VectorEMRobinBC

First order Robin-style Absorbing/Port BC for vector variables.

Overview

The VectorEMRobinBC object is an implementation of the first-order Robin-style boundary condition outlined in (Jin, 2014) Equation 9.60 and (Jin, 2010) Equation 9.3.51 for vector field variables.

General (vector field) form

The generic Jin condition is, in turn, based on the Sommerfeld radiation condition for scattered fields. Given that any scattered field can be made up of a combination of the scattered field and the incident field ( $var element = document.getElementById("moose-equation-fc7cc577-9340-4637-aead-e59ed49120cc");katex.render("\\vec{E} = \\vec{E}_{sc} + \\vec{E}_{inc}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ), then we have

$var element = document.getElementById("moose-equation-b268f276-d5f6-45f8-b142-1006722e6b78");katex.render(" \\hat{\\mathbf{n}} \\times \\left( \\frac{1}{\\mu_r} \\nabla \\times \\vec{E} \\right) + \\frac{j k_0}{\\eta_r} \\hat{\\mathbf{n}} \\times \\left( \\hat{\\mathbf{n}} \\times \\vec{E} \\right) = \\hat{\\mathbf{n}} \\times (\\nabla \\times \\vec{E}_{inc}) + \\frac{j k_0}{\\eta_r} \\hat{\\mathbf{n}} \\times \\left( \\hat{\\mathbf{n}} \\times \\vec{E}_{inc} \\right)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where

$var element = document.getElementById("moose-equation-f699ff18-fcec-4930-bf55-99cd0f12c6e0");katex.render("\\vec{E}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the total (i.e., the solution) electric field vector,
$var element = document.getElementById("moose-equation-ff2dbdee-eec5-442d-a45f-b194f4ed4213");katex.render("\\vec{E}_{inc}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the incoming electric field vector,
$var element = document.getElementById("moose-equation-f903afbe-099c-4de8-8547-4625312a3636");katex.render("\\mu_r", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the relative magnetic permeability,
$var element = document.getElementById("moose-equation-6e2611c5-fec1-4387-a6c7-fcdcb33ed9fa");katex.render("j = \\sqrt{-1}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ,
$var element = document.getElementById("moose-equation-c0a59c2f-deec-4d54-85a9-bec44454862b");katex.render("k_0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the wave number ( $var element = document.getElementById("moose-equation-0b5dac66-7ab5-499b-bd6d-70826e2585b0");katex.render("2 \\pi / \\lambda", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-d05d13ea-bcc8-40b8-a327-d97332ba65f4");katex.render("\\lambda", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the wavelength),
$var element = document.getElementById("moose-equation-d6dbba90-1445-4fc3-b3a3-0aa304ddd0e4");katex.render("\\eta_r", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a radiation condition parameter ( $var element = document.getElementById("moose-equation-8f88f41c-13ec-4abc-aa87-c4a6312c174c");katex.render("\\eta_r = 1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ if the condition is applied in free space), and
$var element = document.getElementById("moose-equation-42be855a-4b51-4be8-a91a-5a14c059a2f8");katex.render("\\hat{\\mathbf{n}}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the boundary normal vector.

Implemented form

In VectorEMRobinBC, this condition is slightly generalized. The ratio of $var element = document.getElementById("moose-equation-e75d817a-3e0c-488a-8919-54a280adb4de");katex.render("k_0 / \\eta_r", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is generalized to a coefficient function $var element = document.getElementById("moose-equation-bb98095a-e365-47c9-b777-17f9f250efb8");katex.render("\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and so the implemented form is

$var element = document.getElementById("moose-equation-505eb365-bc85-4e24-891f-da1b4e81aeed");katex.render(" \\hat{\\mathbf{n}} \\times \\left( \\frac{1}{\\mu_r} \\nabla \\times \\vec{E} \\right) + j\\beta(\\mathbf{r}) \\hat{\\mathbf{n}} \\times \\left( \\hat{\\mathbf{n}} \\times \\vec{E} \\right) = \\hat{\\mathbf{n}} \\times (\\nabla \\times \\vec{E}_{inc}) + j\\beta(\\mathbf{r}) \\hat{\\mathbf{n}} \\times \\left( \\hat{\\mathbf{n}} \\times \\vec{E}_{inc} \\right)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where

$var element = document.getElementById("moose-equation-ea45fd0d-0eb6-484e-b30d-1e2e064c63f8");katex.render("\\beta(\\mathbf{r})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a function containing the condition coefficients.

