SLKKSMultiACBulkC

Multi-phase SLKKS model kernel for the bulk Allen-Cahn. This includes all terms dependent on chemical potential.

Implements the weak form $var element = document.getElementById("moose-equation-965db90f-6b56-4140-82d5-a689bfd829e5");katex.render("\\left( -M\\frac1{a_{jk}}\\frac{\\partial F_j}{\\partial c_ijk} \\sum_j \\frac{\\partial h_j}{\\partial \\eta_j}\\sum_k c_{ijk}a_{jk},\\psi\\right)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where $var element = document.getElementById("moose-equation-78d1ff6b-81bf-4a43-af9b-0de99e862245");katex.render("c_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the phase concentration for phase $var element = document.getElementById("moose-equation-404d6cbb-7140-4e65-8375-08cdaf590740");katex.render("i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-e6f35de7-94ee-42f0-a55e-d37b97c36f26");katex.render("h_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the interpolation function for phase $var element = document.getElementById("moose-equation-a1cd31ca-a50a-4346-a9e2-dbcab2d47059");katex.render("i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ defined in Folch and Plapp (2005) (referred to as $var element = document.getElementById("moose-equation-cc23c4d1-74a6-4c48-82ea-9ddc2f1f15f9");katex.render("g_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ there, but we use $var element = document.getElementById("moose-equation-f3ea5ba9-5983-44b4-9a6a-2ce3cfc4a485");katex.render("h_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ to maintain consistency with other interpolation functions in MOOSE). Since in the KKS model, chemical potentials are constrained to be equal at each position, $var element = document.getElementById("moose-equation-3f469412-d798-4271-b3ff-ad741f602c8d");katex.render("\\frac{\\partial F_1}{\\partial c_1} = \\frac{\\partial F_2}{\\partial c_2} = \\frac{\\partial F_3}{\\partial c_3}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

Input Parameters

FPhase free energy function that is a function of 'c'
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Phase free energy function that is a function of 'c'
asSublattice site fractions n all phases
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:Sublattice site fractions n all phases
cConcentration variable F depends on
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Concentration variable F depends on
csArray of sublattice concentrations phase for all phases
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Array of sublattice concentrations phase for all phases
h_namesSwitching Function Materials for all phases
C++ Type:std::vector<MaterialPropertyName>
Unit:(no unit assumed)
Controllable:No
Description:Switching Function Materials for all phases
nsNumber of sublattices in each phase
C++ Type:std::vector<unsigned int>
Unit:(no unit assumed)
Controllable:No
Description:Number of sublattices in each phase
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>
Unit:(no unit assumed)
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
etaOrder parameters for all phases
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Order parameters for all phases
eta_iOrder parameter that derivatives are taken with respect to (kernel variable is used if this is not specified)
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Order parameter that derivatives are taken with respect to (kernel variable is used if this is not specified)
mob_nameLThe mobility used with the kernel
Default:L
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The mobility used with the kernel
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
Unit:(no unit assumed)
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.

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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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

Input Files

(modules/phase_field/examples/slkks/CrFe.i)
(modules/phase_field/test/tests/slkks/full_solve.i)

References

R. Folch and M. Plapp. Quantitative phase-field modeling of two-phase growth. Phys. Rev. E, 72:011602, Jul 2005. URL: https://link.aps.org/doi/10.1103/PhysRevE.72.011602, doi:10.1103/PhysRevE.72.011602.

@article{Folch05,
    author = "Folch, R. and Plapp, M.",
    title = "Quantitative phase-field modeling of two-phase growth",
    journal = "Phys. Rev. E",
    volume = "72",
    issue = "1",
    pages = "011602",
    numpages = "27",
    year = "2005",
    month = "Jul",
    publisher = "American Physical Society",
    doi = "10.1103/PhysRevE.72.011602",
    url = "https://link.aps.org/doi/10.1103/PhysRevE.72.011602"
}

(modules/phase_field/examples/slkks/CrFe.i)

#
# SLKKS two phase example for the BCC and SIGMA phases. The sigma phase contains
# multiple sublattices. Free energy from
# Jacob, Aurelie, Erwin Povoden-Karadeniz, and Ernst Kozeschnik. "Revised thermodynamic
# description of the Fe-Cr system based on an improved sublattice model of the sigma phase."
# Calphad 60 (2018): 16-28.
#
# In this simulation we consider diffusion (Cahn-Hilliard) and phase transformation.
#
# This example requires CrFe_sigma_out_var_0001.csv file, which generated by first
# running the CrFe_sigma.i input file.


