BISON TRISO Workshop

Implicit, parallel, fully-coupled nuclear fuel performance analysis



Computational Mechanics and Materials Department
Idaho National Laboratory

BISON Team Members

BISON History

NEAMS/CASL Fuels Programs

Historical Overview

  • Overarching objective to deliver an integrated set of predictive computational tools for nuclear fuel performance analysis and design

  • A multiscale approach has been adopted in which engineering-scale simulations are informed by mesoscale simulations of microstructure evolution, which are enabled by parameters obtained from atomistic simulations

  • Primary products are BISON for engineering-scale analysis and Marmot for mesoscale analysis, both built upon the MOOSE computational framework

LWR Missing Pellet Surface Analysis

Defective TRISO Particle

Bison: What is it?

  • Bison once numbered in the tens of millions and ranged over much of North America.

  • Bison can weigh 1000+ kg and stand 1.8+ m high at the shoulder.

  • Bison can jump 1.8 m vertically.

  • Bison can run 60+ km/h.

BISON: What is it?

A finite element, thermo-mechanics code with material models and other customizations to analyze nuclear fuel

  • Accepts user-defined meshes/geometries

    • 1D, 2D, or 3D

  • Runs on one processor or many

  • Analyzes a variety of fuel types

  • Couples to other analysis codes

BISON Fuel Performance Code

  • Finite element-based engineering scale fuel performance code

  • Solves the fully-coupled thermomechanics and species diffusion equations in 1D, 1.5D, 2D axisymmetric or plane-strain, or full 3D

  • Used for LWR, ATF, TRISO, and metallic fuels

  • Applicable to both steady and transient operations and includes LOCA and RIA capability for LWR fuel

  • Readily coupled to lower length scale material models

  • Designed for efficient use on parallel computers

  • Development follows NQA-1 process

2D axisymmetric (or 1.5D)

2D plane strain

3D

BISON Requirements and Limitations

  • BISON requires:

    -An input file that describes thermal and mechanical material models, boundary conditions, initial conditions, power history - A mesh provided either directly in the input file or through a separate mesh file

  • BISON cannot currently model:

    • Very high strain rate analyses (e.g., car crashes)

    • Structural elements (membranes, shells, beams)

    • Melting or flowing material

  • BISON is not:

    • A thermal-hydraulics or CFD code

    • A neutronics code

BISON Governing Equations

  • Energy conservation (transient heat conduction with fission source)

    • is energy per fission. is fission density rate.

  • Species conservation (transient oxygen or fission product diffusion with radioactive decay)

    • Equation includes Fickian diffusion, Soret diffusion, radioactive decay, and a source term.

  • Momentum conservation (Cauchy's equation of equilibrium)

    • is the stress tensor, and is a body force.

Fuel Behavior: Introduction

At beginning of life, a fuel element is quite simple...

Nakajima et al., Nuc. Eng. Des., 148, 41 (1994)

but irradiation brings about substantial complexity...

Michel et al., Eng. Frac. Mech., 75, 3581 (2008)

Fuel Fracture

Olander, p. 584 (1978)

Multidimensional Contact
and Deformation

Olander, p. 584 (1978)

Fission Gas

Bentejac et al., PCI Seminar (2004)

Stress Corrosion
Cracking Cladding
Failure

Fuel Behavior Modeling: Coupled Multiphysics

Multiphysics

  • Fully coupled nonlinear thermo-mechanics

  • Multiple species diffusion

  • Neutronics

  • Thermal-hydraulics

  • Chemistry

Multi-space scale

  • Important physics at the atomistic and micro-structural levels

  • Practical engineering simulations require the continuum level

Multi-time scale

  • Steady operation ( week)

  • Power ramps/accidents
    ( s)

BISON Example - Axisymmetric LWR Fuel Rodlet

BISON Results - Axisymmetric LWR Fuel Rodlet

  • Thermal expansion, fuel densification, clad creep-down, fission gas release, contact, and burnup dependent fuel thermal conductivity all affect fuel temperatures

  • Hourglass shape of pellets is evident in gap closure histories

BISON Results - Axisymmetric LWR Fuel Rodlet

  • Fission gas release begins at a burnup of 10 MWd/kgU

  • Hourglass shape of pellets creates ridges in clad during PCMI

BISON Example - Missing Pellet Surface

  • High-resolution 3D calculation (25,000 elements, 1.1 dof) run on 120 processors

  • Simulation starting from a fresh fuel state with a typical power history, followed by a late-life power ramp

BISON Results - Missing Pellet Surface

Fuel Temperature

Clad Temperature

  • Missing pellet surface has a very significant effect on the temperature and stress state in the rod

  • Model can be used to examine source of rod failures

Clad Stress

MOOSE, BISON, and Marmot

MOOSE, BISON, and Marmot provide an advanced multidimensional, multiphysics, multiscale fuel performance capability

Atomistic/Mesoscale Material Model Development

  • Predicts microstructure evolution in fuel and cladding

  • Used with atomistic methods to develop multiscale material models

Advanced Multidimensional Fuel Performance Code

  • Models a wide variety of fuel types and geometries at an engineering scale

  • Applicable for steady, transient and accident conditions

  • Simulation framework allowing rapid development of FEM-based applications

MOOSE

MOOSE

TRISO in BISON circa 2013 and Today

TRISO in BISON circa 2013

  • Rich Williamson started BISON with INL LDRD funds over ten years ago.

  • In 2012, the first major BISON paper was published.

  • At about that time, the BISON team decided to add a baseline TRISO capability, including thermal, mass diffusion, and mechanical models.

BISON TRISO Paper, JNM 2013

http://dx.doi.org/10.1016/j.jnucmat.2013.07.070

BISON TRISO Paper, JNM 2013

TRISO Stagnation

  • After the 2013 paper, we talked with potential stakeholders about our capability.

  • But, very little interest.

  • No interest -> no funding -> no development.

  • Added an internal 1D TRISO mesh generation capability in 2018.

  • Things started to pick up in FY19.

Recent Interest in TRISO in BISON

Recent TRISO Development

A more complete set of models from PARFUME

  • Elastic properties

  • Thermal properties

  • Mass diffusion properties

  • Kernel

    • Swelling

    • Fission gas release

  • Buffer

    • Creep

    • Irradiation strain

  • PyC

    • Creep

    • Irradiation strain

  • SiC

    • Palladium penetration

  • Matrix

Recent TRISO Development Continued: Monte Carlo Scheme for Predicting Failure

under construction

Organization of BISON Code, Tests, and Examples

BISON Repository Layout

This is the view from https://github.inl.gov/ncrc/bison.

BISON src Directory

The layout of the include directory is very similar. Source files (.C files) will have corresponding include files (.h files).

BISON test/tests Directory


ADMetallicFuelWastage/
Al2O3/
GrainRadiusPorosity_test/
GraphiteMatrixElasticityTensor/
GraphiteMatrixSpecificHeat/
GraphiteMatrixThermalConductivity/
MetallicFuelWastage/
OxideEnergyDeposition/
SiCPdPenetration/
ThermalFuel_error_messages/
ad_arrhenius_material_property/
ad_d9_thermal/
ad_ht9_thermal/
ad_mc_thermal/
ad_ss316_thermal/
ad_upuzr_burnup/
ad_upuzr_fast_neutron_flux/
ad_upuzr_fission_gas_release/
ad_upuzr_fission_rate/
ad_upuzr_sodium_logging/
ad_upuzr_thermal/
anisotropic_swelling/
arrhenius_diffusion_coef/
arrhenius_material_property/
average_axial_position/
axial_relocation/
burnup_action/
carbon_monoxide_production/
check_error/
circular_cross_section_mesh/
constitutive_heat_conduction/
convective_heat_transfer/
coolant_channel_model/
creep_SiC/
creep_U10Mo/
creep_mox/
creep_uo2/
cumulative_damage_index/
decay_heating/
diffusion_limited_reaction/
dislocation_density/
dryCask/
effective_burnup_aux/
element_integral_power/
example_problem_test/
fast_neutron_flux/
fcci_ht9/
fecral_oxidation/
fgr_fraction/
fgr_percent/
fgr_upuzr/
fill_gas_thermal_conductivity/
fission_gas_1d/
sifgrs/uo2/



