MFEMVectorFEInnerProductIntegralPostprocessor

Overview

Postprocessor for calculating the integral of the dot product between two vector variables over a user-specified subdomain of the mesh:

var element = document.getElementById("moose-equation-5621236b-f371-4f99-afaa-9b414009bcee");katex.render("(k \\vec u, \\vec v)_\\Omega", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-3c7370bc-ac88-4a13-a162-acdd871d7f8d");katex.render("\\vec u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-d35e29a2-48fa-4199-acdd-c5af7fa24627");katex.render("\\vec v \\in H(\\mathrm{curl})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ or $var element = document.getElementById("moose-equation-c151a49e-4993-4d41-9dab-20fb3fba1293");katex.render("H(\\mathrm{div})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-231938a8-67ba-49f5-9379-6ee95e2e994f");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is an optional scalar coefficient, and $var element = document.getElementById("moose-equation-c00be465-69d0-4356-918c-6526299d7224");katex.render("\\Omega", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the user-specified mesh subdomain.

Example Input File Syntax

[Postprocessors<<<{"href": "../../../syntax/Postprocessors/index.html"}>>>]
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor<<<{"description": "Calculates the integral of the inner product of two vector variables within a subdomain.", "href": "MFEMVectorFEInnerProductIntegralPostprocessor.html"}>>>
    coefficient<<<{"description": "Name of optional scalar coefficient to scale integrand by. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number."}>>> = conductivity
    dual_variable<<<{"description": "Name of the second vector variable in the inner product."}>>> = e_field
    primal_variable<<<{"description": "Name of the first vector variable in the inner product."}>>> = e_field
    block<<<{"description": "The list of subdomains (names or ids) that this object will be restricted to. Leave empty to apply to all subdomains."}>>> = 'TorusCore TorusSheath'
  []
[]

Input Parameters

dual_variableName of the second vector variable in the inner product.
C++ Type:VariableName
Unit:(no unit assumed)
Controllable:No
Description:Name of the second vector variable in the inner product.
primal_variableName of the first vector variable in the inner product.
C++ Type:VariableName
Unit:(no unit assumed)
Controllable:No
Description:Name of the first vector variable in the inner product.

Required Parameters

blockThe list of subdomains (names or ids) that this object will be restricted to. Leave empty to apply to all subdomains.
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of subdomains (names or ids) that this object will be restricted to. Leave empty to apply to all subdomains.
coefficient1.Name of optional scalar coefficient to scale integrand by. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number.
Default:1.
C++ Type:MFEMScalarCoefficientName
Controllable:No
Description:Name of optional scalar coefficient to scale integrand by. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number.
execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, FORWARD, ADJOINT, HOMOGENEOUS_FORWARD, ADJOINT_TIMESTEP_BEGIN, ADJOINT_TIMESTEP_END, NONE, INITIAL, LINEAR, LINEAR_CONVERGENCE, NONLINEAR, NONLINEAR_CONVERGENCE, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, MULTIAPP_FIXED_POINT_CONVERGENCE, FINAL, CUSTOM
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.

Optional 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.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.
outputsVector of output names where you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Advanced Parameters

Input Files

(test/tests/mfem/submeshes/nl_hphi_magnetodynamic.i)
(test/tests/mfem/submeshes/hphi_magnetostatic.i)
(test/tests/mfem/submeshes/av_magnetostatic.i)
(test/tests/mfem/submeshes/hphi_magnetodynamic.i)

(test/tests/mfem/submeshes/av_magnetostatic.i)

# Magnetostatic problem solved on a closed conductor subject to
# global loop voltage constraint.

[Mesh]
  type = MFEMMesh
  file = ../mesh/embedded_concentric_torus.e
[]

[Problem]
  type = MFEMProblem
[]

[SubMeshes]
  inactive = 'fluxcut'
  [fluxcut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = 'MeasurementPlane'
    block = 'TorusCore TorusSheath'
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = coil_dom
  []
[]

[FESpaces]
  inactive = 'FluxFESpace'
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [FluxFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = fluxcut
  []
[]

[Variables]
  [a_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  inactive = 'flux_e_field'
  [b_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [flux_e_field]
    type = MFEMVariable
    fespace = FluxFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = b_field
    source = a_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_a_field]
    type = ParsedVectorFunction
    expression_x = '0'
    expression_y = '0'
    expression_z = '0'
  []
[]

[BCs]
  [tangential_a_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = a_field
    vector_coefficient = exact_a_field
    boundary = 'Exterior'
  []
[]

