ACGrGrPolyLinearizedInterface

Grain growth model Allen-Cahn Kernel with linearized interface variable transformation

Overview

This kernel implements the bulk portion of the residual for the Allen-Cahn equation from the multiphase grain growth model using the linearized interface variable substitution. It is the linearized interface version of the ACGrGrPoly kernel. The grain growth model comes from Moelans et al. (2008) and the linearized interface from Glasner (2001).

The kernel inherits the same residual function from ACGrGrPoly but uses a linearized interface substitution. The substitution is defined in the material property this_op. The most common one is $var element = document.getElementById("moose-equation-51d70cb4-7e14-410a-aed4-188bc5a627c3");katex.render(" \\phi_i = \\frac{1}{2} \\left[ 1 + \\tanh\\left( \\frac{\\psi_i}{\\sqrt{2}} \\right) \\right],", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ where $var element = document.getElementById("moose-equation-bc667d73-4b7b-41de-9d91-635c2ade7426");katex.render("\\phi_i", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the order parameter and $var element = document.getElementById("moose-equation-5e3f3a12-0a0f-424a-be0b-04f0850650f2");katex.render("\\psi_i", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the transformed variable. The substitution is implemented in the material LinearizedInterfaceFunction.

Example Input File Syntax

The kernel can be used according to

[Kernels]
  [phi0_ACInt]
    type = ACInterfaceChangedVariable
    variable = phi0
    kappa_name = kappa_op
    mob_name = L
    order_parameter = gr0
  []
[]

However, it can also be created automatically using the GrainGrowthLinearizedInterfaceAction.

Input Parameters

other_opsList of properties defining the order parameters for the variables in v
C++ Type:std::vector<MaterialPropertyName>
Unit:(no unit assumed)
Controllable:No
Description:List of properties defining the order parameters for the variables in v
this_opThe material property defining the order parameter for this variable
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:The material property defining the order parameter for this variable
vArray of coupled order parameter names for other order parameters
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Array of coupled order parameter names for other order parameters
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
coupled_variablesVector of nonlinear variable arguments this object depends on
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Vector of nonlinear variable arguments this object depends on
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

Karl Glasner. Nonlinear preconditioning for diffuse interfaces. Journal of Computational Physics, 174(2):695–711, 2001.

@article{glasner2001nonlinear,
    author = "Glasner, Karl",
    title = "Nonlinear preconditioning for diffuse interfaces",
    journal = "Journal of Computational Physics",
    volume = "174",
    number = "2",
    pages = "695--711",
    year = "2001",
    publisher = "Elsevier"
}

N. Moelans, B. Blanpain, and P. Wollants. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems. Physical Review B, 78(2):024113, Jul 2008. URL: http://link.aps.org/doi/10.1103/PhysRevB.78.024113 (visited on 2016-06-02), doi:10.1103/PhysRevB.78.024113.

@article{moelans_quantitative_2008,
    author = "Moelans, N. and Blanpain, B. and Wollants, P.",
    title = "Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems",
    volume = "78",
    url = "http://link.aps.org/doi/10.1103/PhysRevB.78.024113",
    doi = "10.1103/PhysRevB.78.024113",
    number = "2",
    urldate = "2016-06-02",
    journal = "Physical Review B",
    month = "Jul",
    year = "2008",
    pages = "024113",
    file = "APS Snapshot:/Users/i.greenquist/Zotero/storage/Q8854BXX/PhysRevB.78.html:text/html;PhysRevB.78.024113.pdf:/Users/i.greenquist/Zotero/storage/VSMEC28A/PhysRevB.78.024113.pdf:application/pdf;PhysRevB.78.pdf:/Users/i.greenquist/Zotero/storage/VEU7AATJ/PhysRevB.78.pdf:application/pdf"
}

(moose/modules/phase_field/test/tests/grain_growth_w_linearized_interface/grain_growth_linearized_interface.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 50
  ymax = 50
  nx = 10
  ny = 10
[]

[Variables]
  [phi0]
  []
  [phi1]
  []
[]

[AuxVariables]
  [gr0_aux]
  []
  [gr1_aux]
  []
  [bounds_dummy]
  []
[]

[AuxKernels]
  [gr0]
    type = LinearizedInterfaceAux
    variable = gr0_aux
    nonlinear_variable = phi0
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [gr1]
    type = LinearizedInterfaceAux
    variable = gr1_aux
    nonlinear_variable = phi1
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[ICs]
  [phi0_IC]
    type = SmoothCircleICLinearizedInterface
    variable = phi0
    invalue = 1.0
    outvalue = 0.0
    bound_value = 5.0
    radius = 30
    int_width = 10
    x1 = 0.0
    y1 = 0.0
    profile = TANH
  []
  [phi1_IC]
    type = SmoothCircleICLinearizedInterface
    variable = phi1
    invalue = 0.0
    outvalue = 1.0
    bound_value = 5.0
    radius = 30
    int_width = 10
    x1 = 0.0
    y1 = 0.0
    profile = TANH
  []
[]

[Kernels]
  #phi0 Kernels
  [phi0_dot]
    type = ChangedVariableTimeDerivative
    variable = phi0
    order_parameter = gr0
  []
  [phi0_ACInt]
    type = ACInterfaceChangedVariable
    variable = phi0
    kappa_name = kappa_op
    mob_name = L
    order_parameter = gr0
  []
  [gr0_AC]
    type = ACGrGrPolyLinearizedInterface
    variable = phi0
    mob_name = L
    this_op = gr0
    other_ops = gr1
    v = phi1
  []
  #phi1 Kernels
  [phi1_dot]
    type = ChangedVariableTimeDerivative
    variable = phi1
    order_parameter = gr1
  []
  [phi1_ACInt]
    type = ACInterfaceChangedVariable
    variable = phi1
    kappa_name = kappa_op
    mob_name = L
    order_parameter = gr1
  []
  [gr1_AC]
    type = ACGrGrPolyLinearizedInterface
    variable = phi1
    mob_name = L
    this_op = gr1
    other_ops = gr0
    v = phi0
  []
[]

[Materials]
  [gr0]
    type = LinearizedInterfaceFunction
    f_name = gr0
    phi = phi0
  []
  [gr1]
    type = LinearizedInterfaceFunction
    f_name = gr1
    phi = phi1
  []
  [GBEovlution]
    type = GBEvolution
    GBenergy = 0.97
    GBMobility = 0.6e-6
    T = 300
    wGB = 10
  []
[]

[Bounds]
  [phi0_upper_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = phi0
    bound_type = upper
    bound_value = 5.0
  []
  [phi0_lower_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = phi0
    bound_type = lower
    bound_value = -5.0
  []
  [phi1_upper_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = phi1
    bound_type = upper
    bound_value = 5.0
  []
  [phi1_lower_bound]
    type = ConstantBounds
    variable = bounds_dummy
    bounded_variable = phi1
    bound_type = lower
    bound_value = -5.0
  []
[]

[Postprocessors]
  [grain_area_mat]
    type = ElementIntegralMaterialProperty
    mat_prop = gr0
    execute_on = 'initial TIMESTEP_END'
  []
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = NEWTON

  petsc_options_iname = '-pc_type -ksp_type -snes_type'
  petsc_options_value = 'bjacobi gmres vinewtonrsls'

  dt = 0.1
  end_time = 0.6
[]

[Outputs]
  exodus = true
[]