[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 160
    ny = 1
    nz = 0
    xmin = -25
    xmax = 25
    ymin = -2.5
    ymax = 2.5
    elem_type = QUAD4
  []
[]

[AuxVariables]
  [Fglobal]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [sigma_cr0]
    type = PiecewiseLinear
    data_file = CrFe_sigma_out_var_0001.csv
    format = columns
    x_index_in_file = 5
    y_index_in_file = 2
    xy_in_file_only = false
  []
  [sigma_cr1]
    type = PiecewiseLinear
    data_file = CrFe_sigma_out_var_0001.csv
    format = columns
    x_index_in_file = 5
    y_index_in_file = 3
    xy_in_file_only = false
  []
  [sigma_cr2]
    type = PiecewiseLinear
    data_file = CrFe_sigma_out_var_0001.csv
    format = columns
    x_index_in_file = 5
    y_index_in_file = 4
    xy_in_file_only = false
  []
[]

[Variables]
  # order parameters
  [eta1]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  []
  [eta2]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.5
  []

  # solute concentration
  [cCr]
    order = FIRST
    family = LAGRANGE
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.5+0.1'
    []
  []

  # sublattice concentrations
  [BCC_CR]
    initial_condition = 0.45
  []
  [SIGMA_0CR]
    [InitialCondition]
      type = CoupledValueFunctionIC
      function = sigma_cr0
      v = cCr
      variable = SIGMA_0CR
    []
  []
  [SIGMA_1CR]
    [InitialCondition]
      type = CoupledValueFunctionIC
      function = sigma_cr1
      v = cCr
      variable = SIGMA_1CR
    []
  []
  [SIGMA_2CR]
    [InitialCondition]
      type = CoupledValueFunctionIC
      function = sigma_cr2
      v = cCr
      variable = SIGMA_2CR
    []
  []
  # Lagrange multiplier
  [lambda]
  []
[]

[Materials]
  # CALPHAD free energies
  [F_BCC_A2]
    type = DerivativeParsedMaterial
    property_name = F_BCC_A2
    outputs = exodus
    output_properties = F_BCC_A2
    expression = 'BCC_FE:=1-BCC_CR; G := 8.3145*T*(1.0*if(BCC_CR > 1.0e-15,BCC_CR*log(BCC_CR),0) + '
               '1.0*if(BCC_FE > 1.0e-15,BCC_FE*plog(BCC_FE,eps),0) + 3.0*if(BCC_VA > '
               '1.0e-15,BCC_VA*log(BCC_VA),0))/(BCC_CR + BCC_FE) + 8.3145*T*if(T < '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - '
               '1043.0*BCC_FE*BCC_VA,-8.13674105561218e-49*T^15/(0.525599232981783*BCC_CR*BCC_FE*BCC_'
               'VA*(BCC_CR - BCC_FE) - 0.894055608820709*BCC_CR*BCC_FE*BCC_VA + '
               '0.298657718120805*BCC_CR*BCC_VA - BCC_FE*BCC_VA + 9.58772770853308e-13)^15 - '
               '4.65558036243985e-30*T^9/(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^9 - '
               '1.3485349181899e-10*T^3/(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^3 + 1 - '
               '0.905299382744392*(548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '932.5*BCC_CR*BCC_FE*BCC_VA + 311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA + '
               '1.0e-9)/T,if(T < -548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + '
               '1043.0*BCC_FE*BCC_VA,-8.13674105561218e-49*T^15/(-0.525599232981783*BCC_CR*BCC_FE*BCC'
               '_VA*(BCC_CR - BCC_FE) + 0.894055608820709*BCC_CR*BCC_FE*BCC_VA - '
               '0.298657718120805*BCC_CR*BCC_VA + BCC_FE*BCC_VA + 9.58772770853308e-13)^15 - '
               '4.65558036243985e-30*T^9/(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) '
               '+ 0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^9 - '
               '1.3485349181899e-10*T^3/(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^3 + 1 - '
               '0.905299382744392*(-548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + 1043.0*BCC_FE*BCC_VA + '
               '1.0e-9)/T,if(T > -548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + 1043.0*BCC_FE*BCC_VA & '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA < '
               '0,-79209031311018.7*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^5/T^5 - '
               '3.83095660520737e+42*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^15/T^15 - '
               '1.22565886734485e+72*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^25/T^25,if(T > '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA & 548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - '
               'BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + 311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA > '
               '0,-79209031311018.7*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^5/T^5 - '
               '3.83095660520737e+42*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^15/T^15 - '
               '1.22565886734485e+72*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^25/T^25,0))))*log((2.15*BCC_CR*BCC_FE*BCC_VA - '
               '0.008*BCC_CR*BCC_VA + 2.22*BCC_FE*BCC_VA)*if(2.15*BCC_CR*BCC_FE*BCC_VA - '
               '0.008*BCC_CR*BCC_VA + 2.22*BCC_FE*BCC_VA <= 0,-1.0,1.0) + 1)/(BCC_CR + BCC_FE) + '
               '1.0*(BCC_CR*BCC_VA*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + '
               'BCC_FE*BCC_VA*if(T >= 298.15 & T < 1811.0,77358.5*1/T - 23.5143*T*log(T) + 124.134*T '
               '- 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= 1811.0 & T < '
               '6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - 25383.581,0)))/(BCC_CR '
               '+ BCC_FE) + 1.0*(BCC_CR*BCC_FE*BCC_VA*(500.0 - 1.5*T)*(BCC_CR - BCC_FE) + '
               'BCC_CR*BCC_FE*BCC_VA*(24600.0 - 14.98*T) + BCC_CR*BCC_FE*BCC_VA*(9.15*T - '
               '14000.0)*(BCC_CR - BCC_FE)^2)/(BCC_CR + BCC_FE); G/100000'
    coupled_variables = 'BCC_CR'
    constant_names = 'BCC_VA T eps'
    constant_expressions = '1 1000 0.01'
  []

