HomogenizedHeatConduction

Kernel for asymptotic expansion homogenization for thermal conductivity

note:Einstein summation convention

Einstein summation convention is used in this documentation page.

Description

This Kernel computes the right hand side of the equation of the equation

var element = document.getElementById("moose-equation-f96c2ce2-aeba-4563-a92d-87bb721ea21a");katex.render("\\int_Y \\frac{\\partial v}{\\partial y_i}\\lambda_{ij} \\frac{\\partial \\chi^k}{\\partial y_j} = -\\int_Y \\frac{\\partial v}{\\partial y_i}\\lambda_{ik} \\text{d}y", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-ab515d4c-4287-40ba-aa8c-784ec278ca57");katex.render("\\lambda_{ik}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity tensor, $var element = document.getElementById("moose-equation-49ee8b40-601c-48ef-be05-674cb3480c52");katex.render("y_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the coordinate in the unit cell, and $var element = document.getElementById("moose-equation-397184c4-9b70-46be-b51d-01efa2b61c96");katex.render("\\chi^k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the $var element = document.getElementById("moose-equation-9ae4bff5-c8f6-4446-b9a3-2e8ee8347216");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ -th characteristic function used for homogenizing the thermal conductivity. In this case, the thermal conductivity is isotropic, i.e. $var element = document.getElementById("moose-equation-80e93c1d-121a-456e-9b77-86541a79d9fb");katex.render("\\lambda_{ik} = \\lambda \\delta_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ with $var element = document.getElementById("moose-equation-2fef1fc5-067e-4c85-a296-a080296d5c64");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ being the thermal conductivity and $var element = document.getElementById("moose-equation-80c94764-752c-4863-94a6-310a3641e280");katex.render("\\delta_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ being the Kronecker delta. The equation simplifies to:

var element = document.getElementById("moose-equation-b2b05592-e9e0-49cf-859c-828162b27bb7");katex.render("\\int_Y \\frac{\\partial v}{\\partial y_i}\\lambda \\frac{\\partial \\chi^k}{\\partial y_j} = -\\int_Y \\frac{\\partial v}{\\partial y_k}\\lambda \\text{d}y", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

This kernel is used in conjunction with the Heat Conduction Kernel and the Homogenized Thermal Conductivity Postprocessor to compute homogenized thermal conductivity values.

This homogenization is executed for a unit cell with periodic boundary conditions. For any vector $var element = document.getElementById("moose-equation-5c056182-973c-4145-896d-ad8d06aa81f4");katex.render("\\vec{y}_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ on the boundary, the unit cell geometry must satisfy the condition:

var element = document.getElementById("moose-equation-99ffaad7-00b5-4d78-a99f-4a1d513a1a6d");katex.render("\\vec{n}(\\vec{y}_b + \\vec{p}) = -\\vec{n}(\\vec{y}_b),", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-5a8ad1cd-7f0d-41e0-9709-0b6c278cd4e6");katex.render("\\vec{n}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the outward normal vector, and $var element = document.getElementById("moose-equation-874c0e2a-5aab-4296-8da9-a7e58f0d4e22");katex.render("\\vec{p}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the periodicity of the boundary where $var element = document.getElementById("moose-equation-3bc448f6-eee0-4f3f-a95d-d1f5077963aa");katex.render("\\vec{y}_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is located.

See Hales et al. (2015) and Song and Youn (2006) for more details.

Example Input File Syntax

[Kernels<<<{"href": "../../syntax/Kernels/index.html"}>>>]
  [heat_rhs_x]
    type = HomogenizedHeatConduction<<<{"description": "Kernel for asymptotic expansion homogenization for thermal conductivity", "href": "HomogenizedHeatConduction.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = temp_x
    component<<<{"description": "An integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)"}>>> = 0
  []
[]

Input Parameters

componentAn integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

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
diffusion_coefficientthermal_conductivityThe diffusion coefficient for the temperature gradient
Default:thermal_conductivity
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The diffusion coefficient for the temperature gradient
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data 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.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
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
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
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

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

(modules/heat_transfer/test/tests/homogenization/homogenize_tc_hex.i)
(modules/heat_transfer/test/tests/homogenization/heatConduction2D.i)
(modules/combined/examples/effective_properties/effective_th_cond.i)

References

J. D. Hales, M. R. Tonks, K. Chockalingam, D. M. Perez, S. R. Novascone, B. W. Spencer, and R. L. Williamson. Asymptotic expansion homogenization for multiscale nuclear fuel analysis. Computational Materials Science, 99:290–297, March 2015. URL: http://dx.doi.org/10.1016/j.commatsci.2014.12.039, doi:10.1016/j.commatsci.2014.12.039.

