DiscreteNucleationTimeStep

Return a time step limit for nucleation event to be used by IterationAdaptiveDT

Supply this postprocessor to an IterationAdaptiveDT via the timestep_limiting_postprocessor parameter.

The timestep limit computed by this postprocessor is computed according to two different criteria.

Time step limit at nucleus insertion

If a nucleus has just been added to the nucleus list by the DiscreteNucleationInserter the timestep limit is set to the value supplied using the dt_max parameter for one timestep. In conjunction with IterationAdaptiveDT this causes the time step to be cut to dt_max from which it will slowly have to grow back.

Nucleation rate based timestep limit

Between nucleation event onsets the timestep is limited based on the user supplied upper bound on the probability $var element = document.getElementById("moose-equation-f7a31a4c-7d91-4fec-b227-9e14bc94fcff");katex.render("p_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ (p2nucleus) to have _more than two_ nueceation events to occur during a single timestep. This probability is calculated as

$(1)var element = document.getElementById("moose-equation-2d4cf1f0-7f67-47e8-91d4-0a5464cab862");katex.render(" p_{2nuc} = 1-(1+\\lambda_{2nuc})e^{-\\lambda_{2nuc}},", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

where $var element = document.getElementById("moose-equation-d6dbe12a-da73-4288-a0ed-3e5ca4388f3f");katex.render("\\lambda_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the total nucleation rate over the whole simulation cell that results in the probability $var element = document.getElementById("moose-equation-468e8c0f-b188-4a42-b535-20286d4b2ead");katex.render("p_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . To obtain $var element = document.getElementById("moose-equation-f0353b8f-2162-40c1-b0b0-08c5f4d41236");katex.render("\\lambda_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ for a given $var element = document.getElementById("moose-equation-375a6a2f-c6d0-43de-9841-f5ba3d07f0d8");katex.render("p_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ equation Eq. (1) is numerically inverted. $var element = document.getElementById("moose-equation-d85a269a-9945-47cd-8589-c0733edc7f09");katex.render("\\lambda_{2nuc}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is then divided by the integrated nucleation rate per unit time to obtain the largest possible time step that keeps the probability for two or more nuclei to form below the user specified upper bound.

Timestep (dt) in a nucleation simulation with a DiscreteNucleationTimeStep limited time step. The green curve (dtnuc) shows the time step limit. The envelope of that curve is determined by the upper bound on the two or more nucleus probability. The sharp downward spikes are the time step cut-backs during nucleation events.

This addresses two issues with poisson statistics sampling. At every sampling point in the domain the rate is sufficiently low to stay in the rare event regime (i.e where either zero or one event are happening). At higher rates the time resolution is insufficient to capture all possible nucleation events. Controlling the probability of multiple nuclei forming also reduces the chance of overlapping nuclei to form.

The DiscreteNucleationTimeStep postprocessor is part of the Discrete Nucleation system.

Input Parameters

dt_maxTime step to cut back to at the start of a nucleation event
C++ Type:double
Options:
Description:Time step to cut back to at the start of a nucleation event
inserterDiscreteNucleationInserter user object
C++ Type:UserObjectName
Options:
Description:DiscreteNucleationInserter user object

Required Parameters

execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:NONE INITIAL LINEAR NONLINEAR TIMESTEP_END TIMESTEP_BEGIN FINAL CUSTOM TRANSFER
Description:The list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.
p2nucleus0.01Maximum probability for more than one nucleus to appear during a time step. This will limit the time step based on the total nucleation rate for the domain to make sure the probability for two or more nuclei to appear is always below the chosen number.
Default:0.01
C++ Type:double
Options:
Description:Maximum probability for more than one nucleus to appear during a time step. This will limit the time step based on the total nucleation rate for the domain to make sure the probability for two or more nuclei to appear is always below the chosen number.

Optional Parameters

allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Options:
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
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.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
force_preauxFalseForces the GeneralUserObject to be executed in PREAUX
Default:False
C++ Type:bool
Options:
Description:Forces the GeneralUserObject to be executed in PREAUX
outputsVector of output names were you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object
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.

Advanced Parameters

Input Files

modules/phase_field/test/tests/Nucleation/timestep.i
modules/phase_field/examples/nucleation/refine.i

modules/phase_field/test/tests/Nucleation/timestep.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 2
  ny = 2
  nz = 0
  xmin = 0
  xmax = 20
  ymin = 0
  ymax = 20
[]

[Variables]
  [./c]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    boundary = left
    variable = c
    value = 0
  [../]
  [./right]
    type = DirichletBC
    boundary = right
    variable = c
    value = 1
  [../]
  [./Periodic]
    [./all]
      auto_direction = y
    [../]
  [../]
[]

[Kernels]
  [./c]
    type = Diffusion
    variable = c
  [../]
  [./dt]
    type = TimeDerivative
    variable = c
  [../]
[]

[UserObjects]
  [./inserter]
    type = DiscreteNucleationInserter
    hold_time = 1
    probability = 0.0005
  [../]
  [./map]
    type = DiscreteNucleationMap
    radius = 3.27
    periodic = c
    inserter = inserter
  [../]
[]

[Postprocessors]
  [./dt]
    type = TimestepSize
  [../]
  [./dtnuc]
    type = DiscreteNucleationTimeStep
    inserter = inserter
    p2nucleus = 0.1
    dt_max = 0.5
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  num_steps = 20