  [F_SIGMA]
    type = DerivativeParsedMaterial
    property_name = F_SIGMA
    outputs = exodus
    output_properties = F_SIGMA
    expression = 'SIGMA_0FE := 1-SIGMA_0CR; SIGMA_1FE := 1-SIGMA_1CR; SIGMA_2FE := 1-SIGMA_2CR; G := '
               '8.3145*T*(10.0*if(SIGMA_0CR > 1.0e-15,SIGMA_0CR*plog(SIGMA_0CR,eps),0) + '
               '10.0*if(SIGMA_0FE > 1.0e-15,SIGMA_0FE*plog(SIGMA_0FE,eps),0) + 4.0*if(SIGMA_1CR > '
               '1.0e-15,SIGMA_1CR*plog(SIGMA_1CR,eps),0) + 4.0*if(SIGMA_1FE > '
               '1.0e-15,SIGMA_1FE*plog(SIGMA_1FE,eps),0) + 16.0*if(SIGMA_2CR > '
               '1.0e-15,SIGMA_2CR*plog(SIGMA_2CR,eps),0) + 16.0*if(SIGMA_2FE > '
               '1.0e-15,SIGMA_2FE*plog(SIGMA_2FE,eps),0))/(10.0*SIGMA_0CR + 10.0*SIGMA_0FE + '
               '4.0*SIGMA_1CR + 4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE) + '
               '(SIGMA_0FE*SIGMA_1CR*SIGMA_2CR*SIGMA_2FE*(-70.0*T - 170400.0) + '
               'SIGMA_0FE*SIGMA_1FE*SIGMA_2CR*SIGMA_2FE*(-10.0*T - 330839.0))/(10.0*SIGMA_0CR + '
               '10.0*SIGMA_0FE + 4.0*SIGMA_1CR + 4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE) + '
               '(SIGMA_0CR*SIGMA_1CR*SIGMA_2CR*(30.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - '
               '26.908*T*log(T) + 157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= '
               '2180.0 & T < 6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) '
               '+ 132000.0) + SIGMA_0CR*SIGMA_1CR*SIGMA_2FE*(-110.0*T + 16.0*if(T >= 298.15 & T < '
               '1811.0,77358.5*1/T - 23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - '
               '5.89269e-8*T^3.0 + 1225.7,if(T >= 1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - '
               '46.0*T*log(T) + 299.31255*T - 25383.581,0)) + 14.0*if(T >= 298.15 & T < '
               '2180.0,139250.0*1/T - 26.908*T*log(T) + 157.48*T + 0.00189435*T^2.0 - '
               '1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < 6000.0,-2.88526e+32*T^(-9.0) - '
               '50.0*T*log(T) + 344.18*T - 34869.344,0)) + 123500.0) + '
               'SIGMA_0CR*SIGMA_1FE*SIGMA_2CR*(4.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 26.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 140486.0) '
               '+ SIGMA_0CR*SIGMA_1FE*SIGMA_2FE*(20.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 10.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 148800.0) '
               '+ SIGMA_0FE*SIGMA_1CR*SIGMA_2CR*(10.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 20.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 56200.0) + '
               'SIGMA_0FE*SIGMA_1CR*SIGMA_2FE*(26.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 4.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 152700.0) '
               '+ SIGMA_0FE*SIGMA_1FE*SIGMA_2CR*(14.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 16.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 46200.0) + '
               'SIGMA_0FE*SIGMA_1FE*SIGMA_2FE*(30.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 173333.0))/(10.0*SIGMA_0CR + 10.0*SIGMA_0FE + 4.0*SIGMA_1CR + '
               '4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE); G/100000'
    coupled_variables = 'SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    constant_names = 'T eps'
    constant_expressions = '1000 0.01'
  []

  # h(eta)
  [h1]
    type = SwitchingFunctionMaterial
    function_name = h1
    h_order = HIGH
    eta = eta1
  []
  [h2]
    type = SwitchingFunctionMaterial
    function_name = h2
    h_order = HIGH
    eta = eta2
  []