sifgrs/u3si2/
fission_gas_release_formas/
fission_rate_LWR/
fission_rate_MOX/
fission_rate_axial/
fission_rate_from_power_density/
fission_rate_heat_source/
fuelrodlinevaluesampler/
gamma_heating/
gap_heat_transfer/
gap_heat_transfer_fission/
gap_heat_transfer_htonly/
gap_heat_transfer_mixedgas/
gap_heat_transfer_radiation/
gap_jump_distance/
gap_perfect_transfer/
generic_material_failure/
grain_radius_aux/
hydride/
hydrogen/
ifba_he_production/
irradiation_growth/
irradiation_growth_Zr4/
layered2D/
layered_1D/
mechTests/
mechZry/
mechanical_uo2/
meso_thcond_test/
monolithicSiCThermal/
mox_oxygen_to_metal_ratio/
mox_pore_velocity/
oxygen_aux/
oxygen_transport/
partial_sum_heat_flux/
percolation/
performance_outputs_action/
phase_transition_zircaloy/
phase_upuzr/
plate_mesh/
plenum_pressure/
plenum_temp/
power_peaking_function/
radial_avg_fuel_enthalpy/
radial_crack_counter/
radial_power_factor/
radioactive_decay/
radius_aux/
relocation_UO2/
side_ave_incr_tensor_component/
side_int_var_incr_postprocess/
side_integral_mass_flux/
fuel_pin_mesh/
fuel_pin_mesh_fipd/



fuel_pin_mesh_generator/
fuel_pin_mesh_generator_fipd/
sodium_coolant_channel/
solid_mechanics_deprecated/
species_source/
stan_neumann/
standard_lwr_outputs_action/
submodel_end_bc/
temperature_jump_distance/
solid_mechanics/
thermalChromium/
thermalCompositeSiC/
thermalD9/
thermalFastMOX/
thermalFeCrAl/
thermalFuel_Amaya/
thermalFuel_Duriez/
thermalFuel_FinkLucuta/
thermalFuel_Halden/
thermalFuel_HaldenMOX/
thermalFuel_HaldenUO2/
thermalFuel_NFIR/
thermalFuel_NFImod/
thermalFuel_Ronchi/
thermalFuel_Staicu/
thermalFuel_Toptan/
thermalFuel_rimLayer/
thermalHT9/
thermalMAMOX/
thermalMOX/
thermalNa/
thermalSilicideFuel/
thermalTests/
thermalUO2/
thermalZrO2/
thermalZry/
thermal_accommodation_coeff/
thermirrad_creep_zr42/
thermo_mech_oxygen/
triso/
triso_failure/
un_swelling/
upuzr_burnup/
upuzr_dictra/
upuzr_diffusivity/
upuzr_fast_neutron_flux/
upuzr_fission_rate/
upuzr_phase_lookup/
void_volume/
zirconium_diffusion/
zrdiffusivity_upuzr/
zrh_formation/
zry_plasticity/

BISON has about 1800 regression tests

BISON examples Directory


1.5D_restart/
1.5D_rodlet_10pellets/
2D-RZ_rodlet_10pellets/
2D_plane_strain_rod/
3D_rodlet_3pellets/
TRISO/
accident_tolerant_fuel/



axial_relocation/
fast_mox_sifgrs/
hydride_rim/
metal_fuel/
mox_fuel/
multiapp/



non-cylindrical_fuel/
percolation/
pore_migration/
restart/
spent_fuel/
temperature_tables/

BISON examples/TRISO/full_particle/1D Directory


examples
full_particle_1D.i
gold
tests

Example BISON Source File

src/materials/solid_mechanics/UCOVolumetricSwellingEigenstrain.C

/*************************************************/
/*           DO NOT MODIFY THIS HEADER           */
/*                                               */
/*                     BISON                     */
/*                                               */
/*    (c) 2015 Battelle Energy Alliance, LLC     */
/*            ALL RIGHTS RESERVED                */
/*                                               */
/*   Prepared by Battelle Energy Alliance, LLC   */
/*     Under Contract No. DE-AC07-05ID14517      */
/*     With the U. S. Department of Energy       */
/*                                               */
/*     See COPYRIGHT for full restrictions       */
/*************************************************/

#include "UCOVolumetricSwellingEigenstrain.h"

registerMooseObject("BisonApp", UCOVolumetricSwellingEigenstrain);
registerMooseObject("BisonApp", ADUCOVolumetricSwellingEigenstrain);

template <bool is_ad>
InputParameters
UCOVolumetricSwellingEigenstrainTempl<is_ad>::validParams()
{
  InputParameters params = ComputeEigenstrainBaseTempl<is_ad>::validParams();
  params.addClassDescription(
      "Computes fission-induced swelling (percent per percent FIMA) for UCO.");
  params.addParam<Real>("swelling_rate", 2.9, "Swelling rate (%).");
  params.addParam<Real>("swelling_scale_factor", 1.0, "Multiplier for UCO swelling");
  return params;
}

template <bool is_ad>
UCOVolumetricSwellingEigenstrainTempl<is_ad>::UCOVolumetricSwellingEigenstrainTempl(
    const InputParameters & parameters)
  : ComputeEigenstrainBaseTempl<is_ad>(parameters),
    _swelling_rate(parameters.get<Real>("swelling_rate")),
    _swelling_scale_factor(parameters.get<Real>("swelling_scale_factor")),
    _burnup(this->template getGenericMaterialProperty<Real, is_ad>("burnup")),
    _swelling(this->template declareGenericProperty<Real, is_ad>("swelling"))
{
}

template <bool is_ad>
void
UCOVolumetricSwellingEigenstrainTempl<is_ad>::initQpStatefulProperties()
{
  _swelling[_qp] = 0;
  ComputeEigenstrainBaseTempl<is_ad>::initQpStatefulProperties();
}

template <bool is_ad>
void
UCOVolumetricSwellingEigenstrainTempl<is_ad>::computeQpEigenstrain()
{
  GenericReal<is_ad> volumetric_swelling_strain =
      _swelling_scale_factor * _swelling_rate * _burnup[_qp];
  GenericReal<is_ad> strain_component =
      this->computeVolumetricStrainComponent(volumetric_swelling_strain);
  _swelling[_qp] = volumetric_swelling_strain;

  _eigenstrain[_qp].zero();
  _eigenstrain[_qp].addIa(strain_component);
}

template class UCOVolumetricSwellingEigenstrainTempl<false>;
template class UCOVolumetricSwellingEigenstrainTempl<true>;
(src/materials/solid_mechanics/UCOVolumetricSwellingEigenstrain.C)

Example BISON Test

test/tests/triso/UCOVolumetricSwellingEigenstrain/tests

[Tests]
  parallel_scheduling = True
  [UCOVolumetricSwellingEigenstrain]
    type = 'CSVDiff'
    input = 'UCOVolumetricSwellingEigenstrain.i'
    csvdiff = 'UCOVolumetricSwellingEigenstrain_out.csv'
    requirement = "The system shall calculate volumetric swelling of UCO."
    design = 'UCOVolumetricSwellingEigenstrain.md'
    issues = '#1074'
  []
  [ad_UCOVolumetricSwellingEigenstrain]
    type = 'CSVDiff'
    input = 'ad_UCOVolumetricSwellingEigenstrain.i'
    csvdiff = 'ad_UCOVolumetricSwellingEigenstrain_out.csv'
    requirement = "The system shall calculate volumetric swelling of UCO using automatic differentiation."
    design = 'UCOVolumetricSwellingEigenstrain.md'
    issues = '#6003'
  []
[]
(test/tests/triso/UCOVolumetricSwellingEigenstrain/tests)

Example BISON Test Continued

test/tests/triso/UCOVolumetricSwellingEigenstrain/UCOVolumetricSwellingEigenstrain.i

# UCO fission-induced swelling
# The geometry is a unit cube made of UCO subject to burnup-induced swelling.
#
# The swelling is simply 2.9 * burnup.  Burnup is ramped from 0 to 0.125.
# Thus, swelling increases to 0.3625.  The final volume is 1.36256.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
  volumetric_locking_correction = true
[]

[Mesh]
  coord_type = XYZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmax = 1.0
    ymax = 1.0
    zmax = 1.0
  []
[]

[AuxVariables]
  [swelling]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [burnup]
    type = PiecewiseLinear
    x = '0 10'
    y = '0 0.125'
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  add_variables = true
  [stuff]
    block = '0'
    strain = FINITE
    eigenstrain_names = 'UCO_swelling_eigenstrain'
    generate_output = 'vonmises_stress stress_xx stress_yy stress_zz'
  []
[]

