# KKSMultiPhaseConcentration

KKS multi-phase model kernel to enforce . The non-linear variable of this kernel is , the final phase concentration in the list.

### Residual

For a KKS model with phases, the residual of the phase concentration constraint equation is R = \left( h_1 c_1 + h_2 c_2 + h_3 c_3 + \dots + h_n c_n - c \right) where is the phase concentration for phase , is the physical solute concentration, 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).

### Jacobian

#### On-diagonal

Since the non-linear variable for this kernel is , the on-diagonal Jacobian is \begin{aligned} J && \phi_j \frac{\partial R}{\partial c_n} \\ && \phi_j h_n \end{aligned}

#### Off-diagonal

For the physical concentration , the off-diagonal Jacobian is \begin{aligned} J && \phi_j \frac{\partial R}{\partial c} \\ && - \phi_j \end{aligned} For phase concentrations other than ,: \begin{aligned} J && \phi_j \frac{\partial R}{\partial c_i} \\ && \phi_j h_i \end{aligned}

Finally for the order parameters, such as , \begin{aligned} J && \phi_j \frac{\partial R}{\partial \eta_1} \\ && \phi_j \left( \frac{\partial h_1}{\partial \eta_1} c_1 + \frac{\partial h_2}{\partial \eta_1} c_2 + \frac{\partial h_3}{\partial \eta_1} c_3 \right) \end{aligned} For the off-diagonal Jacobians we also need to multiply by , the Allen-Cahn mobility.

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

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

• cjArray of phase concentrations cj. Place in same order as hj_names!

C++ Type:std::vector

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

• cPhysical concentration

C++ Type:std::vector

Description:Physical concentration

• hj_namesSwitching Function Materials that provide $h(\eta_1, \eta_2,\dots)$

C++ Type:std::vector

Description:Switching Function Materials that provide $h(\eta_1, \eta_2,\dots)$

### Required Parameters

• 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

• etasOrder parameters for all phases

C++ Type:std::vector

Description:Order parameters for all phases

### 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

• 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