  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 8
    iteration_window = 2
    timestep_limiting_postprocessor = dtnuc
    dt = 1
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
[]

modules/phase_field/examples/nucleation/refine.i

#
# Example derived from cahn_hilliard.i demonstrating the use of Adaptivity
# with the DiscreteNucleation system. The DiscreteNucleationMarker triggers
# mesh refinement for the nucleus geometry. It is up to the user to specify
# refinement for the physics. In this example this is done using a GradientJumpIndicator
# with a ValueThresholdMarker. The nucleation system marker and the physics marker
# must be combined using a ComboMarker to combine their effect.
#

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
  xmax = 500
  ymax = 500
  elem_type = QUAD
[]

[Modules]
  [./PhaseField]
    [./Conserved]
      [./c]
        free_energy = F
        mobility = M
        kappa = kappa_c
        solve_type = REVERSE_SPLIT
      [../]
    [../]
  [../]
[]

[ICs]
  [./c_IC]
    type = ConstantIC
    variable = c
    value = 0.2
  [../]
[]

[Materials]
  [./pfmobility]
    type = GenericConstantMaterial
    prop_names  = 'M kappa_c'
    prop_values = '1 25'
  [../]
  [./chemical_free_energy]
    # simple double well free energy
    type = DerivativeParsedMaterial
    f_name = Fc
    args = 'c'
    constant_names       = 'barr_height  cv_eq'
    constant_expressions = '0.1          0'
    function = 16*barr_height*c^2*(1-c)^2 # +0.01*(c*plog(c,0.005)+(1-c)*plog(1-c,0.005))
    derivative_order = 2
    outputs = exodus
  [../]
  [./probability]
    # This is a made up toy nucleation rate it should be replaced by
    # classical nucleation theory in a real simulation.
    type = ParsedMaterial
    f_name = P
    args = c
    function = 'if(c<0.21,c*1e-8,0)'
    outputs = exodus
  [../]
  [./nucleation]
    # The nucleation material is configured to insert nuclei into the free energy
    # tht force the concentration to go to 0.95, and holds this enforcement for 500
    # time units.
    type = DiscreteNucleation
    f_name = Fn
    op_names  = c
    op_values = 0.90
    penalty = 5
    penalty_mode = MIN
    map = map
    outputs = exodus
  [../]
  [./free_energy]
    # add the chemical and nucleation free energy contributions together
    type = DerivativeSumMaterial
    derivative_order = 2
    args = c
    sum_materials = 'Fc Fn'
  [../]
[]

[UserObjects]
  [./inserter]
    # The inserter runs at the end of each time step to add nucleation events
    # that happend during the timestep (if it converged) to the list of nuclei
    type = DiscreteNucleationInserter
    hold_time = 50
    probability = P
  [../]
  [./map]
    # The map UO runs at the beginning of a timestep and generates a per-element/qp
    # map of nucleus locations. The map is only regenerated if the mesh changed or
    # the list of nuclei was modified.
    # The map converts the nucleation points into finite area objects with a given radius.
    type = DiscreteNucleationMap
    radius = 10
    periodic = c
    inserter = inserter
  [../]
[]

[Preconditioning]
  [./SMP]
    type = SMP
    full = true
  [../]
[]

[BCs]
  [./Periodic]
    [./all]
      auto_direction = 'x y'
    [../]
  [../]
[]

[Postprocessors]
  [./dt]
    type = TimestepSize
  [../]
  [./ndof]
    type = NumDOFs
  [../]
  [./rate]
    type = DiscreteNucleationData
    value = RATE
    inserter = inserter
  [../]
  [./dtnuc]
    type = DiscreteNucleationTimeStep
    inserter = inserter
    p2nucleus = 0.0005
    dt_max = 10
  [../]
  [./update]
    type = DiscreteNucleationData
    value = UPDATE
    inserter = inserter
  [../]
  [./count]
    type = DiscreteNucleationData
    value = COUNT
    inserter = inserter
  [../]
[]

[Adaptivity]
  [./Indicators]
    [./jump]
      type = GradientJumpIndicator
      variable = c
    [../]
  [../]
  [./Markers]
    [./nuc]
      type = DiscreteNucleationMarker
      map = map
    [../]
    [./grad]
      type = ValueThresholdMarker
      variable = jump
      coarsen = 0.1
      refine = 0.2
    [../]
    [./combo]
      type = ComboMarker
      markers = 'nuc grad'
    [../]
  [../]
  marker = combo
  cycles_per_step = 3
  recompute_markers_during_cycles = true
  max_h_level = 3
[]

[Executioner]
  type = Transient
  scheme = bdf2
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -sub_pc_type'
  petsc_options_value = 'asm      lu          '

  nl_max_its = 20
  l_tol = 1.0e-4
  nl_rel_tol = 1.0e-10
  nl_abs_tol = 1.0e-10
  start_time = 0.0
  num_steps = 120

  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    growth_factor = 1.5
    cutback_factor = 0.5
    optimal_iterations = 8
    iteration_window = 2
    timestep_limiting_postprocessor = dtnuc
  [../]
[]

[Outputs]
  exodus = true
  csv = true
  print_linear_residuals = false
[]

References

Time step limit at nucleus insertion
Nucleation rate based timestep limit
Input Parameters
Input Files
References

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