[AuxKernels]
  [swelling]
    type = MaterialRealAux
    variable = swelling
    property = swelling
    block = '0'
    execute_on = linear
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0
  []

  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom'
    value = 0
  []

  [no_z]
    type = DirichletBC
    variable = disp_z
    boundary = 'back'
    value = 0
  []
[]

[Materials]

  [burnup]
    type = GenericFunctionMaterial
    prop_names = burnup
    prop_values = burnup
  []

  [UCO_VolumetricSwellingEigenstrain]
    type = UCOVolumetricSwellingEigenstrain
    eigenstrain_name = UCO_swelling_eigenstrain
  []

  [stress]
    type = ComputeFiniteStrainElasticStress
  []

  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 10
    poissons_ratio = 0.4
  []

  [UCO_density]
    type = StrainAdjustedDensity
    block = '0'
    strain_free_density = 11250.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10

  start_time = 0.0
  end_time = 10.0
  dt = 1.0

[]

[Postprocessors]
  [_dt]
    type = TimestepSize
  []

  [volume_UCO]
    type = VolumePostprocessor
    use_displaced_mesh = true
  []

  [swelling_UCO_max]
    type = ElementExtremeValue
    value_type = 'max'
    variable = swelling
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]
(test/tests/triso/UCOVolumetricSwellingEigenstrain/UCOVolumetricSwellingEigenstrain.i)

Running and Evaluating a Model in BISON

Running and Evaluating a Model in BISON

  • Let's look at test/tests/triso/UCOElasticityTensor/UCOElasticityTensor.i

  • We will

    1. Walk through the input file step by step.

    2. Run the file and review the information printed to the terminal.

    3. Examine the CSV and Exodus output.

1. Review Input File

In the terminal or an editor, view test/tests/triso/UCOElasticityTensor/UCOElasticityTensor.i

# Elastic Properties of UCO
# The geometry is a unit cube made of UCO material (initial density = 11.25 g/cm^3)
#   subject to elastic strain.
# Displacement boundary conditions are applied to induce a strain in the x-axis
#   only such that the density becomes 8 g/cm^3.
# The temperature is varied from 673.15 to 2073.15 K.
#
# The analytic solution for stress xx is compared to the BISON result in the xlsx file.
#
# Sample from the xlsx file:
#
# Temp (C) | stress_xx | BISON stress_xx
# ---------+-----------+----------------
#  400.00  | 0.00E+00  | 0.00E+00
#  470.00  | 6.24E+09  | 6.24E+09
#  540.00  | 1.15E+10  | 1.15E+10
#  610.00  | 1.59E+10  | 1.59E+10
#  680.00  | 1.96E+10  | 1.96E+10

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
  order = FIRST
  family = LAGRANGE
  initial_enrichment = 0.15 #[wt-]
  O_U = 1.5
  C_U = 0.4
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    xmax = 1.0
    ymax = 1.0
    zmax = 1.0
  []
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [disp_z]
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 673.15
  []
  [density]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  # Define auxiliary kernels for each of the aux variables
  [density]
    type = MaterialRealAux
    variable = density
    property = density
    execute_on = 'initial timestep_end'
  []
  [temperature]
    type = FunctionAux
    variable = temperature
    function = temp_function
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0      1e3'
    y = '673.15 2073.15'
  []

  [disp_x]
    type = PiecewiseLinear
    x = '0.0 1e3'
    y = '0.0 0.40625'
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [perm_UCO]
    strain = SMALL
    generate_output = 'vonmises_stress stress_xx stress_yy stress_zz strain_xx'
  []
[]

[BCs]
  # Define boundary conditions
  [no_z]
    type = DirichletBC
    variable = disp_z
    boundary = 'back front'
    value = 0
  []

  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom top'
    value = 0
  []

  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0
  []

  [x]
    type = FunctionDirichletBC
    variable = disp_x
    boundary = 'right'
    function = disp_x
  []
[]

[Materials]

  [stress]
    type = ComputeLinearElasticStress
  []

  [elasticity_tensor]
    type = UCOElasticityTensor
    temperature = temperature
  []

  [density]
    type = StrainAdjustedDensity
    strain_free_density = 11250.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10

  start_time = 0.0
  end_time = 1e3

  dtmax = 2e6
  dtmin = 1.0
  dt = 50

[]

[Postprocessors]
  [temp]
    type = ElementExtremeValue
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [sigma_xx]
    type = ElementExtremeValue
    value_type = 'max'
    variable = stress_xx
    execute_on = 'initial timestep_end'
  []

  [sigma_yy]
    type = ElementExtremeValue
    value_type = 'max'
    variable = stress_yy
    execute_on = 'initial timestep_end'
  []

  [strain_xx]
    type = ElementExtremeValue
    value_type = 'max'
    variable = strain_xx
    execute_on = 'initial timestep_end'
  []

  [density]
    type = ElementExtremeValue
    value_type = 'max'
    variable = density
    execute_on = 'initial timestep_end'
  []

  [disp_x]
    type = NodalExtremeValue
    value_type = 'max'
    variable = disp_x
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]
(test/tests/triso/UCOElasticityTensor/UCOElasticityTensor.i)

2. Run Input File

  • First, compile bison-opt.

  • Next, go to the directory containing the input file.

  • Finally, run the input file.

    
make
cd test/tests/triso/UCOElasticityTensor
../../../../bison-opt -i UCOElasticityTensor.i

  • Information to the terminal: BISON header and version, problem information, solve progress, and Postprocessor values.

3. Examine Results

  • First, view the CSV file.

    
cat UCOElasticityTensor_out.csv

  • Next, view the Exodus output file.

  • There isn't one!

  • Use the command line to add the Exodus output option.

    
../../../../bison-opt -i UCOElasticityTensor.i Outputs/exodus=true

  • Load the Exodus output file in Paraview.

    
paraview UCOElasticityTensor_out.e

    (Requires paraview to be in your PATH)

IAEA Benchmark Cases

IAEA Benchmark Cases

IAEA cases studied in the 2013 BISON TRISO paper (IAEA, 1997).

CaseGeometryDescription
1SiC layerElastic only
2IPyC layerElastic only
3IPyC/SiC layerElastic with no fluence
4aIPyC/SiC layerSwelling and no creep
4bIPyC/SiC layerCreep and no swelling
4cIPyC/SiC layerCreep and swelling
4dIPyC/SiC layerCreep- and fluence-dependent swelling
5TRISO350 m kernel, real conditions
6TRISO500 m kernel, real conditions
7TRISOSame as 6 with high BAF pyC
8TRISOSame as 6 with cyclic temperature
10HFR-K310% FIMA, n/m fluence
11HFR-P410% FIMA, n/m fluence

Cases 1-3 have analytical solutions. Cases 4a-4d involve more complex behavior but are fully prescribed. Cases 5-8 apply realistic conditions to a single particle. Cases 10-11 leave internal pressure to be set by the fission gas release and CO production models in the analysis codes.

IAEA Benchmarks from BISON TRISO JNM Paper

Explore the Benchmark Cases

IAEA TECDOC 978

BISON TRISO JNM 2013


cd assessment/TRISO/benchmark/IAEA_CRP-6

Meshing

Mesh Generation

  • Clearly mesh generation depends on the type of the analysis to be run.

    • 1D (fast, spherically symmetric)

    • 2D (medium, axisymmetric)

    • 3D (slow, symmetry across planes or not symmetric)

BISON

BISON/CUBIT

CUBIT

Example BISON Documentation File

  • The documentation for BISON is developed in Markdown (.md) and viewed in an internet browser.

  • Both theory and user information are captured in the files.

  • The documentation is linked directly with the code and therefore is automatically updated.

  • See an example for TRISO1DMeshGenerator.

1D Mesh Generation in BISON

  • Mesh generation for 1D TRISO in BISON is done using TRISO1DMeshGenerator.

  • TRISO1DMeshGenerator supports an arbitrary number of layers.

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 2.485e-4 3.425e-4 3.425e-4 3.835e-4 4.195e-4 4.595e-4'
    mesh_density = '6 6 0 6 8 6'
    block_names = 'fuel buffer IPyC SiC OPyC'
  []
[]
(assessment/TRISO/benchmark/IAEA_CRP-6/fuel_performance/case_11/case_11_1D.i)

1D Mesh Generation in BISON

  • Run Case 11.


