# 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 , 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 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 \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 (similar for ): \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 (similar for ): \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 as: \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 , the Allen-Cahn mobility.

## Input Parameters

• eta_iOrder parameter that derivatives are taken with respect to

C++ Type:std::vector

Description:Order parameter that derivatives are taken with respect to

• wiDouble well height parameter

C++ Type:double

Description:Double well height parameter

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

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

• gi_nameBase name for the double well function g_i(eta_i)

C++ Type:MaterialPropertyName

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

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!

### Required 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

• 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