[FunctorMaterials]
  inactive = 'ConductorBoundary'
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = reluctivity
    prop_values = 1.0
  []
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity
    prop_values = 1.0
    block = 'TorusCore TorusSheath'
  []
  [ConductorBoundary]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity_boundary
    prop_values = 1.0
    boundary = 'MeasurementPlane'
  []
[]

[Kernels]
  [mass]
    type = MFEMVectorFEMassKernel
    variable = a_field
    coefficient = 1e-10
  []
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = a_field
    coefficient = reluctivity
  []
  [source]
    type = MFEMMixedVectorMassKernel
    variable = a_field
    trial_variable = e_field
    coefficient = conductivity
    block = 'TorusCore TorusSheath'
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-14
  l_max_its = 1000
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[MultiApps]
  [coil]
    type = FullSolveMultiApp
    input_files = cut_closed_coil.i
    execute_on = INITIAL
  []
[]

[Transfers]
  inactive = 'submesh_transfer_to_fluxsurface'
  [from_coil]
    type = MultiAppMFEMCopyTransfer
    source_variables = e_field
    variables = e_field
    from_multi_app = coil
  []
  [submesh_transfer_to_fluxsurface]
    type = MFEMSubMeshTransfer
    from_variable = e_field
    to_variable = flux_e_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  inactive = 'CoilCurrent'
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = conductivity
    dual_variable = e_field
    primal_variable = e_field
    block = 'TorusCore TorusSheath'
  []
  [CoilCurrent]
    type = MFEMVectorBoundaryFluxIntegralPostprocessor
    coefficient = conductivity_boundary
    variable = flux_e_field
    boundary = 'MeasurementPlane'
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/MagnetostaticClosedCoil
    vtk_format = ASCII
  []
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/AVMagnetostaticClosedCoilCSV
  []
[]

(test/tests/mfem/submeshes/nl_hphi_magnetodynamic.i)

# Solve for the magnetic field around a closed superconductor subject to
# global current constraint.

conductor_domains = 'TorusCore TorusSheath'
vacuum_permeability = 1.0
n_value = 20.0
j_c = 1.0
e_c = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/split_embedded_concentric_torus.e
[]

[SubMeshes]
  [conductor]
    type = MFEMDomainSubMesh
    block = ${conductor_domains}
    submesh_boundary = conductor_surface
  []
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
    submesh = conductor
  []
  [CoilHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = conductor
  []
[]

[Variables]
  [coil_induced_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
[]

[AuxVariables]
  [coil_external_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
  [j_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [update_j_field]
    type = MFEMCurlAux
    variable = j_field
    source = coil_induced_h_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [resistivity]
    type = MFEMParsedFunction
    expression = '(e_c/j_c) * (j/j_c)^(n_val-1)'
    symbol_names = 'j e_c j_c n_val'
    symbol_values = 'coil_induced_h_field_curl_mag ${e_c} ${j_c} ${n_value}'
  []
  [j_dresistivity_dj]
    type = MFEMParsedFunction
    expression = '(n_val-1) * resistivity'
    symbol_names = 'n_val resistivity'
    symbol_values = '${n_value} resistivity'
  []
[]

[FunctorMaterials]
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = 'permeability'
    prop_values = '${vacuum_permeability}'
  []
[]


[BCs]
  [conductor_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = coil_induced_h_field
    vector_coefficient = coil_external_h_field
    boundary = conductor_surface
  []
[]

[Kernels]
  [dBdt]
    type = MFEMTimeDerivativeVectorFEMassKernel
    variable = coil_induced_h_field
    coefficient = permeability
  []
  [curlE]
    type = MFEMNLCurlCurlKernel
    variable = coil_induced_h_field
    k_coefficient = resistivity
    curlu_dk_dcurlu_coefficient = j_dresistivity_dj
    block = ${conductor_domains}
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = CoilHCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-12
  l_max_its = 100
  print_level = 0
[]

[Executioner]
  type = MFEMTransient
  dt = 0.5
  start_time = 0.0
  end_time = 2.0
  nl_max_its = 150
  nl_abs_tol = 1e-5
  print_level = 1
[]

[MultiApps]
  [hphi_magnetostatic]
    type = FullSolveMultiApp
    input_files = hphi_magnetostatic.i
    execute_on = INITIAL
  []
[]

[Transfers]
  [from_external_field]
    type = MultiAppMFEMShapeEvaluationTransfer
    source_variables = h_field
    variables = coil_external_h_field
    from_multi_app = hphi_magnetostatic
  []
[]

[Postprocessors]
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = resistivity
    dual_variable = j_field
    primal_variable = j_field
    block = 'TorusCore TorusSheath'
  []
[]

