- Fj_namesList of free energies for each phase. Place in same order as hj_names!
C++ Type:std::vector<MaterialPropertyName>
Controllable:No
Description:List of free energies for each phase. Place in same order as hj_names!
- eta_iOrder parameter that derivatives are taken with respect to
C++ Type:std::vector<VariableName>
Controllable:No
Description:Order parameter that derivatives are taken with respect to
- gi_nameBase name for the double well function g_i(eta_i)
C++ Type:MaterialPropertyName
Controllable:No
Description:Base name for the double well function g_i(eta_i)
- hj_namesSwitching Function Materials that provide h. Place in same order as Fj_names!
C++ Type:std::vector<MaterialPropertyName>
Controllable:No
Description:Switching Function Materials that provide h. Place in same order as Fj_names!
- variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this residual object operates on
- wiDouble well height parameter
C++ Type:double
Controllable:No
Description:Double well height parameter
KKSMultiACBulkF
KKS model kernel (part 1 of 2) for the Bulk Allen-Cahn. This includes all terms NOT dependent on chemical potential.
Residual
For the 3-phase KKS model, if the non-linear variable is ,
where is the phase concentration for phase and is the interpolation function for phase defined in Folch and Plapp (2005) (referred to as there, but we use to maintain consistency with other interpolation functions in MOOSE). Here , also for consistency with notation in MOOSE. is the free energy barrier height.
Jacobian
On-diagonal
If the non-linear variable is , the on-diagonal Jacobian is
Off-diagonal
Off-diagonal Jacobian for (similar for ):
Off-diagonal Jacobian for (similar for ):
These statements can be generalized for non-linear variable as:
For the off-diagonal Jacobians we also need to multiply by , the Allen-Cahn mobility.
Input Parameters
- blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
- coupled_variables
C++ Type:std::vector<VariableName>
Controllable:No
- displacementsThe displacements
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements
- mob_nameLThe mobility used with the kernel
Default:L
C++ Type:MaterialPropertyName
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
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
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>
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>
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>
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
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
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>
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>
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
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
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>
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
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
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/combined/examples/publications/rapid_dev/fig6.i)
- (modules/phase_field/test/tests/slkks/full_solve.i)
- (modules/phase_field/test/tests/KKS_system/lagrange_multiplier.i)
- (modules/phase_field/test/tests/KKS_system/nonlinear.i)
- (modules/phase_field/test/tests/KKS_system/auxkernel.i)
- (modules/phase_field/examples/slkks/CrFe.i)
- (modules/phase_field/test/tests/KKS_system/kks_multiphase.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.[BibTeX]
@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/combined/examples/publications/rapid_dev/fig6.i)
#
# Fig. 6 input for 10.1016/j.commatsci.2017.02.017
# D. Schwen et al./Computational Materials Science 132 (2017) 36-45
# Three phase interface simulation demonstrating the interfacial stability
# w.r.t. formation of a tspurious third phase
#
[Mesh]
type = GeneratedMesh
dim = 2
nx = 120
ny = 120
nz = 0
xmin = 0
xmax = 40
ymin = 0
ymax = 40
zmin = 0
zmax = 0
elem_type = QUAD4
[]
[Variables]
# concentration
[./c]
[../]
# order parameter 1
[./eta1]
[../]
# order parameter 2
[./eta2]
[../]
# order parameter 3
[./eta3]
[../]
# phase concentration 1
[./c1]
initial_condition = 0.4
[../]
# phase concentration 2
[./c2]
initial_condition = 0.5
[../]
