UPuZr Lanthanide Diffusivity

Calculates lanthanide diffusivity in U-Zr and U-Pu-Zr fuel.

Description

The UPuZrLanthanideDiffusivity class is used to calculate the diffusivity of lanthanides (and its derivatives) in U-Zr and U-Pu-Zr fuels. It is part of a mechanistic fuel-cladding chemical interaction (FCCI) modeling framework. See UPuZrLanthanideFlux and UPuZrLanthanideWastage for more information.

The diffusivity of lanthanides in the fuel is assumed to be similar to that of Nd. The Nd diffusivity $var element = document.getElementById("moose-equation-2e18323f-eccd-4b8a-aab9-0a87472b3ffd");katex.render("D_{eff}", element, {displayMode:false,throwOnError:false});$ is taken from Aagesen et al. (2023) and Hirschhorn et al. (2025). It accounts for the effects of fuel swelling and sodium infiltration by considering the separate contributions of bulk diffusion, surface diffusion, and diffusion through interconnected fuel pores logged with liquid bond sodium. $var element = document.getElementById("moose-equation-398ae89d-e824-4eb2-ab17-9ccc28e49a21");katex.render("D_{eff}", element, {displayMode:false,throwOnError:false});$ is given by: $(1)var element = document.getElementById("moose-equation-06f246ca-0edf-47a9-9ba8-c37d4eb5df2a");katex.render(" D_{eff} = D_{bulk}\\frac{1 + 2p}{1 - p} + D_{surf}^p I a |p - p_{start}|^\\alpha + \\frac{D_{Na} p_{Na} I}{\\tau}, ", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-040e6178-0521-4d8d-9b29-eac3340d0efd");katex.render("D_{bulk}", element, {displayMode:false,throwOnError:false});$ is the temperature-dependent diffusivity in the matrix, $var element = document.getElementById("moose-equation-47b89869-f19b-4062-94f8-c91888275c46");katex.render("p", element, {displayMode:false,throwOnError:false});$ is the porosity, $var element = document.getElementById("moose-equation-43d7940e-865b-47a7-9707-63fb429b67ac");katex.render("D_{surf}", element, {displayMode:false,throwOnError:false});$ is the temperature-dependent diffusivity on the surfaces of interconnected pores, and $var element = document.getElementById("moose-equation-dd77370f-f4b8-4b01-beca-b7493efe432f");katex.render("I", element, {displayMode:false,throwOnError:false});$ is the pore interconnectivity. Here also, $var element = document.getElementById("moose-equation-82a944b9-28f0-4830-a8d1-19b2f1cc2f22");katex.render("a", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-44f311c3-c9af-4543-a607-a941ecb876b7");katex.render("p_{start}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-01ec67ef-f917-421b-868a-afb29f366321");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ are fitting parameters, $var element = document.getElementById("moose-equation-78999cb6-85f7-4eb2-8d66-84b55ac286a5");katex.render("D_{Na}", element, {displayMode:false,throwOnError:false});$ is the temperature-dependent diffusivity in liquid sodium, $var element = document.getElementById("moose-equation-70b358cc-fc65-4901-b509-6d5df63e94e9");katex.render("p_{Na}", element, {displayMode:false,throwOnError:false});$ is the sodium logged porosity, and $var element = document.getElementById("moose-equation-89cbe5da-7061-4280-be94-1a56b591e59d");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is the tortuosity.

Parameters $var element = document.getElementById("moose-equation-f7f7c03f-1390-4305-b046-803f99b90f74");katex.render("p", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-c3e1924c-99f9-475f-b5a1-1db386fb558c");katex.render("p_{Na}", element, {displayMode:false,throwOnError:false});$ can be coupled to time- and space-varying material properties (such as those calculated by UPuZrGaseousEigenstrain and UPuZrSodiumLogging) to capture the impact of microstructure evolution on FCCI. See Aagesen et al. (2023) and Hirschhorn et al. (2025) for the derivation of $var element = document.getElementById("moose-equation-55ca29c6-a420-4ed1-a039-683de82edff6");katex.render("D_{eff}", element, {displayMode:false,throwOnError:false});$ and more information on its underlying parameters.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [diffusivity]
    type = UPuZrLanthanideDiffusivity<<<{"description": "Calculates lanthanide diffusivity in U-Zr and U-Pu-Zr fuel.", "href": "UPuZrLanthanideDiffusivity.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = diffusivity
    temperature<<<{"description": "The temperature"}>>> = temperature
    porosity_name<<<{"description": "Name of the material property storing the porosity"}>>> = porosity
    sodium_logged_porosity_name<<<{"description": "Name of the material property storing the sodium-logged porosity"}>>> = sodium_logged_porosity
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = exodus
  []
[]

