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 η1\eta_1,

R=(∂h1∂η1F1+∂h2∂η1F2+∂h3∂η1F3+W1∂g1∂η1)R = \left(\frac{\partial h_1}{\partial \eta_1} F_1 + \frac{\partial h_2}{\partial \eta_1} F_2 + \frac{\partial h_3}{\partial \eta_1} F_3 + W_1 \frac{\partial g_1}{\partial \eta_1} \right)

where cic_i is the phase concentration for phase ii and hih_i is the interpolation function for phase ii defined in Folch and Plapp (2005) (referred to as gig_i there, but we use hih_i to maintain consistency with other interpolation functions in MOOSE). Here gi=ηi2(1−ηi)2g_i = \eta_i^2 (1-\eta_i)^2, also for consistency with notation in MOOSE. W1W_1 is the free energy barrier height.

Jacobian

On-diagonal

If the non-linear variable is η1\eta_1, the on-diagonal Jacobian is

J=ϕj∂R∂η1=ϕj(∂2h1∂η12F1+∂2h2∂η12F2+∂2h3∂η12F3+W1∂2g∂η12)\begin{aligned} J &=& \phi_j \frac{\partial R}{\partial \eta_1} \\ &=& \phi_j \left( \frac{\partial ^2 h_1}{\partial \eta_1^2} F_1 + \frac{\partial ^2 h_2}{\partial \eta_1^2} F_2 + \frac{\partial ^2 h_3}{\partial \eta_1^2} F_3 + W_1 \frac{\partial ^2 g}{\partial \eta_1^2} \right) \end{aligned}

Off-diagonal

Off-diagonal Jacobian for η2\eta_2 (similar for η3\eta_3):

J=ϕj∂R∂η2=ϕj(∂2h1∂η1∂η2F1+∂2h2∂η1∂η2F2+∂2h3∂η1∂η2F3)\begin{aligned} J &=& \phi_j \frac{\partial R}{\partial \eta_2} \\ &=& \phi_j \left( \frac{\partial ^2 h_1}{\partial \eta_1 \partial \eta_2} F_1 + \frac{\partial ^2 h_2}{\partial \eta_1 \partial \eta_2} F_2 + \frac{\partial ^2 h_3}{\partial \eta_1 \partial \eta_2} F_3 \right) \end{aligned}

Off-diagonal Jacobian for c1c_1 (similar for c2,c3c_2, c_3):

J=ϕj∂R∂c1=ϕj∂h1∂η1∂F1∂c1\begin{aligned} J &=& \phi_j \frac{\partial R}{\partial c_1} \\ &=& \phi_j \frac{\partial h_1}{\partial \eta_1} \frac{\partial F_1}{\partial c_1} \end{aligned}

These statements can be generalized for non-linear variable vv as:

J=ϕj∂R∂v=(∂2h1∂η1∂vF1+∂2h2∂η1∂vF2+∂2h3∂η1∂vF3+∂h1∂η1∂F1∂v+∂h2∂η1∂F2∂v+∂h3∂η1∂F3∂v)\begin{aligned} J &=& \phi_j \frac{\partial R}{\partial v} \\ &=& \left( \frac{\partial ^2 h_1}{\partial \eta_1 \partial v} F_1 + \frac{\partial ^2 h_2}{\partial \eta_1 \partial v} F_2 + \frac{\partial ^2 h_3}{\partial \eta_1 \partial v} F_3 + \frac{\partial h_1}{\partial \eta_1} \frac{\partial F_1}{\partial v} + \frac{\partial h_2}{\partial \eta_1} \frac{\partial F_2}{\partial v} + \frac{\partial h_3}{\partial \eta_1} \frac{\partial F_3}{\partial v}\right) \end{aligned}

For the off-diagonal Jacobians we also need to multiply by LL, the Allen-Cahn mobility.

Input Parameters

  • Fj_namesList of free energies for each phase. Place in same order as hj_names!

    C++ Type:std::vector<MaterialPropertyName>

    Unit:(no unit assumed)

    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>

    Unit:(no unit assumed)

    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

    Unit:(no unit assumed)

    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>

    Unit:(no unit assumed)

    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

    Unit:(no unit assumed)

    Controllable:No

    Description:The name of the variable that this residual object operates on

  • wiDouble well height parameter

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Double well height parameter

Required 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_variablesVector of nonlinear variable arguments this object depends on

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Vector of nonlinear variable arguments this object depends on

  • displacementsThe displacements

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The displacements

  • matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether this object is only doing assembly to matrices (no vectors)

  • mob_nameLThe mobility used with the kernel

    Default:L

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:The mobility used with the kernel

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

Contribution To Tagged Field Data 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>

    Unit:(no unit assumed)

    Controllable:No

    Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    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>

    Unit:(no unit assumed)

    Controllable:No

    Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    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

  • prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

  • use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

    Default:False

    C++ Type:bool

    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.

Material Property Retrieval Parameters

Input Files

References

  1. 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]