# phase concentration 3
[./c3]
initial_condition = 0.8
[../]
# Lagrange multiplier
[./lambda]
initial_condition = 0.0
[../]
[]
[AuxVariables]
[./T]
[./InitialCondition]
type = FunctionIC
function = 'x-10'
[../]
[../]
[]
[Functions]
[./ic_func_eta1]
type = ParsedFunction
expression = '0.5*(1.0+tanh((x-10)/sqrt(2.0))) * 0.5*(1.0+tanh((y-10)/sqrt(2.0)))'
[../]
[./ic_func_eta2]
type = ParsedFunction
expression = '0.5*(1.0-tanh((x-10)/sqrt(2.0)))'
[../]
[./ic_func_eta3]
type = ParsedFunction
expression = '1 - 0.5*(1.0-tanh((x-10)/sqrt(2.0)))
- 0.5*(1.0+tanh((x-10)/sqrt(2.0))) * 0.5*(1.0+tanh((y-10)/sqrt(2.0)))'
[../]
[./ic_func_c]
type = ParsedFunction
expression = '0.5 * 0.5*(1.0-tanh((x-10)/sqrt(2.0)))
+ 0.4 * 0.5*(1.0+tanh((x-10)/sqrt(2.0))) * 0.5*(1.0+tanh((y-10)/sqrt(2.0)))
+ 0.8 * (1 - 0.5*(1.0-tanh((x-10)/sqrt(2.0)))
- 0.5*(1.0+tanh((x-10)/sqrt(2.0))) * 0.5*(1.0+tanh((y-10)/sqrt(2.0))))'
[../]
[]
[ICs]
[./eta1]
variable = eta1
type = FunctionIC
function = ic_func_eta1
[../]
[./eta2]
variable = eta2
type = FunctionIC
function = ic_func_eta2
[../]
[./eta3]
variable = eta3
type = FunctionIC
function = ic_func_eta3
[../]
[./c]
variable = c
type = FunctionIC
function = ic_func_c
[../]
[]
[Materials]
# simple toy free energies
[./f1]
type = DerivativeParsedMaterial
property_name = F1
coupled_variables = 'c1'
expression = '20*(c1-0.4)^2'
[../]
[./f2]
type = DerivativeParsedMaterial
property_name = F2
coupled_variables = 'c2 T'
expression = '20*(c2-0.5)^2 + 0.01*T'
[../]
[./f3]
type = DerivativeParsedMaterial
property_name = F3
coupled_variables = 'c3'
expression = '20*(c3-0.8)^2'
[../]
# Switching functions for each phase
# h1(eta1, eta2, eta3)
[./h1]
type = SwitchingFunction3PhaseMaterial
eta_i = eta1
eta_j = eta2
eta_k = eta3
f_name = h1
[../]
# h2(eta1, eta2, eta3)
[./h2]
type = SwitchingFunction3PhaseMaterial
eta_i = eta2
eta_j = eta3
eta_k = eta1
f_name = h2
[../]
# h3(eta1, eta2, eta3)
[./h3]
type = SwitchingFunction3PhaseMaterial
eta_i = eta3
eta_j = eta1
eta_k = eta2
f_name = h3
[../]
# Coefficients for diffusion equation
[./Dh1]
type = DerivativeParsedMaterial
material_property_names = 'D h1'
expression = D*h1
property_name = Dh1
[../]
[./Dh2]
type = DerivativeParsedMaterial
material_property_names = 'D h2'
expression = D*h2
property_name = Dh2
[../]
[./Dh3]
type = DerivativeParsedMaterial
material_property_names = 'D h3'
expression = D*h3
property_name = Dh3
[../]
# Barrier functions for each phase
[./g1]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta1
function_name = g1
[../]
[./g2]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta2
function_name = g2
[../]
[./g3]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta3
function_name = g3
[../]
# constant properties
[./constants]
type = GenericConstantMaterial
prop_names = 'L kappa D'
prop_values = '1.0 1.0 1'
[../]
[]
[Kernels]
#Kernels for diffusion equation
[./diff_time]
type = TimeDerivative
variable = c
[../]
[./diff_c1]
type = MatDiffusion
variable = c
diffusivity = Dh1
v = c1
[../]
[./diff_c2]
type = MatDiffusion
variable = c
diffusivity = Dh2
v = c2
[../]
[./diff_c3]
type = MatDiffusion
variable = c
diffusivity = Dh3
v = c3
[../]
# Kernels for Allen-Cahn equation for eta1
[./deta1dt]
type = TimeDerivative
variable = eta1
[../]
[./ACBulkF1]
type = KKSMultiACBulkF
variable = eta1
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g1
eta_i = eta1
wi = 1.0
args = 'c1 c2 c3 eta2 eta3'
[../]
[./ACBulkC1]
type = KKSMultiACBulkC
variable = eta1
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta1
args = 'eta2 eta3'
[../]
[./ACInterface1]
type = ACInterface
variable = eta1
kappa_name = kappa
[../]
[./multipler1]
type = MatReaction
variable = eta1
v = lambda
mob_name = L
[../]
# Kernels for Allen-Cahn equation for eta2
[./deta2dt]
type = TimeDerivative
variable = eta2
[../]
[./ACBulkF2]
type = KKSMultiACBulkF
variable = eta2
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g2
eta_i = eta2
wi = 1.0
args = 'c1 c2 c3 eta1 eta3'
[../]
[./ACBulkC2]
type = KKSMultiACBulkC
variable = eta2
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta2
args = 'eta1 eta3'
[../]
[./ACInterface2]
type = ACInterface
variable = eta2
kappa_name = kappa
[../]
[./multipler2]
