Definite Maxwell Problem

Summary

Solves a 3D electromagnetic diffusion problem for the electric field on a cube missing an octant, discretized using $var element = document.getElementById("moose-equation-f8c6dd21-7d3b-4bff-890d-7043968b07df");katex.render("H(\\mathrm{curl})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ conforming Nédélec elements. This example is based on MFEM Example 3.

Description

This problem solves a definite Maxwell equation with strong form:

$var element = document.getElementById("moose-equation-ebbc186e-b42c-4578-8c36-af112732bba6");katex.render("\\begin{split} \\vec\\nabla \\times \\vec\\nabla \\times \\vec E + \\vec E = \\vec f \\,\\,\\,&\\mathrm{on}\\,\\, \\Omega \\\\ \\vec E \\times \\hat n= \\vec g \\,\\,\\, &\\mathrm{on}\\,\\, \\partial\\Omega \\end{split}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where

$var element = document.getElementById("moose-equation-2a2d861f-72f3-45f1-8c8e-ee0a84cc3a12");katex.render("\\vec f = \\begin{pmatrix} (1+\\pi^2)\\sin(\\pi y)\\\\ (1+\\pi^2)\\sin(\\pi z)\\\\ (1+\\pi^2)\\sin(\\pi x) \\end{pmatrix},\\,\\,\\, \\vec g = \\begin{pmatrix} \\sin(\\pi y)\\\\ \\sin(\\pi z)\\\\ \\sin(\\pi x) \\end{pmatrix}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

In this example, we solve this using the weak form

var element = document.getElementById("moose-equation-3febbf5b-2ad2-4e8b-81f7-694a6b39eeaa");katex.render("(\\vec\\nabla \\times \\vec E, \\vec\\nabla \\times \\vec v)_\\Omega + (\\vec E, \\vec v)_\\Omega = (\\vec f, \\vec v)_\\Omega \\,\\,\\, \\forall \\vec 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

$var element = document.getElementById("moose-equation-3e522d3f-d7da-43b7-af6d-c84d10431885");katex.render("\\begin{split} \\vec E \\in H(\\mathrm{curl})(\\Omega) &: \\vec E \\times \\hat n= \\vec g \\,\\,\\, \\mathrm{on}\\,\\, \\partial\\Omega \\\\ \\vec v \\in H(\\mathrm{curl})(\\Omega) &: \\vec v \\times \\hat n= \\vec 0 \\,\\,\\, \\mathrm{on}\\,\\, \\partial\\Omega \\end{split}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Example File

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3.

[Mesh<<<{"href": "../Mesh/index.html"}>>>]
  type = MFEMMesh
  file = ../mesh/small_fichera.mesh
  dim = 3
[]

[Problem<<<{"href": "../Problem/index.html"}>>>]
  type = MFEMProblem
[]

[FESpaces<<<{"href": "../FESpaces/index.html"}>>>]
  [HCurlFESpace]
    type = MFEMVectorFESpace<<<{"description": "Convenience class to construct vector finite element spaces, abstracting away some of the mathematical complexity of specifying the dimensions.", "href": "../../source/mfem/fespaces/MFEMVectorFESpace.html"}>>>
    fec_type<<<{"description": "Specifies the family of FE shape functions."}>>> = ND
    fec_order<<<{"description": "Order of the FE shape function to use."}>>> = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace<<<{"description": "Convenience class to construct vector finite element spaces, abstracting away some of the mathematical complexity of specifying the dimensions.", "href": "../../source/mfem/fespaces/MFEMVectorFESpace.html"}>>>
    fec_type<<<{"description": "Specifies the family of FE shape functions."}>>> = RT
    fec_order<<<{"description": "Order of the FE shape function to use."}>>> = CONSTANT
  []
[]

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [e_field]
    type = MFEMVariable<<<{"description": "Class for adding MFEM variables to the problem (`mfem::ParGridFunction`s).", "href": "../../source/mfem/variables/MFEMVariable.html"}>>>
    fespace<<<{"description": "The finite element space this variable is defined on."}>>> = HCurlFESpace
  []
[]

[AuxVariables<<<{"href": "../AuxVariables/index.html"}>>>]
  [db_dt_field]
    type = MFEMVariable<<<{"description": "Class for adding MFEM variables to the problem (`mfem::ParGridFunction`s).", "href": "../../source/mfem/variables/MFEMVariable.html"}>>>
    fespace<<<{"description": "The finite element space this variable is defined on."}>>> = HDivFESpace
  []
[]

[AuxKernels<<<{"href": "../AuxKernels/index.html"}>>>]
  [curl]
    type = MFEMCurlAux<<<{"description": "Calculates the curl of an H(curl) conforming ND source variable and stores the result on an H(div) conforming RT result auxvariable", "href": "../../source/mfem/auxkernels/MFEMCurlAux.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = db_dt_field
    source<<<{"description": "Vector H(curl) MFEMVariable to take the curl of."}>>> = e_field
    scale_factor<<<{"description": "Factor to scale result auxvariable by."}>>> = -1.0
    execute_on<<<{"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."}>>> = TIMESTEP_END
  []
[]