  # g(eta)
  [g1]
    type = BarrierFunctionMaterial
    function_name = g1
    g_order = SIMPLE
    eta = eta1
  []
  [g2]
    type = BarrierFunctionMaterial
    function_name = g2
    g_order = SIMPLE
    eta = eta2
  []

  # constant properties
  [constants]
    type = GenericConstantMaterial
    prop_names = 'D   L   kappa'
    prop_values = '10  1   0.1  '
  []

  # Coefficients for diffusion equation
  [Dh1]
    type = DerivativeParsedMaterial
    material_property_names = 'D h1(eta1)'
    expression = D*h1
    property_name = Dh1
    coupled_variables = eta1
    derivative_order = 1
  []
  [Dh2a]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*10/30
    property_name = Dh2a
    coupled_variables = eta2
    derivative_order = 1
  []
  [Dh2b]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*4/30
    property_name = Dh2b
    coupled_variables = eta2
    derivative_order = 1
  []
  [Dh2c]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*16/30
    property_name = Dh2c
    coupled_variables = eta2
    derivative_order = 1
  []
[]

[Kernels]
  #Kernels for diffusion equation
  [diff_time]
    type = TimeDerivative
    variable = cCr
  []
  [diff_c1]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh1
    v = BCC_CR
    args = eta1
  []
  [diff_c2a]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2a
    v = SIGMA_0CR
    args = eta2
  []
  [diff_c2b]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2b
    v = SIGMA_1CR
    args = eta2
  []
  [diff_c2c]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2c
    v = SIGMA_2CR
    args = eta2
  []

  # enforce pointwise equality of chemical potentials
  [chempot1a2a]
    # The BCC phase has only one sublattice
    # we tie it to the first sublattice with site fraction 10/(10+4+16) in the sigma phase
    type = KKSPhaseChemicalPotential
    variable = BCC_CR
    cb = SIGMA_0CR
    kb = '${fparse 10/30}'
    fa_name = F_BCC_A2
    fb_name = F_SIGMA
    args_b = 'SIGMA_1CR SIGMA_2CR'
  []
  [chempot2a2b]
    # This kernel ties the first two sublattices in the sigma phase together
    type = SLKKSChemicalPotential
    variable = SIGMA_0CR
    a = 10
    cs = SIGMA_1CR
    as = 4
    F = F_SIGMA
    coupled_variables = 'SIGMA_2CR'
  []
  [chempot2b2c]
    # This kernel ties the remaining two sublattices in the sigma phase together
    type = SLKKSChemicalPotential
    variable = SIGMA_1CR
    a = 4
    cs = SIGMA_2CR
    as = 16
    F = F_SIGMA
    coupled_variables = 'SIGMA_0CR'
  []

  [phaseconcentration]
    # This kernel ties the sum of the sublattice concentrations to the global concentration cCr
    type = SLKKSMultiPhaseConcentration
    variable = SIGMA_2CR
    c = cCr
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []

  # Kernels for Allen-Cahn equation for eta1
  [deta1dt]
    type = TimeDerivative
    variable = eta1
  []
  [ACBulkF1]
    type = KKSMultiACBulkF
    variable = eta1
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1    h2'
    gi_name = g1
    eta_i = eta1
    wi = 0.1
    coupled_variables = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR eta2'
  []
  [ACBulkC1]
    type = SLKKSMultiACBulkC
    variable = eta1
    F = F_BCC_A2
    c = BCC_CR
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []
  [ACInterface1]
    type = ACInterface
    variable = eta1
    kappa_name = kappa
  []
  [lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name = h1
    lambda = lambda
    coupled_variables = 'eta2'
  []

  # Kernels for Allen-Cahn equation for eta1
  [deta2dt]
    type = TimeDerivative
    variable = eta2
  []
  [ACBulkF2]
    type = KKSMultiACBulkF
    variable = eta2
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1    h2'
    gi_name = g2
    eta_i = eta2
    wi = 0.1
    coupled_variables = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR eta1'
  []
  [ACBulkC2]
    type = SLKKSMultiACBulkC
    variable = eta2
    F = F_BCC_A2
    c = BCC_CR
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []
  [ACInterface2]
    type = ACInterface
    variable = eta2
    kappa_name = kappa
  []
  [lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name = h2
    lambda = lambda
    coupled_variables = 'eta1'
  []

  # Lagrange-multiplier constraint kernel for lambda
  [lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    h_names = 'h1   h2'
    etas = 'eta1 eta2'
    epsilon = 1e-6
  []
[]

