HomogenizedHeatConduction

Kernel for asymptotic expansion homogenization for thermal conductivity

Description

This Kernel computes $var element = document.getElementById("moose-equation-fcbd1115-9a37-46de-950b-84e2e46efb45");katex.render("\\int_Y \\frac{\\partial v}{\\partial y_i}k_{ik} \\text{d}y", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-8c38ac05-8e32-4a98-ab7f-0775a5363e70");katex.render("k_{ik}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the thermal conductivity. It is used in conjunction with the Heat Conduction Kernel and the Homogenized Thermal Conductivity Postprocessor to compute homogenized thermal conductivity values according to $var element = document.getElementById("moose-equation-ce361d43-a4f4-406d-8728-f0eaa594b255");katex.render("\\int_Y \\frac{\\partial v}{\\partial y_i}k_{ik} \\frac{\\partial \\psi^k}{\\partial y_j} = \\int_Y \\frac{\\partial v}{\\partial y_i}k_{ik} \\text{d}y", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-0cfc8fef-698f-4b7e-b09f-ee5abf792b4b");katex.render("k_{ij}^\\text{H}=\\frac{1}{\\left|\\text{Y}\\right|}\\int_\\text{Y}k_{ij}\\left(\\bm{I} + \\frac{\\partial\\psi^k}{\\partial y_j}\\right)\\;\\text{d}\\bm{y}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-705fe62d-983a-4ac1-b443-04fb52c83791");katex.render("k^\\text{H}_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the homogenized thermal conductivity. See Hales et al. (2015).

Example Input File Syntax

[./heat_rhs_x]
  type = HomogenizedHeatConduction
  variable = temp_x
  component = 0
[../]

/opt/civet/build_0/moose/modules/heat_conduction/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
[] # Mesh

[Variables]

  [./temp_x]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  [../]
  [./temp_y]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  [../]

[] # Variables

[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
  [../]

[] # Kernels

[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
 [../]

[] # BCs

[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
  [../]

[] # Materials

[Executioner]

  type = Steady

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'

  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

[] # Executioner

[Outputs]
  exodus = true
[] # Outputs

[Postprocessors]
  [./k_xx]
    type = HomogenizedThermalConductivity
    variable = temp_x
    temp_x = temp_x
    temp_y = temp_y
    component = 0
    execute_on = 'initial timestep_end'
  [../]
  [./k_yy]
    type = HomogenizedThermalConductivity
    variable = temp_y
    temp_x = temp_x
    temp_y = temp_y
    component = 1
    execute_on = 'initial timestep_end'
  [../]
[]

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
Options:
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 Kernel operates on
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this Kernel operates on

Required Parameters

blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied
diffusion_coefficientthermal_conductivityThe diffusion coefficient for the temperature gradient (Default: thermal_conductivity)
Default:thermal_conductivity
C++ Type:MaterialPropertyName
Options:
Description:The diffusion coefficient for the temperature gradient (Default: thermal_conductivity)
displacementsThe displacements
C++ Type:std::vector
Options:
Description:The displacements

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
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
Options:
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
Options:
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
Options:
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
Options:
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
Options:
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
Options:
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

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector
Options:
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
Options:
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
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
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

modules/heat_conduction/test/tests/homogenization/heatConduction2D.i
modules/combined/examples/effective_properties/effective_th_cond.i

modules/heat_conduction/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
[] # Mesh

[Variables]

  [./temp_x]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  [../]
  [./temp_y]
    order = FIRST
    family = LAGRANGE
    initial_condition = 100
  [../]

[] # Variables

[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
  [../]

[] # Kernels

[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
 [../]

[] # BCs

[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
  [../]

[] # Materials

[Executioner]

  type = Steady

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'



  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

[] # Executioner

[Outputs]
  exodus = true
[] # Outputs

[Postprocessors]
  [./k_xx]
    type = HomogenizedThermalConductivity
    variable = temp_x
    temp_x = temp_x
    temp_y = temp_y
    component = 0
    execute_on = 'initial timestep_end'
  [../]
  [./k_yy]
    type = HomogenizedThermalConductivity
    variable = temp_y
    temp_x = temp_x
    temp_y = temp_y
    component = 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'
    function = '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
    args = 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
    variable = Tx_AEH
    temp_x = Tx_AEH
    temp_y = Ty_AEH
    component = 0
    scale_factor = 1e6 #Scale due to length scale of problem
  [../]
  [./k_y_AEH] #Effective thermal conductivity in x-direction from AEH
    type = HomogenizedThermalConductivity
    variable = Ty_AEH
    temp_x = Tx_AEH
    temp_y = Ty_AEH
    component = 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
[]

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"
}

Description
Example Input File Syntax
Input Parameters
Input Files
References

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