type = MatReaction
variable = eta2
v = lambda
mob_name = L
[../]
# Kernels for the Lagrange multiplier equation
[./mult_lambda]
type = MatReaction
variable = lambda
mob_name = 3
[../]
[./mult_ACBulkF_1]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g1
eta_i = eta1
wi = 1.0
mob_name = 1
args = 'c1 c2 c3 eta2 eta3'
[../]
[./mult_ACBulkC_1]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta1
args = 'eta2 eta3'
mob_name = 1
[../]
[./mult_CoupledACint_1]
type = SimpleCoupledACInterface
variable = lambda
v = eta1
kappa_name = kappa
mob_name = 1
[../]
[./mult_ACBulkF_2]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g2
eta_i = eta2
wi = 1.0
mob_name = 1
args = 'c1 c2 c3 eta1 eta3'
[../]
[./mult_ACBulkC_2]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta2
args = 'eta1 eta3'
mob_name = 1
[../]
[./mult_CoupledACint_2]
type = SimpleCoupledACInterface
variable = lambda
v = eta2
kappa_name = kappa
mob_name = 1
[../]
[./mult_ACBulkF_3]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g3
eta_i = eta3
wi = 1.0
mob_name = 1
args = 'c1 c2 c3 eta1 eta2'
[../]
[./mult_ACBulkC_3]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta3
args = 'eta1 eta2'
mob_name = 1
[../]
[./mult_CoupledACint_3]
type = SimpleCoupledACInterface
variable = lambda
v = eta3
kappa_name = kappa
mob_name = 1
[../]
# Kernels for constraint equation eta1 + eta2 + eta3 = 1
# eta3 is the nonlinear variable for the constraint equation
[./eta3reaction]
type = MatReaction
variable = eta3
mob_name = 1
[../]
[./eta1reaction]
type = MatReaction
variable = eta3
v = eta1
mob_name = 1
[../]
[./eta2reaction]
type = MatReaction
variable = eta3
v = eta2
mob_name = 1
[../]
[./one]
type = BodyForce
variable = eta3
value = -1.0
[../]
# Phase concentration constraints
[./chempot12]
type = KKSPhaseChemicalPotential
variable = c1
cb = c2
fa_name = F1
fb_name = F2
[../]
[./chempot23]
type = KKSPhaseChemicalPotential
variable = c2
cb = c3
fa_name = F2
fb_name = F3
[../]
[./phaseconcentration]
type = KKSMultiPhaseConcentration
variable = c3
cj = 'c1 c2 c3'
hj_names = 'h1 h2 h3'
etas = 'eta1 eta2 eta3'
c = c
[../]
[]
[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 = 30
nl_max_its = 10
l_tol = 1.0e-4
nl_rel_tol = 1.0e-10
nl_abs_tol = 1.0e-11
num_steps = 1000
[./TimeStepper]
type = IterationAdaptiveDT
dt = 0.2
optimal_iterations = 10
iteration_window = 2
[../]
[]
[Preconditioning]
active = 'full'
[./full]
type = SMP
full = true
[../]
[./mydebug]
type = FDP
full = true
[../]
[]
[Outputs]
exodus = true
checkpoint = true
print_linear_residuals = false
[./csv]
type = CSV
execute_on = 'final'
[../]
[]
#[VectorPostprocessors]
# [./c]
# type = LineValueSampler
# start_point = '-25 0 0'
# end_point = '25 0 0'
# variable = c
# num_points = 151
# sort_by = id
# execute_on = timestep_end
# [../]
# [./eta1]
# type = LineValueSampler
# start_point = '-25 0 0'
# end_point = '25 0 0'
# variable = eta1
# num_points = 151
# sort_by = id
# execute_on = timestep_end
# [../]
# [./eta2]
# type = LineValueSampler
# start_point = '-25 0 0'
# end_point = '25 0 0'
# variable = eta2
# num_points = 151
# sort_by = id
# execute_on = timestep_end
# [../]
# [./eta3]
# type = LineValueSampler
# start_point = '-25 0 0'
# end_point = '25 0 0'
# variable = eta3
# num_points = 151
# sort_by = id
# execute_on = timestep_end
# [../]
#[]
(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
[]
(modules/phase_field/test/tests/KKS_system/lagrange_multiplier.i)
#
# This test ensures that the equilibrium solution using two order parameters with a
# Lagrange multiplier constraint is identical to the dedicated two phase formulation
# in two_phase.i
#
[Mesh]
type = GeneratedMesh
dim = 1
nx = 20
xmax = 5
[]
[AuxVariables]
[Fglobal]
order = CONSTANT
family = MONOMIAL
[]
[]
[Variables]
# concentration
[c]
order = FIRST
family = LAGRANGE
[InitialCondition]
type = FunctionIC
function = x/5
[]
[]
# order parameter 1
[eta1]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# order parameter 2
[eta2]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# phase concentration 1
[c1]
order = FIRST
family = LAGRANGE
initial_condition = 0.2
[]
# phase concentration 2
[c2]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# Lagrange multiplier
