AsymptoticExpansionHomogenizationKernel

Kernel for asymptotic expansion homogenization for elasticity

Description

This Kernel computes $var element = document.getElementById("moose-equation-0b983365-aa9e-4d11-8238-c173fda571ab");katex.render(" \\int_\\text{Y} \\frac{\\partial v_i}{\\partial y_j}D_{ijkl}\\;\\text{d}\\bm{y}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-868b1699-69c0-4c87-908a-3ed3bd596034");katex.render("D_{ijkl}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the elasticity tensor. It is used in conjunction with the Stress Divergence Kernel and the Asymptotic Expansion Homogenization Elastic Constants PostProcessor to compute homogenized elasticity tensor values according to $var element = document.getElementById("moose-equation-8368d0da-c780-4360-9c24-d531010efca2");katex.render("\\int_\\text{Y}\\frac{\\partial v_i}{\\partial y_j} D_{ijkl} \\frac{\\partial\\chi_\\kappa^{mn}}{\\partial y_l}\\text{d}\\bm{y} = \\int_\\text{Y} \\frac{\\partial v_i}{\\partial y_j}D_{ijkl}\\;\\text{d}\\bm{y}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-5dd2adbd-d9ba-4ccb-8234-223e3c8547d6");katex.render("D_{ijkl}^\\text{H} = \\frac{1}{\\left|\\text{Y}\\right|}\\int_\\text{Y}D_{ijkl} \\left(\\bm{I}+\\frac{\\partial\\chi^{mn}_\\kappa}{\\partial y_l}\\right)\\;d\\bm{y}.", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-ae0e664d-3145-4363-a07f-e76afa72992a");katex.render("D_{ijkl}^\\text{H}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the homogenized elasticity tensor. See Hales et al. (2015).

Example Input File Syntax

[./aeh_dx_xx]
  type = AsymptoticExpansionHomogenizationKernel
  variable = dx_xx
  component = 0
  column = xx
  base_name = xx
[../]

/opt/civet/build_0/moose/modules/tensor_mechanics/test/tests/homogenization/anisoShortFiber.i

#
# Test from:
#   Multiple Scale Analysis of Heterogeneous Elastic Structures Using
#   Homogenization Theory and Voronoi Cell Finite Element Method
#   by S.Ghosh et. al, Int J. Solids Structures, Vol. 32, No. 1,
#   pp. 27-62, 1995.
#
# From that paper, elastic constants should be:
# E1111: 122.4
# E2222: 151.2
# E1212:  42.1
# E1122:  36.23
#
# Note: this is for plane stress conditions
#

[Mesh]
  file = anisoShortFiber.e
  # To calculate matching values, refine the mesh one time.
  # We use a coarse mesh for speed in this test.
  # uniform_refine = 1
[]

[Variables]
  [./dx_xx]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xx]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_yy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_yy]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_xy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xy]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]

  [./div_x_xx]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_y_xx]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_x_yy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_y_yy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_x_xy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]
  [./div_y_xy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]

  [./aeh_dx_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xx
    component = 0
    column = xx
    base_name = xx
  [../]
  [./aeh_dy_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xx
    component = 1
    column = xx
    base_name = xx
  [../]

  [./aeh_dx_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_yy
    component = 0
    column = yy
    base_name = yy
  [../]
  [./aeh_dy_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_yy
    component = 1
    column = yy
    base_name = yy
  [../]

  [./aeh_dx_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xy
    component = 0
    column = xy
    base_name = xy
  [../]
  [./aeh_dy_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xy
    component = 1
    column = xy
    base_name = xy
  [../]
[]

[BCs]

  [./Periodic]
    [./top_bottom]
      primary = 30
      secondary = 40
      translation = '0 1 0'
    [../]

    [./left_right]
      primary = 10
      secondary = 20
      translation = '1 0 0'
    [../]
  [../]

  [./dx_xx_anchor]
    type = DirichletBC
    variable = dx_xx
    boundary = 1
    value = 0.0
  [../]

  [./dy_xx_anchor]
    type = DirichletBC
    variable = dy_xx
    boundary = 1
    value = 0.0
  [../]

  [./dx_yy_anchor]
    type = DirichletBC
    variable = dx_yy
    boundary = 1
    value = 0.0
  [../]

  [./dy_yy_anchor]
    type = DirichletBC
    variable = dy_yy
    boundary = 1
    value = 0.0
  [../]

  [./dx_xy_anchor]
    type = DirichletBC
    variable = dx_xy
    boundary = 1
    value = 0.0
  [../]

