warning

This kernel will be deprecated in the near future (10/01/2025) in favor of exclusively using the ADJouleHeatingSource within the Heat Transfer module, because ADJouleHeatingSource can calculate both electrostatic and electromagnetic Joule heating. For more information, please see ADJouleHeatingSource.

EMJouleHeatingSource

Supplies the heating due to the electic field in the form of $var element = document.getElementById("moose-equation-b6d03162-71be-441f-8d58-a0714f73aa01");katex.render("(0.5Re(\\sigma E \\cdot E^{*} ))", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Overview

The EMJouleHeatingSource object implements a heating term imparted to the medium based on the conduction current. The term is defined as:

$var element = document.getElementById("moose-equation-7cef3d33-00eb-443e-98d7-e405e345f33b");katex.render(" -0.5 * Re \\left(\\sigma \\; \\vec{E} \\cdot \\vec{E}^{*} \\right) * \\text{Scaling}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where

$var element = document.getElementById("moose-equation-264ec488-84ae-4c88-9c42-291fcfddcfad");katex.render("\\sigma", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the conductivity of the medium,
$var element = document.getElementById("moose-equation-eaf10a0c-2daa-4807-93e6-606208a7ea89");katex.render("\\vec{E}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the electric field
$var element = document.getElementById("moose-equation-b1b1d866-df75-4e7d-81b6-a007da4f7151");katex.render("\\vec{E}^{*}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the complex conjugate of the electric field,
and $var element = document.getElementById("moose-equation-137cbeeb-2db9-4d55-84f7-f342e86aba0a");katex.render("\\text{Scaling}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a scaling factor (usually used to convert the units of the heating term).

Example Input File Syntax

[Kernels<<<{"href": "../../syntax/Kernels/index.html"}>>>]
  [microwave_heating]
    type = EMJouleHeatingSource<<<{"description": "Supplies the heating due to the electic field in the form of $(0.5Re(\\sigma E \\cdot E^{*} ))$", "href": "EMJouleHeatingSource.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = n
    E_imag<<<{"description": "The imaginary component of the electric field."}>>> = E_imag
    E_real<<<{"description": "The real component of the electric field."}>>> = E_real
    conductivity<<<{"description": "The real component of the material conductivity."}>>> = cond_real
  []
[]

Input Parameters

E_imagThe imaginary component of the electric field.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The imaginary component of the electric field.
E_realThe real component of the electric field.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The real component of the electric field.
conductivityThe real component of the material conductivity.
C++ Type:std::string
Controllable:No
Description:The real component of the material conductivity.
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>
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
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
value1Coefficient to multiply by heating term.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Coefficient to multiply by heating term.

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>
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>
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>
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
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
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data 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.
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
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
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
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
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

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
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.

Material Property Retrieval Parameters

Input Files

(modules/electromagnetics/test/tests/auxkernels/heating/aux_microwave_heating.i)
(modules/electromagnetics/test/tests/kernels/vector_helmholtz/microwave_heating.i)

(modules/electromagnetics/test/tests/kernels/vector_helmholtz/microwave_heating.i)

# Test for EMJouleHeatingSource
# Manufactured solution: E_real = cos(pi*y) * x_hat - cos(pi*x) * y_hat
#                        E_imag = sin(pi*y) * x_hat - sin(pi*x) * y_hat
#                        n = x^2*y^2

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 5
    ny = 5
    xmin = -1
    ymin = -1
    elem_type = QUAD9
  []
[]

[Functions]
  #The exact solution for the heated species and electric field real and imag. component
  [exact_real]
    type = ParsedVectorFunction
    expression_x = 'cos(pi*y)'
    expression_y = '-cos(pi*x)'
  []
  [exact_imag]
    type = ParsedVectorFunction
    expression_x = 'sin(pi*y)'
    expression_y = '-sin(pi*x)'
  []
  [exact_n]
    type = ParsedFunction
    expression = 'x^2*y^2'
  []