Input Parameters

property_nameName of the parsed material property
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of the parsed material property
temperatureThe temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The temperature

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
porosity_nameporosityName of the material property storing the porosity
Default:porosity
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of the material property storing the porosity
sodium_logged_porosity_namesodium_logged_porosityName of the material property storing the sodium-logged porosity
Default:sodium_logged_porosity
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of the material property storing the sodium-logged porosity

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(examples/metal_fuel/mechanistic_fcci/fcci.i)
(test/tests/upuzr_lanthanide_diffusivity/test.i)

References

L. Aagesen, J. Hirschhorn, and C. Jiang. Mechanistic FCCI model for BISON. Technical Report INL/RPT-23-74500, Idaho National Laboratory, August 2023.

@TECHREPORT{INL/RPT-23-74500,
    author = "Aagesen, L. and Hirschhorn, J. and Jiang, C.",
    title = "Mechanistic {FCCI} model for {BISON}",
    year = "2023",
    number = "INL/RPT-23-74500",
    month = "August",
    publisher = "INL Technical Report",
    institution = "Idaho National Laboratory"
}

J.A. Hirschhorn, L.K. Aagesen, C. Jiang, and G.L. Beausoleil. Development and preliminary validation of a mechanistic multiscale model for fuel-cladding chemical interaction in metallic nuclear fuels. Nuclear Engineering and Design, 432:113811, 2025. doi:10.1016/j.nucengdes.2024.113811.

@article{HIRSCHHORN2025113811,
    author = "Hirschhorn, J.A. and Aagesen, L.K. and Jiang, C. and Beausoleil, G.L.",
    title = "Development and preliminary validation of a mechanistic multiscale model for fuel-cladding chemical interaction in metallic nuclear fuels",
    journal = "Nuclear Engineering and Design",
    volume = "432",
    pages = "113811",
    year = "2025",
    issn = "0029-5493",
    doi = "10.1016/j.nucengdes.2024.113811"
}

(test/tests/upuzr_lanthanide_diffusivity/test.i)

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 4
    ny = 4
    nz = 4
  []
[]

[Variables]
  [test]
  []
[]

[AuxVariables]
  [temperature]
  []
[]

[ICs]
  [temperature]
    type = FunctionIC
    variable = temperature
    function = '800 * x + 300'
  []
[]

[Kernels]
  [test]
    type = TimeDerivative
    variable = test
  []
[]

[Functions]
  [porosity]
    type = ParsedFunction
    expression = '0.4 * y'
  []
  [sodium_logged_porosity]
    type = ParsedFunction
    expression = '0.2 * z'
  []
[]

[Materials]
  [properties]
    type = GenericFunctionMaterial
    prop_names = 'porosity sodium_logged_porosity'
    prop_values = 'porosity sodium_logged_porosity'
    outputs = exodus
  []
  [diffusivity]
    type = UPuZrLanthanideDiffusivity
    property_name = diffusivity
    temperature = temperature
    porosity_name = porosity
    sodium_logged_porosity_name = sodium_logged_porosity
    outputs = exodus
  []
  [diffusivity_exact]
    type = DerivativeParsedMaterial
    property_name = diffusivity_exact
    coupled_variables = temperature
    material_property_names = 'porosity sodium_logged_porosity'
    constant_names =       'D0_bulk  Ea_bulk  D0_surf  Ea_surf  D0_Na    Ea_Na  kB            a           p_start alpha   p_cen delta   tortuosity'
    constant_expressions = '4.007e-8 1.4076   5.916e-8 0.079234 7.86e-9  0.0421 8.6173324e-5  2.17695e-3  0.2508  1.71912 0.269 0.0392  1.5'
    expression = 'D_bulk := D0_bulk * exp(-Ea_bulk / (kB * temperature));
                  D_surf := D0_surf * exp(-Ea_surf / (kB * temperature));
                  D_Na := D0_Na * exp(-Ea_Na / (kB * temperature));
                  interconnectivity := 0.5 * (1 + erf((porosity - p_cen) / delta));
                  D_bulk * (1 + 2 * porosity) / (1 - porosity)
                  + D_surf * a * interconnectivity * (abs(porosity - p_start))^alpha
                  + D_Na * sodium_logged_porosity * interconnectivity * porosity / tortuosity'
    derivative_order = 1
    outputs = exodus
  []
  [value_error]
    type = ParsedMaterial
    property_name = value_error
    material_property_names = 'test:=diffusivity exact:=diffusivity_exact'
    expression = 'abs((test - exact) / exact)'
    outputs = exodus
  []
  [derivative_error]
    type = ParsedMaterial
    property_name = derivative_error
    material_property_names = 'test:=ddiffusivity/dtemperature exact:=ddiffusivity_exact/dtemperature'
    expression = 'abs((test - exact) / exact)'
    outputs = exodus
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
  dtmin = 0.5
[]

