Kim-Kim-Suzuki Example for three or more components

#
# KKS ternary (3 chemical component) system example in the split form
# We track c1 and c2 only, since c1 + c2 + c3 = 1
#

[Mesh<<<{"href": "../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 2
  nx = 150
  ny = 15
  nz = 0
  xmin = -25
  xmax = 25
  ymin = -2.5
  ymax = 2.5
  zmin = 0
  zmax = 0
  elem_type = QUAD4
[]

[AuxVariables<<<{"href": "../../../syntax/AuxVariables/index.html"}>>>]
  [./Fglobal]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
  [../]
[]

[Variables<<<{"href": "../../../syntax/Variables/index.html"}>>>]
  # order parameter
  [./eta]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  [../]

  # solute 1 concentration
  [./c1]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  [../]

  # solute 2 concentration
  [./c2]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  [../]

  # chemical potential solute 1
  [./w1]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  [../]

  # chemical potential solute 2
  [./w2]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  [../]

  # Liquid phase solute 1 concentration
  [./c1l]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 0.1
  [../]

  # Liquid phase solute 2 concentration
  [./c2l]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 0.05
  [../]

  # Solid phase solute 1 concentration
  [./c1s]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 0.8
  [../]

  # Solid phase solute 2 concentration
  [./c2s]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 0.1
  [../]

[]

[Functions<<<{"href": "../../../syntax/Functions/index.html"}>>>]
  [./ic_func_eta]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = '0.5*(1.0-tanh((x)/sqrt(2.0)))'
  [../]
  [./ic_func_c1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = '0.8*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.1*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
  [../]
  [./ic_func_c2]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = '0.1*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.05*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
  [../]

[]

[ICs<<<{"href": "../../../syntax/ICs/index.html"}>>>]
  [./eta]
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = eta
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../source/ics/FunctionIC.html"}>>>
    function<<<{"description": "The initial condition function."}>>> = ic_func_eta
  [../]
  [./c1]
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = c1
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../source/ics/FunctionIC.html"}>>>
    function<<<{"description": "The initial condition function."}>>> = ic_func_c1
  [../]
  [./c2]
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = c2
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../source/ics/FunctionIC.html"}>>>
    function<<<{"description": "The initial condition function."}>>> = ic_func_c2
  [../]

[]

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  # Free energy of the liquid
  [./fl]
    type = DerivativeParsedMaterial<<<{"description": "Parsed Function Material with automatic derivatives.", "href": "../../../source/materials/DerivativeParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = fl
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'c1l c2l'
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '(0.1-c1l)^2+(0.05-c2l)^2'
  [../]

  # Free energy of the solid
  [./fs]
    type = DerivativeParsedMaterial<<<{"description": "Parsed Function Material with automatic derivatives.", "href": "../../../source/materials/DerivativeParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = fs
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'c1s c2s'
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '(0.8-c1s)^2+(0.1-c2s)^2'
  [../]

  # h(eta)
  [./h_eta]
    type = SwitchingFunctionMaterial<<<{"description": "Helper material to provide $h(\\eta)$ and its derivative in one of two polynomial forms.\nSIMPLE: $3\\eta^2-2\\eta^3$\nHIGH: $\\eta^3(6\\eta^2-15\\eta+10)$", "href": "../../../source/materials/SwitchingFunctionMaterial.html"}>>>
    h_order<<<{"description": "Polynomial order of the switching function h(eta)"}>>> = HIGH
    eta<<<{"description": "Order parameter variable"}>>> = eta
  [../]

  # g(eta)
  [./g_eta]
    type = BarrierFunctionMaterial<<<{"description": "Helper material to provide $g(\\eta)$ and its derivative in a polynomial.\nSIMPLE: $\\eta^2(1-\\eta)^2$\nLOW: $\\eta(1-\\eta)$\nHIGH: $\\eta^2(1-\\eta^2)^2$", "href": "../../../source/materials/BarrierFunctionMaterial.html"}>>>
    g_order<<<{"description": "Polynomial order of the barrier function g(eta)"}>>> = SIMPLE
    eta<<<{"description": "Order parameter variable"}>>> = eta
  [../]

  # constant properties
  [./constants]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>>  = 'M   L   eps_sq'
    prop_values<<<{"description": "The values associated with the named properties"}>>> = '0.7 0.7 1.0  '
  [../]
[]