  #The forcing terms for the heated species and electric field real and imag. component
  [source_real]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*cos(pi*y) - 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*y) + epsilon_i*mu_i*omega_r^2*cos(pi*y) - epsilon_i*mu_r*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_r*omega_r^2*sin(pi*y) - epsilon_r*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_r*mu_i*omega_r^2*sin(pi*y) + epsilon_r*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_r*omega_r^2*cos(pi*y) + mu_i*omega_i*sigma_i*cos(pi*y) + mu_i*omega_i*sigma_r*sin(pi*y) + mu_i*omega_r*sigma_i*sin(pi*y) - mu_i*omega_r*sigma_r*cos(pi*y) + mu_r*omega_i*sigma_i*sin(pi*y) - mu_r*omega_i*sigma_r*cos(pi*y) - mu_r*omega_r*sigma_i*cos(pi*y) - mu_r*omega_r*sigma_r*sin(pi*y) + pi^2*cos(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*cos(pi*x) + 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*x) - epsilon_i*mu_i*omega_r^2*cos(pi*x) + epsilon_i*mu_r*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_r*omega_r^2*sin(pi*x) + epsilon_r*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_r*mu_i*omega_r^2*sin(pi*x) - epsilon_r*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_r*omega_r^2*cos(pi*x) - mu_i*omega_i*sigma_i*cos(pi*x) - mu_i*omega_i*sigma_r*sin(pi*x) - mu_i*omega_r*sigma_i*sin(pi*x) + mu_i*omega_r*sigma_r*cos(pi*x) - mu_r*omega_i*sigma_i*sin(pi*x) + mu_r*omega_i*sigma_r*cos(pi*x) + mu_r*omega_r*sigma_i*cos(pi*x) + mu_r*omega_r*sigma_r*sin(pi*x) - pi^2*cos(pi*x)'
  []
  [source_imag]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_i*omega_r^2*sin(pi*y) + epsilon_i*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_i*mu_r*omega_r^2*cos(pi*y) + epsilon_r*mu_i*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_i*omega_r^2*cos(pi*y) + epsilon_r*mu_r*omega_i^2*sin(pi*y) - 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*y) - epsilon_r*mu_r*omega_r^2*sin(pi*y) + mu_i*omega_i*sigma_i*sin(pi*y) - mu_i*omega_i*sigma_r*cos(pi*y) - mu_i*omega_r*sigma_i*cos(pi*y) - mu_i*omega_r*sigma_r*sin(pi*y) - mu_r*omega_i*sigma_i*cos(pi*y) - mu_r*omega_i*sigma_r*sin(pi*y) - mu_r*omega_r*sigma_i*sin(pi*y) + mu_r*omega_r*sigma_r*cos(pi*y) + pi^2*sin(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_i*omega_r^2*sin(pi*x) - epsilon_i*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_i*mu_r*omega_r^2*cos(pi*x) - epsilon_r*mu_i*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_i*omega_r^2*cos(pi*x) - epsilon_r*mu_r*omega_i^2*sin(pi*x) + 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*x) + epsilon_r*mu_r*omega_r^2*sin(pi*x) - mu_i*omega_i*sigma_i*sin(pi*x) + mu_i*omega_i*sigma_r*cos(pi*x) + mu_i*omega_r*sigma_i*cos(pi*x) + mu_i*omega_r*sigma_r*sin(pi*x) + mu_r*omega_i*sigma_i*cos(pi*x) + mu_r*omega_i*sigma_r*sin(pi*x) + mu_r*omega_r*sigma_i*sin(pi*x) - mu_r*omega_r*sigma_r*cos(pi*x) - pi^2*sin(pi*x)'
  []
  [source_n]
    type = ParsedFunction
    symbol_names = 'sigma_r'
    symbol_values = 'sigma'
    expression = '-2*x^2 - 2*y^2 - 0.5*sigma_r*(sin(x*pi)^2 + sin(y*pi)^2 + cos(x*pi)^2 + cos(y*pi)^2)'
  []

