# ACMultiInterface

Gradient energy Allen-Cahn Kernel with cross terms

Implements Allen-Cahn interface terms for a multiphase system. This includes cross terms of the form

(1)

where is the non-linear variable the kernel is acting on, (etas are all non-conserved order parameters in the system, (kappa_name) are the gradient energy coefficents, and (mob_name) is the scalar (isotropic) mobility associated with the order parameter.

## Derivation

The interfacial free energy density is implemented following Nestler and Wheeler (1998) equations (7) and (8) (also see footnote 1)

(2)

Where the sum is taken over unique tuples _a,b_ (i.e. without the permutations _b,a_). We take the functional derivative taken using the lemma

(3)

We obtain a one dimensional sum for each of the -derivatives.

(4)

We transform this expression into the weak form and see that the derivative order on the _order 2_ term has to be reduced by shifting a gradient onto the test function by applying the product rule

(5)

after multiplying with the test function and integrating over the volume . We identify and as follows

(6)

(7)

We get rid of the last two terms by applying the divergence theorem and obtain

(8)

to convert them from volume to surface/boundary integrals. We again apply the product rule to expand the gradient of the product in the _volume terms_ and obtain

(9)

### Residual

The total residual is then

(10)

### On-diagonal Jacobian

The on-diagonal jacobian is obtained by taking the derivative with respect to , where and

(11)

### Off-diagonal jacobian

For the off diagonal Jacobian entry we take the derivative and obtain

\begin{aligned} J_{ab} = &\,& L_a\kappa_{ab}\int_\Omega2\psi\left[ (\eta_a\nabla\phi_j - \phi_j\nabla\eta_a)\nabla\eta_b + (\eta_a\nabla\eta_b - \eta_b\nabla\eta_a)\nabla\phi_j \right]
&+& \int_\Omega\left[ -\left( \eta_a\phi_j\nabla\psi + \psi\phi_j\nabla\eta_a + \psi\eta_a\nabla\phi_j \right) \cdot\nabla\eta_b -\left( \eta_a\eta_b\nabla\psi + \psi\eta_b\nabla\eta_a + \psi\eta_a\nabla\eta_b \right) \cdot\nabla\phi_j \right]
&-& \int_\Omega\left[ %-\left( \eta_b^2\nabla\psi + 2\psi\eta_b\nabla\eta_b \right)\cdot\nabla\eta_a -\left( 2\eta_b\phi_j\nabla\psi + 2\psi(\phi_j\nabla\eta_b + \eta_b\nabla\phi_j) \right)\cdot\nabla\eta_a \right] \end{aligned}

----

1) Note, that in the two-phase case with this reduces to

(12)

which is the familiar form implemented by ACInterface.

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

Options:

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

• kappa_namesThe kappa used with the kernel

C++ Type:std::vector

Options:

Description:The kappa used with the kernel

• etasAll eta_i order parameters of the multiphase problem

C++ Type:std::vector

Options:

Description:All eta_i order parameters of the multiphase problem

### Required Parameters

• mob_nameLThe mobility used with the kernel

Default:L

C++ Type:MaterialPropertyName

Options:

Description:The mobility used with the kernel

• displacementsThe displacements

C++ Type:std::vector

Options:

Description:The displacements

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Options:

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

Options:

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

Options:

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

Options:

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

Options:

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

Options:

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

Options:

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

Options:

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

Options:nontime time

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

Options:

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

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

Options:

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

## Input Files

1. B. Nestler and A. A. Wheeler. Anisotropic multi-phase-field model: interfaces and junctions. Phys. Rev. E, 57:2602–2609, Mar 1998. URL: https://link.aps.org/doi/10.1103/PhysRevE.57.2602, doi:10.1103/PhysRevE.57.2602.[BibTeX]