[Kernels<<<{"href": "../../../syntax/Kernels/index.html"}>>>]
  # enforce c1 = (1-h(eta))*c1l + h(eta)*c1s
  [./PhaseConc1]
    type = KKSPhaseConcentration<<<{"description": "KKS model kernel to enforce the decomposition of concentration into phase concentration  $(1-h(\\eta))c_a + h(\\eta)c_b - c = 0$. The non-linear variable of this kernel is $c_b$.", "href": "../../../source/kernels/KKSPhaseConcentration.html"}>>>
    ca<<<{"description": "Phase a concentration"}>>>       = c1l
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c1s
    c<<<{"description": "Real concentration"}>>>        = c1
    eta<<<{"description": "Phase a/b order parameter"}>>>      = eta
  [../]

  # enforce c2 = (1-h(eta))*c2l + h(eta)*c2s
  [./PhaseConc2]
    type = KKSPhaseConcentration<<<{"description": "KKS model kernel to enforce the decomposition of concentration into phase concentration  $(1-h(\\eta))c_a + h(\\eta)c_b - c = 0$. The non-linear variable of this kernel is $c_b$.", "href": "../../../source/kernels/KKSPhaseConcentration.html"}>>>
    ca<<<{"description": "Phase a concentration"}>>>       = c2l
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c2s
    c<<<{"description": "Real concentration"}>>>        = c2
    eta<<<{"description": "Phase a/b order parameter"}>>>      = eta
  [../]

  # enforce pointwise equality of chemical potentials
  [./ChemPotSolute1]
    type = KKSPhaseChemicalPotential<<<{"description": "KKS model kernel to enforce the pointwise equality of phase chemical potentials $dF_a/dc_a = dF_b/dc_b$. The non-linear variable of this kernel is $c_a$.", "href": "../../../source/kernels/KKSPhaseChemicalPotential.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c1l
    cb<<<{"description": "Phase b concentration"}>>>       = c1s
    fa_name<<<{"description": "Base name of the free energy function Fa (f_name in the corresponding derivative function material)"}>>>  = fl
    fb_name<<<{"description": "Base name of the free energy function Fb (f_name in the corresponding derivative function material)"}>>>  = fs
    args_a<<<{"description": "Vector of further parameters to Fa (optional, to add in second cross derivatives of Fa)"}>>>   = 'c2l'
    args_b<<<{"description": "Vector of further parameters to Fb (optional, to add in second cross derivatives of Fb)"}>>>   = 'c2s'
  [../]
  [./ChemPotSolute2]
    type = KKSPhaseChemicalPotential<<<{"description": "KKS model kernel to enforce the pointwise equality of phase chemical potentials $dF_a/dc_a = dF_b/dc_b$. The non-linear variable of this kernel is $c_a$.", "href": "../../../source/kernels/KKSPhaseChemicalPotential.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c2l
    cb<<<{"description": "Phase b concentration"}>>>       = c2s
    fa_name<<<{"description": "Base name of the free energy function Fa (f_name in the corresponding derivative function material)"}>>>  = fl
    fb_name<<<{"description": "Base name of the free energy function Fb (f_name in the corresponding derivative function material)"}>>>  = fs
    args_a<<<{"description": "Vector of further parameters to Fa (optional, to add in second cross derivatives of Fa)"}>>>   = 'c1l'
    args_b<<<{"description": "Vector of further parameters to Fb (optional, to add in second cross derivatives of Fb)"}>>>   = 'c1s'

  [../]

  #
  # Cahn-Hilliard Equations
  #
  [./CHBulk1]
    type = KKSSplitCHCRes<<<{"description": "KKS model kernel for the split Bulk Cahn-Hilliard term. This kernel operates on the physical concentration 'c' as the non-linear variable", "href": "../../../source/kernels/KKSSplitCHCRes.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c1
    ca<<<{"description": "phase concentration corresponding to the non-linear variable of this kernel"}>>>       = c1l
    fa_name<<<{"description": "Base name of an arbitrary phase free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fl
    w<<<{"description": "Chemical potential non-linear helper variable for the split solve"}>>>        = w1
    args_a<<<{"description": "Vector of additional arguments to Fa"}>>>   = 'c2l'
  [../]
  [./CHBulk2]
    type = KKSSplitCHCRes<<<{"description": "KKS model kernel for the split Bulk Cahn-Hilliard term. This kernel operates on the physical concentration 'c' as the non-linear variable", "href": "../../../source/kernels/KKSSplitCHCRes.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = c2
    ca<<<{"description": "phase concentration corresponding to the non-linear variable of this kernel"}>>>       = c2l
    fa_name<<<{"description": "Base name of an arbitrary phase free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fl
    w<<<{"description": "Chemical potential non-linear helper variable for the split solve"}>>>        = w2
    args_a<<<{"description": "Vector of additional arguments to Fa"}>>>   = 'c1l'
  [../]