[AuxKernels]
  [GlobalFreeEnergy]
    type = KKSMultiFreeEnergy
    variable = Fglobal
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1 h2'
    gj_names = 'g1 g2'
    interfacial_vars = 'eta1 eta2'
    kappa_names = 'kappa kappa'
    w = 0.1
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  line_search = none

  petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -ksp_gmres_restart'
  petsc_options_value = 'asm      lu          nonzero                    30'

  l_max_its = 100
  nl_max_its = 20
  nl_abs_tol = 1e-10

  end_time = 10000

  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 12
    iteration_window = 2
    growth_factor = 1.5
    cutback_factor = 0.7
    dt = 0.1
  []
[]

[VectorPostprocessors]
  [var]
    type = LineValueSampler
    start_point = '-25 0 0'
    end_point = '25 0 0'
    variable = 'cCr eta1 eta2 SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    num_points = 151
    sort_by = id
    execute_on = 'initial timestep_end'
  []
  [mat]
    type = LineMaterialRealSampler
    start = '-25 0 0'
    end = '25 0 0'
    property = 'F_BCC_A2 F_SIGMA'
    sort_by = id
    execute_on = 'initial timestep_end'
  []
[]

[Postprocessors]
  [F]
    type = ElementIntegralVariablePostprocessor
    variable = Fglobal
    execute_on = 'initial timestep_end'
  []
  [cmin]
    type = NodalExtremeValue
    value_type = min
    variable = cCr
    execute_on = 'initial timestep_end'
  []
  [cmax]
    type = NodalExtremeValue
    value_type = max
    variable = cCr
    execute_on = 'initial timestep_end'
  []
  [ctotal]
    type = ElementIntegralVariablePostprocessor
    variable = cCr
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  exodus = true
  print_linear_residuals = false
  csv = true
  perf_graph = true
[]

(modules/phase_field/test/tests/slkks/full_solve.i)

#
# SLKKS two phase example for the BCC and SIGMA phases. The sigma phase contains
# multiple sublattices. Free energy from
# Jacob, Aurelie, Erwin Povoden-Karadeniz, and Ernst Kozeschnik. "Revised thermodynamic
# description of the Fe-Cr system based on an improved sublattice model of the sigma phase."
# Calphad 60 (2018): 16-28.
#
# In this simulation we consider diffusion (Cahn-Hilliard) and phase transformation.
#

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 30
    ny = 1
    xmin = -25
    xmax = 25
  []
[]

[AuxVariables]
  [Fglobal]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Variables]
  # order parameters
  [eta1]
    initial_condition = 0.5
  []
  [eta2]
    initial_condition = 0.5
  []

  # solute concentration
  [cCr]
    order = FIRST
    family = LAGRANGE
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.5+0.1'
    []
  []

  # sublattice concentrations (good guesses are needed here! - they can be obtained
  # form a static solve like in sublattice_concentrations.i)
  [BCC_CR]
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.5+0.1'
    []
  []
  [SIGMA_0CR]
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.17+0.01'
    []
  []
  [SIGMA_1CR]
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.36+0.02'
    []
  []
  [SIGMA_2CR]
    [InitialCondition]
      type = FunctionIC
      function = '(x+25)/50*0.33+0.20'
    []
  []

