LevelSetOlssonReinitialization

This kernel implements the non-time portion of the conservative "re-initialization" algorithm of Olsson et al. (2007) that transforms $var element = document.getElementById("moose-equation-7b3a9437-50f0-465e-9ea5-48b71ca58446");katex.render("u_h", element, {displayMode:false,throwOnError:false});$ into a smooth function in the range $var element = document.getElementById("moose-equation-7604d740-6212-4598-a03b-e2b29cd1d7d5");katex.render("[0, 1]", element, {displayMode:false,throwOnError:false});$ rather than a signed distance, which is useful for certain types of problems such as phase identification. The weak form of the original re-initialization equation is: find $var element = document.getElementById("moose-equation-400d1318-d504-4e8d-9336-7f6236cc60ea");katex.render("U_h", element, {displayMode:false,throwOnError:false});$ such that

$var element = document.getElementById("moose-equation-27775ea8-bf0e-4d4a-ad10-d44c9c783886");katex.render(" \\left(\\frac{\\partial U_h}{\\partial \\tau}, \\psi_i\\right) + \\underbrace{\\left(\\nabla\\psi_i, -f\\hat{n}_{\\ast} + \\epsilon\\nabla U_h\\right)}_{\\textrm{LevelSetOlssonReinitialization}} = 0", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-fe10e0da-862e-4d26-b1f1-e8fa4797944a");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is the pseudo time during the re-initialization, $var element = document.getElementById("moose-equation-d3224cd5-3c9c-4a60-8ac9-1c152290972a");katex.render("\\hat{n}_{\\ast}", element, {displayMode:false,throwOnError:false});$ is the normal vector computed from the level set variable $var element = document.getElementById("moose-equation-e39a2c9b-8d8a-4c9e-a911-8f03e4daba6d");katex.render("u_h", element, {displayMode:false,throwOnError:false});$ at pseudo time $var element = document.getElementById("moose-equation-b35df697-8579-4353-bc3c-2bb08d2396b4");katex.render("\\tau=0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-c383d6b9-14bf-4974-b911-3c5d3dcc16ce");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ is the interface thickness, $var element = document.getElementById("moose-equation-68c2fa34-4762-46a6-a73e-2771ffa0c34d");katex.render("f\\equiv U_h(1-U_h)", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-f3d0659e-0ffe-4bc3-877a-95203cebd3b3");katex.render("U_h(\\tau=0) = u_h", element, {displayMode:false,throwOnError:false});$ .

To avoid any tangential diffusion, a modified re-initialization formulation is proposed by restricting compressive flux to normal direction. The weak form of the modified re-initialization equation is: find $var element = document.getElementById("moose-equation-8b01895f-cf2c-420e-aa66-d6eb793a31dc");katex.render("U_h", element, {displayMode:false,throwOnError:false});$ such that

$var element = document.getElementById("moose-equation-28db05eb-40f4-41bc-a014-ee3201307355");katex.render(" \\left(\\frac{\\partial U_h}{\\partial \\tau}, \\psi_i\\right) + \\underbrace{\\left(\\nabla\\psi_i, -f\\hat{n}_{\\ast} + \\epsilon\\nabla U_h\\cdot\\hat{n}_{\\ast}\\hat{n}_{\\ast}\\right)}_{\\textrm{LevelSetOlssonReinitialization}} = 0,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-de29c57c-06cd-41dc-a062-63d9eeb7a1f3");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is the pseudo time during the re-initialization, $var element = document.getElementById("moose-equation-29fe99a6-ff7a-4969-985b-8779dbec7511");katex.render("\\hat{n}_{\\ast}", element, {displayMode:false,throwOnError:false});$ is the normal vector computed from the level set variable $var element = document.getElementById("moose-equation-6a763816-7927-4bf3-9c11-a413dcc5787f");katex.render("u_h", element, {displayMode:false,throwOnError:false});$ at pseudo time $var element = document.getElementById("moose-equation-f85b9f37-caaf-448b-8d3b-48852a2c7855");katex.render("\\tau=0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-abe7a822-35b9-4f7c-89b7-7e02d0ce6ba3");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ is the interface thickness, $var element = document.getElementById("moose-equation-4211420f-29b9-4a24-9c56-00e0cb7bbe1f");katex.render("f\\equiv U_h(1-U_h)", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-809ec02a-2c5d-4f2c-a732-62bffbabfce5");katex.render("U_h(\\tau=0) = u_h", element, {displayMode:false,throwOnError:false});$ .