> cd assessment/TRISO/benchmark/IAEA_CRP-6/fuel_performance/case_11
> ../../../../../bison-opt -i case_11_1D.i

1D Mesh Generation in BISON

  • View mesh for Case 11.


> paraview case_11_1D_out.e

  • Orient -Z.

  • Change line width.

  • Note block names and sideset names.

2D Mesh Generation in BISON

  • CircularCrossSectionMeshGenerator will create quarter- or half- circle meshes for axisymmetric analysis.

  • This tool was built with LWR fuel in mind but can be used for TRISO meshes.

  • This tool is a bit more involved. See the documentation.

Mesh Generation with CUBIT

TRISO Thermal Models

TRISO Thermal Models

  • Thermal modeling for TRISO fuel follows thet same pattern as for any other fuel:

  • That is, we need to

    1. Define density, specific heat, and thermal conductivity, which may be functions of temperature or other parameters.

    2. Ef is the energy released in a single fission event and F dot is the volumetric fission rate.

    3. Invoke the heat conduction and heat conduction time derivative kernels.

  • Often, thermal properties are specified as constants.

  [IPyC_thermal]
    type = HeatConductionMaterial
    block = IPyC
    thermal_conductivity = 4.0
    specific_heat = 720.0
  []

  [IPyC_density]
    type = GenericConstantMaterial
    block = IPyC
    prop_names = 'density'
    prop_values = 1900.0
  []
(examples/TRISO/parfume/parfume.i)

  • Thermal properties may also be computed in a Material object.

[Materials]
  [UCO_thermal]
    type = UCOThermal
    block = fuel
    temperature = temperature
  []

  [buffer_thermal]
    type = BufferThermal
    block = buffer
    initial_density = 1050.0
  []

  [SiC_thermal]
    type = MonolithicSiCThermal
    block = SiC
    temperature = temperature
    thermal_conductivity_model = miller
  []
[]
(examples/TRISO/parfume/parfume.i)

  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
    extra_vector_tags = 'ref'
  []
  [heat]
    type = HeatConduction
    variable = temperature
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = NeutronHeatSource
    variable = temperature
    block = fuel
    fission_rate = fission_rate
    extra_vector_tags = 'ref'
  []
(examples/TRISO/parfume/parfume.i)
[ThermalContact]
  [thermal_contact]
    type = GasGapHeatTransfer
    variable = temperature
    primary = IPyC_inner_boundary
    secondary = buffer_outer_boundary
    initial_moles = initial_moles # coupling to a postprocessor which supplies the initial plenum/gap gas mass
    initial_gas_types = 'Kr Xe'
    initial_fractions = '0.185 0.815'
    gas_released = 'fis_gas_released'
    released_gas_types = 'Kr Xe'
    released_fractions = '0.185 0.815'
    tangential_tolerance = 1e-6
    roughness_primary = 0e-6
    roughness_secondary = 0e-6
    jumpdistance_primary = 0
    jumpdistance_secondary = 0
    quadrature = true
    emissivity_secondary = 0.0
    emissivity_primary = 0.0
    min_gap = 1e-7
    max_gap = 50e-6
    gap_geometry_type = sphere
  []
(examples/TRISO/parfume/parfume.i)

TRISO Mechanical Models

TRISO Mechanical Models

  • Mechanical modeling for TRISO fuel follows the same pattern as for any other fuel:

  • This is more involved than thermal modeling.

    1. Define constitutive response (define, e.g., an elasticity tensor and an object to convert strain to stress). 2. Define so-called eigenstrains (thermal strain, irradiation strain)

  • The code provides input shortcuts to invoke the computation of strain and the divergence of stress. This shortcut may or may not be used.

Simple Elasticity

  [SiC_elasticity_tensor]
    type = MonolithicSiCElasticityTensor
    block = SiC
    temperature = temperature
    elastic_modulus_model = miller
  []

  [SiC_stress]
    type = ComputeFiniteStrainElasticStress
    block = SiC
  []
(examples/TRISO/parfume/parfume.i)

Creep

  [IPyC_elasticity_tensor]
    type = PyCElasticityTensor
    block = IPyC
    temperature = temperature
    initial_BAF = 1.045
  []

  [IPyC_stress]
    type = PyCCEGACreep
    block = IPyC
    temperature = temperature
  []
(examples/TRISO/parfume/parfume.i)

Creep + irradiation Strain

[Materials]
  [IPyC_elasticity_tensor]
    type = PyCElasticityTensor
    block = IPyC
    temperature = temperature
    initial_BAF = 1.045
  []

  [IPyC_stress]
    type = PyCCEGACreep
    block = IPyC
    temperature = temperature
  []

  [IPyC_IIDC_strain]
    type = PyCCEGAIrradiationEigenstrain
    block = IPyC
    eigenstrain_name = IPyC_IIDC_strain
    temperature = temperature
  []
[]
(examples/TRISO/parfume/parfume.i)

Strain and Divergence of Stress

[Physics]
  [SolidMechanics]
    [QuasiStatic]
      [fuel]
        block = fuel
        add_variables = true
        strain = FINITE
        incremental = true
        generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
        eigenstrain_names = 'UCO_swelling_eigenstrain UCO_thermal_strain'
        extra_vector_tags = 'ref'
      []
      [buffer]
        block = buffer
        add_variables = true
        strain = FINITE
        incremental = true
        generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
        eigenstrain_names = 'buffer_IIDC_strain buffer_thermal_strain'
        extra_vector_tags = 'ref'
      []
      [IPyC]
        block = IPyC
        add_variables = true
        strain = FINITE
        incremental = true
        generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
        eigenstrain_names = 'IPyC_IIDC_strain IPyC_thermal_strain'
        extra_vector_tags = 'ref'
      []
      [SiC]
        block = SiC
        add_variables = true
        strain = FINITE
        incremental = true
        generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
        eigenstrain_names = 'SiC_thermal_eigenstrain'
        extra_vector_tags = 'ref'
      []
      [OPyC]
        block = OPyC
        add_variables = true
        strain = FINITE
        incremental = true
        generate_output = 'hydrostatic_stress stress_xx stress_yy stress_zz strain_xx strain_yy strain_zz'
        eigenstrain_names = 'OPyC_IIDC_strain OPyC_thermal_strain'
        extra_vector_tags = 'ref'
      []
    []
  []
[]
(examples/TRISO/parfume/parfume.i)

Mechanical Contact

[Contact]
  [mechanical]
    primary = IPyC_inner_boundary
    secondary = buffer_outer_boundary
    penalty = 1e5
    model = frictionless
    formulation = kinematic
  []
[]
(examples/TRISO/parfume/parfume.i)

TRISO Mechanical Models in BISON

  • If the material requires only linear elasticity and thermal expansion, no specialized models are required.

Fission Gas and Internal Pressure

Fission Gas Production and Release for TRISO Fuel

  • BISON provides two models for fission gas production and release:

    1. Sifgrs (Simple Integrated Fission Gas Release and Swelling)

    - For use with UO. - Used in all our LWR cases.

    2. UCOFGR (Booth model)

    - For use with UCO - New model developed in cooperation with Kairos Power.

Sifgrs

[Materials]
  [fission_gas_release]
    type = UO2Sifgrs
    block = fuel
    temperature = temperature
    fission_rate = fission_rate
    grain_radius_const = 5.0e-6
  []
[]
(assessment/TRISO/benchmark/IAEA_CRP-6/fuel_performance/case_11/case_11_1D.i)
[Postprocessors]
  [fis_gas_produced]
    type = ElementIntegralFisGasGeneratedSifgrs
    block = fuel
  []

  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = fuel
  []
[]
(assessment/TRISO/benchmark/IAEA_CRP-6/fuel_performance/case_11/case_11_1D.i)

UCOFGR

[Materials]
  [fission_gas_release]
    type = UCOFGR
    block = fuel
    average_grain_radius = 10e-6
    temperature = temperature
    triso_geometry = particle_geometry
    cutoff_neutron_flux = 0.0
  []
[]
(examples/TRISO/parfume/parfume.i)
[Postprocessors]
  [fis_gas_produced]
    type = ElementIntegralMaterialProperty
    mat_prop = fis_gas_produced
    block = fuel
    execute_on = 'initial timestep_end'
  []

  [fis_gas_released]
    type = ElementIntegralMaterialProperty
    mat_prop = fis_gas_released
    block = fuel
    execute_on = 'initial timestep_end'
  []
[]
(examples/TRISO/parfume/parfume.i)

Internal Pressure

  • BISON uses the ideal gas law to compute internal pressure.