  # Lagrange multiplier
  [lambda]
  []
[]

[Materials]
  # CALPHAD free energies
  [F_BCC_A2]
    type = DerivativeParsedMaterial
    property_name = F_BCC_A2
    outputs = exodus
    output_properties = F_BCC_A2
    expression = 'BCC_FE:=1-BCC_CR; G := 8.3145*T*(1.0*if(BCC_CR > 1.0e-15,BCC_CR*log(BCC_CR),0) + '
               '1.0*if(BCC_FE > 1.0e-15,BCC_FE*plog(BCC_FE,eps),0) + 3.0*if(BCC_VA > '
               '1.0e-15,BCC_VA*log(BCC_VA),0))/(BCC_CR + BCC_FE) + 8.3145*T*if(T < '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - '
               '1043.0*BCC_FE*BCC_VA,-8.13674105561218e-49*T^15/(0.525599232981783*BCC_CR*BCC_FE*BCC_'
               'VA*(BCC_CR - BCC_FE) - 0.894055608820709*BCC_CR*BCC_FE*BCC_VA + '
               '0.298657718120805*BCC_CR*BCC_VA - BCC_FE*BCC_VA + 9.58772770853308e-13)^15 - '
               '4.65558036243985e-30*T^9/(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^9 - '
               '1.3485349181899e-10*T^3/(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^3 + 1 - '
               '0.905299382744392*(548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '932.5*BCC_CR*BCC_FE*BCC_VA + 311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA + '
               '1.0e-9)/T,if(T < -548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + '
               '1043.0*BCC_FE*BCC_VA,-8.13674105561218e-49*T^15/(-0.525599232981783*BCC_CR*BCC_FE*BCC'
               '_VA*(BCC_CR - BCC_FE) + 0.894055608820709*BCC_CR*BCC_FE*BCC_VA - '
               '0.298657718120805*BCC_CR*BCC_VA + BCC_FE*BCC_VA + 9.58772770853308e-13)^15 - '
               '4.65558036243985e-30*T^9/(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) '
               '+ 0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^9 - '
               '1.3485349181899e-10*T^3/(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^3 + 1 - '
               '0.905299382744392*(-548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + 1043.0*BCC_FE*BCC_VA + '
               '1.0e-9)/T,if(T > -548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '932.5*BCC_CR*BCC_FE*BCC_VA - 311.5*BCC_CR*BCC_VA + 1043.0*BCC_FE*BCC_VA & '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA < '
               '0,-79209031311018.7*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^5/T^5 - '
               '3.83095660520737e+42*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^15/T^15 - '
               '1.22565886734485e+72*(-0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) + '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA - 0.298657718120805*BCC_CR*BCC_VA + '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^25/T^25,if(T > '
               '548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + '
               '311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA & 548.2*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - '
               'BCC_FE) - 932.5*BCC_CR*BCC_FE*BCC_VA + 311.5*BCC_CR*BCC_VA - 1043.0*BCC_FE*BCC_VA > '
               '0,-79209031311018.7*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^5/T^5 - '
               '3.83095660520737e+42*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^15/T^15 - '
               '1.22565886734485e+72*(0.525599232981783*BCC_CR*BCC_FE*BCC_VA*(BCC_CR - BCC_FE) - '
               '0.894055608820709*BCC_CR*BCC_FE*BCC_VA + 0.298657718120805*BCC_CR*BCC_VA - '
               'BCC_FE*BCC_VA + 9.58772770853308e-13)^25/T^25,0))))*log((2.15*BCC_CR*BCC_FE*BCC_VA - '
               '0.008*BCC_CR*BCC_VA + 2.22*BCC_FE*BCC_VA)*if(2.15*BCC_CR*BCC_FE*BCC_VA - '
               '0.008*BCC_CR*BCC_VA + 2.22*BCC_FE*BCC_VA <= 0,-1.0,1.0) + 1)/(BCC_CR + BCC_FE) + '
               '1.0*(BCC_CR*BCC_VA*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + '
               'BCC_FE*BCC_VA*if(T >= 298.15 & T < 1811.0,77358.5*1/T - 23.5143*T*log(T) + 124.134*T '
               '- 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= 1811.0 & T < '
               '6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - 25383.581,0)))/(BCC_CR '
               '+ BCC_FE) + 1.0*(BCC_CR*BCC_FE*BCC_VA*(500.0 - 1.5*T)*(BCC_CR - BCC_FE) + '
               'BCC_CR*BCC_FE*BCC_VA*(24600.0 - 14.98*T) + BCC_CR*BCC_FE*BCC_VA*(9.15*T - '
               '14000.0)*(BCC_CR - BCC_FE)^2)/(BCC_CR + BCC_FE); G/100000'
    coupled_variables = 'BCC_CR'
    constant_names = 'BCC_VA T eps'
    constant_expressions = '1 1000 0.01'
  []