[lambda]
order = FIRST
family = LAGRANGE
initial_condition = 0.0
[]
[]
[Materials]
# simple toy free energies
[f1] # = fd
type = DerivativeParsedMaterial
property_name = F1
coupled_variables = 'c1'
expression = '(0.9-c1)^2'
[]
[f2] # = fm
type = DerivativeParsedMaterial
property_name = F2
coupled_variables = 'c2'
expression = '(0.1-c2)^2'
[]
# Switching functions for each phase
[h1_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta1
function_name = h1
[]
[h2_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta2
function_name = h2
[]
# Coefficients for diffusion equation
[Dh1]
type = DerivativeParsedMaterial
material_property_names = 'D h1'
expression = D*h1
property_name = Dh1
[]
[Dh2]
type = DerivativeParsedMaterial
material_property_names = 'D h2'
expression = D*h2
property_name = Dh2
[]
# Barrier functions for each phase
[g1]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta1
function_name = g1
[]
[g2]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta2
function_name = g2
[]
# constant properties
[constants]
type = GenericConstantMaterial
prop_names = 'D L kappa'
prop_values = '0.7 0.7 0.2'
[]
[]
[Kernels]
#Kernels for diffusion equation
[diff_time]
type = TimeDerivative
variable = c
[]
[diff_c1]
type = MatDiffusion
variable = c
diffusivity = Dh1
v = c1
[]
[diff_c2]
type = MatDiffusion
variable = c
diffusivity = Dh2
v = c2
[]
# Kernels for Allen-Cahn equation for eta1
[deta1dt]
type = TimeDerivative
variable = eta1
[]
[ACBulkF1]
type = KKSMultiACBulkF
variable = eta1
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gi_name = g1
eta_i = eta1
wi = 0.2
coupled_variables = 'c1 c2 eta2'
[]
[ACBulkC1]
type = KKSMultiACBulkC
variable = eta1
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
cj_names = 'c1 c2'
eta_i = eta1
coupled_variables = 'eta2'
[]
[ACInterface1]
type = ACInterface
variable = eta1
kappa_name = kappa
[]
[multipler1]
type = MatReaction
variable = eta1
v = lambda
mob_name = L
[]
# Kernels for the Lagrange multiplier equation
[mult_lambda]
type = MatReaction
variable = lambda
mob_name = 2
[]
[mult_ACBulkF_1]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gi_name = g1
eta_i = eta1
wi = 0.2
mob_name = 1
coupled_variables = 'c1 c2 eta2 '
[]
[mult_ACBulkC_1]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
cj_names = 'c1 c2'
eta_i = eta1
coupled_variables = 'eta2 '
mob_name = 1
[]
[mult_CoupledACint_1]
type = SimpleCoupledACInterface
variable = lambda
v = eta1
kappa_name = kappa
mob_name = 1
[]
[mult_ACBulkF_2]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gi_name = g2
eta_i = eta2
wi = 0.2
mob_name = 1
coupled_variables = 'c1 c2 eta1 '
[]
[mult_ACBulkC_2]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
cj_names = 'c1 c2'
eta_i = eta2
coupled_variables = 'eta1 '
mob_name = 1
[]
[mult_CoupledACint_2]
type = SimpleCoupledACInterface
variable = lambda
v = eta2
kappa_name = kappa
mob_name = 1
[]
# Kernels for constraint equation eta1 + eta2 = 1
# eta2 is the nonlinear variable for the constraint equation
[eta2reaction]
type = MatReaction
variable = eta2
mob_name = 1
[]
[eta1reaction]
type = MatReaction
variable = eta2
v = eta1
mob_name = 1
[]
[one]
type = BodyForce
variable = eta2
value = -1.0
[]
# Phase concentration constraints
[chempot12]
type = KKSPhaseChemicalPotential
variable = c1
cb = c2
fa_name = F1
fb_name = F2
[]
[phaseconcentration]
type = KKSMultiPhaseConcentration
variable = c2
cj = 'c1 c2'
hj_names = 'h1 h2'
etas = 'eta1 eta2'
c = c
[]
[]
[AuxKernels]
[Fglobal_total]
type = KKSMultiFreeEnergy
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gj_names = 'g1 g2 '
variable = Fglobal
w = 0.2
interfacial_vars = 'eta1 eta2 '
kappa_names = 'kappa kappa'
[]
[]
[Executioner]
type = Transient
solve_type = NEWTON
petsc_options_iname = '-pc_type -sub_pc_factor_shift_type'
petsc_options_value = 'lu nonzero'
l_max_its = 30
nl_max_its = 10
l_tol = 1.0e-4
nl_rel_tol = 1.0e-10
nl_abs_tol = 1.0e-11
num_steps = 35
dt = 10
[]
[VectorPostprocessors]
[c]
type = LineValueSampler
variable = c
start_point = '0 0 0'
end_point = '5 0 0'
num_points = 21
sort_by = x
[]
[]
[Outputs]
csv = true
execute_on = FINAL
[]
(modules/phase_field/test/tests/KKS_system/nonlinear.i)