  • The PlenumPressure object was built for use with LWRs, but it works just as well for TRISO.

  • It is a boundary condition and appears in the BCs area of the input file.

  • We must supply:

    • Volume (a Postprocessor value)

    • Gas temperature (a Postprocessor value)

    • Initial pressure (a raw number; used to compute initial moles of gas)

    • Added moles of gas over time (one or more Postprocessor values)

PlenumPressure

[BCs]
  [PlenumPressure]
    #  apply gas pressure on buffer and IPyC boundaries
    [plenumPressure]
      boundary = buffer_IPyC_boundary
      initial_pressure = 100.0
      startup_time = 0
      R = 8.3145
      output_initial_moles = initial_moles
      temperature = ave_gas_temp
      volume = 'gap_volume buffer_void_volume kernel_void_volume'
      material_input = 'fis_gas_released'
      output = gas_pressure
    []
  []
[]
(examples/TRISO/parfume/parfume.i)

InternalVolume

  • InternalVolume is a Postprocessor

  [gap_volume]
    type = InternalVolume
    boundary = buffer_IPyC_boundary
    execute_on = 'initial linear'
    use_displaced_mesh = true
  []

  [buffer_void_volume]
    type = VoidVolume
    block = buffer
    theoretical_density = 2250
    execute_on = 'initial timestep_end'
    use_displaced_mesh = true
  []

  [kernel_th_density]
    type = UCOTheoreticalDensity
    execute_on = initial
  []

  [kernel_void_volume]
    type = VoidVolume
    block = fuel
    theoretical_density = kernel_th_density
    execute_on = 'initial timestep_end'
    use_displaced_mesh = true
  []
(examples/TRISO/parfume/parfume.i)

Gas Temperature

  • For the gas temperature, we use SideAverageValue

[Postprocessors]
  [ave_temp_interior]
    type = SideAverageValue
    boundary = buffer_outer_boundary
    variable = temperature
    execute_on = 'initial timestep_end'
  []
[]
(examples/TRISO/parfume/parfume.i)

Added Moles of Gas

  • If using Sifgrs:

[Postprocessors]
  [fis_gas_released]
    type = ElementIntegralFisGasReleasedSifgrs
    block = fuel
  []
[]
(assessment/TRISO/benchmark/IAEA_CRP-6/fuel_performance/case_11/case_11_1D.i)

  • For CO production with UO fuel:

[Postprocessors]
  [co_production]
    type = CarbonMonoxideProduction
    total_fissions = total_fissions
    initial_enrichment = 0.14029
    execute_on = 'initial timestep_end'
  []
[]
(test/tests/carbon_monoxide_production/carbon_monoxide_production_test.i)

Fission Product Diffusion

TRISO Fission Product Diffusion Models

  • Fission product diffusion modeling for TRISO fuel follows the same pattern as for any other fuel:

  • That is, we need to

    1. Define diffusion coefficient D.

    2. Define decay (C) and source (S) terms.

    3. Invoke the diffusion kernels.

Define Diffusion Coefficient Material Properties

  • Be sure to define coefficient for each material.

[Materials]
  [fuel_conc]
    type = ArrheniusDiffusionCoef
    block = fuel
    d1 = 5.6e-8 # m^2/s
    q1 = 209.0e+3 # J/mol
    d2 = 5.2e-4 # m^2/s
    q2 = 362.0e+3 # J/mol
    gas_constant = 8.3143 # J/K-mol
    temperature = temp
  []

  [SiC_conc]
    type = ArrheniusDiffusionCoef
    block = SiC
    d1 = 5.5e-14 # m^2/s
    d1_function = d1_function
    d1_function_variable = fluence
    q1 = 125.0e+3 # J/mol
    d2 = 1.6e-2 # m^2/s
    q2 = 514.0e+3 # J/mol
    gas_constant = 8.3143 # J/K-mol
    temperature = temp
  []
[]
(examples/TRISO/accident_simulation/triso2D_accident.i)

Invoke Decay, Source, and Diffusion Kernels

[Kernels]
  [mass_ie]
    type = TimeDerivative
    variable = conc
    extra_vector_tags = 'ref'
  []

  [mass]
    type = ArrheniusDiffusion
    variable = conc
    extra_vector_tags = 'ref'
  []

  [mass_source]
    type = BodyForce
    variable = conc
    function = power_history
    value = 1.22e-5 # units of moles/m**3-s
    block = fuel
    extra_vector_tags = 'ref'
  []

  [mass_decay]
    type = Decay
    variable = conc
    radioactive_decay_constant = 7.297e-10 # units:(1/sec)  The constant for Cesium
    extra_vector_tags = 'ref'
  []
[]
(examples/TRISO/accident_simulation/triso2D_accident.i)

Mass transfer across gap

[ThermalContact]
  [cesium_contact]
    type = GapHeatTransfer
    variable = conc
    primary = IPyC_inner_boundary
    secondary = buffer_outer_boundary
    tangential_tolerance = 1e-6
    gap_conductivity_function = d_gap
    gap_conductivity_function_variable = temperature
    appended_property_name = _conc
    quadrature = true
    gap_geometry_type = sphere
    emissivity_primary = 0.0
    emissivity_secondary = 0.0
    min_gap = 1e-7
  []
[]
(examples/TRISO/parfume/parfume.i)

Failure Analysis

TRISO Failure modes

Basic fuel particle behavior

  • Several physical phenomena influence the behavior of the particles including fission gas production and irradiation effects.

Fuel Failure Mechanisms

  • Mechanical

    • Pressure vessel failure

    • Irradiation-induced PyC failure leading to SiC cracking

    • IPyC-SiC partial debonding

  • Thermomechanical

    • Kernel migration

    • SiC thermal decomposition

    • Fission product attack of SiC

    • Corrosion of SiC by CO

TRISO single layer failure probability

In the Weibull theory, the failure probability is

The stress integration above is performed using the principle of independent action (PIA) model as follows:

[Postprocessors]
  [Weibull_failure_probability_IPyC]
    type = WeibullFailureProbability
    block = IPyC
    weibull_modulus = 6
    characteristic_strength = characteristic_strength_IPyC
  []
[]
(test/tests/triso_failure/triso_1d_weibull_probability.i)
[Materials]
  [characteristic_strength_PyC]
    type = PyCCharacteristicStrength
    temperature = temp
    X = 1.02
    flux_conversion_factor = 0.85
    block = 'IPyC OPyC'
  []
[]
(test/tests/triso_failure/triso_ipyc_characteristic_strength.i)

Effective mean strength for the layer

The effective mean strength for the layer is defined to be:

[Postprocessors]
  [strength_SiC]
    type = WeibullEffectiveMeanStrength
    block = SiC
    weibull_modulus = 6
  []
[]
(test/tests/triso_failure/triso_1d_asphericity_failure.i)
[Materials]
  [characteristic_strength]
    type = GenericConstantMaterial
    prop_values = '9640000'
    prop_names = 'characteristic_strength'
    block = SiC
  []
[]
(test/tests/triso_failure/triso_1d_asphericity_failure.i)

TRISO failure determination

Whether or not particle failure occurs is determined by comparing the maximum stress and a strength that is sampled from a Weibull distribution

[Postprocessors]
  [failure_indicator_SiC]
    type = WeibullFailureOutputUsingCorrelation
    block = SiC
    weibull_modulus = 6
    stress_name = max_principal_stress
    effective_mean_strength = strength_SiC
  []
[]
(test/tests/triso_failure/triso_ipyc_characteristic_strength.i)
[Postprocessors]
  [triso_failure]
    type = TRISOFailureEvaluation
    SiC_failure = failure_indicator_SiC
  []
[]
(test/tests/triso_failure/triso_1d_asphericity_failure.i)

High-fidelity analysis and stress correlation

High-Fidelity Analysis

Perform high-fidelity analysis on cracked particles using parameters at their mean values.

