ShootMethodLog

An acceleration scheme based on the shooting method

Overview

To reduce the run time of simulations of radio frequency discharges involving neutral particles (such as metastable species), acceleration by shooting method can be implemented. Zapdos' version of the shooting method scheme is based on work presented by Lymberopoulos and Economou (1993), but a more general description of the method can be found by Gogolides et al. (1992). This scheme takes advantage of the fact that a RF discharge will eventually have a periodic steady-state, in the form of:

$var element = document.getElementById("moose-equation-7da9ca21-3245-4ecd-8ba8-a88d4af8a633");katex.render(" N_{j}(0)-N_{j}(T)=0", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-e4233f30-919b-423c-8364-5d336dd0c081");katex.render("N_{j}(0)", element, {displayMode:false,throwOnError:false});$ is the density at the beginning of the steady-state cycle and $var element = document.getElementById("moose-equation-1cfad440-14a6-4dec-97e5-13d11c51ab00");katex.render("N_{j}(T)", element, {displayMode:false,throwOnError:false});$ is the density end of the same cycle. To obtain a guess for the periodic solution of $var element = document.getElementById("moose-equation-32f64a3f-231f-445c-9026-073c0d88a901");katex.render("N_{j}(0)", element, {displayMode:false,throwOnError:false});$ , Newton method can be applied and results in:

$var element = document.getElementById("moose-equation-d37c24d1-8764-4942-b3c7-e601d733e9b0");katex.render(" N_{j}(0)^\\text{new}=N_{j}(0)^\\text{old}-J^{-1}(N_{j}(0)^\\text{old}-N_{j}(T))", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-5bd926fd-5884-431c-8e37-a2d77b0fbe41");katex.render(" J=I-\\left( \\frac{\\partial N_{j}}{\\partial N_{j}(0)} \\right)", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-9a89c9f5-4b60-45b5-83cf-e10ef46e2fec");katex.render("I", element, {displayMode:false,throwOnError:false});$ is the identity matrix and $var element = document.getElementById("moose-equation-4eebf308-c1fb-45c0-8e17-3d949d6f4683");katex.render("\\frac{\\partial N_{j}}{\\partial N_{j}(0)}", element, {displayMode:false,throwOnError:false});$ is known as the sensitivity matrix. The sensitivity matrix can be calculated by:

$var element = document.getElementById("moose-equation-6587b67b-6a8a-4908-9756-4d1ef68acd20");katex.render(" \\frac{d}{dt}\\left( \\frac{\\partial N_{j}}{\\partial N_{j}(0)} \\right) = \\frac{\\partial F}{\\partial N_{j}} \\frac{\\partial N_{j}}{\\partial N_{j}(0)}", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-c7e16226-409b-422d-b77b-5ef20459ed17");katex.render("\\frac{\\partial N_{j}}{\\partial N_{j}(0)} = I", element, {displayMode:false,throwOnError:false});$ at the beginning of the cycle. With this method, the acceleration scheme is as follows:

The main simulation runs for X RF cycles.
After X cycles, a sub-app runs for one RF cycle to calculate $var element = document.getElementById("moose-equation-dc2d4b58-18fe-4878-93af-707d86b50740");katex.render("N_{j}(T)", element, {displayMode:false,throwOnError:false});$ and the sensitivity matrix.
The metastable density is then accelerated and sends the new density to the main run.
The main simulation runs for another X RF cycles and then accelerates again.

Example Input File Syntax

An example of how to use ShootMethodLog can be found in the test file Acceleration_By_Shooting_Method_Shooting.i.

