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

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 coefficients, 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)

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

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

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

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

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

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

Residual

The total residual is then

On-diagonal Jacobian

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

Off-diagonal jacobian

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

\begin{equation} \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}\end{equation}

----

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

which is the familiar form implemented by ACInterface.

Input Parameters

  • etasAll eta_i order parameters of the multiphase problem

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:All eta_i order parameters of the multiphase problem

  • kappa_namesThe kappa used with the kernel

    C++ Type:std::vector<MaterialPropertyName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The kappa used with the kernel

  • variableThe name of the variable that this residual object operates on

    C++ Type:NonlinearVariableName

    Unit:(no unit assumed)

    Controllable:No

    Description:The name of the variable that this residual object operates on

Required Parameters

  • blockThe list of blocks (ids or names) that this object will be applied

    C++ Type:std::vector<SubdomainName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of blocks (ids or names) that this object will be applied

  • displacementsThe displacements

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The displacements

  • mob_nameLThe mobility used with the kernel

    Default:L

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:The mobility used with the kernel

  • prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

  • use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Optional Parameters

  • absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution

  • extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

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

  • extra_vector_tagsThe extra tags for the vectors this Kernel should fill

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

    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

    Unit:(no unit assumed)

    Options:nontime, system

    Controllable:No

    Description:The tag for the matrices this Kernel should fill

  • vector_tagsnontimeThe tag for the vectors this Kernel should fill

    Default:nontime

    C++ Type:MultiMooseEnum

    Unit:(no unit assumed)

    Options:nontime, time

    Controllable:No

    Description:The tag for the vectors this Kernel should fill

Tagging Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

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

  • 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<AuxVariableName>

    Unit:(no unit assumed)

    Controllable:No

    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.)

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:Yes

    Description:Set the enabled status of the MooseObject.

  • implicitTrueDetermines whether this object is calculated using an implicit or explicit form

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

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

  • 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<AuxVariableName>

    Unit:(no unit assumed)

    Controllable:No

    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.)

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    Unit:(no unit assumed)

    Controllable:No

    Description:The seed for the master random number generator

  • 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

    Unit:(no unit assumed)

    Controllable:No

    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.

Advanced Parameters

References

  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]