[Postprocessors]
  [min_temperature]
    type = ElementExtremeValue
    variable = temperature
    value_type = min
    outputs = console
  []
  [max_porosity]
    type = ElementExtremeMaterialProperty
    mat_prop = porosity
    value_type = max
    outputs = console
  []
  [val_err_max]
    type = ElementExtremeMaterialProperty
    mat_prop = value_error
    value_type = max
    outputs = console
  []
  [deriv_err_max]
    type = ElementExtremeMaterialProperty
    mat_prop = derivative_error
    value_type = max
    outputs = console
  []
[]

[Outputs]
  exodus = true
[]

(examples/metal_fuel/mechanistic_fcci/fcci.i)

# This example exercises the mechanistic Fuel-Cladding Chemical Interaction
# (FCCI) modeling approach for U-Zr and U-Pu-Zr metallic nuclear fuels. It is
# run as a SubApp, which is called by the combination of an existing assessment
# case and XXXX_multiapp.i.

# It is a collection of BISON classes and objects needed to apply a mechanistic
# FCCI model for U-Zr and U-Pu-Zr metallic nuclear fuels. It is designed to
# complement the modeling approaches adopted by the FIPD-powered X447 assessment
# cases and manually-developed X430 and IFR1 assessment cases. Thermomechanical
# behaviors are modeled in the existing assessment cases whereas lanthanide (Ln)
# transport, FCCI, and wastage growth are modeled in the SubApp. XXXX_multiapp.i
# then overrides the empirical FCCI model currently used in the assessments and
# replaces it with the mechanistic FCCI model. Examples of how to do this are
# included in the test and example specifications.

# Note that, in this model, c indicates the stable concentration of Lns. The
# model relies on an effective diffusion coefficient and flux terms informed by
# lower length scale modeling and simulations. Decay terms are included for
# illustration purposes but do not affect the solution.

# combined effective Ln production constant in mol/fission
ln_yield = 0.3 # atom/fission
avogadro = 6.0221e23 # atom/mol
production_constant = '${fparse ln_yield / avogadro}' # mol/fission

# combined effective Ln decay constant in 1/s
decay_constant = 0

# variable order, block and boundary names, and end time used in the assessment
order = placeholder
fuel_block_name = placeholder
fuel_surface_name = placeholder
end_time = placeholder

# fuel rod geometry and material properties used in the assessment
cladding_density = placeholder
fuel_outer_radius = placeholder
gap_thickness = placeholder

[GlobalParams]
  order = ${order}
  family = LAGRANGE
[]

[Problem]
  # nothing is being solved for on the cladding
  kernel_coverage_check = false
  material_coverage_check = false
[]

[Mesh]
[]

[Variables]
  [c] # stable Ln concentration in mol/m^3
    block = ${fuel_block_name}
  []
[]

[AuxVariables]
  [T]
    initial_condition = 298
  []
  [fission_rate]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
  [porosity_aux]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
  [sodium_logged_porosity_aux]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
  [diffusivity]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
  [dcdx]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
  [c_flux]
    block = ${fuel_block_name}
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [diffusivity]
    type = MaterialRealAux
    boundary = ${fuel_surface_name}
    variable = diffusivity
    property = D
  []
  [dcdx]
    type = VariableGradientComponent
    boundary = ${fuel_surface_name}
    variable = dcdx
    gradient_variable = c
    component = x
  []
  [c_flux]
    type = ParsedAux
    boundary = ${fuel_surface_name}
    variable = c_flux
    coupled_variables = 'diffusivity dcdx'
    expression = '-diffusivity * dcdx'
  []
[]