#
# This test checks if the thwo phase and lagrange multiplier solutions can be replicated
# with a two order parameter approach, where the second order parameter eta2 is a
# nonlinear variable that is set as eta2 := 1 - eta1 (using Reaction, CoupledForce, and BodyForce)
# The solution is reproduced.
#
[Mesh]
type = GeneratedMesh
dim = 1
nx = 20
xmax = 5
[]
[AuxVariables]
[Fglobal]
order = CONSTANT
family = MONOMIAL
[]
[]
[Variables]
# concentration
[c]
order = FIRST
family = LAGRANGE
[InitialCondition]
type = FunctionIC
function = x/5
[]
[]
# order parameter 1
[eta1]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# order parameter 2
[eta2]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# phase concentration 1
[c1]
order = FIRST
family = LAGRANGE
initial_condition = 0.9
[]
# phase concentration 2
[c2]
order = FIRST
family = LAGRANGE
initial_condition = 0.1
[]
[]
[Materials]
# simple toy free energies
[f1] # = fd
type = DerivativeParsedMaterial
property_name = F1
coupled_variables = 'c1'
expression = '(0.9-c1)^2'
[]
[f2] # = fm
type = DerivativeParsedMaterial
property_name = F2
coupled_variables = 'c2'
expression = '(0.1-c2)^2'
[]
# Switching functions for each phase
[h1_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta1
function_name = h1
[]
[h2_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta2
function_name = h2
[]
# Coefficients for diffusion equation
[Dh1]
type = DerivativeParsedMaterial
material_property_names = 'D h1(eta1)'
expression = D*h1
property_name = Dh1
coupled_variables = eta1
[]
[Dh2]
type = DerivativeParsedMaterial
material_property_names = 'D h2(eta2)'
expression = D*h2
property_name = Dh2
coupled_variables = eta2
[]
# Barrier functions for each phase
[g1]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta1
function_name = g1
[]
[g2]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta2
function_name = g2
[]
# constant properties
[constants]
type = GenericConstantMaterial
prop_names = 'D L kappa'
prop_values = '0.7 0.7 0.2'
[]
[]
[Kernels]
#Kernels for diffusion equation
[diff_time]
type = TimeDerivative
variable = c
[]
[diff_c1]
type = MatDiffusion
variable = c
diffusivity = Dh1
v = c1
args = 'eta1'
[]
[diff_c2]
type = MatDiffusion
variable = c
diffusivity = Dh2
v = c2
args = 'eta2'
[]
# Kernels for Allen-Cahn equation for eta1
[deta1dt]
type = TimeDerivative
variable = eta1
[]
[ACBulkF1]
type = KKSMultiACBulkF
variable = eta1
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gi_name = g1
eta_i = eta1
wi = 0.2
coupled_variables = 'c1 c2 eta2'
[]
[ACBulkC1]
type = KKSMultiACBulkC
variable = eta1
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
cj_names = 'c1 c2'
eta_i = eta1
coupled_variables = 'eta2'
[]
[ACInterface1]
type = ACInterface
variable = eta1
kappa_name = kappa
[]
# Phase concentration constraints
[chempot12]
type = KKSPhaseChemicalPotential
variable = c1
cb = c2
fa_name = F1
fb_name = F2
[]
[phaseconcentration]
type = KKSMultiPhaseConcentration
variable = c2
cj = 'c1 c2'
hj_names = 'h1 h2'
etas = 'eta1 eta2'
c = c
[]
# equation for eta2 = 1 - eta1
# 0 = eta2 + eta1 -1
[constraint_eta1] # eta2
type = Reaction
variable = eta2
[]
[constraint_eta2] # + eta1
type = CoupledForce
variable = eta2
coef = -1
v = eta1
[]
[constraint_one] # - 1
type = BodyForce
variable = eta2
[]
[]
[AuxKernels]
[Fglobal_total]
type = KKSMultiFreeEnergy
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gj_names = 'g1 g2 '
variable = Fglobal
w = 0.2
interfacial_vars = 'eta1 eta2 '
kappa_names = 'kappa kappa'
[]
[]
[Executioner]
type = Transient
solve_type = NEWTON
petsc_options_iname = '-pc_type -sub_pc_factor_shift_type'
petsc_options_value = 'lu nonzero'
l_max_its = 30
nl_max_its = 10
l_tol = 1.0e-4
nl_rel_tol = 1.0e-10
nl_abs_tol = 1.0e-11
end_time = 350
dt = 10
[]
[VectorPostprocessors]
[c]
type = LineValueSampler
variable = c
start_point = '0 0 0'
end_point = '5 0 0'
num_points = 21
sort_by = x
[]
[]
[Outputs]
csv = true
execute_on = FINAL
[]
(modules/phase_field/test/tests/KKS_system/auxkernel.i)