Stress Correlation

  1. Perform similar one-dim analysis with same parameters.

  2. Produce correlation functions to be used for calculating maximum stress in SiC layer.

Failure Probability Determination

  1. Perform 1D simulation to calculate stresses in Monte Carlo sampling

  2. Use Weibull statistics to determine failure probability for particles.

IPyC cracking

[Mesh]
  coord_type = RZ
  [mesh]
    type = TRISO2DMeshGenerator
    elem_type = quad4
    coordinates = '0 ${coordinates1} ${coordinates2} ${coordinates2} ${coordinates3} ${coordinates4} ${coordinates5}'
    mesh_density = '20 8 0 4 4 4'
    block_names = 'fuel buffer IPyC SiC OPyC'
    num_sectors = 60
    aspect_ratio = ${aspect_ratio}
    all_bottom_left = True
  []
[]
[BCs]
  [no_disp_x]
    type = DirichletBC
    variable = disp_x
    boundary = xzero
    value = 0.0
  []

  [no_disp_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []
[]
(examples/TRISO/correlation_function/h_asphericity/triso_asphericity.i)

IPyC cracking

Stress correlation function: 

[Materials]
  [characteristic_strength_SiC]
    type = GenericConstantMaterial
    prop_values = '9640000'
    block = SiC
    prop_names = 'characteristic_strength'
  []

  [characteristic_strength_PyC]
    type = PyCCharacteristicStrength
    temperature = temperature
    X = 1.02
    block = 'IPyC OPyC'
  []
[]

[Postprocessors]
  [SiC_stress]
    type = ElementalVariableValue
    elementid = 6300
    variable = tangential_stress
  []

  [strength_SiC]
    type = WeibullEffectiveMeanStrength
    block = SiC
    weibull_modulus = 6
  []
[]
(examples/TRISO/correlation_function/h_asphericity/triso_asphericity.i)

Asphericity

Stress correlation function: 

[Mesh]
  coord_type = RZ
  [mesh]
    type = TRISO2DMeshGenerator
    elem_type = quad4
    coordinates = '0 ${coordinates1} ${coordinates2} ${coordinates2} ${coordinates3} ${coordinates4} ${coordinates5}'
    mesh_density = '20 8 0 4 4 4'
    block_names = 'fuel buffer IPyC SiC OPyC'
    num_sectors = 60
    aspect_ratio = ${aspect_ratio}
    all_bottom_left = True
  []
[]
(examples/TRISO/correlation_function/h_asphericity/triso_asphericity.i)

[Postprocessors]
  [SiC_stress]
    type = ElementalVariableValue
    elementid = 6300
    variable = tangential_stress
  []

  [strength_SiC]
    type = WeibullEffectiveMeanStrength
    block = SiC
    weibull_modulus = 6
  []
[]
(examples/TRISO/correlation_function/h_asphericity/triso_asphericity.i)

Monte Carlo

Monte Carlo uses MOOSE's stochastic_tools modules

  • Master input file: set distribution, random variables, etc.

  • Sub input file: standard TRISO 1D input file.

Monte Carlo: Distributions and Samplers

[Distributions]
  [normal_kernel_r]
    type = TruncatedNormal
    mean = 213.35e-6
    standard_deviation = 4.4e-6
    lower_bound = 1.9575e-04
    upper_bound = 2.3095e-04
  []
  [normal_buffer_t]
    type = TruncatedNormal
    mean = 98.9e-6
    standard_deviation = 8.4e-6
    lower_bound = 6.53e-05
    upper_bound = 1.325e-04
  []
  [normal_ipyc_t]
    type = TruncatedNormal
    mean = 40.4e-6
    standard_deviation = 2.5e-6
    lower_bound = 3.0400e-05
    upper_bound = 5.0400e-05
  []
  [normal_sic_t]
    type = TruncatedNormal
    mean = 35.2e-6
    standard_deviation = 1.2e-6
    lower_bound = 3.0400e-05
    upper_bound = 4.0000e-05
  []
  [normal_opyc_t]
    type = TruncatedNormal
    mean = 43.4e-6
    standard_deviation = 2.9e-6
    lower_bound = 3.1800e-05
    upper_bound = 5.5000e-05
  []
  [uniform]
    type = Uniform
  []
  [normal_bond_strength]
    type = Normal
    mean = 20e6
    standard_deviation = 1e5
  []
[]

[Samplers]
  [sample]
    type = MonteCarlo
    num_rows = 100 # Number of Monte Carlo samples
    distributions = 'normal_kernel_r normal_buffer_t normal_ipyc_t normal_sic_t normal_opyc_t uniform uniform uniform normal_bond_strength'
    execute_on = 'PRE_MULTIAPP_SETUP'
  []
[]
(examples/TRISO/failure_probability_monte_carlo/monte_carlo.i)

Monte Carlo: Multiapps and Transfers

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = triso_1d_function.i
    sampler = sample
    execute_on = 'TIMESTEP_BEGIN'
    mode = batch-reset
  []
[]

[Transfers]
  [sic_failure_overall]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = sic_failure_overall
    from_postprocessor = sic_failure_overall
  []
  [ipyc_cracking]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = ipyc_cracking
    from_postprocessor = ipyc_cracking
  []
  [debonding]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = debonding
    from_postprocessor = debonding
  []
  [sic_failure_due_to_pressure]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = sic_failure_due_to_pressure
    from_postprocessor = sic_failure_due_to_pressure
  []
  [sic_failure_due_to_ipyc_cracking]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = sic_failure_due_to_ipyc_cracking
    from_postprocessor = sic_failure_due_to_ipyc_cracking
  []
  [max_fluence]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = max_fluence
    from_postprocessor = max_fluence
  []
  [fluence_at_failure]
    type = SamplerPostprocessorTransfer
    from_multi_app = sub
    sampler = sample
    to_vector_postprocessor = fluence_at_failure
    from_postprocessor = fluence_at_failure
  []
[]
(examples/TRISO/failure_probability_monte_carlo/monte_carlo.i)

Monte Carlo: Controls and VectorPostprocessors

[Controls]
  [cmdline]
    type = MultiAppSamplerControl
    multi_app = sub
    sampler = sample
    param_names = 'Mesh/gen/kernel_radius Mesh/gen/buffer_thickness Mesh/gen/IPyC_thickness Mesh/gen/SiC_thickness Mesh/gen/OPyC_thickness Postprocessors/failure_indicator_IPyC/quantile Postprocessors/failure_indicator_SiC_crackedIPyC/quantile Postprocessors/failure_indicator_SiC/quantile Postprocessors/failure_indicator_debonding/bond_strength'
  []
[]

[VectorPostprocessors]
  [sic_failure_overall]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [ipyc_cracking]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [debonding]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [sic_failure_due_to_pressure]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [sic_failure_due_to_ipyc_cracking]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [sampler_data]
    type = SamplerData
    sampler = sample
    execute_on = 'TIMESTEP_END'
  []
  [max_fluence]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
  [fluence_at_failure]
    type = StochasticResults
    execute_on = 'TIMESTEP_END'
  []
[]
(examples/TRISO/failure_probability_monte_carlo/monte_carlo.i)