Note that $var element = document.getElementById("moose-equation-61722087-07cc-4d73-b42b-a30c2da13d7f");katex.render("\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ generally is just the wavenumber $var element = document.getElementById("moose-equation-645de62b-a68a-4b15-ad74-ff5cfdde57cc");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , since this condition is generally applied in free space, but any function can be applied through the "beta" parameter.

Usage notes

It is important to note that when used as an absorber (strictly absorbing means "mode" is set to absorbing and zero incoming wave but a port also has an absorbing component), care must be taken in setting the shape of the truncation boundary as well as the distance from the scattering object. Boundaries as close as $var element = document.getElementById("moose-equation-d286e529-bc0e-4bb7-937a-24fac3626dff");katex.render("0.3 \\lambda", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ away from the object was shown in (Jin, 2014), and several wavelengths were used in the Dipole Antenna Benchmark for VectorEMRobinBC. Also, this boundary condition is best applied on spherical boundaries (as a result of its origin from the Sommerfeld condition, which was derived for spherical boundaries). Of course, it is also valid on any non-spherical smooth surface with a trade-off in accuracy.

Example Input File Syntax

As a Port

[BCs]
  [sides_real]
    type = VectorEMRobinBC
    variable = u_real
    component = real
    coupled_field = u_imaginary
    imag_incoming = mms_imaginary
    real_incoming = mms_real
    boundary = 'left right top bottom'
  []
[]

As an Absorber

[BCs]
  [radiation_condition_real]
    type = VectorEMRobinBC
    variable = E_real
    coupled_field = E_imag
    boundary = boundary
    component = real
    mode = absorbing
    beta = 20.9439510239 # wave number at 1 GHz
  []
[]

Input Parameters

boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
coupled_fieldCoupled field variable.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled field variable.
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

beta1Scalar wave number.
Default:1
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Scalar wave number.
componentVariable field component (real or imaginary).
C++ Type:MooseEnum
Unit:(no unit assumed)
Options:real, imaginary
Controllable:No
Description:Variable field component (real or imaginary).
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
imag_incoming0Imaginary incoming field vector.
Default:0
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Imaginary incoming field vector.
modeportMode of operation for VectorEMRobinBC.
Default:port
C++ Type:MooseEnum
Unit:(no unit assumed)
Options:absorbing, port
Controllable:No
Description:Mode of operation for VectorEMRobinBC.
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.
real_incoming0Real incoming field vector.
Default:0
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Real incoming field vector.
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 BC'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 BC'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 BC'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 BC'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

(modules/electromagnetics/test/tests/benchmarks/evanescent_wave/evanescent_wave.i)
(modules/electromagnetics/test/tests/benchmarks/dipole_antenna/dipole.i)
(modules/electromagnetics/test/tests/bcs/vector_robin_bc/portbc_waves.i)

References

Jian-Ming Jin. Theory and Computation of Electromagnetic Fields. John Wiley & Sons, Hoboken, New Jersey, USA, 1st edition, 2010.

@book{jin-computation,
    author = "Jin, Jian-Ming",
    title = "Theory and Computation of Electromagnetic Fields",
    edition = "1st",
    year = "2010",
    publisher = "John Wiley \\& Sons",
    address = "Hoboken, New Jersey, USA"
}

Jian-Ming Jin. The Finite Element Method in Electromagnetics. John Wiley & Sons, Hoboken, New Jersey, USA, 3rd edition, 2014.