  #Material Coefficients
  [omega]
    type = ParsedFunction
    expression = '2.0'
  []
  [mu]
    type = ParsedFunction
    expression = '1.0'
  []
  [epsilon]
    type = ParsedFunction
    expression = '3.0'
  []
  [sigma]
    type = ParsedFunction
    expression = '4.0'
    #expression = 'x^2*y^2'
  []
[]

[Materials]
  [WaveCoeff]
    type = WaveEquationCoefficient
    eps_rel_imag = eps_imag
    eps_rel_real = eps_real
    k_real = k_real
    k_imag = k_imag
    mu_rel_imag = mu_imag
    mu_rel_real = mu_real
  []
  [eps_real]
    type = ADGenericFunctionMaterial
    prop_names = eps_real
    prop_values = epsilon
  []
  [eps_imag]
    type = ADGenericFunctionMaterial
    prop_names = eps_imag
    prop_values = epsilon
  []
  [mu_real]
    type = ADGenericFunctionMaterial
    prop_names = mu_real
    prop_values = mu
  []
  [mu_imag]
    type = ADGenericFunctionMaterial
    prop_names = mu_imag
    prop_values = mu
  []
  [k_real]
    type = ADGenericFunctionMaterial
    prop_names = k_real
    prop_values = omega
  []
  [k_imag]
    type = ADGenericFunctionMaterial
    prop_names = k_imag
    prop_values = omega
  []
  [cond_real]
    type = ADGenericFunctionMaterial
    prop_names = cond_real
    prop_values = sigma
  []
  [cond_imag]
    type = ADGenericFunctionMaterial
    prop_names = cond_imag
    prop_values = sigma
  []
[]

[Variables]
  [n]
    family = LAGRANGE
    order = FIRST
  []

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

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = ADMatWaveReaction
    variable = E_real
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = real
  []
  [conduction_real]
    type = ADConductionCurrent
    variable = E_real
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = real
  []
  [body_force_real]
    type = VectorBodyForce
    variable = E_real
    function = source_real
  []

  [curl_curl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = ADMatWaveReaction
    variable = E_imag
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = imaginary
  []
  [conduction_imag]
    type = ADConductionCurrent
    variable = E_imag
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = imaginary
  []
  [body_force_imag]
    type = VectorBodyForce
    variable = E_imag
    function = source_imag
  []

  [n_diffusion]
    type = Diffusion
    variable = n
  []
  [microwave_heating]
    type = EMJouleHeatingSource
    variable = n
    E_imag = E_imag
    E_real = E_real
    conductivity = cond_real
  []
  [body_force_n]
    type = BodyForce
    variable = n
    function = source_n
  []
[]

[BCs]
  [sides_real]
    type = VectorCurlPenaltyDirichletBC
    variable = E_real
    function = exact_real
    penalty = 1e8
    boundary = 'left right top bottom'
  []
  [sides_imag]
    type = VectorCurlPenaltyDirichletBC
    variable = E_imag
    function = exact_imag
    penalty = 1e8
    boundary = 'left right top bottom'
  []

  [sides_n]
    type = FunctorDirichletBC
    variable = n
    boundary = 'left right top bottom'
    functor = exact_n
    preset = false
  []
[]

[Postprocessors]
  [error_real]
    type = ElementVectorL2Error
    variable = E_real
    function = exact_real
  []
  [error_imag]
    type = ElementVectorL2Error
    variable = E_imag
    function = exact_imag
  []

  [error_n]
    type = ElementL2Error
    variable = n
    function = exact_n
  []

  [h]
    type = AverageElementSize
  []
  [h_squared]
    type = ParsedPostprocessor
    pp_names = 'h'
    expression = 'h * h'
  []
[]

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

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

  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

(modules/electromagnetics/test/tests/auxkernels/heating/aux_microwave_heating.i)