[Kernels]
  [Shoot_Method]
    type = ShootMethodLog
    variable = Ar*
    density_at_start_cycle = Ar*S
    density_at_end_cycle = Ar*T
    sensitivity_variable = SMDeriv
    growth_limit = 100.0
  []
[]

Input Parameters

density_at_end_cycleThe accelerated density at the end of the cycle in logarithmic form
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The accelerated density at the end of the cycle in logarithmic form
density_at_start_cycleThe accelerated density at the start of the cycle in logarithmic form
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The accelerated density at the start of the cycle in logarithmic form
sensitivity_variableThe variable that represents the sensitivity of accelerationas defined for the shooting method
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The variable that represents the sensitivity of accelerationas defined for the shooting method
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
growth_limit0A limit of the growth factor(growth_limit = 0.0 means no limit)
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:A limit of the growth factor(growth_limit = 0.0 means no limit)
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

Input Files

(test/tests/accelerations/Acceleration_By_Shooting_Method_Shooting.i)

References

E. Gogolides, H.H. Sawin, and R.A. Brown. Direct calculation of time-periodic states of continuum models of radio-frequency plasmas. Chemical Engineering Science, 47(15):3839–3855, 1992. doi:10.1016/0009-2509(92)85133-V.

@article{Gogolides1992,
    author = "Gogolides, E. and Sawin, H.H. and Brown, R.A.",
    title = "Direct calculation of time-periodic states of continuum models of radio-frequency plasmas",
    journal = "Chemical Engineering Science",
    volume = "47",
    number = "15",
    pages = "3839-3855",
    year = "1992",
    issn = "0009-2509",
    doi = "10.1016/0009-2509(92)85133-V"
}

Dimitris P. Lymberopoulos and Demetre J. Economou. Fluid simulations of glow discharges: effect of metastable atoms in argon. Journal of Applied Physics, 73(8):3668–3679, 04 1993. doi:10.1063/1.352926.

@article{Lymberopoulos1993,
    author = "Lymberopoulos, Dimitris P. and Economou, Demetre J.",
    title = "Fluid simulations of glow discharges: Effect of metastable atoms in argon",
    journal = "Journal of Applied Physics",
    volume = "73",
    number = "8",
    pages = "3668-3679",
    year = "1993",
    month = "04",
    issn = "0021-8979",
    doi = "10.1063/1.352926"
}

(test/tests/accelerations/Acceleration_By_Shooting_Method_Shooting.i)

[Mesh]
  [file]
    type = FileMeshGenerator
    file = 'Lymberopoulos_paper.msh'
  []
  [left]
    type = SideSetsFromNormalsGenerator
    normals = '-1 0 0'
    new_boundary = 'left'
    input = file
  []
  [right]
    type = SideSetsFromNormalsGenerator
    normals = '1 0 0'
    new_boundary = 'right'
    input = left
  []
[]

[Problem]
  type = FEProblem
[]

[Variables]
  [Ar*]
  []
[]

[AuxVariables]
  [em]
  []
  [Ar+]
  []
  [mean_en]
  []
  [potential]
  []
  [Ar*S]
  []

  [Ar*T]
  []
  [SMDeriv]
  []

  [SMDerivReset]
    initial_condition = 1.0
  []
[]

[Kernels]
  [Shoot_Method]
    type = ShootMethodLog
    variable = Ar*
    density_at_start_cycle = Ar*S
    density_at_end_cycle = Ar*T
    sensitivity_variable = SMDeriv
    growth_limit = 100.0
  []
[]

[BCs]
  [Ar*_physical_right_diffusion]
    type = LogDensityDirichletBC
    variable = Ar*
    boundary = 'right'
    value = 1e-5
  []
  [Ar*_physical_left_diffusion]
    type = LogDensityDirichletBC
    variable = Ar*
    boundary = 'left'
    value = 1e-5
  []
[]

#MultiApp used to calculate the non-accelerated density for 1RF cycle.
#This is used to get A*T (density at end of cycle) and
##the sensitivity variable needed for the shoot method.
[MultiApps]
  [SensitivityMatrix]
    type = FullSolveMultiApp
    input_files = 'Acceleration_By_Shooting_Method_SensitivityMatrix.i'
  []
[]

[Transfers]
  [em_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = em
    variable = em
  []
  [Ar+_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = Ar+
    variable = Ar+
  []
  [mean_en_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = mean_en
    variable = mean_en
  []
  [potential_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = potential
    variable = potential
  []
  [Ar*_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = Ar*
    variable = Ar*
  []
  [Deriv_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = SMDerivReset
    variable = SMDeriv
  []