@book{jin-fem,
    author = "Jin, Jian-Ming",
    title = "The Finite Element Method in Electromagnetics",
    edition = "3rd",
    year = "2014",
    publisher = "John Wiley \\& Sons",
    address = "Hoboken, New Jersey, USA"
}

(modules/electromagnetics/test/tests/bcs/vector_robin_bc/portbc_waves.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = -1
    ymin = -1
    elem_type = QUAD9
  []
  uniform_refine = 1
[]

[Functions]
  [mms_real] # Manufactured solution, real component
    type = ParsedVectorFunction
    expression_x = 'cos(pi*x)*sin(pi*y)'
    expression_y = '-cos(pi*x)*sin(pi*y)'
    curl_z = 'pi*sin(pi*x)*sin(pi*y) - pi*cos(pi*x)*cos(pi*y)'
  []
  [mms_imaginary] # Manufactured solution, imaginary component
    type = ParsedVectorFunction
    expression_x = 'cos(pi*x + pi/2)*sin(pi*y)'
    expression_y = '-cos(pi*x + pi/2)*sin(pi*x)'
    curl_z = 'pi*sin(pi*x)*cos(pi*y) + pi*sin(pi*y)*cos(pi*x)'
  []
[]

[Variables]
  [u_real]
    family = NEDELEC_ONE
    order = FIRST
  []
  [u_imaginary]
    family = NEDELEC_ONE
    order = FIRST
  []
[]

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = u_real
  []
  [coeff_real]
    type = VectorFunctionReaction
    variable = u_real
  []
  [rhs_real]
    type = VectorBodyForce
    variable = u_real
    function_x = 'pi*pi*sin(pi*x)*cos(pi*y) + sin(pi*y)*cos(pi*x) + pi*pi*sin(pi*y)*cos(pi*x)'
    function_y = '-pi*pi*sin(pi*x)*cos(pi*y) - pi*pi*sin(pi*y)*cos(pi*x) - sin(pi*y)*cos(pi*x)'
  []
  [curl_curl_imaginary]
    type = CurlCurlField
    variable = u_imaginary
  []
  [coeff_imaginary]
    type = VectorFunctionReaction
    variable = u_imaginary
  []
  [rhs_imaginary]
    type = VectorBodyForce
    variable = u_imaginary
    function_x = '-pi*pi*sin(pi*x)*sin(pi*y) - sin(pi*x)*sin(pi*y) + pi*pi*cos(pi*x)*cos(pi*y)'
    function_y = 'sin(pi*x)*sin(pi*y) + pi*pi*sin(pi*x)*sin(pi*y) - pi*pi*cos(pi*x)*cos(pi*y)'
  []
[]

[BCs]
  [sides_real]
    type = VectorEMRobinBC
    variable = u_real
    component = real
    coupled_field = u_imaginary
    imag_incoming = mms_imaginary
    real_incoming = mms_real
    boundary = 'left right top bottom'
  []
  [sides_imaginary]
    type = VectorEMRobinBC
    variable = u_imaginary
    component = imaginary
    coupled_field = u_real
    imag_incoming = mms_imaginary
    real_incoming = mms_real
    boundary = 'left right top bottom'
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
[]

(modules/electromagnetics/test/tests/benchmarks/dipole_antenna/dipole.i)

# Verification Benchmark - Half-wave Dipole Antenna (Frequency Domain)
# Resonant Frequency = 1 GHz
# Wave Propagation Medium: Vacuum

[Mesh]
  [file_mesh]
    type = FileMeshGenerator
    file = dipole_antenna_1G.msh
  []
  [refine]
    type = RefineBlockGenerator
    input = file_mesh
    block = 'vacuum'
    refinement = 2
  []
[]

[Variables]
  [E_real]
    order = FIRST
    family = NEDELEC_ONE
  []
  [E_imag]
    order = FIRST
    family = NEDELEC_ONE
  []
[]

[Functions]
  [WaveNumberSquared]
    type = ParsedFunction
    expression = '(2*pi*1e9/3e8)*(2*pi*1e9/3e8)'
  []
[]

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = VectorFunctionReaction
    variable = E_real
    function = WaveNumberSquared
    sign = negative
  []
  [curl_curl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = VectorFunctionReaction
    variable = E_imag
    function = WaveNumberSquared
    sign = negative
  []
[]