# Test for EMJouleHeatingHeatGeneratedAux
# Manufactured solution: E_real = cos(pi*y) * x_hat - cos(pi*x) * y_hat
#                        E_imag = sin(pi*y) * x_hat - sin(pi*x) * y_hat
#                        n = x^2*y^2
#                        heating = '0.5*sigma_r*(sin(x*pi)^2 + sin(y*pi)^2 + cos(x*pi)^2 + cos(y*pi)^2)'

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 5
    ny = 5
    xmin = -1
    ymin = -1
    elem_type = QUAD9
  []
[]

[Functions]
  #The exact solution for the heated species and electric field real and imag. component
  [exact_real]
    type = ParsedVectorFunction
    expression_x = 'cos(pi*y)'
    expression_y = '-cos(pi*x)'
  []
  [exact_imag]
    type = ParsedVectorFunction
    expression_x = 'sin(pi*y)'
    expression_y = '-sin(pi*x)'
  []
  [exact_n]
    type = ParsedFunction
    expression = 'x^2*y^2'
  []

  #The forcing terms for the heated species and electric field real and imag. component
  [source_real]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*cos(pi*y) - 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*y) + epsilon_i*mu_i*omega_r^2*cos(pi*y) - epsilon_i*mu_r*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_r*omega_r^2*sin(pi*y) - epsilon_r*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_r*mu_i*omega_r^2*sin(pi*y) + epsilon_r*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_r*omega_r^2*cos(pi*y) + mu_i*omega_i*sigma_i*cos(pi*y) + mu_i*omega_i*sigma_r*sin(pi*y) + mu_i*omega_r*sigma_i*sin(pi*y) - mu_i*omega_r*sigma_r*cos(pi*y) + mu_r*omega_i*sigma_i*sin(pi*y) - mu_r*omega_i*sigma_r*cos(pi*y) - mu_r*omega_r*sigma_i*cos(pi*y) - mu_r*omega_r*sigma_r*sin(pi*y) + pi^2*cos(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*cos(pi*x) + 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*x) - epsilon_i*mu_i*omega_r^2*cos(pi*x) + epsilon_i*mu_r*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_r*omega_r^2*sin(pi*x) + epsilon_r*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_r*mu_i*omega_r^2*sin(pi*x) - epsilon_r*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_r*omega_r^2*cos(pi*x) - mu_i*omega_i*sigma_i*cos(pi*x) - mu_i*omega_i*sigma_r*sin(pi*x) - mu_i*omega_r*sigma_i*sin(pi*x) + mu_i*omega_r*sigma_r*cos(pi*x) - mu_r*omega_i*sigma_i*sin(pi*x) + mu_r*omega_i*sigma_r*cos(pi*x) + mu_r*omega_r*sigma_i*cos(pi*x) + mu_r*omega_r*sigma_r*sin(pi*x) - pi^2*cos(pi*x)'
  []
  [source_imag]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_i*omega_r^2*sin(pi*y) + epsilon_i*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_i*mu_r*omega_r^2*cos(pi*y) + epsilon_r*mu_i*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_i*omega_r^2*cos(pi*y) + epsilon_r*mu_r*omega_i^2*sin(pi*y) - 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*y) - epsilon_r*mu_r*omega_r^2*sin(pi*y) + mu_i*omega_i*sigma_i*sin(pi*y) - mu_i*omega_i*sigma_r*cos(pi*y) - mu_i*omega_r*sigma_i*cos(pi*y) - mu_i*omega_r*sigma_r*sin(pi*y) - mu_r*omega_i*sigma_i*cos(pi*y) - mu_r*omega_i*sigma_r*sin(pi*y) - mu_r*omega_r*sigma_i*sin(pi*y) + mu_r*omega_r*sigma_r*cos(pi*y) + pi^2*sin(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_i*omega_r^2*sin(pi*x) - epsilon_i*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_i*mu_r*omega_r^2*cos(pi*x) - epsilon_r*mu_i*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_i*omega_r^2*cos(pi*x) - epsilon_r*mu_r*omega_i^2*sin(pi*x) + 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*x) + epsilon_r*mu_r*omega_r^2*sin(pi*x) - mu_i*omega_i*sigma_i*sin(pi*x) + mu_i*omega_i*sigma_r*cos(pi*x) + mu_i*omega_r*sigma_i*cos(pi*x) + mu_i*omega_r*sigma_r*sin(pi*x) + mu_r*omega_i*sigma_i*cos(pi*x) + mu_r*omega_i*sigma_r*sin(pi*x) + mu_r*omega_r*sigma_i*sin(pi*x) - mu_r*omega_r*sigma_r*cos(pi*x) - pi^2*sin(pi*x)'
  []
  [source_n]
    type = ParsedFunction
    symbol_names = 'sigma_r'
    symbol_values = 'sigma'
    expression = '-2*x^2 - 2*y^2 - 0.5*sigma_r*(sin(x*pi)^2 + sin(y*pi)^2 + cos(x*pi)^2 + cos(y*pi)^2)'
  []