[Outputs]
  inactive = ConductorParaViewDataCollection
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/HPhiMagnetodynamicNLClosedCoilCSV
  []
  [ConductorParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HPhiMagnetodynamicNLClosedCoil
    vtk_format = ASCII
    submesh = conductor
  []
[]

(test/tests/mfem/submeshes/hphi_magnetostatic.i)

# Solve for the magnetic field around a closed conductor subject to
# global current constraint.

initial_vacuum_domains = 'Exterior'
vacuum_cut_surface = 'Cut'
conductor_current = 1.0
vacuum_permeability = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/split_embedded_concentric_torus.e
[]

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = permeability
    prop_values = ${vacuum_permeability}
  []
[]

[ICs]
  [vacuum_cut_potential_ic]
    type = MFEMScalarBoundaryIC
    variable = vacuum_cut_potential
    boundary = ${vacuum_cut_surface}
    coefficient = ${conductor_current}
  []
[]

[SubMeshes]
  [cut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = ${vacuum_cut_surface}
    block = ${initial_vacuum_domains}
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = vacuum_dom
  []
  [vacuum]
    type = MFEMDomainSubMesh
    block = vacuum_dom
  []
[]

[FESpaces]
  [VacuumH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = vacuum
  []
  [VacuumHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = vacuum
  []
  [TransitionH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = cut
  []
  [TransitionHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = cut
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [vacuum_magnetic_potential]
    type = MFEMVariable
    fespace = VacuumH1FESpace
  []
[]

[AuxVariables]
  [vacuum_cut_potential]
    type = MFEMVariable
    fespace = VacuumH1FESpace
  []
  [transition_cut_potential]
    type = MFEMVariable
    fespace = TransitionH1FESpace
  []
  [transition_cut_function_field]
    type = MFEMVariable
    fespace = TransitionHCurlFESpace
  []
  [background_h_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [cut_function_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [vacuum_h_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [h_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [update_background_h_field]
    type = MFEMGradAux
    variable = background_h_field
    source = vacuum_magnetic_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_transition_cut_function_field]
    type = MFEMGradAux
    variable = transition_cut_function_field
    source = transition_cut_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_total_h_field]
    type = MFEMSumAux
    variable = vacuum_h_field
    source_variables = 'background_h_field cut_function_field'
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  # Set zero of magnetic potential on symmetry plane
  [Exterior]
    type = MFEMScalarDirichletBC
    variable = vacuum_magnetic_potential
    boundary = 'Cut'
    coefficient = 0.0
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = vacuum_magnetic_potential
    coefficient = permeability
  []
  [source]
    type = MFEMMixedGradGradKernel
    trial_variable = vacuum_cut_potential
    variable = vacuum_magnetic_potential
    coefficient = permeability
    block = 'transition_dom'
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-8
  l_max_its = 100
[]

[Executioner]
  type = MFEMSteady
[]

[Transfers]
  [submesh_transfer_to_transition]
    type = MFEMSubMeshTransfer
    from_variable = vacuum_cut_potential
    to_variable = transition_cut_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_transition]
    type = MFEMSubMeshTransfer
    from_variable = transition_cut_function_field
    to_variable = cut_function_field
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_vacuum]
    type = MFEMSubMeshTransfer
    from_variable = vacuum_h_field
    to_variable = h_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [MagneticEnergy]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = ${fparse 0.5*vacuum_permeability}
    dual_variable = vacuum_h_field
    primal_variable = vacuum_h_field
    block = 'Exterior'
  []
[]

[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/HPhiMagnetostaticClosedCoilCSV
  []
  [VacuumParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HPhiMagnetostaticClosedCoil
    vtk_format = ASCII
    submesh = vacuum
  []
[]

(test/tests/mfem/submeshes/av_magnetostatic.i)

# Magnetostatic problem solved on a closed conductor subject to
# global loop voltage constraint.

[Mesh]
  type = MFEMMesh
  file = ../mesh/embedded_concentric_torus.e
[]

[Problem]
  type = MFEMProblem
[]

[SubMeshes]
  inactive = 'fluxcut'
  [fluxcut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = 'MeasurementPlane'
    block = 'TorusCore TorusSheath'
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = coil_dom
  []
[]

[FESpaces]
  inactive = 'FluxFESpace'
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [FluxFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = fluxcut
  []
[]

[Variables]
  [a_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  inactive = 'flux_e_field'
  [b_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [flux_e_field]
    type = MFEMVariable
    fespace = FluxFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = b_field
    source = a_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_a_field]
    type = ParsedVectorFunction
    expression_x = '0'
    expression_y = '0'
    expression_z = '0'
  []
[]