@Article{hales15homogenization,
    author = "Hales, J. D. and Tonks, M. R. and Chockalingam, K. and Perez, D. M. and Novascone, S. R. and Spencer, B. W. and Williamson, R. L.",
    title = "Asymptotic expansion homogenization for multiscale nuclear fuel analysis",
    journal = "Computational Materials Science",
    year = "2015",
    volume = "99",
    month = "March",
    pages = "290--297",
    doi = "10.1016/j.commatsci.2014.12.039",
    url = "http://dx.doi.org/10.1016/j.commatsci.2014.12.039"
}

Young Seok Song and Jae Ryoun Youn. Evaluation of effective thermal conductivity for carbon nanotube/polymer composites using control volume finite element method. Carbon, 44(4):710–717, 2006. URL: https://www.sciencedirect.com/science/article/pii/S0008622305005609, doi:https://doi.org/10.1016/j.carbon.2005.09.034.

@article{SONG2006710,
    author = "Song, Young Seok and Youn, Jae Ryoun",
    title = "Evaluation of effective thermal conductivity for carbon nanotube/polymer composites using control volume finite element method",
    journal = "Carbon",
    volume = "44",
    number = "4",
    pages = "710-717",
    year = "2006",
    issn = "0008-6223",
    doi = "https://doi.org/10.1016/j.carbon.2005.09.034",
    url = "https://www.sciencedirect.com/science/article/pii/S0008622305005609"
}

(modules/heat_transfer/test/tests/homogenization/heatConduction2D.i)

#
# Homogenization of thermal conductivity according to
#   Homogenization of Temperature-Dependent Thermal Conductivity in Composite
#   Materials, Journal of Thermophysics and Heat Transfer, Vol. 15, No. 1,
#   January-March 2001.
#
# The problem solved here is a simple square with two blocks.  The square is
#   divided vertically between the blocks.  One block has a thermal conductivity
#   of 10.  The other block's thermal conductivity is 100.
#
# The analytic solution for the homogenized thermal conductivity in the
#   horizontal direction is found by summing the thermal resistance, recognizing
#   that the blocks are in series:
#
#   R = L/A/k = R1 + R2 = L1/A1/k1 + L2/A2/k2 = .5/1/10 + .5/1/100
#   Since L = A = 1, k_xx = 18.1818.
#
# The analytic solution for the homogenized thermal conductivity in the vertical
#   direction is found by summing reciprocals of resistance, recognizing that
#   the blocks are in parallel:
#
#   1/R = k*A/L = 1/R1 + 1/R2 = 10*.5/1 + 100*.5/1
#   Since L = A = 1, k_yy = 55.0.
#

[Mesh]
  file = heatConduction2D.e
[]

[Variables]
  [temp_x]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  []
  [temp_y]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  []
[]

[Kernels]
  [heat_x]
    type = HeatConduction
    variable = temp_x
  []
  [heat_y]
    type = HeatConduction
    variable = temp_y
  []
  [heat_rhs_x]
    type = HomogenizedHeatConduction
    variable = temp_x
    component = 0
  []
  [heat_rhs_y]
    type = HomogenizedHeatConduction
    variable = temp_y
    component = 1
  []
[]

[BCs]
 [Periodic]
   [left_right]
     primary = 10
     secondary = 20
     translation = '1 0 0'
   []
   [bottom_top]
     primary = 30
     secondary = 40
     translation = '0 1 0'
   []
 []
 [fix_center_x]
   type = DirichletBC
   variable = temp_x
   value = 100
   boundary = 1
 []
 [fix_center_y]
   type = DirichletBC
   variable = temp_y
   value = 100
   boundary = 1
 []
[]

[Materials]
  [heat1]
    type = HeatConductionMaterial
    block = 1
    specific_heat = 0.116
    thermal_conductivity = 10
  []
  [heat2]
    type = HeatConductionMaterial
    block = 2
    specific_heat = 0.116
    thermal_conductivity = 100
  []
[]