  [heating_func]
    type = ParsedFunction
    symbol_names = 'sigma_r'
    symbol_values = 'sigma'
    expression = '0.5*sigma_r*(sin(x*pi)^2 + sin(y*pi)^2 + cos(x*pi)^2 + cos(y*pi)^2)'
  []

  #Material Coefficients
  [omega]
    type = ParsedFunction
    expression = '2.0'
  []
  [mu]
    type = ParsedFunction
    expression = '1.0'
  []
  [epsilon]
    type = ParsedFunction
    expression = '3.0'
  []
  [sigma]
    type = ParsedFunction
    expression = '4.0'
    #expression = 'x^2*y^2'
  []
[]

[Materials]
  [WaveCoeff]
    type = WaveEquationCoefficient
    eps_rel_imag = eps_imag
    eps_rel_real = eps_real
    k_real = k_real
    k_imag = k_imag
    mu_rel_imag = mu_imag
    mu_rel_real = mu_real
  []
  [eps_real]
    type = ADGenericFunctionMaterial
    prop_names = eps_real
    prop_values = epsilon
  []
  [eps_imag]
    type = ADGenericFunctionMaterial
    prop_names = eps_imag
    prop_values = epsilon
  []
  [mu_real]
    type = ADGenericFunctionMaterial
    prop_names = mu_real
    prop_values = mu
  []
  [mu_imag]
    type = ADGenericFunctionMaterial
    prop_names = mu_imag
    prop_values = mu
  []
  [k_real]
    type = ADGenericFunctionMaterial
    prop_names = k_real
    prop_values = omega
  []
  [k_imag]
    type = ADGenericFunctionMaterial
    prop_names = k_imag
    prop_values = omega
  []
  [cond_real]
    type = ADGenericFunctionMaterial
    prop_names = cond_real
    prop_values = sigma
  []
  [cond_imag]
    type = ADGenericFunctionMaterial
    prop_names = cond_imag
    prop_values = sigma
  []
[]

[Variables]
  [n]
    family = LAGRANGE
    order = FIRST
  []

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

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = ADMatWaveReaction
    variable = E_real
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = real
  []
  [conduction_real]
    type = ADConductionCurrent
    variable = E_real
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = real
  []
  [body_force_real]
    type = VectorBodyForce
    variable = E_real
    function = source_real
  []

  [curl_curl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = ADMatWaveReaction
    variable = E_imag
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = imaginary
  []
  [conduction_imag]
    type = ADConductionCurrent
    variable = E_imag
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = imaginary
  []
  [body_force_imag]
    type = VectorBodyForce
    variable = E_imag
    function = source_imag
  []

  [n_diffusion]
    type = Diffusion
    variable = n
  []
  [microwave_heating]
    type = EMJouleHeatingSource
    variable = n
    E_imag = E_imag
    E_real = E_real
    conductivity = cond_real
  []
  [body_force_n]
    type = BodyForce
    variable = n
    function = source_n
  []
[]

[AuxVariables]
  [heating_term]
    family = MONOMIAL
    order = FIRST
  []
[]