  [./dc1dt]
    type = CoupledTimeDerivative<<<{"description": "Time derivative Kernel that acts on a coupled variable. Weak form: $(\\psi_i, \\frac{\\partial v_h}{\\partial t})$.", "href": "../../../source/kernels/CoupledTimeDerivative.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = w1
    v<<<{"description": "Coupled variable"}>>> = c1
  [../]
  [./dc2dt]
    type = CoupledTimeDerivative<<<{"description": "Time derivative Kernel that acts on a coupled variable. Weak form: $(\\psi_i, \\frac{\\partial v_h}{\\partial t})$.", "href": "../../../source/kernels/CoupledTimeDerivative.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = w2
    v<<<{"description": "Coupled variable"}>>> = c2
  [../]

  [./w1kernel]
    type = SplitCHWRes<<<{"description": "Split formulation Cahn-Hilliard Kernel for the chemical potential variable with a scalar (isotropic) mobility", "href": "../../../source/kernels/SplitCHWRes.html"}>>>
    mob_name<<<{"description": "The mobility used with the kernel"}>>> = M
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = w1
  [../]
  [./w2kernel]
    type = SplitCHWRes<<<{"description": "Split formulation Cahn-Hilliard Kernel for the chemical potential variable with a scalar (isotropic) mobility", "href": "../../../source/kernels/SplitCHWRes.html"}>>>
    mob_name<<<{"description": "The mobility used with the kernel"}>>> = M
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = w2
  [../]

  #
  # Allen-Cahn Equation
  #
  [./ACBulkF]
    type = KKSACBulkF<<<{"description": "KKS model kernel (part 1 of 2) for the Bulk Allen-Cahn. This includes all terms NOT dependent on chemical potential.", "href": "../../../source/kernels/KKSACBulkF.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = eta
    fa_name<<<{"description": "Base name of the free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fl
    fb_name<<<{"description": "Base name of the free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fs
    w<<<{"description": "Double well height parameter"}>>>        = 1.0
    coupled_variables<<<{"description": "Vector of nonlinear variable arguments this object depends on"}>>> = 'c1l c1s c2l c2s'
  [../]
  [./ACBulkC1]
    type = KKSACBulkC<<<{"description": "KKS model kernel (part 2 of 2) for the Bulk Allen-Cahn. This includes all terms dependent on chemical potential.", "href": "../../../source/kernels/KKSACBulkC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = eta
    ca<<<{"description": "a-phase concentration"}>>>       = c1l
    cb<<<{"description": "b-phase concentration"}>>>       = c1s
    fa_name<<<{"description": "Base name of the free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fl
    coupled_variables<<<{"description": "Vector of nonlinear variable arguments this object depends on"}>>>     = 'c2l'
  [../]
  [./ACBulkC2]
    type = KKSACBulkC<<<{"description": "KKS model kernel (part 2 of 2) for the Bulk Allen-Cahn. This includes all terms dependent on chemical potential.", "href": "../../../source/kernels/KKSACBulkC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = eta
    ca<<<{"description": "a-phase concentration"}>>>       = c2l
    cb<<<{"description": "b-phase concentration"}>>>       = c2s
    fa_name<<<{"description": "Base name of the free energy function F (f_base in the corresponding KKSBaseMaterial)"}>>>  = fl
    coupled_variables<<<{"description": "Vector of nonlinear variable arguments this object depends on"}>>>     = 'c1l'
  [../]
  [./ACInterface]
    type = ACInterface<<<{"description": "Gradient energy Allen-Cahn Kernel", "href": "../../../source/kernels/ACInterface.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = eta
    kappa_name<<<{"description": "The kappa used with the kernel"}>>> = eps_sq
  [../]
  [./detadt]
    type = TimeDerivative<<<{"description": "The time derivative operator with the weak form of $(\\psi_i, \\frac{\\partial u_h}{\\partial t})$.", "href": "../../../source/kernels/TimeDerivative.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = eta
  [../]
[]

[AuxKernels<<<{"href": "../../../syntax/AuxKernels/index.html"}>>>]
  [./GlobalFreeEnergy]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Fglobal
    type = KKSGlobalFreeEnergy<<<{"description": "Total free energy in KKS system, including chemical, barrier and gradient terms", "href": "../../../source/auxkernels/KKSGlobalFreeEnergy.html"}>>>
    fa_name<<<{"description": "Base name of the free energy function F (f_name in the corresponding derivative function material)"}>>> = fl
    fb_name<<<{"description": "Base name of the free energy function F (f_name in the corresponding derivative function material)"}>>> = fs
    w<<<{"description": "Double well height parameter"}>>> = 1.0
  [../]
[]