[Kernels]
  [c_dt]
    type = TimeDerivative
    variable = c
  []
  [c_diffusion]
    type = MatDiffusion
    block = ${fuel_block_name}
    variable = c
    diffusivity = D
    args = 'c T'
  []
  [c_source]
    type = NeutronHeatSource
    block = ${fuel_block_name}
    variable = c
    fission_rate = fission_rate # fission/m^3-s
    energy_per_fission = ${production_constant} # mol/fission giving mol/m^3-s
  []
  [c_decay]
    type = Decay
    block = ${fuel_block_name}
    variable = c
    radioactive_decay_constant = ${decay_constant} # 1/s
  []
[]

[Materials]
  [porosity]
    type = ParsedMaterial
    block = ${fuel_block_name}
    property_name = porosity
    coupled_variables = porosity_aux
    expression = porosity_aux
  []
  [sodium_logged_porosity]
    type = ParsedMaterial
    block = ${fuel_block_name}
    property_name = sodium_logged_porosity
    coupled_variables = sodium_logged_porosity_aux
    expression = sodium_logged_porosity_aux
  []
  [diffusivity]
    # informed by lower length scale simulations
    type = UPuZrLanthanideDiffusivity
    block = ${fuel_block_name}
    property_name = D
    temperature = T
    porosity_name = porosity
    sodium_logged_porosity_name = sodium_logged_porosity
    outputs = exodus
    output_properties = D
  []
  [flux]
    # informed by lower length scale simulations
    type = UPuZrLanthanideFlux
    boundary = ${fuel_surface_name}
    property_name = F
    temperature = T
    concentration = c
    outputs = exodus
    output_properties = F
  []
  [wastage_thickness]
    # informed by lower length scale simulations
    type = UPuZrLanthanideWastage
    boundary = ${fuel_surface_name}
    cladding_density = ${cladding_density}
    fuel_outer_radius = ${fuel_outer_radius}
    gap_thickness =${gap_thickness}
    lanthanide_flux = F
    outputs = exodus
  []
[]

[BCs]
  [c_right]
    type = MatNeumannBC
    boundary = ${fuel_surface_name}
    variable = c
    boundary_material = F
    value = -1
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  automatic_scaling = true
  compute_scaling_once = false
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       superlu_dist                  NONZERO               1e-15'
  line_search = NONE
  l_max_its = 30
  l_tol = 1e-3
  nl_max_its = 30
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-6
  start_time = 0
  dt = 1e7
  end_time = ${end_time}
  verbose = true
[]

[Postprocessors]
  [c_in_fuel] # mol
    type = ElementIntegralVariablePostprocessor
    block = ${fuel_block_name}
    variable = c
    execute_on = 'initial timestep_end'
    outputs = 'console csv'
  []
  [fission_rate_total] # fission/s
    type = ElementIntegralVariablePostprocessor
    block = ${fuel_block_name}
    variable = fission_rate
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [fission_total] # fission
    type = TimeIntegratedPostprocessor
    value = fission_rate_total
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_produced] # mol
    type = LinearCombinationPostprocessor
    pp_names = fission_total
    pp_coefs = ${production_constant} # mol/fission giving mol
    execute_on = 'initial timestep_end'
    outputs = 'console csv'
  []
  [c_decay_rate] # mol/s
    type = LinearCombinationPostprocessor
    pp_names = c_in_fuel # mol
    pp_coefs = ${decay_constant} # 1/s
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_decayed] # mol
    type = TimeIntegratedPostprocessor
    value = c_decay_rate
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_reaction_rate] # mol/s
    type = SideDiffusiveFluxIntegral
    boundary = ${fuel_surface_name}
    variable = c
    diffusivity = D
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_reacted] # mol
    type = TimeIntegratedPostprocessor
    value = c_reaction_rate
    execute_on = 'initial timestep_end'
    outputs = 'console csv'
  []
  [c_flux_max]
    type = SideExtremeValue
    boundary = ${fuel_surface_name}
    variable = c_flux
    value_type = max
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_flux_min]
    type = SideExtremeValue
    boundary = ${fuel_surface_name}
    variable = c_flux
    value_type = min
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [wastage_thickness_max] # m
    type = ElementExtremeValue
    variable = wastage_thickness
    value_type = max
    execute_on = 'initial timestep_end'
    outputs = 'console csv'
  []
  [c_surface_max] # mol
    type = SideExtremeValue
    boundary = ${fuel_surface_name}
    variable = c
    value_type = max
    execute_on = 'initial timestep_end'
    outputs = csv
  []
  [c_surface_min] # mol
    type = SideExtremeValue
    boundary = ${fuel_surface_name}
    variable = c
    value_type = min
    execute_on = 'initial timestep_end'
    outputs = csv
  []
[]