#
# This test checks if the two phase and lagrange multiplier solutions can be replicated
# with a two order parameter approach, where the second order parameter eta2 is an
# auxiliary variable that is set as eta2 := 1 - eta1
# The solution is reproduced, but convergence is suboptimal, as important Jacobian
# terms for eta1 (that should come indirectly from eta2) are missing.
#
[Mesh]
type = GeneratedMesh
dim = 1
nx = 20
xmax = 5
[]
[AuxVariables]
[Fglobal]
order = CONSTANT
family = MONOMIAL
[]
# order parameter 2
[eta2]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
[]
#
# With this approach the derivative w.r.t. eta1 is lost in all terms depending on
# eta2 a potential fix would be to make eta2 a material property with derivatives.
# This would require a major rewrite of the phase field kernels, though.
#
[AuxKernels]
[eta2]
type = ParsedAux
variable = eta2
expression = '1-eta1'
coupled_variables = eta1
[]
[]
[Variables]
# concentration
[c]
order = FIRST
family = LAGRANGE
[InitialCondition]
type = FunctionIC
function = x/5
[]
[]
# order parameter 1
[eta1]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[]
# phase concentration 1
[c1]
order = FIRST
family = LAGRANGE
initial_condition = 0.9
[]
# phase concentration 2
[c2]
order = FIRST
family = LAGRANGE
initial_condition = 0.1
[]
[]
[Materials]
# simple toy free energies
[f1] # = fd
type = DerivativeParsedMaterial
property_name = F1
coupled_variables = 'c1'
expression = '(0.9-c1)^2'
[]
[f2] # = fm
type = DerivativeParsedMaterial
property_name = F2
coupled_variables = 'c2'
expression = '(0.1-c2)^2'
[]
# Switching functions for each phase
[h1_eta]
type = SwitchingFunctionMaterial
h_order = HIGH
eta = eta1
function_name = h1
[]
[h2_eta]
type = DerivativeParsedMaterial
material_property_names = 'h1(eta1)'
expression = '1-h1'
property_name = h2
coupled_variables = eta1
[]
# Coefficients for diffusion equation
[Dh1]
type = DerivativeParsedMaterial
material_property_names = 'D h1(eta1)'
expression = D*h1
property_name = Dh1
coupled_variables = eta1
[]
[Dh2]
type = DerivativeParsedMaterial
material_property_names = 'D h2(eta1)'
expression = 'D*h2'
property_name = Dh2
coupled_variables = eta1
[]
# Barrier functions for each phase
[g1]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta1
function_name = g1
[]
[g2]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta2
function_name = g2
[]
# constant properties
[constants]
type = GenericConstantMaterial
prop_names = 'D L kappa'
prop_values = '0.7 0.7 0.2'
[]
[]
[Kernels]
#Kernels for diffusion equation
[diff_time]
type = TimeDerivative
variable = c
[]
[diff_c1]
type = MatDiffusion
variable = c
diffusivity = Dh1
v = c1
args = eta1
[]
[diff_c2]
type = MatDiffusion
variable = c
diffusivity = Dh2
v = c2
args = eta1
[]
# Kernels for Allen-Cahn equation for eta1
[deta1dt]
type = TimeDerivative
variable = eta1
[]
[ACBulkF1]
type = KKSMultiACBulkF
variable = eta1
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
gi_name = g1
eta_i = eta1
wi = 0.2
coupled_variables = 'c1 c2 eta2'
[]
[ACBulkC1]
type = KKSMultiACBulkC
variable = eta1
Fj_names = 'F1 F2'
hj_names = 'h1 h2'
cj_names = 'c1 c2'
eta_i = eta1
coupled_variables = 'eta2'
[]
[ACInterface1]
type = ACInterface
variable = eta1
kappa_name = kappa
[]
# Phase concentration constraints
[chempot12]
type = KKSPhaseChemicalPotential
variable = c1
cb = c2
fa_name = F1
fb_name = F2
[]
[phaseconcentration]
type = KKSMultiPhaseConcentration
variable = c2
cj = 'c1 c2'
hj_names = 'h1 h2'
etas = 'eta1 eta2'
c = c
[]
[]
[AuxKernels]
[Fglobal_total]
type = KKSMultiFreeEnergy
Fj_names = 'F1 F2 '
hj_names = 'h1 h2 '
gj_names = 'g1 g2 '
variable = Fglobal
w = 0.2
interfacial_vars = 'eta1 eta2 '
kappa_names = 'kappa kappa'
[]
[]
[Executioner]
type = Transient
solve_type = PJFNK
petsc_options_iname = '-pc_type'
petsc_options_value = 'lu '
l_max_its = 30
nl_max_its = 10
l_tol = 1.0e-4
nl_rel_tol = 1.0e-10
nl_abs_tol = 1.0e-11
end_time = 350
dt = 10
[]
[Preconditioning]
[full]
type = SMP
full = true
[]
[]
[VectorPostprocessors]
[c]
type = LineValueSampler
variable = c
start_point = '0 0 0'
end_point = '5 0 0'
num_points = 21
sort_by = x
[]
[]
[Outputs]
csv = true
execute_on = FINAL
[]
(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.