[Executioner<<<{"href": "../../../syntax/Executioner/index.html"}>>>]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      ilu          nonzero'

  l_max_its = 100
  nl_max_its = 100

  num_steps = 50
  dt = 0.1
[]

#
# Precondition using handcoded off-diagonal terms
#
[Preconditioning<<<{"href": "../../../syntax/Preconditioning/index.html"}>>>]
  [./full]
    type = SMP<<<{"description": "Single matrix preconditioner (SMP) builds a preconditioner using user defined off-diagonal parts of the Jacobian.", "href": "../../../source/preconditioners/SingleMatrixPreconditioner.html"}>>>
    full<<<{"description": "Set to true if you want the full set of couplings between variables simply for convenience so you don't have to set every off_diag_row and off_diag_column combination."}>>> = true
  [../]
[]

[Outputs<<<{"href": "../../../syntax/Outputs/index.html"}>>>]
  exodus<<<{"description": "Output the results using the default settings for Exodus output."}>>> = true
[]

When additional chemical components are added to the KKS model, a Cahn-Hilliard equation must be added for each additional component. (For $var element = document.getElementById("moose-equation-2d8e7927-d8aa-4725-b81e-af10e1327b04");katex.render("n", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ components, $var element = document.getElementById("moose-equation-f22364d8-c6c6-4cfd-90fd-e7ac0080be1c");katex.render("n-1", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ Cahn-Hilliard equations are required). Each additional Cahn-Hilliard equation requires the kernels:

To enforce the composition and chemical potential constraints, each additional component also requires the kernels

The Allen-Cahn equation is also modified when additional components are added. The residual becomes

$var element = document.getElementById("moose-equation-d00c4f8d-e05d-442e-b661-c31441db3a1c");katex.render("R=-\\frac{dh}{d\\eta} \\left(F_a-F_b- \\sum_{i=1}^{n-1} \\frac{dF_a}{dc_{ia}}(c_{ia}-c_{ib})\\right) + w\\frac{dg}{d\\eta}.", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

where $var element = document.getElementById("moose-equation-789a1ace-fdf4-4f9e-b43b-51b544cc9f77");katex.render("n", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the number of components. A single KKSACBulkF kernel is needed as in the 2-component case, and an additional KKSACBulkC kernel must added for each additional component.

(moose/modules/phase_field/examples/kim-kim-suzuki/kks_example_ternary.i)

#
# KKS ternary (3 chemical component) system example in the split form
# We track c1 and c2 only, since c1 + c2 + c3 = 1
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 150
  ny = 15
  nz = 0
  xmin = -25
  xmax = 25
  ymin = -2.5
  ymax = 2.5
  zmin = 0
  zmax = 0
  elem_type = QUAD4
[]

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

[Variables]
  # order parameter
  [./eta]
    order = FIRST
    family = LAGRANGE
  [../]

  # solute 1 concentration
  [./c1]
    order = FIRST
    family = LAGRANGE
  [../]

  # solute 2 concentration
  [./c2]
    order = FIRST
    family = LAGRANGE
  [../]


  # chemical potential solute 1
  [./w1]
    order = FIRST
    family = LAGRANGE
  [../]

  # chemical potential solute 2
  [./w2]
    order = FIRST
    family = LAGRANGE
  [../]


  # Liquid phase solute 1 concentration
  [./c1l]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]

  # Liquid phase solute 2 concentration
  [./c2l]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.05
  [../]

  # Solid phase solute 1 concentration
  [./c1s]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.8
  [../]

  # Solid phase solute 2 concentration
  [./c2s]
    order = FIRST
    family = LAGRANGE
    initial_condition = 0.1
  [../]

[]

[Functions]
  [./ic_func_eta]
    type = ParsedFunction
    expression = '0.5*(1.0-tanh((x)/sqrt(2.0)))'
  [../]
  [./ic_func_c1]
    type = ParsedFunction
    expression = '0.8*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.1*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
  [../]
  [./ic_func_c2]
    type = ParsedFunction
    expression = '0.1*(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10)+0.05*(1-(0.5*(1.0-tanh(x/sqrt(2.0))))^3*(6*(0.5*(1.0-tanh(x/sqrt(2.0))))^2-15*(0.5*(1.0-tanh(x/sqrt(2.0))))+10))'
  [../]

[]