[BCs]
  [tangential_a_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = a_field
    vector_coefficient = exact_a_field
    boundary = 'Exterior'
  []
[]

[FunctorMaterials]
  inactive = 'ConductorBoundary'
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = reluctivity
    prop_values = 1.0
  []
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity
    prop_values = 1.0
    block = 'TorusCore TorusSheath'
  []
  [ConductorBoundary]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity_boundary
    prop_values = 1.0
    boundary = 'MeasurementPlane'
  []
[]

[Kernels]
  [mass]
    type = MFEMVectorFEMassKernel
    variable = a_field
    coefficient = 1e-10
  []
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = a_field
    coefficient = reluctivity
  []
  [source]
    type = MFEMMixedVectorMassKernel
    variable = a_field
    trial_variable = e_field
    coefficient = conductivity
    block = 'TorusCore TorusSheath'
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-14
  l_max_its = 1000
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[MultiApps]
  [coil]
    type = FullSolveMultiApp
    input_files = cut_closed_coil.i
    execute_on = INITIAL
  []
[]

[Transfers]
  inactive = 'submesh_transfer_to_fluxsurface'
  [from_coil]
    type = MultiAppMFEMCopyTransfer
    source_variables = e_field
    variables = e_field
    from_multi_app = coil
  []
  [submesh_transfer_to_fluxsurface]
    type = MFEMSubMeshTransfer
    from_variable = e_field
    to_variable = flux_e_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  inactive = 'CoilCurrent'
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = conductivity
    dual_variable = e_field
    primal_variable = e_field
    block = 'TorusCore TorusSheath'
  []
  [CoilCurrent]
    type = MFEMVectorBoundaryFluxIntegralPostprocessor
    coefficient = conductivity_boundary
    variable = flux_e_field
    boundary = 'MeasurementPlane'
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/MagnetostaticClosedCoil
    vtk_format = ASCII
  []
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/AVMagnetostaticClosedCoilCSV
  []
[]

(test/tests/mfem/submeshes/hphi_magnetodynamic.i)

# Solve for the magnetic field around a closed conductor subject to
# global current constraint.

conductor_domains = 'TorusCore TorusSheath'
conductor_resistivity = 1.0
vacuum_permeability = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/split_embedded_concentric_torus.e
[]

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = 'resistivity'
    prop_values = ${conductor_resistivity}
    block = ${conductor_domains}
  []
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = 'permeability'
    prop_values = '${vacuum_permeability}'
  []
[]

[SubMeshes]
  [conductor]
    type = MFEMDomainSubMesh
    block = ${conductor_domains}
    submesh_boundary = conductor_surface
  []
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [CoilHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = conductor
  []
[]

[Variables]
  [coil_induced_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
[]

[AuxVariables]
  [h_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [coil_external_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
  [j_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [update_j_field]
    type = MFEMCurlAux
    variable = j_field
    source = h_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  [conductor_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = coil_induced_h_field
    vector_coefficient = coil_external_h_field
    boundary = conductor_surface
  []
[]

[Kernels]
  [dBdt]
    type = MFEMTimeDerivativeVectorFEMassKernel
    variable = coil_induced_h_field
    coefficient = permeability
  []
  [curlE]
    type = MFEMCurlCurlKernel
    variable = coil_induced_h_field
    coefficient = resistivity
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = CoilHCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-9
  l_max_its = 100
[]

[Executioner]
  type = MFEMTransient
  dt = 0.5
  start_time = 0.0
  end_time = 2.0
[]

[MultiApps]
  [hphi_magnetostatic]
    type = FullSolveMultiApp
    input_files = hphi_magnetostatic.i
    execute_on = INITIAL
  []
[]

[Transfers]
  [from_external_field]
    type = MultiAppMFEMCopyTransfer
    source_variables = h_field
    variables = h_field
    from_multi_app = hphi_magnetostatic
  []
  [submesh_transfer_to_coil]
    type = MFEMSubMeshTransfer
    from_variable = h_field
    to_variable = coil_external_h_field
    execute_on = TIMESTEP_BEGIN
  []
  [submesh_transfer_from_coil]
    type = MFEMSubMeshTransfer
    from_variable = coil_induced_h_field
    to_variable = h_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = resistivity
    dual_variable = j_field
    primal_variable = j_field
    block = 'TorusCore TorusSheath'
  []
[]

[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/HPhiMagnetodynamicClosedCoilCSV
  []
  [VacuumParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HPhiMagnetodynamicClosedCoil
    vtk_format = ASCII
  []
[]

Overview
Example Input File Syntax
Input Parameters
Input Files