[Functions<<<{"href": "../Functions/index.html"}>>>]
  [exact_e_field]
    type = ParsedVectorFunction<<<{"description": "Returns a vector function based on string descriptions for each component.", "href": "../../source/functions/MooseParsedVectorFunction.html"}>>>
    expression_x<<<{"description": "x-component of function."}>>> = 'sin(kappa * y)'
    expression_y<<<{"description": "y-component of function."}>>> = 'sin(kappa * z)'
    expression_z<<<{"description": "z-component of function."}>>> = 'sin(kappa * x)'

    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the vals vector."}>>> = kappa
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction<<<{"description": "Returns a vector function based on string descriptions for each component.", "href": "../../source/functions/MooseParsedVectorFunction.html"}>>>
    expression_x<<<{"description": "x-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y<<<{"description": "y-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z<<<{"description": "z-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the vals vector."}>>> = kappa
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 3.1415926535
  []
[]

[BCs<<<{"href": "../BCs/index.html"}>>>]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC<<<{"description": "Applies a Dirichlet condition to the tangential components of a vector variable.", "href": "../../source/mfem/bcs/MFEMVectorTangentialDirichletBC.html"}>>>
    variable<<<{"description": "Variable on which to apply the boundary condition"}>>> = e_field
    vector_coefficient<<<{"description": "Vector coefficient specifying the values the variable takes on the boundary. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a numeric vector value (enclosed in curly braces)."}>>> = exact_e_field
  []
[]

[Kernels<<<{"href": "../Kernels/index.html"}>>>]
  [curlcurl]
    type = MFEMCurlCurlKernel<<<{"description": "Adds the domain integrator to an MFEM problem for the bilinear form $(k\\vec\\nabla \\times \\vec u, \\vec\\nabla \\times \\vec v)_\\Omega$ arising from the weak form of the curl curl operator $k\\vec\\nabla \\times \\vec\\nabla \\times \\vec u$.", "href": "../../source/mfem/kernels/MFEMCurlCurlKernel.html"}>>>
    variable<<<{"description": "Variable labelling the weak form this kernel is added to"}>>> = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel<<<{"description": "Adds the domain integrator to an MFEM problem for the bilinear form $(k \\vec u, \\vec v)_\\Omega$ arising from the weak form of the mass operator $k \\vec u$.", "href": "../../source/mfem/kernels/MFEMVectorFEMassKernel.html"}>>>
    variable<<<{"description": "Variable labelling the weak form this kernel is added to"}>>> = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel<<<{"description": "Adds the domain integrator to an MFEM problem for the linear form $(\\vec f, \\vec v)_\\Omega$ arising from the weak form of the forcing term $\\vec f$.", "href": "../../source/mfem/kernels/MFEMVectorFEDomainLFKernel.html"}>>>
    variable<<<{"description": "Variable labelling the weak form this kernel is added to"}>>> = e_field
    vector_coefficient<<<{"description": "The name of the vector coefficient f. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a numeric vector value (enclosed in curly braces)."}>>> = forcing_field
  []
[]

[Preconditioner<<<{"href": "../Preconditioner/index.html"}>>>]
  [ams]
    type = MFEMHypreAMS<<<{"description": "Hypre auxiliary-space Maxwell solver and preconditioner for the iterative solution of MFEM equation systems.", "href": "../../source/mfem/solvers/MFEMHypreAMS.html"}>>>
    fespace<<<{"description": "H(curl) FESpace to use in HypreAMS setup."}>>> = HCurlFESpace
  []
[]

[Solver<<<{"href": "../Solver/index.html"}>>>]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-6
[]

[Executioner<<<{"href": "../Executioner/index.html"}>>>]
  type = MFEMSteady
  device = cpu
[]

[Outputs<<<{"href": "../Outputs/index.html"}>>>]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection<<<{"description": "Output for controlling export to an mfem::ParaViewDataCollection.", "href": "../../source/mfem/outputs/MFEMParaViewDataCollection.html"}>>>
    file_base<<<{"description": "The desired solution output name without an extension. If not provided, MOOSE sets it with Outputs/file_base when available. Otherwise, MOOSE uses input file name and this object name for a master input or uses master file_base, the subapp name and this object name for a subapp input to set it."}>>> = OutputData/CurlCurl
    vtk_format<<<{"description": "Select VTK data format to use, choosing between BINARY, BINARY32, and ASCII."}>>> = ASCII
  []
[]

(test/tests/mfem/kernels/curlcurl.i)

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3.

[Mesh]
  type = MFEMMesh
  file = ../mesh/small_fichera.mesh
  dim = 3
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
[]

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

[AuxVariables]
  [db_dt_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = db_dt_field
    source = e_field
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_e_field]
    type = ParsedVectorFunction
    expression_x = 'sin(kappa * y)'
    expression_y = 'sin(kappa * z)'
    expression_z = 'sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction
    expression_x = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[BCs]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = e_field
    vector_coefficient = exact_e_field
  []
[]

[Kernels]
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = e_field
    vector_coefficient = forcing_field
  []
[]

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

[Solver]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-6
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/CurlCurl
    vtk_format = ASCII
  []
[]

Summary
Description
Example File