[ICs]
  [./eta]
    variable = eta
    type = FunctionIC
    function = ic_func_eta
  [../]
  [./c1]
    variable = c1
    type = FunctionIC
    function = ic_func_c1
  [../]
  [./c2]
    variable = c2
    type = FunctionIC
    function = ic_func_c2
  [../]

[]

[Materials]
  # Free energy of the liquid
  [./fl]
    type = DerivativeParsedMaterial
    property_name = fl
    coupled_variables = 'c1l c2l'
    expression = '(0.1-c1l)^2+(0.05-c2l)^2'
  [../]

  # Free energy of the solid
  [./fs]
    type = DerivativeParsedMaterial
    property_name = fs
    coupled_variables = 'c1s c2s'
    expression = '(0.8-c1s)^2+(0.1-c2s)^2'
  [../]

  # h(eta)
  [./h_eta]
    type = SwitchingFunctionMaterial
    h_order = HIGH
    eta = eta
  [../]

  # g(eta)
  [./g_eta]
    type = BarrierFunctionMaterial
    g_order = SIMPLE
    eta = eta
  [../]

  # constant properties
  [./constants]
    type = GenericConstantMaterial
    prop_names  = 'M   L   eps_sq'
    prop_values = '0.7 0.7 1.0  '
  [../]
[]

[Kernels]
  # enforce c1 = (1-h(eta))*c1l + h(eta)*c1s
  [./PhaseConc1]
    type = KKSPhaseConcentration
    ca       = c1l
    variable = c1s
    c        = c1
    eta      = eta
  [../]

  # enforce c2 = (1-h(eta))*c2l + h(eta)*c2s
  [./PhaseConc2]
    type = KKSPhaseConcentration
    ca       = c2l
    variable = c2s
    c        = c2
    eta      = eta
  [../]

  # enforce pointwise equality of chemical potentials
  [./ChemPotSolute1]
    type = KKSPhaseChemicalPotential
    variable = c1l
    cb       = c1s
    fa_name  = fl
    fb_name  = fs
    args_a   = 'c2l'
    args_b   = 'c2s'
  [../]
  [./ChemPotSolute2]
    type = KKSPhaseChemicalPotential
    variable = c2l
    cb       = c2s
    fa_name  = fl
    fb_name  = fs
    args_a   = 'c1l'
    args_b   = 'c1s'

  [../]

  #
  # Cahn-Hilliard Equations
  #
  [./CHBulk1]
    type = KKSSplitCHCRes
    variable = c1
    ca       = c1l
    fa_name  = fl
    w        = w1
    args_a   = 'c2l'
  [../]
  [./CHBulk2]
    type = KKSSplitCHCRes
    variable = c2
    ca       = c2l
    fa_name  = fl
    w        = w2
    args_a   = 'c1l'
  [../]

  [./dc1dt]
    type = CoupledTimeDerivative
    variable = w1
    v = c1
  [../]
  [./dc2dt]
    type = CoupledTimeDerivative
    variable = w2
    v = c2
  [../]

  [./w1kernel]
    type = SplitCHWRes
    mob_name = M
    variable = w1
  [../]
  [./w2kernel]
    type = SplitCHWRes
    mob_name = M
    variable = w2
  [../]


  #
  # Allen-Cahn Equation
  #
  [./ACBulkF]
    type = KKSACBulkF
    variable = eta
    fa_name  = fl
    fb_name  = fs
    w        = 1.0
    coupled_variables = 'c1l c1s c2l c2s'
  [../]
  [./ACBulkC1]
    type = KKSACBulkC
    variable = eta
    ca       = c1l
    cb       = c1s
    fa_name  = fl
    coupled_variables     = 'c2l'
  [../]
  [./ACBulkC2]
    type = KKSACBulkC
    variable = eta
    ca       = c2l
    cb       = c2s
    fa_name  = fl
    coupled_variables     = 'c1l'
  [../]
  [./ACInterface]
    type = ACInterface
    variable = eta
    kappa_name = eps_sq
  [../]
  [./detadt]
    type = TimeDerivative
    variable = eta
  [../]
[]

[AuxKernels]
  [./GlobalFreeEnergy]
    variable = Fglobal
    type = KKSGlobalFreeEnergy
    fa_name = fl
    fb_name = fs
    w = 1.0
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      ilu          nonzero'

  l_max_its = 100
  nl_max_its = 100

  num_steps = 50
  dt = 0.1
[]

#
# Precondition using handcoded off-diagonal terms
#
[Preconditioning]
  [./full]
    type = SMP
    full = true
  [../]
[]

[Outputs]
  exodus = true
[]