- variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

- eta_iOrder parameter that derivatives are taken with respect to
C++ Type:std::vector

Description:Order parameter that derivatives are taken with respect to

- cj_namesArray of phase concentrations cj. Place in same order as Fj_names!
C++ Type:std::vector

Description:Array of phase concentrations cj. Place in same order as Fj_names!

- hj_namesSwitching Function Materials that provide h. Place in same order as Fj_names!
C++ Type:std::vector

Description:Switching Function Materials that provide h. Place in same order as Fj_names!

- Fj_namesList of free energies for each phase. Place in same order as hj_names!
C++ Type:std::vector

Description:List of free energies for each phase. Place in same order as hj_names!

# KKSMultiACBulkC

Multi-phase KKS model kernel (part 2 of 2) for the Bulk Allen-Cahn. This includes all terms dependent on chemical potential.

### Residual

For the 3-phase KKS model, if the non-linear variable is ,

(1)

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). Since in the KKS model, chemical potentials are constrained to be equal at each position, .

### Jacobian

#### On-diagonal

If the non-linear variable is , the on-diagonal Jacobian is

(2)

#### Off-diagonal: Phase Concentrations

Since appears in the residual, the off-diagonal Jacobian for is a little more complicated than for or . (3) For , (4) and similarly for . then , and are handled first in the code. For the off-diagonal Jacobians we also need to multiply by , the Allen-Cahn mobility.

#### Off-diagonal: Other Coupled Variables

If the non-linear variable is , the off-diagonal Jacobian for is (5) and similar for .

For any other coupled variables, for example temperature (6)

What's implemented in the code for the off-diagonal Jacobian for , and any other coupled variables such as is the generalization for the above two equations for non-linear variable : (7) (This handles everything except for , , , which are handled separately first). For the off-diagonal Jacobians we also need to multiply by , the Allen-Cahn mobility.

## Input Parameters

- mob_nameLThe mobility used with the kernel
Default:L

C++ Type:MaterialPropertyName

Description:The mobility used with the kernel

- argsVector of arguments of the mobility
C++ Type:std::vector

Description:Vector of arguments of the mobility

- blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

- enableTrueSet the enabled status of the MooseObject.
Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

- 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

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.)

- 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

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.

- control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

- seed0The seed for the master random number generator
Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

- 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

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.)

- implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

### Advanced Parameters

- vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime

C++ Type:MultiMooseEnum

Description:The tag for the vectors this Kernel should fill

- extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector

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

Description:The tag for the matrices this Kernel should fill

- extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector

Description:The extra tags for the matrices this Kernel should fill