[BCs]
  [antenna_real]                          # Impose exact solution of E-field onto antenna surface.
    type = VectorCurlPenaltyDirichletBC   # Replace with proper antenna surface current condition.
    penalty = 1e5
    function_y = '1'
    boundary = antenna
    variable = E_real
  []
  [antenna_imag]
    type = VectorCurlPenaltyDirichletBC
    penalty = 1e5
    function_y = '1'
    boundary = antenna
    variable = E_imag
  []
  [radiation_condition_real]
    type = VectorEMRobinBC
    variable = E_real
    coupled_field = E_imag
    boundary = boundary
    component = real
    mode = absorbing
    beta = 20.9439510239  # wave number at 1 GHz
  []
  [radiation_condition_imag]
    type = VectorEMRobinBC
    variable = E_imag
    coupled_field = E_real
    boundary = boundary
    component = imaginary
    mode = absorbing
    beta = 20.9439510239  # wave number at 1 GHz
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

(modules/electromagnetics/test/tests/benchmarks/evanescent_wave/evanescent_wave.i)

# Evanescent wave decay benchmark
# frequency = 20 GHz
# eps_R = 1.0
# mu_R = 1.0

[Mesh]
  [fmg]
    type = FileMeshGenerator
    file = waveguide_discontinuous.msh
  []
[]

[Functions]
  [waveNumberSquared]
    type = ParsedFunction
    expression = '(2*pi*20e9/3e8)^2'
  []
  [omegaMu]
    type = ParsedFunction
    expression = '2*pi*20e9*4*pi*1e-7'
  []
  [beta]
    type = ParsedFunction
    expression = '2*pi*20e9/3e8'
  []
  [curr_real]
    type = ParsedVectorFunction
    expression_y = 1.0
  []
  [curr_imag] # defaults to '0.0 0.0 0.0'
    type = ParsedVectorFunction
  []
[]

[Variables]
  [E_real]
    family = NEDELEC_ONE
    order = FIRST
  []
  [E_imag]
    family = NEDELEC_ONE
    order = FIRST
  []
[]

[Kernels]
  [curlCurl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = VectorFunctionReaction
    variable = E_real
    function = waveNumberSquared
    sign = negative
  []
  [source_real]
    type = VectorCurrentSource
    variable = E_real
    component = real
    source_real = curr_real
    source_imag = curr_imag
    function_coefficient = omegaMu
    block = source
  []
  [curlCurl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = VectorFunctionReaction
    variable = E_imag
    function = waveNumberSquared
    sign = negative
  []
  [source_imaginary]
    type = VectorCurrentSource
    variable = E_imag
    component = imaginary
    source_real = curr_real
    source_imag = curr_imag
    function_coefficient = omegaMu
    block = source
  []
[]

[BCs]
  [absorbing_left_real]
    type = VectorEMRobinBC
    variable = E_real
    component = real
    beta = beta
    coupled_field = E_imag
    mode = absorbing
    boundary = 'port'
  []
  [absorbing_right_real]
    type = VectorEMRobinBC
    variable = E_real
    component = real
    beta = beta
    coupled_field = E_imag
    mode = absorbing
    boundary = 'exit'
  []
  [absorbing_left_imag]
    type = VectorEMRobinBC
    variable = E_imag
    component = imaginary
    beta = beta
    coupled_field = E_real
    mode = absorbing
    boundary = 'port'
  []
  [absorbing_right_imag]
    type = VectorEMRobinBC
    variable = E_imag
    component = imaginary
    beta = beta
    coupled_field = E_real
    mode = absorbing
    boundary = 'exit'
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
  print_linear_residuals = true
[]

[Debug]
  show_var_residual_norms = true
[]

(modules/electromagnetics/test/tests/benchmarks/dipole_antenna/dipole.i)

# Verification Benchmark - Half-wave Dipole Antenna (Frequency Domain)
# Resonant Frequency = 1 GHz
# Wave Propagation Medium: Vacuum

[Mesh]
  [file_mesh]
    type = FileMeshGenerator
    file = dipole_antenna_1G.msh
  []
  [refine]
    type = RefineBlockGenerator
    input = file_mesh
    block = 'vacuum'
    refinement = 2
  []
[]

[Variables]
  [E_real]
    order = FIRST
    family = NEDELEC_ONE
  []
  [E_imag]
    order = FIRST
    family = NEDELEC_ONE
  []
[]

[Functions]
  [WaveNumberSquared]
    type = ParsedFunction
    expression = '(2*pi*1e9/3e8)*(2*pi*1e9/3e8)'
  []
[]