[Executioner]
  type = Steady
  petsc_options_iname = '-pc_type -ksp_gmres_restart'
  petsc_options_value = 'lu       101'
  line_search = 'none'
  nl_abs_tol = 1e-11
  nl_rel_tol = 1e-10
  l_max_its = 20
[]

[Outputs]
  exodus = true
[]

[Postprocessors]
  [k_xx]
    type = HomogenizedThermalConductivity
    chi = 'temp_x temp_y'
    row = 0
    col = 0
    execute_on = 'initial timestep_end'
  []

  [k_yy]
    type = HomogenizedThermalConductivity
    chi = 'temp_x temp_y'
    row = 1
    col = 1
    execute_on = 'initial timestep_end'
  []
[]

(modules/heat_transfer/test/tests/homogenization/homogenize_tc_hex.i)

[Mesh]
  [fmg]
    type = FileMeshGenerator
    file = homogenize_tc_hex.e
  []
[]

[Variables]
  [chi_x]
  []

  [chi_y]
  []

  [chi_z]
  []
[]

[Kernels]
  [conduction_x]
    type = HeatConduction
    variable = chi_x
  []

  [conduction_y]
    type = HeatConduction
    variable = chi_y
  []

  [conduction_z]
    type = HeatConduction
    variable = chi_z
  []

  [rhs_hom_x]
    type = HomogenizedHeatConduction
    variable = chi_x
    component = 0
  []

  [rhs_hom_y]
    type = HomogenizedHeatConduction
    variable = chi_y
    component = 1
  []

  [rhs_hom_z]
    type = HomogenizedHeatConduction
    variable = chi_z
    component = 2
  []
[]

[Materials]
  [thermal_conductivity1]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '20'
    block = 'mat1'
  []

  [thermal_conductivity2]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '10'
    block = 'mat2'
  []

  [thermal_conductivity3]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '4'
    block = 'mat3 mat4 mat5'
  []

  [thermal_conductivity4]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '100'
    block = 'mat6 mat7'
  []

  [thermal_conductivity5]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '0.001'
    block = 'mat8'
  []

  [thermal_conductivity6]
    type = GenericConstantMaterial
    prop_names = 'thermal_conductivity'
    prop_values = '12'
    block = 'mat9'
  []
[]

[BCs]
  [fix_chi_x]
    type = DirichletBC
    variable = chi_x
    value = 0
    boundary = fix_chi
  []

  [fix_chi_y]
    type = DirichletBC
    variable = chi_y
    value = 0
    boundary = fix_chi
  []

  [fix_chi_z]
    type = DirichletBC
    variable = chi_z
    value = 0
    boundary = fix_chi
  []

  [Periodic]
    [pair_1]
      primary = pb_1a
      secondary = pb_1b
      translation = '0 7 0'
    []

    F2F = 7
    dx = ${fparse 3 * F2F / 2 / sqrt(3)}
    [pair_2]
      primary = pb_2a
      secondary = pb_2b
      translation = '${fparse -dx} ${fparse F2F / 2} 0'
    []

    [pair_3]
      primary = pb_3a
      secondary = pb_3b
      translation = '${fparse dx} ${fparse F2F / 2} 0'
    []

    [pair_4]
      primary = pb_4a
      secondary = pb_4b
      translation = '0 0 -2'
    []
  []
[]

[Executioner]
  type = Steady
  petsc_options_iname = '-pc_type -pc_hypre_type -pc_hypre_boomeramg_strong_threshold -ksp_gmres_restart -pc_hypre_boomeramg_max_iter -pc_hypre_boomeramg_tol'
  petsc_options_value = 'hypre boomeramg 0.7 100 30 1e-5'
[]

[Postprocessors]
  [k_xx]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 0
    col = 0
    execute_on = 'initial timestep_end'
  []
  [k_xy]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 0
    col = 1
    execute_on = 'initial timestep_end'
  []
  [k_xz]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 0
    col = 2
    execute_on = 'initial timestep_end'
  []
  [k_yx]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 1
    col = 0
    execute_on = 'initial timestep_end'
  []
  [k_yy]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 1
    col = 1
    execute_on = 'initial timestep_end'
  []
  [k_yz]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 1
    col = 2
    execute_on = 'initial timestep_end'
  []
  [k_zx]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 2
    col = 0
    execute_on = 'initial timestep_end'
  []
  [k_zy]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 2
    col = 1
    execute_on = 'initial timestep_end'
  []
  [k_zz]
    type = HomogenizedThermalConductivity
    chi = 'chi_x chi_y chi_z'
    row = 2
    col = 2
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  [outp_csv]
    type = CSV
    # these are too small and will cause issues
    hide = 'k_xz k_yz k_zx k_zy'
  []
[]