Monte Carlo: Output Files (Sample Data)

sample_0,sample_1,sample_2,sample_3,sample_4,sample_5,sample_6,sample_7,sample_8,sample_9,sample_10
0.00021637407664929,9.0665843266826e-05,3.9206960548527e-05,3.7765444364501e-05,4.1150990919452e-05,1902.9762845819,1910.2445726526,1.043903284486,0.36559025109643,0.83405431911942,0.19324791394218
0.00021751667216784,8.9393031054998e-05,4.2964996396561e-05,3.3697727860768e-05,4.0857634877945e-05,1900.8283230996,1910.4878603364,1.0574563648234,0.35820538838607,0.17884348788218,0.10892050788373
0.00021545969586394,8.7994557883001e-05,3.9938992403407e-05,3.6891003432006e-05,3.4532253887986e-05,1872.6921139432,1916.5770611868,1.0458123116343,0.041004797258524,0.44282376681833,0.67162977044723
0.00021346374559576,0.00011458781456504,4.1767630604059e-05,3.3393721040404e-05,3.7575821368445e-05,1930.3726376962,1905.1508299719,1.0477514802606,0.82395491493935,0.43074619764046,0.73081781082655
0.00020917844513318,9.6376849284529e-05,4.1667439327391e-05,3.7850725034576e-05,3.8688695661387e-05,1913.9688606718,1890.2091451491,1.0418011393073,0.68723634971943,0.4459364227904,0.35999042058209
0.00021960990911511,0.0001148189784978,3.6315844514982e-05,3.6351085807994e-05,4.5198706788899e-05,1893.9052214614,1937.2395585961,1.0504862269043,0.58848931556147,0.51112818304236,0.64623323767235
0.00021113098705503,0.00010467648024848,3.8886501674112e-05,3.3538170032432e-05,4.5526677798013e-05,1896.2188863875,1912.9277209197,1.0498363161117,0.79263878879797,0.75216980451615,0.63259580700936
0.00021418998090287,8.8377421335382e-05,3.6341910732714e-05,3.5894783256591e-05,4.1143931344128e-05,1876.4603866841,1919.9993025071,1.0412353326822,0.99397221034491,0.055007717443307,0.25922612542033
0.00021015863602917,0.00010197728675811,4.0442158241633e-05,3.2912616811623e-05,3.637002785012e-05,1865.926597935,1897.945150619,1.0471468040478,0.098470559938411,0.25871203851172,0.08720546033451
0.00020717664414964,0.00010814647769256,3.5759398875267e-05,3.7432118258049e-05,4.2042385906956e-05,1891.3815045874,1892.7312888448,1.0492311395109,0.51894025654813,0.10742587457047,0.47432230790532
0.00021286232458187,8.8873111019891e-05,3.5442396532755e-05,3.4149969286569e-05,3.8033522971351e-05,1897.5768754415,1891.9214353358,1.0555386738498,0.3725596403175,0.12148786664822,0.99730718916737
0.00021439384343386,9.0778646209651e-05,3.801992544064e-05,3.7301972402598e-05,4.0236923589445e-05,1940.6716666758,1898.1775289361,1.0498822682043,0.23601109498261,0.68049782016736,0.53235501616889
0.00021275321244524,0.00010486565093629,3.8760531641886e-05,3.1889179568117e-05,4.1550865299327e-05,1906.0295459483,1902.1876826297,1.0456134047778,0.37996923639042,0.48435988270735,0.47865172841508
0.00021039092261463,0.0001011539169834,3.4217844242274e-05,3.754845939591e-05,4.2496758614696e-05,1900.1910120462,1892.1534966553,1.0551200144249,0.026282623764998,0.38096372024001,0.020447137540497
0.00020521048946964,9.7022314709875e-05,3.8378336211066e-05,3.2629738285283e-05,3.8910736646013e-05,1893.52099783,1928.6895591694,1.0503850552309,0.12489400991981,0.19329963759718,0.051005932823001
0.00020790232396898,9.2947852909839e-05,3.8002426913888e-05,3.5723805207023e-05,3.5567535304588e-05,1903.8170303499,1882.2953178751,1.0474712303408,0.30438050651964,0.64931781179008,0.3073643978831
0.00020803260984517,9.5337127011165e-05,4.1712799885772e-05,3.5070800257705e-05,3.8965699716991e-05,1876.4188851698,1910.001583432,1.0440135192707,0.58154453287355,0.29864544000591,0.8345355736922
0.00020841893104101,9.6631560578141e-05,4.2202101699427e-05,3.380152632004e-05,3.8813144685972e-05,1899.8196161747,1916.2815464807,1.0473085968172,0.14981119812282,0.22993706315161,0.38106968965054
0.00021737766513872,8.8158264532508e-05,3.5153546730745e-05,3.4128221418823e-05,4.5099565158512e-05,1897.2447227459,1903.0833043178,1.0473111495923,0.7002536218619,0.56012111830447,0.12994422158617
0.00021048522936763,8.1083320930163e-05,3.7215912939902e-05,3.2532317757957e-05,4.1494394598651e-05,1925.280025742,1909.2840283292,1.0456622085534,0.40842579294811,0.65133778613123,0.601637047984
0.00021125910696843,0.00010562011801763,4.1070105237062e-05,3.6649843877134e-05,3.9678192535298e-05,1878.1801020843,1920.1238265036,1.055024341865,0.9679746893921,0.29315403768834,0.2175429494991
0.00021575085374553,0.00010412454648476,3.8990919947376e-05,3.2250072364983e-05,3.5606804057112e-05,1894.610230137,1879.3614191126,1.0547907918859,0.38195745471269,0.32980548143122,0.28261448712375
0.00021256405926474,0.00010353196678073,3.9709913699364e-05,3.6711883043785e-05,4.0772756660712e-05,1883.6398716725,1897.8659888871,1.0480668209837,0.90112593813904,0.47338839614188,0.8253359680258
0.00022232267177052,9.3815384963033e-05,3.6104725115404e-05,3.5087002639934e-05,3.8094420796066e-05,1899.7116196631,1897.2689381125,1.0581348797492,0.034608430708271,0.22442144433382,0.15437811962466
0.00021383080335259,0.00010041464033315,4.0197352842149e-05,3.3265211827049e-05,3.6875880547396e-05,1886.8827755606,1904.3253614375,1.0475523579903,0.89345734236269,0.86119897797151,0.46265257766826
0.00021543035570559,0.00010903227823575,4.1849837831973e-05,3.5891082368272e-05,4.3868193698344e-05,1883.4465565213,1884.4806529668,1.0595195042156,0.95239511792334,0.9073324947907,0.086883876331278
0.00021357921998213,9.991647939129e-05,4.0341917385895e-05,3.2892657506087e-05,3.6233499783344e-05,1910.055923218,1838.2256176049,1.0480604571155,0.0070499797314517,0.86825618113578,0.94854101551074
0.00020923624915276,9.4054061548247e-05,3.6866480559697e-05,3.1232422962176e-05,4.2243638607878e-05,1901.2219258758,1891.727102537,1.0567167342993,0.86923193546583,0.8866263444973,0.34478016887071
0.00021168772563546,9.8328837822567e-05,3.9767692058928e-05,3.2822073301439e-05,3.5094663587356e-05,1877.480044399,1884.1036598282,1.055578939545,0.91058429936092,0.049011953519517,0.94373648604372
0.00021695054093395,0.00011038497236656,4.0681753783783e-05,3.6532406673621e-05,4.1729448636085e-05,1876.6378259258,1915.5956841898,1.0556848667982,0.77419237672158,0.74287350244049,0.25813241878449
0.00020878820621239,0.00010234901958904,4.047477519771e-05,3.4679947575337e-05,3.715701842427e-05,1884.2047426534,1908.7283661636,1.0574908257666,0.35300574155917,0.38934777342916,0.84339080264298
0.00021460281611525,9.9645698402439e-05,4.1015054865719e-05,3.3845101015718e-05,4.1119665585894e-05,1878.3537943993,1912.3219347918,1.0437276181571,0.51598436670702,0.74862314623373,0.14043886292551
0.00021405835531446,8.968070838554e-05,4.0525327816185e-05,3.2766966993294e-05,3.9149303052967e-05,1878.0214796237,1912.4527060713,1.0443411011531,0.17666919644913,0.94491169237549,0.38280574427536
0.0002093977707835,9.5929860090776e-05,4.1141034687078e-05,3.6151178326861e-05,3.9835636050713e-05,1883.4955875605,1948.3137236565,1.0637387314392,0.33572281665558,0.88541284974289,0.36502689922379
0.00021627207542983,8.9614314978176e-05,4.4049086176071e-05,3.5666990289792e-05,3.884232987594e-05,1899.1625737372,1919.3074694742,1.0552035439973,0.38650556584916,0.66362144332231,0.50008670625551
0.00021939519208179,9.5320243498849e-05,4.100333786084e-05,3.383195901158e-05,4.4017727266917e-05,1884.9212245843,1899.4976967505,1.0493741516518,0.27654839073155,0.010961859583138,0.6272905621202