  [F_SIGMA]
    type = DerivativeParsedMaterial
    property_name = F_SIGMA
    outputs = exodus
    output_properties = F_SIGMA
    expression = 'SIGMA_0FE := 1-SIGMA_0CR; SIGMA_1FE := 1-SIGMA_1CR; SIGMA_2FE := 1-SIGMA_2CR; G := '
               '8.3145*T*(10.0*if(SIGMA_0CR > 1.0e-15,SIGMA_0CR*plog(SIGMA_0CR,eps),0) + '
               '10.0*if(SIGMA_0FE > 1.0e-15,SIGMA_0FE*plog(SIGMA_0FE,eps),0) + 4.0*if(SIGMA_1CR > '
               '1.0e-15,SIGMA_1CR*plog(SIGMA_1CR,eps),0) + 4.0*if(SIGMA_1FE > '
               '1.0e-15,SIGMA_1FE*plog(SIGMA_1FE,eps),0) + 16.0*if(SIGMA_2CR > '
               '1.0e-15,SIGMA_2CR*plog(SIGMA_2CR,eps),0) + 16.0*if(SIGMA_2FE > '
               '1.0e-15,SIGMA_2FE*plog(SIGMA_2FE,eps),0))/(10.0*SIGMA_0CR + 10.0*SIGMA_0FE + '
               '4.0*SIGMA_1CR + 4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE) + '
               '(SIGMA_0FE*SIGMA_1CR*SIGMA_2CR*SIGMA_2FE*(-70.0*T - 170400.0) + '
               'SIGMA_0FE*SIGMA_1FE*SIGMA_2CR*SIGMA_2FE*(-10.0*T - 330839.0))/(10.0*SIGMA_0CR + '
               '10.0*SIGMA_0FE + 4.0*SIGMA_1CR + 4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE) + '
               '(SIGMA_0CR*SIGMA_1CR*SIGMA_2CR*(30.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - '
               '26.908*T*log(T) + 157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= '
               '2180.0 & T < 6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) '
               '+ 132000.0) + SIGMA_0CR*SIGMA_1CR*SIGMA_2FE*(-110.0*T + 16.0*if(T >= 298.15 & T < '
               '1811.0,77358.5*1/T - 23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - '
               '5.89269e-8*T^3.0 + 1225.7,if(T >= 1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - '
               '46.0*T*log(T) + 299.31255*T - 25383.581,0)) + 14.0*if(T >= 298.15 & T < '
               '2180.0,139250.0*1/T - 26.908*T*log(T) + 157.48*T + 0.00189435*T^2.0 - '
               '1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < 6000.0,-2.88526e+32*T^(-9.0) - '
               '50.0*T*log(T) + 344.18*T - 34869.344,0)) + 123500.0) + '
               'SIGMA_0CR*SIGMA_1FE*SIGMA_2CR*(4.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 26.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 140486.0) '
               '+ SIGMA_0CR*SIGMA_1FE*SIGMA_2FE*(20.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 10.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 148800.0) '
               '+ SIGMA_0FE*SIGMA_1CR*SIGMA_2CR*(10.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 20.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 56200.0) + '
               'SIGMA_0FE*SIGMA_1CR*SIGMA_2FE*(26.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 4.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 152700.0) '
               '+ SIGMA_0FE*SIGMA_1FE*SIGMA_2CR*(14.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 16.0*if(T >= 298.15 & T < 2180.0,139250.0*1/T - 26.908*T*log(T) + '
               '157.48*T + 0.00189435*T^2.0 - 1.47721e-6*T^3.0 - 8856.94,if(T >= 2180.0 & T < '
               '6000.0,-2.88526e+32*T^(-9.0) - 50.0*T*log(T) + 344.18*T - 34869.344,0)) + 46200.0) + '
               'SIGMA_0FE*SIGMA_1FE*SIGMA_2FE*(30.0*if(T >= 298.15 & T < 1811.0,77358.5*1/T - '
               '23.5143*T*log(T) + 124.134*T - 0.00439752*T^2.0 - 5.89269e-8*T^3.0 + 1225.7,if(T >= '
               '1811.0 & T < 6000.0,2.2960305e+31*T^(-9.0) - 46.0*T*log(T) + 299.31255*T - '
               '25383.581,0)) + 173333.0))/(10.0*SIGMA_0CR + 10.0*SIGMA_0FE + 4.0*SIGMA_1CR + '
               '4.0*SIGMA_1FE + 16.0*SIGMA_2CR + 16.0*SIGMA_2FE); G/100000'
    coupled_variables = 'SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    constant_names = 'T eps'
    constant_expressions = '1000 0.01'
  []

  # h(eta)
  [h1]
    type = SwitchingFunctionMaterial
    function_name = h1
    h_order = HIGH
    eta = eta1
  []
  [h2]
    type = SwitchingFunctionMaterial
    function_name = h2
    h_order = HIGH
    eta = eta2
  []

  # g(eta)
  [g1]
    type = BarrierFunctionMaterial
    function_name = g1
    g_order = SIMPLE
    eta = eta1
  []
  [g2]
    type = BarrierFunctionMaterial
    function_name = g2
    g_order = SIMPLE
    eta = eta2
  []

  # constant properties
  [constants]
    type = GenericConstantMaterial
    prop_names = 'D   L   kappa'
    prop_values = '10  1   0.1  '
  []

  # Coefficients for diffusion equation
  [Dh1]
    type = DerivativeParsedMaterial
    material_property_names = 'D h1(eta1)'
    expression = D*h1
    property_name = Dh1
    coupled_variables = eta1
    derivative_order = 1
  []
  [Dh2a]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*10/30
    property_name = Dh2a
    coupled_variables = eta2
    derivative_order = 1
  []
  [Dh2b]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*4/30
    property_name = Dh2b
    coupled_variables = eta2
    derivative_order = 1
  []
  [Dh2c]
    type = DerivativeParsedMaterial
    material_property_names = 'D h2(eta2)'
    expression = D*h2*16/30
    property_name = Dh2c
    coupled_variables = eta2
    derivative_order = 1
  []
[]

[Kernels]
  #Kernels for diffusion equation
  [diff_time]
    type = TimeDerivative
    variable = cCr
  []
  [diff_c1]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh1
    v = BCC_CR
    args = eta1
  []
  [diff_c2a]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2a
    v = SIGMA_0CR
    args = eta2
  []
  [diff_c2b]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2b
    v = SIGMA_1CR
    args = eta2
  []
  [diff_c2c]
    type = MatDiffusion
    variable = cCr
    diffusivity = Dh2c
    v = SIGMA_2CR
    args = eta2
  []