  [./dy_xy_anchor]
    type = DirichletBC
    variable = dy_xy
    boundary = 1
    value = 0.0
  [../]
[]

[Materials]

  [./elastic_stress_xx]
    type = ComputeLinearElasticStress
    base_name = xx
  [../]
  [./elastic_stress_yy]
    type = ComputeLinearElasticStress
    base_name = yy
  [../]
  [./elastic_stress_xy]
    type = ComputeLinearElasticStress
    base_name = xy
  [../]
  [./strain_xx]
    type = ComputeSmallStrain
    displacements = 'dx_xx dy_xx'
    base_name = xx
  [../]
  [./strain_yy]
    type = ComputeSmallStrain
    displacements = 'dx_yy dy_yy'
    base_name = yy
  [../]
  [./strain_xy]
    type = ComputeSmallStrain
    displacements = 'dx_xy dy_xy'
    base_name = xy
  [../]

  [./block1]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xx
  [../]

  [./block2]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = yy
  [../]

  [./block3]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xy
  [../]

  [./block4]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xx
  [../]

  [./block5]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = yy
  [../]

  [./block6]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xy
 [../]

[]

[Postprocessors]

  [./E1111]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E2222]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = yy
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E1122]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E2211]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xy
    row = yy
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E1212]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xy
    column = xy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]
[]

[Preconditioning]
 [./SMP]
  type = SMP
  full = true
 [../]
[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-ksp_gmres_modifiedgramschmidt'
  petsc_options_iname = '-ksp_gmres_restart -pc_type   -pc_hypre_type -pc_hypre_boomeramg_max_iter -pc_hypre_boomeramg_grid_sweeps_all -ksp_type -mat_mffd_type'
  petsc_options_value = '201                 hypre       boomeramg      2                            2                                   fgmres    ds'

  line_search = 'none'

  l_tol = 1e-4
  l_max_its = 40
  nl_max_its = 40
  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-10

  start_time = 0.0
  end_time = 10.0
  num_steps = 1
  dt = 10

[]

[Outputs]
  exodus = true
  csv = true
[]

Input Parameters

columnThe column of the material matrix this kernel acts in. (xx, yy, zz, yz, xz, or xy)
C++ Type:MooseEnum
Options:xx yy zz yz xz xy
Description:The column of the material matrix this kernel acts in. (xx, yy, zz, yz, xz, or xy)
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

base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Options:
Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
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
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/tensor_mechanics/test/tests/homogenization/anisoLongFiber.i
modules/tensor_mechanics/test/tests/homogenization/anisoShortFiber.i

modules/tensor_mechanics/test/tests/homogenization/anisoLongFiber.i

#
# Test from:
#   Multiple Scale Analysis of Heterogeneous Elastic Structures Using
#   Homogenization Theory and Voronoi Cell Finite Element Method
#   by S.Ghosh et. al, Int J. Solids Structures, Vol. 32, No. 1,
#   pp. 27-62, 1995.
#
# From that paper, elastic constants should be:
# E1111: 136.1
# E2222: 245.8
# E1212:  46.85
# E1122:  36.08
#
# Note: this is for plane stress conditions
#

[Mesh]
  file = anisoLongFiber.e
[]

[Variables]
  [./dx_xx]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xx]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_yy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_yy]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_xy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xy]
    order = FIRST
    family = LAGRANGE
  [../]
[]


[Kernels]

  [./div_x_xx]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_y_xx]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_x_yy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_y_yy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_x_xy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]
  [./div_y_xy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]

  [./homo_dx_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xx
    component = 0
    column = xx
    base_name = xx
  [../]
  [./homo_dy_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xx
    component = 1
    column = xx
    base_name = xx
  [../]

  [./homo_dx_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_yy
    component = 0
    column = yy
    base_name = yy
  [../]
  [./homo_dy_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_yy
    component = 1
    column = yy
    base_name = yy
  [../]

  [./homo_dx_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xy
    component = 0
    column = xy
    base_name = xy
  [../]
  [./homo_dy_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xy
    component = 1
    column = xy
    base_name = xy
  [../]
[]

[BCs]

  [./Periodic]
    [./top_bottom]
      primary = 30
      secondary = 40
      translation = '0 1 0'
    [../]

    [./left_right]
      primary = 10
      secondary = 20
      translation = '1 0 0'
    [../]
  [../]

  [./dx_xx_anchor]
    type = DirichletBC
    variable = dx_xx
    boundary = 1
    value = 0.0
  [../]

