MFEMMixedGradGradKernel

Overview

Adds the domain integrator for integrating the mixed bilinear form

var element = document.getElementById("moose-equation-55d82c5d-3f84-4095-bb33-0977e55e6223");katex.render("(k\\vec\\nabla u, \\vec\\nabla v)_\\Omega \\,\\,\\, \\forall v \\in V", 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 $var element = document.getElementById("moose-equation-0fe629d9-8e05-4810-a8a2-73ce1d240480");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-bafcd064-9a23-4979-8863-58e2a919d7e3");katex.render("v", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are both $var element = document.getElementById("moose-equation-cdcdeda2-dd30-422e-89be-ad78fd8106b7");katex.render("\\in H^1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and $var element = document.getElementById("moose-equation-1389bff7-cdf8-4f61-9440-2b45365cf4e3");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a scalar coefficient.

This term arises from the weak form of the Laplacian operator

var element = document.getElementById("moose-equation-313652a0-aa06-4222-b8bd-21afb81e5813");katex.render("\\nabla\\cdot k \\vec\\nabla u", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Example Input File Syntax

Input 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 property k to use. 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 property k to use. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number.
transposeFalseIf true, adds the transpose of the integrator to the system instead.
Default:False
C++ Type:bool
Controllable:No
Description:If true, adds the transpose of the integrator to the system instead.
trial_variableThe trial variable this kernel is acting on and which will be solved for. If empty (default), it will be the same as the test variable.
C++ Type:VariableName
Unit:(no unit assumed)
Controllable:No
Description:The trial variable this kernel is acting on and which will be solved for. If empty (default), it will be the same as the test variable.
variableVariable labelling the weak form this kernel is added to
C++ Type:VariableName
Unit:(no unit assumed)
Controllable:No
Description:Variable labelling the weak form this kernel is added to

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.

Advanced Parameters

Input Files

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

(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/cut_closed_coil.i)

# Solve for the electric field on a closed conductor subject to
# global loop voltage constraint.

initial_coil_domains = 'TorusCore TorusSheath'
coil_cut_surface = 'Cut'
coil_loop_voltage = -1.0
coil_conductivity = 1.0

[Problem]
  type = MFEMProblem
[]

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

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity
    prop_values = ${coil_conductivity}
  []
[]

[ICs]
  [coil_external_potential_ic]
    type = MFEMScalarBoundaryIC
    variable = coil_external_potential
    boundary = ${coil_cut_surface}
    coefficient = ${coil_loop_voltage}
  []
[]

[SubMeshes]
  [cut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = ${coil_cut_surface}
    block = ${initial_coil_domains}
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = coil_dom
  []
  [coil]
    type = MFEMDomainSubMesh
    block = coil_dom
  []
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [CoilH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = coil
  []
  [CoilHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = coil
  []
  [TransitionH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = cut
  []
  [TransitionHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = cut
  []
[]

[Variables]
  [coil_induced_potential]
    type = MFEMVariable
    fespace = CoilH1FESpace
  []
[]

[AuxVariables]
  [coil_external_potential]
    type = MFEMVariable
    fespace = CoilH1FESpace
  []
  [transition_external_potential]
    type = MFEMVariable
    fespace = TransitionH1FESpace
  []
  [transition_external_e_field]
    type = MFEMVariable
    fespace = TransitionHCurlFESpace
  []
  [induced_potential]
    type = MFEMVariable
    fespace = H1FESpace
  []
  [induced_e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [external_e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [update_induced_e_field]
    type = MFEMGradAux
    variable = induced_e_field
    source = induced_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_external_e_field]
    type = MFEMGradAux
    variable = transition_external_e_field
    source = transition_external_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_total_e_field]
    type = MFEMSumAux
    variable = e_field
    source_variables = 'induced_e_field external_e_field'
    execute_on = TIMESTEP_END
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = coil_induced_potential
    coefficient = conductivity
  []
  [source]
    type = MFEMMixedGradGradKernel
    trial_variable = coil_external_potential
    variable = coil_induced_potential
    coefficient = conductivity
    block = 'transition_dom'
  []
[]

[Solver]
  type = MFEMSuperLU
[]

[Executioner]
  type = MFEMSteady
[]

[Transfers]
  [submesh_transfer_from_coil]
    type = MFEMSubMeshTransfer
    from_variable = coil_induced_potential
    to_variable = induced_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_to_transition]
    type = MFEMSubMeshTransfer
    from_variable = coil_external_potential
    to_variable = transition_external_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_transition]
    type = MFEMSubMeshTransfer
    from_variable = transition_external_e_field
    to_variable = external_e_field
    execute_on = TIMESTEP_END
  []
[]

[Outputs]
  [GlobalParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/WholePotentialCoil
    vtk_format = ASCII
  []
  [TransitionParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/CutPotentialCoil
    vtk_format = ASCII
    submesh = cut
  []
  [CoilParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Coil
    vtk_format = ASCII
    submesh = coil
  []
[]

Overview
Example Input File Syntax
Input Parameters
Input Files