(modules/heat_transfer/test/tests/homogenization/heatConduction2D.i)

#
# Homogenization of thermal conductivity according to
#   Homogenization of Temperature-Dependent Thermal Conductivity in Composite
#   Materials, Journal of Thermophysics and Heat Transfer, Vol. 15, No. 1,
#   January-March 2001.
#
# The problem solved here is a simple square with two blocks.  The square is
#   divided vertically between the blocks.  One block has a thermal conductivity
#   of 10.  The other block's thermal conductivity is 100.
#
# The analytic solution for the homogenized thermal conductivity in the
#   horizontal direction is found by summing the thermal resistance, recognizing
#   that the blocks are in series:
#
#   R = L/A/k = R1 + R2 = L1/A1/k1 + L2/A2/k2 = .5/1/10 + .5/1/100
#   Since L = A = 1, k_xx = 18.1818.
#
# The analytic solution for the homogenized thermal conductivity in the vertical
#   direction is found by summing reciprocals of resistance, recognizing that
#   the blocks are in parallel:
#
#   1/R = k*A/L = 1/R1 + 1/R2 = 10*.5/1 + 100*.5/1
#   Since L = A = 1, k_yy = 55.0.
#

[Mesh]
  file = heatConduction2D.e
[]

[Variables]
  [temp_x]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  []
  [temp_y]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  []
[]

[Kernels]
  [heat_x]
    type = HeatConduction
    variable = temp_x
  []
  [heat_y]
    type = HeatConduction
    variable = temp_y
  []
  [heat_rhs_x]
    type = HomogenizedHeatConduction
    variable = temp_x
    component = 0
  []
  [heat_rhs_y]
    type = HomogenizedHeatConduction
    variable = temp_y
    component = 1
  []
[]

[BCs]
 [Periodic]
   [left_right]
     primary = 10
     secondary = 20
     translation = '1 0 0'
   []
   [bottom_top]
     primary = 30
     secondary = 40
     translation = '0 1 0'
   []
 []
 [fix_center_x]
   type = DirichletBC
   variable = temp_x
   value = 100
   boundary = 1
 []
 [fix_center_y]
   type = DirichletBC
   variable = temp_y
   value = 100
   boundary = 1
 []
[]

[Materials]
  [heat1]
    type = HeatConductionMaterial
    block = 1
    specific_heat = 0.116
    thermal_conductivity = 10
  []
  [heat2]
    type = HeatConductionMaterial
    block = 2
    specific_heat = 0.116
    thermal_conductivity = 100
  []
[]

[Executioner]
  type = Steady
  petsc_options_iname = '-pc_type -ksp_gmres_restart'
  petsc_options_value = 'lu       101'
  line_search = 'none'
  nl_abs_tol = 1e-11
  nl_rel_tol = 1e-10
  l_max_its = 20
[]

[Outputs]
  exodus = true
[]

[Postprocessors]
  [k_xx]
    type = HomogenizedThermalConductivity
    chi = 'temp_x temp_y'
    row = 0
    col = 0
    execute_on = 'initial timestep_end'
  []

  [k_yy]
    type = HomogenizedThermalConductivity
    chi = 'temp_x temp_y'
    row = 1
    col = 1
    execute_on = 'initial timestep_end'
  []
[]

(modules/combined/examples/effective_properties/effective_th_cond.i)

# This example calculates the effective thermal conductivity across a microstructure
# with circular second phase precipitates. Two methods are used to calculate the effective thermal conductivity,
# the direct method that applies a temperature to one side and a heat flux to the other,
# and the AEH method.
[Mesh] #Sets mesh size to 10 microns by 10 microns
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 100
    ny = 100
    xmax = 10
    ymax = 10
  []
  [new_nodeset]
    input = gen
    type = ExtraNodesetGenerator
    coord = '5 5'
    new_boundary = 100
  []
[]

