gap heat transfer, mechanical contact, and the evolution of the gap/plenum pressure with plenum volume, gas temperature, and fission gas addition. Bison is based on the MOOSE framework (Gaston et al., 2009) and can therefore efficiently solve problems using standard workstations or very large high-performance computers.

## High-Level Description of a Bison Simulation

The primary purpose of Bison is to solve coupled systems of partial differential equations (PDEs), where the equations represent important physics related to engineering scale nuclear fuel behavior. Fuel simulations typically consist of solving the following energy, momentum, and mass (or species) conservation equations:

(1)

(2)

(3)

In Equation Eq. 1, , and are the temperature, density and specific heat, respectively, is the energy released in a single fission event, and is the volumetric fission rate.

Momentum conservation (Eq. 2) is prescribed assuming static equilibrium at each time increment where is the Cauchy stress tensor and is the body force per unit mass (e.g. gravity). The displacement field , which is the primary solution variable, is connected to the stress field via the strain, through a constitutive relation.

In the equation for species conservation (Eq. 3) , , and are the concentration, radioactive decay constant, and source rate of a given species, respectively.

Often, fuels performance problems are limited to thermomechanics, where only Equations Eq. 1 and Eq. 2 are solved.

Each term in Eq. 1 - Eq. 3 (time derivatives, divergence, source, sinks, etc.) are referred to as Kernels and are discussed in greater detail here.

These equations are solved simultaneously using the finite element method (FEM) and JFNK approach (Knoll and Keyes, 2004) on a discretized domain. The domain (also referred to as a mesh) may represent uranium dioxide fuel pellets and zirconium clad in a light water reactor (LWR) simulation. Blocks, side sets, and node sets are defined on the mesh such that material models and boundary conditions can be assigned to different parts of the model. Details regarding the mesh, material models, and boundary conditions can be found in the hyperlinks provided on each.

## SI Units

Because Bison uses several empirical models, Bison input expects SI units. This convention simplifies model input by eliminating the possibility of one set of units for one model and another set of units for a different model. Any needed unit conversions are done inside Bison.

## References

1. D. Gaston, C. Newman, G. Hansen, and D. Lebrun-GrandiÃ©. MOOSE: a parallel computational framework for coupled systems of nonlinear equations. Nuclear Engineering and Design, 239:1768â€“1778, 2009. URL: http://dx.doi.org/10.1016/j.nucengdes.2009.05.021.[BibTeX]
2. J. D. Hales, R. L. Williamson, S. R. Novascone, D. M. Perez, B. W. Spencer, and G. Pastore. Multidimensional multiphysics simulation of TRISO particle fuel. Journal of Nuclear Materials, 443:531â€“543, November 2013. URL: http://dx.doi.org/10.1016/j.jnucmat.2013.07.070, doi:10.1016/j.jnucmat.2013.07.070.[BibTeX]
3. D. A. Knoll and D. E. Keyes. Jacobian-free Newton-Krylov methods: a survey of approaches and applications. Journal of Computational Physics, 193(2):357â€“397, 2004.[BibTeX]
4. Pavel Medvedev. Fuel performance modeling results for representative FCRD irradiation experiments: projected deformation in the annular AFC-3A U-10Zr fuel pins and comparison to alternative designs. Technical Report INL/EXT-12-27183 Revision 1, Idaho National Laboratory, 2012.[BibTeX]
5. R. L. Williamson, J. D. Hales, S. R. Novascone, M. R. Tonks, D. R. Gaston, C. J. Permann, D. Andrs, and R. C. Martineau. Multidimensional multiphysics simulation of nuclear fuel behavior. Journal of Nuclear Materials, 423:149â€“163, 2012. URL: http://dx.doi.org/10.1016/j.jnucmat.2012.01.012, doi:10.1016/j.jnucmat.2012.01.012.[BibTeX]