[AuxKernels]
  [aux_microwave_heating]
    type = EMJouleHeatingHeatGeneratedAux
    variable = heating_term
    E_imag = E_imag
    E_real = E_real
    conductivity = cond_real
  []
[]

[BCs]
  [sides_real]
    type = VectorCurlPenaltyDirichletBC
    variable = E_real
    function = exact_real
    penalty = 1e8
    boundary = 'left right top bottom'
  []
  [sides_imag]
    type = VectorCurlPenaltyDirichletBC
    variable = E_imag
    function = exact_imag
    penalty = 1e8
    boundary = 'left right top bottom'
  []

  [sides_n]
    type = FunctorDirichletBC
    variable = n
    boundary = 'left right top bottom'
    functor = exact_n
    preset = false
  []
[]

[Postprocessors]
  [error_real]
    type = ElementVectorL2Error
    variable = E_real
    function = exact_real
  []
  [error_imag]
    type = ElementVectorL2Error
    variable = E_imag
    function = exact_imag
  []

  [error_n]
    type = ElementL2Error
    variable = n
    function = exact_n
  []

  [error_aux_heating]
    type = ElementL2Error
    variable = heating_term
    function = heating_func
  []

  [h]
    type = AverageElementSize
  []
  [h_squared]
    type = ParsedPostprocessor
    pp_names = 'h'
    expression = 'h * h'
  []
[]

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

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

  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

(modules/electromagnetics/test/tests/kernels/vector_helmholtz/microwave_heating.i)

# Test for EMJouleHeatingSource
# Manufactured solution: E_real = cos(pi*y) * x_hat - cos(pi*x) * y_hat
#                        E_imag = sin(pi*y) * x_hat - sin(pi*x) * y_hat
#                        n = x^2*y^2

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 5
    ny = 5
    xmin = -1
    ymin = -1
    elem_type = QUAD9
  []
[]

[Functions]
  #The exact solution for the heated species and electric field real and imag. component
  [exact_real]
    type = ParsedVectorFunction
    expression_x = 'cos(pi*y)'
    expression_y = '-cos(pi*x)'
  []
  [exact_imag]
    type = ParsedVectorFunction
    expression_x = 'sin(pi*y)'
    expression_y = '-sin(pi*x)'
  []
  [exact_n]
    type = ParsedFunction
    expression = 'x^2*y^2'
  []

