HomogenizedThermalConductivity

Postprocessor for asymptotic expansion homogenization for thermal conductivity

Description

This PostProcessor computes $var element = document.getElementById("moose-equation-b6aa2dcc-be05-40f1-826d-3a2a1c053fa7");katex.render("k_{ij}^\\text{H}=\\frac{1}{\\left|\\text{Y}\\right|}\\int_\\text{Y}k_{ij}\\left(\\bm{I} + \\frac{\\partial\\psi^k}{\\partial y_j}\\right)\\;\\text{d}\\bm{y}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-78be4b62-954f-422c-91a5-a59682c2b09a");katex.render("k^\\text{H}_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the homogenized thermal conductivity. It is used in conjunction with the Heat Conduction Kernel and the Homogenized Heat Conduction Kernel to compute homogenized thermal conductivity values according to $var element = document.getElementById("moose-equation-df570b20-8b92-486c-899e-36903faa8ea2");katex.render("\\int_Y \\frac{\\partial v}{\\partial y_i}k_{ik} \\frac{\\partial \\psi^k}{\\partial y_j} = \\int_Y \\frac{\\partial v}{\\partial y_i}k_{ik} \\text{d}y", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-ffcda4a9-9346-4c18-955c-d9c982beaf48");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the diffusion coefficient (thermal conductivity). See Hales et al. (2015).

Example Input File Syntax

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

/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 this pp acts in (0 for x, 1 for y, 2 for z)
C++ Type:unsigned int
Options:
Description:An integer corresponding to the direction this pp acts in (0 for x, 1 for y, 2 for z)
temp_xsolution in x
C++ Type:std::vector
Options:
Description:solution in x
variableThe name of the variable that this object operates on
C++ Type:std::vector
Options:
Description:The name of the variable that this object 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_conductivityProperty name of the diffusivity (Default: thermal_conductivity)
Default:thermal_conductivity
C++ Type:MaterialPropertyName
Options:
Description:Property name of the diffusivity (Default: thermal_conductivity)
execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:NONE INITIAL LINEAR NONLINEAR TIMESTEP_END TIMESTEP_BEGIN FINAL CUSTOM TRANSFER
Description:The list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.
scale_factor1Scale factor
Default:1
C++ Type:double
Options:
Description:Scale factor
temp_ysolution in y
C++ Type:std::vector
Options:
Description:solution in y
temp_zsolution in z
C++ Type:std::vector
Options:
Description:solution in z

Optional Parameters

allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Options:
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
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.
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
outputsVector of output names were you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object
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

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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