When steady-state for $var element = document.getElementById("moose-equation-96c988b2-c9a5-480e-aed0-185df4ec8b93");katex.render("U_h", element, {displayMode:false,throwOnError:false});$ is detected when (see LevelSetOlssonTerminator the entire process is repeated at time $var element = document.getElementById("moose-equation-4e614b32-c986-4211-b29f-49cbda5560b3");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ .

Our preliminary numerical tests indicate that the original re-initialization formulation works better to maintain the smoothness of an interface, so the original re-initialization formulation is set to be default. The modified re-initialization formulation can be used by setting use_modified_reinitilization_step = true although additional work needs to be done to investigate its numerical instability issue.

Example Syntax

The LevelSetOlssonReinitialization kernel is typically employed in a reinitialization sub-app (see the MOOSE MultiApp system), within this context this Kernel is invoked with the [Kernels] as follows.

[Kernels<<<{"href": "../../syntax/Kernels/index.html"}>>>]
  [./time]
    type = TimeDerivative<<<{"description": "The time derivative operator with the weak form of $(\\psi_i, \\frac{\\partial u_h}{\\partial t})$.", "href": "TimeDerivative.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = phi
  [../]

  [./reinit]
    type = LevelSetOlssonReinitialization<<<{"description": "The re-initialization equation defined by Olsson et. al. (2007).", "href": "LevelSetOlssonReinitialization.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = phi
    phi_0<<<{"description": "The level set variable to be reinitialized as signed distance function."}>>> = phi_0
    epsilon<<<{"description": "The epsilon coefficient to be used in the reinitialization calculation."}>>> = 0.05
  [../]
[]

Input Parameters

epsilonThe epsilon coefficient to be used in the reinitialization calculation.
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:The epsilon coefficient to be used in the reinitialization calculation.
phi_0The level set variable to be reinitialized as signed distance function.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The level set variable to be reinitialized as signed distance function.
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>
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
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
use_modified_reinitilization_formulationFalseUse the modified reinitilization formulation (Olsson et. al. (2007), section 2.2.1).
Default:False
C++ Type:bool
Controllable:No
Description:Use the modified reinitilization formulation (Olsson et. al. (2007), section 2.2.1).

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>
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>
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>
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
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
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
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
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
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
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
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

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
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.

Material Property Retrieval Parameters

References

Elin Olsson, Gunilla Kreiss, and Sara Zahedi. A conservative level set method for two phase flow ii. Journal of Computational Physics, 225(1):785–807, 2007. URL: http://dx.doi.org/10.1016/j.jcp.2006.12.027.

@article{olsson2007conservative,
    author = "Olsson, Elin and Kreiss, Gunilla and Zahedi, Sara",
    title = "A conservative level set method for two phase flow II",
    journal = "Journal of Computational Physics",
    volume = "225",
    number = "1",
    pages = "785--807",
    year = "2007",
    publisher = "Elsevier",
    url = "http://dx.doi.org/10.1016/j.jcp.2006.12.027"
}

(moose/modules/level_set/test/tests/reinitialization/reinit.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 8
  ny = 8
  uniform_refine = 3
[]

[Variables]
  [./phi]
  [../]
[]

[AuxVariables]
  [./phi_0]
  [../]
[]

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

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./reinit]
    type = LevelSetOlssonReinitialization
    variable = phi
    phi_0 = phi_0
    epsilon = 0.05
  [../]
[]

[Problem]
  type = LevelSetReinitializationProblem
[]

[UserObjects]
  [./arnold]
    type = LevelSetOlssonTerminator
    tol = 1
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  num_steps = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart'
  petsc_options_value = 'hypre    boomeramg      300'
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 0.001
    optimal_iterations = 5
    growth_factor = 5
  [../]
[]

Example Syntax
Input Parameters
References