  [Ar*T_from_sub]
    type = MultiAppCopyTransfer
    from_multi_app = SensitivityMatrix
    source_variable = Ar*
    variable = Ar*T
  []
  [Deriv_from_sub]
    type = MultiAppCopyTransfer
    from_multi_app = SensitivityMatrix
    source_variable = SMDeriv
    variable = SMDeriv
  []
[]

[Preconditioning]
  active = 'smp'
  [smp]
    type = SMP
    full = true
  []

  [fdp]
    type = FDP
    full = true
  []
[]

[Executioner]
  type = Steady

  petsc_options = '-snes_converged_reason -snes_linesearch_monitor'
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -ksp_type'
  petsc_options_value = 'lu NONZERO 1.e-10 fgmres'
[]

[Outputs]
  perf_graph = true
  [out]
    type = Exodus
  []
[]

(test/tests/accelerations/Acceleration_By_Shooting_Method_Shooting.i)

[Mesh]
  [file]
    type = FileMeshGenerator
    file = 'Lymberopoulos_paper.msh'
  []
  [left]
    type = SideSetsFromNormalsGenerator
    normals = '-1 0 0'
    new_boundary = 'left'
    input = file
  []
  [right]
    type = SideSetsFromNormalsGenerator
    normals = '1 0 0'
    new_boundary = 'right'
    input = left
  []
[]

[Problem]
  type = FEProblem
[]

[Variables]
  [Ar*]
  []
[]

[AuxVariables]
  [em]
  []
  [Ar+]
  []
  [mean_en]
  []
  [potential]
  []
  [Ar*S]
  []

  [Ar*T]
  []
  [SMDeriv]
  []

  [SMDerivReset]
    initial_condition = 1.0
  []
[]

[Kernels]
  [Shoot_Method]
    type = ShootMethodLog
    variable = Ar*
    density_at_start_cycle = Ar*S
    density_at_end_cycle = Ar*T
    sensitivity_variable = SMDeriv
    growth_limit = 100.0
  []
[]

[BCs]
  [Ar*_physical_right_diffusion]
    type = LogDensityDirichletBC
    variable = Ar*
    boundary = 'right'
    value = 1e-5
  []
  [Ar*_physical_left_diffusion]
    type = LogDensityDirichletBC
    variable = Ar*
    boundary = 'left'
    value = 1e-5
  []
[]

#MultiApp used to calculate the non-accelerated density for 1RF cycle.
#This is used to get A*T (density at end of cycle) and
##the sensitivity variable needed for the shoot method.
[MultiApps]
  [SensitivityMatrix]
    type = FullSolveMultiApp
    input_files = 'Acceleration_By_Shooting_Method_SensitivityMatrix.i'
  []
[]

[Transfers]
  [em_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = em
    variable = em
  []
  [Ar+_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = Ar+
    variable = Ar+
  []
  [mean_en_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = mean_en
    variable = mean_en
  []
  [potential_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = potential
    variable = potential
  []
  [Ar*_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = Ar*
    variable = Ar*
  []
  [Deriv_to_sub]
    type = MultiAppCopyTransfer
    to_multi_app = SensitivityMatrix
    source_variable = SMDerivReset
    variable = SMDeriv
  []

  [Ar*T_from_sub]
    type = MultiAppCopyTransfer
    from_multi_app = SensitivityMatrix
    source_variable = Ar*
    variable = Ar*T
  []
  [Deriv_from_sub]
    type = MultiAppCopyTransfer
    from_multi_app = SensitivityMatrix
    source_variable = SMDeriv
    variable = SMDeriv
  []
[]

[Preconditioning]
  active = 'smp'
  [smp]
    type = SMP
    full = true
  []

  [fdp]
    type = FDP
    full = true
  []
[]

[Executioner]
  type = Steady

  petsc_options = '-snes_converged_reason -snes_linesearch_monitor'
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -ksp_type'
  petsc_options_value = 'lu NONZERO 1.e-10 fgmres'
[]

[Outputs]
  perf_graph = true
  [out]
    type = Exodus
  []
[]

Overview
Example Input File Syntax
Input Parameters
Input Files
References