  [./dy_xx_anchor]
    type = DirichletBC
    variable = dy_xx
    boundary = 1
    value = 0.0
  [../]

  [./dx_yy_anchor]
    type = DirichletBC
    variable = dx_yy
    boundary = 1
    value = 0.0
  [../]

  [./dy_yy_anchor]
    type = DirichletBC
    variable = dy_yy
    boundary = 1
    value = 0.0
  [../]

  [./dx_xy_anchor]
    type = DirichletBC
    variable = dx_xy
    boundary = 1
    value = 0.0
  [../]

  [./dy_xy_anchor]
    type = DirichletBC
    variable = dy_xy
    boundary = 1
    value = 0.0
  [../]
[]

[Materials]

  [./elastic_stress_xx]
    type = ComputeLinearElasticStress
    base_name = xx
  [../]
  [./elastic_stress_yy]
    type = ComputeLinearElasticStress
    base_name = yy
  [../]
  [./elastic_stress_xy]
    type = ComputeLinearElasticStress
    base_name = xy
  [../]
  [./strain_xx]
    type = ComputeSmallStrain
    displacements = 'dx_xx dy_xx'
    base_name = xx
  [../]
  [./strain_yy]
    type = ComputeSmallStrain
    displacements = 'dx_yy dy_yy'
    base_name = yy
  [../]
  [./strain_xy]
    type = ComputeSmallStrain
    displacements = 'dx_xy dy_xy'
    base_name = xy
  [../]


  [./block1]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xx
  [../]

  [./block2]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = yy
  [../]

  [./block3]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xy
  [../]

  [./block4]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xx
  [../]

  [./block5]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = yy
  [../]

  [./block6]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xy
 [../]

[]

[Postprocessors]

  [./E1111]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E2222]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = yy
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E1122]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]


  [./E2211]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = yy
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]


  [./E1212]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xy
    column = xy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]
[]

[Preconditioning]
  [./SMP]
    type = SMP
   full = true
  [../]
[]


[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-ksp_gmres_modifiedgramschmidt'
  petsc_options_iname = '-ksp_gmres_restart -pc_type   -pc_hypre_type -pc_hypre_boomeramg_max_iter -pc_hypre_boomeramg_grid_sweeps_all -ksp_type -mat_mffd_type'
  petsc_options_value = '201                 hypre       boomeramg      2                            2                                   fgmres    ds'

  line_search = 'none'


  l_tol = 1e-4
  l_max_its = 40
  nl_max_its = 40
  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-10

  start_time = 0.0
  end_time = 10.0
  num_steps = 1
  dt = 10

[]


[Outputs]
  exodus = true
  csv = true
[]

modules/tensor_mechanics/test/tests/homogenization/anisoShortFiber.i

#
# Test from:
#   Multiple Scale Analysis of Heterogeneous Elastic Structures Using
#   Homogenization Theory and Voronoi Cell Finite Element Method
#   by S.Ghosh et. al, Int J. Solids Structures, Vol. 32, No. 1,
#   pp. 27-62, 1995.
#
# From that paper, elastic constants should be:
# E1111: 122.4
# E2222: 151.2
# E1212:  42.1
# E1122:  36.23
#
# Note: this is for plane stress conditions
#

[Mesh]
  file = anisoShortFiber.e
  # To calculate matching values, refine the mesh one time.
  # We use a coarse mesh for speed in this test.
  # uniform_refine = 1
[]

[Variables]
  [./dx_xx]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xx]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_yy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_yy]
    order = FIRST
    family = LAGRANGE
  [../]

  [./dx_xy]
    order = FIRST
    family = LAGRANGE
  [../]
  [./dy_xy]
    order = FIRST
    family = LAGRANGE
  [../]
[]


[Kernels]

  [./div_x_xx]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_y_xx]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xx
    displacements = 'dx_xx dy_xx'
    use_displaced_mesh = false
    base_name = xx
  [../]
  [./div_x_yy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_y_yy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_yy
    displacements = 'dx_yy dy_yy'
    use_displaced_mesh = false
    base_name = yy
  [../]
  [./div_x_xy]
    type = StressDivergenceTensors
    component = 0
    variable = dx_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]
  [./div_y_xy]
    type = StressDivergenceTensors
    component = 1
    variable = dy_xy
    displacements = 'dx_xy dy_xy'
    use_displaced_mesh = false
    base_name = xy
  [../]

  [./aeh_dx_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xx
    component = 0
    column = xx
    base_name = xx
  [../]
  [./aeh_dy_xx]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xx
    component = 1
    column = xx
    base_name = xx
  [../]

