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

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

- cbphase concentration corresponding to the non-linear variable of this kernel
C++ Type:std::vector

Description:phase concentration corresponding to the non-linear variable of this kernel

- caphase concentration corresponding to the non-linear variable of this kernel
C++ Type:std::vector

Description:phase concentration corresponding to the non-linear variable of this kernel

- fa_nameBase name of the free energy function F (f_name in the corresponding derivative function material)
C++ Type:MaterialPropertyName

Description:Base name of the free energy function F (f_name in the corresponding derivative function material)

- fb_nameBase name of the free energy function F (f_name in the corresponding derivative function material)
C++ Type:MaterialPropertyName

Description:Base name of the free energy function F (f_name in the corresponding derivative function material)

# KKSCHBulk

KKS model kernel for the Bulk Cahn-Hilliard term. This operates on the concentration 'c' as the non-linear variable

Non-split KKS Cahn-Hilliard bulk kernel, which is not fully implemented**. The non-linear variable for this Kernel is the concentration .

### Residual

In the residual routine we need to calculate the term . We exploit the KKS identity and arbitrarily use the a-phase instead. The gradient can be calculated through the chain rule (note that is potentially a function of many variables).

(1)

With being the vector of all arguments to this simplifies to

(2)

using as a shorthand for the term (and represented in the code as the array `_second_derivatives[i]`

). We do have access to the gradients of through MOOSE (stored in `_grad\_args[i]`

).

### Jacobian

The calculation of the Jacobian involves the derivative of the Residual term w.r.t. the individual coefficients of all parameters of . Here can stand for any variable .

(3)

In the code is given by `jvar`

for the off diagonal case, and (not or !) in the on diagonal case.

#### Off-diagonal

Let's focus on off diagonal first. Here is zero, if `jvar`

is not equal . Allowing us to remove the sum over and replace it with the single non-zero summand

(4)

In the first term in the square brackets the derivative is only non-zero if is `jvar`

. We can therefore pull this term out of the sum.

(5)

With the rules for derivatives we get

(6)

where is `_j`

in the code.

#### On-diagonal

For the on diagonal terms we look at the derivative w.r.t. the components of the non-linear variable of this kernel. Note, that is only indirectly a function of . We assume the dependence is given through . The chain rule will thus yield terms of this form

(7)

which is given as equation (23) in KKS. Following the off-diagonal derivation we get

(8)

#### On-diagonal second approach

Let's get back to the original residual with . Then

(9)

## Input Parameters

- mob_nameMThe mobility used with the kernel
Default:M

C++ Type:MaterialPropertyName

Description:The mobility used with the kernel

- args_aVector of additional arguments to Fa
C++ Type:std::vector

Description:Vector of additional arguments to Fa

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

Description:Vector of arguments of the mobility

- h_namehBase name for the switching function h(eta)
Default:h

C++ Type:MaterialPropertyName

Description:Base name for the switching function h(eta)

- 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