  #The forcing terms for the heated species and electric field real and imag. component
  [source_real]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*cos(pi*y) - 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*y) + epsilon_i*mu_i*omega_r^2*cos(pi*y) - epsilon_i*mu_r*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_r*omega_r^2*sin(pi*y) - epsilon_r*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_r*mu_i*omega_r^2*sin(pi*y) + epsilon_r*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_r*omega_r^2*cos(pi*y) + mu_i*omega_i*sigma_i*cos(pi*y) + mu_i*omega_i*sigma_r*sin(pi*y) + mu_i*omega_r*sigma_i*sin(pi*y) - mu_i*omega_r*sigma_r*cos(pi*y) + mu_r*omega_i*sigma_i*sin(pi*y) - mu_r*omega_i*sigma_r*cos(pi*y) - mu_r*omega_r*sigma_i*cos(pi*y) - mu_r*omega_r*sigma_r*sin(pi*y) + pi^2*cos(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*cos(pi*x) + 2*epsilon_i*mu_i*omega_i*omega_r*sin(pi*x) - epsilon_i*mu_i*omega_r^2*cos(pi*x) + epsilon_i*mu_r*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_r*omega_r^2*sin(pi*x) + epsilon_r*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_r*mu_i*omega_r^2*sin(pi*x) - epsilon_r*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_r*omega_r^2*cos(pi*x) - mu_i*omega_i*sigma_i*cos(pi*x) - mu_i*omega_i*sigma_r*sin(pi*x) - mu_i*omega_r*sigma_i*sin(pi*x) + mu_i*omega_r*sigma_r*cos(pi*x) - mu_r*omega_i*sigma_i*sin(pi*x) + mu_r*omega_i*sigma_r*cos(pi*x) + mu_r*omega_r*sigma_i*cos(pi*x) + mu_r*omega_r*sigma_r*sin(pi*x) - pi^2*cos(pi*x)'
  []
  [source_imag]
    type = ParsedVectorFunction
    symbol_names = 'omega_r mu_r epsilon_r sigma_r omega_i mu_i epsilon_i sigma_i'
    symbol_values = 'omega   mu   epsilon   sigma   omega   mu   epsilon   sigma'
    expression_x = '-epsilon_i*mu_i*omega_i^2*sin(pi*y) + 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*y) + epsilon_i*mu_i*omega_r^2*sin(pi*y) + epsilon_i*mu_r*omega_i^2*cos(pi*y) + 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*y) - epsilon_i*mu_r*omega_r^2*cos(pi*y) + epsilon_r*mu_i*omega_i^2*cos(pi*y) + 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*y) - epsilon_r*mu_i*omega_r^2*cos(pi*y) + epsilon_r*mu_r*omega_i^2*sin(pi*y) - 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*y) - epsilon_r*mu_r*omega_r^2*sin(pi*y) + mu_i*omega_i*sigma_i*sin(pi*y) - mu_i*omega_i*sigma_r*cos(pi*y) - mu_i*omega_r*sigma_i*cos(pi*y) - mu_i*omega_r*sigma_r*sin(pi*y) - mu_r*omega_i*sigma_i*cos(pi*y) - mu_r*omega_i*sigma_r*sin(pi*y) - mu_r*omega_r*sigma_i*sin(pi*y) + mu_r*omega_r*sigma_r*cos(pi*y) + pi^2*sin(pi*y)'
    expression_y = 'epsilon_i*mu_i*omega_i^2*sin(pi*x) - 2*epsilon_i*mu_i*omega_i*omega_r*cos(pi*x) - epsilon_i*mu_i*omega_r^2*sin(pi*x) - epsilon_i*mu_r*omega_i^2*cos(pi*x) - 2*epsilon_i*mu_r*omega_i*omega_r*sin(pi*x) + epsilon_i*mu_r*omega_r^2*cos(pi*x) - epsilon_r*mu_i*omega_i^2*cos(pi*x) - 2*epsilon_r*mu_i*omega_i*omega_r*sin(pi*x) + epsilon_r*mu_i*omega_r^2*cos(pi*x) - epsilon_r*mu_r*omega_i^2*sin(pi*x) + 2*epsilon_r*mu_r*omega_i*omega_r*cos(pi*x) + epsilon_r*mu_r*omega_r^2*sin(pi*x) - mu_i*omega_i*sigma_i*sin(pi*x) + mu_i*omega_i*sigma_r*cos(pi*x) + mu_i*omega_r*sigma_i*cos(pi*x) + mu_i*omega_r*sigma_r*sin(pi*x) + mu_r*omega_i*sigma_i*cos(pi*x) + mu_r*omega_i*sigma_r*sin(pi*x) + mu_r*omega_r*sigma_i*sin(pi*x) - mu_r*omega_r*sigma_r*cos(pi*x) - pi^2*sin(pi*x)'
  []
  [source_n]
    type = ParsedFunction
    symbol_names = 'sigma_r'
    symbol_values = 'sigma'
    expression = '-2*x^2 - 2*y^2 - 0.5*sigma_r*(sin(x*pi)^2 + sin(y*pi)^2 + cos(x*pi)^2 + cos(y*pi)^2)'
  []

  #Material Coefficients
  [omega]
    type = ParsedFunction
    expression = '2.0'
  []
  [mu]
    type = ParsedFunction
    expression = '1.0'
  []
  [epsilon]
    type = ParsedFunction
    expression = '3.0'
  []
  [sigma]
    type = ParsedFunction
    expression = '4.0'
    #expression = 'x^2*y^2'
  []
[]