  [./aeh_dx_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_yy
    component = 0
    column = yy
    base_name = yy
  [../]
  [./aeh_dy_yy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_yy
    component = 1
    column = yy
    base_name = yy
  [../]

  [./aeh_dx_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dx_xy
    component = 0
    column = xy
    base_name = xy
  [../]
  [./aeh_dy_xy]
    type = AsymptoticExpansionHomogenizationKernel
    variable = dy_xy
    component = 1
    column = xy
    base_name = xy
  [../]
[]

[BCs]

  [./Periodic]
    [./top_bottom]
      primary = 30
      secondary = 40
      translation = '0 1 0'
    [../]

    [./left_right]
      primary = 10
      secondary = 20
      translation = '1 0 0'
    [../]
  [../]

  [./dx_xx_anchor]
    type = DirichletBC
    variable = dx_xx
    boundary = 1
    value = 0.0
  [../]

  [./dy_xx_anchor]
    type = DirichletBC
    variable = dy_xx
    boundary = 1
    value = 0.0
  [../]

  [./dx_yy_anchor]
    type = DirichletBC
    variable = dx_yy
    boundary = 1
    value = 0.0
  [../]

  [./dy_yy_anchor]
    type = DirichletBC
    variable = dy_yy
    boundary = 1
    value = 0.0
  [../]

  [./dx_xy_anchor]
    type = DirichletBC
    variable = dx_xy
    boundary = 1
    value = 0.0
  [../]

  [./dy_xy_anchor]
    type = DirichletBC
    variable = dy_xy
    boundary = 1
    value = 0.0
  [../]
[]

[Materials]

  [./elastic_stress_xx]
    type = ComputeLinearElasticStress
    base_name = xx
  [../]
  [./elastic_stress_yy]
    type = ComputeLinearElasticStress
    base_name = yy
  [../]
  [./elastic_stress_xy]
    type = ComputeLinearElasticStress
    base_name = xy
  [../]
  [./strain_xx]
    type = ComputeSmallStrain
    displacements = 'dx_xx dy_xx'
    base_name = xx
  [../]
  [./strain_yy]
    type = ComputeSmallStrain
    displacements = 'dx_yy dy_yy'
    base_name = yy
  [../]
  [./strain_xy]
    type = ComputeSmallStrain
    displacements = 'dx_xy dy_xy'
    base_name = xy
  [../]


  [./block1]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xx
  [../]

  [./block2]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = yy
  [../]

  [./block3]
    type =  ComputeElasticityTensor
    block = 1
    fill_method = symmetric9
    C_ijkl = '81.360117 26.848839 26.848839 81.360117 26.848839 81.360117 27.255639 27.255639 27.255639'
    base_name = xy
  [../]

  [./block4]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xx
  [../]

  [./block5]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = yy
  [../]

  [./block6]
    type =  ComputeElasticityTensor
    block = 2
    fill_method = symmetric9
    C_ijkl = '416.66667 83.33333 83.33333 416.6667 83.33333 416.66667 166.66667 166.66667 166.66667'
    base_name = xy
 [../]

[]

[Postprocessors]

  [./E1111]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E2222]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = yy
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]

  [./E1122]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xx
    column = yy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]


  [./E2211]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xy
    row = yy
    column = xx
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]


  [./E1212]
    type = AsymptoticExpansionHomogenizationElasticConstants
    base_name = xx
    row = xy
    column = xy
    dx_xx = dx_xx
    dy_xx = dy_xx
    dx_yy = dx_yy
    dy_yy = dy_yy
    dx_xy = dx_xy
    dy_xy = dy_xy
    execute_on = 'initial timestep_end'
  [../]
[]

[Preconditioning]
 [./SMP]
  type = SMP
  full = true
 [../]
[]


[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options = '-ksp_gmres_modifiedgramschmidt'
  petsc_options_iname = '-ksp_gmres_restart -pc_type   -pc_hypre_type -pc_hypre_boomeramg_max_iter -pc_hypre_boomeramg_grid_sweeps_all -ksp_type -mat_mffd_type'
  petsc_options_value = '201                 hypre       boomeramg      2                            2                                   fgmres    ds'

  line_search = 'none'


  l_tol = 1e-4
  l_max_its = 40
  nl_max_its = 40
  nl_abs_tol = 1e-10
  nl_rel_tol = 1e-10

  start_time = 0.0
  end_time = 10.0
  num_steps = 1
  dt = 10

[]


[Outputs]
  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"
}

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