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = VectorFunctionReaction
    variable = E_real
    function = WaveNumberSquared
    sign = negative
  []
  [curl_curl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = VectorFunctionReaction
    variable = E_imag
    function = WaveNumberSquared
    sign = negative
  []
[]

[BCs]
  [antenna_real]                          # Impose exact solution of E-field onto antenna surface.
    type = VectorCurlPenaltyDirichletBC   # Replace with proper antenna surface current condition.
    penalty = 1e5
    function_y = '1'
    boundary = antenna
    variable = E_real
  []
  [antenna_imag]
    type = VectorCurlPenaltyDirichletBC
    penalty = 1e5
    function_y = '1'
    boundary = antenna
    variable = E_imag
  []
  [radiation_condition_real]
    type = VectorEMRobinBC
    variable = E_real
    coupled_field = E_imag
    boundary = boundary
    component = real
    mode = absorbing
    beta = 20.9439510239  # wave number at 1 GHz
  []
  [radiation_condition_imag]
    type = VectorEMRobinBC
    variable = E_imag
    coupled_field = E_real
    boundary = boundary
    component = imaginary
    mode = absorbing
    beta = 20.9439510239  # wave number at 1 GHz
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

(modules/electromagnetics/test/tests/bcs/vector_robin_bc/portbc_waves.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = -1
    ymin = -1
    elem_type = QUAD9
  []
  uniform_refine = 1
[]

[Functions]
  [mms_real] # Manufactured solution, real component
    type = ParsedVectorFunction
    expression_x = 'cos(pi*x)*sin(pi*y)'
    expression_y = '-cos(pi*x)*sin(pi*y)'
    curl_z = 'pi*sin(pi*x)*sin(pi*y) - pi*cos(pi*x)*cos(pi*y)'
  []
  [mms_imaginary] # Manufactured solution, imaginary component
    type = ParsedVectorFunction
    expression_x = 'cos(pi*x + pi/2)*sin(pi*y)'
    expression_y = '-cos(pi*x + pi/2)*sin(pi*x)'
    curl_z = 'pi*sin(pi*x)*cos(pi*y) + pi*sin(pi*y)*cos(pi*x)'
  []
[]

[Variables]
  [u_real]
    family = NEDELEC_ONE
    order = FIRST
  []
  [u_imaginary]
    family = NEDELEC_ONE
    order = FIRST
  []
[]

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = u_real
  []
  [coeff_real]
    type = VectorFunctionReaction
    variable = u_real
  []
  [rhs_real]
    type = VectorBodyForce
    variable = u_real
    function_x = 'pi*pi*sin(pi*x)*cos(pi*y) + sin(pi*y)*cos(pi*x) + pi*pi*sin(pi*y)*cos(pi*x)'
    function_y = '-pi*pi*sin(pi*x)*cos(pi*y) - pi*pi*sin(pi*y)*cos(pi*x) - sin(pi*y)*cos(pi*x)'
  []
  [curl_curl_imaginary]
    type = CurlCurlField
    variable = u_imaginary
  []
  [coeff_imaginary]
    type = VectorFunctionReaction
    variable = u_imaginary
  []
  [rhs_imaginary]
    type = VectorBodyForce
    variable = u_imaginary
    function_x = '-pi*pi*sin(pi*x)*sin(pi*y) - sin(pi*x)*sin(pi*y) + pi*pi*cos(pi*x)*cos(pi*y)'
    function_y = 'sin(pi*x)*sin(pi*y) + pi*pi*sin(pi*x)*sin(pi*y) - pi*pi*cos(pi*x)*cos(pi*y)'
  []
[]

[BCs]
  [sides_real]
    type = VectorEMRobinBC
    variable = u_real
    component = real
    coupled_field = u_imaginary
    imag_incoming = mms_imaginary
    real_incoming = mms_real
    boundary = 'left right top bottom'
  []
  [sides_imaginary]
    type = VectorEMRobinBC
    variable = u_imaginary
    component = imaginary
    coupled_field = u_real
    imag_incoming = mms_imaginary
    real_incoming = mms_real
    boundary = 'left right top bottom'
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
[]

Overview
Example Input File Syntax
Input Parameters
Input Files
References