[Materials]
  [WaveCoeff]
    type = WaveEquationCoefficient
    eps_rel_imag = eps_imag
    eps_rel_real = eps_real
    k_real = k_real
    k_imag = k_imag
    mu_rel_imag = mu_imag
    mu_rel_real = mu_real
  []
  [eps_real]
    type = ADGenericFunctionMaterial
    prop_names = eps_real
    prop_values = epsilon
  []
  [eps_imag]
    type = ADGenericFunctionMaterial
    prop_names = eps_imag
    prop_values = epsilon
  []
  [mu_real]
    type = ADGenericFunctionMaterial
    prop_names = mu_real
    prop_values = mu
  []
  [mu_imag]
    type = ADGenericFunctionMaterial
    prop_names = mu_imag
    prop_values = mu
  []
  [k_real]
    type = ADGenericFunctionMaterial
    prop_names = k_real
    prop_values = omega
  []
  [k_imag]
    type = ADGenericFunctionMaterial
    prop_names = k_imag
    prop_values = omega
  []
  [cond_real]
    type = ADGenericFunctionMaterial
    prop_names = cond_real
    prop_values = sigma
  []
  [cond_imag]
    type = ADGenericFunctionMaterial
    prop_names = cond_imag
    prop_values = sigma
  []
[]

[Variables]
  [n]
    family = LAGRANGE
    order = FIRST
  []

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

[Kernels]
  [curl_curl_real]
    type = CurlCurlField
    variable = E_real
  []
  [coeff_real]
    type = ADMatWaveReaction
    variable = E_real
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = real
  []
  [conduction_real]
    type = ADConductionCurrent
    variable = E_real
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = real
  []
  [body_force_real]
    type = VectorBodyForce
    variable = E_real
    function = source_real
  []

  [curl_curl_imag]
    type = CurlCurlField
    variable = E_imag
  []
  [coeff_imag]
    type = ADMatWaveReaction
    variable = E_imag
    field_real =  E_real
    field_imag =  E_imag
    wave_coef_real = wave_equation_coefficient_real
    wave_coef_imag = wave_equation_coefficient_imaginary
    component = imaginary
  []
  [conduction_imag]
    type = ADConductionCurrent
    variable = E_imag
    field_imag =  E_imag
    field_real =  E_real
    conductivity_real = cond_real
    conductivity_imag = cond_imag
    ang_freq_real = k_real
    ang_freq_imag = k_imag
    permeability_real = mu_real
    permeability_imag = mu_imag
    component = imaginary
  []
  [body_force_imag]
    type = VectorBodyForce
    variable = E_imag
    function = source_imag
  []

  [n_diffusion]
    type = Diffusion
    variable = n
  []
  [microwave_heating]
    type = EMJouleHeatingSource
    variable = n
    E_imag = E_imag
    E_real = E_real
    conductivity = cond_real
  []
  [body_force_n]
    type = BodyForce
    variable = n
    function = source_n
  []
[]

[BCs]
  [sides_real]
    type = VectorCurlPenaltyDirichletBC
    variable = E_real
    function = exact_real
    penalty = 1e8
    boundary = 'left right top bottom'
  []
  [sides_imag]
    type = VectorCurlPenaltyDirichletBC
    variable = E_imag
    function = exact_imag
    penalty = 1e8
    boundary = 'left right top bottom'
  []

  [sides_n]
    type = FunctorDirichletBC
    variable = n
    boundary = 'left right top bottom'
    functor = exact_n
    preset = false
  []
[]

[Postprocessors]
  [error_real]
    type = ElementVectorL2Error
    variable = E_real
    function = exact_real
  []
  [error_imag]
    type = ElementVectorL2Error
    variable = E_imag
    function = exact_imag
  []

  [error_n]
    type = ElementL2Error
    variable = n
    function = exact_n
  []

  [h]
    type = AverageElementSize
  []
  [h_squared]
    type = ParsedPostprocessor
    pp_names = 'h'
    expression = 'h * h'
  []
[]

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

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

  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

Overview
Example Input File Syntax
Input Parameters
Input Files