  # enforce pointwise equality of chemical potentials
  [chempot1a2a]
    # The BCC phase has only one sublattice
    # we tie it to the first sublattice with site fraction 10/(10+4+16) in the sigma phase
    type = KKSPhaseChemicalPotential
    variable = BCC_CR
    cb = SIGMA_0CR
    kb = '${fparse 10/30}'
    fa_name = F_BCC_A2
    fb_name = F_SIGMA
    args_b = 'SIGMA_1CR SIGMA_2CR'
  []
  [chempot2a2b]
    # This kernel ties the first two sublattices in the sigma phase together
    type = SLKKSChemicalPotential
    variable = SIGMA_0CR
    a = 10
    cs = SIGMA_1CR
    as = 4
    F = F_SIGMA
    coupled_variables = 'SIGMA_2CR'
  []
  [chempot2b2c]
    # This kernel ties the remaining two sublattices in the sigma phase together
    type = SLKKSChemicalPotential
    variable = SIGMA_1CR
    a = 4
    cs = SIGMA_2CR
    as = 16
    F = F_SIGMA
    coupled_variables = 'SIGMA_0CR'
  []

  [phaseconcentration]
    # This kernel ties the sum of the sublattice concentrations to the global concentration cCr
    type = SLKKSMultiPhaseConcentration
    variable = SIGMA_2CR
    c = cCr
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []

  # Kernels for Allen-Cahn equation for eta1
  [deta1dt]
    type = TimeDerivative
    variable = eta1
  []
  [ACBulkF1]
    type = KKSMultiACBulkF
    variable = eta1
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1    h2'
    gi_name = g1
    eta_i = eta1
    wi = 0.1
    coupled_variables = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR eta2'
  []
  [ACBulkC1]
    type = SLKKSMultiACBulkC
    variable = eta1
    F = F_BCC_A2
    c = BCC_CR
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []
  [ACInterface1]
    type = ACInterface
    variable = eta1
    kappa_name = kappa
  []
  [lagrange1]
    type = SwitchingFunctionConstraintEta
    variable = eta1
    h_name = h1
    lambda = lambda
    coupled_variables = 'eta2'
  []

  # Kernels for Allen-Cahn equation for eta1
  [deta2dt]
    type = TimeDerivative
    variable = eta2
  []
  [ACBulkF2]
    type = KKSMultiACBulkF
    variable = eta2
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1    h2'
    gi_name = g2
    eta_i = eta2
    wi = 0.1
    coupled_variables = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR eta1'
  []
  [ACBulkC2]
    type = SLKKSMultiACBulkC
    variable = eta2
    F = F_BCC_A2
    c = BCC_CR
    ns = '1      3'
    as = '1      10        4         16'
    cs = 'BCC_CR SIGMA_0CR SIGMA_1CR SIGMA_2CR'
    h_names = 'h1   h2'
    eta = 'eta1 eta2'
  []
  [ACInterface2]
    type = ACInterface
    variable = eta2
    kappa_name = kappa
  []
  [lagrange2]
    type = SwitchingFunctionConstraintEta
    variable = eta2
    h_name = h2
    lambda = lambda
    coupled_variables = 'eta1'
  []

  # Lagrange-multiplier constraint kernel for lambda
  [lagrange]
    type = SwitchingFunctionConstraintLagrange
    variable = lambda
    h_names = 'h1   h2'
    etas = 'eta1 eta2'
    epsilon = 1e-6
  []
[]

[AuxKernels]
  [GlobalFreeEnergy]
    type = KKSMultiFreeEnergy
    variable = Fglobal
    Fj_names = 'F_BCC_A2 F_SIGMA'
    hj_names = 'h1 h2'
    gj_names = 'g1 g2'
    interfacial_vars = 'eta1 eta2'
    kappa_names = 'kappa kappa'
    w = 0.1
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  line_search = none

  petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -ksp_gmres_restart'
  petsc_options_value = 'asm      lu          nonzero                    30'

  l_max_its = 100
  nl_max_its = 20
  nl_abs_tol = 1e-10

  end_time = 1000

  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 12
    iteration_window = 2
    growth_factor = 2
    cutback_factor = 0.5
    dt = 0.1
  []
[]

[Postprocessors]
  [F]
    type = ElementIntegralVariablePostprocessor
    variable = Fglobal
  []
  [cmin]
    type = NodalExtremeValue
    value_type = min
    variable = cCr
  []
  [cmax]
    type = NodalExtremeValue
    value_type = max
    variable = cCr
  []
[]

[Outputs]
  exodus = true
  print_linear_residuals = false
  # exclude lagrange multiplier from output, it can diff more easily
  hide = lambda
[]

Input Parameters
Input Files
References