#
[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
coupled_variables = eta1
[]
[diff_c2a]
type = MatDiffusion
variable = cCr
diffusivity = Dh2a
v = SIGMA_0CR
coupled_variables = eta2
[]
[diff_c2b]
type = MatDiffusion
variable = cCr
diffusivity = Dh2b
v = SIGMA_1CR
coupled_variables = eta2
[]
[diff_c2c]
type = MatDiffusion
variable = cCr
diffusivity = Dh2c
v = SIGMA_2CR
coupled_variables = 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/KKS_system/kks_multiphase.i)
#
# This test is for the 3-phase KKS model
#
[Mesh]
type = GeneratedMesh
dim = 2
nx = 20
ny = 20
nz = 0
xmin = 0
xmax = 40
ymin = 0
ymax = 40
zmin = 0
zmax = 0
elem_type = QUAD4
[]
[BCs]
[./Periodic]
[./all]
auto_direction = 'x y'
[../]
[../]
[]
[AuxVariables]
[./Energy]
order = CONSTANT
family = MONOMIAL
[../]
[]
[Variables]
# concentration
[./c]
order = FIRST
family = LAGRANGE
[../]
# order parameter 1
[./eta1]
order = FIRST
family = LAGRANGE
[../]
# order parameter 2
[./eta2]
order = FIRST
family = LAGRANGE
[../]
# order parameter 3
[./eta3]
order = FIRST
family = LAGRANGE
initial_condition = 0.0
[../]
# phase concentration 1
[./c1]
order = FIRST
family = LAGRANGE
initial_condition = 0.2
[../]
# phase concentration 2
[./c2]
order = FIRST
family = LAGRANGE
initial_condition = 0.5
[../]
# phase concentration 3
[./c3]
order = FIRST
family = LAGRANGE
initial_condition = 0.8
[../]
# Lagrange multiplier
[./lambda]
order = FIRST
family = LAGRANGE
initial_condition = 0.0
[../]
[]
[ICs]
[./eta1]
variable = eta1
type = SmoothCircleIC
x1 = 20.0
y1 = 20.0
radius = 10
invalue = 0.9
outvalue = 0.1
int_width = 4
[../]
[./eta2]
variable = eta2
type = SmoothCircleIC
x1 = 20.0
y1 = 20.0
radius = 10
invalue = 0.1
outvalue = 0.9
int_width = 4
[../]
[./c]
variable = c
type = SmoothCircleIC
x1 = 20.0
y1 = 20.0
radius = 10
invalue = 0.2
outvalue = 0.5
int_width = 2
[../]
[]
[Materials]
# simple toy free energies
[./f1]
type = DerivativeParsedMaterial
property_name = F1
coupled_variables = 'c1'
expression = '20*(c1-0.2)^2'
[../]
[./f2]
type = DerivativeParsedMaterial
property_name = F2
coupled_variables = 'c2'
expression = '20*(c2-0.5)^2'
[../]
[./f3]
type = DerivativeParsedMaterial
property_name = F3
coupled_variables = 'c3'
expression = '20*(c3-0.8)^2'
[../]
# Switching functions for each phase
# h1(eta1, eta2, eta3)
[./h1]
type = SwitchingFunction3PhaseMaterial
eta_i = eta1
eta_j = eta2
eta_k = eta3
property_name = h1
[../]
# h2(eta1, eta2, eta3)
[./h2]
type = SwitchingFunction3PhaseMaterial
eta_i = eta2
eta_j = eta3
eta_k = eta1
property_name = h2
[../]
# h3(eta1, eta2, eta3)
[./h3]
type = SwitchingFunction3PhaseMaterial
eta_i = eta3
eta_j = eta1
eta_k = eta2
property_name = h3
[../]
# Coefficients for diffusion equation
[./Dh1]
type = DerivativeParsedMaterial
material_property_names = 'D h1'
expression = D*h1
property_name = Dh1
[../]
[./Dh2]
type = DerivativeParsedMaterial
material_property_names = 'D h2'
expression = D*h2
property_name = Dh2
[../]
[./Dh3]
type = DerivativeParsedMaterial
material_property_names = 'D h3'
expression = D*h3
property_name = Dh3
[../]
# Barrier functions for each phase
[./g1]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta1
function_name = g1
[../]
[./g2]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta2
function_name = g2
[../]
[./g3]
type = BarrierFunctionMaterial
g_order = SIMPLE
eta = eta3
function_name = g3
[../]
# constant properties
[./constants]
type = GenericConstantMaterial
prop_names = 'L kappa D'
prop_values = '0.7 1.0 1'
[../]
[]
[Kernels]
#Kernels for diffusion equation
[./diff_time]
type = TimeDerivative
variable = c
[../]
[./diff_c1]
type = MatDiffusion
variable = c
diffusivity = Dh1
v = c1
[../]
[./diff_c2]
type = MatDiffusion
variable = c
diffusivity = Dh2
v = c2
[../]
[./diff_c3]
type = MatDiffusion
variable = c
diffusivity = Dh3
v = c3
[../]
# Kernels for Allen-Cahn equation for eta1
[./deta1dt]