0.00021646549130163,0.00011400050755281,4.1869419408821e-05,3.6125408874093e-05,4.2442500252346e-05,1904.4717492497,1933.1323370518,1.0440122522734,0.17540635125335,0.13351531627025,0.407133191351
0.00021687224716186,9.1372982705859e-05,4.0156593368216e-05,3.5714709271955e-05,4.3225849022471e-05,1926.7030396256,1917.2720788339,1.0460322638226,0.0045863444065441,0.19649984000651,0.39376996823598
0.00021072913687077,8.8920865698668e-05,3.9865053221136e-05,3.6662212483362e-05,4.0505038129629e-05,1902.6187164215,1856.92253613,1.0507455131327,0.87588714183301,0.89614657454179,0.6816599120636
0.0002178922204202,8.9166323189706e-05,3.8451297008674e-05,3.7108513935361e-05,4.3259860772523e-05,1870.0605892369,1898.8132253845,1.0569584772321,0.51898428519068,0.5948036307406,0.36848974111973
0.00021369708150236,0.00011337345090711,4.0983676561027e-05,3.5456079141823e-05,3.4148062503304e-05,1926.175660429,1904.2304955527,1.0508359755162,0.12812797972644,0.59924327783051,0.81168098769434
0.00021212117064659,8.5055563818595e-05,4.1215654158526e-05,3.6610989667712e-05,3.5728554305864e-05,1927.7260679752,1908.5231320533,1.057685463426,0.55370543865466,0.64484017949751,0.3301799088466
0.00020805159129625,0.00010528692170499,3.8959429948689e-05,3.4718772155339e-05,4.1097593602255e-05,1910.6126075689,1914.1024106959,1.0496302080824,0.2359600761004,0.5046304740174,0.27373025730783
0.00021405315182056,0.00010099150402564,4.5057316264428e-05,3.7092173599108e-05,4.2223018871506e-05,1871.1801851753,1901.8513495008,1.0519147835817,0.72309514786351,0.20667575701831,0.86398243749853
0.00021823015577034,0.00010179312504322,3.4857728326978e-05,3.374006915187e-05,3.8107959136109e-05,1902.8252415876,1895.6007398969,1.0565623945892,0.8267198399628,0.26989211933623,0.4724557266605
0.00021904114981531,9.6990775322301e-05,4.5081910513944e-05,3.7001446020558e-05,4.1614634853531e-05,1867.4880200904,1903.5136176562,1.0483511186877,0.38005822228888,0.63814373042885,0.4977564681465
0.0002178768472868,0.00010189537275935,3.9905653249728e-05,3.5151124229787e-05,3.7376781445445e-05,1888.2775079003,1924.5318432718,1.0436274403816,0.58358778760054,0.13280714099893,0.49974790831085
0.00021633470390484,9.5509210045277e-05,3.8249538147764e-05,3.436745561269e-05,3.9210467222525e-05,1895.8356689139,1930.1279151495,1.0414227210238,0.86125864384281,0.5842249057165,0.40145734518956
0.00021003632889278,0.00010867109190165,3.9665025795692e-05,3.5995126902045e-05,4.2938796758557e-05,1883.1023384052,1921.2467390948,1.0503820369591,0.67270328674441,0.52887351609066,0.12520766712873
0.00021118537730156,9.1104813363285e-05,3.879939238708e-05,3.8627652088504e-05,4.253028321749e-05,1891.1284930567,1911.9478993337,1.0545717973444,0.43398786267257,0.7136784347172,0.93670987675293
0.00021143986118105,9.4459688433443e-05,4.0511218079828e-05,3.7784282232138e-05,3.4881662977925e-05,1882.4828261804,1876.2348072711,1.0428041200251,0.85860321253878,0.28633019660745,0.86805558411464
0.00021479340645403,9.9144864585172e-05,3.9555193182502e-05,3.5735981911246e-05,4.3431372102698e-05,1881.8449363293,1962.813201324,1.0625235681743,0.51634423301366,0.88971139823139,0.084337840672853
0.00021056824652347,0.00010639378202498,4.1707853405567e-05,3.3012828780145e-05,4.5169396948577e-05,1945.3507204792,1889.6870253954,1.0490397595366,0.92489797659817,0.91907748615717,0.47501741606392
0.00020262403168757,8.8796800253677e-05,3.5159136348765e-05,3.735235848182e-05,3.9886577721486e-05,1884.2815279074,1900.4200002906,1.0453754064001,0.42692040465267,0.60942777835222,0.10681207412146
0.00021617989514722,0.00010777581784151,4.033261300102e-05,3.7918524077618e-05,4.5055123169266e-05,1882.5542350029,1897.0966974152,1.0457721736786,0.86163561001951,0.50361510096962,0.51104456235673
0.00021190055521211,0.00010262134569495,3.8202063654513e-05,3.7764049072113e-05,3.662109984742e-05,1897.048888562,1918.6646239386,1.0507520829208,0.86502994645409,0.96308612472328,0.76704729483114
0.00021227229392204,9.6991786175882e-05,4.2202076848615e-05,3.6627455719811e-05,4.0976322690718e-05,1895.4516545226,1923.5900703415,1.0540734575088,0.72545654132818,0.97369298622784,0.19482644778146
0.00020863133285396,8.9779014891075e-05,4.1582892948338e-05,3.7849217128553e-05,4.3814064004093e-05,1872.6874398132,1895.2003478365,1.0469244214991,0.090652718182083,0.15115404430636,0.58402996457524
0.00021965770148213,0.00010642015231803,4.1144808008908e-05,3.618713668504e-05,3.9017264062964e-05,1915.0911460766,1901.062630448,1.0580785298583,0.27932685526588,0.01356573738345,0.74965485613371
0.00021081491775326,9.1761255853802e-05,3.9983424680442e-05,3.7433129306759e-05,3.6743908762693e-05,1893.5360483265,1901.4230472083,1.0532913113927,0.21780523002445,0.007806734574679,0.75179368682359
0.00021232046374794,9.0030148864917e-05,4.1588887903628e-05,3.6069466810426e-05,4.4461587715269e-05,1934.9309313925,1911.0527637984,1.0498981172573,0.2905124736917,0.77605030014118,0.2417628496081
0.00021247569175195,9.1418979482315e-05,3.8496869550193e-05,3.5796367555951e-05,4.2370581548088e-05,1921.2364739067,1900.7286895587,1.0507010865188,0.89263965380905,0.13399932271232,0.77491317935244
0.00020963369493444,0.00010475398367818,3.6457829586801e-05,3.4259973466195e-05,4.186468531062e-05,1901.9892496501,1912.9435152226,1.0542732245088,0.76738172915432,0.44488992994603,0.90617803754277
0.00021515886516462,0.00010749572238335,3.9170155935897e-05,3.5251545254447e-05,3.7023426535889e-05,1908.8391813257,1885.3091124914,1.0532268995341,0.85627335728713,0.87034850839411,0.76650459428354
0.0002094175744645,8.2960905673176e-05,4.1131329833923e-05,3.2266632186855e-05,3.9233174387353e-05,1913.9924029841,1937.2116328459,1.0458704680621,0.48016557749425,0.4340863377307,0.60550199184906
0.00021531259487069,9.3804653733516e-05,4.5515328695399e-05,3.4158432828231e-05,3.9531592263291e-05,1897.4940000485,1926.3037325277,1.0507933731003,0.25704871655417,0.41200761169033,0.6236247974357
0.00021280038706345,9.3358547671575e-05,3.9394052982687e-05,3.3149021797012e-05,3.6470175687247e-05,1903.8361138641,1903.9943317491,1.0570535150426,0.11948262468699,0.35113542591458,0.54430518051755
0.00021296293779076,0.00010183156433382,3.8808464355678e-05,3.5437833476469e-05,4.1213278423376e-05,1896.3159481163,1948.6398476364,1.053630772961,0.0296331261085,0.94867675486332,0.084400280492879
0.00022206369452916,0.00011594858883821,3.579686882421e-05,3.5726029270366e-05,4.2959089248255e-05,1919.9441817313,1899.3052675051,1.0527606806738,0.049262394456485,0.71995295769185,0.44757124696444
0.00020807961788271,8.6801919787419e-05,4.3028097411553e-05,3.3978526776395e-05,3.5866748658152e-05,1896.4575001279,1898.5818262448,1.0479972627589,0.45686363668973,0.010747294153506,0.75835467688262
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(workshop/monte_carlo_example_results/monte_carlo_out_sampler_data_0001.csv)

Monte Carlo: Output Files (Failure Data)

result:triso_failure
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(workshop/monte_carlo_example_results/monte_carlo_out_failure_results_0001.csv)

References

  1. IAEA. Fuel performance and fission product behavior in gas cooled reactors. Technical Report IAEA-TECDOC-978, IAEA, 1997.