[Variables] #Adds variables needed for two ways of calculating effective thermal cond.
  [T] #Temperature used for the direct calculation
    initial_condition = 800
  []
  [Tx_AEH] #Temperature used for the x-component of the AEH solve
    initial_condition = 800
    scaling = 1.0e4 #Scales residual to improve convergence
  []
  [Ty_AEH] #Temperature used for the y-component of the AEH solve
    initial_condition = 800
    scaling = 1.0e4  #Scales residual to improve convergence
  []
[]

[AuxVariables] #Creates second constant phase
  [phase2]
  []
[]

[ICs] #Sets the IC for the second constant phase
  [phase2_IC] #Creates circles with smooth interfaces at random locations
    variable = phase2
    type = MultiSmoothCircleIC
    int_width = 0.3
    numbub = 20
    bubspac = 1.5
    radius = 0.5
    outvalue = 0
    invalue = 1
    block = 0
  []
[]

[Kernels]
  [HtCond] #Kernel for direct calculation of thermal cond
    type = HeatConduction
    variable = T
  []
  [heat_x] #All other kernels are for AEH approach to calculate thermal cond.
    type = HeatConduction
    variable = Tx_AEH
  []
  [heat_rhs_x]
    type = HomogenizedHeatConduction
    variable = Tx_AEH
    component = 0
  []
  [heat_y]
    type = HeatConduction
    variable = Ty_AEH
  []
  [heat_rhs_y]
    type = HomogenizedHeatConduction
    variable = Ty_AEH
    component = 1
  []
[]

[BCs]
  [Periodic]
    [all]
      auto_direction = 'x y'
      variable = 'Tx_AEH Ty_AEH'
    []
  []
  [left_T] #Fix temperature on the left side
    type = DirichletBC
    variable = T
    boundary = left
    value = 800
  []
  [right_flux] #Set heat flux on the right side
    type = NeumannBC
    variable = T
    boundary = right
    value = 5e-6
  []
  [fix_x] #Fix Tx_AEH at a single point
    type = DirichletBC
    variable = Tx_AEH
    value = 800
    boundary = 100
  []
  [fix_y] #Fix Ty_AEH at a single point
    type = DirichletBC
    variable = Ty_AEH
    value = 800
    boundary = 100
  []
[]

[Materials]
  [thcond] #The equation defining the thermal conductivity is defined here, using two ifs
    # The k in the bulk is k_b, in the precipitate k_p2, and across the interaface k_int
    type = ParsedMaterial
    block = 0
    constant_names = 'length_scale k_b k_p2 k_int'
    constant_expressions = '1e-6 5 1 0.1'
    expression = 'sk_b:= length_scale*k_b; sk_p2:= length_scale*k_p2; sk_int:= k_int*length_scale; if(phase2>0.1,if(phase2>0.95,sk_p2,sk_int),sk_b)'
    outputs = exodus
    f_name = thermal_conductivity
    coupled_variables = phase2
  []
[]

[Postprocessors]
  [right_T]
    type = SideAverageValue
    variable = T
    boundary = right
  []
  [k_x_direct] #Effective thermal conductivity from direct method
    # This value is lower than the AEH value because it is impacted by second phase
    # on the right boundary
    type = ThermalConductivity
    variable = T
    flux = 5e-6
    length_scale = 1e-06
    T_hot = 800
    dx = 10
    boundary = right
  []
  [k_x_AEH] #Effective thermal conductivity in x-direction from AEH
    type = HomogenizedThermalConductivity
    chi = 'Tx_AEH Ty_AEH'
    row = 0
    col = 0
    scale_factor = 1e6 #Scale due to length scale of problem
  []
  [k_y_AEH] #Effective thermal conductivity in x-direction from AEH
    type = HomogenizedThermalConductivity
    chi = 'Tx_AEH Ty_AEH'
    row = 1
    col = 1
    scale_factor = 1e6 #Scale due to length scale of problem
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    off_diag_row = 'Tx_AEH Ty_AEH'
    off_diag_column = 'Ty_AEH Tx_AEH'
  []
[]

[Executioner]
  type = Steady
  l_max_its = 15
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart -pc_hypre_boomeramg_strong_threshold'
  petsc_options_value = 'hypre boomeramg 31 0.7'
  l_tol = 1e-04
[]

[Outputs]
  execute_on = 'timestep_end'
  exodus = true
  csv = true
[]

Description
Example Input File Syntax
Input Parameters
Input Files
References