type = TimeDerivative
variable = eta1
[../]
[./ACBulkF1]
type = KKSMultiACBulkF
variable = eta1
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g1
eta_i = eta1
wi = 1.0
coupled_variables = 'c1 c2 c3 eta2 eta3'
[../]
[./ACBulkC1]
type = KKSMultiACBulkC
variable = eta1
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta1
coupled_variables = 'eta2 eta3'
[../]
[./ACInterface1]
type = ACInterface
variable = eta1
kappa_name = kappa
[../]
[./multipler1]
type = MatReaction
variable = eta1
v = lambda
mob_name = L
[../]
# Kernels for Allen-Cahn equation for eta2
[./deta2dt]
type = TimeDerivative
variable = eta2
[../]
[./ACBulkF2]
type = KKSMultiACBulkF
variable = eta2
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g2
eta_i = eta2
wi = 1.0
coupled_variables = 'c1 c2 c3 eta1 eta3'
[../]
[./ACBulkC2]
type = KKSMultiACBulkC
variable = eta2
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta2
coupled_variables = 'eta1 eta3'
[../]
[./ACInterface2]
type = ACInterface
variable = eta2
kappa_name = kappa
[../]
[./multipler2]
type = MatReaction
variable = eta2
v = lambda
mob_name = L
[../]
# Kernels for the Lagrange multiplier equation
[./mult_lambda]
type = MatReaction
variable = lambda
mob_name = 3
[../]
[./mult_ACBulkF_1]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g1
eta_i = eta1
wi = 1.0
mob_name = 1
coupled_variables = 'c1 c2 c3 eta2 eta3'
[../]
[./mult_ACBulkC_1]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta1
coupled_variables = 'eta2 eta3'
mob_name = 1
[../]
[./mult_CoupledACint_1]
type = SimpleCoupledACInterface
variable = lambda
v = eta1
kappa_name = kappa
mob_name = 1
[../]
[./mult_ACBulkF_2]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g2
eta_i = eta2
wi = 1.0
mob_name = 1
coupled_variables = 'c1 c2 c3 eta1 eta3'
[../]
[./mult_ACBulkC_2]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta2
coupled_variables = 'eta1 eta3'
mob_name = 1
[../]
[./mult_CoupledACint_2]
type = SimpleCoupledACInterface
variable = lambda
v = eta2
kappa_name = kappa
mob_name = 1
[../]
[./mult_ACBulkF_3]
type = KKSMultiACBulkF
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gi_name = g3
eta_i = eta3
wi = 1.0
mob_name = 1
coupled_variables = 'c1 c2 c3 eta1 eta2'
[../]
[./mult_ACBulkC_3]
type = KKSMultiACBulkC
variable = lambda
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
cj_names = 'c1 c2 c3'
eta_i = eta3
coupled_variables = 'eta1 eta2'
mob_name = 1
[../]
[./mult_CoupledACint_3]
type = SimpleCoupledACInterface
variable = lambda
v = eta3
kappa_name = kappa
mob_name = 1
[../]
# Kernels for constraint equation eta1 + eta2 + eta3 = 1
# eta3 is the nonlinear variable for the constraint equation
[./eta3reaction]
type = MatReaction
variable = eta3
mob_name = 1
[../]
[./eta1reaction]
type = MatReaction
variable = eta3
v = eta1
mob_name = 1
[../]
[./eta2reaction]
type = MatReaction
variable = eta3
v = eta2
mob_name = 1
[../]
[./one]
type = BodyForce
variable = eta3
value = -1.0
[../]
# Phase concentration constraints
[./chempot12]
type = KKSPhaseChemicalPotential
variable = c1
cb = c2
fa_name = F1
fb_name = F2
[../]
[./chempot23]
type = KKSPhaseChemicalPotential
variable = c2
cb = c3
fa_name = F2
fb_name = F3
[../]
[./phaseconcentration]
type = KKSMultiPhaseConcentration
variable = c3
cj = 'c1 c2 c3'
hj_names = 'h1 h2 h3'
etas = 'eta1 eta2 eta3'
c = c
[../]
[]
[AuxKernels]
[./Energy_total]
type = KKSMultiFreeEnergy
Fj_names = 'F1 F2 F3'
hj_names = 'h1 h2 h3'
gj_names = 'g1 g2 g3'
variable = Energy
w = 1
interfacial_vars = 'eta1 eta2 eta3'
kappa_names = 'kappa kappa kappa'
[../]
[]
[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 = 30
nl_max_its = 10
l_tol = 1.0e-4
nl_rel_tol = 1.0e-10
nl_abs_tol = 1.0e-11
num_steps = 2
dt = 0.5
[]
[Preconditioning]
active = 'full'
[./full]
type = SMP
full = true
[../]
[./mydebug]
type = FDP
full = true
[../]
[]
[Outputs]
exodus = true
[]