# [VectorPostprocessors]
#   [surface_conditions]
#     type = SideValueSampler
#     variable = 'T D F c c_flux wastage_thickness'
#     boundary = ${fuel_surface_name}
#     sort_by = y
#     execute_on = 'initial timestep_end'
#     outputs = csv
#   []
# []

[Outputs]
  csv = true
  exodus = true
[]

(test/tests/upuzr_lanthanide_diffusivity/test.i)

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 4
    ny = 4
    nz = 4
  []
[]

[Variables]
  [test]
  []
[]

[AuxVariables]
  [temperature]
  []
[]

[ICs]
  [temperature]
    type = FunctionIC
    variable = temperature
    function = '800 * x + 300'
  []
[]

[Kernels]
  [test]
    type = TimeDerivative
    variable = test
  []
[]

[Functions]
  [porosity]
    type = ParsedFunction
    expression = '0.4 * y'
  []
  [sodium_logged_porosity]
    type = ParsedFunction
    expression = '0.2 * z'
  []
[]

[Materials]
  [properties]
    type = GenericFunctionMaterial
    prop_names = 'porosity sodium_logged_porosity'
    prop_values = 'porosity sodium_logged_porosity'
    outputs = exodus
  []
  [diffusivity]
    type = UPuZrLanthanideDiffusivity
    property_name = diffusivity
    temperature = temperature
    porosity_name = porosity
    sodium_logged_porosity_name = sodium_logged_porosity
    outputs = exodus
  []
  [diffusivity_exact]
    type = DerivativeParsedMaterial
    property_name = diffusivity_exact
    coupled_variables = temperature
    material_property_names = 'porosity sodium_logged_porosity'
    constant_names =       'D0_bulk  Ea_bulk  D0_surf  Ea_surf  D0_Na    Ea_Na  kB            a           p_start alpha   p_cen delta   tortuosity'
    constant_expressions = '4.007e-8 1.4076   5.916e-8 0.079234 7.86e-9  0.0421 8.6173324e-5  2.17695e-3  0.2508  1.71912 0.269 0.0392  1.5'
    expression = 'D_bulk := D0_bulk * exp(-Ea_bulk / (kB * temperature));
                  D_surf := D0_surf * exp(-Ea_surf / (kB * temperature));
                  D_Na := D0_Na * exp(-Ea_Na / (kB * temperature));
                  interconnectivity := 0.5 * (1 + erf((porosity - p_cen) / delta));
                  D_bulk * (1 + 2 * porosity) / (1 - porosity)
                  + D_surf * a * interconnectivity * (abs(porosity - p_start))^alpha
                  + D_Na * sodium_logged_porosity * interconnectivity * porosity / tortuosity'
    derivative_order = 1
    outputs = exodus
  []
  [value_error]
    type = ParsedMaterial
    property_name = value_error
    material_property_names = 'test:=diffusivity exact:=diffusivity_exact'
    expression = 'abs((test - exact) / exact)'
    outputs = exodus
  []
  [derivative_error]
    type = ParsedMaterial
    property_name = derivative_error
    material_property_names = 'test:=ddiffusivity/dtemperature exact:=ddiffusivity_exact/dtemperature'
    expression = 'abs((test - exact) / exact)'
    outputs = exodus
  []
[]

[Executioner]
  type = Transient
  num_steps = 1
  dtmin = 0.5
[]

[Postprocessors]
  [min_temperature]
    type = ElementExtremeValue
    variable = temperature
    value_type = min
    outputs = console
  []
  [max_porosity]
    type = ElementExtremeMaterialProperty
    mat_prop = porosity
    value_type = max
    outputs = console
  []
  [val_err_max]
    type = ElementExtremeMaterialProperty
    mat_prop = value_error
    value_type = max
    outputs = console
  []
  [deriv_err_max]
    type = ElementExtremeMaterialProperty
    mat_prop = derivative_error
    value_type = max
    outputs = console
  []
[]

[Outputs]
  exodus = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References